Session 1R - Ideal and Resistive MHD.
POSTER session, Monday morning, November 11
Exhibit Hall - Concourse Level, Adam's Mark

MHD generalization of the fluid ``tilting'' instability (M.N. Rosenbluth, V.D. Shapiro, Physics of Plasmas, 1, 222, 1994) is investigated taking into account the source generating small scale magnetic field fluctuations and dissipation due to magnetic viscosity. A linear stability analysis is presented describing generation of the mean magnetic flux and fluid convection from the magnetic field fluctuations in the form of magnetic rolls. Nonlinear evolution of the instability is investigated numerically using Fouric spectral methods. Because of the inverse cascade in two-dimensional MHD, the mechanism under consideration presents itself as a potential method for the generation of large scale magnetic fields from small scale turbulent fluid flow in the solar dynamo problem.

[1R.02] Collisionless magnetic reconnection and the structure of thin current layers

It is now widely accepted that magnetic reconnection in both laboratory (during sawtooth crashes) and space and astrophysical systems is dominated by collisionless processes, independent of classical resistivity. Current layers which form near the magnetic x-line during collisionless reconnection are extrememly narrow, comparable in width to the electron skin depth, c/ømega_pe. Such narrow layers are violently unstable to whistler-like disturbances driven unstable by the local current gradient. 3-D hybrid simulations (particle ions and finite mass fluid electrons) reveal that narrow current layers of width c/ømega_pe rapidly become fully turbulent and broaden out to a width intermediate between the electron and ion skin depths. Computer generated movies reveal that the current layer consists of continuously-evolving, sheet-like beams of electrons. The turbulence persists even after the layer has broadened significantly. It is therefore expected that thin current layers are typically turbulent rather than laminar structures. Satellite measurements of electric and magnetic field fluctuations at the magnetopause support this conclusion. Implications for predicting the rate of collisionless reconnection are discussed.

[1R.03] Theory of the Reconnection Layer in the MRX Experiment.

The physics of reconnection in the MRX experiment has been resolved into a global problem, and a local problem.( D. Uzdensky, R. Kulsrud, and M. Yamada, Phys. Plasmas 3), 1220, April 1996 In the global problem the plasma in the region outside the reconnection layer has been shown to evolve through a sequence of magnetostatic equilibria, which, in general, is uniquely determined independent of the physics inside the reconnection region. The rate of progress through the sequence is given by the reconnection rate. In this paper we analyze the local problem. We rescale the time dependent resistive MHD equations inside the layer in the limit of large magnetic Reynold's number, making use of the boundary conditions given by the global solution. The solution of these equations enables us to resolve the structure and dynamics of the layer. We intend to determine whether the reconnection rate can be faster than the Sweet-Parker rate, and whether the solution can be consistent with the Petschek model.

[1R.04] The Magnetic Reconnection Experiment (MRX), a Fundamental Study of Magnetic Reconnection in Merging Plasmas

Since Oct.\ '95, the MRX device\footnote M. Yamada, et al., Bull.\ APS. 44, 1877 (1995). has been operated at PPPL to investigate the fundamental physics of magnetic reconnection, a key physics process in solar flare physics, magnetospheric physics, astrophysics, and laboratory experiments. Experiments with full 3-D magnetic reconnection are now possible. The MRX device generates two annular plasmas by inducing currents around two flux cores. The two toroidal plasmas can be made to merge together on a common axis with or without breaking off from the flux cores. Lundquist number S ranges from 100 to 1000 and the plasma sizes are approximately 30-100 ion gyroradii. Diagnostics include magnetic and Langmuir probes, spectroscopy, and measurement of flows and topology. The important results include (i) identification of X-, Y-, O-shaped diffusion regions, the origins of which depend on the angle of the field lines, (ii) dependence of the reconnection rate on the merging angle of field lines, (iii) helicity inventory during the merging of plasmas, and (iv) evolution of plasma parameters of the diffusion region during reconnection. The results are analyzed quantitatively in light of basic MHD equations. An overview of our results to date will be presented and compared with the leading 2-D theoretical models.

[1R.05] MHD Analysis of Magnetic Reconnection in MRX

One of the possible operation modes in MRX is the double annular plasma configuration, in which two annular plasmas are formed independently around two flux cores.(M. Yamada, et al.), this conference (1996). Once the annular plasmas are created, the poloidal field (PF) coil current can be increased or decreased. In the case of increasing PF coil current, the poloidal flux in each plasma is \lq\lq pushed" toward the X point (push mode). In the case of decreasing PF coil current, the poloidal flux in the common plasma is \lq\lq pulled" back toward the X point (pull mode). Pull mode reconnection has been intensively investigated in both the co-helicity case, where the third magnetic component (B_T) is large, and the counter-helicity case, where B_T is essentially zero. The major findings are an O-shaped (Y-shaped) diffusion region has been identified in the co-helicity (counter-helicity) case; (2) the current sheet in the counter-helicity case is much narrower than in the co-helicity case; (3) each term of the induction equation and simple Ohm's law has been examined across the current sheet, and an enhancement of resistivity over its classical value by a factor of 5-20 has been measured. The cause of this resistivity enhancement will be discussed. Time evolution of the global magnetic helicity and magnetic energy during the merging process will be also presented.

[1R.06] Measurements of Ion Temperature and Plasma Flows During Magnetic Reconnection in MRX

We present diagnostics to measure local ion temperature, T_i, and plasma flows inside and near the reconnection layer in MRX (Magnetic Reconnection Experiment). Local measurements of T_i and ion drift speed, v_id, have never been performed before during controlled magnetic reconnection. T_i is measured with a retarding grid ion energy analyzer, and v_id is measured with Mach probes. The ion energy analyzer consists of a negatively biased electron repeller and a collector biased either with a DC or fast swept (100 kHz) voltage from which an I-V characteristic emerges. The many subtleties of data interpretation for this probe will be discussed. The Mach probe yields v_id/c_s, where c_s \equiv \sqrtk T_i/M is the ion acoustic speed, by taking the difference of ion saturation currents collected by back-to-back, negatively biased, conductors. It is also possible, as will be discussed, to measure electron flow with the Mach probe. Mach probe measurements of local v_id are qualitatively consistent with conventional 2-D fluid models, and in initial experiments, the measured upstream v_id is typically of order 10 km/s, compared to a c_s of approximately 30 km/s. Preliminary ion energy analyzer measurements of local T_i in the reconnection layer will be presented and discussed in the context of ion heating during magnetic reconnection.

[1R.07] Electron Temperature and Density Measurements in MRX

We present measurements of electron temperature, T_e, and density, n_e, during co- and counter-helicity reconnection and compact toroid formation in the Magnetic Reconnection Experiment (MRX). A triple Langmuir probe has been used to measure instantaneous T_e, n_e, and floating potential, V_f, in MRX discharges. Initial results from the probes show 5 < T_e < 40 eV and 5\times10^12 < n_e < 1\times10^14 cm^-3 in reconnecting plasmas, suggesting a Lundqvist number 100 < S < 1000. Langmuir probe measurements of n_e and T_e profiles across the reconnection region are seen to be peaked in the neutral sheet and to rise and fall in sync with the formation and dissolution of the reconnection layer. These results suggest that compressibility and non-uniform resistivity, \eta, may be important during reconnection in MRX. We will discuss the measurement of T_e and n_e in MRX using a fast-swept (100 kHz) single Langmuir probe, obtaining full I-V characteristics. Electron flow velocity measurements in the device using a simple Mach probe biased to collect electrons will be discussed. We will also present temperature and density profile measurements of compact toroids (spheromaks and FRC's) formed in MRX.

[1R.08] Response of a Rotating Plasma to a Resonant Boundary Perturbation

The response of a plasma to a resonant perturbation at the boundary is studied in a sheared slab geometry for a case where the plasma is flowing relative to perfectly conducting walls. Unlike the case when no flow is present (T.S. Hahm and R.M. Kulsrud, Phys. Fl. 28), 2412 (1985), (X. Wang and A. Bhattacharjee, Phys. Fl. B 4), 1795 (1992), it is found that no significant reconnection occurs at the rational surface for a sufficiently small perturbation. However, there is a singular current layer formed at the Alfven resonance, where the local Alfven frequency equals the Doppler shift due to the equilibrium flow velocity. The boundary perturbation produces a torque at the Alfven resonances which slows the plasma down. Implications for the problem of field error penetration in rotating toroidal plasmas will be discussed.

[1R.09] Secondary Ballooning Instabilities in the Vicinity of an m/n=1/1 Island

Recent experimental results in TFTR indicate an asymmetric structure of tokamak sawteeth to the bad curvature side( Y.\ Nagayama, et al., Phys.\ Plasmas \mbox3), 1647 (1996). and concomitant thermal transport during the sawtooth crash.

It is suggested that secondary pressure gradient driven ballooning modes play a role in describing this phenomena. Previous analysis of secondary ballooning instabilities for m \ge 2 tearing modes( C.\ C.\ Hegna and J.\ D.\ Callen, Phys.\ Fluids B \mbox4), 3031 (1992). suggest that an additional toroidal localization can reduce the effective magnetic shear and destabilize the modes below the threshold for the associated axisymmetric equilibria. In this work, a similar calculation is performed for the m/n=1/1 instability which accounts for the presence of two Y-points( F.\ L.\ Waelbroeck, Phys.\ Fluids B \mbox1), 2372 (1989). and the connected sheet ribbon. Additionally, a 3D nonlinear MHD simulations of the m/n=1/1 mode with the FAR code is used to study the enhanced parallel thermal transport. The effects of the local pressure gradient across the q=1 surface is extensively studied. Stochasticity in the q=1 annular region is studied ! quantitatively.

[1R.10] Mode Coupling as a Trigger for Neoclassical-MHD Driven Magnetic Islands in a Tokamak Plasma

A set of incompressible, reduced-MHD equations, which include neoclassical viscous-stress effects (\nabla\cdot\vec\vec\pi) based on analytical neoclassical closures(J.D.\ Callen, et al., Plasma Phys. and Controlled Nuclear Fusion Research \mbox2) 156 (1986). have been implemented in the three dimensional, toroidal, initial value neofar code(T.A.\ Gianakon, PhD Thesis (Univeristy of Wisconsin-Madison, 1996)). The addition of neoclassical effects through the viscous force introduces the bootstrap current into Ohm's law, which can then lead to destabilization of tearing modes which are \Delta^' stable( C.C.\ Hegna and J.D.\ Callen, Phys.\ Fluids B \mbox4), 1855 (1992).. To access the instability a minimum island width must be exceeded---a threshold for the instability exists. The results of an investigation of a possible mechanism for producing a seed magnetic island above the threshold based upon toroidal mode-coupling to a pre-existing instability is presented.

[1R.11] The Role of Parallel Electron Viscosity on m = 1 Instabilities in High Temperature Tokamaks*

In high temperature tokamaks, the linear and nonlinear properties of m = 1 instabilities are characterized by two length scales. The Hall term in Ohm's Law, while it is incapabale of breaking field lines, introduces a length scale of the order of the ion gyroradius which is the spatial scale of the parallel electric field. The singular current layer is typically much smaller, of the order of the skin depth. In the collisionless limit, where viscosity and resitivity are ignored and an isothermal assumption is used, the mode structure and growth rates of the reconnection are described by reversible equations. In the present work, we assess the role of parallel electron viscosity (Landau damping) in the current layer. While this term does not dramatically alter the growth rate scaling, it importantly does introduce dissipation and a degree of irreversibility to the system. *This work is supported by the U. S. DOE under grant no. DE-FG02-86ER53218.

[1R.12] Formation of Plane Current Sheets under High-Pressure He-filling. Some Peculiarities of Magnetic Reconnection.

Formation of plane current sheets (CS) under high-pressure (300 mTorr) He-filling in 2-D magnetic fields with null-lines was studied for the first time. Contrary to the previous experiments [1], the pressure increase results in slowing down of the CS formation, in the relative increase of the role of hydrodynamic effects, in the thermo-conductivity suppression so that plasma radiation becomes important.

It was shown that CS formation occurs during 2-3 mks if the magnetic field gradient exceeded 500 G/cm. Both the electric current and plasma are compressed into a plane sheet with an elongation of (14 cm: 1 cm). The maximum intensity in the HeII 468,6 nm spectral line is emitted from a bright kern that is formed in the very middle of CS, where both electric current and plasma density are of the maximum values. The peculiarities of magnetic reconnection are discussed.

Supported by INTAS-93-2836-ext and RBSF-96-02-18546a grants.

1. Bogdanov S.Yu. et al.//Sov. J. Plasma Phys. 1992. V.18. P.654.

[1R.13] Nonlinear Resistive Wall Tearing Modes and Locking

Nonlinear simulations in 2D of tearing modes with a resistive wall and plasma rotation will be presented. The regimes of interest include: unstable tearing mode with locking; (2) resistive wall tearing mode slowed but not stabilized by rotation; (3) stable resistive wall mode in the presence of field errors. In (1), the tearing mode can begin a locking process, during which the plasma flow and the mode both slow, allowing flux diffusion through the wall. There is a bifurcation between multiple states, called fast and slow modes. In a certain range of parameters, another state is found, in which there is a relaxation oscillation between the two modes. With field errors, the field (but not the plasma rotation) in the slow mode can become completely locked. In (3), field errors can slow the rotation sufficiently that the resistive wall mode is destabilized and begins the locking process. An important control parameter for the bifurcation is \bar\nu, which is the anomalous decay rate of total momentum. With \bar\nu large and a source to balance the decay, the range of parameters over which there are multiple states or oscillating states is much smaller.

[1R.14] Magnetic Helicity Scan Experiment in the TS-3 Merging Device

A novel merging formation of FRC has been developed in the TS-3 merging experiment[1], leading us to a helicity scan experiment to study the helicity conservation and its extension to high-\beta regime. When two spheromaks with opposing magnetic helicities (and toroidal field) are axially collided, their total magnetic helicity K given initially is varied from zero to the value for the Taylor state by adjusting their flux ratio from zero to one. The merging spheromaks are observed to relax either to a high-\beta FRC or to a new low-\beta spheromak, depending on whether K is smaller or larger than a threshold value, when the initially-given total magnetic energy is fixed. In the former low-K case, their counterhelicity reconnection accelerates and heats selectively plasma ions from 10eV up to 200eV, increasing the plasma \beta from 0.1 to 0.7-1 within 15\mu sec. The efficient conversion of the toroidal magnetic energy to the ion thermal energy is found to conserve their total energy better than their total magnetic helicity. In the latter high-K case, the magnetic energy decays about five time faster than the magnetic helicity in consistent with the Taylor relaxation. The ion temperature stays around 20-40eV, indicating that the large loss of the converted thermal energy is related with the low-\beta relaxation: the flux-conversion from poloidal to toroidal. [1] Y.Ono et al., Phys.Rev.Lett. 76, 3328 (1996).

[1R.15] Experimental Investigation of Driven Magnetic Reconnection on TS-3 device

Mechanism of the externally driven magnetic reconnection has been investigated by measuring detailed structures of the neutral current sheet. Two spheromaks or tokamaks are merged together in the axial direction and the magnetic pressure is varied by the poloidal field coil current. A thin current sheet is clearly observed near the contact region of the two merging toroids. The large external pressure is observed to cause the rapid dissipation of the current sheet, increasing the magnetic reconnection speed. The measured resistivity \eta at the X point increases from 1 \times 10^-4 Ømega m to 1 \times 10^-3 Ømega m , that is about 20 times larger than the classical resistivity (T_e = 10eV, Z=2). The small external pressure no longer increases the resistivity of the current sheet, causing the slow reconnection. These results agree well with the driven reconnection model. Comparison of tokamak and spheromak mergings will indicate that the magnetic field parallel to the X-line decreases the reconnection speed. The effect of various topological changes in the current sheet will be presented in connection with the reconnection speed.

[1R.16] Plasma Reconnection in HI-1 Helicity Injector

The intensive research on the spheromaks in steady state conditions are studied in 1980's as well as the fundamental numerical simulations. However, the experimental understanding of the plasma production phase with the drastic reconnection process are out of DOE's main focus. In this meeting an experimental findings closely related to the electrode polarity effect are introduced as well as the fundamental results especially on the stagnation point. The fundamental parameters are already shown in the last meeting in LUISVILLE. This device is a simulator of future FBX-III BURNER EXPERIMENT and operated under a 30kA,10Pa coaxial discharge up to 0.1 Tesla axial magnetic field.

Bull.American Phys.Soc., 40,11(1995) OCT, 1764

[1R.17] Asymmetry and Thermal Effects of Parallel Electron Motion and Ion Larmor Radius Effect in Collisionless Magnetic Reconnection

The macro-particle simulation of fast collisionless reconnection of the flux loops, which is typical in the solar flares and the flux-core merging experiment, shows remarkable asymmetry of the plasma flow when the toroidal magnetic field is present [Tanaka, Phys.Plasmas, 2, 2920 (1995)]. It has been found that the parallel motion of electrons induced by the reconnection electric field produces significant density and toroidal magnetic field inhomogeneities of a quadrupole shape, \delta n/n_0 \sim 0.3 , around the X-point unlike the m=1 mode case. The divergence of the flow is locally not identical to zero, \nabla \cdot V^(s) \ne 0 ( s= el, ion ), due to the parallel motion (spatial displacement) of the electrons even if B_t \ne 0 . This internal structure results in virtually a thick current layer, L > c/ømega_pe . A plasmoid that impedes magnetic reconnection is created in the current layer if the parallel particle diffusivity of electrons is suppressed (the fluid limit). This proves that the parallel electron motion, both bulk and thermal ones, greatly promotes the collisionless reconnection process. The reconnection rate becomes a smoothly increasing function of the ion mass and an inverse of the toroidal magnetic field (compressional effect), whereas the rate decreases drastically for the ion Larmor radius exceeding the ion skin depth (for fixed B_t \ne 0 ).

[1R.18] Forced Reconnection and Mode-Locking in Rotating Cylindrical Plasmas

The problem of mode-locking due to forced magnetic reconnection in rotating cylindrical plasmas is revisited. Forced reconnection is generally characterized by very large values of the tearing stability parameter \Delta^\prime. This makes the constant \Psi approximation generally inapplicable in the linear and nonlinear regimes. The nonlinear dynamics of rotating non-constant \Psi islands is characterized by the persistence of current sheets. Mode-locking due to sudden as well as slowly time-dependent error fields is discussed. A new locking threshold is obtained for sudden error fields that differ significantly from the theory of mode-locking based on the constant \Psi approximation [R. Fitzpatrick and T. C. Hender, Phys. Fluids B 3, 644 (1991)]. For error fields that are ramped up with a slow time-dependence, it is found that the constant \Psi phase can be attained under certain conditions, leading to the threshold obtained by Fitzpatrick [R. Fitzpatrick, Nucl. Fusion 33, 1075 (1993)]. The predictions of the theory are compared with experimental tokamak observations. \rule2in1pt ^\astWork supported by NSF and AFOSR.

[1R.19] The Tearing and Ballooning Modes in the Tail Current Sheet

We have analytically investigated the evolution of resistive MHD tearing and ballooning modes, by assuming that the dissipation is anomalous in the current sheet region. We have displayed a generalized technique for obtaining the solutions for both these modes near the singular layer, where the B_x (z) field reverses sign. We have found that the stability of tearing modes with two-dimensional characteristics is controlled by the compressibility effect and the Lundquist number, S, where S is the ratio of anomalous diffusion time to Alfvén time. For S \ll 5 \times 10^3, the fluid compressibility plays more stabilizing role than the normal component, B_z. For k_y\neq 0 (k_y being the wavenumber), we have shown that the magnetic field curvature causes the excitation of a new class of unstable tearing and ballooning modes with their growth rates scaling as fractional powers of anomalous resistivity and the Alfvénic frequency. We have also shown that the ballooning modes enhance the current and the magnetic field gradients in the center of the plasma sheet and thereby activates the excitation of tearing modes in a period of ten seconds. Finally, we have briefly discussed the relevance of the newly excited modes to recent AMPTE/IRM and GEOS 2 results.

[1R.20] Pseudo-MHD Ballooning Modes In Tokamak Plasmas

The magnetohydrodynamic (MHD) description of a plasma is extended to allow electrons to have both fluid-like (ømega /k_\|>v_Te) and adiabatic-regime (ømega /k_\| s^2 (2^7/3/9) (r_p/R_0) or -d\sqrt\beta/dr>(2^1/6/3)(s/R_0q) where \alpha \equiv R_0 q^2 \beta/r_p, \beta \equiv 2\mu_0P/B^2, 1/r_p \equiv (1/P) (dP/dr), s\equiv (r/q)(dq/dr). The marginally stable pressure profile is similar to that observed in ohmically heated tokamak plasmas. A resestive form of these modes exists for electron temperatures below a critical value and may be operative in the edge of tokamak plasmas.

[1R.21] Destabilization of Ballooning Modes by Sheared Rotation in the Supersonic Regime

Sheared toroidal plasma rotation was shown recently( 1. R.L. Miller, et al.), Phys. Plasmas 2, 3676 (1995). to reduce the growth rate of ballooning modes in a tokamak, and ultimately to stabilize them when dØmega /dq becomes a significant fraction of the Alfven wave toroidal frequency. We analyze the case of a much lower (but realistic) flow shear, with dØmega /dq on the order of the slow magnetosonic wave toroidal frequency. We consider both subsonic and supersonic values of dØmega /dq. The supersonic case is novel in that the energy integral \delta W is not bounded from below. The integral can nevertheless be modified and we prove that when the modified version becomes negative there is an odd number of unstable ballooning modes. When it is positive, there can be an even number of such modes (but stability in the subsonic case). These results are borne out by both analytical and numerical calculations.

[1R.22] Shear Flows in Toroidal Anisotropic Plasmas: Steady State and Stability

An adequate description of equilibrium and stability of anisotropic plasma with macroscopic flows in toroidal confinement systems is presented. Chew-Goldberger-Low (CGL) approximation is consistently used to analyze an anisotropic plasma dynamics. The structure of a stationary flow is prescribed by a topology of nested magnetic surfaces. An allowance of the same relabeling symmetry as in ideal magnetohydrodynamics (MHD) with isotropic plasma pressure is proved. We generalized the Grad-Shafranov equation for anisotropic plasma with flows in axisymmetric magnetic field. We also obtained a variational principle, which allows for a stability analysis of anisotropic pressure plasma with flows and takes into account the conservation laws (V. I. Ilgisonis, V. P. Pastukhov, Plasma Phys. Reports, 22), 228 (1996) resulting from the relabeling symmetry. The way of how to avoid the difficulty related to non-self-ajointness of force operator accounting the flow inertia is discussed. Our results cover the previous ones for static CGL plasma and for ideal MHD flows in isotropic plasma as well.

[1R.23] Coupling the \sc pest and \sc spark Codes for \delta W Stability Analyses with 3\small D Wall Configurations

To obtain estimates of \sc mhd stability from the non-axisymmetric portions of the enclosing shell in tokamaks we interleave our 2\small D \sc vacuum code with the \sc spark code which calculates the eddy currents generated in non-axisymmetric shell configurations. The magnetic field from the eddy currents is combined with that of the generating perturbations themselves to obtain a modified vacuum energy which contains the 3\small D nature of the shell: \( \delta W_v = - \int_S_p \cal J d\theta\, d\phi\,[\chi^p*B\!\cdot\! \nabla\xi_\psi + \xi_\psi \hatB\!\cdot\!\delta B_s ]. \) Here, \chi^p is the magnetic scalar potential due to the plasma displacement, \xi_\psi, and \hatB\!\cdot\!\delta B_s originates from the eddy current contribution of \sc spark. \deltaB_s\!\cdot\!n is also needed for the boundary condition on the magnetic field. Several harmonics in the toroidal angle, \phi, are required for a 3\small D shell. The interfacing of the codes requires obtaining the vector potential form the scalar potential and involves intensive checks against semi-analytic long wavelength circular and concentric elliptic cylindrical models. It is then checked against 2\small D configurations. The eddy current patterns of the \sc pest and \sc spark codes are compared, and finally, 3\small D calculations are obtained. \rule[1.ex]1.9in.005in ^*\small This work supported by DoE contract No.\ DE--AC02--76--CHO--3073.

[1R.24] Converting the VACUUM code from a Fourier basis to a Finite-Element Basis

To incorporate the features of the Fourier-based \scvacuum code into the \scgato code, the former is modified to include an option to calculate the vacuum matrix, V, in a finite-element basis. Two distinct methods of conversion are compared: directly calculating V_kk' in the a finite element expansion, and calculating the orthogonal transformation, T_kl which takes the Fourier-based V_ll' into V_kk'. These are compared with each other and with the present vacuum calculations of \scgato. Checks are also made against the analytical or semi-analytical large-aspect-ratio circular cases, with and without an enclosing conducting shell. Interesting observations about the transformation and the relation between the two representations will be presented, as well as some of the features of the \scvacuum code generated from \scgato, such as the Mirnov loop and eddy current simulations. \rule[1.ex]1.9in.005in \small This work supported by DoE contract Nos.\ DE--AC02--76--CHO--3073 and DE-AC03-89ER51114.

[1R.25] MH3D Simulations of Three-Dimensional Halo Currents

During plasma disruptions, especially those which accompany vertical displacement events (VDE's), halo currents can flow along open field lines connecting the plasma edge region to the vacuum vessel or conducting plate structures. A substantial contribution to the J \times B forces on conducting structures during disruptions can be due to the halo currents. Most theoretical models of halo currents have assumed toroidal axisymmetry for the plasma. However experimental results (eg., in Alcator C-mod, JET and JT-60U) indicate that non-axisymmetric halo currents may be significant. Using the MH3D computer code, we have begun an attempt to model 3-D halo currents resulting from non-linear helical deformations of the plasma caused by external kink instabilities. We assume an external kink has triggered a \beta- collapse. MH3D follows the nonlinear development of this external kink as the plasma + halo interacts with the conducting wall. Our generic studies assume a circular cross-section toroidal plasma surrounded by a concentric circular cross-section toroidal conducting shell. We evolve the plasma in fully 3D geometry, and impose a horizontal magnetic field to force a positional instability for the circular cross section plasma. \rule[1.ex]1.9in.005in \small This work supported by DoE contract No.\ DE--AC02--76--CHO--3073.

[1R.26] Nonlinear Magnetohydrodynamic Detonation: Part I

The sudden release of magnetic free energy, as occurs in spectacular solar flare events, tokamak disruptions, and enigmatic magnetospheric substorms, has long defied any acceptable explanation. Recently, Cowley et al. [Physics Reports, 1996] have demonstrated a new mechanism for nonlinear explosive magnetohydrodynamic (MHD) destabilization of a line tied Rayleigh-Taylor model. They call this process detonation. In this paper, this picture is generalized to arbitrary magnetic field geometries. As an intermediate step, the ballooning equation in generalized coordinates is derived including the effects of magnetic field curvature, shear, and gravity. This equation determines the linear stability of the plasma configuration and the behavior of the plasma displacement along the magnetic field line. Finally, the nonlinear equation which determines the time and spatial dependence, transverse to the equilibrium magnetic field, of the plasma displacement is obtained. Thus, we prove that explosive detonation is a natural and generic property of a marginally ballooning linearly stable MHD plasma. Application to the stability of solar prominances will be discussed.

[1R.27] Nonlinear Magnetohydrodynamic Detonation, Part II

Both laboratory and astrophysical plasmas have been observed to evolve on an Alfvén time scale from previously stable states: tokamak disruptions and solar prominence eruptions are standard examples. Recently, Cowley and Artun [Physics Reports, 1996] have described a theoretical ``detonation'' mechanism for the rapid release of a plasma's stored energy in the context of the line-tied Rayleigh-Taylor instability: a plasma locally crossing the stability boundary nonlinearly evolves to destabilize previously stable regions. In Nonlinear Magnetohydrodynamic Detonation Part I the detonation mechanism is extended to general magnetic geometries, and a nonlinear equation describing the evolution of a perturbation transverse to a marginally unstable field line is derived. The nonlinear equation is similar in structure to that obtained by Cowley and Artun, with the exception of an additional nonlinearity associated with field line shear. In this paper, the analysis and numerical simulations of the nonlinear equation are presented. The explosive detonation of the plasma is found to depend crucially on the magnitude of the new nonlinearity. Applications will be discussed.

[1R.28] Simulation of ELMs Initiated by Pellet Injection in Tokamaks

Three dimensional MHD simulations in a divertor tokamak geometry have been carried out in order to study the effects of pellet injection. An initial equilibrium is prepared which is MHD stable. Superposed on this is a spatially localized pressure blob. The blob is assumed to form when a pellet is injected in the plasma, and it ionizes and heats in a time much shorter than the time it takes the blob to move. Nonlinear simulations show that the blob spreads out along the field, while producing large cross - field pressure disturbances. These appear to trigger subcritical ballooning modes: the poloidal velocity field has a typical ballooning structure. Plasma pressure flows across the separatrix into the divertor, similar to nonlinear MHD simulations (H. Strauss, BAPS 40, 1875 (1995)) of the evolution of pressure driven giant ELMs. The computations were carried out using FEM3D, a finite element RMHD code, and with MH3D++, an unstructured mesh version of the MH3D code.

[1R.29] Stability of Two-Dimensional Vortices in Incompressible Fluids and Plasmas

The three-dimensional instability of two-dimensional vortices with elliptical streamlines has attracted much attention in hydrodynamics. It has been shown [Pierrehumbert 1986; Bayly 1986] that the vortices are unstable to an infinite number of wavenumbers with the point of accumulation at infinity. For uniform vortices without axial flow, an exact solution of the linear ideal problem is also an exact solution of the nonlinear viscous problem. The equivalence between eigenmodes and Floquet-type solutions for these vortices is analogous to that found for ballooning modes with sheared flow in a torus. Using the recent approach of Taylor and Wilson (1996), we show that this equivalence breaks down in the presence of nonlinear shear because the scale-invariance of the equations is lost. The amplitude functions associated the Floquet solutions are shown to be modified over long time scales. Implications for the stability of such vortices in MHD will be discussed. \rule2in1pt ^\astWork supported by NSF and AFOSR.

[1R.30] Comparison of Tokamak Axisymmetric Mode Growth Rates from Linear MHD and Equilibrium Evolution Approaches

The axisymmetric displacement dynamics of free boundary plasma equilibrium configurations surrounded by conductors in an external magnetic field are described by both the linear MHD and equilibrium evolution approaches. A comparison of these models was made for \hboxDIII--D like free boundary equilibria. The equilibria were ideally stable to allow comparison of the resistive growth rates. Computed growth rates from both approaches are in good agreement for configurations without perturbed surface currents (up-down symmetric equilibria). However, considerable differences were found for configurations with a perturbed surface current (up-down asymmetric equilibria), and absence of the perturbed surface current in the equilibrium evolution model can lead to overestimation of vertical displacement growth rates. Common and specific features of the two approaches and the results of the comparison are presented and discussed.

[1R.31] Aspect-Ratio and Profile Effects on Resistive Wall Stabilization by Toroidal Rotation in Tokamaks

Numerical calculations(A. Bondeson and D. J. Ward, Phys.\ Rev.\ Lett.\ 72), 2709 (1994)^,(D. J. Ward and A. Bondeson, Phys. Plasmas 2) 1570 (1995) have shown that it is possible to stabilize pressure-driven kink modes, where the value for \beta is above the Troyon limit, in a tokamak plasma surrounded by a resistive wall given sufficient toroidal rotation. Experimental results(E. J. Strait, et al., Phys.\ Rev.\ Lett.\ 74) 2483 (1995)^,(M. Okabayashi, et al., to appear in Nucl. Fusion (1996)) have confirmed this stabilizing effect.

The effect of rotation on pressure-driven modes is complicated because of the toroidal coupling between different poloidal harmonics and between the Alfvén and sound waves. Rotation modifies the eigenfunction at resonances near the rational surfaces, and calculations indicate^2,4 that additional rational surfaces, particularly those in regions of relatively high pressure, make the stabilization more effective. Results will be presented here examining the effect of multiple rational surfaces in the equilibrium. The effect of aspect ratio on resistive wall stabilization, especially for cases with low aspect ratio in which many rational surfaces reside in the plasma, will be examined.

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[1R.32] Variational principle for rotating MHD stability

The recent experimental discovery of reversed magnetic shear(F. M. Levinton et al.), Phys.\ Rev.\ Lett.\ 75, 4417 (1995)^,(E. J. Strait et al.), Phys.\ Rev.\ Lett.\ 75, 4421 (1995). modes with fluctuation-free core and high plasma rotation velocity highlighted the importance of the MHD stability of rotating plasmas. The standard MHD energy principle loses self-adjointness and usefulness in the presence of an equilibrium plasma flow v, because it ignores the conservation laws associated with both the magnetic field and the fluid velocity. We argue that, in addition to the usual frozen-in law restricting the magnetic field variation to \deltaB =\nabla\times(\bf\xi\timesB), one also needs to include the local conservation of the cross helicity \intv\cdotB\,d^3x between nested magnetic flux surfaces. This leads to a new restriction on the velocity field variation required to build a necessary and sufficient nonlinear energy principle for stability of rotating plasmas. In terms of the vorticity \bfømega=\nabla\timesv, the flow variation is written, \delta\bfømega=\nabla\times(\bf\xi\times\bfømega+ \bf\eta\timesB), where \bf\xi and \bf\eta are arbitrary ``displacements.'' The corresponding energy principle and new methods of MHD stability analysis will be discussed.

Part 1 of program listing