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

[1] 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: (rho(0,1)rho(0,2))(V-k,V-1 - V-k,V-2)(2) < (4 pi)(-1)(rho(0,1) + rho(0,2))[(B-k,B-1)(2) + (B-k,B-2)(2)]( a). Here the indices 1 and 2 refer to quantities on either side of the magnetopause, rho(0) is the plasma density, and V-k, B-k are the projections of the plasma velocity <(V-o)over right arrow> and magnetic field (B-o) over right arrow on the direction of the wave vector (k) over right arrow, respectively. An example of a continuous velocity profile with finite thickness D 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 Delta --> 0. Using hyperbolic tangent profiles for rho(o), (V-o) over right arrow , and (B-o) over right arrow , 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.