One‐Dimensional Convective Thermal Evolution Calculation Using a Modified Mixing Length Theory: Application to Saturnian Icy Satellites

Abstract

Solid‐state thermal convection plays a major role in the thermal evolution of solid planetary bodies. Solving the equation system for thermal evolution considering convection requires 2‐D or 3‐D modeling, resulting in large calculation costs. A 1‐D calculation scheme based on mixing length theory (MLT) requires a much lower calculation cost and is suitable for parameter studies. A major concern for the MLT scheme is its accuracy due to a lack of detailed comparisons with higher dimensional schemes. In this study, I quantify its accuracy via comparisons of thermal profiles obtained by 1‐D MLT and 3‐D numerical schemes. To improve the accuracy, I propose a new definition of the mixing length (l), which is a parameter controlling the efficiency of heat transportation due to convection, for a bottom‐heated convective layer. Adopting this new definition of l, I investigate the thermal evolution of Saturnian icy satellites, Dione and Enceladus, under a wide variety of parameter conditions. Calculation results indicate that each satellite requires several tens of GW of heat to possess a thick global subsurface ocean suggested from geophysical analyses. Dynamical tides may be able to account for such an amount of heat, though the reference viscosity of Dione's ice and the ammonia content of Dione's ocean need to be very high. Otherwise, a thick global ocean in Dione cannot be maintained, implying that its shell is not in a minimum stress state.

Methane hydrate creates the thick oceans in Mimas and Enceladus

“…To calculate the evolution of heat transport with time, including conduction and convection, the modified mixing length theory, the one-dimensional calculation scheme developed by (Kamata (2018)), is used. The governing equation of heat transfer is given…”

“…where is the gravitational acceleration, is the viscosity, and is the mixing length. depends on the computational grid temperature, shell thickness, and other parameters (Kamata (2018)). The viscosities of ice and methane hydrate are as follows:…”

“…Our code can reproduce the results for Kamata et al (2019), Kamata (2018), and Kimura and Kamata (2020). The icy shell and rocky core are divided into thin layers.…”

“…Assuming p is now 2000, the equilibrium tidal heating rate for Enceladus is ∼10 GW. This tidal heating rate is insufficient for the survival of a thick ocean (Kamata (2018)). Mimas is not currently in any eccentricity resonance, and it is difficult to determine its equilibrium tidal heating rate.…”

Numerical solution of a non-linear conservation law applicable to the interior dynamics of partially molten planets

“…Modelling the radial structure is often the most scientifically informative approach, particularly when astrophysical data constraints are limited or model parameters poorly constrained. Using MLT, 1-D models can replicate the general results of 3-D models (e.g., Kamata et al, 2015) with significantly less calculation cost, thus permitting an extensive search of the parameter space. An MLT scheme allows physical properties to be determined locally which enables properties to vary substantially throughout the 1-D domain.…”