Boundary layer instability at the top of the Earths outer core
Benoit Desjardinsa, Emmanuel Dormyb,
Emmanuel Grenierc
aC.E.A./D.I.F., B.P. 12, Bruyères le Chatel, 91680 France
bI.P.G.P./C.N.R.S., 4 place Jussieu, Paris Cedex 05, 75252 France
cU.M.P.A., E.N.S. Lyon, 46 allée dItalie, Lyon Cedex 07,
69364 France
Journal of Computational and Applied Mathematics 166 (2004) 123-131
Abstract.
We investigate the instability of mixed EkmanHartmann boundary layers
arising in rotating incompressible
magnetohydrodynamics 6ows 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 theta n.e. pi/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 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 as far as 45o 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 6ows 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.
Keywords: Fluid dynamics; Magnetohydrodynamics; Boundary layers