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author:"Goedbloed, J.P."
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Title: Advanced Magnetohydrodynamics
Authors: Goedbloed, J. P.; Keppens, Rony; Poedts, Stefaan
2010adma.book.....G Altcode:
Preface; Part III. Flow and Dissipation: 12. Waves and instabilities of
stationary plasmas; 13. Shear flow and rotation; 14. Resistive plasma
dynamics; 15. Computational linear MHD; Part IV. Toroidal Plasmas:
16. Static equilibrium of toroidal plasmas; 17. Linear dynamics of
static toroidal plasmas; 18. Linear dynamics of stationary toroidal
plasmas; Part V. Nonlinear Dynamics: 19. Computational nonlinear MHD;
20. Transonic MHD flows and shocks; 21. Ideal MHD in special relativity;
Appendices; References; Index.
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Title: PHOENIX: MHD spectral code for rotating laboratory and
gravitating astrophysical plasmas
Authors: Blokland, J. W. S.; van der Holst, B.; Keppens, R.; Goedbloed,
J. P.
2007JCoPh.226..509B Altcode:
The new PHOENIX code is discussed together with a sample of many
new results that are obtained concerning magnetohydrodynamic
(MHD) spectra of axisymmetric plasmas where flow and gravity are
consistently taken into account. PHOENIX, developed from the CASTOR code
[W. Kerner, J.P. Goedbloed, G.T.A. Huysmans, S. Poedts, E. Schwarz,
J. Comput. Phys. 142 (1998) 271], incorporates purely toroidal, or both
toroidal and poloidal flow and external gravitational fields to compute
the entire ideal or resistive MHD spectrum for general tokamak or
accretion disk configurations. These equilibria are computed by means of
FINESSE [A.J.C. Beliën, M.A. Botchev, J.P. Goedbloed, B. van der Holst,
R. Keppens, J. Comp. Physics 182 (2002) 91], which discriminates between
the different elliptic flow regimes that may occur. PHOENIX makes use
of a finite element method in combination with a spectral method for
the discretization. This leads to a large generalized eigenvalue
problem, which is solved by means of Jacobi-Davidson algorithm
[G.L.G. Sleijpen, H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17
(1996) 401]. PHOENIX is compared with CASTOR, PEST-1 and ERATO for
an internal mode of Soloviev equilibria. Furthermore, the resistive
internal kink mode has been computed to demonstrate that the code can
accurately handle small values for the resistivity. A new reference
test case for a Soloviev-like equilibrium with toroidal flow shows that,
on a particular unstable mode, the flow has a quantifiable stabilizing
effect regardless of the direction of the flow. PHOENIX reproduces
the Toroidal Flow induced Alfvén Eigenmode (TFAE, [B. van der Holst,
A.J.C. Beliën, J.P. Goedbloed, Phys. Rev. Lett. 84 (2000) 2865]) where
finite resistivity in combination with equilibrium flow effects causes
resonant damping. Localized ideal gap modes are presented for tokamak
plasmas with toroidal and poloidal flow. Finally, we demonstrate the
ability to spectrally diagnose magnetized accretion disk equilibria
where gravity acts together with either purely toroidal flow or both
toroidal and poloidal flow. These cases show that the MHD continua
can be unstable or overstable due to the presence of a gravitational
field together with equilibrium flow-driven dynamics [J.P. Goedbloed,
A.J.C. Beliën, B. van der Holst, R. Keppens, Phys. Plasmas 11
(2004) 28].
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Title: Unstable magnetohydrodynamical continuous spectrum of accretion
disks. A new route to magnetohydrodynamical turbulence in accretion
disks
Authors: Blokland, J. W. S.; Keppens, R.; Goedbloed, J. P.
2007A&A...467...21B Altcode: 2007astro.ph..3581B
Context: We present a detailed study of localised magnetohydrodynamical
(MHD) instabilities occurring in two-dimensional magnetized accretion
disks. <BR />Aims: We model axisymmetric MHD disk tori, and solve
the equations governing a two-dimensional magnetized accretion disk
equilibrium and linear wave modes about this equilibrium. We show
the existence of novel MHD instabilities in these two-dimensional
equilibria which do not occur in an accretion disk in the cylindrical
limit. <BR />Methods: The disk equilibria are numerically computed by
the FINESSE code. The stability of accretion disks is investigated
analytically as well as numerically. We use the PHOENIX code to
compute all the waves and instabilities accessible to the computed
disk equilibrium. <BR />Results: We concentrate on strongly magnetized
disks and sub-Keplerian rotation in a large part of the disk. These disk
equilibria show that the thermal pressure of the disk can only decrease
outwards if there is a strong gravitational potential. Our theoretical
stability analysis shows that convective continuum instabilities
can only appear if the density contours coincide with the poloidal
magnetic flux contours. Our numerical results confirm and complement
this theoretical analysis. Furthermore, these results show that the
influence of gravity can either be stabilizing or destabilizing on
this new kind of MHD instability. In the likely case of a non-constant
density, the height of the disk should exceed a threshold before this
type of instability can play a role. <BR />Conclusions: This localised
MHD instability provides an ideal, linear route to MHD turbulence in
strongly magnetized accretion disk tori.
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Title: Stationary field-aligned MHD flows at astropauses and in
astrotails. Principles of a counterflow configuration between a
stellar wind and its interstellar medium wind
Authors: Nickeler, D. H.; Goedbloed, J. P.; Fahr, H. -J.
2006A&A...454..797N Altcode: 2012arXiv1203.5500N
Context: .A stellar wind passing through the reverse shock is deflected
into the astrospheric tail and leaves the stellar system either
as a sub-Alfvénic or as a super-Alfvénic tail flow. An example
is our own heliosphere and its heliotail.<BR /> Aims: . We present
an analytical method of calculating stationary, incompressible, and
field-aligned plasma flows in the astrotail of a star. We present a
recipe for constructing an astrosphere with the help of only a few
governing parameters, like the inner Alfvén Mach number and the outer
Alfvén Mach number, the magnetic field strength within and outside the
stellar wind cavity, and the distribution of singular points (neutral
points) of the magnetic field within these flows. <BR /> Methods:
. Within the framework of a one-fluid approximation, it is possible
to obtain solutions of the governing MHD equations for stationary
flows from corresponding static MHD equilibria, by using noncanonical
mappings of the canonical variables. The canonical variables are
the Euler potentials of the magnetic field of magnetohydrostatic
equilibria. Thus we start from static equilibria determined by the
distribution of magnetic neutral points, and assume that the Alfvén
Mach number for the corresponding stationary equilibria is finite.<BR
/> Results: .The topological structure, i.e. the distribution of
magnetic neutral points, determines the geometrical structure of
the interstellar gas - stellar wind interface. Additional boundary
conditions like the outer magnetic field and the jump of the magnetic
field across the astropause allow determination of the noncanonical
transformations. This delivers the strength of the magnetic field at
every point in the astrotail/astrosheath region beyond the reverse
shock. <BR /> Conclusions: .The mathematical technique for describing
such a scenario is applied to astrospheres in general, but is also
relevant for the heliosphere. It shows the restrictions of the outer and
the inner magnetic field strength in comparison with the corresponding
Alfvén Mach numbers in the case of subalfvénic flows.
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Title: Magneto-rotational overstability in accretion disks
Authors: Blokland, J. W. S.; van der Swaluw, E.; Keppens, R.;
Goedbloed, J. P.
2005A&A...444..337B Altcode: 2005astro.ph..4381B
We present analytical and numerical studies of magnetorotational
instabilities occuring in magnetized accretion disks. These calculations
are performed for general radially stratified disks in the cylindrical
limit. We elaborate on earlier analytical results and confirm and
expand them with numerical computations of unstable eigenmodes of the
full set of linearised compressible MHD equations. We compare these
solutions with those found from approximate local dispersion equations
from WKB analysis. In particular, we investigate the influence of a
nonvanishing toroidal magnetic field component on the growth rate and
oscillation frequency of magnetorotational instabilities in Keplerian
disks. These calculations are performed for a constant axial magnetic
field strength. We find the persistence of these instabilities in
accretion disks close to equipartition. Our calculations show that
these eigenmodes become overstable (complex eigenvalue), due to the
presence of a toroidal magnetic field component, while their growth
rate reduces slightly. Furthermore, we demonstrate the presence of
magneto-rotational overstabilities in weakly magnetized sub-Keplerian
rotating disks. We show that the growth rate scales with the rotation
frequency of the disk. These eigenmodes also have a nonzero oscillation
frequency, due to the presence of the dominant toroidal magnetic field
component. The overstable character of the MRI increases as the rotation
frequency of the disk decreases.
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Title: Transonic instabilities in accretion disks
Authors: Goedbloed, J. P.; Keppens, R.
2005AIPC..784..639G Altcode:
In two previous publications, we have demonstrated that stationary
rotation of magnetized plasma about a compact central object
permits an enormous number of different MHD instabilities, with the
well-known magneto-rotational instability as just one of them. We
here concentrate on the new instabilities found that are driven by
transonic transitions of the poloidal flow. A particularly promising
class of instabilities, from the point of view of MHD turbulence in
accretion disks, is the class of trans-slow Alfvén continuum modes,
that occur when the poloidal flow exceeds a critical value of the slow
magnetosonic speed. When this happens, virtually every magnetic/flow
surface of the disk becomes unstable with respect to highly localized
modes of the continuous spectrum. The mode structures rotate, in turn,
about the rotating disk. These structures lock and become explosively
unstable when the mass of the central object is increased beyond a
certain critical value. Their growth rates then become huge, of the
order of the Alfvén transit time. These instabilities appear to have
all requisite properties to facilitate accretion flows across magnetic
surfaces and jet formation.
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Title: Response to “Comment on `Variational principles for
stationary one- and two-fluid equilibria of axisymmetric laboratory
and astrophysical plasmas' ” [Phys. Plasmas 12, 064701 (2005)]
Authors: Goedbloed, J. P.
2005PhPl...12f4702G Altcode:
Contrary to the Comment by McClements and Thyagaraja that the two-fluid
equations for stationary axisymmetric equilibria are easier to deal
with numerically than the corresponding ideal magnetohydrodynamics
(MHD) equations, since they resolve the Alfvén singularity of
the latter whereas transonic transitions do not create substantial
numerical difficulties, the opposite proposition is maintained. The
numerical solution of the single (MHD) or two (two-fluid) Bernoulli
equations already eliminates the Alfvén singularity, but it presents
major complications (such as the possible nonexistence, multiplicity,
and hyperbolicity of the solutions) in the construction of stationary
equilibria that are accurate enough to permit spectral analysis of
the waves and instabilities of those equilibria. Furthermore, it is
shown that imposing charge neutrality on the two-fluid equations not
only obscures the solution procedure of the two independent Bernoulli
equations but also eliminates the possibility of a self-consistent
description of the charge imbalances that occur in rotating and
gravitating astrophysical plasmas, such as Goldreich-Julian charges
in magnetospheres of pulsars and massive black holes.
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Title: Variational principles for stationary one- and two-fluid
equilibria of axisymmetric laboratory and astrophysical plasmas
Authors: Goedbloed, J. P.
2004PhPl...11L..81G Altcode:
It is shown that the core equations of both the magnetohydrodynamics
and the two-fluid description of stationary axisymmetric equilibrium
flows may be derived from variational principles in terms of the
core variables of the respective descriptions. The latter replace
the primitive variables because of the stream function constraints
associated with axisymmetry. This yields a concise representation of
stationary flows in tokamaks, accretion disks, and jets, and permits
accurate numerical implementation. Since hyperbolic flows occur in
both descriptions, the limitation of the variational principles to
elliptic flow regimes presents an intricate problem.
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Title: Principles of Magnetohydrodynamics
Authors: Goedbloed, J. P. Hans; Poedts, Stefaan
2004prma.book.....G Altcode:
Part I. Plasma Physics Preliminaries: 1. Introduction; 2. Elements
of plasma physics; 3. 'Derivation' of the macroscopic equations;
Part II. Basic Magnetohydrodynamics: 4. The MHD model; 5. Waves and
characteristics; 6. Spectral theory; 7. Waves and instabilities
on inhomogeneous plasmas; 8. Magnetic structures and dynamics;
9. Cylindrical plasmas; 10. Initial value problem and wave damping;
11. Resonant absorption and wave heating; Appendices; References; Index.
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Title: Transsonic instabilities in tokamaks and astrophysical
accretion flows
Authors: Goedbloed, J. P. (Hans); Beliën, A. J. C.; van der Holst,
B.; Keppens, R.
2004AIPC..703...42G Altcode:
Waves and instabilities of transonically rotating toroidal
plasmas present a very complex problem of interest for the two
unrelated fields of magnetically-dominated laboratory plasmas and
gravitationally-dominated astrophysical plasmas. The complexity
originates from the transonic transitions of the poloidal flow which
causes the character of the rotating equilibrium states to change
dramatically, from elliptic to hyperbolic or vice versa, when the
poloidal velocity surpasses certain critical speeds. Associated with
these transitions the different types of magnetohydrodynamic (MHD)
shocks may appear. Obviously, at such transitions the possible waves
and instabilities of the system also change dramatically. We have
investigated these changes for the two mentioned physical systems,
starting from the point of view that the continuous spectrum of
magnetohydrodynamics presents the best organizing principle for the
structure of the complete spectrum since it is the most robust part of
it. We found a new class of local MHD instabilities, that we called
trans-slow Alfvén continuum modes, which are due to poloidal flows
exceeding the critical slow magnetosonic speed. They operate both in
laboratory plasmas (tokamaks), in the absence of gravitational effects,
and in astrophysical plasmas (accretion tori), when the gravitational
field of a compact object dominates the flow. They become extremely
violent when the mass of the central object is large, providing a new
route to MHD turbulence in plasmas rotating about a massive central
object.
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Title: Transsonic Instabilities in Laboratory and Astrophysical
Plasmas
Authors: Goedbloed, J. P. >
2004PhST..107..159G Altcode:
Waves and instabilities of transsonically rotating axisymmetric plasmas
present a highly complex problem that is of interest for two unrelated
fields of research, viz. laboratory tokamak confinement for the eventual
thermonuclear energy production and the dynamics of a vast number of
astrophysical plasmas rotating about compact objects, broadly indicated
as accretion disks. The complexity originates from the transsonic
transitions of the poloidal flow which causes the character of the
rotating equilibrium states to change dramatically, from elliptic to
hyperbolic or vice versa, when the poloidal velocity surpasses certain
critical speeds. Associated with these transitions the different
types of magnetohydrodynamic (MHD) shocks may appear. Obviously, at
such transitions the possible waves and instabilities of the system
also change dramatically. We investigate these changes for the two
mentioned classes of physical systems, starting from the point of
view that the continuous spectrum of magnetohydrodynamics presents the
best organizing principle for the structure of the complete spectrum
of waves and instabilities since it is the most robust part of it, for
that reason called the “essential spectrum” by mathematicians. The
physical importance of the problem is that it provides the simplest
approach to local waves and instabilities of the system and, possibly,
to the onset of MHD turbulence in accretion disks.
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Title: Three-dimensional magnetohydrodynamic simulations of in situ
shock formation in the coronal streamer belt
Authors: Zaliznyak, Yu.; Keppens, R.; Goedbloed, J. P.
2003PhPl...10.4478Z Altcode: 2004astro.ph..3122Z
A numerical study of an idealized magnetohydrodynamic (MHD)
configuration consisting of a planar wake flow embedded into a
three-dimensional (3D) sheared magnetic field is presented. The
simulations investigate the possibility for in situ development
of large-scale compressive disturbances at cospatial current
sheet-velocity shear regions in the heliosphere. Using a linear
MHD solver, the systematical investigation of the destabilized
wavenumbers, corresponding growth rates, and physical parameter
ranges for dominant 3D sinuous-type instabilities in an equilibrium
wake-current sheet system was done. Wakes bounded by sufficiently
supersonic (Mach number M<SUB>s</SUB>>2.6) flow streams are found
to support dominant fully 3D sinuous instabilities when the plasma
beta is of order unity. Fully nonlinear, compressible 2.5D and 3D
MHD simulations show the self-consistent formation of shock fronts
of fast magnetosonic type. They carry density perturbations far away
from the wake's center. Shock formation conditions are identified
in sonic and Alfvénic Mach number parameter space. Depending on the
wake velocity contrast and magnetic field magnitude, as well as on the
initial perturbation, the emerging shock patterns can be plane-parallel
as well as fully three-dimensionally structured. Similar large-scale
transients could therefore originate at distances far above coronal
helmet streamers or at the location of the ecliptic current sheet.
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Title: Adaptive Mesh Refinement for conservative systems:
multi-dimensional efficiency evaluation
Authors: Keppens, R.; Nool, M.; Tóth, G.; Goedbloed, J. P.
2003CoPhC.153..317K Altcode: 2004astro.ph..3124K
Obtainable computational efficiency is evaluated when using an Adaptive
Mesh Refinement (AMR) strategy in time accurate simulations governed
by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro-
and magnetohydrodynamic simulations, AMR is used in combination with
several shock-capturing, conservative discretization schemes. Solution
accuracy and execution times are compared with static grid simulations
at the corresponding high resolution and time spent on AMR overhead
is reported. Our examples reach corresponding efficiencies of 5
to 20 in multi-dimensional calculations and only 1.5-8% overhead is
observed. For AMR calculations of multi-dimensional magnetohydrodynamic
problems, several strategies for controlling the ∇.B=0 constraint
are examined. Three source term approaches suitable for cell-centered B
representations are shown to be effective. For 2D and 3D calculations
where a transition to a more globally turbulent state takes place, it
is advocated to use an approximate Riemann solver based discretization
at the highest allowed level(s), in combination with the robust
Total Variation Diminishing Lax-Friedrichs method on the coarser
levels. This level-dependent use of the spatial discretization acts
as a computationally efficient, hybrid scheme.
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Title: Analogies of Rapidly Rotating Tokamaks and Accretion Disks
Authors: Goedbloed, J. P.; Belien, A. J. C.; van der Holst, B.
2003AIPC..669..642G Altcode:
Equilibrium, waves, and instabilities of tokamaks and accretion disks
that are rotating with arbitrary transonic velocities have been solved
by means of advanced numerical and analytical techniques. The different
transonic flow regimes yield a surprisingly large number of new MHD
waves and instabilities that (1) are relevant for turbulent processes in
accretion disks, (2) provide a clear correspondence between tokamaks and
accretion disk dynamics, with different influence of rotation profiles,
gravity, and magnetic pressure, (3) provide a new angle on rapid
transition phenomena in transonic MHD flows of rotating astrophysical
plasmas. The new angle entails a complete revision of all previously
obtained spectral results. The reason is that transonic flows upset
the standard theoretical approach to plasma dynamics, consisting of
a separate study of the equilibrium state and of the perturbations
of this background. We will discuss a new approach to this dichotomy
consisting of a study of the similarities of the nonlinear stationary
flow patterns and the different linear wave structures that occur when
the background speed traverses the full range of critical speeds (from
`slow magnetosonic' to `Alfvén' to `fast magnetosonic'). This has
required the development of new computational tools that yield the
mentioned plethora of new waves and instabilities.
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Title: Computer simulations of solar plasmas
Authors: Goedbloed, J. P.; Keppens, R.; Poedts, S.
2003SSRv..107...63G Altcode:
Plasma dynamics has been investigated intensively for toroidal
magnetic confinement in tokamaks with the aim to develop a controlled
thermonuclear energy source. On the other hand, it is known that
more than 90% of visible matter in the universe consists of plasma,
so that the discipline of plasma-astrophysics has an enormous
scope. Magnetohydrodynamics (MHD) provides a common theoretical
description of these two research areas where the hugely different
scales do not play a role. It describes the interaction of electrically
conducting fluids with magnetic fields that are, in turn, produced by
the dynamics of the plasma itself. Since this theory is scale invariant
with respect to lengths, times, and magnetic field strengths, for
the nonlinear dynamics it makes no difference whether tokamaks, solar
coronal magnetic loops, magnetospheres of neutron stars, or galactic
plasmas are described. Important is the magnetic geometry determined
by the magnetic field lines lying on magnetic surfaces where also the
flows are concentrated. Yet, transfer of methods and results obtained
in tokamak research to solar coronal plasma dynamics immediately
runs into severe problems with trans‘sonic’ (surpassing any one
of the three critical MHD speeds) stationary flows. For those flows,
the standard paradigm for the analysis of waves and instabilities,
viz. a split of the dynamics in equilibrium and perturbations, appears
to break down. This problem is resolved by a detailed analysis of the
singularities and discontinuities that appear in the trans‘sonic’
transitions, resulting in a unique characterization of the permissible
flow regimes. It then becomes possible to initiate MHD spectroscopy of
axi-symmetric transonic astrophysical plasmas, like accretion disks or
solar magnetic loops, by computing the complete wave and instability
spectra by means of the same methods (with unprecedented accuracy)
exploited for tokamak plasmas. These large-scale linear programs are
executed in tandem with the non-linear (shock-capturing, massively
parallel) Versatile Advection Code to describe both the linear and
the nonlinear phases of the instabilities.
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Title: Stability and waves of transonic laboratory and space plasmas
Authors: Goedbloed, J. P.
2003SSRv..107..353G Altcode:
The properties of magnetohydrodynamic waves and instabilities
of laboratory and space plasmas are determined by the overall
magnetic confinement geometry and by the detailed distributions of
the density, pressure, magnetic field, and background velocity of
the plasma. Consequently, measurement of the spectrum of MHD waves
(MHD spectroscopy) gives direct information on the internal state of
the plasma, provided a theoretical model is available to solve the
forward as well as the inverse spectral problems. This terminology
entails a program, viz. to improve the accuracy of our knowledge of
plasmas, both in the laboratory and in space. Here, helioseismology
(which could be considered as one of the forms of MHD spectroscopy) may
serve as a luminous example. The required study of magnetohydrodynamic
waves and instabilities of both laboratory and space plasmas has
been conducted for many years starting from the assumption of static
equilibrium. Recently, there is a outburst of interest for plasma states
where this assumption is violated. In fusion research, this interest
is due to the importance of neutral beam heating and pumped divertor
action for the extraction of heat and exhaust needed in future tokamak
reactors. Both result in rotation of the plasma with speeds that do not
permit the assumption of static equilibrium anymore. In astrophysics,
observations in the full range of electromagnetic radiation has revealed
the primary importance of plasma flows in such diverse situations
as coronal flux tubes, stellar winds, rotating accretion disks,
and jets emitted from radio galaxies. These flows have speeds which
substantially influence the background stationary equilibrium state,
if such a state exists at all. Consequently, it is important to study
both the stationary states of magnetized plasmas with flow and the waves
and instabilities they exhibit. We will present new results along these
lines, extending from the discovery of gaps in the continuous spectrum
and low-frequency Alfvén waves driven by rotation to the nonlinear
flow patterns that occur when the background speed traverses the full
range from sub-slow to super-fast.
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Title: Waves and Instabilities in Accretion Disks: Magnetohydrodynamic
Spectroscopic Analysis
Authors: Keppens, R.; Casse, F.; Goedbloed, J. P.
2002ApJ...569L.121K Altcode: 2002astro.ph..3237K
A complete analytical and numerical treatment of all magnetohydrodynamic
waves and instabilities for radially stratified, magnetized accretion
disks is presented. The instabilities are a possible source of
anomalous transport. While recovering results on known hydrodynamic
and both weak- and strong-field magnetohydrodynamic perturbations,
the full magnetohydrodynamic spectra for a realistic accretion
disk model demonstrate a much richer variety of instabilities
accessible to the plasma than previously realized. We show that both
weakly and strongly magnetized accretion disks are prone to strong
nonaxisymmetric instabilities. The ability to characterize all waves
arising in accretion disks holds great promise for magnetohydrodynamic
spectroscopic analysis.
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Title: Transonic Magnetohydrodynamic Flows in Laboratory and
Astrophysical Plasmas
Authors: Goedbloed, J. P.
2001PhST...98...43G Altcode:
Magnetohydrodynamic (MHD) waves control the dynamics of plasma, the
main constituent of the universe. They occur as the natural response
to global excitation. Frequency and wave forms are determined by
the magnetic confinement geometry and distribution of background
equilibrium variables. Hence, measurement of the spectrum of MHD waves
gives direct information on the internal state of the plasma, provided
a theoretical model is available to solve the forward and inverse
spectral problems. This activity has been called MHD spectroscopy,
[1, Goedbloed et al. Phys. Control. Fusion 35 B277 (1993)] in analogy
with quantum mechanical spectroscopy which also involves eigenvalue
problems of linear operators. The terminology also entails a program,
viz. to improve the accuracy of our knowledge of plasmas, both in the
laboratory and in astrophysics.
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Title: Magnetohydrodynamic waves in laboratory and astrophysical
plasmas
Authors: Goedbloed, J. P.
2000AIPC..537..109G Altcode: 2000wdss.conf..109G
The study of magnetohydrodynamic waves and instabilities of both
laboratory and astrophysical plasmas has been conducted for many years
starting from the assumption of static equilibrium. Recently, there
is an outburst of interest for plasma states where this assumption is
violated. In fusion research, this interest is due to the importance
of neutral beam heating and pumped divertor action for the extraction
of heat and exhaust needed in future tokamak reactors. Both result in
rotation of the plasma with speeds that do not permit the assumption of
static equilibrium anymore. In astrophysics, observations in the full
range of electromagnetic radiation has revealed the primary importance
of plasma flows in such diverse situations as coronal flux tubes,
stellar winds, rotating accretion disks, and jets emitted from radio
galaxies. These flows have speeds which substantially influence the
background stationary equilibrium state, if such a state exists at
all. Consequently, it is important to study both the stationary states
of magnetized plasmas with flow and the waves and instabilities they
exhibit. We will present new results along these lines, extending from
the discovery of gaps in the continuous spectrum and low-frequency
Alfvén waves driven by rotation to the nonlinear flow patterns that
occur when the background speed traverses the full range from sub-slow
to super-fast. The solutions obtained may bridge the gap between
insights from linear and nonlinear analyses. .
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Title: Stellar Winds, Dead Zones, and Coronal Mass Ejections
Authors: Keppens, R.; Goedbloed, J. P.
2000ApJ...530.1036K Altcode: 1999astro.ph.10152K
Axisymmetric stellar wind solutions are presented that were
obtained by numerically solving the ideal magnetohydrodynamic (MHD)
equations. Stationary solutions are critically analyzed using
the knowledge of the flux functions. These flux functions enter
in the general variational principle governing all axisymmetric
stationary ideal MHD equilibria. The magnetized wind solutions for
(differentially) rotating stars contain both a “wind” and a “dead”
zone. We illustrate the influence of the magnetic field topology on the
wind acceleration pattern by varying the coronal field strength and the
extent of the dead zone. This is evident from the resulting variations
in the location and appearance of the critical curves for which the wind
speed equals the slow, Alfvén, and fast speed. Larger dead zones cause
effective, fairly isotropic acceleration to super-Alfvénic velocities
as the polar, open field lines are forced to fan out rapidly with
radial distance. A higher field strength moves the Alfvén transition
outward. In the ecliptic, the wind outflow is clearly modulated by
the extent of the dead zone. The combined effect of a fast stellar
rotation and an equatorial dead zone in a bipolar field configuration
can lead to efficient thermocentrifugal equatorial winds. Such winds
show both a strong poleward collimation and some equatorward streamline
bending due to significant toroidal field pressure at midlatitudes. We
discuss how coronal mass ejections are then simulated on top of the
transonic outflows.
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Title: Stationary and Time-Dependent MHD Simulations of the Solar Wind
Authors: Keppens, R.; Goedbloed, J. P.
1999ESASP.448.1177K Altcode: 1999ESPM....9.1177K; 1999mfsp.conf.1177K
No abstract at ADS
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Title: Numerical simulations of stellar winds: polytropic models
Authors: Keppens, R.; Goedbloed, J. P.
1999A&A...343..251K Altcode: 1999astro.ph..1380K
We discuss steady-state transonic outflows obtained by direct numerical
solution of the hydrodynamic and magnetohydrodynamic equations. We make
use of the Versatile Advection Code, a software package for solving
systems of (hyperbolic) partial differential equations. We proceed
stepwise from a spherically symmetric, isothermal, unmagnetized,
non-rotating Parker wind to arrive at axisymmetric, polytropic,
magnetized, rotating models. These represent 2D generalisations of
the analytical 1D Weber-Davis wind solution, which we obtain in the
process. Axisymmetric wind solutions containing both a `wind' and a
`dead' zone are presented. Since we are solving for steady-state
solutions, we efficiently exploit fully implicit time stepping. The
method allows us to model thermally and/or magneto-centrifugally driven
stellar outflows. We particularly emphasize the boundary conditions
imposed at the stellar surface. For these axisymmetric, steady-state
solutions, we can use the knowledge of the flux functions to verify
the physical correctness of the numerical solutions.
---------------------------------------------------------
Title: Growth and saturation of the Kelvin-Helmholtz instability
with parallel and antiparallel magnetic fields
Authors: Keppens, Rony; Tóth, G.; Westermann, R. H. J.; Goedbloed,
J. P.
1999JPlPh..61....1K Altcode: 1999astro.ph..1166K
Available from <A
href="http://journals.cambridge.org/bin/bladerunner?REQUNIQ=1105385252&REQSESS=958582&118000REQEVENT=&REQINT1=18471&REQAUTH=0">http://journals.cambridge.org/bin/bladerunner?REQUNIQ=1105385252&REQSESS=958582&118000REQEVENT=&REQINT1=18471&REQAUTH=0</A>
---------------------------------------------------------
Title: Numerical Simulations of Stellar Winds
Authors: Keppens, R.; Goedbloed, J. P.
1999SSRv...87..223K Altcode:
We discuss steady-state transonic outflows obtained by direct numerical
solution of the hydrodynamic and magnetohydrodynamic equations. We
make use of the Versatile Advection Code, a software package for
solving systems of (hyperbolic) partial differential equations. We
model thermally and magneto-centrifugally driven stellar outflows
as generalizations of the well-known Parker and Weber-Davis wind
solutions. To obtain steady-state solutions efficiently, we exploit
fully implicit time stepping.
---------------------------------------------------------
Title: Two-dimensional equilibrium in coronal magnetostatic flux
tubes: an accurate equilibrium solver
Authors: Beliën, A. J. C.; Poedts, S.; Goedbloed, J. P.
1997CoPhC.106...21B Altcode:
To study linearized magnetohydrodynamic (MHD) waves, continuous spectra,
and instabilities in coronal magnetic flux tubes that are anchored
in dense chromospheric and photospheric regions, a two-dimensional
numerical code, called PARIS, has been developed. PARIS solves the
pertinent nonlinear Grad-Shafranov type, partial differential equation
for the magnetic flux on a flux coordinate grid. Both a straight field
line coordinate system and an orthogonal flux coordinate system are
exploited. Isoparametric bicubic Hermite finite elements have been
adopted to solve the Grad-Shafranov-like equation. These elements
allow for a continuous representation of the flux and the gradient
of the flux throughout the tube and can be aligned conveniently along
the boundary of the tube. These properties are important to obtain an
accurate representation of the solution on flux coordinate grids. An
analytical test case is used to show that accurate solutions have
been obtained, even for a small number of grid points. The equilibria
calculated by PARIS are used to study the continuous spectra of
two-dimensional magnetic flux tubes. One illustrative example is
given here; extensive results are presented elsewhere (A.J.C. Beliën,
S. Poedts and J.P. Goedbloed, Astron. Astrophys. 322 (1997) 995). The
equilibria obtained by PARIS are also well suited to calculate the
stability and the normal mode MHD spectrum.
---------------------------------------------------------
Title: Continuous magnetohydrodynamic spectra of two-dimensional
coronal magnetostatic flux tubes.
Authors: Belieen, A. J. C.; Poedts, S.; Goedbloed, J. P.
1997A&A...322..995B Altcode:
In this paper we derive the equations for the continuous ideal
magnetohydrodynamic (MHD) spectrum of two-dimensional coronal
loops, including gravity and expansion, in general curvilinear
coordinates. The equations clearly show the coupling between Alfven
and slow magnetosonic continuum waves when both pressure and geodesic
curvature of the magnetic field lines are present. Gravity always
gives rise to Alfven-slow mode coupling when the magnetic field is
twisted. Numerical calculations show that the coupling of Alfven
and slow magnetosonic continuum waves can be strong, especially for
Alfven-like continuum waves, when the magnetic flux concentration
near the bases of flux tubes is taken into account. Amplitude ratios
of the parallel and perpendicular displacement components of 0.4 were
obtained for concentration of the flux with a factor of 4. Gravity has
less effect on the coupling of Alfven and slow magnetosonic continuum
waves than the concentration of flux but it has a large influence on
the low frequency slow magnetosonic-like continuum branches.
---------------------------------------------------------
Title: Nonlinear MHD Simulations of Wave Dissipation in Flux Tubes
Authors: Poedts, S.; Tóth, G.; Beliën, A. J. C.; Goedbloed, J. P.
1997SoPh..172...45P Altcode: 1997ESPM....8...45P
The phase mixing and resonant dissipation of Alfvén waves is studied in
both the 'closed' magnetic loops and the 'open' coronal holes observed
in the hot solar corona. The resulting energy transfer from large
to small length scales contributes to the heating of these magnetic
structures. The nonlinear simulations show that the periodically varying
shear flows that occur in the resonant layers are unstable. In coronal
holes, the phase mixing of running Alfvén waves is speeded up by the
'flaring out' of the magnetic field lines in the lower chromosphere.
---------------------------------------------------------
Title: Nonlinear wave heating of solar coronal loops.
Authors: Poedts, S.; Goedbloed, J. P.
1997A&A...321..935P Altcode:
The heating of magnetically closed structures (loops) in the solar
corona by the resonant absorption of incident waves is studied
by means of numerical simulations in the framework of nonlinear
resistive magnetohydrodynamics (MHD). It is shown that the dynamics
in the resonant layer is indeed very nonlinear for typical coronal
parameters. The effect of the nonlinearity on the efficiency of the
resonant heating mechanism is investigated. It turns out that this
heating mechanism may be less efficient than concluded from the linear
MHD studies. As a matter of fact, the modification of the background
magnetic field results in a shift of the resonance positions in time
which in turn yields broader dissipation layers.
---------------------------------------------------------
Title: Critical Issues in Transonic Magnetohydrodynamic Flows
Authors: Goedbloed, J. P.; Lifschitz, A. E.
1997ESASP.404..417G Altcode: 1997cswn.conf..417G
No abstract at ADS
---------------------------------------------------------
Title: Slow Magnetosonic Waves and Instabilities in Expanded Flux
Tubes Anchored in Chromospheric/Photospheric Regions
Authors: Beliën, A. J. C.; Poedts, S.; Goedbloed, J. P.
1997ESASP.404..193B Altcode: 1997cswn.conf..193B
No abstract at ADS
---------------------------------------------------------
Title: Visualization of resonant absorption in solar coronal loops
by simulation of soft X-ray images.
Authors: Belien, A. J. C.; Poedts, S.; Spoelder, H. J. W.; Leenders,
R.; Goedbloed, J. P.
1996ComPh..10..573B Altcode: 1996CoPhy..10..573B
One of the proposed mechanisms to explain the heating of the solar
corona is resonant absorption of magnetic Alfven waves. Numerical
studies of this mechanism often involve large scale computations
and produce large amounts of data that need to be visualized. In
this article the authors present a method to visualize numerically
calculated density and temperature evolutions of heating calculations
by simulating the soft X-ray observations of the soft X-ray telescope
aboard the Yohkoh satellite. The visualization method is applied to
two different model calculations of the heating of coronal magnetic
loops by the resonant absorption of Alfven waves. For these two
cases, information on the variations of temperature and density can be
extracted conveniently from the generated images. The resulting images
reveal features that are characteristic of the resonant absorption
process. This suggests that signatures of resonant absorption can be
extracted from real soft X-ray observations of coronal loops.
---------------------------------------------------------
Title: Calculation of Soft X-ray Images from MHD Simulations of
Heating of Coronal Loops
Authors: Belien, A. J. C.; Poedts, S.; Goedbloed, J. P.
1996mpsa.conf..423B Altcode: 1996IAUCo.153..423B
No abstract at ADS
---------------------------------------------------------
Title: Symmetry of Magnetohydrodynamic Flows
Authors: Goedbloed, J. P.; Lifschitz, A.
1996ApL&C..34..261G Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Magnetohydrodynamic Continua and Stratification Induced
Alfvén Eigenmodes in Coronal Magnetic Loops
Authors: Beliën, A. J. C.; Poedts, S.; Goedbloed, J. P.
1996PhRvL..76..567B Altcode:
The continuous spectra of a 2D inhomogeneous, cylindrical magnetic flux
tube are studied and applied to solar coronal loops. The density is
stratified radially as well as longitudinally, while other equilibrium
quantities only vary in the radial direction. Stratification causes
gaps to appear in the continuous spectrum, and it is shown that
discrete global, stratification-induced Alfvén eigenmodes occur in
these gaps. These global modes may be important for the heating of
coronal loops.
---------------------------------------------------------
Title: 2D and 3D Nonlinear MHD Simulations of Coronal Loop Heating
by Alfven Waves
Authors: Poedts, S.; Goedbloed, J. P.
1996mpsa.conf..425P Altcode: 1996IAUCo.153..425P
No abstract at ADS
---------------------------------------------------------
Title: Alfven wave heating of coronal loops: photospheric excitation.
Authors: Halberstadt, G.; Goedbloed, J. P.
1995A&A...301..559H Altcode:
Alfven resonant heating of bounded coronal loops is investigated
numerically in a two-dimensional model. A coronal loop is modelled as
a cylindrical, magnetized plasma column, that is bounded by the high
density photospheric plasma. Alfven wave excitation is assumed to be
due to the convective motion of the photosphere. The excitation of
coronal loops at the foot points by this photospheric motion, which
is not incorporated in most existing models, is the main topic in the
work described here. The dynamics of the heating of coronal loops due
to the excited Alfven waves is treated by numerically solving the fully
resistive linearized equations of magnetohydrodynamics for the described
model. In previous work (Goedbloed & Halberstadt 1994), it was shown
that pure Alfven and pure fast magnetosonic waves no longer exist in
bounded loops. In the present paper the general role of the fast wave
component for Alfven heating is investigated. In particular, it is
shown that the coupling of Alfven and fast waves gives rise to a new
kind of eigenmodes, that consist of a global fast wave contribution and
a localized and damped Alfven tail. These waves are efficiently excited
by compressional fluid motion, and yield high Ohmic dissipation rates.
---------------------------------------------------------
Title: Alfven heating of line-tied coronal loops. Surface excitation
revisited.
Authors: Halberstadt, G.; Goedbloed, J. P.
1995A&A...301..577H Altcode:
Alfven resonant heating of closed coronal loops is investigated by
linear magnetohydrodynamics simulations. The main subject of the
presented work is the excitation of waves by the motion of the high
density photospheric plasma. In recent two-dimensional calculations,
the photospheric energy was introduced directly at the foot points
of the loop (Strauss & Lawson 1989; Halberstadt 1994a). This
enables the direct excitation of Alfven waves by purely incompressible
motion, and a high rate of energy transfer from the photosphere to
the dissipative layers is assured. In this paper it is emphasized
that a realistic model of loop excitation by convective motion mainly
involves plasma compression and therefore the introduced photospheric
energy must be transferred across the magnetic field lines to reach
the resonant layers. This is simulated by an excitation source which
is localized towards the end points of the loop and which has an energy
flux in the cross field line direction only. Both the dissipation rate,
and the coupling between the source and the loop are investigated. The
presence of global modes in the Alfven spectrum simultaneously enhances
the coupling and the dissipation rate. An estimate of the attainable
X-ray flux is made.
---------------------------------------------------------
Title: Magnetohydrodynamic waves in fusion and astrophysical plasmas.
Authors: Goedbloed, J. P.
1995AIPC..345..465G Altcode:
Macroscopic plasma dynamics in both controlled thermonuclear confinement
machines and in the atmospheres of X-ray emitting stars is described
by the equations of magnetohydrodynamics. This provides a vast area of
overlapping research activities which is presently actively pursued. In
this lecture the author concentrates on some important differences
in the dynamics of the two confined plasma systems related to the
very different geometries that are encountered and, thus, the role of
the different boundary conditions that have to be posed. As a result,
the basic MHD waves in a tokamak are quite different from those found
in a solar magnetic flux tube. The result is that, whereas the three
well-known MHD waves can be traced stepwise in the curved geometry of a
tokamak, their separate existence is eliminated right from the start in
a line-tied coronal loop because line-tying in general conflicts with
the phase relationships between the vector components of the three
velocity fields. The consequences are far-reaching, viz. completely
different resonant frequencies and continuous spectra, absence of
rational magnetic surfaces, and irrelevance of local marginal stability
theory for coronal magnetic loops.
---------------------------------------------------------
Title: The Influence of Line-Tying on Coronal Perturbations in a
Gravitationally Stratified Equilibrium
Authors: van der Linden, R. A. M.; Hood, A. W.; Goedbloed, J. P.
1994SoPh..154...69V Altcode:
We study the influence of gravitational stratification of the
solar atmosphere on the stability of coronal magnetic structures. In
particular we question whether the (presumably stabilizing) influence
of the anchoring of the magnetic field lines in the solar photosphere
(`line-tying') can be adequately modelled by either `rigid wall'
or `flow-through' boundary conditions on the coronal perturbations,
as is commonly done. Using the ideal MHD model without gravitational
effects,inertial line-tying alone cannot lead to afull stabilization,
as marginal stability cannot be crossed by including only the rapid
density increase at the photospheric interface.
---------------------------------------------------------
Title: Magnetohydrodynamic waves in coronal flux tubes
Authors: Goedbloed, J. P.; Halberstadt, G.
1994A&A...286..275G Altcode:
The problem of the basic MHD waves of a coronal flux loop is
investigated for the simplest configuration conceivable, viz. a plasma
in a rectangular box, with an oblique magnetic field, and line-tied at
the ends. The basic waves found are completely different from those
found in a periodic box, representative for tokamak plasmas. They
consist of an interwoven structure of Alfven and slow components with
a ballooning factor, favouring minimal field line bending, and fast
components without such a factor. The implications for stability are
the existence of a global excess of stability in coronal loops, as
opposed to local marginal stability at rational magnetic surfaces in
tokamaks. The relationship to flares is pointed out. Pure Alfven and
pure slow modes are only found as singular limiting cases of cluster
spectra of Alfven-fast or slow-fast waves, where the fast components
are localised in a photospheric boundary layer which is dictated by the
requirements of line-tying. This justifies the assumption of continuous
spectra in coronal loops, required for the mechanism of resonant Alfven
wave heating. The waves consist of large amplitude Alfven components
in the corona and fast components with a small but rapidly varying
amplitude in the photospheric boundary layer, so that they appear to
have all the right characteristics for effective transfer of energy
from the photosphere to the corona.
---------------------------------------------------------
Title: On the Quality of Resonant Absorption as a Coronal Loop
Heating Mechanism
Authors: Poedts, S.; Belien, A. J. C.; Goedbloed, J. P.
1994SoPh..151..271P Altcode:
The qualityQ of a resonance is defined as the ratio of the total energy
contained in the system to the dissipation per driving cycle. Hence,
a `good quality' resonance is one with little losses, i.e., little
dissipation per driving cycle. However, for heating coronal plasmas by
means of resonant absorption of waves, `bad' quality resonances are
required. Here, the quality of the MHD resonances that occur when an
inhomogeneous coronal loop is excited by incident waves is investigated
for typical coronal loop parameter values in the frame work of linear,
resistive MHD. It is shown that the resonances in coronal loops have
bad quality and, hence, yield a lot of Ohmic heating per driving cycle
compared to the total energy stored in the loop. As a consequence, the
time scales of the heating process are relatively short and resonant
absorption turns out to be a viable candidate for the heating of the
magnetic loops observed in the solar corona.
---------------------------------------------------------
Title: 3D nonlinear wave heating of coronal loops
Authors: Poedts, S.; Goedbloed, J. P.
1994SSRv...68..103P Altcode:
The heating of solar coronal loops by the resonant absorption or
phase-mixing of incident wave energy is investigated in the framework
of 3D nonlinear magnetohydrodynamics (MHD) by means of numerical
simulations.
---------------------------------------------------------
Title: MHD waves in coronal flux tubes
Authors: Goedbloed, J. P.; Halberstadt, G.
1994SSRv...68..121G Altcode:
The basic MHD waves of a coronal flux loop are investigated for the
rectangular box model of a plasma with oblique magnetic field and
line-tied at the ends. The waves found are completely different
from those in a periodic box, representative for tokamaks. They
consist of a mixture of Alfvén components with a ballooning factor,
favouring minimal field line bending, and fast components without such
a factor. Pure Alfvén modes are only found as singular limiting cases
of cluster spectra of Alfvén-fast waves, where the fast components
are localised in a photospheric boundary layer which is dictated
by the requirements of line-tying. This justifies the assumption of
continuous spectra in coronal loops, required for the mechanism of
resonant Alfvén wave heating. The waves consist of large amplitude
Alfvén components in the corona and fast components with a small but
rapidly varying amplitude in the boundary layer, so that they appear
to have the right signature for effective transfer of energy from the
photosphere to the corona.
---------------------------------------------------------
Title: Nonlinear wave heating of the solar corona
Authors: Poedts, S.; Goedbloed, J. P.
1994smf..conf..396P Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The continuous ALfven spectrum of line-tied coronal loops
Authors: Halberstadt, G.; Goedbloed, J. P.
1993A&A...280..647H Altcode:
The effect of the photospheric boundary conditions on Alfven
continuum waves in coronal magnetic loops is considered. A coronal
loop is modeled as effectively line-tied to the photospheric plasma,
such that the foot points of the loop are forced to follow the
photospheric velocity perturbations. Starting from recent work on the
nature of magnetohydrodynamic (MHD) waves in line-tied magnetic loops
(Goedbloed & Halberstadt 1993), we derive an expression for the
Alfven continuum frequencies in a line-tied cylindrical plasma, which
reveals that the line-tied Alfven continuum no longer depends on the
poloidal magnetic field and that the corresponding eigenmodes have a
global ballooning character. Subsequently, we derive a variational
principle by which the Alfven and slow continuum frequencies in a
line-tied cylinder with density variation along the field lines can be
obtained. It is shown that the line-tied Alfven continuum determines
the coronal heating due to resonant absorption in coronal loops that
are excited at the foot points.
---------------------------------------------------------
Title: Resonant Heating of Line-Tied Coronal Loops
Authors: Halberstadt, G.; Goedbloed, J. P.
1993ASSL..183..583H Altcode: 1993pssc.symp..583H
No abstract at ADS
---------------------------------------------------------
Title: Coronal heating: the role of resonant absorption.
Authors: Poedts, Stefaan; Goedbloed, J. P.
1992ESASP.348..253P Altcode: 1992cscl.work..253P
The efficiency and time scales of Alfvén wave heating of solar coronal
loops is investigated by means of numerical simulations in the framework
of both linear and nonlinear dissipative magnetohydrodynamics. The
coronal loops are modeled by cylindrical plasma columns that are
excited by waves that are incident on them. Parameter studies are
presented of the efficiency of the coupling of the external source to
the coronal loop plasma, the fraction of the power supplied by the
external source that is actually absorbed and converted into heat,
the quality of the resonances that occur, the basic time scales of
the resonant absorption mechanism, and the temporal evolution of the
energetics of the driven dissipative system. The results of these
investigations indicate that resonant absorption is a viable heating
mechanism for solar coronal loops.
---------------------------------------------------------
Title: Line-Tying Effects on Stability and Heating of Solar Coronal
Loops (With 2 Figures)
Authors: Halberstadt, G.; Goedbloed, J. P.; Poedts, S. M.; van der
Linden, R. A. M.
1991mcch.conf..489H Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Stability of solar coronal loops
Authors: Goedbloed, J. P.
1990CoPhC..59...39G Altcode:
The equations of magnetohydrodynamics do not contain an intrinsic
length scale determining the size of phenomena. Hence, size only enters
through the external geometrical properties of the configurations
considered. This is one of the reasons why tokamaks and solar coronal
loops may be considered as similar objects. The equations of MHD do
not distinguish between the two. It is only the geometry and, hence,
the boundary conditions that discriminate between them. Whereas for
tokamaks toroidal periodicity and normal confinement provide the
appropriate boundary conditions, for coronal loops line-tying at the
photosphere and some prescription for the behavior across the “edge”
of the loop determine the solutions. The latter is a more complicated
problem and gives rise to even more complex dynamics than encountered
in tokamaks. Here, we consider the influence of the two mentioned
groups of boundary conditions for the problem of the stability and
disruption of a solar coronal loop. <P />We consider the stability
properties of a single loop with twisted magnetic field lines under the
simultaneous influence of photospheric line-tying and constraining by
neighboring flux loops. The loops would be violently unstable without
these two ingredients (i.e. for the corresponding tokamak problem). It
is shown that line-tying alone in not sufficient for stability, but
the neighboring flux tubes provide a normal boundary condition similar
to a conducting shell in tokamaks. This stabilization gets lost on
the time scale associated with reconnection of the tangetial magnetic
field discontinuities present in the many-loop system. On this time
scale the magnetic energy, which has been built up during the twisting
of the field lines, gets released, resulting in a disruption of the
loop. This process may be considered as the single loop variant of
Parker's solar flare model.
---------------------------------------------------------
Title: Kink modes in coronal loops.
Authors: Goedbloed, J. P.; Goossens, M.; Poedts, S.
1989plap.work..103G Altcode:
Spectral theory of magnetohydrodynamic waves and instabilities has been
extensively developed. With proper modifications results obtained for
tokamaks can be transferred to the study of stability of coronal flux
loops and heating of the corona by means of Alfvén waves. In tokamaks
external kink modes are stabilized by the geometric constraint that
the modes should fit into the torus. In current-carrying coronal
loops the opposite problem arises, viz. the apparent absence of
external kink modes, as evidenced by their long life-time, spanning
many orders of magnitude of the characteristic growth-time of these
instabilities. Anchoring of the foot points of the field lines in the
photosphere is generally considered to be the responsible agent for
stabilization. Given the overall MHD stability of a coronal magnetic
loop structure, the subtle analysis of Alfvén wave heating by means
of the continuous spectrum may be undertaken. Here, an additional
complication is encountered which turns out to be quite beneficial
though from the point of view of heating efficiency. This gives rise to
improper modes which have both a global character and a non-integrable
part which admits quasi-dissipation.
---------------------------------------------------------
Title: Stabilization of external kink modes by means of a limiter
Authors: Freidberg, J. P.; Goedbloed, J. P.; Rohatgi, R.
1983PhRvL..51.2105F Altcode:
It is shown that poloidal ring limiters are very effective in
stabilizing ideal external kink modes in a tokamak. With one poloidal
limiter all external modes are stable for q<SUB>a</SUB>>1. However,
toroidal limiters have negligible influence on stability. Reversed-field
pinches require a finite number of poloidal limiters (typically six)
to stabilize the strong external kink modes that would result if the
conducting wall were removed.
---------------------------------------------------------
Title: Stabilization of magnetohydrodynamic instabilities by
force-free magnetic fields. II. Linear pinch.
Authors: Goedbloed, J. P.
1971Phy....53..501G Altcode:
The marginal-stability analysis, developed in a previous paper
<SUP>1</SUP>), is applied to the stability problem of a linear
pinch with a distributed current. Complete stability criteria
are derived from the marginal equation of motion and it is shown
that this method is equivalent to the application of the energy
principle. For a sharp-pinch model of a dense plasma surrounded by a
force-free magnetic field of constant α kinks, unstable modes of the
surface-layer, and of the force-free region are shown to be absent
if α is properly chosen. Explicit expressions for the growth rates
of unstable configurations are derived. An appendix is devoted to the
stability with respect to resistive tearing modes.
---------------------------------------------------------
Title: Stabilization of magnetohydrodynamic instabilities by
force-free magnetic fields. I. Plane plasma layer.
Authors: Goedbloed, J. P.
1971Phy....53..412G Altcode:
A marginal-stability analysis is applied to the stability problem of a
plane plasma layer under the influence of gravity. Complete stability
criteria are derived from the marginal equation of motion and it is
shown that this method is equivalent to the application of the energy
principle. Ideal magnetohydrodynamic instabilities, as occurring in
simple plasma-vacuum systems, can be suppressed by replacing the vacuum
by a force-free magnetic field, that is a field satisfying the relation
∇ × B = α B. Force-free fields of constant α are investigated in
particular. By proper choice of α the gravitational instabilities,
of nonlocal and of surface-layer type, are absent in a plane plasma
layer supported from below by a horizontal force-free magnetic field. An
appendix is devoted to a rigorous treatment of the influence of singular
points of the marginal equation of motion on the stability analysis.
---------------------------------------------------------
Title: Stabilization of magnetohydrodynamic instabilities by
force-free magnetic fields. III: Shearless magnetic fields.
Authors: Goedbloed, J. P.
1971Phy....53..535G Altcode:
The marginal-stability analysis, as given in two previous papers,
is further elaborated for magnetic fields of constant direction
in the plane case and for magnetic fields of constant pitch in the
cylindrical case. For these shearless magnetic fields discontinuities
arise in the stability criteria. The physical significance of these
discontinuities is discussed. From the marginal-stability analysis
the principle of exchange of stabilities is derived. This theory is
applied to constant-pitch magnetic fields, which are shown to be
necessarily unstable. Growth rates of the instabilities of these
fields are calculated, correcting earlier results of Ware. Simple
modifications of some constant-pitch models, namely Van der Laan's
model of a constant-pitch force-free field and Alfvén's model of a
constant-pitch field with parabolic pressure profile, prove to yield
completely stable pinch configurations of a sharp or diffuse kind. An
appendix is devoted to toroidal effects.
