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magnetohydrodynamics; Alfvén waves; Fourier analysis; singularity; small perturbations; equilibrium plasma; mixed elliptic-hyperbolic system
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$\left(A_1\Delta_u\right) + A_2u =0$, where $A_1$ may have singularities in the domaind $U$ 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 absorptions 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 $U$.
[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. DOI 10.1063/1.1694904
[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. DOI 10.1029/JA079i007p01024
[5] Kivelson M. G., Southwood D. J.: Resonant ULF waves: A new interpretation. Geoph. Res. Letters 12 (1985), 49-52. DOI 10.1029/GL012i001p00049
[6] Freidberg J. P.: Ideal Magnetohydrodynamic theory of magnetic fusion systems. Rev. of Modern Phys. 54 (1982), 801-902. DOI 10.1103/RevModPhys.54.801
[7] Tataronis J. A.: Energy absorption in the continuous spectrum of ideal Magnetohydrodynamics. J. Plasma Phys. 13 (1975), 87-105. DOI 10.1017/S0022377800025897
[8] Sedlacek Z.: Electrostatic oscillations in cold inhomogeneous plasma. J. Plasma Phys. 5 (1971), 239-263. DOI 10.1017/S0022377800005754
[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. DOI 10.1007/BF01391914
[10] Bender C. M., Orszag S. A.: Advanced Mathematical Methods for Scientist and Engineers. McGraw-Hill, 1984. MR 0538168
[11] Olver F. W. J.: Asymptotics and Special Functions. Academic Press, 1974. MR 0435697 | Zbl 0308.41023
[12] Stein E. M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1975. MR 0304972
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