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 flow is observed off the coast of Alaska. The physical model for the thawing subsea permafrost is that of Harrison and Swift [8], with the field variables: the brine velocity, the temperature and the salinity. We simplify the problem by imposing a temperature field that is linear in the depth variable. For this simplified 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 ord...

We propose an efficient implementation of the Chebyshev Galerkin spectral method for the biharmonic operator. This discretization leads to banded matrices which, compared with other methods of the same type, are also better conditioned. The efficiency of the method is illustrated on the Orr-Sommerfeld eigenvalue problem, where an improved convergence can be observed and the spurious eigenvalues are removed.

The role of nonnormality in the overreflection of gravity waves is investigated. In the limit of weak stratification, wave packets having a critical level inside a shear layer of finite depth are reflected with amplified energy. This process, which exhibits the characteristics of stimulated emission, occurs in three stages: first, the incoming wave enters the shear layer and excites nonpropagating perturbations leaning with and against the shear. Subsequently, the energy of perturbations leaning against the shear grows in a manner similar to energy growth of perturbations in constant shear flows, indicating that the Orr mechanism that is slightly modified by stratification underlies the observed growth. Finally, the amplified perturbations excite propagating waves originating from the vicinity of the shear layer boundary. The role of nonnormality in this process is also investigated from the perspective of the associated nonorthogonality of the modes of the dynamical system. It is found that the incident wave packet projects on nonorthogonal analytic modes having the structure of a downward propagating wave in the far field below the shear layer and overreflection expressed by the interaction among these nonorthogonal modes. 1.

A spectral Galerkin method for the the coupled Orr–Sommerfeld and induction equations for free-surface MHD

We compute solutions for the Orr-Sommerfeld equations for the case of an asymmetric Poiseuille-like parallel flow. The calculations show that very small asymmetry has little effect on the prediction for linear instability of Poiseuille-like flow but that moderate asymmetry, such as found in channel flow near an elongated wall vortex, has a large effect and that instability can occur at much lower (less than 100) Reynolds numbers. We give some characterisation of the instability.