Poro-viscoelastic tidal heating of Io

TL;DR

This work develops a self-consistent poro-viscoelastic framework for Io's partially molten mantle by coupling tidal deformation with melt segregation and porous-flow dissipation. The authors derive governing equations for a self-gravitating two-phase Maxwell material, linearise them in the frequency domain, and compute tidal heating from shear, compaction, and Darcy flow across a four-layer Io model. They find that Darcy heating could match or exceed shear heating only for large melt fractions and grain sizes, while compaction heating remains modest (often <1% of Io’s observed heating), indicating that two-phase effects modestly alter the heat budget unless extreme porosity and permeability conditions apply. Overall, the study provides a first-principles, self-consistent basis for the two-phase dynamics of Io’s interior and offers guidance on where Darcy and compaction processes might influence mantle convection and eruption timing, while highlighting the need for laboratory data on poroviscoelastic properties at tidal frequencies.

Abstract

Io's tidally driven global volcanism indicates widespread partial melting in its mantle. How this melt participates in the interior dynamics, and, in particular, the role it plays in tidal dissipation, is poorly understood. We model Io's tidal deformation by treating its mantle as a two-phase (solid and melt) system. By combining poro-viscous and poro-elastic compaction theories in a Maxwell framework with a consistent model of tidal and self-gravitation, we produce the first self-consistent evaluation of Io's tidal heating rate due to shearing, compaction, and Darcy flow. We find that Darcy dissipation can potentially exceed shear heating, but only for large (0.05 to 0.2) melt fractions, and if the grain size is large or melt viscosity ultra-low. Since grain sizes larger than 1cm are unlikely, this suggests that Darcy dissipation is secondary to shear dissipation. Compaction dissipation is maximised when the asthenosphere is highly resistive to isotropic stresses, but contributes at most 1% of Io's observed heating rate. This work represents a crucial step toward a self-consistent quantitative theory for the dynamics of Io's partially molten interior.

Figures (9)

• Figure 1: Qualitative tidal deformation solutions of a 300 km thick asthenosphere at perijove ($t=0$) for deformation due to a) shear, b) compaction, and c) melt segregation. The arrows in a) and c) indicate the solid and segregation displacements, respectively ($\bm{u}_s$ and $\bm{u}_{rel}$), tangent to the surface in which they are plotted. The tidal axis (red arrow) points towards Jupiter. The inner and outer sphere are the base and surface of the asthenosphere, respectively, and the colours represent the period-averaged volumetric heating rate within each surface plotted. Panels d), e) and f) show the total heating rate for shear, compaction, and Darcy flow, respectively, as a function of each mechanism's primary control parameter. Red stars correspond to the solution shown in the top row. Io's observed heating rate (red line) is from ref. lainey2009StrongTidal.
• Figure 2: a) Global- and time-averaged shear dissipation rate in the solid, $\dot{E}_{S}$, as a function of shear viscosity, $\eta$, for two different Biot's coefficient $\alpha$. The black dotted line indicates the heating rate for a model with an entirely solid asthenosphere. b) Normalised volumetric heating rate depth profiles for $\alpha=0.99$ and $\kappa_l =$ 1GPa, which correspond to the open circles in panel a). Normalisation is taken relative to the mean heating rate. In both panels, the solid is taken to be elastic in shear ($\eta \rightarrow \infty$), $\zeta$ is taken to be independent of porosity (which is held at $\phi = 0.1$), the asthenosphere thickness is $H =$ 300, and mobility is $M_\phi=$ 5e-7□mPas.
• Figure 3: a) Global- and time-averaged compaction dissipation rate in the solid, $\dot{E}_C$, as a function of compaction viscosity, $\zeta$, for different liquid bulk moduli, $\kappa_l$, and Biot's coefficient $\alpha$. b) Normalised- and spherically averaged volumetric heating rate depth profiles for $\alpha=0.01$ and $\kappa_l =$ 1GPa, which correspond to the open circles in panel a). Normalisation is taken relative to the mean heating rate. In both panels, the solid skeleton is taken to be elastic in shear ($\eta \rightarrow \infty$), $\zeta$ is taken to be independent of melt fraction (which is held at $\phi = 0.1$), the asthenosphere thickness is $H =$ 50, and mobility is $M_\phi =$ e-7□mPas.
• Figure 4: a) Global- and time-averaged Darcy dissipation rate, $\dot{E}_D$, as a function of mobility, $M_{\phi}$, for different liquid bulk moduli, $\kappa_l$, and Biot's coefficient $\alpha$. The colours represent the bulk modulus (compressibility) of the melt, with warmer colours denoting a more incompressible melt. Note that the lower $\kappa_l$ curves do not extend to the smallest mobilities due to numerical instability. b) Normalised- and spherically averaged volumetric heating rate depth profiles, which correspond to the open circles in panel a) for $\kappa_l \rightarrow \infty$. Normalisation is taken relative to the mean heating rate. In both panels, the solid is taken to be elastic, $M_{\phi}$ is taken to be independent of melt fraction (which is held at $\phi = 0.1$), the asthenosphere is $H =$ 50 thick, and compaction dissipation is ignored.
• Figure 5: a) Asthenosphere permeability as a function of melt fraction $\phi$ and grain size $a$, calculated using Eq. \ref{['eq:permbeability_phi']} with $K_0 = 50$. For the liquid viscosity of $\eta_l =$ 1Pas used in the manuscript, the colour scale can equivalently be interpreted in terms of mobility $\log_{10}(M_\phi)$. b) Compaction and shear viscosity ($\zeta$ and $\eta$, black), and the corresponding nondimensional Maxwell times ($\omega \eta/ \mu$ and $\omega \zeta / \kappa_d$, blue), as a function of melt fraction, calculated with Eqs. \ref{['eq:bulk_viscosity_phi']} and \ref{['eq:shear_viscosity_phi']}. c) Drained bulk modulus (black) and Biot's coefficient (blue) as a function of melt fraction, calculated with Eqs. \ref{['eq:bulk_modulus_phi']} and \ref{['eq:biot_coeff_phi']}. The power-law exponent used to calculate $\kappa_d$ in panels b) and c) is set to $b=$ 0.5 (solid lines), 1 (long dashes), and 1.5 (short dashes).