A new approach to modeling the tidal dissipation in subsurface oceans of icy moons of Jupiter and Saturn
Authors: Aygün Burak , Čadek Ondřej 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Introduction&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Tidal dissipation is an important source of heat for several icy moons of Jupiter and Saturn. &amp;#160;While the evaluation of tidal heating in the solid parts of the moons is straightforward, with the only limitation being the lack of knowledge about anelastic properties of ice and silicates, modeling the tidal dissipation in subsurface oceans is challenging and the issue of how much it contributes to the total heat production has not yet been fully resolved. The standard approach to modeling the tidal dissipation in a subsurface ocean is based on the solution of hydrodynamic equations in the shallow water approximation [e.g., 1&amp;#8211; 4]. Numerical studies addressing the tidal deformation of the ocean in a three-dimensional geometry have appeared only recently [5,6]. Although these studies represent significant progress, they have one important limitation: they do not take into account the mechanical coupling between the flow in the ocean and the viscoelastic deformation of the ice shell and the core. To overcome this limitation, we have developed a new method that allows the tidal deformation to be solved simultaneously in all three parts of the icy moon. &amp;#160;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Method&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;The problem of tidal deformation is solved in a spherical computational domain corresponding in size to Saturn&amp;#8217;s moon Enceladus and consisting of three sub-domains (the viscoelastic ice shell, the viscous ocean, and the viscoelastic core), which are coupled by the boundary conditions. The Navier-Stokes equation is integrated in time and solved using a pseudo-spectral method based on spectral decomposition in angular coordinates and finite differences in radius. &amp;#160;The spherical harmonic coefficients are discretized in 800 unevenly spaced grid points guaranteeing a sufficiently high sampling density in the boundary layers.&amp;#160; Self-gravitation and the Coriolis force are included. The thickness of the ocean is varied from 10 m to about 50 km and its viscosity is considered in the range 1&amp;#8211;10&lt;sup&gt;6&lt;/sup&gt; Pa s.&amp;#160; Calculations are performed for two end-member cases, recently discussed in literature: a weakly deformable elastic core and a highly deformable Maxwell viscoelastic core [7].&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Results and Conclusion&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Our results confirm the conclusions of previous studies that tidal dissipation in the present day ocean on Enceladus is negligible compared to the expected heat production [e.g., 8]. However, unlike some previous models, we find less pronounced resonance peaks, suggesting that the mechanical coupling between the ocean and the solid parts of the moon tends to reduce the resonance effects. Our results indicate that dissipation in the ocean strongly depends on the ocean thickness, reaching the maximum value for a thickness of 1 km when the ocean viscosity is 10&lt;sup&gt;6&lt;/sup&gt; Pa s and less than 0.1 km when the viscosity is smaller 10 Pa s. The maximum dissipation predicted in the ocean (1 GW for the weakly deformable core and 20 GW for the highly deformable core) is independent of the viscosity prescribed in the ocean. This suggests that although the tidal dissipation currently plays a negligible role in Enceladus&amp;#8217; heat budget, it can effectively hinder the freezing of the ocean when the ocean is thin.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;[1] Tyler, 2011, Icarus 211, 770&amp;#8211;779.&lt;/p&gt;
&lt;p&gt;[2] Chen et al., 2014, Icarus 229, 11-30.&lt;/p&gt;
&lt;p&gt;[3] Matsuyama, 2014, Icarus 242, 11&amp;#8211;18.&lt;/p&gt;
&lt;p&gt;[4] Hay and Matsuyama, 2019, Icarus 319, 68&amp;#8211;85.&lt;/p&gt;
&lt;p&gt;[5] Rovira-Navarro et al., 2019, Icarus 321, 126&amp;#8211;140.&lt;/p&gt;
&lt;p&gt;[6] Rekier et al., 2019, J. Geophys. Res.: Planets 124, 2198&amp;#8211;2212. &amp;#160;&lt;/p&gt;
&lt;p&gt;[7] Choblet et al., Nat. Astr. 1, 841&amp;#8211;847.&lt;/p&gt;
&lt;p&gt;[8] &amp;#268;adek et al., Icarus 319, 476&amp;#8211;484.&lt;/p&gt;