Penetrative 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. The thermal equation of state relating the density, temperature and salinity is assumed to be that of ocean water as proposed by the UNESCO formula. Employing the Boussinesq approximated Darcy-flow equations with such a realistic density formula in the buoyancy term, the problem of convective motion of brine is studied. Such convection flow is observed off the coast of Alaska. The field variables in question are the brine-velocity field, the temperature and the salinity, although we simplify the problem by imposing a temperature field that is linear in the depth variable. For this simplified system we study the continuous dependence of the velocity and salinity on the initial data, develop a linear instability analysis and, additionally, present a fully nonlinear three-dimensional stability analysis. This nonlinear analysis necessitates the introduction of a generalised energy (or Lyapunov function) due to the extra terms present in the realistic equation of state. Numerical results indicate that values of the critical Rayleigh numbers are smaller than when these extra terms are omitted.