# Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

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## Abstract

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfvén and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.

## Keywords

Non-uniform current-vortex sheet Richtmyer–Meshkov instability Alfvén number Surface Alfvén wave MHD interfacial instability## Mathematics Subject Classification

76W05 76E17 76B47 76E30## Notes

### Acknowledgments

This work was supported by a Grant-in-Aid for Scientific Research (B) (Grant No. 26287147) and (C) (Grant No. 23540453) from the Japan Society for the Promotion of Science, a Grant-in-Aid for Research Promotion, Ehime University, and joint research project of ILE, Osaka University. The authors would like to thank Professor K. Hiraide and Professor S. Yanagi for their mathematical advice. We are deeply grateful to Professor A. Kageyama, Professor K. Kusano, Professor J. G. Wouchuk, and Professor Z. Yoshida for their valuable comments and discussions on plasma physics. We are also particularly indebted to Professor Y. Kaneda for his advice and suggestions.

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