Study of Combined Effect of Kelvin?Helmholtz Instability and Gravitational Instability on a Self?Gravitating Fluid Layer

Abstract

The instability of a self-gravitating fluid layer of finite thickness surrounded by another fluid of different density has been studied recently by Uberoi (1963) and Tassoul (1967) § under varying conditions. Now the condition can arise when the fluid inside the layer and the surrounding material are in relative horizontal motion. It is interesting to study the combined effects of the Kelvin-Helmholtz (KH) instability associated with short wavelengths and the gravitational instability associated with long wavelengths on this layer.We consider a homogeneous distribution of gravitating ideally conducting fluid mass with constant density p in the form of a plane layer of thickness 2h. The X 0 Y plane is taken to coincide with the unperturbed middle level of the layer and the positive z axis in the upward direction normal to the unperturbed fluid surfaces. This layer is surrounded by a nonconducting fluid of uniform density po. In the equilibrium state we assume that the system is pervaded by a uniform magnetic field HI in the conducting layer and H 2 in the nonconducting fluid, both directed in the x direction. We further assume that initially the conducting and nonconducting fluids are moving with velocities VI and V2 respectively in the x direction. As the disturbances along the direction of the streaming velocities of the fluids and the magnetic field are most sensitive to the KH instability (Chandrasekhar 1961), we shall consider the wave propagation in the x direction only. Hence, we assume that the perturbed quantities depend on time t and spatial coordinate x as exp{i(wt+kx)}. As the mathematical procedure is well known (e.g. Uberoi 1963; Tassoul 1967), we shall not give the details here. Following the procedure given in Uberoi (1963), we obtain the dispersion relation, which factorizes into the following two factors corresponding to the asymmetric and symmetric perturbations:

Effects of Hall currents on the instability of a gravitating partially ionized plasma in a two dimensional horizontal magnetic field

“…Analysing the disturbance of the normal modes we assume that the perturbed quantities have a dependence on the space co-ordinate (x, y, z) and time t of the form F(z) exp (ik, x + ik, y + nt), (7) where F(z) is some function of z and k, and k, (k2 = k i + k;) are the horizontal and vertical wave numbers and n denotes the rate at which a system departs from equilibrium.…”

“…Using (7) and eliminating some of the variables from the above equations and writing we obtain the equations…”

“…The instability of a self-gravitating plasma layer of varying density has been investigated by among others, Uberoi [SI, Sundaram [6], Oberoi [7], Bajwa and Srivastava [8]. All these investigations of plasma instabilities have been carried out in the absence of Hall currents.…”