Isostasy with Love: I Elastic equilibrium

Abstract

Isostasy explains why observed gravity anomalies are generally much weaker than what is expected from topography alone, and why planetary crusts can support high topography without breaking up. Classical isostasy, however, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies. Isostasy should instead be defined either by minimizing deviatoric elastic stresses within the elastic shell, or by studying the dynamic response of the body in the long-time limit. I implement here the first option by formulating Airy isostatic equilibrium as the response of an elastic shell to surface and internal loads. Isostatic ratios are defined in terms of deviatoric Love numbers which quantify deviations with respect to a fluid state. The Love number approach separates the physics of isostasy from the technicalities of elastic-gravitational spherical deformations, and provides flexibility in the choice of the interior structure. Since elastic isostasy is invariant under a global rescaling of the shell shear modulus, it can be defined in the fluid shell limit, which reveals a deep connection with viscous isostasy. If the shell is homogeneous, minimum stress isostasy is dual to a variant of elastic isostasy called zero deflection isostasy, which is less physical but simpler to compute. Each isostatic model is combined with general boundary conditions applied at the surface and bottom of the shell, resulting in one-parameter isostatic families. At long wavelength, the influence of boundary conditions disappears as all isostatic families members yield the same isostatic ratios. At short wavelength, topography is supported by shallow stresses so that Airy isostasy becomes similar to either pure top or bottom loading. The isostatic ratios of incompressible bodies with three homogeneous layers are implemented in freely available software.