Long time behavior of Alfvén waves in flowing plasma: the destruction of the magnetic island. (English) Zbl 1475.76050
Summary: In this paper, we study the linearized MHD equations around a steady flowing plasma with the velocity field \((k_fy,0)\) and magnetic field \((k_m y,0)\). We prove the spectral stability of the linearized operator conjectured in [M. Hirota et al., “Resonance between continuous spectra: secular behavior of Alfvén waves in a flowing plasma”, Phys. Plasmas 12, No. 1, Article ID 012107, 11 p. (2005; doi:10.1063/1.1834591)] and confirm the destruction phenomena of a magnetic island observed in that work in the case when flow shear exceeds magnetic shear (\(0<|k_m|<|k_f|\)). The key ingredient of the proof is the resolvent estimate of the pressure. To this end, we have to overcome two main difficulties: coalescence of singularities and exclusion of embedding eigenvalues.
MSC:
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
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