Concerning a problem on the Kelvin‐Helmholtz stability of the thin magnetopause

Abstract

According to incompressible MHD theory, when the magnetopause is modeled as a tangential discontinuity with jumps in the field and flow parameters, it is Kelvin‐Helmholtz (KH) stable when the following inequality is satisfied: (ρ 0,1 ρ 0,2 )( V ,1 − V ,2) 2 < (4π) −1 (ρ 0,1 + ρ 0,2 )[( B ,1) 2 + ( B ,2) 2 ] ( a ). Here the indices 1 and 2 refer to quantities on either side of the magnetopause, ρ 0 is the plasma density, and V , B κ are the projections of the plasma velocity and magnetic field on the direction of the wave vector , respectively. An example of a continuous velocity profile with finite thickness Δ that can be solved in closed form is presented for which condition ( a ) is satisfied. Yet the configuration can be shown to be KH unstable, and it approaches stability only in the limit Δ → 0. Using hyperbolic tangent profiles for ρ 0 , , and , and solving the stability problem numerically with parameters typical of the dayside magnetopause, we show cases of unstable configurations, all of which are stable according to ( a ). This possibility, as far as we know, has passed unnoticed in the literature. Being incompressible, the theory applies to subsonic regions of the dayside magnetopause. We conclude that condition ( a ) must be used with care in data analysis work.