Calibrating mixing-length theory for thermal convection in rocky planets

Kurzfassung

We present a calibration of the mixing-length parameter in the local mixing-length theory used to model thermal convection inside rocky planets. The parametrization is derived from a comparison of 3-D numerical simulation experiments to predictions by the 1-D mixing-length theory with a depth-dependent mixing length. The comparison encompasses a wide variety of parameters, that is Rayleigh numbers ranging from 104 to 1010 and viscosity contrasts up to 106 have been investigated for a convective system heated either solely from below or from both below and within. We find that the mixing length is sensitive to both viscosity contrast and Rayleigh number. With the commonly used scaling relationship, mixing-length theory is unable to reproduce simultaneously heat flux and temperature of the numerical simulation experiments with satisfactory accuracy. According to the calibration, we identify that the mixing length depends on the convection regime. In particular, stagnant-lid convection should be separated from sluggish-lid and mobile-lid convection. In the mobile-lid regime, the mixing length is generally much larger than in the stagnant-lid regime. The presented parametrization yields thermal profiles in excellent agreement with those obtained from fully dynamic convection simulations and important physical properties such as surface heat flux and average mantle temperature can be computed with small errors. This establishes the mixing-length theory as a viable alternative to other parametrized convection models including the widely used boundary-layer theory and offers the possibility to study the thermal state of a large number of Earth-like planets.