Effect of parallel magnetic field on the compressible Kelvin-Helmholtz problem

Abstract

In this paper, we present an analysis of the KH instability in 2D magnetohydrodynamic (MHD) flows, providing rigorous confirmation that a parallel magnetic field can have a destabilizing effect on this instability. When the Mach number M := v₊ / c lies strictly between M_low := sqrt((1 - q)/(1 + beta)) * (1 + epsilon_0) and M_upp := sqrt((1 + q)/(1 - beta)) * (1 - epsilon_0), where beta := c_A^2 / c^2 and epsilon_0 > 0 is a small but fixed constant, we prove the linear and nonlinear ill-posedness of the KH problem for compressible MHD flows.

Keywords

free surface / Kelvin-Helmholtz instability / two-dimensional MHD flows / parallel magnetic field

References

[1] Y. G. Wang, F. Yu, Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 208 (2013), 341-389. https://doi.org/10.1007/s00205-012-0601-9

[2] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Oxford University Press, New York, 1961. https://doi.org/10.1063/1.3058072

[3] A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, M. Lopez-Fernandez, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math., 175 (2012), 909-948. https://doi.org/10.4007/annals.2012.175.2.9

[4] Y. Guo, I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-712. https://doi.org/10.1512/iumj.2011.60.4193

[5] Y. Guo, I. Tice, Linear Rayleigh-Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2011), 1688-1720. https://doi.org/10.1137/090777438

[6] H. J. Hwang, Y. Guo, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal., 167 (2003), 235-253. https://doi.org/10.1007/s00205-003-0243-z

[7] F. Jiang, S. Jiang, W. Zhan, Instability of the abstract rayleigh-taylor problem and applications, Math. Models Methods Appl. Sci., 30 (2020), 2299-2388. https://doi.org/10.1142/S021820252050044X

[8] D. G. Ebin, Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids, Comm. Partial Differential Equations, 13 (1988), 1265-1295. https://doi.org/10.1080/03605308808820576

[9] H. L. F. Helmholtz, On the discontinuous movements of fluids, Sitz.ber. Preuss. Akad.Wiss. Berl. Philos.-Hist. Kl., 23 (1868), 215-228.

[10] W. Thomson, Hydrokinetic solutions and observations, Philos. Mag., 42 (1871), 362-377.

[11] H. Hasegawa, M. Fujimoto, T. D. Phan, H. Reme, A. Balogh, M. W. Dunlop, et al., Transport of solar wind into earth's magnetosphere through rolled-up kelvin-helmholtz vortices, Nature, 430 (2004), 755-758. https://doi.org/10.1038/nature02799

[12] J. W. Dungey, Electrodynamics of the outer atmosphere, In: Proceedings of the Ionosphere, The Physical Society of London, London, 1955, 255-265.

[13] E. N. Parker, Dynamics of the interplanetary gas and magnetic fields, Astrophys. J., 128 (1958), 664-678. https://doi.org/10.1086/146579

[14] P. A. Sturrock, R. E. Hartle, Two-fluid model of the solar wind, Phys. Rev. Lett., 16 (1966), 628-636. https://doi.org/10.1103/physrevlett.16.628

[15] A. I. Ershkovich, Solar wind interaction with the tail of comet kohoutek, Planet. Space Sci., 24 (1976), 287-290. https://doi.org/10.1016/0032-0633(76)90025-8

[16] J. F. Coulombel, P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012.

[17] J. F. Coulombel, P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. cole Norm. Supr., 41 (2008), 85-139. https://doi.org/10.24033/asens.2064

[18] A. Morando, P. Trebeschi, Two-dimensional vortex sheets for the nonisentropic euler equations: linear stability, J. Hyperbolic Differ. Equ., 5 (2008), 487-518. https://doi.org/10.1142/S021989160800157X

[19] A. Morando, P. Trebeschi, T. Wang, Two-dimensional vortex sheets for the nonisentropic euler equations: nonlinear stability, J. Differential Equations, 266 (2019), 5397-5430. https://doi.org/10.1016/j.jde.2018.10.029

[20] R. M. Chen, J. Hu, D. Wang, Linear stability of compressible vortex sheets in 2d elastodynamics: variable coefficients, Math. Ann., 376 (2020), 863-912. https://doi.org/10.1007/s00208-018-01798-w

[21] R. M. Chen, J. Hu, D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math., 311 (2017), 18-60. https://doi.org/10.1016/j.aim.2017.02.014

[22] R. M. Chen, F. M. Huang, D. Wang, D. F. Yuan, Stabilization effect of elasticity on three-dimensional compressible vortex sheets, J. Math. Pures Appl., 172 (2023), 105-138. https://doi.org/10.1016/j.matpur.2023.01.005

[23] B. Q. Xie, B. Zhao, D. W. Huang, Ill-posedness of the Kelvin-Helmholtz problem for incompressible MHD fluids, Preprint.

[24] J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech., 4 (1958), 538-552. https://doi.org/10.1017/S0022112058000653

[25] J. A. Fejer, J. W. Miles, On the stability of a plane vortex sheet with respect to three-dimensional disturbances, J. Fluid Mech., 15 (1963), 335-336. https://doi.org/10.1017/S002211206300029X

[26] B. Q. Xie, B. Zhao, Kelvin-Helmholtz instability for compressible Euler fluids, Math. Methods Appl. Sci., accepted. https://doi.org/10.1002/mma.70669

[27] B. Q. Xie, B. L. Guo, B. Zhao, Effect of weak elasticity on the Kelvin-Helmholtz instability, J.Math. Fluid Mech., 28 (2026), 20. https://doi.org/10.1007/s00021-025-00983-7

[28] Y. Z. Sun, W. Wang, Z. F. Zhang, Nonlinear stability of the current-vortex sheet to the incompressible mhd equations, Commun. Pure Appl. Math., 71 (2018), 356-403. https://doi.org/10.1002/cpa.21710

[29] B. Q. Xie, B. L. Guo, B. Zhao, Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field, J. Math. Phys., 66 (2025), 121506. https://doi.org/10.1063/5.0268159

[30] Y. Trakhinin, Existence of compressible current-vortex sheets: variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331-366. https://doi.org/10.1007/s00205-005-0364-7

[31] Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. https://doi.org/10.1007/s00205-008-0124-6

[32] G. Q. Chen, Y. G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369-408. https://doi.org/10.1007/s00205-007-0070-8

[33] B. Q. Xie, B. Zhao, Effect of transverse magnetic field on the compressible Kelvin-Helmholtz problem, J. Geom. Anal., 36 (2026), 7. https://doi.org/10.1007/s12220-025-02247-2

[34] A. Morando, P. Secchi, P. Trebeschi, On the evolution equation of compressible vortex sheets, Math. Nachr., 293 (2020), 945-969. https://doi.org/10.1002/mana.201800162

[35] Z. Y. Pu, M. G. Kivelson, Kelvin-helmholtz instability at the magnetopause solution for compressible plasmas, J. Geophysical Res., 88 (1983), 841-853. https://doi.org/10.1029/ja088ia02p00841

[36] S. J. Wu, Mathematical analysis of vortex sheets, Comm. Pure. Appl. Math, 59 (2006), 1065-1206. https://doi.org/10.1002/cpa.20110

[37] G. Lebeau, Regularite du probleme de Kelvin-Helmholtz pour l'equation d'euler 2d, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 801-825. https://doi.org/10.1051/cocv:2002052

[38] Y. J. Wang, Z. P. Xin, Existence of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics, Comm. Pure Appl. Math., 77 (2024), 583-629. https://doi.org/10.1002/cpa.22148

[39] J. Simon, Compact sets in the space L^p(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96.