Boundary layer instability at the top of the Earth's outer core

We investigate the instability of mixed Ekman-Hartmann boundary layers arising in rotating incompressible magnetohydrodynamics flows in a parameter regime relevant to the Earth liquid core. Relying on the small depth of the boundary layer, we perform a local study in a half-space at a given co-latitude θ≠π/2, and assume a mean dipolar axial magnetic field with internal sources (the geodynamo). Instabilities are driven, for high enough Reynolds number, by the quadratic term in the momentum equation. Nonlinear stability can be proven using energy methods in the neighborhood of the poles (Nonlinearity 12 (2) (1999) 181). Next, following the work of Lilly (J. Atmos. Sci. 23 (1966) 481), we restrict our analysis to the linear growth phase. We describe the dependence of the critical Reynolds number in terms of θ and Elsasser number (measuring the relative strength of Lorentz and Coriolis forces). It turns out that no matter how large the Elsasser number is, there exists a critical band centered on the equator in which instabilities can occur. For geophysically relevant values of parameters, this band could extend as far as 45° away from the equator. This establishes the possibility of boundary layer instabilities near the Earth core-mantle boundary (CMB). We finally present a first attempt of interaction with field maps at the CMB and core flows derived from the secular variation of the field, and discuss the sensitivity of the instability onset not only on the boundary layer Reynolds number, but also on the direction of the flow.