Stability of an MHD shear flow with a piecewise linear velocity profile

Abstract

In this paper we present the results of the stability analysis of a simple shear flow of an incompressible fluid with a piecewise linear velocity profile in the presence of a magnetic field. In the flow, a finite transitional magnetic-free layer with a linear velocity profile is sandwiched by two semi-infinite regions. One of these regions is magnetic-free and the flow velocity in the region is constant. The other region is magnetic and the fluid in it is quiescent. The magnetic field is constant and parallel to the flow in the transitional layer. The fluid density is constant both in the magnetic as well as the magnetic-free regions, while it has a jump-type discontinuity at the boundary between the transitional layer and the magnetic region. The effect of gravity is included in the model, and it is assumed that the lighter fluid is overlaying the heavier one, thus no Rayleigh-Taylor instability is present. The dispersion equation governing the normal-mode stability of the flow is derived and its properties are analysed. We study stability of two cases: (i) magnetic-free flow in the presence of gravity, and (ii) magnetic flow without gravity. In the first case, the flow stability is controlled by the Rayleigh number, R. In the second case, the control parameter is the inverse squared Alfvénic Mach number, H. Stability of a particular monochromatic perturbation also depends on its dimensionless wavenumber α. We combine the analytical and numerical approaches to obtain the neutral stability curves in the (α, R)-plane in the case of the magnetic-free flow, and in the (α, H)-plane in the case of the magnetic flow. The dependence of the instability increment on R in the first case, and on H in the second case is treated. We apply the results of the analysis to the stability of a strongly subsonic portion of the heliopause. Our main conclusion is as follows: The inclusion of a transitional layer near the heliopause into the model increases by an order of magnitude the strength of the interstellar magnetic field required to stabilize this portion of the heliopause in comparison with the corresponding stabilizing strength of the magnetic field required when modelling the heliopause as a tangential discontinuity.

MHD Waves and Instabilities in Space Plasma Flows

“…These results were reconsidered by Ruderman and Brevdo (2006). In accordance with their analysis the transitional layer with a continuous velocity variation at the solar wind side of the heliopause would destabilise the flow.…”

MHD-shock structures of astrospheres: λ Cephei -like astrospheres

“…In a 3D HD simulations by Reyes-Iturbide et al (2019) these features can also be found. A detailed analysis of the KH in the case of the heliosphere is given by Ruderman & Brevdo (2006), where the authors show that the magnetic field stabilizes the astropause. The feature of a smooth astropause is mainly seen in heliospheric simulations (for example Pogorelov et al 2017a) and in the 3D MHD astrosphere simulations by Katushkina et al (2018) and Gvaramadze et al (2018).…”

Dispersive shock waves in the solar wind

“…Observations of periodic motion in the last decade through high resolution space and ground-based telescopes have changed our view about the dynamics of solar plasma in many ways. The observed waves and oscillations are used to describe the stability of the plasma (Ruderman & Brevdo 2006;Simpson, Ruderman & Erdélyi 2006), to tackle the complicated problem of coronal heating and plasma acceleration in the solar wind (Ballai, Thelen & Roberts 2003), to prescribe the dynamical behaviour of various solar structures (see, e.g. De Pontieu, Erdélyi & James 2004;Marcu, Ballai & Pintér 2006) or to sample the magnetic field, both in coronal loops and quiet Sun (Nakariakov et al 1999;Ballai 2007).…”