We investigate the tidal dissipation in Io’s hypothetical fluid magma ocean using a new approach based on the solution of the 3D Navier-Stokes equations. Our results indicate that Io may have experienced a period of intense tidal heating (104 TW) accompanied by excessive volcanism in the equatorial region, leading to catastrophic resurfacing of the pre-existing terrain. Tidal heating in Io’s magma ocean does not correlate with the distribution of hot spots, and is maximum for an ocean thickness of about 1 km and a viscosity of less than 104 Pa s. Due to the Coriolis effect, the k2 Love number can depend on the harmonic order. We show that the analysis of k2 may not reveal the presence of a fluid magma ocean if the ocean thickness is less than 2 km. If the fluid layer is thicker than 2 km, k20 ≈ k22/2 ≈ 0.7.

The volcanic activity of Jupiter's moon Io is driven by the heat generated by tidal deformation induced by its orbital resonance with Europa and Ganymede. The question of whether tidal dissipation primarily occurs in a partially molten solid layer ("magmatic sponge'') or in a hypothetical liquid magma ocean has long been a subject of debate. One way to answer this question is to measure the \(k_2\) Love number, characterizing the gravitational response of the body to tidal forcing. Previous modeling studies have shown that \(k_2 \leq 0.1\) if Io's silicate interior is solid. It has been suggested that \(k_2 \gtrsim 0.5\) if Io has a magma ocean but, to date, there has been no systematic study focusing on how much  \(k_2\) can be affected by the presence of a liquid layer in Io's interior. Here we address this issue by solving the Navier-Stokes equations in the magma ocean and the standard equations governing the viscoelastic deformation in the rest of the mantle. Our results indicate that, in the presence of a magma ocean, \(k_2\) can strongly vary depending on three parameters: the depth of the ocean below the surface, \(l\), the thickness of the fluid layer, \(d\), and the viscosity of the magma, \(\eta\). Varying \(l\), \(d\), and \(\eta\) in the range of 50-200 km, 0.1-20 km, and \(10^2\)- \(10^7\)Pa s, respectively, and considering only the models with a dissipation power of about 100 TW, we find that \(k_2\) can be either less than 0.1 (thus about the same as for the model without a magma ocean) or greater than 0.2. The first branch of the solution      (\(0.05 \leq k_2 \leq 0.1\)) is typical of models with \( d < 1\) km and is almost independent of parameters \(l\) and \(\eta\), while the second branch (\(0.2\leq k_2 \leq 1\)) corresponds to models with \(d > 1\) km and is sensitive to the depth of the magma ocean. The tidal response in the second branch can be affected by the Coriolis effect. As a consequence, \(k_2\) may depend on the harmonic order, with the Love numbers of order 2 being significantly larger than those of order 0.