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 _mathrm{f} after flooding with a salt solution of higher density rho _mathrm{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 {{{mathbf {{small {uppercase {case~A}}}}}}} and the laterally bounded {{{mathbf {{small {uppercase {case~B}}}}}}}. The ratio of the diffusivity and the density difference (rho _mathrm{s} - rho _mathrm{f}) induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded {{{mathbf {{small {uppercase {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 {{{mathbf {{small {uppercase {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 {{{mathbf {{small {uppercase {case~A}}}}}}} and {{{mathbf {{small {uppercase {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 L^2-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 perturbations, 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 transiently unstable. These analytical predictions are confirmed by direct two-dimensional numerical simulations, which also show the details of the transient instabilities as function of the wave number for {{{mathbf {{small {uppercase {case~A}}}}}}} and the wave number and Rayleigh number for {{{mathbf {{small {uppercase {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.
Highlights
Stability of density stratified fluids has been studied for more than 100 years
Van Duijn et al (2002) analyzed gravitational stability of a saline boundary layer below an evaporating salt lake. They compared stability criteria obtained by three different methods: the method of linearized stability; the traditional energy method using as constraint the integrated Darcy equation; an alternative energy method using as constraint the point-wise Darcy equation
Continuity of Ψ is the only condition imposed at those interfaces. Numerical simulations for this three-layer setup are shown in Fig. 10 for a system Rayleigh number RS = 20. This Rayleigh number corresponds to a system with a transient instability, see Table 3: The perturbation develops in layer I until the fingertip reaches z = d1, the start of the layer II
Summary
Stability of density stratified fluids has been studied for more than 100 years. In the book ’The self-made tapestry’, Ball (2001) gives in chapter 7 a lucid description of the history of the analysis of pattern formation in such fluids. Later it became feasible to use analytical and numerical methods to study the full nonlinear equations Employing those modern methods, van Duijn et al (2002) analyzed gravitational stability of a saline boundary layer below an evaporating salt lake. Post and Simmons (2009) illustrate by means of sand tank laboratory experiments and numerical modeling how low-permeability lenses protect freshwater from mixing with overlying saline water This is in line with the simulations presented in this paper. Webster et al (1996) used a computational model and laboratory experiments to demonstrate that gravitational convection can make an important contribution to the exchange of water and solutes between sediments and a supernatant water column in regions subject to significant temporal variations in salinity, such as estuaries In effect, they verified the suggestion put forward by Smetacek et al (1976). They are given in a comprehensive version of this work that appeared as the technical report Pieters et al (2018)
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