Stability Results for Convection in Thawing Subsea Permafrost

Penetrative convection is investigated in a porous medium bounded above by the ocean bed and below by the interface of the thawing permafrost ground. Such convection ow is observed oo the coast of Alaska. The physical model for the thawing subsea permafrost is that of Harrison and Swift 8], with the eld variables: the brine velocity, the temperature and the salinity. We simplify the problem by imposing a temperature eld that is linear in the depth variable. For this simpliied model values are obtained for the critical Rayleigh number, for both linear and nonlinear stability. From the mathematical point of view the analysis reduces to studying convection in a porous medium with a nonlinear boundary condition. Initially we consider the linear instability analysis which provides us with a linear critical Rayleigh number. If this linear critical Rayleigh number boundary is exceeded, this ensures instability. It does not preclude the possibility of subcritical instabilities. In order to complete the stability analysis, we use the energy method to determine a nonlinear critical Rayleigh number below which convection cannot develop. This critical value is close to that of linear theory. The nonlinear critical Rayleigh number guarantees unconditional nonlinear stability.