Spatial patterns of tidal heating

In a body periodically strained by tides, heating produced by viscous friction is far from homogeneous. The spatial distribution of tidal heating depends in a complicated way on the tidal potential and on the internal structure of the body. I show here that the distribution of the dissipated power within a spherically stratified body is a linear combination of three angular functions. These angular functions depend only on the tidal potential whereas the radial weights are specified by the internal structure of the body. The 3D problem of predicting spatial patterns of dissipation at all radii is thus reduced to the 1D problem of computing weight functions. I compute spatial patterns in various toy models without assuming a specific rheology: a viscoelastic thin shell stratified in conductive and convective layers, an incompressible homogeneous body and a two-layer model of uniform density with a liquid or rigid core. For a body in synchronous rotation undergoing eccentricity tides, dissipation in a mantle surrounding a liquid core is highest at the poles. Within a soft layer (or asthenosphere) in contact with a more rigid layer, the same tides generate maximum heating in the equatorial region with a significant degree-four structure if the soft layer is thin. The asthenosphere can be a layer of partial melting in the upper mantle or, very differently, an icy layer in contact with a silicate mantle or solid core. Tidal heating patterns are thus of three main types: mantle dissipation (with the icy shell above an ocean as a particular case), dissipation in a thin soft layer and dissipation in a thick soft layer. Finally, I show that the toy models predict well patterns of dissipation in Europa, Titan and Io. The formalism described in this paper applies to dissipation within solid layers of planets and satellites for which internal spherical symmetry and viscoelastic linear rheology are good approximations.