Hydromagnetic edge waves and instability in reduction cells

To clarify the function of gravity in the shallow-water theory of the interfacial instability in aluminium reduction cells, we analyse the existing long-wave theory of the instability in the limit of vanishing gravitational acceleration g. The flow then has an inner and outer structure, with gravity remaining essential within thin layers coating the cell walls. In those thin wall layers, the growing disturbance takes the form of a trapped magnetogravity wave propagating horizontally on the internal interface, and the growth rate (sigma) is determined by the coupling of that edge wave to the large-scale flow in the core of the cell; that coupling is expressed as an oblique-derivative problem for the core flow. Although(sigma) is asymptotically independent of g, gravity is essential to the long-wave instability because a correlation imposed by the magneto-gravity waves is essential for the disturbance to extract power from the mean state.