Mikael Beuthe

East–west faults due to planetary contraction

Abstract Contraction, expansion and despinning have been common in the past evolution of Solar System bodies. These processes deform the lithosphere until it breaks along faults. Their characteristic tectonic patterns have thus been sought for on all planets and large satellites with an ancient surface. While the search for despinning tectonics has not been conclusive, there is good observational evidence on several bodies for the global faulting pattern associated with contraction or expansion, though the pattern is seldom isotropic as predicted. The cause of the non-random orientation of the faults has been attributed either to regional stresses or to the combined action of contraction/expansion with another deformation (despinning, tidal deformation, reorientation). Another cause of the mismatch may be the neglect of the lithospheric thinning at the equator or at the poles due either to latitudinal variation in solar insolation or to localized tidal dissipation. Using thin elastic shells with variable thickness, I show that the equatorial thinning of the lithosphere transforms the homogeneous and isotropic fault pattern caused by contraction/expansion into a pattern of faults striking east–west, preferably formed in the equatorial region. By contrast, lithospheric thickness variations only weakly affect the despinning faulting pattern consisting of equatorial strike-slip faults and polar normal faults. If contraction is added to despinning, the despinning pattern first shifts to thrust faults striking north–south and then to thrust faults striking east–west. If the lithosphere is thinner at the poles, the tectonic pattern caused by contraction/expansion consists of faults striking north/south. I start by predicting the main characteristics of the stress pattern with symmetry arguments. I further prove that the solutions for contraction and despinning are dual if the inverse elastic thickness is limited to harmonic degree two, making it easy to determine fault orientation for combined contraction and despinning. I give two methods for solving the equations of elasticity, one numerical and the other semi-analytical. The latter method yields explicit formulas for stresses as expansions in Legendre polynomials about the solution for constant shell thickness. Though I only discuss the cases of a lithosphere thinner at the equator or at the poles, the method is applicable for any latitudinal variation of the lithospheric thickness. On Iapetus, contraction or expansion on a lithosphere thinner at the equator explains the location and orientation of the equatorial ridge. On Mercury, the combination of contraction and despinning makes possible the existence of zonal provinces of thrust faults differing in orientation (north–south or east–west), which may be relevant to the orientation of lobate scarps.

Spatial patterns of tidal heating

Abstract 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.

Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets

SUMMARY Planetary topography can either be modelled as a load supported by the lithosphere, or as a dynamic effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However, local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this paper, we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Youngs modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. While being constrained by the static assumption, degree-one loads can deform the shell and generate stresses. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.

Tidal Love numbers of membrane worlds: Europa, Titan, and Co

Abstract Under tidal forcing, icy satellites with subsurface oceans deform as if the surface were a membrane stretched around a fluid layer. ‘Membrane worlds’ is thus a fitting name for these bodies and membrane theory provides the perfect toolbox to predict tidal effects. I describe here a new membrane approach to tidal perturbations based on the general theory of viscoelastic–gravitational deformations of spherically symmetric bodies. The massive membrane approach leads to explicit formulas for viscoelastic tidal Love numbers which are exact in the limit of zero crust thickness. Formulas for load Love numbers come as a bonus. The accuracy on k 2 and h 2 is better than one percent if the crust thickness is less than five percents of the surface radius, which is probably the case for Europa and Titan. The new approach allows for density differences between crust and ocean and correctly includes crust compressibility. This last feature makes it more accurate than the incompressible propagator matrix method. Membrane formulas factorize shallow and deep interior contributions, the latter affecting Love numbers mainly through density stratification. I show that a screening effect explains why ocean stratification typically increases Love numbers instead of reducing them. For Titan, a thin and dense liquid layer at the bottom of a light ocean can raise k 2 by more than ten percents. The membrane approach can also deal with dynamical tides in a non-rotating body. I show that a dynamical resonance significantly decreases the tilt factor and may thus lead to underestimating Europa’s crust thickness. Finally, the dynamical resonance increases tidal deformations and tidal heating in the crust if the ocean thickness is of the order of a few hundred meters.

Tides on Europa: The membrane paradigm

Abstract 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 ν ¯ and the membrane spring constant Λ, 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 Λ , ν ¯ , the ocean-to-bulk density ratio and the number k 2 ∘ 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 ( Λ ) 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 ∘ ) ). 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.

Global contraction/expansion and polar lithospheric thinning on Titan from patterns of tectonism

We investigate the underlying physical processes that govern the formation and evolution of Titans tectonic features. This is done by mapping mountain chains and hills using Cassini RADAR data obtained during Titan flybys T3 to T69. Our mapping of mountain chains and hills reveals a global pattern: east-west orientations within 30° of the equator and north-south between 60° latitude and the poles. This result makes Titan one of the few solar system bodies where global processes, rather than regional processes, dominate tectonism. After comparison with five global stress models showing theoretical mountain chain orientations, we suggest that either global contraction coupled with spin-up or global expansion coupled with despinning could explain our observations if coupled with a lithosphere thinner in Titans polar regions.

Enceladus’s crust as a non-uniform thin shell: I tidal deformations

Abstract The geologic activity at Enceladus’s south pole remains unexplained, though tidal deformations are probably the ultimate cause. Recent gravity and libration data indicate that Enceladus’s icy crust floats on a global ocean, is rather thin, and has a strongly non-uniform thickness. Tidal effects are enhanced by crustal thinning at the south pole, so that realistic models of tidal tectonics and dissipation should take into account the lateral variations of shell structure. I construct here the theory of non-uniform viscoelastic thin shells, allowing for depth-dependent rheology and large lateral variations of shell thickness and rheology. Coupling to tides yields two 2D linear partial differential equations of the fourth order on the sphere which take into account self-gravity, density stratification below the shell, and core viscoelasticity. If the shell is laterally uniform, the solution agrees with analytical formulas for tidal Love numbers; errors on displacements and stresses are less than 5% and 15%, respectively, if the thickness is less than 10% of the radius. If the shell is non-uniform, the tidal thin shell equations are solved as a system of coupled linear equations in a spherical harmonic basis. Compared to finite element models, thin shell predictions are similar for the deformations due to Enceladus’s pressurized ocean, but differ for the tides of Ganymede. If Enceladus’s shell is conductive with isostatic thickness variations, surface stresses are approximately inversely proportional to the local shell thickness. The radial tide is only moderately enhanced at the south pole. The combination of crustal thinning and convection below the poles can amplify south polar stresses by a factor of 10, but it cannot explain the apparent time lag between the maximum plume brightness and the opening of tiger stripes. In a second paper, I will study the impact of a non-uniform crust on tidal dissipation.

Ocean tidal heating in icy satellites with solid shells

Abstract As a long-term energy source, tidal heating in subsurface oceans of icy satellites can influence their thermal, rotational, and orbital evolution, and the sustainability of oceans. We present a new theoretical treatment for tidal heating in thin subsurface oceans with overlying incompressible elastic shells of arbitrary thickness. The stabilizing effect of an overlying shell damps ocean tides, reducing tidal heating. This effect is more pronounced on Enceladus than on Europa because the effective rigidity on a small body like Enceladus is larger. For the range of likely shell and ocean thicknesses of Enceladus and Europa, the thin shell approximation of Beuthe (2016) is generally accurate to less than about 4%. Explaining Enceladus’ endogenic power radiated from the south polar terrain by ocean tidal heating requires ocean and shell thicknesses that are significantly smaller than the values inferred from gravity and topography constraints. The time-averaged surface distribution of ocean tidal heating is distinct from that due to dissipation in the solid shell, with higher dissipation near the equator and poles for eccentricity and obliquity forcing, respectively. This can lead to unique horizontal shell thickness variations if the shell is conductive. The surface displacement driven by eccentricity and obliquity forcing can have a phase lag relative to the forcing tidal potential due to the delayed ocean response. For Europa and Enceladus, eccentricity forcing generally produces greater tidal amplitudes due to the large eccentricity values relative to the obliquity values. Despite the small obliquity values, obliquity forcing generally produces larger phase lags due to the generation of Rossby–Haurwitz waves. If Europa’s shell and ocean are, respectively, 10 and 100 km thick, the tide amplitude and phase lag are 26.5 m and