Tidal flow and dissipation in planetary cores

Tidal forcing at diurnal periods drives a flow in the liquid cores of planets, altering the nutation of these bodies. An approximate description of the flow in an inviscid fluid is comprised of a uniform vorticity and a potential flow to satisfy boundary conditions. This flow (often called Poincare flow) is unable to satisfy boundary conditions when the solid inner core tilts relative to the mantle or when the ellipticity of the inner-core boundary differs from the ellipticity of the core-mantle boundary. A numerical solution is used to show that internal shear layers develop when Poincare flow fails. The thickness of the shear layers scale as E1/3, where E is the Ekman number. The total viscous dissipation also scales as E1/3 when stress-free boundary conditions are assumed. Extrapolating E to realistic values for planetary cores suggests that viscous dissipation is negligible. However, shear layers also induce electric currents when the core is permeated with a magnetic field. Scaling arguments for a weak magnetic field suggest that the resulting ohmic dissipation is proportional to E-2/3. This estimate is supported by numerical calculations with E = 10-4 to 10-6. The singularity at vanishingly small values of E is avoided by the influence of the Lorentz force on the flow. The resulting ohmic dissipation should be detectable in observations of the Earth’s nutation, but it is insufficient to explain all of the dissipation that is actually observed. Strong magnetic coupling at the core-mantle and inner-core boundaries probably accounts for most of the observed dissipation.