# Nonlinear ideal magnetohydrodynamics instabilities

## Abstract

Explosive phenomena such as internal disruptions in toroidal discharges and solar flares are difficult to explain in terms of linear instabilities. A plasma approaching a linear stability limit can, however, become nonlinearly and explosively unstable, with noninfinitesimal perturbations even before the marginal state is reached. For such investigations, a nonlinear extension of the usual MHD (magnetohydrodynamic) energy principle is helpful. (This was obtained by Merkel and Schlueter, Sitzungsberichted. Bayer. Akad. Wiss., Munich, 1976, No. 7, for Cartesian coordinate systems.) A coordinate system independent Eulerian formulation for the Lagrangian allowing for equilibria with flow and with built-in conservation laws for mass, magnetic flux, and entropy is developed in this paper which is similar to Newcomb's Lagrangian method of 1962 [Nucl. Fusion, Suppl., Pt. II, 452 (1962)]. For static equilibria nonlinear stability is completely determined by the potential energy. For a potential energy which contains second- and [ital n]th order or some more general contributions only, it is shown in full generality that linearly unstable and marginally stable systems are explosively unstable even for infinitesimal perturbations; linearly absolutely stable systems require finite initial perturbations. For equilibria with Abelian symmetries symmetry breaking initial perturbations are needed, which should be observed in numerical simulations.more »

- Authors:

- (Max-Planck-Institut fuer Plasmaphysik, EURATOM Association, D-8046 Garching (Germany))
- (Laboratory of Plasma Studies, Cornell University, Ithaca, New York 14853 (United States))

- Publication Date:

- OSTI Identifier:
- 6232275

- Alternate Identifier(s):
- OSTI ID: 6232275

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Fluids B; (United States); Journal Volume: 5:7

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; MHD EQUILIBRIUM; VARIATIONAL METHODS; CONSERVATION LAWS; MAGNETOHYDRODYNAMICS; PLASMA INSTABILITY; POTENTIAL ENERGY; SYMMETRY; CALCULATION METHODS; ENERGY; EQUILIBRIUM; FLUID MECHANICS; HYDRODYNAMICS; INSTABILITY; MECHANICS 700370* -- Plasma Fluid & MHD Properties-- (1992-); 700340 -- Plasma Waves, Oscillations, & Instabilities-- (1992-)

### Citation Formats

```
Pfirsch, D., and Sudan, R.N.
```*Nonlinear ideal magnetohydrodynamics instabilities*. United States: N. p., 1993.
Web. doi:10.1063/1.860792.

```
Pfirsch, D., & Sudan, R.N.
```*Nonlinear ideal magnetohydrodynamics instabilities*. United States. doi:10.1063/1.860792.

```
Pfirsch, D., and Sudan, R.N. Thu .
"Nonlinear ideal magnetohydrodynamics instabilities". United States. doi:10.1063/1.860792.
```

```
@article{osti_6232275,
```

title = {Nonlinear ideal magnetohydrodynamics instabilities},

author = {Pfirsch, D. and Sudan, R.N.},

abstractNote = {Explosive phenomena such as internal disruptions in toroidal discharges and solar flares are difficult to explain in terms of linear instabilities. A plasma approaching a linear stability limit can, however, become nonlinearly and explosively unstable, with noninfinitesimal perturbations even before the marginal state is reached. For such investigations, a nonlinear extension of the usual MHD (magnetohydrodynamic) energy principle is helpful. (This was obtained by Merkel and Schlueter, Sitzungsberichted. Bayer. Akad. Wiss., Munich, 1976, No. 7, for Cartesian coordinate systems.) A coordinate system independent Eulerian formulation for the Lagrangian allowing for equilibria with flow and with built-in conservation laws for mass, magnetic flux, and entropy is developed in this paper which is similar to Newcomb's Lagrangian method of 1962 [Nucl. Fusion, Suppl., Pt. II, 452 (1962)]. For static equilibria nonlinear stability is completely determined by the potential energy. For a potential energy which contains second- and [ital n]th order or some more general contributions only, it is shown in full generality that linearly unstable and marginally stable systems are explosively unstable even for infinitesimal perturbations; linearly absolutely stable systems require finite initial perturbations. For equilibria with Abelian symmetries symmetry breaking initial perturbations are needed, which should be observed in numerical simulations. Nonlinear stability is proved for two simple examples, [ital m]=0 perturbations of a Bennet Z-pinch and [ital z]-independent perturbations of a [theta] pinch. The algebra for treating these cases reduces considerably if symmetries are taken into account from the outset, as suggested by M. N. Rosenbluth (private communication, 1992).},

doi = {10.1063/1.860792},

journal = {Physics of Fluids B; (United States)},

number = ,

volume = 5:7,

place = {United States},

year = {Thu Jul 01 00:00:00 EDT 1993},

month = {Thu Jul 01 00:00:00 EDT 1993}

}