STABILITY OF INTERFACIAL WAVES IN ALUMINUM 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 wh ich the Lorentz force is expressed explicitly in terms of the fluid mo tion. Our wave equation differs from previous models, most notably tha t developed by Urata (1985), in that earlier formulations rested on a more complex, implicit coupling between the fluid motion and the Loren tz force. Our formulation furnishes a number of quite general stabilit y results without the need to resort to Fourier analysis. (The need fo r Fourier analysis typifies previous studies.) Moreover, our equation supports both travelling and standing waves. We investigate each in tu rn. We obtain three new results. First, we show that travelling waves may become unstable in the presence of a uniform, vertical magnetic fi eld. (Our travelling waves are quite different to those discovered by previous investigators (Sneyd 1985 and Moreau & Ziegler 1986) which re quire more complex magnetic fields to become unstable.) Second, in lin e with previous studies we confirm that standing waves may also become unstable. In this context we derive a simple energy criterion which s hows which types of motion may extract energy from the background magn etic held. This indicates that a rotating, tilted interface is particu larly 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. Thi s allows us to place a lower bound on the critical value of the backgr ound magnetic field at which an instability first appears, without sol ving the governing equations of motion.