Linear stability of a layered fluid with mobile surface plates

We develop a general method of calculating the linear stability of a fluid with homogeneous layers that is heated from below. The method employs a propagator technique to obtain expressions for the fluid velocity, stress, and temperature. The principal advantage of the method is the ease with which solutions are adapted to a wide variety of boundary conditions and fluid properties. We demonstrate the utility of the method using three examples which quantify the effects of (1) rheological layering, (2) mobile plates at the surface, and (3) multiple phase transitions. Each example is presented in the context of Earth's mantle. In the first example, we predict that convection becomes confined to the upper mantle once the viscosity increase between the upper and lower mantle exceeds a factor of 2000, consistent with the nonlinear calculations of Davies (1977). In the second example we find that the heat flux variations in a convecting fluid with variably sized, surface plates can be attributed, in part, to changes in the critical Rayleigh number. The linear stability of a fluid with multiple phase transitions is significantly affects by the locations of the transitions. We find that phase transitions have their largest effect when they are located at the center of the fluid layer and become much less important when they are located near the exterior boundaries.