Lorentz Center - Principles of Magnetohydrodynamics from 21 Mar 2005 through 24 Mar 2005
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### Principles of Magnetohydrodynamicsfrom 21 Mar 2005 through 24 Mar 2005

"Dust diffusion in protoplanetary discs by magnetorotational turbulence"

# 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

J.P. Goedbloed

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

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

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. Young1, S. E. Sharapov2 and V. M. Nakariakov1

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

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