Quarterly of Applied Mathematics

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Variational approach to nonlinear gravity-driven instabilities in a MHD setting

Author(s): Hyung Ju Hwang
Journal: Quart. Appl. Math. 66 (2008), 303-324.
MSC (2000): Primary 76E30
Posted: February 8, 2008
MathSciNet review: 2416775
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Abstract | References | Similar articles | Additional information

Abstract: We establish a variational framework for nonlinear instabilities in a setting of the ideal magnetohydrodynamic (MHD) equations. We apply a variational method to unstable smooth steady states for both incompressible and compressible ideal MHD equations. The destabilizing effect of compressibility is justified along with the stabilizing effect of magnetic field lines arising in the MHD dynamics. This generalizes the result of the Rayleigh-Taylor instability for incompressible fluids in the absence of magnetic field lines; see Hwang and Guo, On the dynamical Rayleigh-Taylor instability, Arch. Rational Mech. Anal. 167 (2003), 235-253.

References:

1.: G. Bateman, MHD instabilities, MIT Press, Cambridge, MA, 1980.
2.: I. B. Bernstein, E. A. Frieman, M. D. Kruskal, R. M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. (London) A244 (1958), 17-40. MR 0091737 (19:1009e)
3.: S. Cordier, E. Grenier, Y. Guo, Two-stream instabilities in plasmas, In honor of C. S. Morawetz. Methods Appl. Anal. 7 (2000), no. 2, 391-405. MR 1869291 (2003g:82091)
4.: S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford 1961. MR 0128226 (23:B1270)
5.: S. Chandrasekhar, The stability of the radiative gradients in the interior of a star, Proc. Natl. Acad. Sci. U. S. 23 (1937), 572-577.
6.: C. Cherfils-Cléouin, O. Lafitte, P-A. Raviart, Asymptotic results for the linear stage of the Rayleigh-Taylor instability, Mathematical fluid mechanics, 47-71, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2001. MR 1865049 (2002h:76059)
7.: T. G. Cowling, The stability of gaseous stars, Monthly Notices Roy. Astro. Soc. 94 (1934), 768.
8.: E. Grenier, Semiclassical limit of the nonlinear Schrödinger equations, Proc. AMS. 126 (1998), no. 2, 523-530. MR 1425123 (98d:35204)
9.: Y. Guo, W. Strauss, Instability of periodic BGK equilibria, Comm. Pure. Appl. Math. 48 (1995), 861-894. MR 1361017 (96j:35252)
10.: H. J. Hwang, Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Rational Mech. Anal. 167 (2003), 235-253. MR 1978583 (2004f:76064)
11.: H. Jeffreys, The stability of a layer of fluid heated below, Phil. Mag. 2 (1926), 833-844.
12.: M. D. Kruskal, M. Schwarzschild, Some instabilities of a completely ionized plasma, Proc. Roy. Soc. (London) A223 (1954), 348-360. MR 0061997 (15:914a)
13.: P-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. MR 1637634 (99m:76001)
14.: L. D. Landau, E. M. Lifshitz, Fluid mechanics, 2nd ed., Pergamon, New York, 1987. MR 961259 (89i:00006)
15.: A. Majda, Compressible fluid flows and systems of conservation laws in several variables. Appl. Math. Sci. 53, Springer, 1984.
16.: D. Nicholson, Introduction to plasma theory, John Wiley & Sons, 1983.
17.: E. R. Priest, Solar magnetohydrodynamics, London, 1983.
18.: L. Rayleigh, Analytic solutions of the Rayleigh equation for linear density profiles, Proc. London. Math. Soc. 14 (1883), 170-177.
19.: K. Schwarzchild, Equilibrium of the sun's atmosphere, Nachr. Kgl. Ges. Wiss. Göttingen 1 (1906), 41-53.
20.: Yu. A. Tserkovikov, Convective instability of a rarefied plasma. Doklady Akad. Nauk S. S. S. R. 130 (1960), 295 [translation: Soviet Phys.-Doklady 5 (1960), 87].

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