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Author name code: durney
ADS astronomy entries on 2022-09-14
author:"Durney, Bernard R."
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Title: The Hawking Effect for Massive Particles
Authors: Durney, Bernard R.
2014IJAA....4...11D Altcode:
No abstract at ADS
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Title: Magnetically Preferred Solar Longitudes: Reality?
Authors: Henney, C. J.; Durney, B. R.
2005ASPC..346..381H Altcode: 2007astro.ph..1118H
The observed persistence of specific periodicities detected in time
series associated with solar surface magnetic activity over several
solar cycles has led to numerous papers supporting the existence of
preferred longitudes. Recent analysis of the past 120 years of sunspot
number data showed that no observed periodicity remained coherent for
durations greater than two 11-year solar cycles. Here we address the
question: Could the observed periodicities of solar magnetic signals
on time scales of two decades be the result of a purely stochastic
process? We begin to answer this by comparing phase coherence between
observed periodic signals and signals from a model using longitudinally
random eruptions. A surprisingly non-negligible likelihood is found,
approximately 1 in 3, that observed periodicities from integrated
full-disk solar parameters are a chance occurrence for time series on
the order of 20 years in duration.
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Title: Solar Subsurface Fluid Dynamics Descriptors Derived from
Global Oscillation Network Group and Michelson Doppler Imager Data
Authors: Komm, R.; Corbard, T.; Durney, B. R.; González Hernández,
I.; Hill, F.; Howe, R.; Toner, C.
2004ApJ...605..554K Altcode:
We analyze Global Oscillation Network Group (GONG) and Michelson Doppler
Imager (MDI) observations obtained during Carrington rotation 1988
(2002 March 30-April 26) with a ring-diagram technique in order to
measure the zonal and meridional flow components in the upper solar
convection zone. We derive daily flow maps over a range of depths up
to 16 Mm on a spatial grid of 7.5d in latitude and longitude covering
+/-60° in latitude and central meridian distance and combine them
to make synoptic flow maps. We begin exploring the dynamics of the
near-surface layers and the interaction between flows and magnetic flux
by deriving fluid dynamics descriptors such as divergence and vorticity
from these flow maps. Using these descriptors, we derive the vertical
velocity component and the kinetic helicity density. For this particular
Carrington rotation, we find that the vertical velocity component is
anticorrelated with the unsigned magnetic flux. Strong downflows are
more likely associated with locations of strong magnetic activity. The
vertical vorticity is positive in the northern hemisphere and negative
in the southern hemisphere. At locations of magnetic activity,
we find an excess vorticity of the same sign as that introduced by
differential rotation. The vertical gradient of the zonal flow is
mainly negative except within 2 Mm of the surface at latitudes poleward
of about 20°. The zonal-flow gradient appears to be related to the
unsigned magnetic flux in the sense that locations of strong activity
are also locations of large negative gradients. The vertical gradient
of the meridional flow changes sign near about 7 Mm, marking a clear
distinction between near-surface and deeper layers. GONG and MDI data
show very similar results. Differences occur mainly at high latitudes,
especially in the northern hemisphere, where MDI data show a counter
cell in the meridional flow that is not present in the corresponding
GONG data.
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Title: The Contribution of Sound Waves and Instabilities to the
Penetration of the Solar Differential Rotation Below the Convection
Zone
Authors: Durney, Bernard R.
2004SoPh..219..231D Altcode:
The response of a layer to a horizontal shear flow at its top the
surface was studied numerically as an initial value problem. The
geometry was Cartesian and the conservation equations were solved with
the help of the Zeus-3D code. In the initial state, the pressure, p,
and density, ρ, of the layer were assumed to be related by a polytropic
equation of index 1.14, which best approximates the solar values in
the region of interest. The values of p and ρ at the lower boundary of
the layer, namely r=R<SUB>l</SUB>=0.4 R<SUB>⊙</SUB>, were taken to be
the solar values. The upper boundary was chosen to be the base of the
solar convection zone, r=R<SUB>c</SUB>=0.7 R<SUB>⊙</SUB>. The shear
flow at the surface, v<SUB>φ</SUB>(R<SUB>c</SUB>), was proportional
to the solar differential rotation, and acoustical oscillations were
present in the layer.
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Title: The Energy Equation in the Lower Solar Convection Zone
Authors: Durney, Bernard R.
2003SoPh..217....1D Altcode:
As a consequence of the Taylor-Proudman balance, a balance between the
pressure, Coriolis and buoyancy forces in the radial and latitudinal
momentum equations (that is expected to be amply satisfied in
the lower solar convection zone), the superadiabatic gradient is
determined by the rotation law and by an unspecified function of r,
say, S'<SUB>Ω</SUB>(r), where r is the radial coordinate. If the
rotation law and S'<SUB>Ω</SUB>(r) are known, then the solution
of the energy equation, performed in this paper in the framework of
the MLΩ formalism, leads to a knowledge of the Reynolds stresses,
convective fluxes, and meridional motions. The MLΩ-formalism is an
extension of the mixing length theory to rotating convection zones,
and the calculations also involve the azimuthal momentum equation,
from which an expression for the meridional motions in terms of the
Reynolds stresses can be derived. The meridional motions are expanded as
U<SUB>r</SUB>(r,θ)=P<SUB>2</SUB>(cosθ)ψ<SUB>2</SUB>(r)/r<SUP>2</SUP>ρ+P<SUB>4</SUB>(cosθ)ψ<SUB>4</SUB>(r)/r<SUP>2</SUP>
ρ+..., and a corresponding equation for U<SUB>θ</SUB>(r,θ). Here θ
is the polar angle, ρ is the density, and P<SUB>2</SUB>(cosθ),
P<SUB>4</SUB>(cosθ) are Legendre polynomials. A good
approximation to the meridional motion is obtained by setting
ψ<SUB>4</SUB>(r)=−Hψ<SUB>2</SUB>(r) with H≈−1.6, a constant. The
value of ψ<SUB>2</SUB>(r) is negative, i.e., the P<SUB>2</SUB>
flow rises at the equator and sinks at the poles. For the value of H
obtained in the numerical calculations, the meridional motions have a
narrow countercell at the poles, and the convective flux has a relative
maximum at the poles, a minimum at mid latitudes and a larger maximum
at the equator. Both results are in agreement with the observations.
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Title: Temporal Variation of Angular Momentum in the Solar Convection
Zone
Authors: Komm, R.; Howe, R.; Durney, B. R.; Hill, F.
2003ApJ...586..650K Altcode:
We derive the angular momentum as a function of radius and time with the
help of the rotation rates resulting from inversions of helioseismic
data obtained from the Global Oscillation Network Group (GONG) and
the Michelson Doppler Imager (MDI) and the density distribution from
a model of the Sun. The base of the convection zone can be identified
as a local maximum in the relative angular momentum after subtracting
the contribution of the solid-body rotation. The angular momentum as
a function of radius shows the strongest temporal variation near the
tachocline. This variation extends into the lower convection zone and
into the radiative interior and is related to the 1.3 yr periodicity
found in the equatorial rotation rate of the tachocline. In the upper
convection zone, we find a small systematic variation of the angular
momentum that is related to torsional oscillations. The angular momentum
integrated from the surface to a lower limit in the upper convection
zone provides a hint that the torsional oscillation pattern extends
deep into the convection zone. This is supported by other quantities
such as the coefficients of a fit of Legendre polynomials to the
rotation rates as a function of latitude. The temporal variation of the
coefficient of P<SUB>4</SUB>, indicative of torsional oscillations,
suggests that the signature of these flows in the inversion results
extend to about r~0.83R<SUB>solar</SUB>. With the lower limit of
integration placed in the middle or lower convection zone, the angular
momentum fluctuates about the mean without apparent trend, i.e., the
angular momentum is conserved within the measurement errors. However,
when integrated over the layers slightly below the convection zone
(0.60-0.71R<SUB>solar</SUB>), the angular momentum shows the 1.3 yr
period and hints at a long-term trend that might be related to the
solar activity cycle.
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Title: Temporal variation of angular momentum in the convection zone
Authors: Komm, R.; Howe, R.; Durney, B. R.; Hill, F.
2003ESASP.517...97K Altcode: 2003soho...12...97K
We derive the angular momentum as a function of radius and time with the
help of the rotation rates resulting from inversions of helioseismic
data obtained from the Global Oscillation Network Group (GONG)
and the Michelson Doppler Imager (MDI) and the density distribution
from a model of the Sun. The angular momentum as a function of radius
shows the strongest temporal variation near the base of the convection
zone. This variation extends into the lower convection zone and into
the radiative interior and is related to the 1.3-yr periodicity found in
the equatorial rotation rate of the tachocline. In the upper convection
zone, we find a small systematic variation of the angular momentum that
is related to torsional oscillations. The angular momentum integrated
from the surface to a lower limit in the upper convection zone provides
a hint that the torsional oscillation pattern extends deep into the
convection zone. With the lower limit of integration placed in the
lower half of the convection zone, the angular momentum fluctuates
about the mean without apparent trend, i.e. the angular momentum is
conserved within the measurement errors. However, when integrated over
the layers slightly below the convection zone, the angular momentum
shows the 1.3-yr period and hints at a long-term trend which might be
related to the solar activity cycle.
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Title: Temporal Variation of Angular Momentum in the Solar Convection
Zone
Authors: Komm, R.; Howe, R.; Durney, B.; Hill, F.
2002AAS...200.0404K Altcode: 2002BAAS...34Q.644K
We present the temporal variation of the solar angular momentum
derived from helioseismic observations. In the absence of `true'
angular momentum inversions, we use the rotation rates resulting from
rotation inversions of GONG data and the density distribution from a
model of the Sun. We focus especially on the layers near the base of
the convection zone and the layers near the solar surface. We derive
the angular momentum as a function of depth and the corresponding
solid-body rotation. The angular momentum decreases with increasing
radius following essentially the product of density times the fourth
power of radius. The tachocline can be identified as a local maximum
in the radial gradient of the angular momentum and as a local maximum
in the relative angular momentum after subtracting the contribution
of the solid-body rotation. The angular momentum shows the strongest
temporal variation near the tachocline. This variation is reminiscent
of the 1.3-yr periodicity found in the equatorial rotation rate of the
tachocline, which is not too surprising since the angular momentum of
a spherical shell is heavily weighted toward the equator. We discuss
the extension of this variation into the convection zone and into
the radiative interior. In addition, we fit the rotation rates as
functions of latitude with Legendre polynomials to cross-validate
the numerical results and to draw conclusions about the zonal flows
(`torsional oscillations') in the upper convection zone. This work
was supported by NASA Grant S-92698-F.
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Title: Approximate Isocontours For The Solar Angular Velocity In
The Convection Zone
Authors: Durney, Bernard R.
2001SoPh..202..201D Altcode:
The angular velocity, Ω, in the solar convection zone (SCZ) is expanded
in Legendre polynomials, P<SUB>n</SUB>(cosθ), and the values for Ω at
the equator are assumed to be given by Kosovichev's helioseismic data;
here, r, θ, and φ, label the radial, latitudinal and longitudinal
coordinates, respectively. The isocontours for Ω are calculated for
the following two cases. (i) The angular momentum of a thin spherical
shell of radius r is identical to the shell's angular momentum for solid
body rotation, i.e., rotation just distributes in latitude the angular
momentum of each layer. (ii) Considerations based on the Taylor-Proudman
balance (a balance between the pressure, Coriolis and buoyancy forces
which is expected to be amply satisfied in the SCZ), require that the
radial component of the superadiabatic gradient be strongly dependent
on latitude unless the coefficients in the expansion for Ω defined
above satisfy a first-order differential equation, DE. The isocontours
for the angular velocity determined from DE, compare remarkably well
with the helioseismic data, whereas for case (i) there is a marked
difference at high latitudes. The radial and latitudinal balance of
angular momentum are studied. The meridional motions are determined
mainly (but not entirely) by the radial balance of angular momentum,
and they depend principally on the Reynolds stress, «u<SUB>r</SUB>
u<SUB>φ</SUB> ». Concerning the latitudinal balance, ∂Ω/∂θ
increases until the transport of angular momentum toward the poles by
the meridional motions is able to balance the transport of angular
momentum towards the equator by « u<SUB>θ</SUB> u<SUB>φ</SUB> »
( = « u<SUB>θ</SUB> u<SUB>φ</SUB> »<SUB>0</SUB> - ν<SUB>t</SUB>
sinθ∂Ω/∂θ)). Here the subscript 0 stands for solid body rotation,
and ν<SUB>t</SUB> is a turbulent viscosity coefficient. In contrast
to the radial balance, the viscosity term plays a fundamental role in
the latitudinal balance of angular momentum.
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Title: The Energy Lost by Differential Rotation in the Generation
of the Solar Toroidal Magnetic Field
Authors: Durney, Bernard R.
2000SoPh..197..215D Altcode:
The integrals, I<SUB>i</SUB>(t) = ∫<SUB>GL</SUB> u<SUB>i</SUB>j
× B<SUB>i</SUB>dv over the volume GL are calculated in a dynamo
model of the Babcock-Leighton type studied earlier. Here, GL is the
generating layer for the solar toroidal magnetic field, located at
the base of the solar convection zone (SCZ); i=r, θ, φ, stands for
the radial, latitudinal, and azimuthal coordinates respectively;
j = (4π)<SUP>-1</SUP>∇ × B, where B is the magnetic field;
u<SUB>r</SUB>,u<SUB>θ</SUB> are the components of the meridional
motion, and u<SUB>φ</SUB> is the differential rotation. During a
ten-year cycle the energy ∫<SUB>cycle</SUB> I<SUB>φ</SUB>(t)dt needs
to be supplied to the azimuthal flow in the GL to compensate for the
energy losses due to the Lorentz force. The calculations proceed as
follows: for every time step, the maximum value of |B<SUB>φ</SUB>|
in the GL is computed. If this value exceeds B<SUB>cr</SUB> (a
prescribed field) then there is eruption of a flux tube that rises
radially, and reaches the surface at a latitude corresponding to the
maximum of |B<SUB>φ</SUB>| (the time of rise is neglected). This flux
tube generates a bipolar magnetic region, which is replaced by its
equivalent axisymmetric configuration, a magnetic ring doublet. The
erupted flux can be multiplied by a factor F<SUB>t</SUB>, i.e., by
the number of eruptions per time step. The model is marginally stable
and the ensemble of eruptions acts as the source for the poloidal
field. The arbitrary parameters B<SUB>cr</SUB> and F<SUB>t</SUB> are
determined by matching the flux of a typical solar active region,
and of the total erupted flux in a cycle, respectively. If E(B) is
the energy, in the GL, of the toroidal magnetic field B<SUB>φ</SUB>
= B sin θ cos θ, B (constant), then the numerical calculations show
that the energy that needs to be supplied to the differential rotation
during a ten-year cycle is of the order of E(B<SUB>cr</SUB>), which is
considerably smaller than the kinetic energy of differential rotation
in the GL. Assuming that these results can be extrapolated to larger
values of B<SUB>cr</SUB>, magnetic fields ≈10<SUP>4</SUP> G, could be
generated in the upper section of the tachocline that lies below the SCZ
(designated by UT). The energy required to generate these 10<SUP>4</SUP>
G fields during a cycle is of the order of the kinetic energy in the UT.
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Title: On the Differences Between Odd and Even Solar Cycles
Authors: Durney, Bernard R.
2000SoPh..196..421D Altcode:
It is proposed that the observed differences between odd and even
solar cycles are a consequence of the nonlinear interactions that
provide the stabilizing mechanism for the cycle's amplitude. If, for
example, the magnetic field is larger than average for a given cycle
(say odd), the nonlinear feedback mechanism can generate a magnetic
field that is smaller than average for the next cycle (even), and then
one that is larger than average for the following cycle (odd), etc. As
a consequence the odd cycles have larger amplitudes than even cycles. A
very simply model having a nonlinear interaction that reproduces this
behavior is discussed.
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Title: On The Torsional Oscillations In Babcock-Leighton Solar
Dynamo Models
Authors: Durney, Bernard R.
2000SoPh..196....1D Altcode:
The torsional oscillations at the solar surface have been interpreted
by Schüssler and Yoshimura as being generated by the Lorentz force
associated with the solar dynamo. It has been shown recently that
they are also present in the upper half of the solar convection zone
(SCZ). With the help of a solar dynamo model of the Babcock-Leighton
type studied earlier, the longitudinal component of the Lorentz force,
L<SUB>φ</SUB>, is calculated, and its sign or isocontours, are plotted
vs. time, t, and polar angle, θ (the horizontal and vertical axis
respectively). Two cases are considered, (1) differential rotation
differs from zero only in the tachocline, (2) differential rotation
as in (1) in the tachocline, and purely latitudinal and independent
of depth in the bulk of the SCZ. In the first case the sign of
L<SUB>φ</SUB> is roughly independent of latitude (corresponding to
vertical bands in the t,θ plot), whereas in the second case the bands
show a pole-equator slope of the correct sign. The pattern of the bands
still differs, however, considerably from that of the helioseismic
observations, and the values of the Lorentz force are too small at low
latitudes. It is all but certain that the toroidal field that lies
at the origin of the large bipolar magnetic regions observed at the
surface, must be generated in the tachocline by differential rotation;
the regeneration of the corresponding poloidal field, B<SUB>p</SUB>
has not yet been fully clarified. B<SUB>p</SUB> could be regenerated,
for example, at the surface (as in Babcock-Leighton models), or slightly
above the tachocline, (as in interface dynamos). In the framework of
the Babcock-Leighton models, the following scenario is suggested:
the dynamo processes that give rise to the large bipolar magnetic
regions are only part of the cyclic solar dynamo (to distinguish it
from the turbulent dynamo). The toroidal field generated locally by
differential rotation must contribute significantly to the torsional
oscillations patterns. As this field becomes buoyant, it should give
rise, at the surface, to the smaller bipolar magnetic regions as, e.g.,
to the ephemeral bipolar magnetic regions. These have a weak non-random
orientation of magnetic axis, and must therefore also contribute to
the source term for the poloidal field. Not only the ephemeral bipolar
regions could be generated in the bulk of the SCZ, but many of the
smaller bipolar regions as well (at depths that increase with their
flux), all contributing to the source term for the poloidal field. In
contrast to the butterfly diagram that provides only a very weak test
of dynamo theories, the pattern of torsional oscillations has the
potential of critically discriminating between different dynamo models.
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Title: Meridional Motions and the Angular Momentum Balance in the
Solar Convection Zone
Authors: Durney, Bernard R.
2000ApJ...528..486D Altcode:
The solar angular velocity, Ω, and meridional motions
in the solar convection zone (SCZ) are expanded
in Legendre polynomials. If the velocity correlations
<u<SUB>r</SUB>u<SUB>φ</SUB>>,<u<SUB>θ</SUB>u<SUB>φ</SUB>>
and the angular velocity are known, then the azimuthal momentum
equation determines the meridional flow; here u stands for the turbulent
convective velocities and the bracket denotes an appropriate average; θ
and φ are the polar angle and longitude. <P />The velocity correlation
<u<SUB>r</SUB>u<SUB>φ</SUB>> transports angular momentum to the
inner regions of the SCZ. This angular momentum can either spin-up the
inner regions, or be removed by a meridional motion that rises at the
equator and sinks at the poles; the stream function for this motion will
be designated by ψ<SUB>2</SUB>. For slowly rotating stars, the inner
regions must spin-up. As the angular velocity increases, a transition
must take place to the second option: in the Sun the angular velocity
does not increase sharply with depth. This transition should occur at
a value for Ω at which the Taylor-Proudman balance (a balance between
the pressure, Coriolis, and buoyancy forces) becomes valid. In the SCZ,
this balance determines the latitudinal variations of the superadiabatic
gradient (\b.nabla ΔT) from the rotation law, and it provides,
therefore, a link between the energy equation and the azimuthal momentum
equation. <P />The solar meridional motion also has a component, with
stream function ψ<SUB>4</SUB>, that rises at the equator and poles
and sinks at midlatitudes; its contribution to the removal of angular
momentum from the inner regions of the SCZ is negligible. In the Sun,
ψ<SUB>2</SUB> depends mainly on <u<SUB>r</SUB>u<SUB>φ</SUB>> and
ψ<SUB>4</SUB>~-4ψ<SUB>2</SUB>/3 (this expression for ψ<SUB>4</SUB>
is not as robust as that of ψ<SUB>2</SUB>, which is an excellent
approximation). Therefore, the meridional motions are essentially
determined by <u<SUB>r</SUB>u<SUB>φ</SUB>>. However, the
ψ<SUB>2</SUB>-meridional circulation transports angular momentum
toward the polar regions of the Sun which must be balanced by
<u<SUB>θ</SUB>u<SUB>φ</SUB>> and ψ<SUB>4</SUB>. Globally, the
conservation of angular momentum in the latitudinal direction requires
that the sum of the terms in <u<SUB>r</SUB>u<SUB>φ</SUB>> and in
<u<SUB>θ</SUB>u<SUB>φ</SUB>> of an integral over the entire SCZ
cancels. For this to be the case, <u<SUB>θ</SUB>u<SUB>φ</SUB>>
must be positive since <u<SUB>r</SUB>u<SUB>φ</SUB>> is
negative (which is a very robust result). For stars satisfying
the Taylor-Proudman balance, a fast rotating equator appears
to be an unavoidable necessity. <P />An equation is derived
that clarifies the reasons for the existence of the relation
ψ<SUB>4</SUB>~-4ψ<SUB>2</SUB>/3 and for the weak dependence of
ψ<SUB>4</SUB> on <u<SUB>θ</SUB>u<SUB>φ</SUB>>. <P />A
simple model for the velocity correlations is studied. In this
simple model, if the latitudinal differential rotation increases,
<u<SUB>θ</SUB>u<SUB>φ</SUB>> must decrease for the
integral relation defined above to remain valid. This dependence of
<u<SUB>θ</SUB>u<SUB>φ</SUB>> on ∂Ω/∂θ agrees with what
can be inferred from physical considerations.
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Title: The Taylor-Proudman Balance and the Solar Rotational Data
Authors: Durney, Bernard R.
1999ApJ...511..945D Altcode:
For the Sun (and because of the presence of buoyancy), the
implications of the Taylor-Proudman balance (TPB, a balance between
the pressure, Coriolis, and buoyancy forces in the radial and
latitudinal momentum equations) differ fundamentally from those of
an incompressible fluid. The TPB now only determines the latitudinal
variations of the solar entropy in terms of the rotation law and
known functions of r. As a consequence of the TPB, the energy
equation is in fact an equation for the angular velocity, Ω(r,
θ)=Ω<SUB>0</SUB>(ω<SUB>0</SUB>(r)+ω<SUB>2</SUB>(r)P<SUB>2</SUB>(cosθ)),
where P<SUB>2</SUB>(cosθ) is the second-order Legendre polynomial. In
agreement with data from the Solar Oscillations Investigations project
(SOI) Michelson Doppler Imager (MDI) on board SOHO, we assume that
ω<SUB>0</SUB>(r) is constant with r, and solve the equation for
ω<SUB>2</SUB>(r) with a simple, heuristic expression for the convective
flux [if ω<SUB>0</SUB>(r) is constant, then ∂Ω/∂r vanishes for
θ<SUB>c</SUB>=54.7d, P<SUB>2</SUB>(cosθ<SUB>c</SUB>)=0, in remarkable
agreement with Kosovichev's results inferred from isocontours for
Ω(r, θ)]. For values of the meridional motions that are not too
large, solutions for ω<SUB>2</SUB>(r) exist that agree with these
isocontours. These solutions are such that the latitudinal variation
of the convective flux, arising from the latitudinal variations of
the entropy, required by the TPB, are significantly reduced.
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Title: On the Power in the Legendre Modes of the Solar Radial
Magnetic Field
Authors: Durney, Bernard R.
1998SoPh..180....1D Altcode:
The power in the different ℓ modes of an expansion of the
solar radial magnetic field at the surface in terms of Legendre
polynomials,P<SUB>ℓ</SUB> , is calculated with the help of a solar
dynamo model studied earlier. The model is of the Babcock-Leighton type,
i.e., the surface eruptions of the toroidal magnetic field - through the
`tilt angle', γ, formed by the magnetic axis of a bipolar magnetic
region with the east-west line - are the sources for the poloidal
field. In this paper it is assumed that the tilt angle is subject to
fluctuations of the form, γ = γ'(σ)+ <γ> where <γ>
is the average value and γ'(σ) is a random normal fluctuation with
standard deviation σ which is taken from Howard's observations
of the distribution of tilt angles. For numerical considerations,
negative values of γ were not allowed. If this occurred, γ was
recalculated. The numerical integrations were started with a toroidal
magnetic field antisymmetric across the equator, large enough to
generate eruptions, and a negligible poloidal field. The fluctuations
in the tilt angle destroy the antisymmetry as time increases. The
power of the antisymmetric modes across the equator (i.e., odd values
of ℓ) is concentrated in frequencies, ν<SUB>p</SUB>, corresponding
to the cycle period. The maximum power lies in the ℓ=3 mode with
considerable power in the ℓ=5 mode, in broad agreement with Stenflo's
results who finds a maximum power at ℓ=5. For the symmetric modes,
there is considerable power in frequencies larger than ν<SUB>p</SUB>,
again in broad agreement with Stenflo's power spectrum.
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Title: On a Babcock-Leighton Solar Dynamo Model with a Deep-seated
Generating Layer for the Toroidal Magnetic Field. IV.
Authors: Durney, Bernard R.
1997ApJ...486.1065D Altcode:
The study is continued of a dynamo model of the Babcock-Leighton
type (i.e., the surface eruptions of toroidal magnetic field are the
source for the poloidal field) with a thin, deep seated layer (GL),
for the generation of the toroidal field, B<SUB>φ</SUB>. The partial
differential equations satisfied by B<SUB>φ</SUB> and by the vector
potential for the poloidal field are integrated in time with the help of
a second order time- and space-centered finite different scheme. Axial
symmetry is assumed; the gradient of the angular velocity in the GL
is such that within this layer a transition to uniform rotation takes
place; the meridional motion, transporting the poloidal field to the GL,
is poleward and about 3 m s<SUP>-1</SUP> at the surface; the radial
diffusivity η<SUB>r</SUB> equals 5 × 10<SUP>9</SUP> cm<SUP>2</SUP>
s<SUP>-1</SUP>, and the horizontal diffusivity η<SUB>θ</SUB> is
adjusted to achieve marginal stability. The initial conditions are:
a negligible poloidal field, and a maximum value of |B<SUB>φ</SUB>|
in the GL equal to 1.5 × B<SUB>cr</SUB>, where B<SUB>cr</SUB>
is a prescribed field. <P />For every time step the maximum value
of |B<SUB>φ</SUB>| in the GL is computed. If this value exceeds
B<SUB>cr</SUB>, then there is eruption of a flux tube (at the latitude
corresponding to this maximum) that rises radially to the surface. Only
one eruption is allowed per time step (Δt) and B<SUB>φ</SUB> in
the GL is unchanged as a consequence of the eruption. The ensemble
of eruptions is the source for the poloidal field, i.e., no use is
made of a mean field equation relating the poloidal with the toroidal
field. For a given value of Δt, and since the problem is linear, the
solutions scale with B<SUB>cr</SUB>. Therefore, the equations need to
be solved for one value of B<SUB>cr</SUB> only. <P />Since only one
eruption is allowed per time step, the dependence of the solutions on
Δt needs to be studied. Let F<SUB>t</SUB> be an arbitrary numerical
factor (= 3 for example) and compare the solutions of the equations for
(B<SUB>cr</SUB>, Δt) and (B<SUB>cr</SUB>, Δt/3). It is clear that
there will be 3 times as many eruptions in the second case (with the
shorter time step) than in the first case. However, if the erupted
flux in case one is multiplied by 3, then the solutions for this
case become nearly identical to those of case two (Δt is shorter
than any typical time of the system, and the difference due to the
unequal time steps is negligible). Therefore, varying the time step is
equivalent to keeping Δt fixed while multiplying the erupted flux by
an appropriate factor. In the numerical calculations Δt was set equal
to 10<SUP>5</SUP> s. The factor F<SUB>t</SUB> can then be interpreted
as the number of eruptions per 10<SUP>5</SUP> s. The integration
of the equations shows that there is a transition in the nature of
the solutions for F<SUB>t</SUB> ~ 2.5. For F<SUB>t</SUB> < 2.5,
the eruptions occur only at high latitudes, whereas for F<SUB>t</SUB>
> 2.5, the eruptions occur for θ greater than ~ π/4, where θ is
the polar angle. Furthermore, for F<SUB>t</SUB> < 2.5, the toroidal
field, |B<SUB>φ</SUB>|, in the GL can become considerably larger than
B<SUB>cr</SUB>, while this ceases to be the case for F<SUB>t</SUB>
> 2.5. <P />The factor F<SUB>t</SUB> is an arbitrary parameter
in the model and an appeal to observations is necessary. We set
B<SUB>cr</SUB> = 10<SUP>3</SUP> G. In the model, the magnetic flux
of erupting magnetic tubes, is then about 3 × 10<SUP>21</SUP> G,
of the order of the solar values. For this value of B<SUB>cr</SUB>
and for the value of F<SUB>t</SUB> (~2.5) at which the transition
takes place, the total erupted flux in 10 years is about 0.85 ×
10<SUP>25</SUP> Mx in remarkable agreement with the total erupted
flux during a solar cycle. Concerning the dynamo models studied here,
a major drawback encountered in previous papers has been the eruptions
at high latitudes, which entail unrealistically large values for the
radial magnetic field at the poles. The results of this paper provide
a major step forward in the resolution of this difficulty.
---------------------------------------------------------
Title: On the Influence of Gradients in the Angular Velocity on the
Solar Meridional Motions
Authors: Durney, Bernard R.
1996SoPh..169....1D Altcode:
If fluctuations in the density are neglected, the large-scale,
axisymmetric azimuthal momentum equation for the solar convection
zone (SCZ) contains only the velocity correlations and where u
are the turbulent convective velocities and the brackets denote a
large-scale average. The angular velocity, Ω, and meridional motions
are expanded in Legendre polynomials and in these expansions only the
two leading terms are retained (for example, where θ is the polar
angle). Per hemisphere, the meridional circulation is, in consequence,
the superposition of two flows, characterized by one, and two cells
in latitude respectively. Two equations can be derived from the
azimuthal momentum equation. The first one expresses the conservation
of angular momentum and essentially determines the stream function of
the one-cell flow in terms of : the convective motions feed angular
momentum to the inner regions of the SCZ and in the steady state a
meridional flow must be present to remove this angular momentum. The
second equation contains also the integral indicative of a transport
of angular momentum towards the equator.
---------------------------------------------------------
Title: On a Babcock-Leighton Dynamo Model with a Deep-Seated
Generating Layer for the Toroidal Magnetic Field, II
Authors: Durney, Bernard R.
1996SoPh..166..231D Altcode:
In a previous paper (Paper I), we studied a dynamo model of the
Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic
field are the source for the poloidal field) that included a thin, deep
seated, generating layer (GL) for the toroidal field, Bφ. Meridional
motions (of the order of 12 m s<SUP>−1</SUP> at the surface),
rising at the equator and sinking at the poles were essential for
the dynamo action. The induction equation was solved by approximating
the latitudinal dependence of the fields by Legendre polynomials. No
solutions were found with Φ<SUB>p</SUB> = Φ<SUB>f</SUB> where
Φ<SUB>p</SUB> and Φ<SUB>f</SUB> are the fluxes for the preceding
and following spot, respectively. The solutions presented in Paper I,
had Φ<SUB>p</SUB> = −0.5 Φ<SUB>f</SUB>, were oscillatory in time,
and large radial fields, Bτ, were present at the surface.
---------------------------------------------------------
Title: On a Babcock-Leighton dynamo model with a deep-seated
generating layer for the toroidal magnetic field
Authors: Durney, Bernard R.
1995SoPh..160..213D Altcode:
A dynamo model of the Babcock-Leighton type having the following
features is studied. The toroidal fieldB<SUB>φ</SUB> is generated
in a thin layer (the GL), located at the lower solar convection
zone, by a shear in the angular velocity acting on the poloidal
fieldB<SUB>p</SUB>(= ∇ × [0, 0,A<SUB>φ</SUB>].) If, in this layer,
and for a certain value of the polar angle,θ, |B<SUB>Ø</SUB> | exceeds
a critical field,B<SUB>cr</SUB>, then the eruption of a flux tube
occurs. This flux tube, which is assumed to rise radially, generates,
when reaching the surface, a bipolar magnetic region (BMR) with fluxes
Φ<SUB>p</SUB> and Φ<SUB>f</SUB> for the preceding and following spot
respectively. For the purpose of the numerical calculations this BMR is
replaced by its equivalent axisymmetrical magnetic ring doublet. The
ensemble of these eruptions acts as the source term for the poloidal
field. This field, generated in the surface layers, reaches the
lower solar convection by transport due to meridional motions and by
diffusion. The meridional motions are the superpositions of a one-cell
velocity field that rises at the equator and sinks at the poles and of
a two-cell circulation that rises at the equator and poles and sinks at
mid latitudes. The toroidal field andA<SUB>Ø</SUB> were expanded in
Legendre polynomials, and the coupled partial differential equations
(int andr; time and radial coordinate) satisfied by the coefficients
in these expansions were solved by a finite difference method. In the
expansions, Legendre polynomials up to order thirty were included.
---------------------------------------------------------
Title: On a Babcock-Leightom dynamo model with a thin, deep seated
generating layer for the toroidal field.
Authors: Durney, Bernard R.
1994AAS...185.9201D Altcode: 1994BAAS...26Q1472D
The following dynamo model will be discussed and hopefully numerical
results will be presented. Let A be the vector potential for
the axisymmetric poloidal field, and B, the toroidal field. B is
generated by a shear in the angular velocity acting on A in a thin
layer located in the lower solar convection zone. If in this layer
B exceeds a critical value for a certain value of theta (the polar
angle), eruption occurs. The flux tube is assumed to rise radially and
to surface as a magnetic ring doublet. The rates of eruption of the
ensemble of these doublets constitute the source term of the equation
for partial A / partial t that regenerates the poloidal field. The
poloidal field generated in the solar surface layers reaches the
lower solar convection by transport due to meridional motions and by
diffusion. The meridional motions being considered are the superposition
of a one-cell velocity field that rises at the equator and sinks at the
poles and of a two-cell motion that rises at the equator and poles and
sinks at mid latitudes. Meridional motions of this type have a strong
theoretical and observational support.
---------------------------------------------------------
Title: On the Generation of the Largescale and Turbulent Magnetic
Fields in the Solar Type Stars
Authors: Durney, Bernard R.; De Young, David S.; Roxburgh, Ian W.
1993SoPh..145..207D Altcode:
It is thought that the large-scale solar-cycle magnetic field is
generated in a thin region at the interface of the radiative core
(RC) and solar convection zone (SCZ). We show that the bulk of the SCZ
virogoursly generates a small-scale turbulent magnetic field. Rotation,
while not essential, increases the generation rate of this field.
---------------------------------------------------------
Title: On the Solar Differential Rotation: Meridional Motions
Associated with a Slowly Varying Angular Velocity
Authors: Durney, Bernard R.
1993ApJ...407..367D Altcode:
The paper calculates the meridional flow in the solar convection
zone from the azimuthal momentum equation and compares the results
with the existing surface observations. The dominant flows are found
to contain only a few cells per hemisphere. Other than the meridional
flow, the azimuthal momentum equation contains the velocity correlations
(u<SUB>r</SUB> u<SUB>phi)</SUB> and (u<SUB>theta</SUB> u<SUB>phi),</SUB>
as well as the angular velocity, Omega. The stream functions psi2 and
psi4 define meridional motions with one and two latitudinal cells,
respectively. The resultant meridional flow has a one-cell component
rising at the equator and a two-cell component sinking at midlatitudes,
in agreement with observations.
---------------------------------------------------------
Title: Observational Constraints on Theories of the Solar Differential
Rotation
Authors: Durney, Bernard R.
1991ApJ...378..378D Altcode:
The dependence of the inner solar angular velocity (ISAV) on the spatial
coordinates is examined through helioseismic observation. If the ratios,
designated RR, are specified and ISAV is known from the observations,
then the azimuthal momentum equation determines the meridional
motions. For all realistic values of RR, the stream function of a
meridional motion is negative, i.e., the flow rises at the equator and
sinks at the poles. Such flows together with the fact that the turbulent
convective velocities are negative suggest a natural explanation for the
helioseismic observations near the solar surface (with increasing depth,
the angular velocity first increases and then decreases). It is proposed
that flows with few cells per hemisphere will dominate. Restrictions
of this type imposed on the azimuthal momentum equation circumscribe
the values of RR. In the simple case under consideration, the eddies
take on a slablike appearance elongated along the axis of rotation.
---------------------------------------------------------
Title: On the Generation of the Solar Magnetic Field in a Region of
Weak Buoyancy
Authors: Durney, Bernard R.; De Young, David S.; Passot, Thierry P.
1990ApJ...362..709D Altcode:
The possibility that the cyclic magnetic field of the sun can be
generated in a layer of weak buoyancy at the lower boundary of the solar
convection zone (SCZ) is addressed using an eddy-damped quasi-normal
Markovian closure model of the turbulent MHD equations. It is concluded
that only models with kinetic energy of turbulent motion larger than
roughly 10 to the 6th g/cm/s are viable. The action of differential
rotation on this poloidal field generates a toroidal field exceeding
the equipartition value which is sufficiently strong to interfere with
the transport of heat and is amplified further in the SCZ.
---------------------------------------------------------
Title: On the Numerical Calculation of the Solar Rotational Splitting
Coefficients
Authors: Durney, Bernard R.
1990ApJ...351..682D Altcode:
An alternative method for numerically calculating the solar rotational
splitting coefficient is developed. The efficacy of the method is
illustrated using South Pole data. The robustness of a(2i+1) is shown
to be excellent for i = 0 and to decrease with increasing i, whereas the
values of b(2i+1) appear to be sensitive to the method of calculation.
---------------------------------------------------------
Title: Some Controversial Issues in Theories of the Solar Differential
Rotation and Dynamo
Authors: Durney, Bernard R.
1989SoPh..123..197D Altcode:
The following points are discussed: The dependence of the angular
velocity, , on the spatial coordinates near the lower boundary,
R<SUB>c</SUB>, of the solar convection zone (SCZ) can be obtained
from an integration with respect to r of a sound approximation
to the azimuthal equation of motion. Here P<SUB>2</SUB>
(cos θ) is the second-order Legendre polynomial and θ is the
polar angle. Estimates of ω'<SUB>0</SUB>, ω'<SUB>2</SUB> (the
primes denote derivatives with respect to r), based on the best
available values for the Reynolds stresses and anisotropic viscosity
coefficients, suggest that ω'<SUB>0</SUB> < 0, ω'<SUB>2</SUB>
≈ 0 for r = R<SUB>c</SUB>. Since a reliable theory of anisotropic
turbulent coefficients does not exist at present, positive values of
ω'<SUB>0</SUB> are conceivable.
---------------------------------------------------------
Title: On the Behavior of the Angular Velocity in the Lower Part of
the Solar Convection Zone
Authors: Durney, Bernard R.
1989ApJ...338..509D Altcode:
The solar angular velocity is expanded in Legendre polynomials. The
meridional motions are restricted to one or two cells per hemisphere,
and an approximation to the azimuthal equation of motion is integrated
with respect to r with the help of the boundary condition at r = Rc,
the lower boundary of the solar convection zone (SCZ). The Reynolds
stresses appearing in the equation are estimated for the lower SCZ,
and approximate expressions are derived for the turbulent viscosity
coefficients (which are due to the influence of the mean flow on the
turbulent velocities). It is shown that the assumption of isotropic
viscosity is always open to criticism. An order-of-magnitude estimate
of the different terms in the E(r) and E(theta) equations suggests that
the Reynolds and viscous stresses are important only near the boundaries
of the SCZ. Away from the boundaries, in the Taylor-Proudman region,
the suggested balance is between Coriolis forces, pressure gradients,
and buoyancy forces.
---------------------------------------------------------
Title: Book-Review - the Internal Solar Angular Velocity - Theory
Observations and Relationship to Solar Magnetic Fields
Authors: Durney, B. R.; Sofia, S.; Gough, D.
1988JBAA...98..261D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Book-Review - the Internal Solar Angular Velocity - Theory
Observations and Relationship to Solar Magnetic Fields
Authors: Durney, B. R.; Sofia, S.
1988S&T....75Q.498D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Expansion of the Rotational Eigenfrequencies in
Legendre Polynomials
Authors: Durney, Bernard R.; Hill, Frank; Goode, Philip R.
1988ApJ...326..486D Altcode:
In the context of helioseismology, it has become customary to fit
data using Δv(n, l, m) ≡ v(n, l, m) - v(n, l) = L Σ<SUP>N</SUP>
<SUB>i=0</SUB> a<SUB>i</SUB> P<SUB>i</SUB>(-m/L) (Duvall, Harvey,
and Pomerantz) where v is the frequency of the nth p-mode averaged
over m, the P<SUB>i</SUB> are Legendre polynomials and L = [(l +
1)l]<SUP>1/2</SUP>. It is shown here that, instead, it is advantageous
to use the following expansion for v(n, l, m) - v(n, l): v(n, l, m) -
v(n, l) = m Σ <SUP>N</SUP> <SUB> i=0</SUB> b<SUB>i</SUB> P<SUB>i</SUB>
(m/L). In this case the b<SUB>i</SUB>'s are simply related to the
coefficients which determine the angular velocity, leading to the
expectation that we can more accurately determine the internal rotation
of the Sun from the extant helioseismological data.
---------------------------------------------------------
Title: Book-Review - the Internal Solar Angular Velocity - Theory
Observations and Relationship to Solar Magnetic Fields
Authors: Durney, B. R.; Sofia, S.
1988Sci...239..926D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: A simple dynamo model and the anisotropic alpha-effect
Authors: Durney, B. R.
1988A&A...191..374D Altcode:
The α-term in dynamo theory is evaluated in the quasi-linear
approximation with no assumption concerning the isotropy of the
turbulent motions. The resulting expression for α is discussed in
the framework of a dynamo model depending only on time and having a
simple buoyancy term. It is argued that the main cyclic solar magnetic
field is amplified in the lower boundary layer of the solar convection
zone whereas the convection zone proper generates a weak, stochastic,
background, magnetic field.
---------------------------------------------------------
Title: Book-Review - the Internal Solar Angular Velocity
Authors: Durney, B. R.; Sofia, S.
1988ApL&C..27R.286D Altcode: 1988ApL....27R.286D
No abstract at ADS
---------------------------------------------------------
Title: Book-Review - the Internal Solar Angular Velocity - Theory
Observations and Relationship to Solar Magnetic Fields
Authors: Durney, B. R.; Sofia, S.
1987JBAA...98Q..48D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Solar Angular Velocity in the Lower Solar Convection
Zone
Authors: Durney, Bernard R.
1987BAAS...19Q.934D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Generalization of Mixing Length Theory to Rotating
Convection Zones and Application to the Sun
Authors: Durney, Bernard R.
1987ASSL..137..235D Altcode: 1987isav.symp..235D
The consequences of a balance between the Coriolis forces, pressure
gradients and buoyancy forces in a compressible medium are investigated
(the Taylor-Proudman theorem). A simple proof is given that if
this balance holds, then the latitudinally dependent part of the
superadiabatic gradient (∇ΔT) is determined by the angular velocity,
Ω, and it is of the order of 2Ω<SUP>2</SUP><SUB>0</SUB>T/7g for
rotation laws other than Ω constant along cylinders (it vanishes
in this case). Here Ω<SUB>0</SUB> is the average angular velocity,
T the temperature and g gravity. In the lower part of the solar
convection zone, 2Ω<SUP>2</SUP><SUB>0</SUB>T/7g is of the order of
∇ΔT<SUB>r</SUB>, itself, i.e., very large.
---------------------------------------------------------
Title: The internal solar angular velocity. Theory, observations
and relationship to solar magnetic fields
Authors: Durney, Bernard R.; Sofia, Sabatino
1987ASSL..137.....D Altcode: 1987isav.symp.....D
The conference presents papers on observations of solar p-mode
rotational splittings, observations of surface velocity fields, the
equatorial rotation rate in the solar convective zone, chromospheric
activity in open clusters, and solar rotation variations from sunspot
group statistics. Other topics include adiabatic nonradial oscillations
of a differentially rotating star, a spherical harmonic decomposition
technique for analyzing steady photospheric flows, turbulent transport
in the radiative zone of a rotating star, and the generation of magnetic
fields in the sun. Consideration is also given to magnetic fields and
the rotation of the solar convection zone, the hydrostatic adjustment
time of the solar subconvective layer, models for a differentially
rotating solar-convection zone, and horizontal Reynolds stress and
the radial rotation law of the sun.
---------------------------------------------------------
Title: On theories of rotating convection zones
Authors: Durney, B. R.
1985ApJ...297..787D Altcode:
It is shown that the time rate of change (brought about by the turbulent
convective motions) in the angular momentum of a thin spherical shell
of radius r is such as to increase the angular velocity of the lower
part (τΩ<SUB>0</SUB> > 1; τ is the dominant eddy's lifetime)
of the solar convection zone (SCZ) and to decrease the angular
velocity of the upper part (τΩ<SUB>0</SUB> < 1). A tentative
model of rotation in the SCZ is proposed: (1) the (τΩ<SUB>0</SUB>
> 1)-region is in weaker differential rotation than the surface
and not constrained by the Taylor-Proudman theorem, (2) the observed
solar differential rotation at the surface is then generated as the SCZ
relaxes from the (τΩ<SUB>0</SUB> > 1)-state in the lower part to
the (τΩ<SUB>0</SUB> < 1)-state at the surface. In the upper and
lower layers of the SCZ, angular-momentum conservation between the
turbulent motions and viscous stresses leads to an angular velocity
increasing inward.
---------------------------------------------------------
Title: A search for long-lived velocity fields at the solar poles
Authors: Durney, B. R.; Lytle, D. M.; Cram, L. E.; Guenther, D. B.;
Keil, S. L.
1985ApJ...292..752D Altcode:
A search has been made in the polar regions of the sun for large-scale
(50-200 Mm) velocity fields with lifetimes of the order of the solar
rotation period (approximately equal to or greater than 30 days). The
observations show that any such large-scale, long-lived velocity
patterns in the polar regions must have an amplitude less than 5
m/s. Marginally significant detections (at the 2-3 sigma level) were
made of two kinds of structures with amplitudes of order 3 m/s. One has
a rotation period approximately 38 days (close to the polar rotation
period at the sun's surface), and a scale approximately 150 Mm; the
other has a period approximately 24 days and a scale approximately
100 Mm. Tentatively, the first structure is interpreted as being of
supergranular origin. The second structure is interpreted as the
overshooting of the dominant convective mode of the lower solar
convection zone - the giant granulation.
---------------------------------------------------------
Title: On the Generalization of the Mixing Length Theory to Rotating
Convection Zones
Authors: Durney, B. R.
1985BAAS...17..644D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the influence of turbulent motions on non-radial
oscillations.
Authors: Durney, B. R.
1984sses.nasa..325D Altcode: 1984sss..conf..325D
The effect of turbulent motions on oscillations is studied, considering
only the coupling between turbulent and oscillatory velocities. In this
case, the turbulence affects the oscillations through the Reynolds
stresses in the momentum equation for the pulsations. A simple model
of turbulence is adopted to evaluate these Reynolds stresses and the
perturbed eigenfrequencies are expressed as a function of certain
averages of the turbulent velocities.
---------------------------------------------------------
Title: On the rotation rate of polar features in the sun
Authors: Durney, B. R.; Lytle, D. M.; Keil, S. L.
1984ApJ...281..455D Altcode:
The authors evaluate the rotation rate of solar features in the
vicinity of the poles with the help of a correlation procedure. The
average rotation rates for both poles are systematically smaller than
those predicted by Howard and Harvey's formula, but not in serious
disagreement with their results.
---------------------------------------------------------
Title: On the large-scale dynamics of rapidly rotating convection
zones
Authors: Durney, B. R.
1983ApJ...269..671D Altcode:
The fact that the values of the eight basic waves present in turbulent
flows in the presence of rotation prohibit a tilt of eddy towards the
axis of rotation is incorporated into a formalism for rapidly rotating
convection zones. Equations for turbulent velocities are defined in a
rotating coordinate system, assuming that gravity and grad delta T act
in a radial direction. An expression is derived for the lifetime of a
basic wave and then for the average velocity vector. A real convective
eddy is formulated and the wave vectors are calculated. The velocity
amplitude and the stress tensor amplitude are integrated over the
eddy domain. Applied to the solar convective zone, it is found that
the convective cells are aligned along the axis of rotation at the
poles and at the equator, a model that conflicts with nonrotating
mixng length theory predictions.
---------------------------------------------------------
Title: On the first-order smoothing expression for the alpha-effect
in dynamo theory
Authors: Durney, B. R.
1983ApJ...267..822D Altcode:
The term regenerating the poloidal magnetic field from the toroidal one
(the alpha-effect) plays a central role in the generation of magnetic
fields by a dynamo process. Using the so-called first order smoothing
approximation, Steenbeck and Krause (1969) derived an expression
for alpha, taking into account the case of homogeneous, isotropic
motions. An equation, which is valid in the Boussinesq approximation,
was found for alpha by making use of the velocity distribution employed
by Durney and Spruit (1979) in the evaluation of the Reynolds stresses
generating the solar differential rotation. It is important to derive an
expression for alpha which is compatible with this equation, because
little is known about the structure of the dominant convective eddy
in the lower solar convection zone. The present investigation is
concerned with the derivation of such an expression, taking into
account a disappearance of alpha in the Boussinesq approximation if
curvature effects are neglected.
---------------------------------------------------------
Title: Preliminary observations of velocity fields at the solar poles
Authors: Cram, L. E.; Durney, B. R.; Guenther, D. B.
1983ApJ...267..442C Altcode:
Using the 13 m Littrow spectrograph at Sacramento Peak Observatory,
the Doppler shift of Fe I 5863 A in the polar regions of the sun over
a 20 day interval is studied. The daily observations were assembled
into a polar projection of the line-of-sight velocity field. The
projection shows a very clear pattern of supergranulation. When a
low-pass spatial filter is run over the data, a pattern of large-scale
(80-100 Mm) velocity features can be seen. Cross-correlation studies
show that the supergranular pattern rotates with a synodic period of
35 days, while there is evidence that the larger features rotate with a
shorter period of about 30 days. At present, it is not possible to say
whether the large-scale patterns represent a new scale of convection
(possibly related to the dominant convective eddy in the lower solar
convection zone) or to the low-wavenumber tail of a distribution of
supergranular cells.
---------------------------------------------------------
Title: Observations of Polar Velocity Fields
Authors: Durney, B. R.; Lytle, D. M.; Cram, L. E.; Guenther, D. B.;
Keil, S. L.
1983BAAS...15..716D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the generation of magnetic fields in late-type stars -
A local time-dependent dynamo model
Authors: Robinson, R. D.; Durney, B. R.
1982A&A...108..322R Altcode:
We assume that the magnetic field of late-type stars is generated in
the lower part of the star's convection zone and study this generation
mechanism with the help of local (in latitude) dynamo equations. For
the spectral types GO, GS, KO, KS, MO, M2, and M5 we evaluate the
magnetic field and period of the cycles as a function of rotation and
compare them with the available observational data.
---------------------------------------------------------
Title: On an estimate of the dynamo-generated magnetic fields in
late-type stars.
Authors: Durney, B. R.; Robinson, R. D.
1982ApJ...253..290D Altcode:
The principal objective of the present investigation is related to a
prediction of the variation of magnetic fields with stellar type and the
role of pertinent variables (such as rotation and differential rotation)
with respect to the field properties. This is accomplished by estimating
a typical amplification time for the magnetic field, and a typical 'time
of rise' for the magnetic field due to magnetic buoyancy. It is assumed
that the magnetic field is generated principally in the lower part of
the stellar convection zone. Local (in latitude) dynamo equations are
considered. The selected approach consists basically in an estimate of
the typical magnitude of the magnetic field as predicted by the local
dynamo equations. The employed approach constitutes only a first step
towards the evaluation of magnetic fields in stars other than the sun.
---------------------------------------------------------
Title: A preliminary interpretation of stellar chromospheric CA II
emission variations within the framework of stellar dynamo theory.
Authors: Durney, B. R.; Mihalas, D.; Robinson, R. D.
1981PASP...93..537D Altcode:
Recent stellar chromospheric Ca II emission data are analyzed
and interpreted within the framework of simple concepts of dynamo
theory. From an examination of the rotation rates and B-V indexes of
26 stars as presented by Vaughn at el. (1981) and the background flux
values derived by Wilson (1978) for 18 reference stars, an empirical
relation is derived between dynamo number, calculated from the B-V
index and rotation rate, and stellar chromospheric emission flux. The
Ca-emission cycle morphology of the sample stars is then examined,
and differences between the four morphological classes identified
are explained in terms of the correlation of large dynamo numbers
with the presence of several interfering magnetic modes of different
spatial scales, which do not exhibit a marked cyclic behavior, and
small numbers with the excitation of only a single mode. The gap
noted by Vaughn and Preston (1980) in the relation between the log
of the emission flux with (B-V) is then interpreted as representing
a transition from a multiple-mode dynamo to a single-mode dynamo as
the dynamo number decreases.
---------------------------------------------------------
Title: On an Estimate of the Dynamo-Generated Magnetic Fields in
Late-Type Stars
Authors: Durney, B. R.; Robinson, R. D.
1981BAAS...13..791D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On Theories of Rotating Convection Zones
Authors: Durney, B.
1981siwn.conf....1D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On an Estimate of the Dynamo-Generated Magnetic Fields in
Late-Type Stars
Authors: Durney, B. R.; Robinson, R. D.
1981BAAS...13..906D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Observations of Velocity Fields at the Solar Poles
Authors: Guenther, D.; Cram, L.; Durney, B.
1981BAAS...13Q.906G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On a model of a slowly rotating solar convection zone
Authors: Durney, B. R.
1981ApJ...244..678D Altcode:
Numerical solutions are evaluated of the equations governing the
large-scale motions of rotating stellar convection zones, as derived
by Durney (1976) and Spruit (1977) (DS). With reference to the solar
convection zone, these equations were solved by a perturbation method
with the uniformly rotating convection zone as the unperturbed state
(approximated by a polytrope). The calculations suggest that (1) large
pole-equator differences in flux in the lower part of the convection
zone are entirely compatible with negligible pole-equator differences
in flux at the surface; (2) in realistic models of the rotating solar
convection zone the energy carried by radiation should be included;
and (3) in the lower part of the convection zone the solar convection
velocities could differ substantially from those evaluated in the
absence of rotation.
---------------------------------------------------------
Title: Sacramento Peak Observatory, Sunspot, New Mexico 88349. Report.
Authors: Zirker, J. B.; Durney, B. R.
1981BAAS...13..389Z Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Effect of Rotation on Solar Convection
Authors: Durney, B. R.
1980BAAS...12..895D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: A Formalism for Differential Rotation
Authors: Durney, B. R.; Spruit, H. C.
1980HiA.....5..121D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Dynamics of the Solar Convection Zone
Authors: Durney, B. R.; Spruit, H. C.
1980LNP...114...15D Altcode: 1980IAUCo..51...15D; 1980sttu.coll...15D
No abstract at ADS
---------------------------------------------------------
Title: On the dynamics of stellar convection zones - The effect of
rotation on the turbulent viscosity and conductivity
Authors: Durney, B. R.; Spruit, H. C.
1979ApJ...234.1067D Altcode:
We derive expressions for the turbulent viscosity and turbulent
conductivity applicable to convection zones of rotating stars. We assume
that the relative dimensions of the dominant convective cell are known
and derive a simple distribution function for the turbulent convective
velocities under the influence of rotation. From this distribution
function (which includes, in particular, the stabilizing effect of
rotation on convection) we calculate in the mixing-length approximation:
(i) the turbulent Reynolds stress tensor and (ii) the expression for the
heat flux in terms of the superadiabatic gradient. The contributions
of the turbulent convective motions to the mean momentum and energy
equation (which determine the large-scale motions in stellar convection
zones) are treated consistently, and assumptions about the turbulent
viscosity and heat transport are replaced by assumptions about the
turbulent flow itself. The free parameters in our formalism are the
relative cell dimensions and their dependence on depth and latitude.
---------------------------------------------------------
Title: A comparison of numerical simulations of eddy generation
performed with a two- and a three-layer quasi-geostrophic model of
oceanic mesoscale eddies
Authors: Durney, Bernard
1978GApFD..10..275D Altcode:
The generation of eddies by a large-scale flow over mesoscale
topography is studied with the help of two- and three-layer nonlinear
quasi-geostrophic models of the open ocean. The equations are integrated
forward in time with no eddies present initially. For a given time, the
displacement of the interface between layers two and three () tends to
a well-defined limit (function of the horizontal spatial coordinates)
as 3- 2 0 (r is the density of layer r). Even for values of α[=
(ρ3 - ρ2)/(ρ2 - ρ1)] as small as 0.01 the potential energy due to
ζ is not negligible and it can reach, in some cases, a considerable
fraction of the total eddy energy.
---------------------------------------------------------
Title: On the angular momentum loss of late-type stars.
Authors: Durney, B. R.; Latour, J.
1978GApFD...9..241D Altcode:
The observed surface angular velocity of main-sequence stars shows a
sharp decrease at about spectral type F6. It is suggested that stars
more massive than F6 cannot experience an appreciable angular-momentum
loss because their convection zones cannot sustain a magnetic dynamo:
without a magnetic field the angular-momentum loss is very small. The
influence of rotation on the convective motions is essential for
the existence of a solar-type dynamo. Rotation can influence these
convective motions only if the typical convective time is larger
than the rotation time. For main-sequence stars of different masses
and chemical compositions the dimensionless parameter (convective
velocity/sum's angular velocity times mixing length in the lower part
of the convection zone) is evaluated. It is shown that this parameter
increases very sharply for stars whose mass exceeds that defined by
the relation log(star mass/solar mass) is of the order of 0.1. Thus
even for large angular velocities, magnetic dynamos are not feasible
if log(star mass/solar mass) appreciably exceeds 0.1.
---------------------------------------------------------
Title: On the angular momentum loss of late-type stars
Authors: Durney, B. R.; Latour, J.
1977GApFD...9..241D Altcode:
The observed surface angular velocity of main-sequence stars shows a
sharp decrease at about spectral type F6. We suggest that stars more
massive than F6 cannot experience an appreciable angular momentum loss
because their convection zones cannot sustain a magnetic dynamo: without
a magnetic field the angular momentum loss is very small. The influence
of rotation on the convective motions is essential for the existence of
a solar type dynamo. Rotation can influence these convective motions
only if the typical convective time is larger than the rotation
time, i.e., if l/uc > 1/, where uc and l are typical values of
the convective velocity and mixing length in the lower part of the
convection zone and is the star's angular velocity. For main-sequence
stars of different masses and chemical compositions we evaluate the
dimensionless parameter uc/Ω⊙ l and show that it increases very
sharply for stars whose mass, M, exceeds that defined by log(M /M⊙ )
eDot 0.1 (Ω⊙, and M⊙, are the sun's angular velocity and mass,
respectively). Thus even for large angular velocities, magnetic dynamos
are not feasible if log(M/M⊙) appreciably exceeds 0.1.
---------------------------------------------------------
Title: The influence of mesoscale topography on the stability and
growth rates of a two-layer model of the open ocean
Authors: Durney, Bernard R.
1977GApFD...9..115D Altcode:
The influence of mesoscale topography on the baroclinic instability
of a two-layer model of the open ocean is considered. For westward
velocities in the top layer (U), and for a sinusoidal topography
independent of x or longitude (a cross-stream topography), the critical
value of U (Uc) leading to instability is the same as when there is no
topography. The wavelength of the unstable perturbation corresponding
to Uc is shortened. For a given wavevector (k) of the perturbation
the system becomes stable (as also in the absence of topography)
for large values of |U|. The minimum value of the shear leading to
stability is, however, significantly reduced by the topography. For
sufficiently large values of the height of the topographic features,
instabilities appear which are localized within a narrow range of the
shear. These instabilities are studied for a topography that depends
both on x and y. <P />For a cross-stream topography the growth rates
are somewhat smaller than those without topography and they depend only
weakly on ky. For the topographies considered here which depend both
on x and y, perturbations with different values of ky can again have
roughly the same growth rate. <P />In the case of stable oscillations,
variations in the eddy energy with very long periods are made possible
by the coexistence of topographic modes with closely lying periods.
---------------------------------------------------------
Title: On Theories of Solar Rotation
Authors: Durney, B. R.
1976IAUS...71..243D 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: A comparison of the meridional flows in the sun's convection
zone, predicted by theories of the solar dynamo and differential
rotation.
Authors: Durney, B. R.
1975ApJ...199..761D Altcode:
Recently Yoshimura has evaluated the gradient of the Sun's angular
velocity (d /Jr) necessary to give a good fit to the observed solar
activity cycle. We estimate the meridional velocities in the convection
zone, implied by this value of . These meridional velocities are
in good agreement with those necessary to explain the Sun's surface
differential rotation. Subject headings: hydromagnetics - interiors,
solar - rotation, solar
---------------------------------------------------------
Title: On Coronal Streamers with T-Type Neutral Points
Authors: Durney, B. R.
1975SoPh...41..233D Altcode:
The gas-magnetic field interaction of an isothermal axisymmetric corona
is considered. A method is suggested for solving the MHD equations
in the case when a uniform gas pressure and the radial component
of the magnetic field (as in a dipole) are specified at the Sun's
surface. The flux of open field lines (φ) can be given arbitrarily,
and no reconnection or opening of field lines can take place. If
configurations in hydrostatic equilibrium between the regions of open
and closed field lines can be found, then the method of solution
converges. The equation of hydrostatic equilibrium at the neutral
point (assumed to be of the T-type) is written in a simple form, and
it is shown that if φ is smaller than a certain φ<SUB>min</SUB>,
this equation cannot be satisfied. Configurations in hydrostatic
equilibrium between the regions of open and closed field lines are
expected to exist for any value of φ larger than φ<SUB>min</SUB>.
---------------------------------------------------------
Title: A Comparison of the Meridional Flows in the Sun's Convection
Zone Predicted by Theories of the Solar Dynamo and Differential
Rotation
Authors: Durney, B. R.
1975BAAS....7Q.364D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Solar-Interplanetary Modeling: 3-D Solar Wind Solutions in
Prescribed Non-Radial Magnetic Field Geometries
Authors: Durney, B. R.; Pneuman, G. W.
1975SoPh...40..461D Altcode:
A model is presented which describes the 3-dimensional non-radial
solar wind expansion between the Sun and the Earth in a specified
magnetic field configuration subject to synoptically observed plasma
properties at the coronal base. In this paper, the field is taken to be
potential in the inner corona based upon the Mt. Wilson magnetograph
observations and radial beyond a certain chosen surface. For plasma
boundary conditions at the Sun, we use deconvoluted density profiles
obtained from synopticK-coronameter brightness observations. The
temperature is taken to be 2 × 10<SUP>6</SUP> K at the base of closed
field lines and 1.6 x 10<SUP>6</SUP>K at the base of open field lines.
---------------------------------------------------------
Title: On the Sun's Differential Rotation. Implications of the
Difference in Angular Velocity between the Sunspots and Photosphere
Authors: Durney, B. R.
1974SoPh...38..301D Altcode:
It is assumed that the meridional motions (U) and angular velocity
(Ω) in the surface layers of the convection zone are given by
simple expressions of the form: U<SUB>r</SUB>= 2<SUB>ψ</SUB>(r)
P<SUB>2</SUB>(cosθ)/ϱr<SUP>2</SUP>, U<SUB>0</SUB> = −ψ'(r)
sinθ cosθ/ϱr, and Ω = Ω<SUB>0</SUB>[(1 + ω<SUB>0</SUB>(r) +
ω<SUB>2</SUB>(r) P<SUB>2</SUB>(cosθ)]. Here ψ(r) is the stream
function, P<SUB>2</SUB>(cosθ) the second order Legendre polynomial,
and θ the polar angle. Allowance is made for a possible difference
in the rate of momentum exchange between the directions parallel and
perpendicular to gravity by introducing an anisotropic turbulent
viscosity coefficient, μ, which is assumed furthermore to be
proportional to the density, ϱ;μ = ϱν, and ν<SUB>θθ</SUB>=
ν<SUB>φφ</SUB>= sν<SUB>rr</SUB>. It is shown that if the sunspots
give an indication of the Sun's angular velocity at a depth h(∽
3 × 10<SUP>4</SUP> km) then the turbulent viscosity is necessarily
anisotropic. The radial variation introduced by this anisotropy seems
to explain well the sunspot data if we assume that the sunspots act
as tracers of the Sun's angular velocity.
---------------------------------------------------------
Title: On the Sun's Differential Rotation: its Maintenance by
Large-Scale Meridional Motions in the Convection Zone
Authors: Durney, Bernard R.
1974ApJ...190..211D Altcode:
It is shown that if the observed differential rotation of the Sun
is generated in the lower, Boussinesq part of the convection zone
(where the interaction of rotation with convection is important), a
large pole-equator difference in flux would also have to be present. The
Sun's angular velocity is evaluated as a function of depth and latitude
under the assumption that the main effect of the interaction of rotation
with convection is the generation of a small pole-equator difference
in temperature in the lower part of the convection zone, which drives
a meridional motion over the entire convection zone. The pole-equator
difference in flux associated with this meridional circulation (which
gives rise to the Sun's differential rotation) is negligible. Subject
headings: interiors, solar - rotation, solar
---------------------------------------------------------
Title: The expansion of a low-density solar corona: A one-fluid
model with magnetically modified thermal conductivity
Authors: Durney, B. R.; Hundhausen, A. J.
1974JGR....79.3711D Altcode:
A one-fluid model of the coronal expansion, including the reduction in
radial heat conduction produced by a spiral interplanetary magnetic
field, is extended to the low coronal densities that may occur in
the regions of open diverging magnetic field lines, or ‘coronal
holes,’ that are regarded as probable sources of the solar wind. At
such densities, the ‘cutoff’ in heat conduction at very large
heliocentric distances (where the magnetic field becomes nearly
azimuthal) has a profound effect on the nature of the expansion. The
corona becomes nearly isothermal out to the distance where the flow
of plasma dominates the transport of energy. This outward extension
of high coronal temperatures leads to large solar wind speeds,
approaching those given by Parker's original isothermal model as the
coronal density becomes vanishingly small. The model predicts expansion
speeds as high as 500 km s<SUP>-1</SUP>, with densities in agreement
with those observed near the orbit of earth, for a reasonable set of
coronal densities and temperatures (e.g., with coronal temperatures
no higher than 2.1×10<SUP>6</SUP> °K). However, the temperatures
(or pressures) predicted at the orbit of earth are substantially higher
than those observed; this deficiency of the model could only be removed
by incorporation of additional physical effects or processes.
---------------------------------------------------------
Title: On the Energetics and Momentum Balance of Pole-Equator
Temperature Differences in the Sun
Authors: Durney, Bernard
1973ApJ...183..665D Altcode:
It is suggested that, as a consequence of the action of magnetic fields,
the acoustic energy heating the chromosphere could be deposited at
different heights at the equator than at the poles. The resultant
pole-equator difference in pressure can be balanced by tilted sinusoidal
motions [in the (r, 0)-plane] having some resemblance to horizontal
Rossby waves. Subject heading: atmospheres, solar
---------------------------------------------------------
Title: Solar-Wind Properties at the Earth as Predicted by the
Two-Fluid Model
Authors: Durney, B. R.
1973SoPh...30..223D Altcode:
The two-fluid equations for the solar wind are written down
in a simplified form, similar to that suggested by Roberts
(1971) for the one-fluid model. The equations are shown to
depend only on one parameter, K = GMκ<SUB>e</SUB>m<SUB>p</SUB>
(ɛ<SUB>&infty</SUB>T<SUP>0</SUP>)<SUP>3/2</SUP>/4k<SUP>2</SUP>
Fe, where G is the gravitational constant, M the mass of the star,
κ<SUB>e</SUB> the thermal electron conductivity, m<SUB>p</SUB> the
proton mass, k the Boltzman constant, kɛ<SUB>∞</SUB>T<SUB>0</SUB>
the residual energy per particle at infinity and F<SUB>e</SUB> the
electron-particle flux. For a variety of values of the density and
temperature at the base of the corona we compute the solutions of
the two-fluid solar wind model and compare the predicted and observed
solar wind parameters at the Earth.
---------------------------------------------------------
Title: On the Energetics and Momentum Balance of Pole-equator
Temperature Differences in the Sun
Authors: Durney, B. R.
1973BAAS....5V.271D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Solar wind properties at the Earth as predicted by the
one-fluid model with helioclassical thermal electron conductivity
Authors: Durney, B. R.
1973JGR....78.7229D Altcode:
The solar wind properties at the earth are computed for a variety of
values of the density and temperature at the base of the corona. For
the electron thermal conductivity the expression derived by Perkins
is adopted. Good agreement with observations is obtained for values
of the density and temperature at the base of the corona equal to
∼9 × 10<SUP>7</SUP> cm<SUP>-3</SUP> and ∼1.7 × 10<SUP>6</SUP>
°K, respectively.
---------------------------------------------------------
Title: On the Sun's Differential Rotation and Pole-Equator Temperature
Difference
Authors: Durney, B.
1972SoPh...26....3D Altcode:
The Sun's differential rotation can be understood in terms of a
preferential stabilization of convection (by rotation) in the polar
regions of the lower part of the convection zone (where the Taylor
number is large). A significant pole-equator difference in flux
(Δℱ) can develop deep inside the convection zone which would be
unobservable at the surface, because ℱ can be very efficiently reduced
by large scale meridional motions rising at the poles and sinking at
the equator. This is the sense of circulation needed to produce the
observed equatorial acceleration of the Sun. Differential rotation is
generated, therefore, in the upper part of the convection zone (where
the interaction of rotation with convection is small) and results as
the convection zone adjusts to a state of negligible Taylor number.
---------------------------------------------------------
Title: Polytropic Subsonic Stellar Winds with Magnetic Fields
Authors: Durney, B.
1972Ap&SS..17..489D Altcode:
In any complex magnetic field configuration it is to be expected that
there will be not only regions of no flow (closed magnetic field lines)
and of supersonic flow, but also regions of subsonic flow. Subsonic
stellar winds could also be of importance in stars of different type
than the Sun. In the present paper the equations for the stellar wind
are examined in the case of a polytropic relation between pressure
and density and for small values of the parameter ɛ=Ω<SUP>2</SUP>
r <SUB> a </SUB> <SUP>2</SUP>/u <SUB> a </SUB> <SUP>2</SUP>. The
radial distance (r) and the velocity at the Alfvénic point (4πϱu
<SUP>2</SUP>/B <SUP>2</SUP>=1) are denoted byr <SUB> a </SUB> andu <SUB>
a </SUB>, and Ω is the angular velocity. It is shown that: solar
breeze solutions (that is, solutions that are subsonic for values
ofr such thatrnot ≫ r_a) exist only if the dimensionless energy
flux is larger thantfrac{3}{2}\varepsilon ^{{raise0.5exhboxriptstyle
2kern-0.1em/kern-0.15emlower0.25exhboxriptstyle 3}}. If ɛ→0 the
flow with magnetic field tends to the flow without magnetic field
forr<R(R→∞ as ɛ→0). Forr→∞ the velocity tends always
to a finite value; this does not introduce, however, a singularity in
the equations.
---------------------------------------------------------
Title: The Effect of Radiative Equilibrium on the Photospheric
Angular Velocity.
Authors: Durney, B. R.
1972ApJ...172..479D Altcode:
The photospheric angular velocity [co = w(r)] is evaluated in the range
of optical depths 0.05 < r 0.8under the assumption of radiative
equilibrium and vanishing von Zeipel currents. The angular velocity
decreases outward, and significant pole-equator temperature differences
develop The von Zeipel currents are estimated for the case of uniform
rotation.
---------------------------------------------------------
Title: On Stellar Activity Cycles
Authors: Durney, B. R.; Stenflo, J. O.
1972Ap&SS..15..307D Altcode:
The relation between the average magnetic fieldB, the angular velocity
Ω, and the periodP of stellar activity cycles is studied. For the
calculations we have used Leighton's (1969) model for the solar cycle
with the additional assumption that the differential rotation and the
cyclonic turbulence (Parker, 1955) (that is the ‘sunspot tilt’ or
the ‘α-effect’) are both proportional to Ω. We then find thatB is
roughly proportional to Ω and thatP decreases with increasing Ω. The
period of the solar cycle increases therefore with the age of the Sun.
---------------------------------------------------------
Title: On the Domains of Existence of the Three Types of Supersonic
Solutions of the Inviscid Solar-Wind Equations
Authors: Durney, B. R.; Werner, N.
1972ApJ...171..609D Altcode:
The approximate energy equation for radial distances larger than the
critical point is solved with "critical point" boundary conditions. It
is shown that if E = (41/24) (12/5)5/3 (35/2A)2/3 = e "' (where E is
the residual energy per particle at infinity, A = 5.8 X 106/C, and C
is the mass flow) then the equation has the solution T . If 6 <
, the asymptotic behavior of the temperature is whereas if e >
6 "' then T for large values of r. This clarifies the domains of
existence of the Parker, Whang and Chang, and supersonic solutions of
the solar-wind equations.
---------------------------------------------------------
Title: Transition From a Supersonic to a Subsonic Solar Wind
Authors: Durney, B.
1972NASSP.308..232D Altcode: 1972sowi.conf..232D
No abstract at ADS
---------------------------------------------------------
Title: Solar-wind properties at the Earth as Predicted by One-Fluid
Models
Authors: Durney, B. R.
1972JGR....77.4042D Altcode:
The spiraling magnetic field of the sun reduces the electron
conductivity κ by the factor cos²θ, where θ is the spiral field
angle. For a variety of values of the density and temperature at the
base of the corona, we compute one-fluid solar-wind models for thermal
conductivities equal to κ and κ cos²θ. For both cases, the values
of the computed solar-wind parameters at the earth are compared with
observed properties.
---------------------------------------------------------
Title: Evidence for Changes in the Angular Velocity of the Surface
Regions of the Sun and Stars - Comments
Authors: Durney, B.
1972NASSP.308..282D Altcode: 1972sowi.conf..282D
No abstract at ADS
---------------------------------------------------------
Title: On the solar oblateness: The combined effect of a pole-equator
difference in effective temperature and mechanical heating
Authors: Durney, B. R.; Werner, N. E.
1971SoPh...21...21D Altcode:
With the help of a model atmosphere of the Sun we evaluate the
pole-equator difference in flux (as measured by Dicke and Goldenberg)
assuming the following type of pole-equator temperature difference
(ΔT=T<SUB>e</SUB>−T<SUB>p</SUB>): (a) ΔT ≈ 2K for τ >
τ<SUB>0</SUB> (τ<SUB>0</SUB> ≈ 0.05); (b) ΔT ≈ 10K for τ <
τ<SUB>0</SUB>.
---------------------------------------------------------
Title: On the Theory of Stellar Winds
Authors: Durney, B. R.; Roberts, P. H.
1971ApJ...170..319D Altcode:
It has recently been shown by Roberts that solutions of the stellar
wind equations depend essentially on a single parameter IC = 12A where
E is the (dimensionless) residual energy per particle at infinity and
A is a nondimensional constant proportional to the reciprocal of the
mass flux C. This transformation makes it a comparatively simple matter
to examine solutions for a wide variety of A and t . The calculations
reported below are for the range 75 <K < 2000, which covers many
cases of astrophysical interest.
---------------------------------------------------------
Title: A New Type of Supersonic Solution for the Inviscid Equations
of the Solar Wind
Authors: Durney, B.
1971ApJ...166..669D Altcode:
The transition from a supersonic to a subsonic corona was investigated
by increasing the density N0 (initially N0 = 9.3 X 10 cm-3) at the
base of the corona while keeping the temperature T0 there constant
(T0 = 2.1 X 108 K). As the density was increased, the energy flux
at infinity due to thermal conductivity, t , steadily decreased
and vanished for N0 1.17 X 108 . However, the total energy flux at
infinity 8 remained different from zero. From somewhat higher values
of the density up to N0 3 X 108 a new type of supersonic solution was
found; the temperature behaving as (1/r)413 for large distances. It
is expected that this type of supersonic solution will exist up to
a value of the density for which subsonic solutions are possible
(N0 > 10 cm-3). Thus for the chosen value of T0, there is not a
direct transition from the Parker-type supersonic solution to the
subsonic solution (as N0 increases from its initial value); instead,
there is first a transition to a supersonic flow characterized by a
(l/r)413 asymptotic behavior for the temperature.
---------------------------------------------------------
Title: Differential Rotation, Meridional Velocities, and Pole-Equator
Difference in Temperature of a Rotating Convective Spherical Shell
Authors: Durney, B.
1971ApJ...163..353D Altcode:
A rotating, convective, spherical layer of fluid is considered in the
Boussinesq approximation. The coupled equations for the axisymmetric
modes of the velocity and temperature fields are solved in the steady
state with the low-order "Legendre components" of the fluctuating
self-interactions (evaluated in the quasilinear approximation) as
the driving terms It is found that (a) the angular velocity increases
inward; (b) there is pole-equator differential rotation with equatorial
acceleration; (c) beneath the surface the equator is hotter than
the poles (at the surface the temperature is given as a boundary
condition); (d) two types of meridional circulation are compatible
with the observed differential rotation of the Sun. For example,
in the northern hemisphere, in one case the flow comprises two cells
in the radial direction, and in the other case each radial cell is
divided latitudinally into two subcells
---------------------------------------------------------
Title: Inhomogeneous convection and the equatorial acceleration of
the sun.
Authors: Durney, B. R.; Roxburgh, I. W.
1971BAAS....3S.260D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Inhomogeneous Convection and the Equatorial Acceleration of
the Sun
Authors: Durney, B. R.; Roxburgh, I. W.
1971SoPh...16....3D Altcode:
The interaction of rotation and turbulent convection is assumed to give
rise to an inhomogeneous, but isotropic, latitude dependent turbulent
energy transport, which is described by a `convective conduction
coefficient κ<SUB>c</SUB>' which varies with latitude. Energy
balance in the convective zone is then possible only with a slow
meridian circulation in the outer convective zone of the sun. The
angular momentum transported by this circulation is balanced in a
steady state by turbulent viscous transport down an angular velocity
gradient. A detailed model is constructed allowing for the transition
from convective transport to radiative transport at the boundaries
of the convective zone, by using a perturbation analysis in which the
latitude variation of κ<SUB>c</SUB> is small. The solution for a thin
compressible shell gives equatorial acceleration and a hotter equator
than pole, assuming that the convection is preferentially stabilised at
the equator. For agreement with the sun's equatorial acceleration the
model predicts an equatorial temperature excess of 70 K and a surface
meridional velocity of 350 cm/sec from pole to equator.
---------------------------------------------------------
Title: The Interaction of Convection with Rotation
Authors: Durney, B. R.
1970BAAS....2S.310D Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Nonaxisymmetric Convection in a Rotating Spherical Shell
Authors: Durney, B.
1970ApJ...161.1115D Altcode:
The problem of a rotating, convective spherical shell is considered
in the Herring approximation. The relevant equations are integrated
in time as an initial-value problem. The main results are: 1. The most
unstable modes correspond to convective cells stretching from pole to
pole. 2. The calculations of the Reynolds stresses show transport of
angular momentum toward the equator. That is, differential rotation
sets in with equatorial acceleration. 3. The convective transport of
heat is maximum at the equator. This would give rise to an equatorpole
difference in flux. 4. If convection is nonaxisymmetric (as in the most
unstable modes), then there are no time-independent solutions. The
time-dependence is oscillatory and of the form A cos (cot + m ) +
B sin (cot + m .
---------------------------------------------------------
Title: Models of close and contact binary stars 1.Polytropic models
Authors: Durney, B. R.; Roxburgh, I. W.
1970MNRAS.148..239D Altcode:
Polytropic models of close and contact binary stars are
constructed using a combination of perturbation techniques and
a Laplace approximation previously applied to uniformly rotating
stars. Synchronism between orbital and intrinsic angular velocity is
assumed. Models are constructed including the effects of distortion
for polytropes with indices fl = 1, , 2, 3 and 4. The conditions for
the two stars to be just in contact are determined and contact models
with a mass ratio of unity are constructed, right up to the limiting
case when the stars fill all the available space inside the critical
Roche surface surrounding the two stars. When the two stars are built
on the same polytropic model contact stars with mass ratios different
from unity are not possible.
---------------------------------------------------------
Title: The Interaction of Rotation with Convection
Authors: Durney, B. R.
1970stro.coll...30D Altcode: 1970IAUCo...4...30D
No abstract at ADS
---------------------------------------------------------
Title: Decrease in the Period of Pulsar PSR 0833-45
Authors: Durney, B.
1969Natur.222.1260D Altcode:
A DECREASE of 196 ns in the period of pulsar PSR 0833-45 has recently
been observed<SUP>1,2</SUP>; I suggest that this decrease results
from the addition of mass to the pulsar. A remnant of the supernova
explosion forming the pulsar may not have escaped the gravitational
field and may now have fallen back on the pulsar.
---------------------------------------------------------
Title: Model Atmosphere Calculation of the Solar Oblateness
Authors: Durney, B. R.
1969Natur.221..646D Altcode:
Dicke and Goldenberg<SUP>1</SUP> measured the difference between
the polar and the equatorial flux coming from the limb of the Sun
(ΔF), and inferred that the surface of equal potential at the limb is
oblate by 35 km. The Sun thus has a quadrupole moment due to a rapidly
rotating interior, producing a perihelion shift of Mercury of 3.4 s
of arc century<SUP>-1</SUP>. Agreement between the value predicted
by general relativity and the observed perihelion shift of Mercury
is thus destroyed. One of us<SUP>2</SUP> has criticized the Dicke and
Goldenberg interpretation and has suggested that the flux difference is
due to a stronger stabilization of convection, by rotation, at the pole.
---------------------------------------------------------
Title: Pulsation Periods of Rotating White Dwarfs
Authors: Durney, B. R.; Faulkner, J.; Gribbin, J. R.; Roxburgh, I. W.
1968Natur.219...20D Altcode:
When uniform rotation is included, the periods of pulsation for
white dwarfs can become as small as 0.9 s. With non-uniform rotation,
periods as short as 0.1 s may be possible.
---------------------------------------------------------
Title: Non-Radial Oscillations of Slowly Rotating Polytropes
Authors: Durney, B.; Skumanich, A.
1968ApJ...152..255D Altcode:
The linearized equations for non-radial adiabatic oscillations of slowly
rotating polytropes are studied for both stable and marginally stable
states. For oscillations in the stable state, the eigenfrequencies
are a continuous function of the parameter = (~ - F)/(m~)2, which
measures the ratio of buoyant to gyroscopic forces; here T = 1 + 1/n,
where n is the polytropic index, ~y is the ratio of the specific
heats, w the non-dimensional angular velocity, and m determines
the azimuthal dependence. The structures of these oscillations,
which could be called gravitational gyroscopic waves, are determined
from two coupled first-order partial differential equations. In the
marginally stable state one finds the solutions to be oscillatory,
thus indicating overstability; the parameter yi takes on discrete
negative values which indicate the stabilizing influence of rotation
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Title: Rotating Massive Stars in General Relativity
Authors: Durney, B. R.; Roxburgh, I. W.
1967RSPSA.296..189D Altcode:
Equilibrium models of uniformly rotating massive stars are investigated,
using a weak field, slow rotation approximation, which is shown to be
adequate for all cases of interest. The fate of radial perturbations
about these equilibrium configurations is investigated using a
linearized stability analysis to determine the oscillation frequency
σ in a peturbation propto e<SUP>1σ t</SUP>. An eigenvalue equation
for σ^2 is obtained which can be made self adjoint with respect to
the spatial metric, and a variational principle to determine σ^2 is
derived. Numerical determinations of σ^2 have been carried out for a
variety of masses, radii and rotational velocities, and these results
are incorporated in a simple formula that gives the dependence of σ^2
on these quantities. The condition for instability, σ^2 negative,
is determined, and it is found that for large masses and maximum
rotation velocity, so that when centrifugal force balances gravity at
the surface, a massive star becomes unstable when its radius is 208
times the Schwarzschild radius 2GM/c^2.
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Title: The effect of a toroidal magnetic field on the radial
oscillations of stars
Authors: Roxburgh, I. W.; Durney, B. R.
1967MNRAS.135..329R Altcode:
The internal structure of a polytrope n =3 containing a toroidal
magnetic field is investigated. For static equilibrium configurations
the general solution for the structure of the field is given and
a particular solution Ht rp sin 0 is investigated in detail. The
linearized equations for small radial motion about the equilibrium
configuration are presented and with a time dependence ei these
equations reduce to an eigenvalue equation for 2 A variational principle
for determiing is derived and 2 is estimated using this principle
as well as by direct numerical iteration, for values of the ratio of
specific heats of the gas F = 4/3,413+ , and 5/3. Results are given
for different field strengths. For F =4/3 the star is neutrally stable
whether or not there is a magnetic field, whereas for the other values
of F the magnetic field decreases the value of a as compared to the
non-magnetic values.
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Title: Structure, Oscillations and Stability of Rotating White Dwarfs
Authors: Roxburgh, I. W.; Durney, B. R.
1966ZA.....64..504R Altcode:
No abstract at ADS
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Title: Stability of Rotating Massive Stars in General Relativity
Authors: Durney, B.; Roxburgh, I. W.
1965Natur.208.1304D Altcode:
THE suggestion by Hoyle and Fowler<SUP>1</SUP> that stars with masses
of 10<SUP>6</SUP>-10<SUP>10</SUP> M<SUB>solar</SUB> may provide the
energy for radio sources, and the subsequent discovery of quasars,
has stimulated considerable interest in the structure of very massive
stars<SUP>2</SUP>. Iben<SUP>3</SUP>, using a binding-energy argument,
showed that within the framework of general relativity a spherical
massive star becomes unstable long before it has contracted to
the stage at which nuclear reactions become important. A similar
conclusion was obtained by Chandrasekhar<SUP>4</SUP>, using a detailed
stability analysis on the spherically symmetric relativistic equations
and calculating the relaxation oscillations from a variational
principle. Similar results have been obtained by Fowler<SUP>5</SUP>
using a virial theorem approach.
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Title: Stresses Induced in a Purely Elastic Earth Model under Various
Tectonic Loads
Authors: Durney, B.
1965GeoJ...10..163D Altcode: 1965GeoJI..10..163D
No abstract at ADS
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Title: Distorted Wave Approximation in the Reaction P + PT
Authors: Durney, B. R.
1958RSPSA..71..654D Altcode: 1958RSLPS..71..654D
No abstract at ADS
