5:15 PM - 7:15 PM
[MIS16-P03] Convective structures and stability of internally heated thermal convection of temperature-dependent viscous fluid
Keywords:Mantle convection, Convective structures, Internally heated convection, Transitional regime, Rayleigh Taylor instability
Thermal convection of highly viscous fluids with temperature-dependent viscosity has been studied to understand the fundamental characteristics of mantle convection in terrestrial planets. The setup of convection driven by bottom heating has been well studied so far. The flow structures with this setup are classified into three types with different top-surface mobility depending on the viscosity contrast of the fluid (e.g., Solomatov, 1995). When the viscosity contrast is small, the structure is almost the smale as that of constant viscosity (small viscosity contrast [SVC] regime). When the viscosity contrast is large, a thermally conductive layer without motion is formed just below the top surface so that convection is restricted in the sublayer (stagnant lid [ST] regime). For moderate viscosity contrast, a conductive lid with motion is formed, which drifts more slowly than the convective flow in the sublayer (transitional [TR] or sluggish lid regime).
On the other hand, thermal convection driven by internal heating has not been fully investigated. Although the SVC and ST regimes have been characterized and their scaling relations have been obtained (e.g., Grasset and Parmentier, 1998), the TR regime has not been identified clearly.
We obtain steady-state solutions of two-dimensional thermal convection in a temperature dependent viscous fluid driven by internal heating for various values of the parameters and investigate their structures, especially the solutions between the SVC and ST regimes. The model assumes a uniform heat source throughout the fluid and an exponential temperature-dependence of the viscosity. The free-slip boundary condition is applied at the upper and lower boundaries. The steady-state finite-amplitude numerical solutions are obtained by the Newton method. The solutions are classified into three regimes by the top surface mobility, and then the scaling relations between the Nusselt number and the Rayleigh number of each regime are examined. Solutions of the SVC and ST regimes are consistent with their scaling relations from previous studies. Solutions of the TR regime are found between the SVC and ST regimes in the parameter space, and they do not follow the classical scaling relations. The viscosity contrast ratio across the active thermal boundary layer in the convective sublayer becomes larger for the TR regime solutions than for the SVC and ST regimes. This feature disappears when the boundary condition of the top surface is replaced by a rigid boundary.
The time integration calculations are performed by selecting some steady solutions with an additional pointwise temperature perturbation for the initial condition. The results show that the stability of steady convective solutions does not depend on their regime, and the solutions of the TR regime are not necessarily unstable. The stability is determined by the Rayleigh number defined using the viscosity at the bottom surface. The solutions with small Rayleigh number are found to be stable where the perturbation decreases, while the solutions with large Rayleigh number are unstable where the perturbation grows and the plumes develop from the thermal boundary layer. These phenomena are explained by applying the theory of the Rayleigh-Taylor instability of two-layer fluids with different viscosities to the active boundary layer.
On the other hand, thermal convection driven by internal heating has not been fully investigated. Although the SVC and ST regimes have been characterized and their scaling relations have been obtained (e.g., Grasset and Parmentier, 1998), the TR regime has not been identified clearly.
We obtain steady-state solutions of two-dimensional thermal convection in a temperature dependent viscous fluid driven by internal heating for various values of the parameters and investigate their structures, especially the solutions between the SVC and ST regimes. The model assumes a uniform heat source throughout the fluid and an exponential temperature-dependence of the viscosity. The free-slip boundary condition is applied at the upper and lower boundaries. The steady-state finite-amplitude numerical solutions are obtained by the Newton method. The solutions are classified into three regimes by the top surface mobility, and then the scaling relations between the Nusselt number and the Rayleigh number of each regime are examined. Solutions of the SVC and ST regimes are consistent with their scaling relations from previous studies. Solutions of the TR regime are found between the SVC and ST regimes in the parameter space, and they do not follow the classical scaling relations. The viscosity contrast ratio across the active thermal boundary layer in the convective sublayer becomes larger for the TR regime solutions than for the SVC and ST regimes. This feature disappears when the boundary condition of the top surface is replaced by a rigid boundary.
The time integration calculations are performed by selecting some steady solutions with an additional pointwise temperature perturbation for the initial condition. The results show that the stability of steady convective solutions does not depend on their regime, and the solutions of the TR regime are not necessarily unstable. The stability is determined by the Rayleigh number defined using the viscosity at the bottom surface. The solutions with small Rayleigh number are found to be stable where the perturbation decreases, while the solutions with large Rayleigh number are unstable where the perturbation grows and the plumes develop from the thermal boundary layer. These phenomena are explained by applying the theory of the Rayleigh-Taylor instability of two-layer fluids with different viscosities to the active boundary layer.