The ‘‘Alfvén paradox’’ is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfvén continuum. To resolve the paradox, the ε‐pseudospectrum of the resistive magnetohydrodynamic (RMHD) operator is considered. It is proven that for any ε, the ε‐pseudospectrum contains the Alfvén continuum for sufficiently small resistivity. Formal ε‐pseudoeigenmodes are constructed using the formal Wentzel–Kramers–Brillouin–Jeffreys solutions, and it is shown that the entire stable half‐annulus of complex frequencies with ρ‖ω‖2=‖k⋅B(x)‖2 is resonant to order ε, i.e., belongs to the ε‐pseudospectrum. The resistive eigenmodes are exponentially ill‐conditioned as a basis and the condition number is proportional to exp(R1/2M), where RM is the magnetic Reynolds number.
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October 1994
Research Article|
October 01 1994
The pseudospectrum of the resistive magnetohydrodynamics operator: Resolving the resistive Alfvén paradox
D. Borba;
D. Borba
JET Joint Undertaking, Abingdon, Oxon, OX14 3EA, England
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K. S. Riedel;
K. S. Riedel
New York University, 251 Mercer St., New York, New York 10012‐1185
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W. Kerner;
W. Kerner
JET Joint Undertaking, Abingdon, Oxon, OX14 3EA, England
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G. T. A. Huysmans;
G. T. A. Huysmans
JET Joint Undertaking, Abingdon, Oxon, OX14 3EA, England
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M. Ottaviani;
M. Ottaviani
JET Joint Undertaking, Abingdon, Oxon, OX14 3EA, England
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P. J. Schmid
P. J. Schmid
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195
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Phys. Plasmas 1, 3151–3160 (1994)
Article history
Received:
January 26 1994
Accepted:
June 23 1994
Citation
D. Borba, K. S. Riedel, W. Kerner, G. T. A. Huysmans, M. Ottaviani, P. J. Schmid; The pseudospectrum of the resistive magnetohydrodynamics operator: Resolving the resistive Alfvén paradox. Phys. Plasmas 1 October 1994; 1 (10): 3151–3160. https://doi.org/10.1063/1.870468
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