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Principles of Magnetohydrodynamics |
Some Key Issues in Solar Plasmas
Eric
Priest An overview of some of the fundamental
unsolved problems about our Sun will be described, including the structure of
the solar interior, the nature of sunspots, the acceleration of the solar wind,
the occurrence of eruptive solar flares and the heating of the corona. In most
cases the subtle nonlinear interaction between solar plasma and the magnetic
field is crucial and new insights and surprises have recently come from
ground-based and space observations. MHD
for Fusion – Where to Next? J.
P. Freidberg MIT Ideal magnetohydrodynamics has evolved over the years to become one of the most important models in fusion research. The model is used and trusted to predict the macroscopic equilibrium and stability properties of virtually all magnetic configurations proposed for fusion. In this sense it provides a necessary justification for the viability of new experiments or new fusion concepts. MHD also strongly couples to plasma engineering by making accurate and realistic predictions of coil locations and coil currents to produce a desired plasma configuration. These applications are so successful as to make one wonder whether MHD has evolved to such an advanced state that it can now be classified as a routine tool with little need for further development. There is a partial element of truth in this
viewpoint although clearly more theory and computational modeling are needed to
understand the MHD of plasma flows and 3-D geometries. However, the future of
MHD in fusion is only partially concerned with improvements in the modeling and
development of computational tools. What is more critical is that there is
still one absolutely crucial, unresolved MHD dominated problem, facing the
development of the mainline tokamak concept as a source of energy. This problem
is the need for steady state operation. The ultimate solution to the problem
will require new ideas concerning the resistive wall mode, a high bootstrap
current, flow stabilization, and edge localized modes all strongly connected to
MHD. These issues will be discussed in the presentation in the context of the goal of fusion
energy, specifically with respect to the leading fusion contenders, the
tokamak, the stellarator, and the spherical torus. MHD
spectral theory of flowing plasmas FOM-Institute
for Plasma Physics 'Rijnhuizen' & Astronomical
Institute, Utrecht University Spectral theory of magnetohydrodynamic waves and instabilities has been developed to a high degree of sophistication, with intricate analytical tools supplemented by accurate multi-dimensional computational methods to compute MHD equilibria and spectra of laboratory fusion experiments, like tokamaks and stellarators. With the extension to astrophysical plasmas, like solar-type coronae and winds and accretion disks and jets about compact objects, an enormous complication is implicit that is generally not appreciated: For none of those plasmas, the assumption of static equilibria (that has been crucial for the unraveling of laboratory spectra) is justified. In fact, if one wishes to extend MHD spectral theory to plasmas with a sizeable background flow, the whole theoretical edifice has to be reconstructed. Not only does the physical system display Doppler shifts and Coriolis forces, so that the spectrum becomes intrinsically complex and the relevant operators become non-selfadjoint, but also 'transonic' transitions occur in the background flows that have major (and largely unknown) effects on the MHD spectrum. These three-fold transitions are the MHD counterparts of the single HD transitions from sub- to supersonic flow. In the reconstruction of MHD spectral theory for flowing plasmas, one basic organizing principle, that was very fruitful for static plasmas, remains effective though: The MHD spectrum still 'hangs' on the continuous spectra so that the structure of the spectrum about these continua has to be elaborated again. With the gradual elucidation of this structure, quite a number of apparently unrelated dynamical properties of laboratory and astrophysical plasmas could be pinpointed: the fate of the HD flow continua in MHD, the monotonicity properties of the discrete spectrum, singularities in the equations governing the magneto-rotational instability (generally assumed to be responsible for accretion onto black holes and neutrons stars), but also new instabilities (that should also be operative in accretion flows) of the continua spectrum itself driven by 'transonic' MHD flows. It turned out to be a tremendous joy to encounter old friends again (like genuine and apparent continuous spectra), together with the excitement and controversies that these subjects invariably accompany. There can be no doubt that a rich reservoir of discoveries is still awaiting us. Mathematical
methods for studying hydrodynamic and magneto-hydrodynamic equilibrium
configurations and their stability Sasha
Lipton-Lifschitz Citadel
Investment Group, L.L.C., Chicago In this talk we discuss both classical and modern methods for studying equilibrium configurations and their stability in hydrodynamics and magneto-hydrodynamics. We make a particular emphasis on the mutually beneficial interactions between the two fields over the recent decades. We present a number of examples which illustrate these interactions. Magnetorotational overstability in accretion disks Jan-Willem
Blokland FOM
Rijnhuizen 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 confront these solutions with
those found from approximate local dispersion equations from WKB analysis. In particular, we investigate the influence of 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, 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 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. Coronal
loop kink-mode oscillations and their role in coronal seismology Jesse
Andries, Inigo Arregui, Tom Van Doorsselaere, Stefaan Poedts and Marcel
Goossens CPA,
Leuven Transverse oscillations of coronal loops have been observed regularly since 1999 and are well documented. In order to use these oscillations to extract information about the coronal loop plasma, i.e. to do coronal loop seismology, a theoretical model of these oscillations needs to be provided. Although, a straight cylindrically symmetric flux tube is certainly not the best equilibrum model of a coronal loop that we can produce, the study of the eigenmodes of such a structure is evidently of great relevance to the subject of coronal loop seismology. Even from the first detection of transverse coronal loop oscillations on, this model and in particular its kink-mode oscillation have been used for coronal loop seismology. An important issue concerning these oscillations is their very rapid damping within a few oscillation periods. The possible role of resonant absorption in this respect is discussed extensively within the framework of the flux tube model. Furthermore, special attention will be given to the recent simultaneous detection of multiple modes in one and the same loop, and the exciting new opportunities for coronal loop seismology that are offered by these observations. Magneto-Hydro-Dynamics in tokamak plasmas G.
Huysmans Association
Euratom-CEA Cadarache, CEA/DSM/DRFC, 13108
St-Paul-lez-Durance, France The theory of linearised ideal MHD is very successful in describing many observed MHD phenomena in tokamak plasmas and is one of the very few theories in tokamak physics that allows a detailed comparison of the stability limits and the mode structures with the experimental observations. Examples are the global MHD stability limits, local stability limits related to the edge localized modes (ELMs) and waves such as Toroidal AlfvÃ©n Eigenmodes (TAEs). Evidently, more extended and non-linear MHD models are required to describe the rich details of the evolution, saturation, crashes, and in some cases even the onset of the instabilities. An important example of the last category is the neo-classical tearing mode, which requires the presence of a (seed) island created by another MHD instability to grow. Other examples include the MHD instabilities in steady state plasmas where all the plasma current is driven by a local source. The interaction of the MHD instabilities (like double tearing modes) with the current deposition profile can lead to a deteriorated operating regime with continuous MHD modes or with giant oscillations in the temperature profile. An overview will be presented of the observed MHD instabilities mentioned above and the status of the modeling of the observations. MHD
in tokamaks - life on rational surfaces Niek
Lopes Cardozo, Roger Jaspers, Egbert Westerhof, Marco de Baar FOM
Rijnhuizen In the control room of a tokamak experiment,
the terms 'MHD' or 'MHD-activity' are often used somewhat loosely for anything
that gives rise to fluctuating signals on magnetic pick-up coils or
time-resolved measurements of e.g. temperature or density. Often, 'MHD' is not
wanted and must be avoided. It is a sign of bad confinement. Even worse, if the
fluctuation enters into a phase of uncontrolled growth, it can lead to loss of
confinement. But this 'MHD' is also one of the most interesting and beautiful
chapters in the science of magnetically confined plasmas. I took the liberty to make a personal
selection of three experiments that I find particularly rewarding. The first concerns an exotic topic: Beams of
runaway electrons trapped in a magnetic island topology - perfectly confined,
whereas all runaway electrons outside the island were lost from the plasma
during a brief period of field stochastization. Interestingly, the electrons
that show the island topology, may physically live outside the island! The second concerns a series of experiments
which elucidated the special role of flux surfaces with rational field line
winding ration. This was done in a tokamak which had completely dominant,
local, electron heating. By scanning this local source through the plasma, a
discrete, stepwise response of the temperature profile was found. This could be
related to the heat source successively crossing 'rational' flux surfaces. Finally, I will discuss recent experiments on
the control of tearing modes by means of local electron heating and current
drive. In these experiments both the drive of the tearing mode (by means of
externally generated magnetic perturbation fields) and the stabilization were
completely under control of the experimentalist. One could say that in this way
different terms in the Rutherford equation can be tested individually. More
practically, the experiment also provides a route to control, i.e. ameliorate
or deteriorate at will, confinement in a fusion reactor. MHD waves and instabilities in the solar corona
Alan
Hood MHD waves are now regularly observed in the
solar corona. A key feature influencing wave propagation is the non uniform
nature of the coronal plasma. This means that physical processes such as
resonnant absorption, phase mixing and mode coupling will be important and
these must be investigated before a complete understanding of coronal
observations is achieved. Damping of waves via such processes may contribute to
the problem of coronal heating. Complex magnetic field structures can become
unstable to a variety MHD instabilities, such as the kink instability. These
instabilities are responsible for a variety of dynamic phenomena that are
driven by a release of stored magnetic energy. This free magnetic energy must
be stored in the coronal field. Thus, the field must be initially stable before
the instability is triggered. A key stabilising feature is the inertial
line-tying due to the dense photosphere that essentially anchors the magnetic
footpoints. Non linear MHD simulations have confirmed the previous linear
stability results but have shown how current sheets may form and release the
free energy. MHD
Modeling of the Solar Corona Zoran
Mikic, SAIC San Diego The MHD model has been invaluable in refining our
understanding of the structure and dynamics of the solar corona and inner
heliosphere. It has been used to describe the detailed magnetic and thermal
structure in active regions and the emission of X-ray and EUV radiation, the
acceleration of the solar wind and its propagation into the inner heliosphere,
and the structure of open and closed magnetic field lines on a global scale. It
has also been used to study the initiation and propagation of coronal mass
ejections. Even though it has limited formal applicability, it is often the
starting point for analyzing the physics of a given situation. Computational
solar MHD has advanced rapidly in the last two decades, to the point where it
is useful for understanding the rich content in present-day solar observations.
We will review some of the recent developments and applications of
computational MHD to the physics of the solar corona. MHD Wave Propagation in the Neighbourhood of a
Null Point
James
McLaughlin The nature of fast magnetoacoustic and Alfvén
waves is investigated in the neighbourhood of a of two-dimensional null point.
This gives an indication of wave propagation in the low $\beta$ solar corona.
The main body of the talk will discuss the findings of McLaughlin \& Hood
(2004), but extra effects will now be taken into account including: 1) Most importantly: how does having a finite
$\beta$ influence the results? With the generation of the slow wave (due to
coupling), can the fast wave now cross the null? 2) Do the results persist to a configuration
of two null points? 3) What is the role played by the inclusion of
non-linear effects? Determination of magnetic
configuration through MHD spectroscopy N. P. Young^{1},
S. E. Sharapov^{2} and
V. M. Nakariakov^{1} 1 Physics Department,
University of Warwick, Coventry CV4 7AL, United Kingdom 2 UKAEA Euratom Fusion
Association, Culham Science Centre, Oxfordshire, United Kingdom An
understanding of the confinement of a plasma within a toroidal magnetic confinement
fusion device such as a tokamak is dependent on knowledge of the toroidal
current contained within the device. This current is measured using a quantity
known as the safety factor q = B _ r_=B _ r_ (where _ and _
are the toroidal and poloidal angles of the torus). Determination of the q(r)
profile for a tokamak plasma is commonly attempted through the use of two
methods: extrapolation from readings by magnetic coils external to the plasma
and through examination of radiation emitted by particles undergoing the
motional Stark effect (MSE). Both methods are well understood but neither can
reproduce the equilibrium accurately enough during the pre-heating phase of the
discharge; the fast ramp-up of the inductive current makes the first method
extremely inaccurate whilst the low plasma density does not allow the
application of neutral beam injection for the use in MSE. Here
we apply a third technique for the determination of q(r): MHD spectroscopy on
modes excited by ions accelerated with ion cyclotron resonance heating. This
technique relies upon the fact that the frequency at which certain MHD modes
are observed is linked to the value and gradient of q at the radial location of
the mode; it was first proposed for use with the toroidal Alfvén eigenmodes
(TAE) [1] and has since been applied to Alfvén cascades (ACs) [2]. Whilst
both the theoretical basis and possible techniques for MHD spectroscopy have
been well known for some time the routine application of these techniques has
been limited. Here we demonstrate MHD spectroscopy by computational mode
discovery using the generalised MHD code MISHKA code which should allow more
general use. We begin with a computational equilibrium
reconstructed from experimental data using the magnetostatic HELENA code on
magnetic and, where available, MSE data. Using this equilibria and the MISHKA
code we reconstruct the evolution of the frequency of an individual mode as we
shift q(r) up or down. We then make a modification to the plasma current
profile input to HELENA and repeat the process thus observing the effect of the
alteration on the frequency evolution of the mode. By iteratively tuning the
parameters governing the alteration of the current profile in order to match
the resulting frequency evolution to that of a TAE or AC observed in
experimental data we are able to produce a more accurate plasma equilibrium. References [1]
J. P. Goedbloed et al., Plasma Physics and Controlled Fusion 35, B277 (1993). [2]
S. E. Sharapov et al., Physics of Plasmas 9, 2027 (2002). Connecting
astrophysical fluid dynamics and plasma physics to the Laboratory Robert
Rosner, University of Chicago I will discuss some recent examples of the
growing connections between astrophysical and laboratory magnetohydrodynamics
and plasma physics, including the central role being played by numerical
simulations. Magnetohydrodynamic modeling of cosmic jets
Kanaris
Tsinganos, University of Athens, Greece Some key results from the use of the
Principles of Ideal Magnetohydrodynamics to model rotating and magnetized
collimated astrophysical outflows (jets) will be reviewed. Analytical steady
solutions crossing the appropriate singularities at the limiting MHD
characteristics and satisfying causality will be compared to corresponding
numerical simulations, for non relativistic as well as to relativistic jets. It
can be shown that outflows from inefficient magnetic rotators are rather weakly
collimated while outflows from efficient magnetic rotators produce tightly
collimated jets, a result which may be used to explain the observed dichotomy
of uncollimated winds and collimated jets. Nevertheless, magnetic
self-collimation can be shown to be inefficient in a single-component model
consisting of a wind from a central object, or an accretion disk in the sense
that the collimated portion of the mass and magnetic fluxes is uncomfortably
low. The theory of magnetic collimation may be then applied to a two-component
model consisting of an outflow from a central source and a wind from the
surrounding disk. It will be shown that in this two-component model it is
possible to collimate into cylindrical jets all the mass and magnetic fluxes
which are available from the central source. In such a case the disk-wind plays
the role of the jet collimator and it may also induce the formation of a
sequence of shock waves as the two components eventually collide. This theory
may be applied, for example, to the slow collimation of the relativistic jet of
M87 which collimates from a wide opening angle of about 60 degrees at sub
parsec scales to a smaller angle of about 10 degrees at the pc scale. Dust diffusion in
protoplanetary discs by magnetorotational turbulence Anders
Johansen We measure the turbulent
diffusion coefficient of dust grains embedded in magnetorotational turbulence
in a protoplanetary disc from direct numerical simulations. The simulations are
done in a local coordinate frame comoving with the gas in Keplerian rotation.
Periodic boundary conditions are used in all directions, and vertical gravity
is not applied to the gas. Using a two-fluid approach small dust grains of
various sizes (with friction times up to $\varOmega \tau_{\rm f}=0.02$) are
allowed to move under the influence of friction with the turbulent gas. We
measure the turbulent diffusion coefficient of the dust grains by applying an
external sinusoidal force field to the dust component only. This concentrates
the dust around the mid-plane, and an equilibrium dust density distribution is
achieved when the vertical settling is counteracted by the turbulent transport
away from the mid-plane. Comparing with an analytical expression for the
equilibrium amplitude we deduce the vertical diffusion coefficient. We find
that the turbulent transport is well-described as a diffusive process, and also
that the diffusion coefficient is independent of the considered grain sizes. A
similar radial force field allows us to measure the radial turbulent diffusion
coefficient as well. We find that the radial diffusion coefficient is around
50\% higher than the vertical, and also that both the vertical and the radial
diffusion coefficients are significantly higher than suggested by the angular
momentum transport by Reynolds stresses. Space Weather Prediction:
Challenges in Computational Magnetohydrodynamics Gabor
Toth Space weather is the interaction of the Sun
and the Earth which can affect human life and technology. Prediction of space
weather can mitigate the harmful effects of solar eruptions and magnetic
storms. Physics based modeling of space weather is one of the most challenging
problems for computational magnetohydrodynamics (MHD). There are disparate spatial scales ranging
from the Sun-Earth distance to 100-s of kilometers resolution required near the
Earth. One of the successful approaches to this problem is the use of adaptive
grids. A predictive model must run faster than real time. This can only be
achieved on massively parallel computers and the code must scale well up to
hundreds of processors. The fast magnetosonic speed can approach the speed of
light near the magnetic poles of the Earth. The stiffness of the equations can
lead to restrictively small time steps, which can be avoided by the use of
local time stepping, Boris correction and implicit time integration schemes.
Huge gradients of the magnetic field near the Earth can lead to large numerical
errors, which can result in negative pressure. The divergence of the
numerically obtained magnetic field should be controlled. Successful modeling of space weather cannot be
achieved with only solving the ideal MHD equations. We have built a framework
comprising of several numerical models spanning from the surface of the Sun to
the surface of the Earth. These models include effects of resistivity, heat
conduction, multi-species plasma, high energy particles. The various models
need to communicate with the framework and with each other, which requires
efficient parallel algorithms. I will discuss the challenges and the successful approaches we developed in the past years which resulted in the Space Weather Modeling Framework (SWMF). Although our main goal is space weather simulation, the algorithms are general and can be used successfully in many applications. Plasmaphysics and Mathematics: mutual
inspiration
Henk
van der Vorst Matrix notation was unknown in 1846. This did
not prevent Jacobi from inventing an algorithm for the computation of
eigenvalues [1]. He did this in the context of linear systems related to the
stability of the orbits of the then known 7 planets. His method was reinvented
100 years later by Von Neumann and his colleagues, and part of Jacobi's method
became popular for linear eigenproblems Ax=cx. The method was some thirty years
later overtaken by more successfull methods: QR and the methods of Lanczos and
Arnoldi. In 1975, the chemist Davidson proposed a new
method that became very popular, specially for applications in Chemistry [2].
Numerical analysts all over the world largely ignored his method, mainly
because of lack of understanding. In 1993 our numerical pride was triggered by chemists
from Utrecht, who impressed us with the apparant superiority of the Davidson
method for their problems. At the same time, a student at our institute did her
Master thesis on the old publications of Jacobi. This led to some remarkable
observations and eventually it led to a happy marriage between parts of the
methods of Jacobi and Davidson [3]. On top of this, we were faced with very
difficult generalized eigenproblems associated with plasma physics. These
problems arose in a project led by Hans Goedbloed. The Jacobi-Davidson method
became mature within this project. It turned out that the method, originally
triggered by standard eigenproblems, was very effective and competitive for
generalized eigenproblems. Impressive results could be obtained for large
and complicated eigenproblems associated with acoustics and with Tokamak plasma
stability. [1] C.G.J. Jacobi, 'Ueber ein leichtes
Verfahren, die in der Theorie der Saecularstoerungen vorkommenden Gleichungen
numerisch aufzuloesen', J. fuer die reine und Angew. Math., 1846, p.51-94 [2] E.R. Davidson, 'The iterative calculation
of a few of the lowest eigenvalues and corresponding eigenvectors of large real
symmetric matrices', J. Comp. Phys, 17, 1975, p.87-94 [3] G.L.G. Sleijpen and H.A. van der Vorst, `A
Jacobi-Davidson iteration method for linear eigenproblems', SIMAX, 17, 1996,
p.401-425 [Back] |