On the MHD boundary of Kelvin-Helmholtz stability diagram at large wavelengths

Abstract

Working within the domain of inviscid incompressible MHD theory, we found that a tangential discontinuity (TD) separating two uniform regions of different density, velocity and magnetic field may be Kelvin-Helmholtz (KH) stable and yet a study of a transition between the same constant regions given by a continuous velocity profile shows the presence of the instability with significant growth rates. Since the cause of the instability stems from the velocity gradient, and since a TD may be considered as the ultimate limit of such gradient, the statement comes as a surprise. In fact, a long wavelength (λ) boundary for the KH instability does not exist in ordinary liquids being instead a consequence of the presence of magnetic shear, a possibility that has passed unnoticed in the literature. It is shown that KH modes of a magnetic field configuration with constant direction do not have the long λ boundary. A theoretical explanation of this feature and examples of the violation of the TD stability condition are given using a model that can be solved in closed form. Stability diagrams in the (kd, MA) plane are given (where kd = 2πd/λ, 2d is the velocity gradient length scale, and MA is the Alfvénic Mach number) that show both the well-known limit at small λs and the boundary for large but finite λs noted here. Consequences of this issue are relevant for stability studies of the dayside magnetopause as the stability condition for a TD should be used with care in data analysis work.

Aspects of magnetopause/magnetosphere response to interplanetary discontinuities, and features of magnetopause Kelvin–Helmholtz waves

“…The ratio l=D of the mode wavelength (l ¼ 2p=jkj) to the width D of the boundary layer is important because a TD model (where D-0) may be stable while a finite-D theory shows instability (Gratton et al, 2004(Gratton et al, , 2005). An important feature of the MP both on the flanks as well as on the frontside is the presence of the low latitude boundary layer (LLBL), a region whose physical values are intermediate between those of the magnetosheath and the magnetosphere proper.…”

“…reviews by Belmont and Chanteur, 1989;Kivelson and Chen, 1995;Farrugia et al, 2001). Studies on KH activity at the dayside MP, too, revealed many interesting features (Farrugia et al, 1998;Gratton et al, 2004). The situation at the MP is complicated by two factors: (i) the magnetized nature of the flow around a magnetized planetary environment, because of the need to overcome restraining magnetic forces for the waves to grow, and, particularly down the flanks, (ii) the effect of compressibility, which is generally stabilizing.…”

Emission of magnetosound from MHD‐unstable shear flow boundaries

“…Some studies have assumed a zero thickness boundary, which is appropriate if the wavelengths of the generated waves are significantly larger than the thickness of the boundary [ Pu and Kivelson , ; Mann et al , ; Turkakin et al , , ]. The effects of a finite boundary thickness have also been investigated in several studies [ Ong and Roderick , ; Walker , ; Farrugia et al , ; Contin et al , ; Gratton et al , ]. The main effect of including a finite thickness boundary is that only perturbations with sufficiently small wave numbers are unstable [ Walker , ; Miura and Pritchett , ; Gratton et al , ].…”

Supersonic mixing layers: Stability of magnetospheric flanks models

“…A feature worth noting is that the limit of very long wavelengths λ » ∆ is stable, while the calculation shows instability at the CLUSTER site for λ ~ ∆. This means that a stability analysis performed with a TD model for the MP would arrive at a wrong conclusion, i.e., a stability verdict while in fact the site is unstable (on this issue see [13] The data, however, show evidence that the MP structure was stratified with a scale-length for B, n, and a different one for V. Another calculation was done, therefore, with a two-scale model. An example of field profiles for this case is shown in figure 8.…”

“…A thin model (tangential discontinuity) for M s = 1.4 and M A = 1.2 fails to predict this instability. Therefore a "pitfall" of the TD model (see [13]) occurs also for supersonic flows. This is worth keeping in mind when the popular TD model is used in the flanks for downstream estimates.…”