Sensitivity of convection with an endothermic phase change to the form of governing equations, initial conditions, boundary conditions, and equation of state

Abstract

Recent convection calculations have demonstrated that an endothermic phase transition can greatly decrease the vertical flow through the transition in a convecting system, in some cases leading to a layered flow. Using reasonable estimates of both the Rayleigh number and Clapeyron slope of the spinel to perovskite plus magnesiowüstite phase change, these results suggest that the 670‐km phase change has a strong effect on mantle convection. This so‐called “dynamic layering” phenomenon is further investigated with a compressible finite element code using a two‐dimensional, Cartesian geometry. We find a weak sensitivity of the pattern of flow to the form of the equations, considering Boussinesq, extended Boussinesq, and anelastic compressible forms of the governing equations, assuming that the thermodynamic properties (thermal expansivity, heat capacity, and latent heat) remain constant. The pattern of flow, however, depends strongly on the initial conditions, boundary conditions and equation of state. We compare the simple equation‐of‐state formulations used in previous work with a self‐consistent equation of state based on Debye and Birch‐Murnaghan finite strain theory under a Mie‐Grüneisen formulation. A thermal expansion coefficient that decreases monotonically with depth and is unaffected by changes in phase or temperature greatly enhances dynamic layering. This trend is reversed when the temperature, pressure, and phase dependence of thermodynamic properties such as thermal expansivity, entropy, and heat capacity is introduced. At moderate Rayleigh numbers, the pattern of the flow is strongly influenced by the pattern of the initial condition (i.e., the location of upwellings and downwellings); however, it is not sensitive to the thickness of the initial thermal boundary layers. The sensitivity of the flow to the pattern of the initial condition can potentially bias mass fluxes, especially for moderate Rayleigh number calculations.

Influence of the Ringwoodite-Perovskite transition on mantle convection in spherical geometry as a function of Clapeyron slope and Rayleigh number

“…While latent heat is known to have an influence at low Ra (Schubert and Turcotte, 1971), with increasing Ra, boundary deflection is known to dominate (Christensen, 1995). The work of Ita and King (1994) and Christensen and Yuen (1985) showing that Boussinesq, extended Boussinesq and compressible simulations give similar results reinforces that latent heat effects are of minor importance at high Ra. Therefore, since we are interested in how the behaviours might extrapolate to high Ra (where influence of latent heat would be expected to be irrelevant), this simplification is sensible.…”

Development of finite strain in the convecting lower mantle and its implications for seismic anisotropy

“…[16] The numerical calculations are performed by solving the nondimensional conservation equations of mass, momentum, and energy in the extended Boussinesq approximation [e.g., Ita and King, 1994]. The equation for mass conservation in incompressible flow is…”

Model for Convective Cooling of Mantle with Phase Changes: Effects of Aspect Ratios and Initial Conditions.

“…Recently, Ita and King (1994) studied the effects of the initial conditions and aspect ratios on the flushing for a steady heat source case and they also found the significant effects of initial conditions and aspect ratios. As shown in this paper, the problem becomes more serious for the non-steady heat source case, which is more appropriate for the real Earth, because of the irreversible nature of the problem.…”