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2006 48th Annual Meeting of the Division of Plasma Physics

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Monday–Friday, October 30–November 3 2006;
Philadelphia, Pennsylvania

### Session VI2: MHD

2:00 PM–5:00 PM,
Thursday, November 2, 2006

Philadelphia Marriott Downtown
Room: Grand Salon CDE

Chair: Darren Craig, Wheaton College

Abstract ID: BAPS.2006.DPP.VI2.5

### Abstract: VI2.00005 : Intermediate Nonlinear Development of a Line-tied $g$-Mode*

4:00 PM–4:30 PM

Preview Abstract
Abstract

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Author:

Ping Zhu

(University of Wisconsin-Madison)

The nonlinear gravitational instability ($g$-mode) of a line-tied
plasma flux tube is a prototypical model for edge localized modes
(ELMs) in tokamaks and magnetotail substorms. Earlier theory
predicted
the explosive nonlinear growth of these modes near marginal
stability~[S.~C. Cowley and M. Artun, Phys. Rep., {\bf 283}, 185-211
(1997)]. Recent direct MHD simulations with both a finite-difference
code and NIMROD indicate that the mode remains bounded in magnitude
throughout, from early to intermediate nonlinear phases~[P. Zhu,
A. Bhattacharjee, and K. Germaschewski, Phys. Rev. Lett. {\bf 96},
065001 (2006); P. Zhu, C.~C. Hegna, and C.~R. Sovinec, submitted to
Phys. Plasmas (2006)]. The mode grows nonlinearly at a rate near or
smaller than the linear growth rate, producing shock-like
discontinuities and large sheared flows. To understand these
simulation results, a new theoretical framework has been
developed. The theory is based on an expansion using two small
parameters, $\epsilon\sim |{\mbox{\boldmath $\xi$}}|/L_{\rm
eq}\ll 1$,
and $n^{-1}\sim k_\parallel/k_\perp\ll 1$, where ${\mbox{\boldmath
$\xi$}}$ denotes the plasma displacement, $L_{\rm eq}$ is the
characteristic equilibrium scale, and $k_\parallel$ and $k_\perp$ are
the dominant wavenumbers of the perturbation parallel and
perpendicular to equilibrium magnetic field lines, respectively. When
$\epsilon\sim n^{-1}$, the Cowley-Artun regime is recovered where the
plasma is incompressible to the lowest order and the Lagrangian
compression is very small [$\nabla_0\cdot{\mbox{\boldmath
$\xi$}}\sim{\cal O}(n^{-1})$]. In this regime, the nonlinearities
only
modify the development of the global mode envelop across field lines
whereas the local eigenmode structure along field lines remains
intact. The detonation regime where the nonlinear growth of the mode
tends to a finite-time singularity is a narrower subset of the
Cowley-Artun regime. However, this regime is not generic and breaks
down when $\epsilon\gg n^{-1}$. In the intermediate nonlinear phase
when $\epsilon\sim n^{-1/2}$, the lowest order Lagrangian
compression is significant [$\nabla_0\cdot{\mbox{\boldmath
$\xi$}}\sim{\cal O}(1)$]. During this phase, the nonlinearities
directly influence
the growth of the local eigenmodes, and couple the global mode
structure across and along field lines. The corresponding governing
equations for this intermediate nonlinear phase are
derived. Comparison of the predictions of analytic theory and
numerical simulations will be discussed.

*In collaboration with C. C. Hegna and C. R. Sovinec, University of Wisconsin-Madison; A. Bhattacharjee and K. Germaschewski, University of New Hampshire. Research supported by U.S. DOE and by U.S. NSF.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2006.DPP.VI2.5