  Two main seismic features characterize the Earth's inner core: a North-South
polar anisotropy and an East-West asymmetry of P-wave velocity and attenuation.
Anisotropy is expected if shear deformation is induced by convective motions.
Translation has recently been put forward as an important mode of convection of
the inner core. Combined with a simple diffusive grain growth model, this
mechanism is able to explain the observed seismic asymmetry, but not the bulk
anisotropy. The source of anisotropy has therefore to be sought in the shear
motions caused by higher modes of convection. Using a hybrid finite-difference
spherical harmonics Navier-Stokes solver, we investigate the interplay between
translation and convection in a 3D spherical model with permeable boundary
conditions at the inner core boundary. Three parameters act independently:
viscosity, internal heating and convection velocity in the outer core. Our
numerical simulations show the dominance of pure translation for viscosities of
the inner core higher than $10^{20}$ Pas. Translation is almost completely
hampered by convective motions for viscosities lower than $10^{18}$ Pas.
Between these values, translation and convection develop, but convective
downwellings are restricted to the coldest hemisphere where crystallization
occurs. On the opposite side, shear is almost absent, thereby allowing grain
growth. We propose that the coexistence of translation and convection observed
in our numerical models leads to a seismic asymmetry but localizes deformation
only in one hemisphere.
