JET-P(98)14

MHD Flow Instability in Presence of Resistive Wall

A conducting fluid flowing along a magnetic field parallel to a conducting wall is unstable if the flow velocity exceeds a critical value. The uniform flow of an ideally conducting mhd fluid along a uniform magnetic field is stable at all velocities. This can be seen by transforming to the frame of the fluid. In this frame the fluid is stationary and only the familiar stable waves can occur. If the flow is sheared a variant of the Kelvin-Kelmholtz instability is possible, the relative velocity of the fluid layers then providing a source of free energy. If, for the uniform flow case, an ideally conducting wall parallel to the flow is introduced the flow remains stable, the boundary condition for perturbations at the wall being independent of the flow velocity. However, if the finite conductivity of the wall is allowed for, the physics is changed. Now the flow velocity affects the magnetic field perturbation at the wall and, conversely, the wall provides a frame with respect to which the fluid flow has a free energy. Configurations of this type can be unstable, and this is illustrated here by considering the simplest case, that of the uniform flow of an incompressible fluid in slab geometry. For low flow velocities there are three stable waves, two forward waves propagating in the direction of the flow and a backward wave propagating against the flow. If the flow velocity is increased, instability appears for a given wave number when the phase velocity of the “backward” wave changes sign and the wave propagates in the direction of the flow. The critical velocity is a function of the wave number, the Alfvén velocity, the fluid-wall separation and the electrical resistance of the wall.
Name Size  
JETP98014 65.80 Kb