11:15 AM - 11:30 AM
[MIS21-03] Convetive structures and scaling relations of internally heated convection with temperature-dependent viscosity
Keywords:Mantle convection, Scaling relations, Convective patterns, Internally heated convection
Thermal convection of Boussinesq fluid has long been investigated to reveal the fundamental characteristics of mantle dynamics in the terrestrial planets. The problem of Rayleigh-Benard convection is a typical setup to be studied, where convective flow is driven by heating from the bottom and cooling from the top of the fluid layer. However, for the application to mantle convection, it is also important to study the problem of thermal convection driven by internal heating, since, for example, the mantle convection in the Earth is mainly driven by internal heating rather than by heating from the bottom. The heat flow through the core-mantle boundary is about 10 or 20 % of the total heat flow through the Earth's surface (Jaupart et al., 2015).
On the other hand, thermal convection of fluids with temperature-dependent viscosity has also been investigated, taking into account the rheological properties of the mantle materials. Previous studies showed that convection of such fluids driven by bottom heating can be classified into three convective structures depending on the strength of the viscosity variation (e.g., Solomatov, 1995). When viscosity has a very weak dependence on temperature, the convective structure is almost the same as that of constant viscosity (Small viscosity contrast [SVC] regime). When temperature-dependence is strong enough for the layer below the top cold boundary to be motionless, the convective flow is restricted in the lower layer below the stagnant layer (Stagnant lid [ST] regime). In the cases of moderate temperature-dependence, the upper layer becomes highly viscous but still drifts slowly while active convection emerges in the lower layer (Transitional [TR], or sluggish lid regime). These regimes are distinguished by the scaling relations of the Nusselt number as functions of the Rayleigh number, the strength of temperature-dependence of viscosity, and horizontal wavelength, which were derived theoretically and confirmed numerically. The scaling theory showed that the Nusselt number follows the Rayleigh number to the power of 1/3 in all three regimes (Solomatov, 1995; Okuda and Takehiro, 2023).
Internally heated thermal convection of temperature-dependent viscous fluids has been studied so far by numerical and laboratory experiments only for the cases of constant viscosity and strongly temperature-dependent viscosity (e.g., Davaille and Jaupart, 1993; Grasset and Parmentier, 1998). It was shown that the temperature of the isothermal core follows the Rayleigh number to the power of -1/4 in both cases, where the Rayleigh number here is defined by using the internal heating rate for the temperature scale. However, the cases of moderate temperature-dependence have not been investigated in detail, and the classification into convective regimes are not yet examined.
Therefore, we investigate the variety of convective structures and scaling relations of internally heated convection of temperature-dependent viscous fluids for different values of the Rayleigh number and the strength of temperature-dependence of viscosity, using two-dimensional numerical calculations of steady convective solutions. The heat source is uniformly distributed throughout the fluids. The viscosity is assumed to depend on temperature exponentially. In the weakly temperature-dependent regime (SVC regime), the scaling relation of the internal temperature follows the Rayleigh number to the power of -1/4, while the solutions in the strongly temperature dependent regime (ST regime) follow the -1/4 power law, which are consistent with the previous studies. Between the SVC and ST regimes, a new intermediate regime is found where the convective solutions follow higher power relations than in the other regimes.
On the other hand, thermal convection of fluids with temperature-dependent viscosity has also been investigated, taking into account the rheological properties of the mantle materials. Previous studies showed that convection of such fluids driven by bottom heating can be classified into three convective structures depending on the strength of the viscosity variation (e.g., Solomatov, 1995). When viscosity has a very weak dependence on temperature, the convective structure is almost the same as that of constant viscosity (Small viscosity contrast [SVC] regime). When temperature-dependence is strong enough for the layer below the top cold boundary to be motionless, the convective flow is restricted in the lower layer below the stagnant layer (Stagnant lid [ST] regime). In the cases of moderate temperature-dependence, the upper layer becomes highly viscous but still drifts slowly while active convection emerges in the lower layer (Transitional [TR], or sluggish lid regime). These regimes are distinguished by the scaling relations of the Nusselt number as functions of the Rayleigh number, the strength of temperature-dependence of viscosity, and horizontal wavelength, which were derived theoretically and confirmed numerically. The scaling theory showed that the Nusselt number follows the Rayleigh number to the power of 1/3 in all three regimes (Solomatov, 1995; Okuda and Takehiro, 2023).
Internally heated thermal convection of temperature-dependent viscous fluids has been studied so far by numerical and laboratory experiments only for the cases of constant viscosity and strongly temperature-dependent viscosity (e.g., Davaille and Jaupart, 1993; Grasset and Parmentier, 1998). It was shown that the temperature of the isothermal core follows the Rayleigh number to the power of -1/4 in both cases, where the Rayleigh number here is defined by using the internal heating rate for the temperature scale. However, the cases of moderate temperature-dependence have not been investigated in detail, and the classification into convective regimes are not yet examined.
Therefore, we investigate the variety of convective structures and scaling relations of internally heated convection of temperature-dependent viscous fluids for different values of the Rayleigh number and the strength of temperature-dependence of viscosity, using two-dimensional numerical calculations of steady convective solutions. The heat source is uniformly distributed throughout the fluids. The viscosity is assumed to depend on temperature exponentially. In the weakly temperature-dependent regime (SVC regime), the scaling relation of the internal temperature follows the Rayleigh number to the power of -1/4, while the solutions in the strongly temperature dependent regime (ST regime) follow the -1/4 power law, which are consistent with the previous studies. Between the SVC and ST regimes, a new intermediate regime is found where the convective solutions follow higher power relations than in the other regimes.