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A WKB analysis of the Alfvén spectrum of the linearized magnetohydrodynamics equations. (English) Zbl 0778.76100

Summary: Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation \(div(A_ 1\bigtriangledown u)+A_ 2u=0\), where \(A_ 1\) may have singularities in the domain \(U\) of definition. We study the case when \(U\) is a half-plane and \(u\) possesses high Fourier components, analyzing the changes brought about by the singularity \(A_ 1=\infty\). We show that absorption of energy takes place precisely at this singularity, that the solutions have a near harmonic character, and the integrability characteristics of the boundary data are kept throughout \(U\).

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
35Q60 PDEs in connection with optics and electromagnetic theory
34E05 Asymptotic expansions of solutions to ordinary differential equations
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References:

[1] Grad H.: Phys. Today 22 (1969), 34. · Zbl 0181.28501
[2] Chen L., Hasegawa A.: Plasma heating by spatial resonance of Alfvén wave. Phys. of Fluids 17(1974), 1399-1403.
[3] Tataronis J., Talmadge J. N., Shohet J. L.: Alfvén wave heating in general toroidal geometry. Univ. of Wisconsin Report (1978).
[4] Chen L., Hasegawa A.: Steady State excitation of field line resonance. J. of Geoph. Res. 79 (1974), 1024-1037.
[5] Kivelson M. G., Southwood D. J.: Resonant ULF waves: A new interpretation. Geoph. Res. Letters 12 (1985), 49-52.
[6] Freidberg J. P.: Ideal Magnetohydrodynamic theory of magnetic fusion systems. Rev. of Modern Phys. 54 (1982), 801-902.
[7] Tataronis J. A.: Energy absorption in the continuous spectrum of ideal Magnetohydrodynamics. J. Plasma Phys. 13 (1975), 87-105.
[8] Sedlacek Z.: Electrostatic oscillations in cold inhomogeneous plasma. J. Plasma Phys. 5 (1971), 239-263.
[9] Grossmann W.,Tataronis J. A.: Decay of MHD waves by phase mixing II: the Theta-Pinch in cylindrical geometry. Z. Physik 261 (1973), 217-236.
[10] Bender C. M., Orszag S. A.: Advanced Mathematical Methods for Scientist and Engineers. McGraw-Hill, 1984.
[11] Olver F. W. J.: Asymptotics and Special Functions. Academic Press, 1974. · Zbl 0308.41023
[12] Stein E. M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1975. · Zbl 0232.42007
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