Instability of Ekman-Hartmann Boundary Layers
near the Core Mantle Boundary
Benoit Desjardinsa, Emmanuel Dormyb,c, Emmanuel Grenierd
a C.E.A./D.I.F., B.P. 12, 91680 Bruyères le Châtel, France.
b IPGP, 4 place Jussieu, F-75252, Paris, France
c C.N.R.S., France.
d U.M.P.A., E.N.S. Lyon, 46 allée d'Italie,
69364 Lyon Cedex 07, France.
Proceedings of BAIL2002.
Abstract.
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
layer, we perform a local study in a half space
at a given co-latitude $\theta \neq \pi/2$, and assume a mean dipolar axial
magnetic field with internal sources. 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 [2]. Next, following the work of D. Lilly [6],
we restrict our analysis to the linear
growth phase. We describe the dependence of the critical Reynolds number
in terms of $\theta$ 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 until some 45 degrees
away from the equator. This establishes the possibility of boundary layer
instabilities near the 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 [4].
We investigate a possible relation between boundary layer instabilities
and rapid geomagnetic impulses (also called ``jerks'') observed some eight
times over the last century.