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Plasma dynamics in thin domains

Darryl D. Holm, Ruiao Hu, Oliver D. Street

Abstract

In the present work, we study the geometric structures of the Rotating Shallow Water Magnetohydrodynamics (RSW-MHD) equations through a Lie group invariant Euler-Poincaré variational principle. In this geometric framework, we derive new, structure-preserving stochastic RSW-MHD models by introducing stochastic perturbations to the Lie-Poisson structure of the deterministic RSW-MHD equations. The resulting stochastic RSW-MHD equations provide new capabilities for potential application to uncertainty quantification and data assimilation, for example, in space plasma (space weather) and solar physics, particularly in solar tachocline dynamics.

Plasma dynamics in thin domains

Abstract

In the present work, we study the geometric structures of the Rotating Shallow Water Magnetohydrodynamics (RSW-MHD) equations through a Lie group invariant Euler-Poincaré variational principle. In this geometric framework, we derive new, structure-preserving stochastic RSW-MHD models by introducing stochastic perturbations to the Lie-Poisson structure of the deterministic RSW-MHD equations. The resulting stochastic RSW-MHD equations provide new capabilities for potential application to uncertainty quantification and data assimilation, for example, in space plasma (space weather) and solar physics, particularly in solar tachocline dynamics.

Paper Structure

This paper contains 15 sections, 2 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of the paper.
  3. Rotating Shallow Water MHD (RSW-MHD)
  4. A variational principle for the RSW-MHD equations
  5. Stream function version of Lagrangian
  6. Hamiltonian structure of RSW-MHD equations
  7. A stream function for the magnetic field.
  8. Thermal effects in shallow water magnetohydrodynamics
  9. Lagrangian formulation.
  10. Hamiltonian formulation.
  11. Stochastic rotating shallow water magnetohydrodynamics
  12. Stochastic Advection by Lie Transport (SALT).
  13. Stochastic Forcing by Lie Transport (SFLT).
  14. Conclusion and outlook
  15. Open problems and future work.

Key Result

Proposition 2.1

The shallow water MHD equations possesses the following Casimir invariances, for arbitrary smooth functions $\phi$ and $\varphi$. The quantity $q$ denotes the potential vorticity (PV) associated with the flow, which is defined as

Figures (1)

  • Figure 1: Schematic of the notation for RSW-MHD.

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2: The coordinate free action
  • Remark 2.3: The form of the Lagrangian
  • Remark 2.4: Weakly versus strongly magnetized regimes
  • Remark 2.5
  • Proposition 2.1
  • proof
  • Remark 2.6: Equvalence of Poisson brackets
  • Remark 3.1
  • Proposition 3.1
  • ...and 6 more