Convection in thawing subsea permafrost

Abstract

Detailed quantitative values are obtained for the critical values of the salt Rayleigh number for both linear and nonlinear stability, for a simplified model appropriate to the onset of buoyant, relatively fresh water motion in a layer of salty subsea sediments. The geophysical problem that motivates this work arises because of the formation of substantial permafrost around the Earth’s shores some 18000 years ago. With the rise of sea levels the permafrost has responded to the relatively warm and salty sea, which has created a thawing front and a layer of salty sediments beneath the sea bed. This phenomenon has been studied extensively off the coast of Alaska by W. Harrison and coworkers and our analysis is based on a model developed by W. Harrison and D. Swift. From the mathematical viewpoint the analysis reduces to studying convection in a porous medium with a nonlinear boundary condition. We find the critical Rayleigh number for convection according to linear theory, but our main thrust is directed toward the nonlinear problem. Here we use an energy method to determine a critical Rayleigh number below which convection cannot develop. We first show there is a critical Rayleigh number close to that of linear theory, which guarantees unconditional nonlinear stability. Then we demonstrate conditional nonlinear stability (i. e. conditional upon the existence of some finite threshold amplitude, which we calculate) provided the critical Rayleigh number of linear theory is not exceeded. The latter analysis requires two approaches according to whether the two-dimensional or three-dimensional problem is considered. In particular, a novel energy has to be introduced to make the three-dimensional problem tractable.