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Author name code: gilman
ADS astronomy entries on 2022-09-14
author:"Gilman, Peter"
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Title: Simulating Solar Near-surface Rossby Waves by Inverse Cascade
from Supergranule Energy
Authors: Dikpati, Mausumi; Gilman, Peter A.; Guerrero, Gustavo
A.; Kosovichev, Alexander G.; McIntosh, Scott W.; Sreenivasan,
Katepalli. R.; Warnecke, Jörn; Zaqarashvili, Teimuraz V.
2022ApJ...931..117D Altcode:
Rossby waves are found at several levels in the Sun, most recently in
its supergranule layer. We show that Rossby waves in the supergranule
layer can be excited by an inverse cascade of kinetic energy from the
nearly horizontal motions in supergranules. We illustrate how this
excitation occurs using a hydrodynamic shallow-water model for a 3D
thin rotating spherical shell. We find that initial kinetic energy
at small spatial scales inverse cascades quickly to global scales,
exciting Rossby waves whose phase velocities are similar to linear
Rossby waves on the sphere originally derived by Haurwitz. Modest
departures from the Haurwitz formula originate from nonlinear finite
amplitude effects and/or the presence of differential rotation. Like
supergranules, the initial small-scale motions in our model contain
very little vorticity compared to their horizontal divergence, but the
resulting Rossby waves are almost all vortical motions. Supergranule
kinetic energy could have mainly gone into gravity waves, but we find
that most energy inverse cascades to global Rossby waves. Since kinetic
energy in supergranules is three or four orders of magnitude larger
than that of the observed Rossby waves in the supergranule layer,
there is plenty of energy available to drive the inverse-cascade
mechanism. Tachocline Rossby waves have previously been shown to
play crucial roles in causing seasons of space weather through their
nonlinear interactions with global flows and magnetic fields. We briefly
discuss how various Rossby waves in the tachocline, convection zone,
supergranule layer, and corona can be reconciled in a unified framework.
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Title: Origin of Rossby waves observed near the solar surface
Authors: Gilman, Peter; Dikpati, Mausumi; Guerrero, Gustavo;
Kosovichev, Alexander; McIntosh, Scott; Sreenivasan, Katepalli;
Warnecke, Joern; Zaqarashvili, Teimuraz
2021AGUFMSH53C..04G Altcode:
Differential rotation and toroidal magnetic bands in the tachocline
are unstable to MHD Rossby waves and may be responsible for patterns
of solar activity seen in the photosphere. Helioseismic and surface
velocity measurements reveal energetically neutral Rossby waves in the
supergranulation layer. To explore plausible sources of energy for
these Rossby waves, we study nonlinear dynamics of horizontal flows
in the supergranular layer in thepresence of rotation and differential
rotation. With a shallow-water model we show that kinetic energy, put
into smallest resolved spatial scales, very quickly 'reverse cascades'
to largest scales, exciting energetically neutral Rossby-Haurwitz type
waves, as well as energetically active ones with low longitudinal
spectral modes, depending on differential rotation. Horizontal
velocities in supergranules are known to be much larger than their
vertical motions; our shallow-water system includes a similar ratio. If
supergranules are responsible for Rossby waves seen in photosphere, it
paradoxically follows that (i) stable stratification of a thin rotating
spherical shell may be a sufficient but not a necessary condition for
Rossby waves, and (ii) small-scale convection producing global Rossby
waves in a thin differentially rotating fluid may be the first ever
example found in a celestial body.
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Title: Observational evidence of spot-producing magnetic ring's
split during MHD evolution
Authors: Norton, Aimee; Dikpati, Mausumi; McIntosh, Scott; Gilman,
Peter
2021AGUFMSH55D1876N Altcode:
Spot-producing toroidal rings of 6-degree latitudinal width, with peak
field of 15 kG, have been found to undergo dynamical splitting due
to nonlinear MHD. Split-time depends on the latitude-location of the
ring. Ring-splitting occurs fastest, within a few weeks, at latitudes
20-25 degrees. Rossby waves work as perturbations to drive instability
of spot-producing toroidal rings. The ring-split is caused by the `mixed
stress' or cross correlations of perturbation velocities and magnetic
fields, which arise due to the interaction of Rossby Waves. Mixed
stress carries magnetic energy and flux from the ring-peak to its
shoulders, eventually leading to the ring-split. The two split-rings
migrate away from each other, the high latitude counterpart slipping
poleward faster, due to migrating mixed stress and magnetic curvature
stress. Broader toroidal bands do not split. Much stronger rings of 35
kG, despite being narrow, don't split, due to rigidity from stronger
magnetic fields within the ring. The analysis of magnetograms from MDI
during solar cycle 23 indicates emergence of active regions sometimes
at the same longitudes but separated in latitude by 20-degrees or more,
which could be evidence of active regions emerging from split-rings,
which consistently contribute to occasional high latitude excursions of
observed butterfly wings during ascending, peak and descending phases of
a solar cycle. In the future, observational studies using much longer
term magnetograms including GONG and SDO/HMI can determine how often
new spots are found at higher latitudes than their lower latitude
counterparts, and how the combinations influence solar eruptions and
space weather events.
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Title: Dynamical Splitting of Spot-producing Magnetic Rings in a
Nonlinear Shallow-water Model
Authors: Dikpati, Mausumi; Norton, Aimee A.; McIntosh, Scott W.;
Gilman, Peter A.
2021ApJ...922...46D Altcode:
We explore the fundamental physics of narrow toroidal rings during their
nonlinear magnetohydrodynamic evolution at tachocline depths. Using
a shallow-water model, we simulate the nonlinear evolution of
spot-producing toroidal rings of 6° latitudinal width and a peak field
of 15 kG. We find that the rings split; the split time depends on the
latitude of each ring. Ring splitting occurs fastest, within a few
weeks, at latitudes 20°-25°. Rossby waves work as perturbations to
drive the instability of spot-producing toroidal rings; the ring split
is caused by the "mixed stress" or cross-correlations of perturbation
velocities and magnetic fields, which carry magnetic energy and flux
from the ring peak to its shoulders, leading to the ring split. The two
split rings migrate away from each other, the high-latitude counterpart
slipping poleward faster due to migrating mixed stress and magnetic
curvature stress. Broader toroidal bands do not split. Much stronger
rings, despite being narrow, do not split due to rigidity from stronger
magnetic fields within the ring. Magnetogram analysis indicates the
emergence of active regions sometimes at the same longitudes but
separated in latitude by 20° or more, which could be evidence of
active regions emerging from split rings, which consistently contribute
to observed high-latitude excursions of butterfly wings during the
ascending, peak, and descending phases of a solar cycle. Observational
studies in the future can determine how often new spots are found at
higher latitudes than their lower-latitude counterparts and how the
combinations influence solar eruptions and space weather events.
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Title: Reconciling various solar Rossby wave observations using global
(M)HD models
Authors: Dikpati, M.; Gilman, P. A.; McIntosh, S. W.; Zaqarashvili,
T. V.
2021AAS...23811319D Altcode:
Various observational analyses indicate that the Sun has Rossby
waves, which include hydrodynamic Rossby waves like that on the
Earth's atmosphere as well as magnetohydrodynamic ones, which do
not have their counterparts on the Earth. Many of us are showing,
from observations and model-calculations, that solar hydrodynamic
Rossby waves have retrograde speed and they follow Rossby-Haurwitz
type dispersion relation. However, it has also recently been shown
MHD Rossby waves can have both retrograde as well as prograde speed;
if MHD Rossby waves are retrograde they are more retrograde than their
HD counterparts, whereas if they are prograde, they are relatively
slow. Helioseismically determined Rossby waves detected so far are HD
waves that are energetically neutral. Coronal bright points as well as
long-lived coronal holes' longitudinal drift-patterns with time show
evidence of HD as well as MHD Rossby waves. We know from nonlinear
simulations of global MHD waves and instabilities that Rossby waves can
be energetically active and hence, can nonlinearly interact with the
solar differential rotation and spot-producing toroidal magnetic fields,
very much like nonlinear Orr mechanism in fluid dynamics. Is it possible
to reconcile the various solar Rossby waves observations? Certainly
different observational techniques applied to different elevations
on the Sun may be measuring different Rossby waves. Furthermore,
the different measurements may be detecting waves that originate at
different depths inside the Sun. Solar atmosphere magnetic features may
have roots deep in the convection zone and reflect MHD Rossby waves at
those depths. We will present model-simulations to show what physical
conditions are responsible for producing sectoral modes and what are
responsible for generating energetically active Rossby waves, which
have important implications in causing short-term variability in solar
activity, and in turn, in space weather. This work is supported by the
NCAR, sponsored by the NSF under cooperative agreement 1852977. MD
acknowledges support from several NASA grants, namely the LWS award
80NSSC20K0355 to NCAR, subaward to NCAR from NASA's DRIVE Center award
80NSSC20K0602 (originally awarded to Stanford) and NASA-HSR subaward
80NSSC18K1206 (originally awarded to NSO).
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Title: Deciphering the Deep Origin of Active Regions via Analysis
of Magnetograms
Authors: Dikpati, Mausumi; McIntosh, Scott W.; Chatterjee, Subhamoy;
Norton, Aimee A.; Ambroz, Pavel; Gilman, Peter A.; Jain, Kiran;
Munoz-Jaramillo, Andres
2021ApJ...910...91D Altcode:
In this work, we derive magnetic toroids from surface magnetograms
by employing a novel optimization method, based on the trust region
reflective algorithm. The toroids obtained in this way are combinations
of Fourier modes (amplitudes and phases) with low longitudinal
wavenumbers. The optimization also estimates the latitudinal width of
the toroids. We validate the method using synthetic data, generated
as random numbers along a specified toroid. We compute the shapes and
latitudinal widths of the toroids via magnetograms, generally requiring
several m's to minimize residuals. A threshold field strength is
chosen to include all active regions in the magnetograms for toroid
derivation, while avoiding non-contributing weaker fields. Higher
thresholds yield narrower toroids, with an m = 1 dominant pattern. We
determine the spatiotemporal evolution of toroids by optimally weighting
the amplitudes and phases of each Fourier mode for a sequence of five
Carrington Rotations (CRs) to achieve the best amplitude and phases for
the middle CR in the sequence. Taking more than five causes "smearing"
or degradation of the toroid structure. While this method applies no
matter the depth at which the toroids actually reside inside the Sun,
by comparing their global shape and width with analogous patterns
derived from magnetohydrodynamic (MHD) tachocline shallow water model
simulations, we infer that their origin is at/near the convection zone
base. By analyzing the "Halloween" storms as an example, we describe
features of toroids that may have caused the series of space weather
events in 2003 October-November. Calculations of toroids for several
sunspot cycles will enable us to find similarities/differences in
toroids for different major space weather events.
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Title: Derivation of Toroid Patterns from Analysis of Magnetograms
And Inferring Their Deep-origin
Authors: Chatterjee, S.; Dikpati, M.; McIntosh, S. W.; Norton, A. A.;
Ambroz, P.; Gilman, P.; Jain, K.; Munoz-Jaramillo, A.
2020AGUFMSH0020013C Altcode:
We employ a novel optimization method based on Trust Region Reflective
algorithm to derive magnetic toroids from surface magnetograms. Toroids
obtained are combinations of Fourier modes (amplitudes and phases)
with low longitudinal wavenumbers. After validating the method using
synthetic data generated as random numbers along a specified toroid,
we compute shapes and latitudinal-widths of toroids from magnetograms,
usually requiring several m 's to minimize residuals. By comparing
properties of these toroids with patterns produced in the bottom
toroidal band undergoing MHD evolution in a 3D thin-shell shallow-water
type model, we infer their deep origin at/near convention-zone's base
or tachocline. A threshold field-strength is chosen to include all
active regions in magnetograms for toroid derivation, while avoiding
non-contributing weaker fields. Higher thresholds yield narrower
toroids, with m = 1 dominant, implying that stronger active regions
are erupting from the core of the toroids at bottom. We determine the
spatio-temporal evolution of toroids by optimally weighting amplitudes
and phases of each Fourier mode for a sequence of 5 Carrington Rotations
(CRs) to get the best amplitude and phases for the middle CR in the
sequence. Taking more than 5 causes 'smearing' or degradation of toroid
structure. As an example case, we analyze 'Halloween' storms toroids,
and describe the features that might have caused the series of space
weather events in October-November of 2003. We compare features of
these toroids with analogous patterns derived from model-output. To find
similarities/differences in toroids for different major space weather
events, we will analyze long-term magnetograms for several solar cycles.
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Title: Tachocline Instabilities and Solar Activity: a Fifty-Six-Year
Personal Perspective
Authors: Gilman, P.
2020AGUFMSH006..01G Altcode:
There are several hydrodynamic and magnetohydrodynamic instabilities
likely to be at work in the solar tachocline. Theoretical models
of these instabilities have been developed over the past 25 years,
ever since the tachocline was discovered. My primary focus will be on
instabilities that arise from rotation and differential rotation, and/or
toroidal magnetic fields, and have global scale at least in longitude. I
will give a personal perspective on these instabilities, their physical
significance, and how they may help determine evidence of magnetic
activity we observe at the solar surface. I will also discuss their
possible helioseismic signatures, and what they might contribute to the
workings of the solar dynamo. A few possible paradoxes will be outlined,
as well as how the theories could be made physically more realistic.
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Title: Physics of Magnetohydrodynamic Rossby Waves in the Sun
Authors: Dikpati, Mausumi; Gilman, Peter A.; Chatterjee, Subhamoy;
McIntosh, Scott W.; Zaqarashvili, Teimuraz V.
2020ApJ...896..141D Altcode:
Evidence of the existence of hydrodynamic and MHD Rossby waves in
the Sun is accumulating rapidly. We employ an MHD Rossby wave model
for the Sun in simplified Cartesian geometry, with a uniform toroidal
field and no differential rotation, to analyze the role of each force
that contributes to Rossby wave dynamics, and compute fluid particle
trajectories followed in these waves. This analysis goes well beyond
the traditional formulation of Rossby waves in terms of conservation
of vorticity. Hydrodynamic Rossby waves propagate retrograde relative
to the rotation of the reference frame, while MHD Rossby waves can be
both prograde and retrograde. Fluid particle trajectories are either
clockwise or counterclockwise spirals, depending on where in the wave
pattern they are initiated, that track generally in the direction
of wave propagation. Retrograde propagating MHD Rossby waves move
faster than their hydrodynamic counterparts of the same wavelength,
becoming Alfvén waves at very high field strengths. Prograde MHD
Rossby waves, which have no hydrodynamic counterpart, move more slowly
eastward than retrograde MHD Rossby waves for the same toroidal field,
but with a speed that increases with toroidal field, in the high
field limit again becoming Alfvén waves. The longitude and latitude
structures of all these waves, as seen in their velocity streamlines
and perturbation field lines as well as fluid particle trajectories,
are remarkably similar for different toroidal fields, rotation,
longitudinal wavelength, and direction of propagation.
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Title: The solar benchmark: rotational modulation of the Sun
reconstructed from archival sunspot records
Authors: Morris, Brett M.; Davenport, James R. A.; Giles, Helen A. C.;
Hebb, Leslie; Hawley, Suzanne L.; Angus, Ruth; Gilman, Peter A.;
Agol, Eric
2019MNRAS.484.3244M Altcode: 2019arXiv190104557M; 2019MNRAS.tmp..205M
We use archival daily spot coverage measurements from Howard et
al. to study the rotational modulation of the Sun as though it
were a distant star. A quasi-periodic Gaussian process measures the
solar rotation period P<SUB>rot</SUB> = 26.3 ± 0.1 d, and activity
cycle period P<SUB>cyc</SUB> = 10.7 ± 0.3 yr. We attempt to search
for evidence of differential rotation in variations of the apparent
rotation period throughout the activity cycle and do not detect a clear
signal of differential rotation, consistent with the null results of
the hare-and-hounds exercise of Aigrain et al. The full reconstructed
solar light curve is available online.
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Title: Magnetic Buoyancy and Magnetorotational Instabilities in
Stellar Tachoclines for Solar- and Antisolar-type Differential
Rotation
Authors: Gilman, Peter A.
2018ApJ...867...45G Altcode:
We present results from an analytical model for magnetic buoyancy and
rotational instabilities in full spherical shell stellar tachoclines
that include rotation, differential rotation of either solar or
antisolar type, and toroidal field. We find that in all cases,
for latitudes where the tachocline vertical rotation gradient is
positive, toroidal fields can be stored against magnetic buoyancy
up to a limit that is proportional to the square root of the local
vertical rotation gradient. For solar magnitude differential rotation,
this limit is about 9 kG. For fixed percentage differential rotation,
storage capacity varies linearly with the rotation rate. Faster rotators
with the same percentage differential rotation can store larger fields,
and slower rotators can store smaller fields. At latitudes where the
vertical rotation gradient is negative, vigorous magnetorotational
instability for even weak (≪1 kG) toroidal fields prevents such
storage. We infer from these results that for stars with solar-type
latitudinal differential rotation (fast equator, slow poles), any
starspots present should be found in low latitudes, similar to the
Sun. For antisolar differential rotation, any spots present should be
found in mid- and high latitudes, perhaps with a peak of occurrence
near 55°. These results hopefully provide some guidance for making
and interpreting observations of stellar activity and differential
rotation on stars with convection zones and tachoclines.
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Title: Phase Speed of Magnetized Rossby Waves that Cause Solar Seasons
Authors: Dikpati, Mausumi; Belucz, Bernadett; Gilman, Peter A.;
McIntosh, Scott W.
2018ApJ...862..159D Altcode:
Motivated by recent analysis of solar observations that show evidence of
propagating Rossby waves in coronal holes and bright points, we compute
the longitudinal phase velocities of unstable MHD Rossby waves found in
an MHD shallow-water model of the solar tachocline (both overshoot and
radiative parts). We demonstrate that phase propagation is a typical
characteristic of tachocline nonlinear oscillations that are created by
unstable MHD Rossby waves, responsible for producing solar seasons. For
toroidal field bands placed at latitudes between 5° and 75°, we find
that phase velocities occur in a range similar to the observations,
with more retrograde speeds (relative to the solar core rotation rate)
for bands placed at higher latitudes, just as coronal holes have at high
latitudes compared to low ones. The phase speeds of these waves are
relatively insensitive to the toroidal field peak amplitude. Rossby
waves for single bands at 25° are slightly prograde. However, at
latitudes lower than 25° they are very retrograde, but much less so if
a second band is included at a much higher latitude. This double-band
configuration is suggested by evidence of an extended solar cycle,
containing a high-latitude band in its early stages that does not
yet produce spots, while the spot-producing low-latitude band is
active. Collectively, our results indicate a strong connection between
longitudinally propagating MHD Rossby waves in the tachocline and
surface manifestations in the form of similarly propagating coronal
holes and patterns of bright points.
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Title: Role of Interaction between Magnetic Rossby Waves and
Tachocline Differential Rotation in Producing Solar Seasons
Authors: Dikpati, Mausumi; McIntosh, Scott W.; Bothun, Gregory; Cally,
Paul S.; Ghosh, Siddhartha S.; Gilman, Peter A.; Umurhan, Orkan M.
2018ApJ...853..144D Altcode:
We present a nonlinear magnetohydrodynamic shallow-water model
for the solar tachocline (MHD-SWT) that generates quasi-periodic
tachocline nonlinear oscillations (TNOs) that can be identified with
the recently discovered solar “seasons.” We discuss the properties
of the hydrodynamic and magnetohydrodynamic Rossby waves that interact
with the differential rotation and toroidal fields to sustain these
oscillations, which occur due to back-and-forth energy exchanges among
potential, kinetic, and magnetic energies. We perform model simulations
for a few years, for selected example cases, in both hydrodynamic and
magnetohydrodynamic regimes and show that the TNOs are robust features
of the MHD-SWT model, occurring with periods of 2-20 months. We find
that in certain cases multiple unstable shallow-water modes govern
the dynamics, and TNO periods vary with time. In hydrodynamically
governed TNOs, the energy exchange mechanism is simple, occurring
between the Rossby waves and differential rotation. But in MHD cases,
energy exchange becomes much more complex, involving energy flow among
six energy reservoirs by means of eight different energy conversion
processes. For toroidal magnetic bands of 5 and 35 kG peak amplitudes,
both placed at 45° latitude and oppositely directed in north and south
hemispheres, we show that the energy transfers responsible for TNO, as
well as westward phase propagation, are evident in synoptic maps of the
flow, magnetic field, and tachocline top-surface deformations. Nonlinear
mode-mode interaction is particularly dramatic in the strong-field
case. We also find that the TNO period increases with a decrease in
rotation rate, implying that the younger Sun had more frequent seasons.
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Title: Magnetic Buoyancy and Rotational Instabilities in the
Tachocline
Authors: Gilman, Peter A.
2018ApJ...853...65G Altcode:
We present results from an analytical model for magnetic buoyancy
and rotational instabilities in a full spherical shell tachocline
that includes rotation, differential rotation close to that observed
helioseismically, and toroidal field. Perturbation solutions are
found for the limit of large latitudinal wave number, a limit commonly
used to maximize instability due to magnetic buoyancy. We find that
at all middle and high latitudes vigorous rotational instability is
induced by weak toroidal fields, particularly for high longitudinal
wave number, even when the vertical rotation gradient is marginally
stable without toroidal field. We infer that this instability will
prevent much storage of toroidal fields in the tachocline at these
latitudes, but could be responsible for the appearance of ephemeral
active regions there. By contrast, the low-latitude vertical rotation
gradient, opposite in sign to that at high latitudes, is not only stable
itself but also prevents magnetic buoyancy instability until the peak
toroidal field is raised above a threshold of about 9 kG at the equator,
declining to zero where the vertical rotation gradient changes sign, at
32\buildrel{\circ}\over{.} 3 in our model. Thus this rotation gradient
provides a previously unnoticed mechanism for storage of toroidal
fields until they amplify by dynamo action to order 10 kG, whereupon
they can overcome the rotation gradient to emerge as sunspots. These
results provide a new explanation for why sunspots are seen only at low
latitudes. The purely rotational instability at latitudes above 50°,
even without toroidal fields, also suggests that the high-latitude
tachocline should be much thicker, due to HD turbulence, than has been
inferred for lower latitudes from helioseismic measurements.
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Title: Baroclinic Instability in the Solar Tachocline for Continuous
Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field
Authors: Gilman, Peter A.
2017ApJ...842..130G Altcode:
We present results from an MHD model for baroclinic instability
in the solar tachocline that includes rotation, effective gravity,
and toroidal field that vary continuously with height. We solve the
perturbation equations using a shooting method. Without toroidal fields
but with an effective gravity declining linearly from a maximum at the
bottom to much smaller values at the top, we find instability at all
latitudes except at the poles, at the equator, and where the vertical
rotation gradient vanishes (32.°3) for longitude wavenumbers m from 1
to >10. High latitudes are much more unstable than low latitudes,
but both have e-folding times that are much shorter than a sunspot
cycle. The higher the m and the steeper the decline in effective
gravity, the closer the unstable mode peak to the top boundary, where
the energy available to drive instability is greatest. The effect
of the toroidal field is always stabilizing, shrinking the latitude
ranges of instability as the toroidal field is increased. The larger
the toroidal field, the smaller the longitudinal wavenumber of the most
unstable disturbance. All latitudes become stable for a toroidal field
exceeding about 4 kG. The results imply that baroclinic instability
should occur in the tachocline at latitudes where the toroidal field
is weak or is changing sign, but not where the field is strong.
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Title: Baroclinic Instability in the Solar Tachocline. II. The
Eady Problem
Authors: Gilman, Peter A.
2016ApJ...818..170G Altcode:
We solve the nongeostrophic baroclinic instability problem for the
tachocline for a continuous model with a constant vertical rotation
gradient (the Eady problem), using power series generated by the
Frobenius method. The results confirm and greatly extend those from a
previous two-layer model. For effective gravity G independent of height,
growth rates and ranges of unstable longitudinal wavenumbers m and
latitudes increase with decreasing G. As with the two-layer model,
the overshoot tachocline is much more unstable than the radiative
tachocline. The e-folding growth times range from as short as 10
days to as long as several years, depending on latitude, G, and
wavenumber. For a more realistic effective gravity that decreases
linearly from the radiative interior to near zero at the top of the
tachocline, we find that only m = 1, 2 modes are unstable, with growth
rates somewhat larger than for constant G, with the same value as
at the bottom of the tachocline. All results are the same whether we
assume that the vertical velocity or the perturbation pressure is zero
at the top of the layer; this is a direct consquence of not employing
the geostrophic assumption for perturbations. We explain most of
the properties of the instability in terms of the Rossby deformation
radius. We discuss further improvements in the realism of the model,
particularly adding toroidal fields that vary in height, and including
latitudinal gradients of both rotation and toroidal fields.
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Title: Effect of Toroidal Fields On Baroclinic Instability in the
Solar Tachocline
Authors: Gilman, Peter A.
2015ApJ...801...22G Altcode:
Using an MHD generalization of a two-layer hydrostatic but
non-geostrophic model, we show that a toroidal field tends to
stabilize baroclinically unstable modes in the solar tachocline. In
the hydrodynamic (HD) case, baroclinic instability occurs at almost
all latitudes in both the radiative and overshoot tachoclines. The
toroidal field creates stable bands of latitude near where the
vertical rotation gradient changes sign, as well as near the equator
and pole, which widen with increasing field until, by ∼2.25 kG,
all latitudes are stable. The stable bands center on where the local
latitudinal entropy gradient is smallest. This result is independent
of how subadiabatic the local stratification is, provided it is
not so subadiabatic that baroclinic instability is absent in the HD
case. Growth rates and most unstable longitudinal wavenumbers remain
close to their HD values until the toroidal field gets within ∼ 20%
of the value that totally suppresses the instability. The results
are similar to those found in the 1960s from an MHD geostrophic
model, but apply to a much wider range of latitudes and subadiabatic
stratifications. Where tachocline toroidal fields are weak enough to
allow baroclinic instability, magnetic patterns in longitude should be
produced that could be transmitted through the convection zone to be
seen in the photosphere. The results also show it should be possible
to construct a baroclinic wave dynamo for the solar tachocline.
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Title: Baroclinic Instability in the Solar Tachocline
Authors: Gilman, Peter; Dikpati, Mausumi
2014ApJ...787...60G Altcode:
The solar tachocline is likely to be close to a geostrophic "thermal
wind," for which the Coriolis force associated with differential
rotation is closely balanced by a latitudinal pressure gradient,
leading to a tight relation between the vertical gradient of
rotation and the latitudinal entropy gradient. Using a hydrostatic but
nongeostrophic spherical shell model, we examine baroclinic instability
of the tachocline thermal wind. We find that both the overshoot and
radiative parts of the tachocline should be baroclinicly unstable at
most latitudes. Growth rates are roughly five times higher in middle
and high latitudes compared to low latitudes, and much higher in the
overshoot than in the radiative tachocline. They range in e-folding
amplification from 10 days in the high latitude overshoot tachocline,
down to 20 yr for the low latitude radiative tachocline. In the
radiative tachocline only, longitudinal wavenumbers m = 1, 2 are
unstable, while in the overshoot tachocline a much broader range
of m are unstable. At all latitudes and with all stratifications,
the longitudinal scale of the most unstable mode is comparable to the
Rossby deformation radius, while the growth rate is set by the local
latitudinal entropy gradient. Baroclinic instability in the tachocline
competing with instability of the latitude rotation gradient established
in earlier studies should be important for the workings of the solar
dynamo and should be expected to be found in most stars that contain
an interface between radiative and convective domains.
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Title: Theory of Solar Meridional Circulation at High Latitudes
Authors: Dikpati, Mausumi; Gilman, Peter A.
2012ApJ...746...65D Altcode: 2011arXiv1112.1107D
We build a hydrodynamic model for computing and understanding the Sun's
large-scale high-latitude flows, including Coriolis forces, turbulent
diffusion of momentum, and gyroscopic pumping. Side boundaries of
the spherical "polar cap," our computational domain, are located at
latitudes >= 60°. Implementing observed low-latitude flows as
side boundary conditions, we solve the flow equations for a Cartesian
analog of the polar cap. The key parameter that determines whether
there are nodes in the high-latitude meridional flow is epsilon =
2ΩnπH <SUP>2</SUP>/ν, where Ω is the interior rotation rate, n is
the radial wavenumber of the meridional flow, H is the depth of the
convection zone, and ν is the turbulent viscosity. The smaller the
epsilon (larger turbulent viscosity), the fewer the number of nodes in
high latitudes. For all latitudes within the polar cap, we find three
nodes for ν = 10<SUP>12</SUP> cm<SUP>2</SUP> s<SUP>-1</SUP>, two for
10<SUP>13</SUP>, and one or none for 10<SUP>15</SUP> or higher. For
ν near 10<SUP>14</SUP> our model exhibits "node merging": as the
meridional flow speed is increased, two nodes cancel each other, leaving
no nodes. On the other hand, for fixed flow speed at the boundary, as ν
is increased the poleward-most node migrates to the pole and disappears,
ultimately for high enough ν leaving no nodes. These results suggest
that primary poleward surface meridional flow can extend from 60° to
the pole either by node merging or by node migration and disappearance.
---------------------------------------------------------
Title: A Family of Simple Solar Dynamo Models for Testing Data
Assimilation Methods
Authors: Gilman, P.; Dikpati, M.
2011AGUFMSH51B2013G Altcode:
We present a family of simplified flux-transport type dynamo models
that can be used for testing various data assimilation schemes. This
family of models contain one, two or three layers in the vertical,
and are in cartesian geometry. More layers allow more detailed
representation of important dynamo processes, including transport by
meridional flow, production of poloidal fields by the alpha-effect, and
magnetic diffusivity that varies with depth. The models are capable of
simulating resonance effects, which make them sensitive to variations
in inputs by assimilation of data. The one-layer version of this family
is closely related to that used by Kitiashvili & Kosovichev (2008)
(ApJ 688,L49) to test Kalman filter methods on solar dynamo equations.
---------------------------------------------------------
Title: Resonance in Forced Flux-transport Dynamos
Authors: Gilman, Peter A.; Dikpati, Mausumi
2011ApJ...738..108G Altcode: 2011arXiv1107.2436G
We show that simple two- and three-layer flux-transport dynamos,
when forced at the top by a poloidal source term, can produce a
widely varying amplitude of toroidal field at the bottom, depending
on how close the meridional flow speed of the bottom layer is to the
propagation speed of the forcing applied above the top layer, and how
close the amplitude of the α-effect is to two values that give rise
to a resonant response. This effect should be present in this class of
dynamo model no matter how many layers are included. This result could
have implications for the prediction of future solar cycles from the
surface magnetic fields of prior cycles. It could be looked for in
flux-transport dynamos that are more realistic for the Sun, done in
spherical geometry with differential rotation, meridional flow, and
α-effect that vary with latitude and time as well as radius. Because
of these variations, if resonance occurs, it should be more localized
in time, latitude, and radius.
---------------------------------------------------------
Title: Physical Origin of Differences Among Various Measures of
Solar Meridional Circulation
Authors: Dikpati, Mausumi; Gilman, Peter A.; Ulrich, Roger K.
2010ApJ...722..774D Altcode: 2010arXiv1008.2772D
We show that systematic differences between surface Doppler and magnetic
element tracking measures of solar meridional flow can be explained by
the effects of surface turbulent magnetic diffusion. Feature-tracking
speeds are lower than plasma speeds in low and mid latitudes, because
magnetic diffusion opposes poleward plasma flow in low latitudes whereas
it adds to plasma flow at high latitudes. Flux-transport dynamo models
must input plasma flow; the model outputs yield estimates of the surface
magnetic feature tracking speed. We demonstrate that the differences
between plasma speed and magnetic pattern speed in a flux-transport
dynamo are consistent with the observed difference between these speeds.
---------------------------------------------------------
Title: Impact of changes in the Sun's conveyor-belt on recent
solar cycles
Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana;
Ulrich, Roger K.
2010GeoRL..3714107D Altcode:
Plasma flowing poleward at the solar surface and returning equatorward
near the base of the convection zone, called the meridional circulation,
constitutes the Sun's conveyor-belt. Just as the Earth's great oceanic
conveyor-belt carries thermal signatures that determine El Nino events,
the Sun's conveyor-belt determines timing, amplitude and shape of a
solar cycle in flux-transport type dynamos. In cycle 23, the Sun's
surface poleward meridional flow extended all the way to the pole,
while in cycle 22 it switched to equatorward near 60°. Simulations
from a flux-transport dynamo model including these observed differences
in meridional circulation show that the transport of dynamo-generated
magnetic flux via the longer conveyor-belt, with slower return-flow
in cycle 23 compared to that in cycle 22, may have caused the longer
duration of cycle 23.
---------------------------------------------------------
Title: Length of a minimum as predictor of next solar cycle's strength
Authors: Dikpati, Mausumi; Gilman, Peter A.; Kane, Rajaram P.
2010GeoRL..37.6104D Altcode: 2010GeoRL..3706104D
Motivated by a prevailing view that a long minimum leads to a weak
sunspot cycle, we estimate the correlation coefficients between the
length of a cycle minimum and (i) the following cycle's peak, (ii)
the preceding cycle's peak, (iii) following peak minus preceding
peak and (iv) depth of minimum. Using both sunspot number and spot
area data, we find that a long minimum is both followed and preceded
by weak cycles. Similarly short minima are followed and preceded by
strong cycles. Consistent with these results, we find no correlation
between the length of a cycle minimum and the difference in peaks
of the following and preceding cycles. From sunspot number data, for
longer-than-average minima, five following cycle peaks were lower than
that of the preceding cycles' peaks, while four were higher. Following
shorter-than-average minima, seven cycle peaks were higher than the
preceding peaks and seven were lower. Therefore one cannot predict
from the length of a minimum whether the next cycle will be stronger or
weaker than the preceding cycle. Thus we cannot predict whether cycle
24 will be stronger or weaker than 23. We also find that there is a
strong anticorrelation between the length of a solar cycle minimum
and the depth of that minimum. We define the depth as the least spot
number or spot area (13-rotation averaged) within the span of a cycle
minimum. We speculate that this anticorrelation is due to the longer
time available for annihilation of late cycle toroidal flux across
the equator in the case of a longer minimum.
---------------------------------------------------------
Title: Axisymmetric MHD Instabilities in Solar/Stellar Tachoclines
Authors: Dikpati, M.; Cally, P. S.; Gilman, P. A.; Miesch, M. S.
2009ASPC..416..525D Altcode:
We show that banded toroidal fields in the tachoclines of the Sun and
other stars should be unstable to 3-D axisymmetric overturning modes if
the peak toroidal field is ∼100 kG or more. This instability should
fragment and limit the amplitude of toroidal fields in tachoclines.
---------------------------------------------------------
Title: Obituary: John W. Firor (1927-2007)
Authors: Gilman, Peter A.
2009BAAS...41.1204G Altcode:
John W. Firor, a former Director of the High Altitude Observatory and
the National Center for Atmospheric Research, and a founder of the
Solar Physics Division of the American Astronomical Society, died of
Alzheimer's disease in Pullman, Washington on November 5, 2007, he was
80. He was born in Athens Georgia on October 18, 1927, where his father
was a professor of agricultural economics. <P />John had an unusually
diverse scientific career. His interest in physics and astrophysics
began while serving in the army, during which time he was assigned to
the Los Alamos National Laboratory, where he guarded highly radioactive
materials (many have heard him describe how informal the protections
were compared to later times). After his service he returned to college
and graduated in physics from Georgia Tech in 1949. He received his
Ph.D. from the University of Chicago in 1954, writing his thesis on
cosmic rays under John Simpson. John Firor would later remark that:
"If you needed cosmic rays to actually do anything, you are sunk." That
thought, partly in jest, may help explain his motivation for moving to
so many new scientific and management pursuits. <P />John moved from
cosmic ray physics to radio astronomy (particularly of the Sun) when
he began work at the Carnegie Institution of Washington's Department
of Terrestrial Magnetism, where he remained until 1961. During this
time, he met Walter Orr Roberts, then the Director of the High Altitude
Observatory (HAO) in Boulder, Colorado. HAO was then affiliated with
the University of Colorado. In 1959, a movement began to upgrade the
atmospheric sciences in the United States by establishing a National
Center, where the largest, most important atmospheric research problems
could be addressed. Roberts became the first Director of NCAR, as well
as the first president of the University Corporation for Atmospheric
Research (UCAR), the consortium of universities that was commissioned
to manage and staff the new Center. HAO became a scientific division of
NCAR (Roberts made it a requirement if he was to be NCAR Director). He
chose John to be the new HAO Director. John was 33 then; his leadership
potential was recognized very early. <P />While Director of HAO,
John presided over a very active scientific program that focused on
non-equilibrium thermodynamics, radiative transfer, plasma physics,
coronal spectroscopy, geomagnetism, physics of the Earth's upper
atmosphere, and solar activity. Prominent scientists such as Jack Evans,
Sydney Chapman, Gene Parker, Leo Goldberg and Donald Menzel visited to
share their insights and enthusiasm with the HAO staff. John was also
an active participant in HAO's well-known solar eclipse expeditions,
traveling to New Guinea and to Lake Chad in Africa. Late in his
tenure as HAO Director, he took the lead in helping to improve the
involvement of the AAS in solar physics, and solar physicists in
the AAS. Having led the AAS committee that planned its birth, he
was the founding chair of the new Solar Physics Division. Much less
importantly, he gave me a summer position in 1966 when I was joining
the faculty at CU--that is when I first met John. <P />In 1968, John
became the Director of NCAR when Walter Roberts decided to split
the directorship from the Presidency of UCAR. As NCAR Director,
John had responsibility for a vastly broadened program compared
to his HAO days. NCAR, the largest NSF supported research center,
conducted research covering all of atmospheric sciences, as well as
oceanography and solar-terrestrial physics. It also provided major
observational and computational facilities to the atmospheric sciences
community. John showed very quickly he understood and could guide
all of it. I came to know John best during this period, because in
1971 he chose me to head the Advanced Study Program, which supported
(and still supports) postdoctoral and graduate education. <P />In
that time, budgets were good, and NCAR was a relatively collegial,
informal place. But in December 1973, that all changed suddenly. NCAR
management was reviewed by an NSF-appointed Joint Evaluation Committee
(JEC), whose report raised many questions. Walter Roberts resigned
as UCAR President, and John had the task of working with the UCAR
Board to make major changes. During this time, John asked me to put
aside my ASP duties to assist him. Over the next 18 months, NCAR was
restructured, several Division Directors replaced, a new, more rigorous
scientific appointment system was developed, and a major budget cut was
absorbed. John did all of this; in my view, it was heroic. Then the
UCAR Board appointed a new UCAR President and NCAR Director, Francis
Bretherton, and John became the NCAR Executive Director. John used to
say that many people asked him how he felt about being 'demoted'--always
gracious, he said he never saw it that way. In fact, he was mentoring
Bretherton, who was very junior to John in experience, never having
held a major management position. The Bretherton-Firor 'team' managed
UCAR until 1980. <P />Then a new UCAR and NCAR management was chosen,
and John had to decide what to do next. He took a sabbatical and became
more deeply involved in environmental issues. But this interest had been
kindled much before. As early as 1970, he was writing and speaking about
environmental questions while NCAR Director. When he returned to NCAR,
he became the Director of the Advanced Study Program (ASP), what I had
been a decade before. He served NCAR in that role until 1996. When
he stepped down, he remarked that he had been a member of the NCAR
Directors Committee for 34 years; a record that he judged should never
be broken. <P />While John was attentive to the needs of the post-docs
and graduate students his program supported, ASP was a platform for
him to pursue his environmental interests. He spoke and wrote clearly
and influentially on environmental issues. He served on the Boards
of several important environmental organizations and wrote two well
received environmental books: "The Changing Atmosphere: A Global
Challenge" (1990), and, with his wife Judith Jacobsen "The Crowded
Greenhouse: Population, Climate Change and Creating a Sustainable World"
(2002). After ASP, he continued his focus on environmental issues as a
member of the Environmental and Societal Impacts group at NCAR. John
retired from NCAR in 2005. <P />John had many active pursuits beyond
his professional work. He was an accomplished pilot, with licenses for
flying single and multiengine aircraft, sailplanes, and balloons. He
piloted a sailplane in at least one meteorological field program. He
also was an avid river rafter. <P />John faced the disease that took
his life as he did all events in his life, with grace and dignity. He
endured the loss of two spouses to cancer, Merle Jenkins Firor in 1979,
and Judith Jacobsen in 2004. <P />John is survived by his four children
with his first wife, Daniel Firor of Seattle, Washington; Kay Firor
of Cove, Oregon; James Firor of Hotchkiss, Colorado, and Susan Firor
of Moscow, Idaho; a sister; a brother; and three grandchildren. His
children and his many friends in Boulder and elsewhere gave him loving
support during his days battling Alzheimer's. <P />John used to define
a 'southern gentleman' as a man dressed in white linen suit on a hot
dusty summer day in a small Georgia town who could cross the street
without breaking a sweat. John and his intellect and his management
ability were like that; he could deal gracefully and successfully with
almost anything that came his way. A man of great accomplishment,
he rarely showed an ego to match. In the darkest days following the
JEC Report, he almost single-handedly invented a new NCAR scientific
appointment system. He chose the first cadre of 'senior scientists'
to populate the top rank. There were about eighteen members in this
group, but there was one name conspicuously absent - his own. This
'error' was quietly corrected by the UCAR Board.
---------------------------------------------------------
Title: Sequential Data-assimilation in a Flux-transport Dynamo Model
Authors: Dikpati, Mausumi; de Toma, G.; Gilman, P. A.; Anderson,
J. L.; Ulrich, R. K.; Boyden, J. E.
2009SPD....40.1114D Altcode:
Applying a very simplified data-nudging technique in a flux-transport
dynamo, Dikpati, de Toma and Gilman predicted solar cycle amplitude
and onset-timing of cycle 24 seperately. In order to simultaneously
predict cycle amplitude and timing we have developed a sequential
data-assimilation technique, in a similar way used in atmospheric and
oceanic prediction models. However two major difficulties in applying
this technique in solar dynamo models are, (i) equatorward return
meridional circulation is unknown, (ii) time-varying surface flow
measurements have not been available for years prior to 1996. With
recent progress of Mount Wilson Observatory's flow-data analysis by
Ulrich and colleagues, we can now go back to 1985. We build sequential
data-assimilation into a flux-transport dynamo model by (i) solving mean
and perturbation equations by incorporating time-varying meridional
flow since 1985; (ii) investigating transport of assimilated poloidal
magnetic fields from surface to tachocline, where they are sheared by
differential rotation to create spot-producing fields; (iii) updating
model after a finite time-interval, by comparing model-output with
observations; (iv) forecasting simultaneously cycle-amplitude, duration
and shape. We form an ensemble of model-runs whose outputs calibrate
best with surface magnetic observations. The ensemble-average gives the
simultaneous prediction of solar cycle timing, amplitude and shape. <P
/>This work is partially supported by NASA grant NNX08AQ34G.
---------------------------------------------------------
Title: Flux-Transport Solar Dynamos
Authors: Dikpati, Mausumi; Gilman, Peter A.
2009SSRv..144...67D Altcode:
Large-scale solar dynamo models were first built by Parker (1955). Over
the past half a century these models have evolved significantly. We
discuss here the development of a class of large-scale dynamo models
which include, along with the α-effect and Ω-effect, an important
third process, flux transport by meridional circulation. We present
the properties of this ‘flux-transport’ dynamo, including the
crucial role meridional circulation plays in giving this dynamo
predictive power.
---------------------------------------------------------
Title: Axisymmetric MHD Instabilities in Solar/Stellar Tachoclines
Authors: Dikpati, Mausumi; Gilman, Peter A.; Cally, Paul S.; Miesch,
Mark S.
2009ApJ...692.1421D Altcode:
Extensive studies over the past decade showed that HD and MHD
nonaxisymmetric instabilities exist in the solar tachocline for
a wide range of toroidal field profiles, amplitudes, and latitude
locations. Axisymmetric instabilities (m = 0) do not exist in two
dimensions, and are excited in quasi-three-dimensional shallow-water
systems only for very high field strengths (2 mG). We investigate here
MHD axisymmetric instabilities in a three-dimensional thin-shell model
of the solar/stellar tachocline, employing a hydrostatic, non-Boussinesq
system of equations. We deduce a number of general properties of the
instability by use of an integral theorem, as well as finding detailed
numerical solutions for unstable modes. Toroidal bands become unstable
to axisymmetric perturbations for solar-like field strengths (100
kG). The e-folding time can be months down to a few hours if the field
strength is 1 mG or higher, which might occur in the solar core, white
dwarfs, or neutron stars. These instabilities exist without rotation,
with rotation, and with differential rotation, although both rotation
and differential rotation have stabilizing effects. Broad toroidal
fields are stable. The instability for modes with m = 0 is driven from
the poleward shoulder of banded profiles by a perturbation magnetic
curvature stress that overcomes the stabilizing Coriolis force. The
nonaxisymmetric instability tips or deforms a band; with axisymmetric
instability, the fluid can roll in latitude and radius, and can convert
bands into tubes stacked in radius. The velocity produced by this
instability in the case of low-latitude bands crosses the equator,
and hence can provide a mechanism for interhemispheric coupling.
---------------------------------------------------------
Title: Flux-Transport Solar Dynamos
Authors: Dikpati, Mausumi; Gilman, Peter A.
2009odsm.book...67D Altcode:
Large-scale solar dynamo models were first built by Parker (1955). Over
the past half a century these models have evolved significantly. We
discuss here the development of a class of large-scale dynamo models
which include, along with the α-effect and Ω-effect, an important
third process, flux transport by meridional circulation. We present the
properties of this `flux-transport' dynamo, including the crucial role
meridional circulation plays in giving this dynamo predictive power.
---------------------------------------------------------
Title: Solar Dynamo and Magnetic Self-Organization
Authors: Kosovichev, A. G.; Arlt, R.; Bonanno, A.; Brandenburg,
A.; Brun, A. S.; Busse, F.; Dikpati, M.; Hill, F.; Gilman, P. A.;
Nordlund, A.; Ruediger, G.; Stein, R. F.; Sekii, T.; Stenflo, J. O.;
Ulrich, R. K.; Zhao, J.
2009astro2010S.160K Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Three-dimensional magneto-shear instabilities in the solar
tachocline - II. Axisymmetric case
Authors: Cally, Paul S.; Dikpati, Mausumi; Gilman, Peter A.
2008MNRAS.391..891C Altcode: 2008MNRAS.tmp.1248C
A Boussinesq model of the development of non-axisymmetric (in particular
m = 1) three-dimensional magneto-shear instabilities in the solar
tachocline was presented in Paper I. However, there it was erroneously
concluded that the axisymmetric (m = 0) modes are stable, and they were
not discussed further. Here it is shown that, although m = 0 modes are
indeed stable for broad magnetic profiles, they are strongly unstable to
radial shredding (high radial wavenumber) instabilities on the poleward
shoulders of toroidal magnetic bands at high field strengths (roughly
40-100kG depending on bandwidth and latitude). These instabilities
have growth rates comparable to or greater than those for tipping
instabilities (m = 1) in many cases, but both are strongly stabilized
by gravitational stratification characteristic of the upper radiative
core. Weaker fields are m = 0 stable (though weakly m = 1 unstable),
even in neutral gravitational stratification (convection zone).
---------------------------------------------------------
Title: A solar mean field dynamo benchmark
Authors: Jouve, L.; Brun, A. S.; Arlt, R.; Brandenburg, A.; Dikpati,
M.; Bonanno, A.; Käpylä, P. J.; Moss, D.; Rempel, M.; Gilman, P.;
Korpi, M. J.; Kosovichev, A. G.
2008A&A...483..949J Altcode:
Context: The solar magnetic activity and cycle are linked to an
internal dynamo. Numerical simulations are an efficient and accurate
tool to investigate such intricate dynamical processes. <BR />Aims:
We present the results of an international numerical benchmark
study based on two-dimensional axisymmetric mean field solar dynamo
models in spherical geometry. The purpose of this work is to provide
reference cases that can be analyzed in detail and that can help in
further development and validation of numerical codes that solve such
kinematic problems. <BR />Methods: The results of eight numerical
codes solving the induction equation in the framework of mean field
theory are compared for three increasingly computationally intensive
models of the solar dynamo: an αΩ dynamo with constant magnetic
diffusivity, an αΩ dynamo with magnetic diffusivity sharply varying
with depth and an example of a flux-transport Babcock-Leighton dynamo
which includes a non-local source term and one large single cell of
meridional circulation per hemisphere. All cases include a realistic
profile of differential rotation and thus a sharp tachocline. <BR
/>Results: The most important finding of this study is that all codes
agree quantitatively to within less than a percent for the αΩ dynamo
cases and within a few percent for the flux-transport case. Both
the critical dynamo numbers for the onset of dynamo action and the
corresponding cycle periods are reasonably well recovered by all
codes. Detailed comparisons of butterfly diagrams and specific cuts of
both toroidal and poloidal fields at given latitude and radius confirm
the good quantitative agreement. <BR />Conclusions: We believe that
such a benchmark study will be a very useful tool since it provides
detailed standard cases for comparison and reference.
---------------------------------------------------------
Title: Global solar dynamo models: Simulations and predictions
Authors: Dikpati, Mausumi; Gilman, Peter A.
2008JApA...29...29D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Polar Flux, Cross-equatorial Flux, and Dynamo-generated
Tachocline Toroidal Flux as Predictors of Solar Cycles
Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A.
2008ApJ...675..920D Altcode:
We evaluate the skill of three solar cycle predictors, namely, polar
magnetic flux, flux crossing the equator, and tachocline toroidal
flux, using both observations and a calibrated dynamo model. Polar
flux measurements are available only for the past three sunspot
cycles, implying poor statistics. However, the correlation between
observed north and south polar flux peaks, and peaks of the next
sunspot cycle is r = 0.785. We find that the correlation between
the observed cross-equatorial flux and the observed peak of the next
solar cycle is also high, close to that of Cameron & Schüssler,
and the statistics are more reliable. Thus, the cross-equatorial
flux is a better predictor of the next cycle than is the polar
flux. From the dynamo model, the correlations with observed cycle
peaks for polar flux, cross-equatorial flux, and toroidal flux are
0.48, 0.76, and 0.96, respectively. All these correlations decline
when the northern and southern hemispheres are simulated separately,
as well as with shortening of the averaging length in input data. A
very high correlation between the model polar flux at the end of
a cycle and the observed peak of that cycle implies that, within a
calibrated flux transport dynamo, the polar flux follows the sunspot
cycle, rather than being a precursor to it. With short-term averaging
of input data, the polar and cross-equatorial fluxes retain much more
short-term variability than does the toroidal flux. This is because
the long traversal time of the input poloidal fields to the bottom of
the convection zone in a mean-field model smooths out the short-term
variability in the toroidal flux. The observed slowdown in meridional
flow during 1996-2004 leads to a weaker polar flux, but a stronger
cross-equatorial flux compared to the case with steady meridional
flow. We infer that it is unlikely that both the cross-equatorial and
the polar fluxes can be good predictors of solar cycle peaks.
---------------------------------------------------------
Title: The Waldmeier Effect: An Artifact of the Definition of Wolf
Sunspot Number?
Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana
2008ApJ...673L..99D Altcode:
We show that the well-known "Waldmeier effect," or anticorrelation
between the peak in sunspot number of a cycle and the time from minimum
to reach that peak, is not present in sunspot area data averaged in
the same way. With a 13-rotation Gaussian running average applied to
both sunspot number and sunspot area, we find the correlation between
cycle rise time and cycle peak is r = - 0.71 for sunspot number and r =
10<SUP>-4</SUP> for sunspot area. Hathaway et al. previously showed
that the Waldmeier effect is much weaker (r = - 0.34) in sunspot
group number. Thus the Waldmeier effect may be specific to only Wolf
sunspot number.
---------------------------------------------------------
Title: Simulating Solar Cycles in Northern and Southern Hemispheres
by Assimilating Magnetic Data into a Calibrated Flux-Transport Dynamo
Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana; Ghosh,
Siddhartha S.
2007SoPh..245....1D Altcode:
We use the flux-transport dynamo prediction scheme introduced by
Dikpati, de Toma, and Gilman (Geophys. Res. Lett.33, L05102, 2006) to
make separate simulations and predictions of sunspot cycle peaks for
northern and southern hemispheres. Despite the division of the data,
the skill level achieved is only slightly lower than that achieved
for the sum of both hemispheres. The model shows skill at simulating
and predicting the difference in peaks between North and South,
provided that difference is more than a few percent. The simulation
and prediction skill is achieved without adjustment to any parameters
of the model that were used when peaks for the sum of North and South
sunspot areas was simulated. The results are also very insensitive to
the averaging length applied to the input data, provided the simulations
and predictions are for peaks defined by averaging the observations
over at least 13 rotations. However, in its present form, the model
is not capable of skillfully simulating or predicting short-time-scale
features of individual solar cycles.
---------------------------------------------------------
Title: Global MHD Instabilities in a Three-dimensional Thin-Shell
Model of Solar Tachocline
Authors: Gilman, Peter A.; Dikpati, Mausumi; Miesch, Mark S.
2007ApJS..170..203G Altcode:
We generalize the linear analysis of the global instability of
coexisting differential rotation and toroidal magnetic fields in
the solar tachocline to include continuous radial stratification,
thermodynamics, and finite tachocline thickness, as perturbed by
three-dimensional disturbances of longitudinal wavenumbers m=1, 2. For
radiative tachocline stratification, the instability for both banded
and broad toroidal field profiles is similar to the two-dimensional
and shallow water cases studied previously, even though the unstable
modes have substantial vertical structure. For overshoot tachocline
stratification, instability for banded toroidal fields with peaks
<~20 kG is similar to the corresponding shallow water case, but for
substantially higher (perhaps unrealistic) toroidal fields there is no
low-subadiabaticity cutoff, and modes appear with much higher growth
rate and increasingly negative phase velocities in longitude, analogous
to those found earlier by Cally. All of these results are only modestly
sensitive to the tachocline thickness chosen. For broad toroidal field
profiles, the instability results are similar to the two-dimensional
and shallow water cases for toroidal field peaks up to at least 80 kG,
unless the shell has a thickness that is a substantial fraction of a
pressure scale height. For thinner shells, only above about 94 kG do
the high growth rate, low phase velocity modes appear, with structure
similar to the “polar kink” instability Cally found. But even in
this case, we find that the slower growing “clam-shell” instability
eventually replaces the faster growing polar kink modes. We conclude
that for conditions most likely to occur in the solar tachocline,
such as peak toroidal fields limited by the dynamo to 20 kG or less,
the two-dimensional and shallow water type unstable modes are likely
to predominate even when the tachocline has finite thickness and the
modes can have radial structure.
---------------------------------------------------------
Title: Magneto-Shear Instabilities in the Solar Tachocline
Authors: Miesch, Mark S.; Gilman, P. A.; Dikpati, M.
2007AAS...210.4607M Altcode: 2007BAAS...39..161M
Previously, linear theory has demonstrated that toroidal magnetic fields
in the solar tachocline are destabilized by the presence of latitudinal
differential rotation. For strong fields, such that the magnetic energy
is comparable to the kinetic energy of the differential rotation,
the most unstable modes are those with longitudinal wavenumber m=1 and
these have been referred to as clam-shell or tipping instabilities for
broad and banded toroidal field profiles respectively. Here we present
nonlinear, three-dimensional simulations of these instabilities
under both freely-evolving and forced conditions. Although the
instabilities are allowed to have an arbitrary vertical structure,
the dynamics remain quasi-2D, proceeding on horizontal surfaces which
are nearly decoupled. The clam-shell instability saturates by opening
up completely until horizontal loops of field become perpendicular to
the equatorial plane. By contrast, the tipping instability saturates
at relatively moderate tipping angles of 6-12 degrees by forming a
jet of fluid along the axis of the tipped band. When the rotational
shear and the magnetic field are maintained indefinitely by external
forcing, clam-shell instabilities can operate continually, exhibiting
a quasi-periodic behavior as mean toroidal fields alternately build
up and destabilize.
---------------------------------------------------------
Title: Steps for Building a Calibrated Flux-Transport Dynamo for
the Sun
Authors: Dikpati, M.; Gilman, P. A.
2007SoPh..241....1D Altcode:
Given the complexity involved in a flux-transport-type dynamo driven
by both Babcock - Leighton and tachocline α effects, we present here
a step-by-step procedure for building a flux-transport dynamo model
calibrated to the Sun as a guide for anyone who wishes to build this
kind of model. We show that a plausible sequence of steps to reach
a converged solution in such a dynamo consists of (i) numerical
integration of a classical α - ω dynamo driven by a tachocline
α effect, (ii) continued integration with inclusion of meridional
circulation to convert the model into a flux-transport dynamo driven by
only a tachocline α effect, (iii) final integration with inclusion of
a Babcock - Leighton surface α effect, resulting in a flux-transport
dynamo that can be calibrated to obtain a close fit of model output
with solar observations.
---------------------------------------------------------
Title: Nonlinear Evolution of Global Magnetoshear Instabilities in
a Three-dimensional Thin-Shell Model of the Solar Tachocline
Authors: Miesch, Mark S.; Gilman, Peter A.; Dikpati, Mausumi
2007ApJS..168..337M Altcode:
We investigate global instabilities of toroidal fields and differential
rotation in the solar tachocline using a three-dimensional thin-shell
model. We initiate our nonlinear numerical simulations by superposing
random, high-wavenumber perturbations on an equilibrium state and we
then allow the system to evolve freely as the instabilities develop,
grow exponentially, and saturate. For broad toroidal field profiles
the dominant mode is the clamshell instability previously identified
in two-dimensional and shallow-water investigations in which loops of
field in the northern and southern hemispheres tilt and reconnect,
eventually becoming perpendicular to the equatorial plane. If the
initial toroidal field is instead confined to thin bands of alternating
polarity in the northern and southern hemispheres, the evolution is more
complex. At early times, a previously unidentified instability occurs
near the edges of each band that is characterized by a high longitudinal
wavenumber m. These edge instabilities are later superseded by an m=1
tipping instability whereby field lines tilt in latitude as in the
broad-field case. The tipping instability saturates by forming a jet of
fluid within each band, which provides gyroscopic stabilization. Both
the clamshell and the tipping instabilities are quasi-two-dimensional
and proceed most efficiently near the impenetrable boundaries where the
vertical velocity vanishes. Strong stable stratification enhances this
tendency by decoupling horizontal layers. These simulations demonstrate
the robustness of global m=1 instabilities in differentially rotating
spherical shells threaded by toroidal magnetic fields and as such
have important implications for tachocline dynamics, including dynamo
processes and tachocline confinement.
---------------------------------------------------------
Title: Global MHD instabilities of the tachocline
Authors: Gilman, Peter A.; Cally, Paul S.
2007sota.conf..243G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Dynamo-based simulations of solar activity differences between
north and south hemispheres and forecasts for cycle 24
Authors: Dikpati, M.; Gilman, P. A.; de Toma, G.
2006AGUFMSH22A..06D Altcode:
Recently we (GRL, 2006, vol.33, L05102, doi:10.1029/2005GL025221) built
a predictive tool based on a Babcock-Leighton type flux-transport
dynamo model of solar cycle. By assimilating the observed surface
magnetic source data since cycle 12, we ran the model and showed that
the model can correctly simulate the relative peaks of cycles 16 through
23. Extending the simulation into the future we predicted that cycle 24
will be 30-50 percent stronger than current cycle 23, because a strong
'seed' for cycle 24 is being formed, from the combination of latitudinal
fields from past three cycles, 21, 22 and 23, in the dynamo layer in
our model. We are currently exploring the data assimilation in our model
for the N and S hemispheres separately, in order to simulate hemispheric
asymmetries in the solar cycle features. Preliminary results show that
the model has skill in simulating the N and S hemispheres separately,
as well as the difference between the two hemispheres. We will discuss
which hemisphere of the Sun should become more active in cycle 24 and
which hemisphere should reach cycle peak first. This work is partially
supported by NASA's LWS grant NNH05AB521, SR&T grant NNH06AD51I and
the NCAR Director's Opportunity Fund. National Center for Atmospheric
Research is sponsored by National Science Foundation.
---------------------------------------------------------
Title: Simulating and Predicting Solar Cycles Using a Flux-Transport
Dynamo
Authors: Dikpati, Mausumi; Gilman, Peter A.
2006ApJ...649..498D Altcode:
We construct a predictive tool based on a Babcock-Leighton-type
flux-transport dynamo model of a solar cycle, run the model by updating
the surface magnetic source using old cycles' data since cycle 12,
and show that the model can correctly simulate the relative peaks of
cycles 16-23. The simulations use the first four cycles to load the
meridional circulation conveyor belt to create the Sun's memory about
its past magnetic fields. Extending the simulation into the future, we
predict that cycle 24 will be 30%-50% stronger than the current cycle
23. These simulations and predictions are robust for a wide range of
convection zone magnetic diffusivity values between 3×10<SUP>10</SUP>
and 2×10<SUP>11</SUP> cm<SUP>2</SUP> s<SUP>-1</SUP>. Our model
predictions are the same for three different treatments of the unknown
surface magnetic source for the cycles to be predicted, namely (1)
assuming some cyclic pattern, (2) incorporating “zero” surface source,
or (3) including a surface source derived from the self-excited version
of the dynamo model. Technique 3, for treating the surface source for
cycles to be predicted, also shows significant skill in predicting
two cycles ahead. Analyzing the evolution of magnetic field patterns
over a full magnetic cycle, we show that the key to success of our
prediction model lies in the formation of a “seed” for producing
cycle n from the combination of latitudinal fields at high latitudes
from three past cycles, n-1, n-2, and n-3, instead of the previous
cycle's polar fields. These results have many implications for both
solar and stellar dynamo modeling.
---------------------------------------------------------
Title: Simulating And Predicting Solar Cycles Using A Flux-transport
Dynamo
Authors: Dikpati, Mausumi; Gilman, P. A.
2006AAS...208.6507D Altcode: 2006BAAS...38Q.145D
We construct a predictive tool based on a Babcock-Leighton type
flux-transport dynamo model of solar cycle, run the model by updating
the surface magnetic source using old cycles' data since cycle 12,
and show that the model can correctly simulate the relative peaks
of cycles 16 through 23. The simulations use the first 4 cycles to
load the meridional circulation conveyor belt to create the Sun's
memory about its past magnetic fields. Extending the simulation into
the future we predict that cycle 24 will be 30-50% stronger than
current cycle 23. These simulations and predictions are robust for
a wide range of convection zone diffusivity values. Analyzing the
evolution of magnetic field patterns over a full magnetic cycle,
we show that the key to success of our prediction model lies in the
formation of a 'seed' for producing cycle <i>n</i> from
the combination of latitudinal poloidal fields at high latitudes from
past three cycles, <i>n</i>-1, <i>n</i>-2 and
<i>n</i>-3, instead of previous cycle's polar fields. These
results have many implications for both solar and stellar dynamo
modeling. The quality of our simulations and predictions implies that
the flux-transport models must contain most of the important physical
processes acting in the solar dynamo.
---------------------------------------------------------
Title: A 42 Year Quest to Understand the Solar Dynamo and Predict
Solar Cycles
Authors: Gilman, Peter A.
2006SPD....37.3101G Altcode: 2006BAAS...38..257G
<table border="1" cellpadding="1" class="DisplayTable"
id="{C4167B88-99D7-4F70-A4E7-C3D16AAB71FF}"><caption></caption><tr><td
rowspan="1" colspan="1">For me this quest began in 1964, when
I was a graduate student in meteorology at MIT. Most recently I
have participated in a collaboration led by Mausumi Dikpati that has
successfully simulated and ”predicted” the relative peaks of the past
8 solar cycles, using a ”flux-transport” solar dynamo. We are also
”hot on the trail” of a theory for active longitudes on the sun, that
involves global MHD instability of the solar tachocline. Over the 42
years, I have seen the solar dynamo problem declared ”solved” by mean
field dynamo theory in the 1970”s, followed by near total rejection of
this conclusion in the 1980”s due to helioseismic measurements of solar
rotation at depth, followed by a spectacular comeback in the 1990”s to
the present, in the form of flux transport models in which meridional
circulation is an essential component. I have seen 3D global MHD models
for solar convection and differential rotation work well as dynamos,
but fail as solar dynamos, yielding backward ”butterfly diagrams”,
or no butterfly diagrams at all. In the near future, we will search for
the limits of skill of axisymmetric flux transport models to predict
details of individual cycles, and we will generalize the theory to
include longitude dependence, by which we hope to produce a unified
theory of the solar cycle and active longitudes. Since flux-transport
dynamos work so well for the sun, they ought to work very well for
many other stars.</td></tr></table>
---------------------------------------------------------
Title: A 42 Year Quest To Understand The Solar Dynamo And Predict
Solar Cycles
Authors: Gilman, Peter A.
2006AAS...208.3101G Altcode: 2006BAAS...38..108G
For me this quest began in 1964, when I was a graduate student in
meteorology at MIT. Most recently I have participated in a collaboration
led by Mausumi Dikpati that has successfully simulated and 'predicted'
the relative peaks of the past 8 solar cycles, using a 'flux-transport'
solar dynamo. We are also 'hot on the trail' of a theory for active
longitudes on the sun, that involves global MHD instability of the
solar tachocline. Over the 42 years, I have seen the solar dynamo
problem declared 'solved' by mean field dynamo theory in the 1970's,
followed by near total rejection of this conclusion in the 1980's due
to helioseismic measurements of solar rotation at depth, followed by
a spectacular comeback in the 1990's to the present, in the form of
flux transport models in which meridional circulation is an essential
component. I have seen 3D global MHD models for solar convection and
differential rotation work well as dynamos, but fail as solar dynamos,
yielding backward 'butterfly diagrams', or no butterfly diagrams at
all. In the near future, we will search for the limits of skill of
axisymmetric flux transport models to predict details of individual
cycles, and we will generalize the theory to include longitude
dependence, by which we hope to produce a unified theory of the solar
cycle and active longitudes. Since flux-transport dynamos work so well
for the sun, they ought to work very well for many other stars.
---------------------------------------------------------
Title: Predicting the strength of solar cycle 24 using a
flux-transport dynamo-based tool
Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A.
2006GeoRL..33.5102D Altcode:
We construct a solar cycle strength prediction tool by modifying a
calibrated flux-transport dynamo model, and make predictions of the
amplitude of upcoming solar cycle 24. We predict that cycle 24 will have
a 30-50% higher peak than cycle 23, in contrast to recent predictions by
Svalgaard et al. and Schatten, who used a precursor method to forecast
that cycle 24 will be considerably smaller than 23. The skill of our
approach is supported by the flux transport dynamo model's ability to
correctly 'forecast' the relative peaks of cycles 16-23 using sunspot
area data from previous cycles.
---------------------------------------------------------
Title: Penetration of Dynamo-generated Magnetic Fields into the
Sun's Radiative Interior
Authors: Dikpati, Mausumi; Gilman, Peter A.; MacGregor, Keith B.
2006ApJ...638..564D Altcode:
Any large-scale magnetic fields present in solar/stellar radiative
interiors have so far been thought to be primordial or residuals
from extinct dynamos. We show that a regular cyclic dynamo can
also be the origin of strong magnetic fields in the solar radiative
tachocline and interior below. By exploiting a kinematic, mean-field
flux-transport dynamo, we show that for a wide range of core-diffusivity
values, from 10<SUP>9</SUP> cm<SUP>2</SUP> s<SUP>-1</SUP> down to a
molecular diffusivity of 10<SUP>3</SUP> cm<SUP>2</SUP> s<SUP>-1</SUP>,
oscillatory dynamo fields penetrate below the tachocline. Amplitudes
of these fields are in the range of ~1 kG to 3×10<SUP>3</SUP> kG,
depending on core diffusivity value, when the dynamo produces ~100
kG peak toroidal fields in the overshoot tachocline. For a low enough
core diffusivity (<~10<SUP>7</SUP> cm<SUP>2</SUP> s<SUP>-1</SUP>),
there is also a steady (nonreversing) dynamo in the radiative tachocline
and below, which generates strong toroidal field of amplitude ~1 kG to
3×10<SUP>3</SUP> kG or more there. The key elements in this dynamo
are the low diffusivity, the differential rotation near the bottom
of the tachocline, and an assumed tachocline α-effect. The Lorentz
force feedback may limit oscillatory dynamo fields to ~30 kG, for
which the mean nonreversing toroidal fields is still ~300 kG, for the
lowest core diffusivity value. The presence of strong oscillatory and
steady toroidal fields in the radiative tachocline implies that there
cannot be a slow tachocline; the dynamics should always be fast there,
dominated by MHD. These results are obtained using solar parameters,
but they should also apply generally to stars with convecting shells
and perhaps also with convective cores.
---------------------------------------------------------
Title: Jets in the Solar Tachocline as Diagnostics of Global MHD
Processes
Authors: Christensen-Dalsgaard, J.; Corbard, T.; Dikpati, M.; Gilman,
P. A.; Thompson, M. J.
2005ASPC..346..115C Altcode:
Multiple theories predict the existence of prograde fluid jets in the
solar tachocline. We find helioseismic evidence of a prograde jet near
60° latitude in N and S hemispheres that persists through almost all
of the current solar cycle. This evidence favors a hydrodynamic origin
for the jet, from global instability of the differential rotation of
the tachocline. We see no evidence for jets that migrate toward the
equator with the advancing solar cycle, which tends to rule out jets
associated with toroidal field bands in the tachocline.
---------------------------------------------------------
Title: A Shallow-Water Theory for the Sun's Active Longitudes
Authors: Dikpati, Mausumi; Gilman, Peter A.
2005ApJ...635L.193D Altcode:
We show that the global MHD shallow-water instability of differential
rotation and toroidal field bands in the solar tachocline provides a
possible mechanism for the formation and evolution of active longitudes
seen in synoptic maps of photospheric magnetic fields. The mechanism
involves the production of upward bulges at selected longitudes in
the overshoot tachocline that contain significant toroidal fields.
---------------------------------------------------------
Title: Constraints on the Applicability of an Interface Dynamo to
the Sun
Authors: Dikpati, Mausumi; Gilman, Peter A.; MacGregor, Keith B.
2005ApJ...631..647D Altcode:
Taking into account the helioseismically inferred interior structure,
we show that a pure interface-type dynamo does not work for the
Sun if the skin effect for poloidal fields does not allow them to
penetrate the tachocline. Using a simple mean-field kinematic α-Ω
dynamo model, we demonstrate that, in the absence of tachocline radial
shear participating in the dynamo process, a latitudinal differential
rotation can provide the necessary Ω-effect to drive an oscillation
in an interface dynamo, but it alone cannot produce the latitudinal
migration. We show that to make an interface dynamo work with the
constraints of interior structure and skin depth, a meridional
circulation is essential. We conclude that a flux-transport dynamo
driven by both the Babcock-Leighton and interface/bottom α-effects
is a robust large-scale dynamo for the Sun.
---------------------------------------------------------
Title: Recovering Solar Toroidal Field Dynamics from Sunspot Location
Patterns
Authors: Norton, Aimee A.; Gilman, Peter A.
2005ApJ...630.1194N Altcode: 2005astro.ph..6025N
We analyze both Kitt Peak magnetogram data and MDI continuum
intensity sunspot data to search for the following solar toroidal
band properties: width in latitude and the existence of a tipping
instability (longitudinal m=1 mode) for any time during the solar
cycle. In order to determine the extent to which we can recover the
toroidal field dynamics, we forward-model artificially generated sunspot
distributions from subsurface toroidal fields that we have assigned
certain properties. Sine-curve fitting of Kitt Peak magnetogram data
provided an upper limit of 15° to the tipping amplitude but could not
adequately separate the tip from the width of the toroidal band. We
then analyzed two sunspot distribution parameters using MDI and model
data: the average latitudinal separation of sunspot pairs as a function
of longitudinal separation and the number of sunspot pairs creating
a given angle with respect to the east-west direction. A toroidal
band of 10° width with a constant tipping of 5° best fits MDI data
early in the solar cycle, when the sunspot band is at high latitudes
(>18.5d). A toroidal band of 20° width with a tipping amplitude
decreasing in time from 5<SUP>deg</SUP> to 0<SUP>deg</SUP> best fits MDI
data late in the solar cycle when the sunspot band is at low latitudes
(<18.5d). Model data generated by untipped toroidal bands cannot fit
MDI high-latitude data using χ<SUP>2</SUP> goodness-of-fit criteria and
can fit only one sunspot distribution parameter at low latitudes. Tipped
toroidal bands satisfy χ<SUP>2</SUP> criteria at both high and low
latitudes for both sunspot distribution parameters. We conclude that
this is evidence to reject the null hypothesis-that toroidal bands
in the solar tachocline do not experience a tipping instability-in
favor of the hypothesis that the toroidal band experiences an m=1
tipping instability for a significant portion of the solar cycle. Our
finding that the band widens from 10° early in the solar cycle to 20°
late in the solar cycle may be explained in theory by magnetic drag
spreading the toroidal band due to altered flow along the tipped field
lines. Higher m modes, most notably m=2 and 6, are apparent in MDI data,
but further analysis is needed to determine this property in detail.
---------------------------------------------------------
Title: Concentration of Toroidal Magnetic Field in the Solar
Tachocline by η-Quenching
Authors: Gilman, Peter A.; Rempel, Matthias
2005ApJ...630..615G Altcode: 2005astro.ph..4003G
We show that if the turbulent magnetic diffusivity used in solar dynamos
is assumed to be “quenched” by increasing toroidal fields, much
larger amplitude and more concentrated toroidal fields can be induced
by differential rotation from an assumed poloidal field than if there is
no quenching. This amplification and concentration mechanism is weakened
and bounded by jXB feedbacks on the differential rotation. Nevertheless,
it is strong enough to contribute to the creation of ~100 kG toroidal
fields near the base of the convection zone, perhaps in conjunction
with the “exploding flux tube” process. Such high fields are necessary
for sunspots to occur in low solar latitudes.
---------------------------------------------------------
Title: Comments on "Full-sphere simulations of circulation-dominated
solar dynamo: Exploring the parity issue"
Authors: Dikpati, M.; Rempel, M.; Gilman, P. A.; MacGregor, K. B.
2005A&A...437..699D Altcode:
Using two distinct simulation codes that respectively apply
semi-implicit and fully explicit schemes, we perform calculations
of a 2D kinematic Babcock-Leighton type flux-transport dynamo with
Chatterjee et al.'s parameter settings. We show that their solutions are
diffusion-dominated, rather than circulation-dominated as their title
implies. We also have been unable to reproduce several properties of
their dynamo solutions, namely we obtain a much faster cycle with ~
4 times shorter period than theirs, with highly overlapping cycles;
a polar field value of ∼ 2 kG if one has to produce a ~ 100 kG
toroidal field at convection zone base; and quadrupolar parity as
opposed to Chatterjee et al.'s dipolar parity solutions.
---------------------------------------------------------
Title: The tachocline and the solar dynamo
Authors: Gilman, P. A.
2005AN....326..208G Altcode:
The solar tachocline contains a rich variety of physics, and contributes
in many ways to the workings of the solar dynamo. It includes complex
quasi-equilibrium states supported by the near balances of gas and
magnetic pressure gradients, gravity, magnetic curvature stresses,
Coriolis and other forces. The dynamics include overshooting convection,
waves of several types, hydrodynamic and MHD instabilities on several
spatial and temporal scales, several types of boundary layers and
dynamo action. The dynamo processes present include the in situ
generation of poloidal fields by global MHD and/or magnetic buoyancy
instabilities; the advection of these poloidal fields by meridional
circulation; the generation of toroidal fields from shearing by the
differential rotation; and storage of these fields and their subsequent
eruption into the convection zone. The tachocline probably also plays
a significant role in creating magnetic patterns that are seen in the
photosphere. This talk will of necessity focus on a subset of these
topics, including particularly global MHD instabilities and whether
we can see evidence of them in surface magnetic data, meridional
circulation and the boundary layers that limit it, possible jets in
the tachocline, and certain solar dynamo questions.
---------------------------------------------------------
Title: The solar photograph archive of the Mount Wilson Observatory. A
resource for a century of digital data
Authors: Lefebvre, S.; Ulrich, R. K.; Webster, L. S.; Varadi, F.;
Javaraiah, J.; Bertello, L.; Werden, L.; Boyden, J. E.; Gilman, P.
2005MmSAI..76..862L Altcode:
The solar telescopes and spectroheliographs of the Mount Wilson
Observatory were among the earliest modern facilities for the study of
the solar surface. The photographic collection of the solar program
at Mt. Wilson begins in 1894 and continues to the present day. A
program to digitize and distribute the images in this collection
was begun at UCLA in 2003 and is now making available the first of
the catalogued and catagorized images from the CaK sequence. Most
of the instrumentation with which the images were obtained is still
available although in a disassembled form. Original log books have
been digitized and associated with the images so that a maximum of
scientific return can be obtained from the data base. The present range
of images available from www.astro.ucla.edu/~ulrich extends from late
1915 to 1952. Each image has been digitized with 12-bit precision and
represented in a 16-bit format. These images are each 13 Mbytes in size
and larger than will be the final product images since not all image
defects have been mitigated at this time. The radii and centers of the
solar images have been determined and are included in the available data
files. Optical vignetting by the system introduces an intensity gradient
of known magnitude that can be used to help characterize the photograph
plates. The roll angle of the images has yet to be determined.
---------------------------------------------------------
Title: The Solar Photograph Archive of the Mount Wilson Observatory -
A Resource for a Century of Digital Data
Authors: Ulrich, R. K.; Webster, L. S.; Varadi, F.; Javaraiah, J.;
Lefebvre, S.; Gilman, P.
2004AGUFMSH52A..03U Altcode:
The solar telescopes and spectroheliographs of the Mount Wilson
Observatory were among the earliest modern facilities for the
study of the solar surface. The photographic collection of the
solar program at Mt. Wilson begins in 1894 and continues to the
present day. A program to digitize and distribute some of the
images in this collection was begun at UCLA in 2003 and is now
making available the first of the catalogued and catagorized images
from the Ca K sequence. Most of the instrumentation with which the
images were obtained is still available although in a disassembled
form. Original logbooks have been digitized and associated with
the images so that a maximum of scientific return can be obtained
from the data base. The present range of images available from:
http://www.astro.ucla.edu/&~slash;ulrich/MW&_slash;SPADP/CaK&_slash;fits/
extends from late 1915 to mid-1925. Each image has been digitized
with 12-bit precision and represented in a 16-bit format. These
images are each 13 Mbytes in size and larger than will be the final
product images since not all image defects have been mitigated at this
time. The radii and centers of the solar images have been determined
and are included in the available data files. Optical vignetting by
the system introduces an intensity gradient of known magnitude that
can be used to help characterize the photograph plates. The roll angle
of the images has yet to be determined.
---------------------------------------------------------
Title: Detection of Jets and Associated Toroidal Fields in the
Solar Tachocline
Authors: Christensen-Dalsgaard, J.; Corbard, T.; Dikpati, M.; Gilman,
P. A.; Thompson, M. J.
2004ESASP.559..376C Altcode: 2004soho...14..376C
No abstract at ADS
---------------------------------------------------------
Title: Global MHD Instabilities in a Thin-Shell Model of the Solar
Tachocline
Authors: Gilman, P. A.; Dikpati, M.; Miesch, M. S.
2004ESASP.559..440G Altcode: 2004soho...14..440G
No abstract at ADS
---------------------------------------------------------
Title: Limits to Penetration of Meridional Circulation below the
Solar Convection Zone
Authors: Gilman, Peter A.; Miesch, Mark S.
2004ApJ...611..568G Altcode:
We show that meridional circulation, such as that observed at the
top of and in the solar convection zone (CZ) by direct doppler and
helioseismic techniques, cannot penetrate significantly below the bottom
(~0.7 R<SUB>solar</SUB>) of the overshoot layer at the bottom of the
CZ. Therefore, solar dynamo models that rely on penetration as deep as
to 0.6 R<SUB>solar</SUB> are ruled out. The analysis we carried out to
reach this conclusion elucidates two boundary layers, one of which we
have not seen applied to astrophysical problems before. This analysis
should be relevant to understanding interfaces between convective and
radiative zones in stellar interiors generally.
---------------------------------------------------------
Title: Linear Analysis and Nonlinear Evolution of Two-Dimensional
Global Magnetohydrodynamic Instabilities in a Diffusive Tachocline
Authors: Dikpati, Mausumi; Cally, Paul S.; Gilman, Peter A.
2004ApJ...610..597D Altcode:
We develop a more realistic two-dimensional model for global MHD
instabilities in the solar tachocline, by including diffusion in
the form of kinetic and magnetic drag (following Newton's cooling
law formulation). This instability has previously been studied by us
and others for an idealized tachocline with no kinematic viscosity
and magnetic diffusivity. Since radial diffusion is more important
than latitudinal diffusion in the thin solar tachocline, diffusive
decay of flow and magnetic fields can be considered as proportional
to those variables. We find that, for solar-like toroidal magnetic
fields of ~100 kG, instability exists for a wide range of kinetic
and magnetic drag parameters, providing a mechanism for enhanced
angular momentum transport in latitudes, which could explain how thin
the solar tachocline is. From a detailed parameter space survey,
we set upper limits of 5×10<SUP>11</SUP> and 3×10<SUP>10</SUP>
cm<SUP>2</SUP> s<SUP>-1</SUP> for kinematic viscosity and magnetic
diffusivity, respectively, such that this instability occurs in the
solar tachocline on a timescale shorter than a sunspot cycle. We
find that magnetic drag has much more influence than kinetic drag
in damping this instability. This happens because the sink due to
magnetic drag dissipates perturbation magnetic energy faster than
the vorticity sink from kinetic drag dissipates perturbation kinetic
energy. Consequently, in the presence of a large enough magnetic drag,
the nonsolar-like clamshell pattern, found by Cally to be an inevitable
final state of a broad profile undergoing an ideal MHD tachocline
instability, is suppressed, whereas a banded profile still tips with
no reduction in tip angle. We discuss how tipping may affect various
surface manifestations of magnetic features, such as the latitudes
and orientations of bipolar active regions.
---------------------------------------------------------
Title: Deciphering Toroidal Field Dynamics from Sunspot Statistics
Authors: Norton, A. A.; Gilman, P. A.
2004AAS...204.5304N Altcode: 2004BAAS...36..756N
We are interested in what solar surface magnetism can tell us about
the interior toroidal magnetic fields. Because some fraction of
solar surface magnetism must be a direct result of the dynamics of the
interior toroidal field, we feel it is worthwhile to study the patterns
of flux which emerge and attempt to recover the basic properties of
the toroidal bands and their time dependent behavior. <P />New theory
predicts a global instability resulting in a tipping of the toroidal
bands with respect to the equatorial plane. We search for evidence of a
tipped toroidal field for some phases of the solar cycle by examining
the dominant latitude of emerging flux as a function of longitude. In
order to determine the extent to which we can recover the toroidal
field dynamics from observations, we use a model to artificially
generate sunspot distributions from subsurface toroidal fields that
we have assigned certain properties such as latitudinal width, peak
field strength and degree of tipping with respect to the equatorial
plane. Observational studies set an upper limit of 15 degrees to the
tipping angle. Model results which best fit the observed data are
those having a toroidal band with a tipping angle of 10 degrees at
high latitudes (early in the sunspot cycle) gradually decreasing to
0 degrees as the sunspot band migrates towards the equator.
---------------------------------------------------------
Title: Global MHD Instabilities In a 3D Thin-shell Model Of Solar
Tachocline
Authors: Gilman, P. A.; Dikpati, M.; Miesch, M. S.
2004AAS...204.5303G Altcode: 2004BAAS...36..755G
Previous work has shown that in model 2D and 'shallow water' tachoclines
containing latitudinally varying differential rotation and toroidal
fields, a global MHD instability is present for tachocline type
differential rotations and a wide range of toroidal field profiles (both
broad and narrow in latitude), in which longitudinal wave number m=1
usually predominates, leading to a tipping of the toroidal field away
from a purely E-W orientation. We show here that the same instability
occurs in model tachoclines with similar differential rotation and
toroidal fields, but for which the stratification is continuous in
the vertical. The unstable disturbances here have periodic vertical
structures up to wave numbers of at least 10, depending on the degree of
'subadiabaticity' of the stratification. For overshoot layer conditions,
the modes of shallow water type dominate and may be the only ones
present, requiring vertical displacement of the top boundary of the
tachocline, while in the radiative tachocline, modes with a wide
range of vertical wavenumbers grow, with the same growth rate in the
asymptotic limit of strongly subadiabatic temperature gradient. The
latitude and longitude variations of the unstable modes are identical
to the corresponding structures of 2D or shallow water modes. They also
have similar kinetic helicity profiles in latitude, but with vertical
structure determined by the vertical wavenumber. These modes should
also contribute to driving the solar dynamo. This work is partially
supported by NASA grants W-10107 and W-10175. National Center for
Atmospheric Research is sponsored by National Science Foundation.
---------------------------------------------------------
Title: The Role of Time-varying Meridional Flow Pattern During Past
20 Years In Influencing Upcoming Solar Cycle Features
Authors: Dikpati, M.; de Toma, G.; Gilman, P. A.; Corbard, T.; Rhodes,
E. J.; Haber, D. A.; Bogart, R. S.; Rose, P. J.
2004AAS...204.5305D Altcode: 2004BAAS...36..756D
Given the success of a recently built flux-transport dynamo-based
scheme (ApJ, 2004, 601, 1136) in reproducing observed polar field
features in cycle 23 including a) why polar reversal as well as polar
field build-up after reversal were unusually slow, and b) why S-pole
reversed a year after N-pole did, we apply this scheme to predict some
features of solar cycle 24. It has been demonstrated (ApJ, 2000, 543,
1027) that the duration of the Sun's memory of its own magnetic field
is governed primarily by the meridional flow speed in flux-transport
dynamos, and is no less than two solar cycles. Therefore, observations
of the Sun's magnetic field patterns over at least the past two cycles,
and dynamical changes in the Sun's large-scale mass-flow in which
the solar magnetic fields are partially frozen, should play important
roles in determining certain features in the upcoming solar cycle. We
first demonstrate theoretically how a N-S asymmetry in meridional
flow pattern can produce asymmetry in sunspot maxima in N &
S hemispheres, hence causing double peaks, as observed in cycles 22
and 23. We also show how deceleration in meridional flow, during the
rising phase of cycle 23 produced a slower rise in this cycle compared
to cycles 21 and 22. We then discuss the team-effort for extracting
observed changes in meridional flow over the past 20 years, using
helioseismic archive of MWO. By incorporating this long-term dynamical
variation in flow-pattern in our prediction model, if we can tune the
model to successfully reproduce various "anomalies" in solar cycle 23,
we can comment further that cycle 23 is going to be a longer cycle
if meridional flow does not accelerate during its declining phase,
hence causing onset of cycle 24 around 2007. This work is supported
by NASA grants W-10107 and W-10175. National Center for Atmospheric
Research is sponsored by National Science Foundation.
---------------------------------------------------------
Title: Thin-Shell Magnetohydrodynamic Equations for the Solar
Tachocline
Authors: Miesch, Mark S.; Gilman, Peter A.
2004SoPh..220..287M Altcode:
We present a new system of equations designed to study global-scale
dynamics in the stably-stratified portion of the solar tachocline. This
system is derived from the 3D equations of magnetohydrodynamics in a
rotating spherical shell under the assumption that the shell is thin and
stably-stratified (subadiabatic). The resulting thin-shell model can
be regarded as a magnetic generalization of the hydrostatic primitive
equations often used in meteorology. It is simpler in form than the more
general anelastic or Boussinesq equations, making it more amenable to
analysis and interpretation and more computationally efficient. However,
the thin-shell system is still three-dimensional and as such represents
an important extension to previous 2D and shallow-water approaches. In
this paper we derive the governing equations for our thin-shell model
and discuss its underlying assumptions, its context relative to other
models, and its application to the solar tachocline. We also demonstrate
that the dissipationless thin-shell system conserves energy, angular
momentum and magnetic helicity.
---------------------------------------------------------
Title: Magnetic Field-Minimum Intensity Correlation in Sunspots:
A Tool for Solar Dynamo Diagnostics
Authors: Norton, Aimee A.; Gilman, Peter A.
2004ApJ...603..348N Altcode:
Within a sunspot umbra, the continuum intensity is known to be
inversely proportional to the magnetic field strength. Studied less
is the relationship between the minimum continuum intensity and the
maximum field strength of different sunspots. We conduct a test of this
global relationship, using brightness ratios and magnetic field data
from the Advanced Stokes Polarimeter and the Michelson Doppler Imager
(MDI) for 10 sunspot umbrae of various sizes observed 1998 May-2003
June. We determine that the peak field strengths of sunspots can be
ascertained from a fit to their corresponding brightness ratios with
an accuracy of ~100 G, nearly twice the accuracy that a fit to the MDI
magnetogram values can provide. We then analyze continuum intensity
data from the MDI to characterize the distribution of sunspots as a
function of latitude. We hand-select 331 and 321 umbrae, respectively,
in the northern and southern hemispheres during Carrington rotations
1910-2003. Although the average location of sunspot eruption moves
equatorward throughout the solar cycle, the northern hemisphere shows
darker umbrae located systematically closer to the equator, while
brighter umbrae are found at higher latitudes. These findings confirm
the results of simulations that show strong flux emerging radially
while weak flux emerges nonradially, causing weak flux to emerge
poleward of its original toroidal field position. The average umbral
intensity decreased in the north through the solar cycle, reaching
a minimum intensity around sunspot maximum, possible evidence of the
toroidal field strength peaking at solar maximum. This finding is in
opposition to previous observations suggesting an increase late in
the cycle. The southern hemisphere umbral distribution appears more
disorganized and periodic in nature.
---------------------------------------------------------
Title: Diagnostics of Polar Field Reversal in Solar Cycle 23 Using
a Flux Transport Dynamo Model
Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A.; Arge,
Charles N.; White, Oran R.
2004ApJ...601.1136D Altcode:
Motivated by observed anomalous features in cycle 23, as inferred from
records of photospheric magnetic flux, we develop a flux transport
dynamo-based scheme in order to investigate the physical cause of such
anomalies. In this first study we focus on understanding anomalies
occurring in the polar field evolutionary pattern in cycle 23, namely,
why the polar reversal in cycle 23 was slow, why after reversal the
buildup of the polar field was slow, and why the south pole reversed
approximately a year after the north pole did. We construct a calibrated
flux transport dynamo model that operates with dynamo ingredients such
as differential rotation, meridional circulation, and large-scale
poloidal field source derived from observations. A few other dynamo
ingredients, such as diffusivity and quenching pattern, for which
direct observations are not possible, are fixed by using theoretical
guidance. By showing that this calibrated model can reproduce major
longitude-averaged solar cycle features, we initialize the model at
the beginning of cycle 22 and operate by incorporating the observed
variations in meridional circulation and large-scale surface magnetic
field sources to simulate the polar field evolution in cycle 23. We show
that a 10%-20% weakening in photospheric magnetic flux in cycle 23 with
respect to that in cycle 22 is the primary reason for a ~1 yr slowdown
in polar reversal in cycle 23. Weakening in this flux is also the reason
for slow buildup of polar field after reversal, whereas the observed
north-south asymmetry in meridional circulation in the form of a larger
decrease in flow speed in the northern hemisphere than that in the
southern hemisphere during 1996-2002 and the appearance of a reverse,
high-latitude flow cell in the northern hemisphere during 1998-2001
caused the north polar field to reverse before the south polar field.
---------------------------------------------------------
Title: The Solar Tachocline: Limiting Magneto-Tipping Instabilities
Authors: Cally, P. S.; Dikpati, M.; Gilman, P. A.
2004IAUS..219..541C Altcode: 2003IAUS..219E.172C
Two dimensional magneto-shear instabilities in the solar tachocline
have been extensively explored in recent years. One of their most
notable traits over a wide range of shear and magnetic profiles is
a propensity for the magnetic field to tip substantially from its
initial axisymmetric configuration with possible implications for
patterns of flux emergence. However it is found that modifications
of the standard models to include either kinetic and magnetic drag
or prograde toroidal velocity jets associated with magnetic bands
can suppress the instabilities or considerably reduce their nonlinear
development. In the case of tip reduction by jets for a toroidal field
of ~100kG in the tachocline (required for sunspots to emerge in sunspot
latitudes) simulations indicate that jets capable of reducing tipping
below the limits of detection from sunspot patterns at the surface are
potentially detectable by helioseismic methods and should be looked
for. Establishing an upper limit to the jet may result in a lower
limit for the amount of tipping to be expected.
---------------------------------------------------------
Title: Penetration of Meridional Circulation into a Stably-Stratified
Region
Authors: Miesch, M. S.; Gilman, P. A.
2003AGUFMSH21B0128M Altcode:
Several recent solar dynamo models have been proposed which require
the axisymmetric component of the solar meridional circulation (MC)
to penetrate substantially below the convective envelope, well into
the stably-stratified interior. In this poster we investigate this
possibility using a thin-shell approximation which has recently
been developed to study the stably-stratified portion of the solar
tachocline. Analytic and numerical results are presented for the
extent of MC penetration as a function of amplitude and background
stratification (represented in the nondimensional thin-shell system as
the Rossby and Froude numbers). The results indicate that substantial
penetration of the meridional circulation below the convective envelope
is unlikely. Furthermore, the analysis is closely related to the
classical Ekman layer analysis and may have additional geophysical
applications.
---------------------------------------------------------
Title: Stability Analysis of Tachocline Latitudinal Differential
Rotation and Coexisting Toroidal Band Using a Shallow-Water Model
Authors: Dikpati, Mausumi; Gilman, Peter A.; Rempel, Matthias
2003ApJ...596..680D Altcode:
Recently global, quasi-two-dimensional instabilities of tachocline
latitudinal differential rotation have been studied using a so-called
shallow-water model. While purely hydrodynamic shallow-water type
disturbances were found to destabilize only the overshoot tachocline,
the MHD analysis showed that in the presence of a broad toroidal
field, both the radiative and overshoot parts of the tachocline can
be unstable. We explore here instability in the shallow-water solar
tachocline with concentrated toroidal bands placed at a wide range
of latitudes, emulating different phases of the solar cycle. In
equilibrium, the poleward magnetic curvature stress of the band is
balanced either by an equatorward hydrostatic pressure gradient or
by the Coriolis force from a prograde jet inside the band. We find
that toroidal bands placed almost at all latitudes make the system
unstable to shallow-water disturbances. For bands without prograde
jets, the instability persists well above 100 kG peak field, while
a jet stabilizes the band at a field of ~40 kG. The jet imparts
gyroscopic inertia to the toroidal band inhibiting it from unstably
“tipping” its axis away from rotation axis. Like previously
studied HD and MHD shallow-water instabilities in the tachocline,
unstable shallow-water modes found here produce kinetic helicity and
hence a tachocline α-effect these narrow kinetic helicity profiles
should generate narrowly confined poloidal fields, which will help
formation of the narrow toroidal field. Toroidal bands poleward of
15° latitude suppress midlatitude hydrodynamic α-effects. However,
even strong toroidal bands equatorward of 15° allow this hydrodynamic
α-effect. Such bands should occur during the late declining phase
of a solar cycle and, thus, could help the onset of a new cycle by
switching on the mid latitude α-effect.
---------------------------------------------------------
Title: The Search for a Tipped Toroidal Field
Authors: Norton, A. A.; Gilman, P. A.; Henney, C. J.; Cally, P. S.
2003SPD....34.1903N Altcode: 2003BAAS...35..842N
A magnetic tipping instability of the tachocline toroidal field has
been predicted (Cally et al., 2003) that could produce a toroidal
field tipped with respect to the equatorial axis. One result of a
tipped toroidal band is the eruption of magnetic flux over a range of
latitudes from the same toroidal ring. The longitudinal dependence
of this flux emergence would contribute to non-axisymmetry of the
whole flux emergence pattern by giving it a longitudinal wavenumber
m=1 dependence. We search for evidence (or lack thereof) of a tipped
toroidal field for some phases of the solar cycle by examining the
dominant latitude of emerging flux as a function of longitude. We use
the existing observational data from Kitt Peak synoptic Carrington
Rotation magnetograms to identify the location of strong flux,
independently in each hemisphere, and test whether the location of
the flux reveals a pattern compatible with the tipping instability m=1.
---------------------------------------------------------
Title: Meridional motion and the slope of isorotation contours
Authors: Gilman, P. A.; Howe, R.
2003ESASP.517..283G Altcode: 2003soho...12..283G
As helioseismic inversions have continued to improve in precision,
it has become apparent that in the bulk of the solar convection zone
the contours of rotation, while certainly not parallel to the rotation
axis as predicted in earlier global convection simulations, are also not
radial, as has often been stated based on earlier inversions. Instead,
between 15 and about 55 degrees latitude (measured at r = 0.8R) the
rotation contours make an angle with the rotation axis of about 25
degrees, which does not appear to vary systematically with latitude
in this latitude range. What then determines this angle? By use of an
extremely simple dynamical equilibrium model, we show that Coriolis
forces from the observed meridional circulation could be responsible
for this angle if, without this effect, the contours of rotation would
be radial.
---------------------------------------------------------
Title: Clamshell and Tipping Instabilities in a Two-dimensional
Magnetohydrodynamic Tachocline
Authors: Cally, Paul S.; Dikpati, Mausumi; Gilman, Peter A.
2003ApJ...582.1190C Altcode:
Building on Cally's nonlinear model of two-dimensional MHD tachocline
instability, we further explore the evolution of a wide variety of
toroidal field profiles due to this instability. Cally showed in a
recent study that an initially broad toroidal field opens up into a
“clamshell” pattern because of nonlinear evolution of MHD tachocline
instability. Various other toroidal field profiles-single toroidal
bands, double bands, and mixed profiles with a band in addition to
broad profiles-may also occur in the Sun during various phases of the
solar cycle. Detailed study of the evolution of banded profiles shows
no occurrence of clamshell instability, but the bands commonly tip
relative to the axis of rotation. The higher the latitude location of
the band, the more it tips. Extreme tipping results when the band is
at 60° latitude or higher-the magnetic ring hangs from the pole on one
side of the Sun. For bands of 10° latitude width at sunspot latitudes
(<=40°), the band tip is within +/-10° about the mean latitude
of the band. This tipping could either enhance or reduce the observed
tilt in bipolar active regions. Double bands, or profiles consisting
of a band and a broad profile, may exist at certain phases of the
solar cycle. We find that double-band systems with two oppositely
directed bands separated widely (>15°) in latitude, as well
as two close bands of same polarities, do not interact in the same
hemisphere-the higher latitude band tips, while the lower latitude
band hardly responds. A significant interaction between two individual
bands in one hemisphere takes place only when the band separation is
<=15° and the bands are oppositely directed, which is a nonsolar
case. In this case, the band system either tips or forms the clamshell
pattern depending on the dominant mode symmetry. We also show that
a mixed profile with oppositely directed narrow fields close to the
equator in addition to the broad fields evolve in such a way as to
oppose the reconnection of the broad fields across the equator, and
thus inhibiting the clamshell formation, at least at certain phases
of the solar cycle. Finally, we note that the tipping and clamshell
instabilities strongly inhibit differential rotation.
---------------------------------------------------------
Title: Solar Multiscale Convection and Rotation Gradients Studied
in Shallow Spherical Shells
Authors: De Rosa, Marc L.; Gilman, Peter A.; Toomre, Juri
2002ApJ...581.1356D Altcode: 2002astro.ph..9054D
The differential rotation of the Sun, as deduced from helioseismology,
exhibits a prominent radial shear layer near the top of the convection
zone wherein negative radial gradients of angular velocity are
evident in the low- and midlatitude regions spanning the outer 5%
of the solar radius. Supergranulation and related scales of turbulent
convection are likely to play a significant role in the maintenance
of such radial gradients and may influence dynamics on a global scale
in ways that are not yet understood. To investigate such dynamics, we
have constructed a series of three-dimensional numerical simulations
of turbulent compressible convection within spherical shells, dealing
with shallow domains to make such modeling computationally tractable. In
all but one case, the lower boundary is forced to rotate differentially
in order to approximate the influence that the differential rotation
established within the bulk of the convection zone might have upon a
near-surface shearing layer. These simulations are the first models
of solar convection in a spherical geometry that can explicitly
resolve both the largest dynamical scales of the system (of order the
solar radius) as well as smaller scale convective overturning motions
comparable in size to solar supergranulation (20-40 Mm). We find that
convection within these simulations spans a large range of horizontal
scales, especially near the top of each domain, where convection
on supergranular scales is apparent. The smaller cells are advected
laterally by the larger scales of convection within the simulations,
which take the form of a connected network of narrow downflow lanes that
horizontally divide the domain into regions measuring approximately
100-200 Mm across. We also find that the radial angular velocity
gradient in these models is typically negative, especially in the low-
and midlatitude regions. Analyses of the angular momentum transport
indicate that such gradients are maintained by Reynolds stresses
associated with the convection, transporting angular momentum inward
to balance the outward transport achieved by viscous diffusion and
large-scale flows in the meridional plane, a mechanism first proposed
by Foukal & Jokipii and tested by Gilman & Foukal. We suggest
that similar mechanisms associated with smaller scale convection in
the Sun may contribute to the maintenance of the observed radial shear
layer located immediately below the solar photosphere.
---------------------------------------------------------
Title: Angular momentum transport in the solar tachocline
Authors: Miesch, Mark S.; Gilman, Peter A.
2002ESASP.505..509M Altcode: 2002IAUCo.188..509M; 2002solm.conf..509M
Angular momentum transport by non-axisymmetric motions in the solar
tachocline likely play a key role in maintaining the global rotation
profile in the solar interior. As such, it has important implications
with regard to the solar dynamo and to the amplification and emergence
of magnetic flux. We report results from several types of numerical
experiments designed to study angular momentum transport due to
stratified turbulence and shear instabilities in the tachocline and
we discuss what implications this may have for its latitudinal and
vertical structure.
---------------------------------------------------------
Title: Analysis of Instability of Latitudinal Differential Rotation
and Toroidal Field in the Solar Tachocline Using a Magnetohydrodynamic
Shallow-Water Model. I. Instability for Broad Toroidal Field Profiles
Authors: Gilman, Peter A.; Dikpati, Mausumi
2002ApJ...576.1031G Altcode:
We examine the global MHD instability of solar tachocline latitudinal
differential rotation and the coexisting broad toroidal magnetic
field, using a “shallow-water” model that captures the simplest
effects of subadiabatic stratification. We assume a single fluid shell
that has a fixed bottom but variable thickness. This model is the
MHD generalization of a hydrodynamic model that we have previously
applied to the tachocline, although the solution method is somewhat
different. Stratification in the model is characterized by an
“effective gravity” G (G=0 for adiabatic stratification). The
radiative (lower) part of the tachocline thus has high G
(~10<SUP>2</SUP>) and the overshoot part, low G (less than 1). We obtain
growth rates, phase velocities, and spatial structures of unstable modes
for a wide range of toroidal field strengths and effective gravities,
as well as differential rotations that are consistent with helioseismic
observations. We recover known two-dimensional MHD stability results
in the limit of large G and hydrodynamic instability results in the
limit of vanishing toroidal field. For strong magnetic fields, only
longitudinal wavenumber m=1 is unstable, but for weak fields m=2 is
also. For peak toroidal fields of 20 kG and above, the growth rates
and disturbance structures are essentially independent of the effective
gravity, until it becomes so small that the fluid shell shrinks to zero
in low latitudes, whereupon the instability is cut off. In contrast,
the instability evolves radically at low G when toroidal field is
increased from zero. In both overshoot and radiative parts of the
tachocline, unstable modes grow fastest for toroidal fields of the
order of 10<SUP>2</SUP> kG. The structure of the unstable disturbances
is always governed by the latitude location of singular or critical
points at which the Doppler-shifted phase velocity of the disturbance
equals the local (angular) Alfvén speed. All unstable disturbances
possess kinetic helicity, narrowly concentrated in the neighborhood
of the same critical points. Just as shown by Dikpati & Gilman for
the hydrodynamic case, such disturbances could provide an “α-effect”
for the solar dynamo. But unlike the hydrodynamic case, this α-effect
would be a function of the toroidal field itself.
---------------------------------------------------------
Title: Flux Transport Solar Dynamos with Near-Surface Radial Shear
Authors: Dikpati, Mausumi; Corbard, Thierry; Thompson, Michael J.;
Gilman, Peter A.
2002ApJ...575L..41D Altcode:
Corbard & Thompson analyzed quantitatively the strong radial
differential rotation that exists in a thin layer near the solar
surface. We investigate the role of this radial shear in driving a flux
transport dynamo operating with such a rotation profile. We show that
despite being strong, near-surface radial shear effectively contributes
only ~1 kG (~30% of the total) to the toroidal fields produced there
unless an abnormally high, surface α-effect is included. While 3 kG
spot formation from ~1-2 kG toroidal fields by convective collapse
cannot be ruled out, the evolutionary pattern of these model fields
indicates that the polarities of spots formed from the near-surface
toroidal field would violate the observed polarity relationship with
polar fields. This supports previous results that large-scale solar
dynamos generate intense toroidal fields in the tachocline, from which
buoyant magnetic loops rise to the photosphere to produce spots. Polar
fields generated in flux transport models are commonly much higher
than observed. We show here that by adding enhanced diffusion in the
supergranulation layer (originally proposed by Leighton), near-surface
toroidal fields undergo large diffusive decay preventing spot formation
from them, as well as reducing polar fields closer to the observed
values. However, the weaker polar fields lead to the regeneration of
a toroidal field of less than ~10 kG at the convection zone base,
too weak to produce spots that emerge in low latitudes, unless an
additional poloidal field is produced at the tachocline. This is
achieved by a tachocline α-effect, previously shown to be necessary
for coupling the north and south hemispheres to ensure toroidal and
poloidal fields that are antisymmetric about the equator.
---------------------------------------------------------
Title: Observational constraints on the solar dynamo
Authors: Gilman, Peter A.
2002ESASP.508...25G Altcode: 2002soho...11...25G
There are basically two types of observations that should constrain
dynamo models for the sun. One is observations of the sun's magnetic
field in all its manifestations, which dynamo models should eventually
predict accurately; the other is observations of solar velocities and
thermodynamic structure and variations, which constrain the inductive
processes available for the dynamo. With respect to magnetic properties,
the level of detail in current observations of fields and inferred
field patterns far outstrips current dynamo capabilities to reproduce,
but models have moved beyond comparison to just the "classical" solar
cycle characteristics such as Hale's polarity laws and the butterfly
diagram. With respect to velocity constraints, much more is known now
about differential rotation and meridional circulation in the convection
zone and tachocline, than when early results from helioseismology
overturned the prevailing α-ω dynamos of the 1970's that required
angular velocity increasing inwards. Meridional circulation has
become a particularly important component, since recent models show
it can determine the solar cycle period correctly for observed flow
speeds. By contrast, so-called giant cells, which have been particularly
difficult to detect, may not be as relevant to the dynamo problem as
they were once thought to be. For the future, improved inference from
helioseismic analysis of structure, velocities and magnetic fields in
the solar tachocline would be particularly valuable.
---------------------------------------------------------
Title: Effect of subsurface radial differential rotation on
flux-transport solar dynamo
Authors: Corbard, T.; Dikpati, M.; Gilman, P. A.; Thompson, M. J.
2002ESASP.508...75C Altcode: 2002soho...11...75C
A near-surface radial gradient of rotation was recently inferred and
quantified from MDI f-modes observations by Corbard & Thompson
(2001). We show, from our preliminary simulation of the large-scale
solar magnetic field, by using a flux-transport type dynamo, that
despite being strong, this gradient plays only a small role compared
to the tachocline in shearing the poloidal fields to produce toroidal
fields. This happens primarily because the turbulent diffusivity near
the surface wins in the competition of generation versus decay of
the magnetic fields unless an abnormally high α-effect is considered
there. This supports the results of previous flux-transport as well
as interface and overshoot layer dynamo models that the major toroidal
fields of the Sun are generated in the tachocline.
---------------------------------------------------------
Title: Equilibrium And Instability Of Toroidal Field Bands And
Rotational Jets In The Solar Tachocline
Authors: Gilman, P. A.; Rempel, M.; Dikpati, M.
2002AAS...200.0416G Altcode: 2002BAAS...34..645G
Recently Dikpati & Gilman (2001, ApJ, 552, 348) have shown,
using a shallow-water model of the solar tachocline that allows
the top surface to deform, that a tachocline with the observed broad
differential rotation and a strong toroidal field is prolate. A strong
toroidal field ring requires extra mass on its poleward side to provide
a hydrostatic latitudinal pressure gradient to balance the poleward
curvature stress. In a parallel study using a different approach,
Rempel et al (2000, A&A, 363, 789) have shown that a weakly
subadiabatic stratification leads to a complementary equilibrium state
of the overshoot tachocline in which the magnetic curvature stress is
balanced by a prograde rotational jet inside the toroidal ring. We show
that the shallow water model yields a similar equilibrium state if we
suppress the shell deformation and allow the differential rotation to
be modified. We are analyzing the stability of such an equilibrium
tachocline by using the MHD shallow-water model of Gilman &
Dikpati (2002, ApJ, submitted). We expect to show that the combination
of toroidal band and rotational jet is virtually always unstable to
disturbances with longitudinal wave number m>0, except perhaps when
the band is extremely narrow. This instability could wipe out the jet,
and lead to some poleward migration of the toroidal field, as well as
the excitation of longitudinally periodic magnetic patterns that might
provide sites for magnetic bouyancy to produce spots as well as other
photospheric magnetic features. This work is supported by NASA grants
W-19752 and S-10145-X. The National Center for Atmospheric Research
is sponsored by the National Science Foundation.
---------------------------------------------------------
Title: Solar rotation: observations, theory, modeling, and
implications for solar magnetism
Authors: Gilman, Peter
2002ocnd.confE..12G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Progress and Problems in Solar Dyanmo Theory
Authors: Gilman, P. A.
2001AGUFMGP22B..02G Altcode:
In the 1970's it was widely believed that an application of mean-field
dynamo theory to the bulk of the solar convection zone had 'solved'
the solar dynamo problem, but the advent of helioseismology provided
measures of solar differential rotation within the convection zone that
contradicted the required assumption of an angular velocity increasing
inward. This and requirements for storage of magnetic flux while it
amplified led to new theories focussed on the solar tachocline, a
sharp shear layer seen at the base of the solar convection zone. Most
recently, new bulk convection zone dynamo models, the so-called flux
transport models, have shown considerable success in reproducing many
features of the solar cycle, but such models still contain assumptions
concerning processes that are not subject to observational test. A
crucial improvement in the these models has been the inclusion of
(measured) meridional circulation, which is now seen as determining
the solar cycle period. Despite these successes, many uncertainties
remain. Little is known about the global dynamics and MHD of the solar
tachocline, where toroidal magnetic flux is likely stored while it
amplifies enough to generate sunspots. Global convection theory still
has not yielded a fully convincing theory of the maintenance of the
differential rotation in the convection zone, though all such models
rely on organized transport of angular momentum toward the equator to
make it spin faster in the face of turbulent diffusion. The history of
the solar dynamo problem makes it clear that observational constraints
are crucial to developing a plausible, durable theory; presumably the
same is also true for the geodynamo.
---------------------------------------------------------
Title: Flux-Transport Dynamos with α-Effect from Global Instability
of Tachocline Differential Rotation: A Solution for Magnetic Parity
Selection in the Sun
Authors: Dikpati, Mausumi; Gilman, Peter A.
2001ApJ...559..428D Altcode:
We propose an αΩ flux-transport dynamo for the Sun that is driven
by a tachocline α-effect. This α-effect comes from the global
hydrodynamic instability of latitudinal differential rotation in the
tachocline, as calculated using a shallow-water model. Growing, unstable
shallow-water modes propagating longitudinally in the tachocline create
vortices that correlate with radial motion in the layer to produce a
longitude-averaged net kinetic helicity and, hence, an α-effect. We
show that such a dynamo is equally successful as a Babcock-Leighton-type
flux-transport dynamo in reproducing many large-scale solar cycle
features. The success of both dynamo types depends on the inclusion
of meridional circulation of a sign and magnitude similar to that seen
on the Sun. Both α-effects (the Babcock-Leighton-type and tachocline
α-effect) are likely to exist in the Sun, but it is hard to estimate
their relative magnitudes. By extending the simulation to a full
spherical shell, we show that the flux-transport dynamo driven by the
tachocline α-effect selects a toroidal field that is antisymmetric
about the equator, while the Babcock-Leighton flux-transport dynamo
selects a symmetric toroidal field. Since our present Sun selects
antisymmetric fields, we argue that the tachocline α-effect must be
more important than the Babcock-Leighton α-effect.
---------------------------------------------------------
Title: Erratum: Shallow-Water Magnetohydrodynamic Waves in the
Solar Tachocline
Authors: Schecter, D. A.; Boyd, J. F.; Gilman, P. A.
2001ApJ...556L.127S Altcode:
An error occurred in the printing of Figure 2 in the Letter
“Shallow-Water Magnetohydrodynamic Waves in the Solar
Tachocline” by D. A. Schecter, J. F. Boyd, and P. A. Gilman (<A
href="/abs/2001ApJ...551L.185">ApJ, 551, L185 [2001]</A>). The f in
the abscissa label of Figure 2b was grossly displaced. The corrected
figure is shown here.
---------------------------------------------------------
Title: Latitudinal Transport of Angular Momentum by Cellular Flows
Observed with MDI
Authors: Hathaway, D. H.; Gilman, P. A.; Beck, J. G.
2001AGUSM..SP21C09H Altcode:
We have analyzed Doppler velocity images from the MDI instrument on
SOHO to determine the latitudinal transport of angular momentum by the
cellular photospheric flows. Doppler velocity images from 60-days in
May to July of 1996 were processed to remove the p-mode oscillations,
the convective blue shift, the axisymmetric flows, and any instrumental
artifacts. The remaining cellular flows were examined for evidence of
latitudinal angular momentum transport. Small cells show no evidence
of any such transport. Cells the size of supergranules (30,000 km in
diameter) show strong evidence for a poleward transport of angular
momentum. This would be expected if supergranules are influenced by
the Coriolis force, and if the cells are elongated in an east-west
direction. We find good evidence for just such an east- west elongation
of the supergranules. This elongation may be the result of differential
rotation shearing the cellular structures. Data simulations of this
effect support the conclusion that elongated supergranules transport
angular momentum from the equator toward the poles. Cells somewhat
larger than supergranules do not show evidence for this poleward
transport. Further analysis of the data is planned to determine if
the direction of angular momentum transport reverses for even larger
cellular structures. The Sun's rapidly rotating equator must be
maintained by such transport somewhere within the convection zone.
---------------------------------------------------------
Title: Prolateness of the Solar Tachocline Inferred from Latitudinal
Force Balance in a Magnetohydrodynamic Shallow-Water Model
Authors: Dikpati, Mausumi; Gilman, Peter A.
2001ApJ...552..348D Altcode:
Motivated by recent helioseismic observations concerning solar
tachocline shape and thickness and by the theoretical development
of MHD shallow-water equations for the tachocline, we compute the
prolateness of the tachocline using an MHD shallow-water model, in
which the shape and thickness are determined from the latitudinal force
balance equation. We show that a strong toroidal magnetic field stored
at or below the overshoot part of the tachocline leads to a pileup of
fluid at high latitude, owing to the poleward magnetic curvature stress
which has to be balanced by an equatorward latitudinal hydrostatic
pressure gradient. For toroidal fields of solar amplitude (~100 kG),
results for differentially rotating and uniformly rotating tachoclines
are almost the same. In contrast, the unmagnetized differentially
rotating tachocline would always be weakly oblate. We propose that a
strong toroidal field in the overshoot part of the tachocline should
tend to suppress the overshooting, thereby increasing the magnetic
storage capacity of the layer since the stratification there should
become more subadiabatic. We illustrate the effect of this process on
the shape and thickness of the layer by assuming its effective gravity
is a function of field strength. If toroidal fields are concentrated
in relatively narrow bands which migrate toward the equator with the
advance of the sunspot cycle, then they should be accompanied by a
“thickness front” advancing at the same rate. Applying our model
to the prolateness estimate of Charbonneau et al. yields toroidal
fields of 60-150 kG in the overshoot layer, consistent with other
considerations. Their prolateness in the radiative part of the
tachocline would require ~600 kG fields to be present.
---------------------------------------------------------
Title: Flux-transport Dynamos Driven by a Tachocline α -effect;
a Solution to Magnetic Parity Selection in the Sun
Authors: Dikpati, M.; Gilman, P. A.
2001AGUSM..SP31A19D Altcode:
We propose here an α Ω flux-transport dynamo driven by a tachocline α
-effect, produced by the global hydrodynamic instability of tachocline
differential rotation as calculated using a shallow-water model
(Dikpati & Gilman, 2001, ApJ, Mar.20 issue). Growing, unstable
shallow-water modes propagating longitudinally in the tachocline
create alternate vortices which correlate with upward/downward radial
motion of top boundary, associated with convergence/divergence of the
disturbance flow to produce a longitude-averaged net kinetic helicity,
and hence an α -effect. We show that a flux-transport dynamo driven
by a tachocline α -effect is equally successful as a Babcock-Leighton
flux-transport dynamo (Dikpati & Charbonneau 1999, ApJ, 518, 508)
in reproducing many large-scale solar cycle features, including the most
difficult feature of phase relationship between the subsurface toroidal
field and surface radial field. In view of the success of flux-transport
dynamos, whether the α -effect is at the surface or in the tachocline,
we argue that the solar dynamo should be considered to involve three
basic processes, rather than two (α -effect and Ω -effect only). The
third important process is the advective transport of flux by meridional
circulation. In reality, both α -effects (Babcock-Leighton type and
tachocline α -effect) are likely to exist, but it is hard to estimate
their relative magnitudes. We show, by extending the simulation in
a full spherical shell model that a flux-transport dynamo driven by a
tachocline α -effect selects toroidal field that is antisymmetric about
the equator, while a Babcock-Leighton flux-transport dynamo selects
symmetric toroidal field. Since our present Sun selects antisymmetric
toroidal fields, we argue that the flux-transport solar dynamo is
primarily driven by a tachocline α -effect. Acknowledgements: This
work is supported by NASA grants W-19752 and S-10145-X. National Center
for Atmospheric Research is sponsored by National Science Foundation.
---------------------------------------------------------
Title: “Shallow Water” MHD Waves in the Solar Tachocline
Authors: Schecter, D. A.; Gilman, P. A.; Boyd, J. F.
2001AGUSM..SP31A18S Altcode:
Peter Gilman recently introduced “Shallow water” magnetohydrodynamics
(SMHD) as a simple model for understanding predominantly horizontal
flows in the solar tachocline. Here, we present analytic solutions to
SMHD waves in the tachocline. We show that there are two branches
of such waves: Alfvén and magneto-gravity. Finite Alfvén and
magneto-gravity waves can both propagate without change of shape in
regions of the tachocline where the Coriolis force is finite. At
the equator, where the Coriolis force is zero, magneto-gravity
waves steepen and develop singularities. We show that the time
required for a singularity to develop increases monotonically with
the ambient magnetic field strength. Currently, a numerical model
is being developed for an extensive study of nonlinear SMHD in the
tachocline. This model is based on the β -plane approximation of
geophysical fluid dynamics. The β -plane approximation removes some
of the complications of spherical geometry, but keeps latitudinal
variation in the Coriolis parameter. Early results from this model
should be available for this presentation.
---------------------------------------------------------
Title: Analysis of Global MHD Instability in the Tachocline Using
a Shallow-water Model
Authors: Gilman, P. A.; Dikpati, M.
2001AGUSM..SP22A05G Altcode:
Results from studying 2D models indicate that tachocline differential
rotation is hydrodynamically stable, but is magnetohydrodynamically
unstable with a coexisting toroidal field (Gilman & Fox 1997,
ApJ,484, 439). In a recent study, Dikpati & Gilman (2001, ApJ,
Mar.20 issue) have shown, by including simplified 3D effects through use
of a shallow-water model, that the overshoot part of the tachocline can
be hydrodynamically unstable even without a toroidal field. Hydrodynamic
shallow-water equations have an MHD analog (Gilman 2000, ApJ,
544, L79). We study here the linear MHD shallow-water instability
of tachocline latitudinal differential rotation with a variety of
coexisting toroidal field profiles. We show that both the radiative
and overshoot parts of the tachocline become unstable in presence
of almost all toroidal field profiles, from broad to narrow, with a
wide range of field strengths, generalizing the 2D MHD instability,
as well as the shallow-water HD instability. Instability occurs for
a wide range of subadiabatic tachocline stratifications. Unstable
MHD shallow-water modes possess kinetic helicity which, unlike the
kinetic helicity of HD shallow-water modes, depends on field strength
and latitude-location of the peak toroidal field. Therefore, the α
-effect produced by global MHD shallow-water instability contains its
own quenching, as well as time-dependence, if we identify the latitude
of the peak magnetic field as a proxy for solar-cycle phase. We also
demonstrate how the toroidal field band can be twisted in the tachocline
due to the first order effect of the instability. Acknowledgements: This
work is supported by NASA grants W-19752 and S-10145-X. National Center
for Atmospheric Research is sponsored by National Science Foundation.
---------------------------------------------------------
Title: “Shallow-Water” Magnetohydrodynamic Waves in the Solar
Tachocline
Authors: Schecter, D. A.; Boyd, J. F.; Gilman, P. A.
2001ApJ...551L.185S Altcode:
This Letter discusses waves in a rotating magnetized fluid layer,
governed by “shallow-water” magnetohydrodynamics. Such waves likely
exist in the solar tachocline, and we focus on this application. A
dispersion relation is derived, giving two branches of waves: Alfvén
and magnetogravity. In general, finite Alfvén and magnetogravity waves
can propagate without change of shape. However, if the Coriolis force
is absent, as on the equator of the tachocline, finite magnetogravity
waves steepen and develop singularities in a time τ<SUB>s</SUB>. It
is shown that τ<SUB>s</SUB> increases monotonically with the ambient
magnetic field strength.
---------------------------------------------------------
Title: Analysis of Hydrodynamic Stability of Solar Tachocline
Latitudinal Differential Rotation using a Shallow-Water Model
Authors: Dikpati, Mausumi; Gilman, Peter A.
2001ApJ...551..536D Altcode:
We examine the global, hydrodynamic stability of solar latitudinal
differential rotation in a “shallow-water” model of the
tachocline. Charbonneau, Dikpati, & Gilman have recently shown
that two-dimensional disturbances are stable in the tachocline (which
contains a pole-to-equator differential rotation s<18%). In
our model, the upper boundary of the thin shell is allowed to
deform in latitude, longitude, and time, thus including simplified
three-dimensional effects. We examine the stability of differential
rotation as a function of the effective gravity of the stratification
in the tachocline. High effective gravity corresponds to the radiative
part of the tachocline; for this case, the instability is similar to the
strictly two-dimensional case (appearing only for s>=18%), driven
primarily by the kinetic energy of differential rotation extracted
through the work of the Reynolds stress. For low effective gravity,
which corresponds to the overshoot part of the tachocline, a second mode
of instability occurs, fed again by the kinetic energy of differential
rotation, which is primarily extracted by additional stresses and
correlations of perturbations arising in the deformed shell. In this
case, instability occurs for differential rotation as low as about 11%
between equator and pole. If this mode occurs in the Sun, it should
destabilize the latitudinal differential rotation in the overshoot part
of the tachocline, even without a toroidal field. For the full range
of effective gravity, the vorticity associated with the perturbations,
coupled with radial motion due to horizontal divergence/convergence
of the fluid, gives rise to a longitude-averaged, net kinetic helicity
pattern, and hence a source of α-effect in the tachocline. Thus there
could be a dynamo in the tachocline, driven by this α-effect and the
latitudinal and radial gradients of rotation.
---------------------------------------------------------
Title: Symmetry Selection in Solar Cycle Dynamo Models
Authors: Dikpati, M.; Gilman, P. A.
2001ASPC..248..125D Altcode: 2001mfah.conf..125D
No abstract at ADS
---------------------------------------------------------
Title: The solar tachocline and its variation (?)
Authors: Corbard, T.; Jiménez-Reyes, S. J.; Tomczyk, S.; Dikpati,
M.; Gilman, P.
2001ESASP.464..265C Altcode: 2001soho...10..265C; 2000astro.ph.11367C
The solar tachocline, located at the interface between the
latitude-dependent rotation of the convection zone and the rigid
radiative interior, presents high gradients of angular velocity
which are of particular interest for the models of the solar dynamo
and angular momentum transport. Furthermore, latitudinal and temporal
variations of the tachocline parameters, if any, are also of particular
interest in order to constrain models. We present a review of some of
the theories of the tachocline and their predictions that may be tested
by helioseismology. We describe the methods for inferring the tachocline
parameters from obervations and the associated difficulties. A review of
results previously obtained is given and an analysis of the new 6 years
database of LOWL observations is presented which yields no compelling
evidence of variations or general trend of the tachocline parameters
during the ascending phase of the current solar cycle (1994-2000).
---------------------------------------------------------
Title: Magnetohydrodynamic “Shallow Water” Equations for the
Solar Tachocline
Authors: Gilman, Peter A.
2000ApJ...544L..79G Altcode:
We argue that the classical “shallow water” equations of geophysical
fluid dynamics should be useful for studying the global dynamics of
the solar tachocline and demonstrate the existence of an MHD analog
that would allow taking into account the strong toroidal magnetic
field likely to be present there.
---------------------------------------------------------
Title: Solar and Stellar Convection: A Perspective for Geophysical
Fluid Dynamicists
Authors: Gilman, Peter A.
2000gac..conf...37G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Erratum: Joint Instability of Latitudinal Differential
Rotation and Toroidal Magnetic Fields below the Solar Convection
Zone. II. Instability for Toroidal Fields That Have a Node between
the Equator and Pole
Authors: Gilman, Peter A.; Fox, Peter A.
2000ApJ...534.1020G Altcode:
There is a repeated error in certain equations in the papers by Peter
A. Gilman and Peter A. Fox, “Joint Instability of Latitudinal
Differential Rotation and Toroidal Magnetic Fields below the
Solar Convection Zone. II. Instability for Toroidal Fields That
Have a Node between the Equator and Pole” (510, 1018 [1999])
(GF) Mausumi Dikpati and Peter A. Gilman, “Joint Instability
of Latitudinal Differential Rotation and Concentrated Toroidal
Fields below the Solar Convection Zone” (512, 417 [1999]) (DG)
and Peter A. Gilman and Mausumi Dikpati, “Joint Instability of
Latitudinal Differential Rotation and Concentrated Toroidal Fields
below the Solar Convection Zone. II. Instability of Narrow Bands
at All Latitudes” (528, 552 [2000]) (GD). In GF, equation (3), DG,
equation (24), and GD, equation (3), all are missing an additive term
that multiplies the variable H. The correct total factor multiplying
H is {1/(1-μ<SUP>2</SUP>) ( 2 - m<SUP>2</SUP>/(1-μ<SUP>2</SUP>)
+ 1/S[2c(ω<SUB>o</SUB>-c) + μ d/dμ(S/(1-μ<SUP>2</SUP>)]} The
second term, involving S in the denominator, is missing in these
papers. As a consequence, the expression for k<SUP>2</SUP>, shown
in equation (6) of GF and equation (10) of DG, is also missing
this term. The correct expression for k<SUP>2</SUP> in both cases
is 1/(1-μ<SUP>2</SUP>) ( 2 - m<SUP>2</SUP>/(1-μ<SUP>2</SUP>) +
((1/2S)(d<SUP>2</SUP>S/dμ<SUP>2</SUP>)) + 1/S[2c(ω<SUB>o</SUB>-c) +
μ d/dμ(S/(1-μ<SUP>2</SUP>)] No results or conclusions are affected in
any of these papers, because the equations containing errors are used
only to identify the location of singular points, where the effective
wavenumber increases without bound. All of these singular points are the
same in the erroneous forms as in the correct forms, being determined
principally by the roots of S. But obviously, if the erroneous forms
are used for other purposes, incorrect conclusions might be reached. The
authors are indebted to Paul Cally for discovering this error.
---------------------------------------------------------
Title: Shear Flow Instabilities
Authors: Gilman, Peter
2000astu.confE..15G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Instability of Tachocline Differential Rotation in a
Shallow-Water Model, and the Solar Dynamo
Authors: Dikpati, M.; Gilman, P. A.
2000SPD....31.0115D Altcode: 2000BAAS...32..803D
We examine the global, hydrodynamic stability of solar
latitudinal differential rotation in a "shallow-water" model of the
tachocline. Charbonneau, Dikpati and Gilman (1999, ApJ 526, 523) have
recently shown that 2D disturbances are stable in the tachocline. In
our model, the upper boundary of the thin shell is allowed to deform in
latitude, longitude and time, thus including simplified 3D effects. We
examine the stability of differential rotation as a function of
the effective gravity of the stratification in the tachocline. High
effective gravity corresponds to the radiative part of the tachocline;
for this case, the instability is similar to the strictly 2D case. For
low effective gravity, which corresponds to the overshoot part of
the tachocline, a second mode of instability occurs, fed primarily by
potential energy stored in the deformed shell. In this case, instability
occurs for differential rotation down to about 13% between equator and
pole. If this mode occurs in the Sun, it should destabilize much of
the tachocline, even without a toroidal field. For the full range of
effective gravity, the vorticity associated with the perturbations,
coupled with radial motion due to horizontal divergence/convergence
of the fluid, gives rise to a longitude-averaged, net kinetic helicity
pattern, and hence a source of α -effect in the tachocline. Thus there
could be a dynamo in the tachocline, driven by this α -effect and the
latitudinal and radial gradients of rotation. Preliminary simulations
suggest that an α -ω dynamo with the above α -effect working in
the tachocline is capable of producing a reasonable butterfly diagram,
particularly when a meridional circulation is included in the convective
envelope, as is known to exist. Acknowledgement. This work has been
partially supported by the NASA grant S-10145-X. The National Center for
Atmospheric Research is sponsored by the National Science Foundation.
---------------------------------------------------------
Title: Fluid Dynamics and MHD of the Solar Convection Zone and
Tachocline: Current Understanding and Unsolved Problems - (Invited
Review)
Authors: Gilman, Peter A.
2000SoPh..192...27G Altcode:
We review recent progress and define unanswered scientific questions in
five related topics: granulation- to supergranulation-scale convection
and magnetic structures; global convection and circulation; the rise
of magnetic flux tubes to the photosphere, and their injection into
the base of the convection zone; tachocline fluid dynamics and MHD;
and the solar dynamo. We close with a set of observational `targets'
for helioseismologists to aim for.
---------------------------------------------------------
Title: Three-dimensional Spherical Simulations of Solar
Convection. I. Differential Rotation and Pattern Evolution Achieved
with Laminar and Turbulent States
Authors: Miesch, Mark S.; Elliott, Julian R.; Toomre, Juri; Clune,
Tom L.; Glatzmaier, Gary A.; Gilman, Peter A.
2000ApJ...532..593M Altcode:
Rotationally constrained convection possesses velocity correlations
that transport momentum and drive mean flows such as differential
rotation. The nature of this transport can be very complex in turbulent
flow regimes, where large-scale, coherent vorticity structures and mean
flows can be established by smaller scale turbulence through inverse
cascades. The dynamics of the highly turbulent solar convection
zone therefore may be quite different than in early global-scale
numerical models, which were limited by computational resources to
nearly laminar flows. Recent progress in high-performance computing
technology and ongoing helioseismic investigations of the dynamics of
the solar interior have motivated us to develop more sophisticated
numerical models of global-scale solar convection. Here we report
three-dimensional simulations of compressible, penetrative convection
in rotating spherical shells in both laminar and turbulent parameter
regimes. The convective structure in the laminar case is dominated
by “banana cells,” but the turbulent case is much more complex,
with an intricate, rapidly evolving downflow network in the upper
convection zone and an intermittent, plume-dominated structure in
the lower convection zone and overshoot region. Convective patterns
generally propagate prograde at low latitudes and retrograde at high
latitudes relative to the local rotation. The differential rotation
profiles show some similarity with helioseismic determinations of
the solar rotation but still exhibit significantly more cylindrical
alignment. Strong, intermittent, vortical downflow lanes and plumes
play an important dynamical role in turbulent flow regimes and are
responsible for significant differences relative to laminar flows with
regard to momentum and energy transport and to the structure of the
overshoot region at the base of the convection zone.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation
and Concentrated Toroidal Fields below the Solar Convection
Zone. II. Instability of Narrow Bands at All Latitudes
Authors: Gilman, Peter A.; Dikpati, Mausumi
2000ApJ...528..552G Altcode:
Assuming that concentrated toroidal field bands occur in the solar
tachocline at different latitudes as the solar cycle progresses, we
examine the joint instability of latitudinal differential rotation and
coexisting narrow bands placed at a wide range of latitudes. Following
the basic formalism developed by Gilman & Fox and employing
the numerical technique of Dikpati & Gilman, we show that the
instability exists for almost all phases of the solar cycle, i.e., for
a wide range of latitudinal positions of the bands of field strengths
between 500 G and 200 kG. Modes with longitudinal wavenumber m=1 up
to m=7, depending on parameter values, are unstable for both kinds of
symmetries. Mid-latitude bands are the most unstable; the instability
disappears if the band is at very high or low latitudes. High-latitude
bands are highly unstable, with e-folding growth times of a few months,
even when the field strength is low (about a few hundred gauss)
low-latitude bands are unstable with longer growth times (>=1
yr), but for high field strengths (10<SUP>4</SUP>-2×10<SUP>5</SUP>
G). We argue that, because of the instability, the high-latitude
bands would undergo turbulent mixing in the latitudinal direction on a
timescale that is short compared to their build-up time from shearing
of poloidal field by the differential rotation, and thus they may not
be buoyant enough to appear as active regions at the surface, but the
low-latitude bands can build up simultaneously with this instability
and can eventually manifest as active regions. Instability for modes
with m>=1 could help determine the longitude distribution of active
regions; nonlinear changes in the toroidal field due to the instability
may contribute to their decay.
---------------------------------------------------------
Title: The Solar Cycle and the Tachocline: Theories and Observations
Authors: Corbard, T.; Jimenez-Reyes, S. J.; Tomoczyk, S.; Dikpati,
M.; Gilman, P.
2000ESASP.463...21C Altcode: 2000sctc.proc...21C
No abstract at ADS
---------------------------------------------------------
Title: Stability of the Solar Latitudinal Differential Rotation
Inferred from Helioseismic Data
Authors: Charbonneau, Paul; Dikpati, Mausumi; Gilman, Peter A.
1999ApJ...526..523C Altcode:
We revisit the hydrodynamical stability problem posed by the
observed solar latitudinal differential rotation. Specifically,
we carry out stability analyses on a spherical shell for
solar-like two-dimensional inviscid shear flow profiles of the form
ν=s<SUB>0</SUB>-s<SUB>2</SUB>μ<SUP>2</SUP>-s<SUB>4</SUB>μ<SUP>4</SUP>,
where μ is the sine of latitude. We find that stability is remarkably
sensitive to the magnitude of the μ<SUP>4</SUP> term. This allows us
to reconcile apparently conflicting results found in the published
literature. We then use latitudinal differential rotation profiles
extracted from various helioseismic inversions of the solar internal
rotation and investigate their stability as a function of depth from the
base of the tachocline to the top of the convective envelope. In all
cases considered, we find that the latitudinal differential rotation
in the tachocline is stable while that in the bulk of the convective
envelope is unstable. Under the assumption that the instability is
not impeded by finite Reynolds number or three-dimensional effects
not accounted for in our analysis, we speculate on possible observable
consequences of the occurrence of the instability in the top half of
the convective envelope.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation
and Toroidal Magnetic Fields below the Solar Convection
Zone. III. Unstable Disturbance Phenomenology and the Solar Cycle
Authors: Gilman, P. A.; Fox, P. A.
1999ApJ...522.1167G Altcode:
We analyze additional solutions for the two-dimensional instability of
coexisting differential rotation and toroidal magnetic fields, organized
in families with fixed ratios ER of magnetic to kinetic energy in the
unperturbed state. Solutions are found for a wide range of differential
rotation amplitudes found in the solar tachocline, for toroidal fields
that have a node that ranges in latitude from the pole to the equator,
as we expect to exist in the Sun through a sunspot cycle. Fixed ER is a
proxy for nonlinear saturation of the solar dynamo due to the reaction
of electromagnetic body forces. Since the saturation ratio is not known
from either theory or observations, we find solutions in the range
0.1<=ER<=30, corresponding to peak toroidal fields in the solar
tachocline of between about 8×10<SUP>3</SUP> to 1.4×10<SUP>5</SUP>
G. We focus on properties of the unstable disturbances that could test
the hypothesis that such disturbances in the solar tachocline provide
a template for surface features. We show that the symmetry of magnetic
pattern about the equator could switch at one or more phases of the
magnetic cycle, and for high ER a switch could also occur between two
antisymmetric patterns of different latitudinal profiles. In the former
case, the pattern rotation rate would be unchanged, but there would be
a sudden longitudinal phase shift in one or both hemispheres. In the
latter case, there would be no phase shift but instead a substantial
change in the rotation rate of the observed magnetic pattern. For a
given mode symmetry and type, the rotation rate is the same at all
latitudes, with the rate being close to that of the local rotation of
the plasma at the latitude where the disturbance amplitude peaks. For
ER<~1, the disturbance magnetic patterns have significant tilts
upstream away from the equator, reminiscent of similar patterns in
synoptic magnetograms. Sharp changes with latitude in the differential
rotation and toroidal field are associated with “critical points” in
the system, where the Doppler-shifted disturbance rotation equals the
local (angular) Alfvén speed. These migrate toward the equator with the
toroidal field node but increasingly lag it. The higher the magnetic
energy for a given differential rotation, the closer the equator
is approached. If these sharp changes in differential rotation and
toroidal field are related to the torsional oscillations and latitudes
of sunspots, then these solutions favor large toroidal fields in the
tachocline, of peak amplitude at least 6×10<SUP>4</SUP> G.
---------------------------------------------------------
Title: Interpretation of Solar Activity Features Using Joint
Instability of Differential Rotation and Toroidal Magnetic Fields
Authors: Gilman, P. A.; Dikpati, M.
1999AAS...194.5609G Altcode: 1999BAAS...31..913G
Gilman & Fox found that latitudinal differential rotation and a
broad toroidal field are jointly unstable to 2D horizontal perturbations
in the solar tachocline, while each of them is separately stable
there. They argued this instability could provide enhanced angular
momentum mixing in latitude, as required by Spiegel and Zahn for
the solar tachocline to retain its present thickness. Motivated by
observations of active regions that imply subsurface toroidal fields of
the Sun occur in narrow bands, and assuming that these bands occur in
the tachocline at different latitudes as the solar cycle progresses,
we examine the joint instability of latitudinal differential rotation
and coexisting narrow bands placed at all latitudes. We show that
instability occurs for almost all phases of solar cycle, for a
wide range of differential rotation amplitudes and toroidal field
strengths (between 500 Gauss and 200 kGauss). Mid-latitude bands are
most unstable; instability disappears if the band is at very high or
low latitudes. We argue that a highly unstable weak (<10(3) Gauss)
high-latitude toroidal band would undergo turbulent mixing in a time
short compared to its build-up time due to shearing of poloidal field by
differential rotation, and thus may never be buoyant enough to appear as
active regions at the surface, providing a possible explanation for why
active regions are not seen in high latitudes. Low latitude bands, on
the other hand, are unstable with a longer growth time and instability
persists for high field strengths (up to 2 x 10(5) Gauss). Therefore,
low-latitude bands can still amplify while experiencing instability,
and can still erupt at the surface as active regions. Instability
for modes with longitudinal wavenumber m equal to or greater than 1
could help determine the longitude distribution of active regions, and
initial nonlinear changes in the toroidal field due to this instability
may contribute to their decay. *The National Center for Atmospheric
Research is sponsored by the National Science Foundation.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation and
Concentrated Toroidal Fields below the Solar Convection Zone
Authors: Dikpati, Mausumi; Gilman, Peter A.
1999ApJ...512..417D Altcode:
Motivated by observations of sunspot and active-region latitudes
that suggest that the subsurface toroidal field in the Sun occurs
in narrow latitude belts, we analyze the joint instability of solar
latitudinal differential rotation and the concentrated toroidal field
below the base of the convection zone, extending the work of Gilman
& Fox (hereafter GF). We represent the profile of the toroidal
field by Gaussian functions whose width is a variable parameter and
solve the two-dimensional perturbation equations of GF by relaxation
methods. We reproduce the results of GF for broad profiles, and we find
instability for a wide range of amplitudes of differential rotation and
toroidal fields (10<SUP>3</SUP>-10<SUP>6</SUP> G fields at the base of
the solar convection zone), as well as a wide range of toroidal-field
bandwidths. We show that the combination of concentrated toroidal fields
and solar-type latitudinal differential rotation is again unstable,
not only to longitudinal wavenumber m=1 as in GF, but also to m>1
for sufficiently narrow toroidal-field profiles. For a fixed peak
field strength, the growth rate first increases as the toroidal-field
band is narrowed, reaching a peak for bandwidths between 10° and
20° in latitude, depending on the peak field strength, and then
decreases to a cut-off in the instability for toroidal field bands of
3°-4°. Irrespective of bandwidth, the differential rotation is the
primary energy source for the instability for weak fields, and the
toroidal field is the primary source for strong fields. The weaker
(stronger) the peak toroidal field is, the narrower (broader) is the
bandwidth for which the toroidal field becomes the primary energy
source. The Reynolds, Maxwell, and mixed stresses required to extract
energy from the differential rotation and toroidal field are most
active in the neighborhood of the singular or turning points of the
perturbation equations. This first study focuses on toroidal fields
that peak near 45° latitude, as in GF; later papers will place the
toroidal-field peak at a wide variety of latitudes, as we might expect
to occur at different phases of a sunspot cycle.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation
and Toroidal Magnetic Fields below the Solar Convection
Zone. II. Instability for Toroidal Fields that Have a Node between
the Equator and Pole
Authors: Gilman, Peter A.; Fox, Peter A.
1999ApJ...510.1018G Altcode:
We generalize results of Gilman and Fox to unperturbed toroidal
fields that have a node somewhere between the equator and the
pole as we speculate the Sun's field to have for most phases of
its magnetic cycle. We use the same solution method as in Gilman
and Fox, namely Legendre polynomial expansion and matrix inversion
to solve for the eigenvalues and eigenfunctions. The solutions are
structured around certain singular or critical points of the suitably
transformed and combined vorticity and induction equations. There
are singular points at the poles, and singularities where
ω<SUB>0</SUB>-c<SUB>r</SUB>=+/-α<SUB>0</SUB>, in which ω<SUB>0</SUB>
is the local rotation rate, c<SUB>r</SUB> is the longitudinal phase
speed of an unstable wave, and α<SUB>0</SUB> is an angular measure of
the toroidal field. We survey the instability as a function of toroidal
field profile and amplitude as well as differential rotation amplitude,
thereby examining reference states that could be characteristic of
most phases of the solar cycle, and most depths within the rotational
shear layer just below the base of the solar convection zone. <P />As
found in Gilman and Fox, instability occurs for a wide range of both
toroidal fields and differential rotations. Differential rotation is
again the primary energy source for growing modes when the toroidal
field is weak, and the toroidal field is the primary source when it
is strong. Unlike in Gilman and Fox, here modes of both symmetries
about the equator are unstable for low and high toroidal fields,
and for high fields a second antisymmetric mode appears. Which mode
symmetry is favored for low fields depends in detail on the relative
amplitudes of differential rotation and toroidal field. For low toroidal
fields (unstable) modes of both symmetries are energetically active
(extracting energy from the unperturbed state) only poleward of the
node and an adjacent singularity, but are coupled to energetically
neutral velocity perturbations equatorward of that singular point. In
transition to higher field strengths, those velocity patterns are damped
out when two additional singular points appear in the system, but the
energetically active high-latitude disturbances remain. By contrast
the second antisymmetric mode is energetically active equatorward
of the toroidal field node and closely adjacent singular points,
but is coupled to an energetically neutral pattern of both velocities
and magnetic fields on the poleward side. <P />As in Gilman and Fox,
we find narrow-latitude bands of sharp changes in both differential
rotation and toroidal magnetic field that migrate toward the equator
with increasing field strength, but are bounded in their migration by
the latitude of the toroidal field node. These sharp changes are always
at the locations of the singular points of the system and represent
narrow domains where both kinetic and magnetic energy are being
extracted from the reference state to drive the instability. <P />We
interpret the instability as a form of resonant overreflection between
singular points, analogous to what happens in stratified shear flow,
as described for example by Lindzen. The instability may contribute
to determining the latitudinal and longitudinal distribution of active
regions and other large-scale, magnetic features on the Sun, as well as
enable a degree of synchronization of the evolution of the solar cycle
between low latitudes and high, and between north and south hemispheres.
---------------------------------------------------------
Title: Fluid Dynamics and MHD of the Solar Convection Zone and
Tachocline: Current Understanding and Unsolved Problems
Authors: Gilman, Peter A.
1999soho....9E...1G Altcode:
My assigned topic is extremely broad, so in this abstract I will state
only goals and emphases, not all of which I may be able to discuss in
the time allotted. The focus of the talk will be primarily theoretical,
but guided by observations. Within the fluid dynamical realm, I will
say something about most scales of convection, as well as other classes
of fluid processes, particularly global instabilities and waves. I
will emphasize the differences in dynamics that should characterize
the different domains of differential rotation we now know exist on
the Sun: strong radial rotation gradients just below the photosphere;
the bulk of the convection zone that has little radial gradient; the
tachocline at the bottom; and possible polar vortices. Within the MHD
realm, I will focus on the multiple dynamos that may be operating,
as well as on processes for getting flux from below the convection
zone through it to the solar atmosphere, and the possible role played
by global MHD instabilities in the tachocline. A key question is how
far below the overshoot layer in the tachocline does the solar cycle
magnetic field extend. I will also speculate on the role the equator
near the base of the convection zone could play in annihilation
of solar cycle flux. In closing, I will try to sketch an integrated
picture of the dynamics and MHD of the convection zone and tachocline,
identify gaps in our understanding, and hopefully stimulate discussion
of which gaps we have the best chance of filling, using the tools of
helioseismology and theoretical and numerical calculations.
---------------------------------------------------------
Title: Instability of concentrated toroidal fields in the latitudinal
differential rotation below the solar convection zone
Authors: Dikpati, M.; Gilman, P. A.
1998ASPC..138...89D Altcode: 1998stas.conf...89D
No abstract at ADS
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation and
Toroidal Magnetic Fields below the Solar Convection Zone
Authors: Gilman, Peter A.; Fox, Peter A.
1997ApJ...484..439G Altcode:
Below the convection zone, where the stratification is radiatively
controlled, large-scale motions should be mainly horizontal, i.e., in
spherical shells due to the stabilizing effect of negative buoyancy
on radial displacements. Watson showed that the observed surface
solar differential rotation is at the threshold for instability to
horizontal disturbances. Therefore, since helioseismology tells us
the latitudinal differential rotation below the convection zone is
less than the surface value, the profile should be stable too. We show
that in the presence of a broad, nonuniform toroidal field the solar
differential rotation is unstable. This is true for a wide range of
kinetic and magnetic energies of the unperturbed state, from well below
equipartition to well above it. We find instability for essentially
all values of differential rotation and toroidal fields for which
we are able to find converged solutions. The instability appears to
occur only for longitudinal wave number 1. Disturbance symmetries
about the equator and profiles in latitude depend on the amplitude of
the toridal field. Peak e-folding times are a few months. The primary
energy source for the instability is differential rotation for low
field strengths and the toroidal field for high field strengths. The
mechanism of energy release from the differential rotation is the
poleward transport of angular momentum, by the Maxwell stress rather
than the Reynolds stress. For the profiles studied, the Reynolds stress
is almost always trying to rebuild differential rotation, the exact
opposite of the nonmagnetic case. Second-order perturbation theory
predicts that the unstable modes produce zonal jets and fine structure
in the toroidal field, the latitude of which migrates toward the equator
with increasing magnetic field strength. The instability we have found
may play a role in the solar dynamo, although being two-dimensional,
it cannot produce a dynamo by itself. Mixing of angular momentum caused
by the instability could allow achievement of equilibrium of the solar
tachocline hypothesized by Spiegel & Zahn.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation and
Concentrated Toroidal Fields below the Solar Convection Zone
Authors: Dikpati, M.; Gilman, P. A.
1997SPD....28.0213D Altcode: 1997BAAS...29..895D
It was found by Gilman and Fox (July 20 1997 issue of ApJ) that the
solar differential rotation below the convection zone is unstable
to horizontal disturbances in the presence of a broad, nonuniform
toroidal magnetic field. This instability occurs for longitudinal
wave number 1. The primary energy source for the instability is the
differential rotation for low field strength, and the toroidal field
for high field strength. Since solar toroidal magnetic fields are
inferred from sunspots to occur in narrow latitude belts, we have now
studied the joint instability of solar differential rotation and highly
concentrated toroidal field. The instability still occurs, and it occurs
for both the longitudinal wave number 1 and higher wave numbers. The
growth rates of the instability for some wave numbers greater than 1
are higher, if the toroidal field is in a belt equatorward of 60(deg)
latitude. The primary source of energy for the instability is usually
the magnetic field, but in some cases extraction of energy from
the differential rotation becomes more important particularly for
higher wave numbers with weaker fields. The instability occurs with
high growth rate even for weak toroidal field when the field is at
30(deg) latitude or higher. This enhanced instability at high latitude
may inhibit the build up of toroidal field preventing occurrence of
sunspots at higher latitudes. For low latitudes, the much lower growth
rate of the instability may allow sufficient toroidal field buildup
that sunspots can be formed. The occurrence of this instability for
wave numbers m=1 to 5 may help determine the observed distribution of
active regions with longitude.
---------------------------------------------------------
Title: Joint Instability of Differential Rotation and Toroidal
Magnetic Fields below the Solar Convection Zone, II
Authors: Gilman, P. A.; Fox, P.
1997SPD....28.0212G Altcode: 1997BAAS...29..895G
At the 1996 AAS/SPD meeting in Madison we reported first results for
the joint instability of differential rotation and toroidal magnetic
fields to 2D disturbances (see also Gilman and Fox, Paper I, July 20
1997 issue of ApJ). This analysis was for the toroidal field profile
B=a*sin(LAT)cos(LAT). This paper reports results for the profile
B=(a*sin(LAT)+b*(sin(LAT))(3) ))cos(LAT), which, with b<-a<0,
allows for a node in the toroidal field at latitude arcsin (-a/b). This
generalization is of interest because we should expect such a node to
appear and migrate equatorward as the sun proceeds from one sunspot
cycle to the next. As with the simpler profile, instability occurs
for virtually all differential rotation amplitudes, and all toroidal
field amplitudes and shapes, and remains confined to disturbances
with longitudinal wave number m=1. For a, b>0, the instability is
enhanced for the same a compared to the b=0 case, particularly in high
latitudes. For 0>b>-a (so no node is present) the instability is
similar to the b=0 case but with diminished growth rates, due to the
reduction of toroidal fields at high latitudes. At b=-a, the symmetric
mode of instability vanishes, but the antisymmetric mode remains. For
b<-a<0, both symmetric and antisymmetric modes are unstable, but
with disturbances confined largely to the domain poleward of the node,
unless the toroidal field energy greatly exceeds the kinetic energy of
differential rotation. Unstable disturbances spread and migrate toward
the equator as the field strength is increased and as the node is moved
equatorward. Thus, the instability may still contribute to the existence
of the solar butterfly diagram, and to other solar dynamo presses.
---------------------------------------------------------
Title: GONG Observations of Solar Surface Flows
Authors: Hathaway, D. H.; Gilman, P. A.; Harvey, J. W.; Hill, F.;
Howard, R. F.; Jones, H. P.; Kasher, J. C.; Leibacher, J. W.; Pintar,
J. A.; Simon, G. W.
1996Sci...272.1306H Altcode:
Doppler velocity observations obtained by the Global Oscillation Network
Group (GONG) instruments directly measure the nearly steady flows in
the solar photosphere. The sun's differential rotation is accurately
determined from single observations. The rotation profile with respect
to latitude agrees well with previous measures, but it also shows a
slight north-south asymmetry. Rotation profiles averaged over 27-day
rotations of the sun reveal the torsional oscillation signal-weak,
jetlike features, with amplitudes of 5 meters per second, that are
associated with the sunspot latitude activity belts. A meridional
circulation with a poleward flow of about 20 meters per second is
also evident. Several characteristics of the surface flows suggest
the presence of large convection cells.
---------------------------------------------------------
Title: The Global Oscillation Network Group (GONG) Project
Authors: Harvey, J. W.; Hill, F.; Hubbard, R. P.; Kennedy, J. R.;
Leibacher, J. W.; Pintar, J. A.; Gilman, P. A.; Noyes, R. W.; Title,
A. M.; Toomre, J.; Ulrich, R. K.; Bhatnagar, A.; Kennewell, J. A.;
Marquette, W.; Patron, J.; Saa, O.; Yasukawa, E.
1996Sci...272.1284H Altcode:
Helioseismology requires nearly continuous observations of the
oscillations of the solar surface for long periods of time in
order to obtain precise measurements of the sun's normal modes of
oscillation. The GONG project acquires velocity images from a network
of six identical instruments distributed around the world. The GONG
network began full operation in October 1995. It has achieved a duty
cycle of 89 percent and reduced the magnitude of spectral artifacts by
a factor of 280 in power, compared with single-site observations. The
instrumental noise is less than the observed solar background.
---------------------------------------------------------
Title: Joint Instability of Latitudinal Differential Rotation and
Toroidal Magnetic Fields below the Solar Convection Zone
Authors: Gilman, Peter A.; Fox, Peter
1996AAS...188.6916G Altcode: 1996BAAS...28..938G
Below the convection zone, where the stratification is radiatively
controlled, large-scale motions should be mainly horizontal, i.e. in
spherical shells, due to the negative buoyancy radial displacements
would experience. Watson (G.A.F.D. 16, 285, 1981) showed that
the observed surface solar differential rotation is right at the
boundary for instability to horizontal disturbances. Therefore,
since helioseismology tells us the latitudinal differential rotation
below the convection zone is less than the surface value, it should be
stable. We show that in the presence of a broad, nonuniform toroidal
field this differential rotation is unstable. This is true for a wide
range of kinetic and magnetic energies of the unperturbed state, from
well below equipartition, to values above it. The instability appears
to occur only for longitudinal wave number 1. Its location in latitude
depends on details of the magnetic field profile. Generally, the primary
energy source for the instability is the differential rotation, but
the toroidal field also contributes. The mechanism of energy release
is the poleward transport of angular momentum, in a complex interplay
between the perturbation Reynolds and Maxwell stresses. This instability
may play a role in the solar dynamo, although being two- dimensional,
it cannot produce a dynamo by itself. Mixing of angular momentum caused
by the instability could allow achievement of equilibrium of the solar
tachocline hypothesized by Spiegel and Zahn.
---------------------------------------------------------
Title: GONG Observations of Solar Surface Flows
Authors: Hathaway, D. H.; Gilman, P. A.; Jones, H. P.; Kasher, J.;
Simon, G. W.; GONG Nearly Steady Flows Team; GONG Magnetic Fields Team
1996AAS...188.5304H Altcode: 1996BAAS...28..903H
Doppler velocity observations obtained by the GONG instruments directly
measure the nearly steady flows in the solar photosphere. The Sun's
differential rotation profile is accurately determined from single
observations. This profile is well represented by a fourth order
polynomial which includes a rapidly rotating equator and a slight
north-south asymmetry. Rotation profiles averaged over 27 day rotations
of the Sun are sufficient to reveal the torsional oscillation signal -
weak, 5 m/s, jet-like features associated with the sunspot latitude
activity belts. A meridional circulation with poleward flow of about 20
m/s is also found from single observations and its spatial structure
is well determined. Several of the observed characteristics of the
surface flows suggest the presence of large convection cells. The
convection spectrum is measured and found to have peak power for cells
with wavelengths of about 50,000 km but the spectrum extends to much
larger wavelengths. Day-to-day variations in the observed structure of
the differential rotation and meridional circulation profiles indicate
the presence of large-scale, nonaxisymmetric velocity signals which may
be of solar origin. Studies correlating the convective flow patterns on
consecutive days also indicate the presence of large cellular patterns
that rotate at the Sun's rotation rate.
---------------------------------------------------------
Title: The Dynamo Problem. (Book Reviews: Lectures on Solar and
Planetary Dynamos; Solar and Planetary Dynamos)
Authors: Gilman, Peter
1995Sci...269..860P Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Book Review: Lectures on solar and planetary dynamos [invited
papers] / Cambridge U Press, 1995
Authors: Gilman, P.
1995Sci...269Q.860G Altcode: 1995Sci...269Q.860P
No abstract at ADS
---------------------------------------------------------
Title: Book-Review - Solar and Planetary Dynamos
Authors: Proctor, M. R. E.; Matthews, P. C.; Rucklidge, A. M.;
Gilman, P.
1995Sci...269R.860P Altcode:
No abstract at ADS
---------------------------------------------------------
Title: What Can We Learn About Solar Cycle Mechanisms from Observed
Velocity Fields?
Authors: Gilman, Peter A.
1992ASPC...27..241G Altcode: 1992socy.work..241G
No abstract at ADS
---------------------------------------------------------
Title: The solar dynamo.
Authors: Deluca, E. E.; Gilman, P. A.
1991sia..book..275D Altcode:
The authors discuss the present state of our understanding of the
origin of the Sun's magnetic field. They begin with an introduction
to the theory of magnetic field generation in rotating, conducting
fluids. Next they consider a dilemma that has persisted for some 15
yr, namely the inconsistency of the kinematic and dynamic convection
zone dynamo models. A resolution of this dilemma has been suggested
by the recent helioseismology observations on the rotation rate as a
function of latitude and depth. These observations, together with other
observational and theoretical constraints, suggest that the solar dynamo
operates not in the convection zone, but rather in a thin layer between
the convection zone and the radiative interior. A model of such a dynamo
is presented and discussed. The authors conclude by discussing the
problems posed by the placement of the dynamo below the convection zone.
---------------------------------------------------------
Title: The Physics of an Interface Dynamo
Authors: Deluca, E. E.; Gilman, P. A.
1989BAAS...21R.842D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Angular Momentum Transport and Dynamo Action in the Sun:
Implications of Recent Oscillation Measurements
Authors: Gilman, Peter A.; Morrow, Cherilynn A.; Deluca, Edward E.
1989ApJ...338..528G Altcode:
The implications of a newly proposed picture of the sun's internal
rotation (Brown et al., 1989; Morrow, 1988) for the distribution
and transport of angular momentum and for the solar dynamo are
considered. The new results, derived from an analysis of solar acoustic
oscillations, affect understanding of how momentum is cycled in the sun
and provide clues as to how and where the solar dynamo is driven. The
data imply that the only significant radial gradient of angular velocity
exists in a transitional region between the bottom of the convection
zone, which is rotating like the solar surface, and the top of the deep
interior, which is rotating rigidly at a rate intermediate between the
equatorial and polar rates at the surface. Thus the radial gradient
must change sign at the latitude where the angular velocity of the
surface matches that of the interior. These inferences suggest that
the cycle of angular momentum that produces the observed latitudinal
differential rotation in the convection zone may be coupled to layers
of the interior beneath the convection zone.
---------------------------------------------------------
Title: SYSTEMATIC OBSERVATIONS OF THE SUN (In honour of Helen Dodson
Prince): Modeling implications
Authors: Wilson, P.; Gilman, P.
1989HiA.....8..675W Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Angular momentum transport and dynamo action in the Sun:
a report on implications of a recent heliospheric estimate of solar
rotation.
Authors: Morrow, C. A.; Gilman, P. A.; Deluca, E. E.
1988ESASP.286..109M Altcode: 1988ssls.rept..109M
Gilman, Morrow and DeLuca have recently introduced some ideas about how
an emerging helioseismic picture of the Sun's internal rotation might
affect the understanding of angular momentum transport and turbulent
dynamic action in the solar interior. The present paper offers a
brief report and commentary on these issues. Analysis of the frequency
splittings of solar acoustic oscillations measured by Brown and Morrow
has suggested that the latitudinal differential rotation observed at the
surface persists throughout the convection zone, and that the rotation
is more uniform in the outer layers of the radiative interior. Taking
this picture at face value, the authors discuss qualitatively how
angular momentum must be transported in order to sustain the observed
rotation, the exchange of angular momentum between the convection zone
and radiative interior, and the action of a turbulent dynamo which might
be operating near the base of the convection zone. This new qualitative
scenario for the solar dynamo explains both the equatorward migration
of the zone where sunspots appear and the observed poleward migration
of non-sunspot magnetic field.
---------------------------------------------------------
Title: Distribution of Sunspot Umbral Areas: 1917--1982
Authors: Bogdan, T. J.; Gilman, Peter A.; Lerche, I.; Howard, Robert
1988ApJ...327..451B Altcode:
Over 24,000 measurements of individual sunspot umbral areas taken from
the Mount Wilson white-light plate collection covering the period
1917-1982 are used to determine the relative size distribution
of sunspot umbras. In the range 1.5-141 millionths of a solar
hemisphere, the sunspot umbral areas are found to be distributed
lognormally. Moreover, the same distribution is obtained for all phases
of the solar cycle (maximum, minimum, ascending, descending), as well
as for various individual cycles, between 1917 and 1982. Both the mean
and the geometric logarithmic standard deviation of this distribution
appear to be intrinsically constant over the entire data set; only
the number of spots exhibits the familiar solar cycle variations. If
the observed lognormal umbral size distribution is not a particular
attribute of the sunspot umbras but is instead of a more fundamental
property of emerging magnetic flux, then the data would predict a
maximum in the size spectrum of photospheric magnetic structures
for flux tubes with radii in the range 500-800 km. The absence of
solar cycle variations in the relative distribution of umbral areas
and especially the lognormal character of this distribution may both
argue for the fragmentation of magnetic elements in the solar envelope.
---------------------------------------------------------
Title: Implications of Recent Estimates of Solar Interior Rotation
for Angular Momentum Transport and Dynamo Action in the Sun
Authors: Gilman, P. A.; Morrow, C. A.; Deluca, E. E.
1988BAAS...20..701G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Dynamo theory for the interface between the convection zone
and the radiative interior of a star part
Authors: Deluca, Edward E.; Gilman, Peter A.
1988GApFD..43..119D Altcode:
We discuss numerical solutions of nonlinear equations that model
magnetic field generation in a thin layer beneath the convection zone
of a late type star. The model equations were derived previously in
Paper I (DeLuca and Gilman, 1986b). Three main results are found: first,
the oscillating, dynamo wave solutions discussed in DeLuca and Gilman
(1986a) are shown to be a result of the severe truncation used in those
calculations; second, the induced velocity feld is shown to have an
important role in determining the spatial structure of the magnetic
field solutions; time dependent solutions have been found. These are
not wave-like solutions, rather the amplitude of different horizontal
wave modes vary in time. Further, we show that the exact solutions
found in Paper I are generally unstable, with the exception of those
that are independent of (latitude in our Cartesian geometry), which
are stable if the transient induced velocity field remains small. We
conclude that the induced velocity fields are an important ingredient
in any model of dynamo action below the solar convection zone.
---------------------------------------------------------
Title: Inertial Oscillations in the Solar Convection Zone. III. A
Cylidrical Model for Nonaxisymmetric Oscillations in a Superadiabatic
Gradient
Authors: Gilman, Peter A.
1987ApJ...318..904G Altcode:
The effects of a superadiabatic gradient and differential rotation
upon inertial oscillations that are nonaxisymmetric about the axis
of rotation are examined as a generalization of earlier calculations
published by Gilman and Guenther in 1985 for a cylindrical model
of a stellar convection zone. It is shown that the frequencies of
these oscillations are sensitive to both superadiabatic gradient and
differential rotation. If such oscillations can be detected on the sun,
then they could be used to measure the degree of superadiabaticity
of the convection zone, and also to distinguish fine differences
in the radial gradient of angular velocity, for example, between
rotation independent of depth, and one constant on cylinders parallel
to the rotation axis, matched to the surface profile of latitudinal
differential rotation, such as predicted for the sun by various global
convection models.
---------------------------------------------------------
Title: Distribution of Sunspot Umbral Areas: 1917-1982
Authors: Bogdan, T. J.; Gilman, P. A.; Lerche, I.; Howard, R.
1987BAAS...19..924B Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Magnetic Field Generation in the Overshoot Region Beneath
the Solar Convection Zone
Authors: Deluca, E. E.; Gilman, P. A.
1987BAAS...19..945D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Influence of the Coriolis Force on Flux Tubes Rising
through the Solar Convection Zone
Authors: Choudhuri, Arnab Rai; Gilman, Peter A.
1987ApJ...316..788C Altcode:
In order to study the effect of the Coriolis force due to solar
rotation on rising magnetic flux, the authors consider a flux ring,
azimuthally symmetric around the rotation axis, starting from rest
at the bottom of the convection zone, and then follow the trajectory
of the flux ring as it rises. If it is assumed that the flux ring
remains azimuthally symmetric during its ascent, then the problem
can be described essentially in terms of two parameters: the value
of the initial magnetic field in the ring when it starts, and the
effective drag experienced by it. For field strengths at the bottom
of the convection zone of order 10,000 G or less, it is found that
the Coriolis force plays a dominant role and flux rings starting from
low latitudes at the bottom are deflected and emerge at latitudes
significantly poleward of sunspot zones.
---------------------------------------------------------
Title: A laboratory model of planetary and stellar convection
Authors: Hart, J. E.; Toomre, J.; Deane, A. E.; Hurlburt, N. E.;
Glatzmaier, G. A.; Fichtl, G. H.; Leslie, F.; Fowlis, W. W.; Gilman,
P. A.
1987STIN...8722108H Altcode:
Experiments on thermal convection in a rotating, differentially-heated
spherical shell with a radial buoyancy force were conducted in an
orbiting microgravity laboratory. A variety of convective structures,
or planforms, were observed depending on the magnitude of the rotation
and the nature of the imposed heating distribution. The results are in
agreement with numerical simulations that can be conducted at modest
parameter values, and suggest possible regimes of motion in rotating
planets and stars.
---------------------------------------------------------
Title: Laboratory Experiments on Planetary and Stellar Convection
Performed on Spacelab 3
Authors: Hart, J. E.; Toomre, J.; Deane, A. E.; Hurlburt, N. E.;
Glatzmaier, G. A.; Fichtl, G. H.; Leslie, F.; Fowlis, W. W.; Gilman,
P. A.
1986Sci...234...61H Altcode:
Experiments on thermal convection in a rotating, differentially heated
hemispherical shell with a radial buoyancy force were conducted in an
orbiting microgravity laboratory. A variety of convective structures,
or planforms, were observed, depending on the magnitude of the rotation
and the nature of the imposed heating distribution. The results are
compared with numerical simulations that can be conducted at the more
modest heating rates, and suggest possible regimes of motion in rotating
planets and stars.
---------------------------------------------------------
Title: Meridional Motions of Sunspots and Sunspot Groups
Authors: Howard, R.; Gilman, P. A.
1986ApJ...307..389H Altcode:
Mount Wilson white-light plate data for north-south sunspot motions
are studied, taking both sunspot groups and individual spots into
consideration. The average results as a function of latitude show a
midlatitude northward flow and an amplitude of a few hundredths of a
degree per day in each hemisphere. For sunspot groups, a dependence on
latitude is seen that tends generally toward more poleward motions at
higher latitudes. A previously reported, systematic variation of the
latitude dependence of the meridional motion of sunspot groups with
phase in the solar cycle is not confirmed.
---------------------------------------------------------
Title: Nonlinear Convection of a Compressible Fluid in a Rotating
Spherical Shell
Authors: Gilman, P. A.; Miller, J.
1986ApJS...61..585G Altcode:
The deeper layers of the solar or stellar convection zone are modeled in
view of extensive calculations of nonlinear convection of a compressible
fluid in a rotating spherical shell, using the anelastic approximation
for the integrated equations. Solar values are used for the luminosity,
rotation, and eddy diffusivities; the Rayleigh number for convection
is then determined by the luminosity. In the cases of both model
depths used, the dominant convection patterns in low latitudes are a
broad spectrum of rolls with north-south axes, approximately symmetric
about the equator, with more cellular patterns in high latitudes. The
results obtained are similar to those of Glatzmaier (1984, 1985),
who used substantially different solution techniques, and they agree
with the differential rotation observed at the solar surface.
---------------------------------------------------------
Title: Rotation and Expansion within Sunspot Groups
Authors: Gilman, P. A.; Howard, R.
1986ApJ...303..480G Altcode:
By superposing data for many sunspot groups measured on the Mount
Wilson white-light plate collection, the authors demonstrate that
differential rotation, about equal to the ambient rate, occurs between
sunspots within the group. It is also shown that the relative motions
of leader and follower sunspots can be characterized primarily as a
simple expansion of the group along its major axis, with very little
rotation of the pattern about group center.
---------------------------------------------------------
Title: The Influence of the Coriolis Force on Flux Tubes Rising
through Solar Convection Zone
Authors: Choudhuri, A. R.; Gilman, P. A.
1986BAAS...18..703C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Dynamo Theory for a Thin Layer Between the Convection Zone
and the Radiative Zone of a Star. Formulation and Preliminary Results
Authors: Deluca, Edward E.; Gilman, Peter A.
1986LNP...254..173D Altcode: 1986csss....4..173D
No abstract at ADS
---------------------------------------------------------
Title: Dynamo Theory for the Sun and Stars
Authors: Gilman, Peter A.; Deluca, Edward E.
1986LNP...254..163G Altcode: 1986csss....4..163G
No abstract at ADS
---------------------------------------------------------
Title: The solar dynamo: observations and theories of solar
convection, global circulation, and magnetic fields.
Authors: Gilman, P. A.
1986psun....1...95G Altcode:
Contents: Convective zone characteristics (basic structure of depth,
oblateness and latitudinal temperature differences, luminosity and
radius changes). Observations and theory of solar convection zone
motions (granulation, supergranulation and mesogranulation, related
theories of convection, global circulation). Phenomenology of the Sun's
magnetic field and related features (flux tubes and network; sunspots,
active regions, and global patterns; the solar cycle: manifestations in
sunspots, magnetic fields, and other properties). Theories of the solar
magnetic field (flux tubes and network, active regions and sunspots,
solar dynamo theory).
---------------------------------------------------------
Title: Dynamo theory for the interface between the convection zone
and the radiative interior of a star: Part I model equations and
exact solutions
Authors: Deluca, Edward E.; Gilman, Peter A.
1986GApFD..37...85D Altcode:
In this paper we derive a set of equations which model magnetic
field generation and maintenance in a thin region, ( 104)km thick,
below the solar convection zone, and present some simple exact
solutions. Energy to drive the dynamo is assumed to come from helical
convection that overshoots into this region. Differential rotation
and meridional circulation result only from feedbacks by the induced
fields. The equations are derived for a homogeneous incompressible
fluid in Cartesian geometry. The momentum equation, magnetic induction
equation, and the continuity equation are included in the analysis. We
assume velocity and magnetic field patterns have an aspect ratio of
1/10 (radial to horizontal scale), that the large scale velocities
are smaller than the convection zone velocities, a few meters per
sec, and the large scale magnetic fields are of the order of 104
Gauss. Finally we assume that the time scale of interest is the
advective time scale. Using these assumptions, we derive governing
equations in which the Coriolis force balances the Lorentz force, the
pressure gradient force, and the viscosity, and in which the magnetic
fields are maintained by the effect but significantly modified by the
above velocity fields. The results of this model will be discussed in
following papers.
---------------------------------------------------------
Title: Inertial oscillations in the solar convection zone. II -
A cylindrical model for equatorial regions
Authors: Gilman, P. A.; Guenther, D. B.
1985ApJ...296..685G Altcode:
Axisymmetric inertial oscillations found from a simplified, cylindrical
model applicable to equatorial latitudes are presented and compared
to results obtained previously by numerical methods for a spherical
shell. The asymptotic behavior of inertial mode frequencies are
analytically derived in the limits of very large and very small axial
wavenumbers, and it is shown that inertial oscillations arise only
when the angular momentum per unit mass increases outward. The effects
of convection zone depth and differential rotation on the different
radial orders of inertial modes are demonstrated. In a convection
zone 15 billion cm in depth, rotation rate increasing inward leads
to decreased oscillation frequency for a given radial order and axial
wavenumber, the opposite of the result for a spherical shell. Deeper
zones in the cylindrical case give increased frequencies for rotation
rate either decreasing or increasing inward. The differences in the
results between spherical and cylindrical models are shown to be due
to the different geometries.
---------------------------------------------------------
Title: Inertial oscillations in the solar convection zone. I -
Spherical shell model
Authors: Guenther, D. B.; Gilman, P. A.
1985ApJ...295..195G Altcode:
The solar oscillation spectrum has the potential to provide a great deal
of information about the sun's interior. However, an interpretation
of the oscillation spectrum is an extremely difficult task. Neither
direct nor indirect approaches can give a complete picture of the
solar interior. It appears, therefore, advisable to employ a wide
variety of methods to probe the sun's interior. In the present paper,
a description is provided of a family of solar oscillations, which is
called inertial oscillations. It is shown that these oscillations are
sensitive (independently) to both the depth and the rotation profile of
the convection zone. If these modes can be observed, they will provide
an independent determination of the convection zone depth and rotation
frequency profile, as a function of radius (and latitude).
---------------------------------------------------------
Title: Rotation rates of leader and follower sunspots
Authors: Gilman, P. A.; Howard, R.
1985ApJ...295..233G Altcode:
The rotation rates of leader and follower sunspots found on Mount Wilson
white light plates have been measured for the years 1917-1983. It is
found that at all latitudes, leader spots rotate faster than follower
spots by about 0.1 deg per day, or 14 m/s. It is also found that, when
examined separately, leaders and followers show the same variations in
rotation with cycle phase as do all spots taken together, as reported
earlier in Gilman and Howard (1984). Leaders and followers show similar
variations in rotation rate even on an annual basis. Thus, while leaders
and followers in each group diverge in longitude from each other at an
average rate of about 0.1 deg per day, each is separately speeding up
or slowing down its rotation according to the phase in the cycle, and
by a similar amount. Leaders and followers also give about the same
covariance of longitude and latitude motions, indicating that whole
sunspot groups participate in tracing the apparent angular momentum
transport toward the equator, as previously reported for all spots in
Gilman and Howard (1984).
---------------------------------------------------------
Title: What would a dynamo theorist like to know about the dynamics
of the solar convection zone?
Authors: Gilman, P. A.
1984sses.nasa...41G Altcode: 1984sss..conf...41G
Observations about the current status of solar dynamo theory are
given. The induction equation for magnetic field is solved using
assumed velocities and parametric representations of the inductive
or diffusive effects of velocities. The equations of motion governing
these flow are not solved in parallel. Results from global compressible
convection models are discussed. Differential rotation and convection
are also investigated.
---------------------------------------------------------
Title: Techniques for detecting giant cells using spatially resolved
solar velocity data
Authors: Brown, T. M.; Gilman, P. A.
1984ApJ...286..804B Altcode:
Whether giant cells exist in the convection zone of the sun, and what
their properties might be, are matters of great importance for the
understanding of the dynamics of the solar interior. So far, such cells
have escaped detection, probably because of the small amplitude of their
associated velocity fields and the large amplitudes of shorter-lived
flows. Techniques for improving the detectability of giant cells are
presented. These methods are based on the spatial extent and symmetry
properties of giant cells as seen in self-consistent dynamical models
of the solar convection zone. Simulations suggest that these techniques
allow detection of giant cells with photospheric rms velocities of 2
m/s (0.4 m/s per longitudinal wavenumber), given observations spanning
about one year.
---------------------------------------------------------
Title: Inertial Oscillations in the Solar Convection Zone Part I
Authors: Guenther, D. B.; Gilman, P. A.
1984BAAS...16Q1000G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Inertial Oscillations in the Solar Convection Zone Part II:
A Cylindrical Model for Equatorial Regions
Authors: Gilman, P. A.; Guenther, D. B.
1984BAAS...16R1000G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Variations in solar rotation with the sunspot cycle
Authors: Gilman, P. A.; Howard, R.
1984ApJ...283..385G Altcode:
The positions of sunspots as photographed in white light at Mt. Wilson
from 1921-82 were analyzed to detect any systematic variations,
particularly in relation to the solar cycle. The study analyzed 5 deg sq
bins of sunspots for both hemispheres. The residual rotation rates were
calculated for individual and grouped sunspots. Peaks in the sunspot
rotation rate were detected near the solar maximum and minimum and
in high solar latitudes 3 yr before the end of the cycle. The highest
sunspot rotation peaks were 0.5 deg/day. The ubiquity of the rotation
rate changes over the whole solar disk implied periodic angular momentum
exchanges between the photosphere and deeper layers of the convective
zone. Finally, the interpretations of the differences observed between
variations in sunspot and Doppler rotation are discussed.
---------------------------------------------------------
Title: Rotation of the sun measured from Mount Wilson white-light
images
Authors: Howard, R.; Gilman, P. I.; Gilman, P. A.
1984ApJ...283..373H Altcode:
The instrumentation, data and data reduction procedures used in white
light observations of sunspot rotation rates are described. The study
covered 62 yr of rotation observations. The data were all gathered using
the same Mt. Wilson telescope, which has had three different main lenses
in the interval 1981-82. Details of the exposure calibration and lens
operation procedures are provided. The data were treated in terms of
eight evenly space determinations of the solar limb and account was
taken of all sunspots within 60 deg of the central meridian. Spot
movements were traced in terms of groups of contiguous individual
spots. Large spots rotated slower than small spots, a condition
attributed to greater viscous drag in the larger flux tubes in the
photosphere. The data tend to confirm theories that the photospheric
gas revolves at a different rate than the sunspots.
---------------------------------------------------------
Title: On the Correlation of Longitudinal and Latitudinal Motions
of Sunspots
Authors: Gilman, P. A.; Howard, R.
1984SoPh...93..171G Altcode:
Using new measurements of positions of individual sunspots and sunspot
groups obtained from 62 years of the Mt. Wilson white-light plate
collection, we have recomputed the correlation between longitude and
latitude motion. Our results for groups are similar to those of Ward
(1965a) computed from the Greenwich record, but for individual spots
the covariance is reduced by a factor of about 3 from the Ward values,
though still of the same sign and still statistically significant. We
conclude that there is a real correlation between longitude and latitude
movement of individual spots, implying angular momentum transport
toward the equator as inferred by Ward. The two thirds reduction in
the covariance for individual spots as opposed to groups is probably
due to certain properties of spot groups, as first pointed out in an
unpublished manuscript by Leighton.
---------------------------------------------------------
Title: The Rotation of the Sun from Mount Wilson Sunspot Measurements
Authors: Howard, R. F.; Gilman, P. A.
1984KodOB...4....1H Altcode:
The authors have completed the measurement and reduction of 62 years
of white-light solar images taken at the Mount Wilson Observatory. The
data have been analyzed for differential rotation and time variations
of this quantity. This is a brief review of the work.
---------------------------------------------------------
Title: Dynamically consistent nonlinear dynamos driven by convection
in a rotating spherical shell. II - Dynamos with cycles and strong
feedbacks
Authors: Gilman, P. A.
1983ApJS...53..243G Altcode:
Nonlinear dynamos with cycles and strong feedback are investigated
theoretically. The convectively driven dynamo model of Gilman and Miller
(1981) is applied to cases with viscosity and thermal diffusivity
reduced by a factor of 10, so that weaker global convection drives
an equatorial acceleration of solar magnitude and profile, and
differential rotation (DR) makes up nearly 80 percent of the system
kinetic energy. The schematics of the dynamo process are introduced, and
the design of the calculations is outlined. The results are discussed
in terms of the onset of dynamo action, the phenomenology of cyclic
solutions, symmetry preservation and switching, total energy statistics,
the maintenance of convection and DR against dissipation and feedbacks,
changes in DR profiles, maintenance of toroidal and poloidal fields,
kinetic and magnetic energy spectra, and applications to the sun and
stars. It is found that the model gives cyclic dynamos with periods
one order of magnitude shorter than that of the sun. Drawings and
graphs illustrating the findings are provided.
---------------------------------------------------------
Title: Compressible Convection in a Deep Rotating Spherical Shell
Authors: Gilman, P. A.
1983BAAS...15..715G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Variations in Sunspot Rotation and the Activity Cycle
Authors: Howard, R.; Gilman, P. A.
1983BAAS...15..698H Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Dynamos of the sun and stars, and associated convection
zone dynamics
Authors: Gilman, P. A.
1983IAUS..102..247G Altcode:
Results obtained from a fully 3D nonaxisymmetric nonlinear MHD model
of convection in a deep spherical shell initially in uniform rotation
and heated uniformly from the inside, with central gravity and constant
viscosity and thermal conductivity, are summarized and discussed as they
apply to solar and stellar dynamos. The constraints imposed by solar
observations are examined, the model results are presented graphically
and contrasted with the observations, and general implications for the
types of stellar dynamo action to be expected are indicated. Cyclic
dynamo action is located primarily at low latitudes outside the tangent
cylinder to the inner boundary of the convection zone; this prediction
leads to line-of-sight-dependent differences in the observability of
cyclic-dynamo-induced changes in Ca II emissions.
---------------------------------------------------------
Title: Compressible convection in a rotating spherical shell. V -
Induced differential rotation and meridional circulation
Authors: Glatzmaier, G. A.; Gilman, P. A.
1982ApJ...256..316G Altcode:
Solutions are described for the mean differential rotation and
meridional circulation in a compressible, rotating, spherical
fluid shell which are induced by linear, anelastic solutions for
global convection. The mean solutions strongly depend on the density
stratification, the rotation rate, the convective velocity distribution,
and the amount of viscous diffusion relative to thermal diffusion. It
is noted that at least one of two conditions must be met in order to
obtain an equatorial acceleration that is large in amplitude relative
to the meridional circulation (together with a small equator-pole
temperature difference) when the density stratification is as large
as in the solar convection zone. Either the effect of rotation must
be large compared with the effects of viscous diffusion and buoyancy
or viscous diffusion must be small relative to thermal diffusion. In
either case, the angular velocity increases with depth in the upper
part of the convection zone but decreases with depth and is nearly
constant on cylinders in the lower part when the global convection
reaches deep layers.
---------------------------------------------------------
Title: Convective dynamos for rotating stars.
Authors: Gilman, P. A.
1982SAOSR.392A.165G Altcode: 1982csss....2..165G
Global dynamo theory is applied to the problem of why some stars
have field reversing dynamos, and others do not. It is argued that
convectively driven dynamos are the most likely source of magnetic
fields in stars that have convection zones.
---------------------------------------------------------
Title: Compressible convection in a rotating spherical shell. IV -
Effects of viscosity, conductivity, boundary conditions, and zone
depth
Authors: Glatzmaier, G. A.; Gilman, P. A.
1981ApJS...47..103G Altcode:
The manner in which variations in given assumptions affect the stability
and structure of solutions is demonstrated for the case of a linear,
anelastic model of compressible convection in a rotating spherical
fluid shell described by Glatzmaier and Gilman (1981). Velocity is
shown to be enhanced in the lower part of the zone opposed to the
upper part, when the kinematic viscosity is assumed to be inversely
proportional to the density rather than constant. The velocity is
more uniformly distributed in radius when the kinematic viscosity
increases with radius at a rate lower than the inverse density, or
when viscous diffusion is small relative to thermal diffusion. When
constant temperature boundaries are replaced by constant diffusive
heat flux boundaries, the most unstable modes are more unstable and
have larger longitudinal dimensions; this is in agreement with previous
studies of plane-parallel, Boussinesq convection.
---------------------------------------------------------
Title: An improved search for large-scale convection cells in the
solar atmosphere
Authors: Labonte, B. J.; Howard, R.; Gilman, P. A.
1981ApJ...250..796L Altcode:
A reanalysis of Mount Wilson solar velocity observations was made to
search for giant cellular patterns. The reanalysis avoids several errors
made in a previous search. No cells are detected with sensitivity of
3 to 12 m/s depending upon wavenumber. The observed amplitudes do not
conflict with recent model predictions.
---------------------------------------------------------
Title: Global circulation and the solar dynamo.
Authors: Gilman, P. A.
1981NASSP.450..231G Altcode: 1981suas.nasa..231G
The Sun is apparently rather typical of stars in its spectral class,
with a convection zone of substantial depth and a modest rotation
rate. These two factors are apparently enough to generate a substantial
global circulation, seen so far principally as a differental rotation,
as well as a nearly cyclic magnetohydrodynamic dynamo, seen principally
as the 22 year "solar cycle". It would be expected therefore that many
stars would have such dynamical characteristics. Recent observations
of very large scale velocity fields of small velocity amplitude were
reviewed.
---------------------------------------------------------
Title: Dynamically consistent nonlinear dynamos driven by convection
in a rotating spherical shell
Authors: Gilman, P. A.; Miller, J.
1981ApJS...46..211G Altcode:
Calculations are presented of a convectively driven hydromagnetic dynamo
in a rotating spherical shell in which all of the induction effects
arise from the same, self-consistent solutions of the convection
equations, and the results obtained are compared with observations
of the solar dynamo. A hydromagnetic model for nonlinear convection
in a rotating spherical shell is used in which it is possible to
solve simultaneously for the evolving velocity and magnetic field and
include the full feedbacks of the magnetic field on the motion. The
solutions are obtained in stages, with the preliminary perturbation
of a state of solid rotation to develop a spectrum of convection and
differential rotation, followed by the introduction of a small-seed
magnetic field of various magnetic Prandtl numbers to induce dynamo
behavior. Although the calculated differential rotation is found to
be similar to the observed equatorial acceleration of the sun, the
calculated dynamo behaves quite differently, with no global magnetic
field reversals, equatorial migration of the toroidal magnetic field or
preferred symmetry of the induced magnetic fields, which is attributed
to an excessively large model helicity. It is concluded that there
as yet does not exist a solar dynamo model which both reproduces the
observational characteristics of the solar dynamo and is consistent
with the laws of fluid dynamics.
---------------------------------------------------------
Title: Differntial Rotation Induced by Compressible Convection
Authors: Glatzmaier, G. A.; Gilman, P. A.
1981BAAS...13..527G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: More on Dynamos Driven by Global Convection and Differential
Rotation
Authors: Gilman, P. A.
1981BAAS...13..907G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Compressible Convection in a Rotating Spherical Shell -
Part Two - a Linear Anelastic Model
Authors: Glatzmaier, G. A.; Gilman, P. A.
1981ApJS...45..351G Altcode:
We study the onset of convection for a compressible fluid in a rotating
spherical shell via linear anelastic fluid equations for a depth of
40% of the radius, constant kinematic viscosity and thermometric
diffusivity, Taylor numbers up to l0<SUP>5</SUP>, and density
stratifications up to seven e-folds across the zone. The perturbations
are expanded in spherical harmonics, and the radially dependent
equations are solved with a Newton-Raphson relaxation method. <P />The
most unstable modes are single cells extending from the bottom to the
top of the convection zone. As the density stratification and rotation
rate increase, the horizontal dimension of these cells decreases, while
their prograde longitudinal phase velocity and the enhancement of the
velocity near the top of the zone increase. Cylindrical cells arranged
symmetrically about the equator develop parallel to the rotation axis
as the rotation rate increases, but for large stratifications, their
axes bend toward the pole at midlatitude instead of intersecting the
outer surface, as they do for small stratifications. The buoyancy force
does negative work near the top of the zone, while the pressure force
does a net positive amount of work for large stratifications. Helicity
profiles and convective heat flux profiles for the most unstable modes
do not change significantly as the density stratification increases,
although distinct differences develop for modes that are not the most
unstable. The radial momentum flux for the most unstable modes at high
rotation rates is inward for large stratifications instead of outward
as it is for small stratifications; this may have a significant effect
on differential rotation and magnetic field generation in nonlinear,
compressible models.
---------------------------------------------------------
Title: Compressible Convection in a Rotating Spherical Shell -
Part Three - Analytic Model for Compressible Vorticity Waves
Authors: Glatzmaier, G. A.; Gilman, P. A.
1981ApJS...45..381G Altcode:
A simple analog to compressible convection in a rotating spherical
shell is described and solved analytically to enable us to understand
the basic physics of the prograde phase velocities that resulted
from the numerical calculations of Paper II. The analog is for an
inviscid, adiabatically stratified, rotating, equatorial annulus
of gas for which a form of potential vorticity is conserved. Linear
perturbations in this system take the form of vorticity waves which
propagate prograde relative to the rotating reference frame as long
as the density decreases outward. Their frequencies depend on the
longitudinal wavenumber and density stratification in the same way as
the convective modes of Paper II. Simple physical arguments explain
these dependences.
---------------------------------------------------------
Title: Compressible convection in a rotating spherical shell. I -
Anelastic equations. II - A linear anelastic model. III - Analytic
model for compressible vorticity waves
Authors: Gilman, P. A.; Glatzmaier, G. A.
1981ApJS...45..335G Altcode:
In order to develop equations whose solution will clarify the
role played by large solar convection zone density variations
in differential rotation transports, anelastic equations for
convection of a compressible fluid in a deep, rotating spherical
shell are derived in the first part of the study. The model equations
represent a generalization of a Boussinesq system that has been studied
extensively with the solar differential rotation problem in mind, and
are expected to apply best in the deep part of a convection zone where
departures of the fluid from an adiabatic atmosphere are smallest. The
second part of the study focuses on the onset of convection for a
compressible fluid in a rotating spherical shell via linear inelastic
fluid equations for a depth of 40% of the radius, constant kinematic
viscosity and thermometric diffusivity, Taylor numbers up to 100,000,
and density stratifications up to seven e-folds across the zone. The
perturbations are expanded in spherical harmonics, and the radially
dependent equations are solved with a Newton-Raphson relaxation method.
---------------------------------------------------------
Title: Compressible convection in a rotating spherical shell.
Authors: Glatzmaier, G. A.; Gilman, P. A.
1981ASIC...68..145G Altcode: 1981spss.conf..145G
No abstract at ADS
---------------------------------------------------------
Title: Effects of certain analysis procedures on solar global
velocity signals
Authors: Gilman, P. A.; Glatzmaier, G. A.
1980ApJ...241..793G Altcode:
The effects of data reduction procedures on the signals obtained for
global solar velocities from the Doppler magnetograph at Mount Wilson
are examined. The data reduction procedure used by Howard (Howard
et al., 1980; LaBonte and Howard, 1979; Howard and LaBonte, 1980)
is applied to idealized global velocity data, and it is shown that
the effects of removing daily rotation, ears and zero offset signals
and the construction of synoptic maps can easily reduce an original
periodic east-west flow velocity of peak amplitude 100 m/sec to 10
m/sec or less for any wavenumber. Furthermore, it is shown that a
velocity spectrum obtained from a nonlinear spherical convection model
is attenuated to rms residual velocities close to or within the upper
limits of Howard and LaBonte. It is concluded that there could be a
substantially larger global velocity signal present in the Mount Wilson
magnetograph data than has so far been demonstrated, and data reduction
and observational techniques for obtaining these signals are suggested.
---------------------------------------------------------
Title: Axisymmetric Convection Driven by Latitudinal Temperature
Gradients in Rotating Spherical Shells.
Authors: Hathaway, D. H.; Gilman, P. A.; Miller, J.; Toomre, J.
1980BAAS...12..686H Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Erratum - Model Calculations Concerning Rotation at High
Solar Latitudes and the Depth of the Solar Convection Zone
Authors: Gilman, P. A.
1980ApJ...236..706G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Dynamics of the Solar Interior and the Solar Dynamo
Authors: Gilman, P. A.
1980NASCP2098....3G Altcode: 1980sscs.nasa....3G
The solar convection zone is the origin of most of the variations in
solar output observed or suspected to occur. The Sun's magnetic field
is rooted there, and solar activity and the solar cycle are generated
and maintained there. Changes in the magnetic fields which reach into
the solar atmosphere and beyond to interplanetary space are largely
determined by the dynamo action of velocity fields in the convection
zone. If changes in solar luminosity occur on time scales of months
to millenia, such changes probably have their origin in the changing
dynamics of the convection zone, either as cause of or in response
to long term changes in the level of solar activity. Fluctuations
would occur in the rate at which energy is brought to the surface
by convection, and the solar diameter would be slightly modified. To
describe and ultimately understand the global workings of the solar
dynamo requires simultaneous high quality photospheric observations of
solar velocities, magnetic fields, intensity patterns, luminosity and
various radiative outputs. The observations must be nearly continuous
in time and of long duration-most or all of a solar cycle. Such a
measurement program should be a major part of the proposed Solar Cycle
and Dynamics Mission.
---------------------------------------------------------
Title: Convective instability when the temperature gradient and
rotation vector are oblique to gravity. II. Real fluids with effects
of diffusion
Authors: Hathaway, D. H.; Toomre, J.; Gilman, P. A.
1980GApFD..15....7H Altcode:
The linear stability analysis of Hathaway, Gilman and Toomre (1979)
(hereafter referred to as Paper I) is repeated for Boussinesq fluids
with viscous and thermal diffusion. As in Paper I the fluid is confined
between plane parallel boundaries and the rotation vector is oblique
to gravity. This tilted rotation vector introduces a preference
for roll-like disturbances whose axes are oriented north-south;
the preference is particularly strong in the equatorial region. The
presence of a latitudinal temperature gradient produces a thermal
wind shear which favors axisymmetric convective rolls if the gradient
exceeds some critical value. For vanishingly small diffusivities the
value of this transition temperature gradient approaches the inviscid
value found in Paper I. For larger diffusivities larger gradients are
required particularly in the high latitudes. These results are largely
independent of the Prandtl number. Diffusion tends to stabilize the
large wavenumber rolls with the result that a unique wavenumber can
be found at which the growth rate is maximized. These preferred rolls
have widths comparable to the depth of the layer and tend to be broader
near the equator. The axisymmetric rolls are similar in many respects
to the cloud bands on Jupiter provided they extend to a depth of about
15,000 km.
---------------------------------------------------------
Title: Global Circulation of the Sun: Where Are We and Where Are
We Going?
Authors: Gilman, P. A.
1980HiA.....5...91G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Differential rotation in stars with convection zones
Authors: Gilman, P. A.
1980LNP...114...19G Altcode: 1980sttu.coll...19G; 1980IAUCo..51...19G
The paper discusses quantitatively and qualitatively, differential
rotation at the surface of stars that rotate and have a convection
zone. Convection zone parameters are considered, particularly the
ratio of convection turnover time to the stellar rotation time. In
addition, the redistribution of angular momentum by axisymmetric
meridional circulation and by eddies is discussed. Some models that
have been developed to explain differential rotation in the sun are
reviewed, particularly models developed for nonaxisymmetric convection
of a stratified liquid in a rotating spherical shell; results from
calculations indicate that finite amplitude equatorial acceleration
is produced only when the influence of rotation upon convection is
strong. Qualitative statements about main sequence stars and red giants
are then extrapolated from the results, including the estimation of
the turnover time for the convection zone from a mixing length model
applied to the whole convection zone.
---------------------------------------------------------
Title: Comment (and Response) on Paper "Nonlinear Dynamics of
Boussinesq Convection in a Deep Rotating Shell. III. Effects of
Velocity Boundary Conditions" by P.A. Gilman
Authors: Busse, F. H.; Gilman, P. A.
1980GApFD..14..251B Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Convective Instability in Rotating Layers with Thermal Winds
and Application to Jupiter
Authors: Hathaway, D. H.; Gilman, P. A.; Toomre, J.
1979BAAS...11Q.618H Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Model calculations concerning rotation at high solar latitudes
and the depth of the solar convection zone.
Authors: Gilman, P. A.
1979ApJ...231..284G Altcode:
The author has previously carried out extensive nonlinear numerical
simulations of convection in a rotating spherical shell, motivated
by the problem of understanding the Sun's differential rotation. The
polar vortex found in earlier calculations, but not observed on the
Sun, disappears when shell depths as large as 40% of the radius are
assumed. The equatorial acceleration is also enhanced, producing a
closer fit to solar observations than previous calculations for depths
of 20%. The reasons for the disappearance of the polar vortex are
that in deeper layers the Reynolds stresses, which transport angular
momentum toward the equator to form the equatorial acceleration,
reach to higher latitudes, and the moment of inertia of the polar cap
region is a smaller fraction of the total for the shell. Although the
model used is for a stratified liquid (the Boussinesq approximation),
we argue that deep layers are less likely to have a polar vortex than
shallow ones in the compressible case too. This result favors a solar
convection zone substantially deeper than previously inferred from
stellar structure calculations applied to the Sun, but is consistent
with recent inferences of a deep convection zone made from measurements
of solar oscillations. Subject headings: convection - Sun: interior -
Sun: rotation
---------------------------------------------------------
Title: Angular velocity gradients in the solar convection zone.
Authors: Gilman, P. A.; Foukal, P. V.
1979ApJ...229.1179G Altcode:
Numerical calculations of Boussinesq nonaxisymmetric convection
in a rotating spherical shell are reported which were performed to
study how convection in the supergranule layer redistributes angular
momentum. It is found that supergranules are at best weakly influenced
by rotation and can be largely responsible for the radial gradient
of angular velocity observed in the thin supergranule layer below the
photosphere. The results indicate that convection in a thin spherical
shell weakly influenced by rotation can produce a substantial outward
decrease in rotational velocity that approaches the limit predicted for
radially moving particles that conserve their angular momentum. This
phenomenon is shown to provide a plausible explanation for the observed
difference in angular velocity between sunspots and the photospheric
plasma.
---------------------------------------------------------
Title: Convective Instability when the Temperature Gradient and
Rotation Vector are Oblique to Gravity. I. Fluids without Diffusion
Authors: Hathaway, D. H.; Toomre, J.; Gilman, P. A.
1979GApFD..13..289H Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Long-term variations in the solar dynamo
Authors: Gilman, P. A.
1979LPICo.390...39G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Response
Authors: Gilman, Peter A.
1979GApFD..14..252G Altcode:
This Article does not have an abstract.
---------------------------------------------------------
Title: A review of: "Rotating fluids in geophysics"
Authors: Gilman, Peter A.
1979GApFD..12..183G Altcode:
By P. H. Roberts and A. M. Soward (Editors). Academic Press
Inc. (London) Ltd, $36.25 (£17.50). (ISBN 0 12 589650 6.)
---------------------------------------------------------
Title: Polar Deceleration and Convection Zone Depth for the Sun.
Authors: Gilman, P. A.
1978BAAS...10Q.400G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Nonlinear dynamics of boussinesq convection in a deep rotating
spherical shell III: Effects of velocity boundary conditions
Authors: Gilman, Peter A.
1978GApFD..11..181G Altcode:
We perform numerical experiments for convection in a rotating spherical
shell with either one or two nonslip boundaries, for the same Rayleigh
and Taylor numbers (R = 104 to 105, T = 105) and the same temperature
boundary conditions as used in Part II. We find that both convection and
differential rotation energies are reduced compared with solutions with
stress free boundaries, but the differential rotation is reduced by a
much larger factor, demonstrating the constraining influence of rigidly
rotating boundaries on the angular momentum profile. Differential
rotation ceases to be a prominent, global feature of the flow, and
may be difficult to observe in laboratory experiments. The convection
spectrum with nonslip boundaries is also broader, and shifts much
less toward low longitudinal wave numbers with increased R. What
differential rotation remains is driven primarily by Coriolis torques
from the axisymmetric meridional circulation, rather than by Reynolds
stresses. For R5 × 104, the bouyancy driven axisymmetric meridional
circulation is substantially larger than for stress free boundaries,
due to Ekman boundary layers breaking the rotational constraints which
otherwise suppress this inherently nongeostrophic flow. In the same
Rayleigh number range, a greater fraction of the total heat flux is
carried by convection than with stress free boundaries, also a result
of the destabilizing influence of the Ekman layers. <BR />Solutions
with stress free top and nonslip bottom behave similarly to the
stress free top and bottom case at low Rayleigh number, because the
convection occurs mostly outside the cylinder tangent to the inner
boundary equator, so that the inner velocity boundary condition is
not strongly felt. As R is increased, the convection and differential
rotation feel this boundary much more strongly. <BR /><BR />Despite
the many differences, the convection solutions for different boundary
conditions have a number of similarities. These include location
of peak convection amplitudes (the equator with a secondary peak
at high latitudes), north-south roll orientation near the equator,
Reynolds stress patterns, and the form of axisymmetric meridional
circulation. Of particular interest is that the helicity profile of
the convection is similar for all boundary conditions, but is of
larger amplitude with nonslip boundaries. <BR /><BR />Our results
suggest that differential rotation is likely to be much different,
and of smaller amplitude relative to the convection which drives it,
in a liquid planetary interior as compared to a stellar convection
zone. This difference may result in favoring planetary dynamos of the
α2 variety, and stellar dynamos of the α-ω type. These speculations
need to be tested by model calculations with magnetic field included.
---------------------------------------------------------
Title: Nonlinear Dynamics of Boussinesq Convection in a Deep Rotating
Spherical Shell. II. Effects of Temperature Boundary Conditions
Authors: Gilman, P. A.
1978GApFD..11..157G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Abstracts to forthcoming papers
Authors: Gilman, Peter A.
1978GApFD..11..155G Altcode:
This Article does not have an abstract.
---------------------------------------------------------
Title: Anomalous Solar Rotation in the Early 17th Century
Authors: Eddy, J. A.; Gilman, P. A.; Trotter, D. E.
1977Sci...198..824E Altcode:
The character of solar rotation has been examined for two periods in the
early 17th century for which detailed sunspot drawings are available:
A.D. 1625 through 1626 and 1642 through 1644. The first period occurred
20 years before the start of the Maunder sunspot minimum, 1645 through
1715; the second occurred just at its commencement. Solar rotation in
the earlier period was much like that of today. In the later period,
the equatorial velocity of the sun was faster by 3 to 5 percent and the
differential rotation was enhanced by a factor of 3. The equatorial
acceleration with declining solar activity is in the same sense as
that found in recent Doppler data. It seems likely that the change
in rotation of the solar surface between 1625 and 1645 was associated
with the onset of the Maunder Minimum.
---------------------------------------------------------
Title: A note on estimating the latitudinal angular momentum transport
in the solar photosphere from Doppler velocities.
Authors: Gilman, P. A.
1977A&A....58..315G Altcode:
If it is assumed that global-scale motions on the sun are predominantly
horizontal and their patterns persist largely unchanged during passage
from the east to the west limb, there is a relationship between
the Doppler signal and the latitudinal angular-momentum transport
which can in principle be tested from observations. This relationship
implies that if there is a significant angular-momentum flux towards
the equator from higher latitudes in each hemisphere, as predicted
by convection theory for maintaining the equatorial acceleration,
the mean-square Doppler-velocity signal must be larger east of the
central meridian than west of it at the same longitude, in both the
northern and southern hemispheres. The difference should be largest
at middle latitudes in each hemisphere.
---------------------------------------------------------
Title: Nonlinear Dynamics of Boussinesq Convection in a Deep Rotating
Spherical Shell. I.
Authors: Gilman, P. A.
1977GApFD...8...93G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Theory of Solar Rotation
Authors: Gilman, Peter A.
1977lsms.proc...87G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Coronal holes and the sun's interior.
Authors: Gilman, P. A.
1977chhs.conf..331G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Abstracts to forthcoming papers
Authors: Gilman, Peter A.
1977GApFD...8...91G Altcode:
This Article does not have an abstract.
---------------------------------------------------------
Title: Progress in Modeling Convection in Deep Rotating Spherical
Shells, with Implications for Solar Differential Rotation Theory
Authors: Gilman, P. A.
1976BAAS....8..299G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Solar rotation during the Maunder minimum.
Authors: Eddy, J. A.; Gilman, P. A.; Trotter, D. E.
1976SoPh...46....3E Altcode:
We have measured solar surface rotation from sunspot drawings made
in a.d. 1642-1644 and find probable differences from present-day
rates. The 17th century sunspots rotated faster near the equator by
3 or 4%, and the differential rotation between 0 and ±20° latitude
was enhanced by about a factor 3. These differences are consistent
features in both spots and groups of spots and in both northern and
southern hemispheres. We presume that this apparent change in surface
rotation was related to the ensuing dearth of solar activity (the
Maunder Minimum) which persisted until about 1715.
---------------------------------------------------------
Title: Theory of Convection in a Deep Rotating Spherical Shell,
and its Application to the Sun
Authors: Gilman, P. A.
1976IAUS...71..207G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Summary of the Final Discussion on August 29
Authors: Durney, B. R.; Gilman, P. A.; Stix, M.
1976IAUS...71..479D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Linear Simulations of Boussinesq Convection in a Deep Rotating
Spherical Shell.
Authors: Gilman, Peter A.
1975JAtS...32.1331G Altcode:
We present extensive linear numerical simulations of Boussinesq
convection in a rotating spherical shell of finite depth. The motivation
for the study is the problem of general circulation of the solar
convection zone. We solve the marching equations on a staggered grid
in the meridian plane for the amplitudes of the most unstable Fourier
mode of longitudinal wavenumber m between 0 and 24, for Taylor number
T between 0 and 10<SUP>6</SUP>, at a Prandtl number P=1, for a shell of
depth 20% of the outer radius. Stress-free, fixed-temperature boundary
conditions are used at the inner and outer bounding surfaces. Modes
of two symmetries, symmetric and antisymmetric about the equator,
are studied. The principal results are as follows:Increasing Taylor
number T splits the most unstable solutions for each m into two
classes: a broad band of high m solutions which peak at or near
the equator, and a small number of low m solutions which peak at or
near the poles. The equatorial modes are unstable at lower Rayleigh
number R. The polar modes appear to be similar in many respects to
plane-parallel convection with rotation parallel to gravity. Modes
symmetric about the equator are unstable at lower R than those which
are antisymmetric, by a percentage which increases with T in the
range studied.Equatorial modes of both symmetries propagate prograde
(frequency >0) in longitude at high T and retrograde (<0) at low
T, in agreement with earlier work. Polar modes propagate, too, but very
slowly.Critical (first unstable) equatorial modes are shown to have or
be closely approaching asymptotic dependence R<SUB>c</SUB>T<SUP></SUP>,
m<SUB>c</SUB>T<SUP></SUP>, <SUB>c</SUB>T<SUP></SUP> with increasing
T, in agreement with analytical analyses of Roberts and Busse.With
increasing T, symmetric equatorial modes take on the form of rolls
swirling about an axis parallel to the rotation axis and extending
across both Northern and Southern Hemispheres in agreement with earlier
results. Antisymmetric modes also assume a roll shape, but with swirl
oppositely directed in the two hemispheres, together with fluid pumped
across the equator parallel to the rotation axis. Polar modes become a
ring of vortices more and more tightly arranged around the pole.Outward
radial heat flux peaks at the equator for symmetric equatorial modes,
and at a low latitude for antisymmetric modes. Both are suppressed near
the equator near the outer boundary at high T. Symmetric modes also
transport heat toward the equator, while antisymmetric modes transport
heat poleward at the lowest latitudes, equatorward at somewhat higher
latitudes. Symmetric equatorial modes transport angular momentum
radially inward at low T, radially outward at high T. These modes
transport angular momentum toward the equator from higher latitudes
at all T.
---------------------------------------------------------
Title: Towards Building a Solar General Circulation Model
Authors: Gilman, P. A.
1975BAAS....7..364G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Comments on Schatten's Reply to My Comments on `Solar Polar
Spindown'
Authors: Gilman, Peter A.
1974SoPh...37..491G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Comments on `solar polar spindown', by Kenneth Schatten
Authors: Gilman, Peter A.
1974SoPh...36...61G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Solar rotation.
Authors: Gilman, P. A.
1974ARA&A..12...47G Altcode:
Questions regarding the significance of solar rotation are investigated,
giving attention to its apparently small influence on certain solar
phenomena. The sun (a G star) is a slow rotator with an equatorial
velocity of about 2 km/sec. The rotation of the solar interior is
considered along with the rotation of the solar wind and aspects of
a differential rotation of the solar surface and atmosphere. Theories
for solar rotation are discussed, taking into account the problem of
angular momentum history and the problem of surface and atmospheric
differential rotation. An outline of interesting problems regarding
solar rotation for future studies is also presented.
---------------------------------------------------------
Title: Nonlinear Boussinesq Convective Model for Large Scale Solar
Circulations
Authors: Gilman, Peter A.
1972SoPh...27....3G Altcode:
We present extensive numerical calculations for a model of thermal
convection of a Boussinesq fluid in an equatorial annulus of a
rotating spherical shell. The convection induces and maintains
differential rotation and meridian circulation. The model is solved
for an effective Prandtl number P = 1, with effective Taylor number
T in the range 10<SUP>2</SUP> <T <10<SUP>6</SUP>, and effective
Rayleigh number R between the critical value for onset of convection,
and a few times that value. With Ω = 2.6 × 10<SUP>−6</SUP>
s<SUP>−1</SUP>, d = 1.4 × 10<SUP>10</SUP> cm (roughly the depth of
the solar convection zone) the range of Taylor number is equivalent
to kinematic viscosities between 10<SUP>14</SUP> and 10<SUP>12</SUP>
cm<SUP>2</SUP> s<SUP>−1</SUP>, which encompasses eddy viscosities
estimated from mixing length theory applied to the Sun.
---------------------------------------------------------
Title: A Method for Constructing Streamlines for the Sun's Large
Scale Flow from Doppler Velocities
Authors: Gilman, Peter A.
1971SoPh...19...40G Altcode:
We show that, if the large scale departures from the mean differential
rotation, measured by Howard and Harvey, represent nearly horizontal
flow, we may under certain assumptions deduce a pattern of streamlines
for these motions from the doppler line of sight velocities. This can be
done with data from a single day, without having to construct the total
flow from different projections of the (assumed) same velocity vectors
seen on different days. Mathematically the method involves integrating
a single first order inhomogeneous partial differential equation along
a set of characteristic curves which are circles concentric with the
center of the solar disk.
---------------------------------------------------------
Title: Instability of Magnetohydrostatic Stellar Interiors from
Magnetic Buoyancy. I.
Authors: Gilman, Peter A.
1970ApJ...162.1019G Altcode:
We show that magnetic buoyancy gives rise to a normal-mode instability
in fluids in magnetobydrostatic balance whose diffusivity of heat
is large compared with viscosity and magnetic diffusivity, such
as in stellar interiors. In an unbounded compressible fluid acted
upon by external gravity containing a straight horizontal magnetic
field; the instability occurs if the magnetic field decreases with
height. The unstable modes are narrow cross-section loops of magnetic
field and matter whose finite wavenumber k along the magnetic field
lies in the range 0 < k2 < - (g/S2) a/az ln B, where B(z) is
the basic magnetic field, g is gravity, and S is the isothermal sound
speed. Thus, in a star with a toroidal magnetic field, the analysis for
which we are giving in a later paper, this would be a nonaxisymmetric
instability. A detailed analysis is made of the special case in which
the Alfven and sound speeds are independent of height and which thus
elucidates the role of magnetic buoyancy. Sufficiently large rotation
perpendicular to gravity is shown to be stabilizing in this case. The
instability simultaneously releases potential and magnetic energy,
through downward transport of mass and upward transport of magnetic
flux, respectively. The instability gives a mechanism by which magnetic
flux may be expelled from stellar interiors.
---------------------------------------------------------
Title: Nonaxisymmetric Instabilities in Magnetohydrostatic Stars
from Magnetic Buoyancy. I. Plane-Parallel Model
Authors: Gilman, Peter A.
1970BAAS....2R.317G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Nonaxisymmetric Instabilities in Magnetohydrostatic Stars
from Magnetic Buoyancy. II: Spherical Model
Authors: Cadez, Vladimir; Gilman, Peter A.
1970BAAS....2Q.300C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On Large-Scale Solar Convection
Authors: Davies-Jones, Robert P.; Gilman, Peter A.
1970SoPh...12....3D Altcode:
We examine the effects of rotation about a vertical axis on thermal
convection with a simple model in which an inviscid, incompressible
fluid of zero thermal conductivity and electrical resistivity is
contained in a thin annulus of rectangular cross-section. The initial
steady state assumed is one of no motion relative to the rotating frame
with constant (unstable) vertical temperature gradient and uniform
toroidal magnetic field. Small periodic disturbances are then introduced
and the linearized perturbation equations solved. We also determine
the second-order mean circulations and magnetic fields that are forced
by non-zero Reynolds and thermal stresses and magnetic field transports.
---------------------------------------------------------
Title: Sunspot motion statistics for 1965 67
Authors: Coffey, Helen E.; Gilman, Peter A.
1969SoPh....9..423C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Baroclinic, Alfvén and Rossby Waves in Geostrophic Flow.
Authors: Gilman, P. A.
1969JAtS...26.1003G Altcode:
We present here further results concerning geostrophic baroclinic
flow with a zonal magnetic field, in a -plane channel, as extensions
of earlier work by the writer. We find a necessary condition for
instability that proves the existence of a short wave cutoff of unstable
baroclinic waves for finite magnetic field. We also place bounds on
the phase velocities of neutral waves in the system.For zero basic
baroclinic and a uniform magnetic field, the neutral oscillations
are zonally propagating geostrophic Alfvén waves. The lowest mode is
purely horizontal and travels at the ordinary Alfvén speed, but all
higher modes, which contain vertical as well as horizontal motion,
move more slowly. With 0, we get, in general, combined Alfvén-Rossby
waves, which, in the limit of large rotation, yield a class of waves
very similar to those found by Hide.Finally, we solve the Eady problem
(stability of baroclinic flow with constant shear) in a uniform magnetic
field, without and with the effect. We find, for = 0, that the field
stabilizes the baroclinic wave. Neutral Alfvén-like waves are found
for large field and/or short wavelengths. We also find a pair of
`edge waves.'For = 0, the longest baroclinic waves are stabilized for
small field strengths, as in the nonmagnetic problem, but very weakly
unstable waves occur for larger fields. A set of neutral Rossby waves
also is present for long wavelengths and the short Alfvén-like waves
are modified, but only slightly.
---------------------------------------------------------
Title: A Rossby-Wave Dynamo for the Sun, II
Authors: Gilman, Peter A.
1969SoPh....9....3G Altcode:
To make the analysis more tractable, we simplify the equations of
Part I to apply to two superposed layers of fluid, with horizontal
variations in the motion and magnetic field represented by a small
number of Fourier harmonics. The resulting set of eighteen ordinary
nonlinear differential equations in time for the Fourier amplitudes
is integrated numerically. We analyze in detail the dynamo action from
a typical Rossby wave motion and compare it with the solar cycle.
---------------------------------------------------------
Title: A Rossby-Wave Dynamo for the Sun, I
Authors: Gilman, Peter A.
1969SoPh....8..316G Altcode:
There is increasing interest in the possible existence of large
horizontally flowing eddies or `Rossby waves' in the sun's convection
zone and photosphere. We present here and in Part II a mathematical
model which shows that flows of this type, driven by an assumed
latitudinal temperature gradient, can act as hydromagnetic dynamos to
induce magnetic fields that periodically reverse.
---------------------------------------------------------
Title: Hydromagnetic Solar Spin Down
Authors: Benton, E. R.; Gilman, P. A.; Loper, D.
1969BAAS....1Q.273B Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Interaction of Giant Convection Cells with the Differential
Rotation
Authors: Davies-Jones, Robert; Gilman, Peter
1969BAAS....1..282D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Thermally Driven Rossby-Mode Dynamo for Solar Magnetic-Field
Reversals
Authors: Gilman, Peter A.
1968Sci...160..760G Altcode:
There is increasing interest in the possible existence of large eddies
or "Rossby waves" in Sun's convection zone and photosphere. It is shown
that many flows of this type, driven by an equator-pole temperature
difference, act as hydromagnetic dynamos to produce magnetic fields
that periodically reverse. The periods and field amplitudes agree with
solar phenomena within an order of magnitude.
---------------------------------------------------------
Title: The General Circulation of the Solar Atmosphere: Large
Thermally Driven Rossby Waves.
Authors: Gilman, Peter A.
1968AJS....73R..61G Altcode:
A set of hydromagnetic equations has been derived to describe
very large-scale flow and magnetic fields induced in response to a
latitudinal temperature gradient in the surface layer of a rotating gas
sphere (such as the convective zone of the sun). The flow is assumed
hydrostatic and nearly heliostrophic, and of horizontal scale comparable
to the solar bipolar magnetic regions. Maxwell stresses are assumed
not larger than Reynolds stresses. A linear study has been made of
the wavelike modes (Rossby waves) that transport heat horizontally in
response to the temperature gradient, for the highly idealized case of
an adiabatic, inviscid, perfectly electrically conducting fluid. From
a toroidal magnetic field, these disturbances produce weak, large-scale
vertical magnetic fields, which qualitatively look like and amplify at
a rate comparable to the solar BMR's. The disturbances also transport
these vertical fields meridionally to form a weak axisymmetric poloidal
field, in the same manner as postulated by Babcock and observed by
Bumba and Howard. The disturbances also transport momentum toward the
equator, in the manner deduced by Ward from sunspot statistics. At gas
densities of 10-~ g/cm3 (within the convection zone), growth of the
disturbances in a toroidal field of 100 gauss requires an equatorpole
temperature difference of a few tens of degrees. Currently, a more
general, nonlinear, dissipating dynamo model (i.e., including thermal
and electrical conductivity and viscosity) is being developed. First
results indicate that dynamo cycles with periods roughly comparable
to the solar cycle are included among the solutions.
---------------------------------------------------------
Title: The Circulation of the Sun's Atmosphere
Authors: Starr, Victor P.; Gilman, Peter A.
1968SciAm.218a.100S Altcode: 1968SciAm.218..100S
No abstract at ADS
---------------------------------------------------------
Title: Stability of Baroclinic Flows in a Zonal Magnetic Field:
Part III.
Authors: Gilman, Peter A.
1967JAtS...24..130G Altcode:
The study of the stability of two-layer baroclinic flows in a zonal
magnetic field (Part II) is generalized to include a parabolic
flow profile in the upper layer. Contrary to the nonmagnetic case,
angularities may be present in the equations even when the potential
vorticity gradient does not change sign, if the magnetic field
is increased beyond a certain strength. Below this strength, power
series solutions show that, just as in the nonmagnetic use, horizontal
shear renders shorter waves unstable. Reynolds stresses are seen to
transport momentum up the gradient, while smaller Maxwell stresses
oppose them.Properties of the solutions in this paper and in Part II
are compared to solar observations. At gas densities 10<SUP>4</SUP>
gm cm<SUP>3</SUP>, a 35K equator-pole temperature difference would be
enough to give baroclinically unstable disturbances in a zonal field
of 100 gauss. The perturbation vertical fields produced from this
magnitude zonal field will have magnitude 1 gauss, as well as a zonal
wave number of 6-8, `e folding' times of a few solar rotations, and
a pronounced tilt upstream away from the maximum of zonal flow. All
these characteristics are qualitatively consistent with the solar
observations.The Reynolds and Maxwell stresses are seen to act in
such a way as to be capable of maintaining the solar differential
rotation in the manner proposed by Starr and Gilman.It is suggested
that the same process could be used to maintain differential rotation
in the dynamo theory. The production of perturbation vertical and
mean poloidal fields, and the changes in the toroidal field by the
disturbances are also seen to correspond to dynamo processes. Suitable
generalizations of the equations may allow a complete dynamo cycle to
take place. If so, the system would have the important computational
advantage of putting all the dynamics on essentially the same time and
space scales.Finally, the `magnetic modes' recently suggested by Hide
to be responsible for the slow westward drift of the geomagnetic field
are very unlikely to be baroclinically unstable. They could, however,
be fed energy from baroclinically unstable `inertial modes.'Suggestions
are made for further studies of the scaled equations presented.
---------------------------------------------------------
Title: Stability of Baroclinic Flows in a Zonal Magnetic Field:
Part II.
Authors: Gilman, Peter A.
1967JAtS...24..119G Altcode:
The equations of Part I are approximated to represent a two-layer
model. All the theorems for the continuous case are shown to hold in
the two-layer case as well. In addition, bounds are placed on the phase
velocities of neutral waves.The stability of purely baroclinic flow (no
horizontal shear) in a uniform zonal magnetic field is then studied. The
minimum vertical shear needed for instability no longer depends upon the
-effect or the static stability, but rather is determined by the zonal
field strength. Short waves are destabilized by the magnetic field, long
waves stabilized. Unstable waves convert available potential energy into
kinetic energy of the disturbances, part of which in turn is converted
into disturbance magnetic energy. Nonmagnetic changes in the initial
state are similar to those of Phillips. Perturbation vertical magnetic
fields and a single-celled meridional (poloidal) field are produced.
---------------------------------------------------------
Title: Stability of Baroclinic Flows in a Zonal Magnetic Field:
Part I.
Authors: Gilman, Peter A.
1967JAtS...24..101G Altcode:
Because of its possible applications to the solar differential rotation,
hydromagnetic dynamo theory, and the geomagnetic secular variation, the
dynamics of baroclinic, geostrophic -plane flow in a zonal (toroidal),
magnetic field are studied. The equations describing the flow in
the absence of a magnetic field are the same as those formulated by
Charney and Stern and by Pedlosky. Due to the horizontal magnetic
field, potential vorticity is no longer conserved. Vertical magnetic
fields can be produced, but the scaling excludes their feedback on
the motions and horizontal fields. Thus, the system in its present
simplest form can not complete a dynamo cycle, but suitable relaxation
of some scaling restrictions may overcome this difficulty.The scaled
equations are perturbed about a steady axially symmetric zonal flow
and zonal (toroidal) magnetic field. Changes in the initial state are
inferred from products of perturbation quantities. These include the
growth in the meridional plane of axially symmetric circulations and
magnetic fields (i.e., poloidal fields). The energetics of the system
are examined. It is shown that the potential vorticity theorem of
Charney and Stern no longer holds due to the magnetic field. For short
wavelength disturbances, the field should dominate in determining the
stability properties. Bounds are placed on the complex phase velocities
of normal mode disturbances. For flows with vertical and horizontal
shear, these are the same as found by Pedlosky for the nonmagnetic
case. The bounds on growth rates of disturbances in barotropic flows
are tightened by the magnetic field. Such flows will be stable to all
wavelengths if the field is large enough.
---------------------------------------------------------
Title: Energetics of the Solar Rotation.
Authors: Starr, Victor P.; Gilman, Peter A.
1965ApJ...141.1119S Altcode:
The energetics of the solar differential rotation is considered in the
light of recent observational studies of sunspot motions by Ward. It
is shown that Ward's calculations of the correlation of longitudinal
and latitudinal sunspot displacements imply a systematic conversion of
horizontal-eddy kinetic energy into kinetic energy of the mean zonal
flow This conversion is therefore directly opposed to the effect of
molecular and eddy viscosity. The conversion rate is such that, if the
conversion were totally suppressed while other processes were allowed
to continue, virtually solid-body rotation would be achieved within a
few solar rotations Some of the energy sources available to maintain
the kinetic energy of the large-scale horizontal eddies against these
conversion losses are briefly considered.
---------------------------------------------------------
Title: On the Structure and Energetics of Large Scale Hydromagnetic
Disturbances in the Solar Photosphere
Authors: Starr, V. P.; Gilman, P. A.
1965Tell...17..334S Altcode: 1965TellA..17..334S
It is suggested that the solar differential rotation is maintained
against frictional dissipation by large scale quasi-horizontal
hydromagnetic disturbances, which possess tilted structures similar to
upper level troughs in the westerlies of the earth's atmosphere. On
the sun, these tilts would be directed from the equator toward the
east limb in each hemisphere. Horizontal Reynolds stresses in these
disturbances thus convert eddy kinetic energy into kinetic energy of
the zonal flow, while horizontal Maxwell stresses convert the mean
zonal kinetic energy into "eddy" magnetic energy. In order for the
disturbances as a whole to feed kinetic energy into the differential
rotation, the horizontal eddy magnetic fields must be somewhat less
than 7 gauss at the level considered. Measurements of the large scale
patterns of the line of sight component of the magnetic field, made by
Howard, show clearly the required tilt in the patterns toward the east
limb in each hemisphere. These measurements also show that the patterns
do have a very large horizontal scale, possessing a longitudinal
wave number of about 6, and a latitudinal extent of about 60°. The
magnitudes of these fields appear to be compatible with horizontal eddy
magnetic fields less than 7 gauss. The braking action by the symmetric
steady component of the horizontal Maxwell stress appears from the
observations to be small compared with that due to the eddy component.
