[M-IS21] Geophysical fluid dynamics-Transfield approach to geoscience

[MIS21-01] Horizontal scale and regime transition of thermal convection with temperature-dependent viscosity

*Hisashi Okuda¹, Shin-ichi Takehiro¹ (1.Research Institute for Mathematical Sciences, Kyoto University)

Keywords:mantle convection, scaling analysis, horizonrally elongated cell, Nusselt number

Thermal convection with strongly temperature-dependent high viscosity has been investigated to understand fundamental dynamical properties of mantle convection. Scaling analysis and numerical time integrations have been performed to reveal transition of structures of finite amplitude convection for such fluid with respect to the parameter of the viscosity contrast (e.g., Solomatov, 1995). In the small viscosity contrast regime (SVC), vertically symmetric convection cells similar to those for constant viscosity are formed. As the viscosity contrast increases, convection cells become vertically asymmetric, as the flow in the upper area decreases due to low temperature and high viscosity (Transitional regime: TR). When the viscosity contrast further increases, the upper area becomes stagnant like a lid (Stagnant Lid regime: ST). In ST regime, the convective structure beneath the stagnant lid is similar to that in SVC regime. However, variation of horizontal length scale of convection cells has not been considered so far. Actually, several numerical time integrations show that the horizontal scale of convection cells becomes larger than the thickness of the fluid layer around TR regime. Therefore, we investigate the relation between horizontal length scale and dynamical regimes of thermal convection with temperature-dependent viscosity.
At first, we obtained steady finite amplitude solutions numerically by Newton method for several values of the Rayleigh number and the parameter for viscosity contrast. The horizontal wave length of each solution was controlled by the lateral length of the domain. We classified those solutions into the three regimes from their dependence of Nusselt number on the parameters, and produced a new regime diagram with respect to the horizontal wave length and the viscosity contrast ratio. Furthermore, we examined the stability of horizontal wave length of the convective solutions in order to diagnose possibility of their emergence for time-dependent problems. As a result, we found that the solutions follow SVC-TR boundary on the parameter space with increase of their horizontal length as the viscosity contrast ratio increases.
Next, we modified the scaling analysis in order to introduce horizontal wave length of convective solutions as an additional parameter. We also produced a regime diagram based on the derived scaling relations of Nusselt number, and found that it is consistent with that based on the numerical steady solutions. In addition, we presumed the convective solutions emerging in the time-dependent problems on the assumption that the solutions with the maximum Nusselt number actually appear. As the viscosity contrast ratio increases, the solutions follow the maximum points in SVC, SVC-TR boundary, and the maximum points in ST in that order, which is also consistent with the result with numerical steady solutions. Finally, we estimated the maximum horizontal wave length of convective solutions by the transition point from the laterally elongated solution on SVC-TR boundary to the solution in ST regime.