Tides on Europa: the membrane paradigm

Jupiter's moon Europa has a thin icy crust which is decoupled from the mantle by a subsurface ocean. The crust thus responds to tidal forcing as a deformed membrane, cold at the top and near melting point at the bottom. In this paper I develop the membrane theory of viscoelastic shells with depth-dependent rheology with the dual goal of predicting tidal tectonics and computing tidal dissipation. Two parameters characterize the tidal response of the membrane: the effective Poisson's ratio $\bar\nu$ and the membrane spring constant $\Lambda$, the latter being proportional to the crust thickness and effective shear modulus. I solve membrane theory in terms of tidal Love numbers, for which I derive analytical formulas depending on $\Lambda$, $\bar\nu$, the ocean-to-bulk density ratio and the number $k_2^o$ representing the influence of the deep interior. Membrane formulas predict $h_2$ and $k_2$ with an accuracy of a few tenths of percent if the crust thickness is less than one hundred kilometers, whereas the error on $l_2$ is a few percents. Benchmarking with the thick-shell software SatStress leads to the discovery of an error in the original, uncorrected version of the code that changes stress components by up to 40%. Regarding tectonics, I show that different stress-free states account for the conflicting predictions of thin and thick shell models about the magnitude of tensile stresses due to nonsynchronous rotation. Regarding dissipation, I prove that tidal heating in the crust is proportional to $Im \Lambda$ and that it is equal to the global heat flow (proportional to $Im k_2$) minus the core-mantle heat flow (proportional to $Im k_2^o$). As an illustration, I compute the equilibrium thickness of a convecting crust. More generally, membrane formulas are useful in any application involving tidal Love numbers such as crust thickness estimates, despinning tectonics or true polar wander.