#### 2.1. Surface and Internal Waves

[8] The linear wave theory for surface gravity waves (SGWs) has been studied extensively (see *Debnath* [1994] and *Kundu* [1990] for a full derivation and exposition). The three key aspects of SGWs which we will focus on are that, in comparison to internal waves, (1) SGWs have relatively short wavelengths, (2) SGWs have comparatively fast propagation speeds, and (3) the amplitude of the pressure profile induced by SGWs decays exponentially with depth of the water column. In particular, if we consider a surface wave with a sinusoidal form, an angular frequency *ω*, and a wave number *k*, the phase and group speed of rightward propagating waves are given by

and

while the pressure distribution at the bottom of a lake with uniform depth *H* is given by

Here, *a* is the amplitude of the gravity wave, *H* is the depth of the lake, and *g* is the acceleration due to gravity [see *Kundu*, 1990].

[9] For the theoretical description of ISWs, we consider an incompressible, inviscid fluid in a fixed reference frame under the Boussinesq and rigid lid approximations. The fluid motion is thus governed by the Euler equations, expressing the conservation of momentum,

[10] The incompressibility condition, expressing conservation of mass,

and the density equation,

where represents the fluid velocity, *p* the pressure, *ρ* the total density, *ρ*_{0} a constant reference density, and the acceleration due to gravity. As is customary, we assume that the density can be decomposed into a background profile , which depends only on the vertical coordinate, and a perturbation. The total density can be written in terms of the far upstream profile, , where defines the isopycnal displacement from its far upstream height. The buoyancy, or Brunt-Vaisala, frequency*N*(*z*), which is defined via the relation

gives the frequency of oscillations of a fluid particle infinitesimally displaced from its resting height *z*. In a lake, the density would be determined via an equation of state from the temperature.

[11] While the above equations are suitable for a time-dependent simulation, a far simpler description is possible for either a propagating internal solitary wave (ISW) or a steady internal wave trapped over topography. In both of these situations the Euler equations can be reduced, via a series of somewhat complicated algebraic manipulations (see*Soontiens et al.* [2010] and *Stastna and Lamb* [2002] for a detailed derivation), to a single, scalar equation for the isopycnal displacement . This is the well-known Dubreil-Jacotin-Long (DJL) equation,

where *U*_{0} is the wave propagation speed for ISWs and the upstream velocity for trapped waves. The boundary conditions far upstream and downstream are that *η* tends to 0, while both the rigid lid at the top (at *z* = *H*) and variable bottom (at ) are streamlines.

[12] From a mathematical point of view, the DJL equation is formally elliptic and strongly nonlinear. Not surprisingly, no analytical solutions are known (except for the special situation when is constant). However, numerical methods for the DJL equation are readily available, and have resulted in a great deal of insight into finite amplitude ISWs and trapped waves.

[13] Briefly, both ISWs and trapped waves are “long waves” in the mathematical sense. The practical implication of this fact is that, unlike the exponential decay of the pressure perturbation induced by SGWs (see equation (1)), the pressure perturbation at the lake bottom due to ISWs and internal trapped waves is large. Moreover, the pressure perturbation is steady in a frame of reference moving with the wave speed. This profile can be diagnostically determined from the solution of the DJL equation and used to drive a model of the porous layer. In practice, the solutions of the DJL equation can be computed up to some limiting amplitude. It has been shown [*Lamb*, 2002] that three qualitative types of upper bounds on wave amplitude are possible. The first is the so-called breaking limit, beyond which waves would have closed streamlines; it can be thought of as the internal wave analog of SGWs breaking on a beach. The second is the broadening limit in which waves tend to a limiting state consisting of a central region with flat isopycnals, and is sometimes referred to as the conjugate flow limit [*Lamb and Wan*, 1998]. The third is the so-called stability limit, in which the gradient Richardson number of the wave induced flow dips below 0.25 and shear instability is possible in the wave [*Lamb and Farmer*, 2011]. In the following we will consider large ISWs and trapped waves, but well below the limiting amplitude. Oceanographic measurements of large amplitude internal waves are discussed in the recent review by *Helfrich and Melville* [2006], while in a lake context well-resolved measurements of large amplitude internal waves have been reported for Lake Constance [*Preusse et al.*, 2012] throughout the stratified season. We briefly mention here that when the pycnocline is above the middepth, ISWs of depression will form. Similarly, when the pycnocline is below the middepth, ISWs of elevation form.

[14] Trapped waves are a more recently reported phenomenon [*Stastna and Peltier*, 2005; *Soontiens et al.*, 2010] and can reach amplitudes larger than the limiting ISW amplitude. They require a forcing mechanism, namely a background current. Mathematically, they are a steady solution and hence remain unchanged for all times. In a practical situation, their lifespan would depend on how long a suitable forcing persisted. As a rough estimate, a passing ISW could be expected to influence the pressure at the SWI for a period on the order of a half hour. In contrast, a trapped wave with an amplitude 150% of the ISW (a typical amplitude reported by *Stastna and Peltier* [2005] and *Soontiens et al.* [2010]) can persist for several hours, and would have the potential to increase across SWI transport by a factor of 10.

[15] For trapped waves, we will assume an isolated topography, or in other words that as . The functional form of the topography was selected to be a hyperbolic secant function (though this is largely a matter of mathematical convenience)

where *A*_{0} is the height of the topography and *λ* specifies the width. The parameters were chosen to match those of a trapped internal wave as found by *Soontiens et al.* [2010, their Figure 1]. A number of exploratory simulations indicate that these results are generic for a broad range of parameter space. The solution of the DJL equation was found using publically available software developed in our group (see *Soontiens et al.* [2010] and *Dunphy et al.* [2011] for details).

[16] We will report results in dimensionless form. Velocities will be scaled by the linear long-wave speed of internal waves, spatial extent by the depth of the lake, with a second parameter for the depth of the porous lake bottom, and time by the advective time scale. Due to the linearity of the Darcy theory, the pore pressure forcing will be scaled by its maximum value.

[17] The situation of an internal wave overlying a porous layer is schematized in Figure 2 (the mathematical formulation is discussed in detail below). A corresponding typical pore pressure perturbation at the SWI due to an ISW with a maximum dimensionless isopycnal displacement of 0.215 is shown in Figure 3 in a frame moving with the wave speed.

#### 2.2. Equations and Numerical Methods for the Porous Medium

[18] We assume throughout that the solid matrix of the porous medium is rigid. We further assume that the pore space is saturated. Under these assumptions, Darcy's law provides a relation between the pore pressure gradient within the porous medium and the seepage rate which is in good agreement with experiment [*Riedl et al.*, 1972; *Salehin et al.*, 2004]

where is the seepage rate (in general defined as the relative velocity of the pore fluid with respect to the solid matrix multiplied by the pore fraction), *k* is the permeability (m^{2}), *μ* is the dynamic viscosity (Pa s, assumed constant for a given fluid), *P* is the pressure (Pa), and Π is the gravitational potential energy per unit volume (kg m^{−1} s^{−2}). Note that *k* was previously used to denote the wave number, as is standard, though the use of *k* to denote permeability is standard as well and should be clear from context. We define

to be the gauge pressure. We drop the prime for the remainder of this paper for simplicity.

[19] Darcy's Law, in conjunction with the conservation of mass for an incompressible fluid yields the condition

If the permeability is constant throughout the porous medium, the problem reduces to a potential problem (i.e., a solution of Laplace's equation) for pressure. We can nondimensionalize the problem by introducing characteristic scaling factors for pressure, length, height, and permeability so that

[20] Using these dimensionless variables, we can rewrite the problem as

where

Hence, the geometric distribution of seepage will only depend on the parameter *R* and the functional form of the permeability. *R* gives the ratio of the typical horizontal scale, set by the pressure perturbation at the SWI, to the depth of the porous layer. We will henceforth drop the tildes for simplicity. We estimate the value of *α* to be around 1 × 10^{−13} for lake scales, but this will vary significantly depending on the case (see *Sawyer and Cardenas* [2009] for an example of parameter values).

[22] In the case of unsteady flow in the porous medium the situation is more ambiguous. Many authors [see *Carr*, 2004, and references therein] drop the acceleration terms in the momentum equations for the porous medium and hence assume that the instantaneous flow is governed by Darcy's law. This is done even in problems of convection in porous medium, which can be expected to involve considerably more vigorous motion in the domain than those we discuss below. We thus follow *Carr* [2004] and assume that even in the unsteady case we can switch to a frame moving with the solitary wave and maintain an elliptic problem for pressure and Darcy's law for velocity, leaving a more critical examination of these assumptions to future work.

[23] The numerical solutions presented in the following were found using a Fourier-Chebyshev pseudospectral method in*x* and *z*, as described by *Trefethen* [2000]. This method achieves optimal order given a number of grid points. In practice, for smooth problems, accuracy nearing machine precision can be achieved, even with a modest number of grid points. The nonrectangular geometry was handled via two methods: an embedding method based on a technique for mountain wave computations [*Laprise and Peltier*, 1989] in the atmosphere and a mapping method (as employed by *Soontiens et al.* [2010] for the internal wave problem). Both methods were found to be accurate, but the embedding method yielded a discretized system whose linear algebra properties allowed for a much faster solution. The embedding method was thus used for all computations presented with the mapping method used to confirm the validity of the results.

#### 2.3. Permeability Distributions

[24] Lake bottoms are composed of a large variety of compounds, each with their own characteristic physical properties [*Johnson*, 1984]. Often these settle into stratified layers, with each layer having its own characteristic permeability. We considered a variety of physically motivated situations. The first case, to which all others will be compared is the case where (height is 4% of the width). Following this, we will provide a sensitivity analysis by varying the parameter *R*. Finally, we will consider the effect of the permeability stratification where the permeability sharply increases/decreases within the domain. Here the permeability function was taken to be a hyperbolic tangent function centered at 50% of the total depth of the porous layer with a transition region over 10% of the total depth. Mathematically this permeability profile can be written as

for increasing and decreasing permeability, respectively.

[25] In addition to the work on internal wave driven seepage, we propose a mechanism by which the finite amplitude internal wave increases the near-surface permeability. The basic idea is that an internal solitary wave, or trapped wave, can induce instabilities in the bottom boundary layer and thereby enhance benthic turbulence (see*Diamessis and Redekopp* [2006] and *Stastna and Lamb* [2008]for detailed discussion of this phenomenon). This turbulence lifts the small pieces of detritus that clog the near-surface pores. Based on this, a mathematical formula for the internal wave modified permeability is proposed,

While this permeability function might appear complex, its rather simple geometric form is shown later in Figure 12a.