On the Stability of Density Stratified Flow Below a Ponded Surface

Abstract

Flooding of coastal areas with seawater often leads to density stratification. The stability of the density-depth profile in a porous medium initially saturated with a fluid of density rho_f after flooding with a salt solution of higher density rho_s is analyzed. The standard convection/diffusion equation subject to the so-called Boussinesq approximation is used. The depth of the porous medium is assumed to be infinite in the analytical approaches and finite in the numerical simulations. Two cases are distinguished: the laterally unbounded case a and the laterally bounded case b. The ratio of the diffusivity and the density difference (rho_s - rho_f ) induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded case a, this geometric length scale is the only length scale and using it to write the problem in dimensionless form results in a formulation with Rayleigh number R=1. In the bounded case b, the lateral geometry provides another length scale. Using this geometrical length scale to write the problem in dimensionless form results in a formulation with a Rayleigh number R given by the ratio of the geometric and intrinsic length scales. For both case a and case b, the well-known Boltzmann similarity solution provides the ground state. Three analytical approaches are used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear equation. For the first linear approach, the surface spatial density gradient is used as an approximation of the entire background density profile. This results in a crude estimate of the L2-norm of the concentration showing that the perturbation at first grows, but eventually decays in time. For the other two approaches, the full ground state solution is used, although for the second linear approach subject to the restriction that the ground state slowly evolves in time (the so-called ’frozen profile approximation’). Just like the ground state, the resulting eigenvalue problems can be written in terms of the Boltzmann variable. The linearized stability approach holds only for infinitesimal small perturbutions, whereas the nonlinear, variational energy approach is not subject to such a restriction. The results for all three approaches can be expressed in terms of Boltzmann sqrt{t} transformed relationships between the system Rayleigh number and perturbation wave number. The results of the linear and nonlinear approaches are surprisingly close to each other. Based on the system Rayleigh number, this allows delineation of systems that are unconditionally stable, marginally stable, or episodically unstable. These analytical predictions are confirmed by direct two-dimensional numerical simulations, which also show the details of the episodic instabilities as function of the wave number for case case a and the wave number and Rayleigh number for case b. A numerical example of the effect of a layer with low permeability is also shown. Using typical values of the physical parameters, the analytical and numerical results are interpreted in terms of dimensional length and time scales. In particular an explicit stability criterion is given for vertical column experiments.