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Eigenmode initialisation of 2D (magneto)hydrodynamic simulations

Jordi De Jonghe, Alexander J. B. Russell

Abstract

The early evolution of unstable hydrodynamic and magnetohydrodynamic equilibria is often governed by a few dominant linear eigenmodes. We investigate whether initialising a simulation with a superposition of linear eigenmodes that contains the most unstable mode saves computation time, and how the selection of the included modes affects the non-linear evolution. Using the non-linear (magneto)hydrodynamic simulation code MPI-AMRVAC, the evolutions of a flow-sheared fluid interface, a Harris current sheet, and a flow-sheared plasma interface were simulated for various initial perturbations. The novel initial perturbations were linear eigenmodes of the equilibrium, or superpositions thereof, and calculated with the Legolas code. We benchmarked to initialisation with velocity noise and, in the case of the Harris sheet, initialisation with an analytic magnetic field perturbation. By perturbing an unstable equilibrium with a superposition of linear eigenmodes that contains the most unstable mode, significantly less computation time is spent on the linear stage of the evolution compared to traditional perturbation methods. In the best case, the simulation time needed to reach the non-linear stage is reduced by an order of magnitude. The inclusion or omission of certain modes in the initial perturbation is observed to affect the non-linear evolution to various degrees. The perturbation of equilibria with a superposition of linear eigenmodes that contains the most unstable mode allows simulations to reach a late-evolution stage faster, thus saving computation time. Additional eigenmodes can be included alongside the fastest growing mode to obtain further benefits, for example, to accelerate symmetry breaking in the non-linear stage, or to delay their effect on the non-linear evolution. Coupling spectroscopic codes with (magneto)hydrodynamic codes therefore offers significant advantages.

Eigenmode initialisation of 2D (magneto)hydrodynamic simulations

Abstract

The early evolution of unstable hydrodynamic and magnetohydrodynamic equilibria is often governed by a few dominant linear eigenmodes. We investigate whether initialising a simulation with a superposition of linear eigenmodes that contains the most unstable mode saves computation time, and how the selection of the included modes affects the non-linear evolution. Using the non-linear (magneto)hydrodynamic simulation code MPI-AMRVAC, the evolutions of a flow-sheared fluid interface, a Harris current sheet, and a flow-sheared plasma interface were simulated for various initial perturbations. The novel initial perturbations were linear eigenmodes of the equilibrium, or superpositions thereof, and calculated with the Legolas code. We benchmarked to initialisation with velocity noise and, in the case of the Harris sheet, initialisation with an analytic magnetic field perturbation. By perturbing an unstable equilibrium with a superposition of linear eigenmodes that contains the most unstable mode, significantly less computation time is spent on the linear stage of the evolution compared to traditional perturbation methods. In the best case, the simulation time needed to reach the non-linear stage is reduced by an order of magnitude. The inclusion or omission of certain modes in the initial perturbation is observed to affect the non-linear evolution to various degrees. The perturbation of equilibria with a superposition of linear eigenmodes that contains the most unstable mode allows simulations to reach a late-evolution stage faster, thus saving computation time. Additional eigenmodes can be included alongside the fastest growing mode to obtain further benefits, for example, to accelerate symmetry breaking in the non-linear stage, or to delay their effect on the non-linear evolution. Coupling spectroscopic codes with (magneto)hydrodynamic codes therefore offers significant advantages.
Paper Structure (10 sections, 12 equations, 16 figures, 3 tables)

This paper contains 10 sections, 12 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Results
  4. Turbulent evolution: Flow-sheared fluid interface
  5. Secondary instabilities: Harris current sheet
  6. Non-linear stabilisation: Flow-sheared plasma interface
  7. Conclusion
  8. Legolas convergence tests
  9. Construction and scaling of initial perturbations
  10. MPI-AMRVAC convergence tests

Figures (16)

  • Figure 1: (a) Equilibrium profiles of the flow-sheared fluid interface (Case 1), Eqs. \ref{['eq:fluid1']}-\ref{['eq:fluid2']}. (b) KHI growth rate for varying $\bm{k} = k\,\hat{\bm{e}}_y$. (c) Legolas spectrum showing the $k = 36.077$ eigenfrequencies of Eqs. \ref{['eq:fluid1']}-\ref{['eq:fluid2']} in the complex plane.
  • Figure 2: Snapshots of the density at $t = 0.8$ in Case 1 simulations starting from a perturbation with (a) KHI modes; (b) propagating waves; and (c) velocity noise.
  • Figure 3: Evolution of the mean (a) kinetic and (b) internal energy in Case 1 simulations starting from a perturbation with (1a) KHI modes, (1b) propagating waves, and (1c) velocity noise.
  • Figure 4: Snapshots of the density at $t = 3.0$ in Case 1 simulations starting from a perturbation with (a) KHI modes; (b) propagating waves; and (c) velocity noise. The associated movie is available online.
  • Figure 5: $x$-averaged 1D $k_y$ power spectrum for Case 1 simulations starting from a perturbation with (a) KHI modes; (b) propagating waves; (c) velocity noise; and a power law fit to Case 1a.
  • ...and 11 more figures