PHYSICAL REVIEW A VOLUME 44, NUMBER 10 15 NOVEMBER 1991 Nonlinear magnetohydrodynamics by Galerkin-method computation Xiaowen Shan, David Montgomery, and Hudong Chen Department of Physics and Astronomy, Dartmouth College, Hanover, Neur Hampshire 03755-3528 (Received 26 November 1990; revised manuscript received 24 July 1991) A fully spectral numerical code is used to explore the properties of voltage-driven dissipative magnetoAuids inside a periodic cylinder with circular cross section. The trial functions are orthonormal eigenfunctions of the curl (Chandrasekhar-Kendall functions). Transitions are observed from axisym- metric resistive equilibria without Row to helically deformed laminar states with Aow, and between pairs of helical laminar states with different pairs of poloidal and toroidal m and n numbers. States of minimum energy-dissipation rate seem to be preferred. At high values of the pinch ratio, fully developed magnetohydrodynamic turbulence is observed. PACS number(s): 52. 30. — q, 52. 55.Ez, 52. 55. Fa, 47.65. + a I. INTRODUCTION This article reports some results from a computational method of exploring electrically driven, dissipative states of a conducting Quid. The situation contains much of the essential magnetohydrodynamics (MHD) of confined fusion plasmas in a torus. However, due to the lack of good internal pointwise diagnostics for the MHD fields in the present generation of tokamaks and toroidal Z pinches, numerical computations provide the possibility of much more detailed descriptions of the dynamical be- havior than presently imaginable laboratory measure- ments do. The price is, of course, the necessary omission of some of the non-MHD features of a real plasma. The computations reported here originate in an earlier set performed a few years ago [1] with a pseudospectral code interior to a rigid, perfectly conducting cylinder with square cross section and periodically identified ends. Those investigations started from a position of consider- ably less understanding than we now have of just what to expect. The Dahlburg et al. computations [1] had to be prepared for essentially anything to happen. The history of MHD computation had often been one of freezing out dynamically important phenomena by assuming sym- metries at the outset that were not satisfied by the phe- nomena that turned out eventually most important. The square-cylinder computations led, among other things, to one result that was robust and unexpected: a transition, above certain critical thresholds in the applied toroidal voltage, to a nonaxisymmetric state involving helical distortions of the current channel and a small but nontrivial amount of Aow in the form of paired helical vortices. In the presence of spatially and temporally vari- able temperatures and temperature-dependent transport coe%cients, the helical states could persist even while ex- ecuting superimposed "sawtooth" oscillations [2]. A theoretical explanation of the behavior was sought and the best explanation so far has seemed to be one of attributing the presence of the helical MHD states to their lower rate of energy dissipation due to resistivity and viscosity [3]. An analytical solution was constructed [4] through the first three terms in a perturbation series (with the expansion parameter being formally the ratio of the helical magnetic components to the larger axisym- metric one), which showed that the steady-state solution to the MHD equations bifurcated at the appearance of the first linear (nonideal) instability. The unstable eigen- mode [5] was constructed of Chandrasekhar-Kendall [6] helical eigenfunctions of the curl and formed the variable helical part of the nonaxisymmetric solution. This helical equilibrium was shown to have a lower energy dissipation rate than the axisymmetric one. Visual comparisons of surfaces of constant values of the field components and amplitudes, between analytically calculated and numeri- cally computed cases, were encouraging enough to sug- gest further exploitation of the Chandrasekhar-Kendall functions [7]. A three-mode (I. orenz-like [8]) truncated Galerkin- method corrlputation was explored by Chen, Shan, and Montgomery [9]. Near the threshold at which the phase-space point corresponding to the axisymmetric state ("axisymmetric fixed point") became linearly unsta- ble, the three-mode model exhibited behavior that was qualitatively the same as the well-resolved computations [1, 2] and the minimum-dissipation theory [4, 7] had sug- gested: relaxation to the steady state of lower energy- dissipation rate, represented by a helical fixed point of the three-mode dynamical system. It was also suggested [9] that, because of the ease with which the Chandrasekhar- Kendall functions seemed to incorporate economically the results that had originated in the earlier well-resolved ( — 3 X 10 degrees of freedom) MHD computations [1], they might be a useful expansion basis for a many-mode spectral computation. In Sec. II we describe the construction of such a spec- tral code. A recent completeness theorem due to Yoshi- da and Giga [10] fills in what had been a worrisome gap in the mathematical underpinnings of such an expansion. Though the ultimate operation of the code is rather sim- ple, the analysis involved in its construction is not alto- gether standard. One motivation in Sec. II is to put some of the details on record before proceeding with a set of applications that may extend well into the future. Some of these applications are presented in Sec. III and will 6800 QC 1991 The American Physical Society