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Abstract. The stability of steady, vertically upward and downward flow of water in a homogeneous layer of soil is analyzed. Three equivalent dimensionless forms of the Richards equation are introduced, namely the pressure head, saturation, and matric flux potential forms. To illustrate general results and derive special results, use is made of several representative classes of soils. For all classes of soils with a Lipschitz continuous relationship between the hydraulic conductivity and the matric flux potential, steady flows are shown to be unique. In addition, linear stability of these steady flows is proved. To this end, use is made of the energy method, in which one considers (weighted) L 2 -norms of the perturbations of the steady flows. This gives a general restriction of the dependence of the hydraulic conductivity upon the matric flux potential, yielding linear stability and exponential decay with time of a specific weighted L 2 -norm. It is shown that for other norms the ultimate decay towards the steadysolution is preceded by transient growth. An extension of the Richards equation to take into account dynamic memory effects is also considered. It is shown that the stability condition for the standard Richards equation implies linear stability of the steady solution of the extended model.

In this paper we study gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at the horizontal surface of a porous medium. Van Duijn et al.,[33], derived stability bounds by means of linear stability analysis and an (improved) energy method. These bounds do not coincide, i.e. there exists a subcritical region or stability gap in the system parameter space which is due to the asymmetry of the linear part of the perturbation equations. We show that the linear operator can be symmetrized by means of a similarity transformation. For system parameter values in the stability gap, we show that there exist optimal initial perturbations for which the linearly stable system exhibits transient growth. We show that transient growth is norm dependent by considering weighted norms, which are induced by a one-parameter family of similarity transformations.

T he upper part of a living mire consists of a sponge-like layer of predominantly moss species, the acrotelm (1), with a porosity above 95%. The green and brownish plants near the surface ( Fig. 1) intercept light and fix CO 2 . Further down, the older plants turn yellow and start to decay. Aerobic decay in the acrotelm takes place relatively rapidly and makes nutrients available for recycling. Below the acrotelm, a denser layer, the catotelm, is present, where the hydraulic conductivity is much lower than in the acrotelm (2), and where the decay rate is several orders of magnitude smaller due to the anoxic conditions (3). It is the peat formation (4, 5) in the slowly growing catotelm that represents a sink of atmospheric CO 2 (5, 6).The production of organic matter at the surface largely depends on the recycling of nutrients originating from decomposing plant material. Because decomposition and photosynthesis take place at different depths, the transport of oxygen, carbon compounds, and nutrients forms an important element in the functioning of the mire ecosystem. This transport takes place both inside (7) and outside the plants by diffusion and fluid flow.In this paper, we investigate a mechanism for fluid flow in a water-saturated peat moss layer, which does not depend on capillarity or an external hydraulic pressure. During the night, the surface cools, leading to relatively cold water on top of warm water, and if the temperature drop is sufficiently large, the cold water sinks and the warm water rises. This type of flow is called buoyancy flow, and it implies convective transport of the heat and solutes carried with the water. Buoyancy flow often occurs as ''cells'' consisting of adjacent regions with upward and downward flow. We studied the phenomenon in a peat moss layer by means of a mathematical model, numerical simulation, and laboratory measurements. Model Equations and StabilityThe Mathematical Model. The model describes the heat flow in a water-saturated porous layer that undergoes periodic and sudden temperature changes at its surface.The imposed surface temperature involves the model parameters ⌬T, which is the temperature difference between ''day'' and ''night,'' and t 0 , which is the duration of each of the two periods. Four parameters describe geometrical and physical properties of the layer: the thickness H, the thermal expansion coefficient of the fluid ␣, the thermal diffusivity of the layer ‫ބ‬ eff , and the hydraulic conductivity K. Table 1 lists parameter values for the water-saturated peat moss layer used in the experiment.The equations for heat transport and fluid flow, together with the boundary conditions, are given in Appendix B. Here, we briefly discuss the physical content of the model equations in dimensionless form. Dimensionless temperatures T are expressed in units ⌬T and lie between 0 and ϩ1. Similarly, the time interval t 0 is used as the unit of time, and the distance ͌ ‫ބ‬ eff t 0 , as the unit of length. This length scale (Ϸ0.078 m) characterizes the distance over which a...

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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 ρ f after flooding with a salt solution of higher density ρ 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 (ρ s − ρ 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 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 √ 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 CASE A and the wave number and Rayleigh number for CASE B.

Soil salinization is a major cause of soil degradation and hampers plant growth. For soils saturated with saline water, the evaporation of water induces accumulation of salt near the top of the soil. The remaining liquid gets an increasingly larger density due to the accumulation of salt, giving a gravitationally unstable situation, where instabilities in the form of fingers can form. These fingers can, hence, lead to a net downward transport of salt. We here investigate the appearance of these fingers through a linear stability analysis and through numerical simulations. The linear stability analysis gives criteria for onset of instabilities for a large range of parameters. Simulations using a set of parameters give information also about the development of the fingers after onset. With this knowledge, we can predict whether and when the instabilities occur, and their effect on the salt concentration development near the top boundary.