Stability of interfacial waves in aluminium reduction cells

We investigate the stability of interfacial waves in conducting fluids under the influence of a vertical current density, paying particular attention to aluminium reduction cells in which such instabilities are commonly observed. We develop a wave equation for the interface in which the Lorentz force is expressed explicitly in terms of the fluid motion. Our wave equation differs from previous models, most notably that developed by Urata (1985), in that earlier formulations rested on a more complex, implicit coupling between the fluid motion and the Lorentz force. Our formulation furnishes a number of quite general stability results without the need to resort to Fourier analysis. (The need for Fourier analysis typifies previous studies.) Moreover, our equation supports both travelling and standing waves. We investigate each in turn.We obtain three new results. First, we show that travelling waves may become unstable in the presence of a uniform, vertical magnetic field. (Our travelling waves are quite different to those discovered by previous investigators (Sneyd 1985 and Moreau & Ziegler 1986) which require more complex magnetic fields to become unstable.) Second, in line with previous studies we confirm that standing waves may also become unstable. In this context we derive a simple energy criterion which shows which types of motion may extract energy from the background magnetic field. This indicates that a rotating, tilted interface is particularly prone to instability, and indeed such a motion is often seen in practice. Finally, we use Gershgorin's theorem to produce a sufficient condition for the stability of standing waves in a finite domain. This allows us to place a lower bound on the critical value of the background magnetic field at which an instability first appears, without solving the governing equations of motion.