Resistive wave breaking in the Earth's outer core

The equations for an electrically conducting fluid in cylindrical coordinates are linearized assuming that the inertial terms in the momentum equation can be ignored (small Rossby number), and that the ratio of the Elsasser number and magnetic Reynolds number is one. After these assumptions, the governing equations are linearized about an ambient solution which vanishes at the the equator. Upon assuming large Elsasser and magnetic Reynolds number, the solutions to the linearized equations are approximated by wave trains having very short wave length (relative to the core radius) but which vary slowly (on a scale of the core radius). The period of the waves is much longer than a day but much shorter than the period of the slow hydromagnetic oscillations. These waves are found to be trapped in a region about the equator and away from the axis of rotation. The waves break at a latitudinal wave region boundary, in the sense that the waves become exponentially large in a boundary layer, having as an exponent some positive power of the large azimuthal wave number. This behavior is amplified as the Elsasser number becomes smaller while still remaining relatively large. Waves in more Earth-like parameter regimes are discussed briefly.