## 1. Introduction

[2] Flow through permeable lake, coastal, or river sediment has received significant attention in the literature [*Cardenas and Wilson*, 2007; *Elliot and Brooks*, 1997; *Genereux and Bandopadhyay*, 2001; *Higashino et al.*, 2009; *Rocha*, 2000]. The resultant flow in, and out, of the porous medium can have profound implications on bed water column heat and chemical exchange [*Bailey and Hamilton*, 1997; *Higashino et al.*, 2009; *Huettel et al.*, 1996; *Meysman et al.*, 2007]. In turn, this exchange can have implications for food web dynamics, especially in nutrient limited situations such as those that occur in oligotrophic lakes. The primary means of generating flow through a porous lake bottom is through spatially varying pressure fluctuations at the sediment-water interface (SWI) [*Elliot and Brooks*, 1997]. Natural waters exhibit motions, and hence pressure fluctuations, on a wide variety of length scales ranging from variations in boundary layer turbulence due to bottom ripples on the scale of centimeters to basin-scale motions on the scale of kilometers. A commonly observed cause of bottom pressure fluctuations in shallow water is surface waves. At depths greater than a few meters, the bottom pressure signature of these waves is weak. However, most lakes and coastal waters are stratified for at least some portion of the year and the resulting density stratification can serve as a waveguide for internal wave motion. Internal waves are typically longer than surface waves, are slow moving, and due to the small density differences across the thermocline, can reach large amplitude. Moreover, they are often long lived and hence can induce significant bottom currents and pressure fluctuations with a characteristic geometric pattern that differs from surface wave induced fluctuations.

[3] The range of motions of a density-stratified fluid including interaction with a wind-driven surface wavefield and bottom topography is truly bewildering. No numerical model we are aware of covers all these scales of interest, and even if one were available, the problem of isolating particular phenomena would be extremely difficult. In stratified lakes, a typical situation leading to the generation of large-amplitude internal waves is schematized inFigure 1. In Figure 1a, a persistent wind has induced a downwind tilt of the lake surface and a corresponding larger upwind tilt of the thermocline. In Figure 1b, after the wind slackens, the internal seiche steepens due to finite amplitude effects and leads to the formation of a train of finite amplitude internal waves. These waves are both nonlinear and nonhydrostatic, and thus present a challenge for many lake-scale models. Measurements of this scenario have been reported in the literature for geography as varied as Lake Kinneret, Israel [*Boegman et al.*, 2003]; Lake Constance, Germany [*Preusse et al.*, 2012]; and Toolik Lake, Alaska, USA [*MacIntyre et al.*, 2006]. A significant body of literature is available on coherent aspects of internal wave motions. In particular, the fully nonlinear theory leading to the Dubreil-Jacotin-Long (DJL) equation which describes internal solitary waves (ISWs) and steady trapped waves (see the review*Helfrich and Melville* [2006] and *Soontiens et al.* [2010]for recent discussions) provides an easy to use tool with which to address internal wave-induced motions in the porous bottom. Observations of supercritical internal waves have also been reported in*da Silva and Helfrich* [2008].

[4] Darcy's law is a classical, experimentally determined relationship which relates pore pressure gradients in the porous medium with the seepage rate (often also referred to as filtration, or filtration rate). Given a pressure distribution at the SWI, Darcy's law thus provides an effective way of determining the flow in the porous medium [*Nield and Bejan*, 2006, chapter 1]. Various refinements of Darcy's law have been proposed, including the Brinkman and Forchheimer terms. These attempt to partially account for shear stress and finite Reynolds number, respectively [*Meysman et al.*, 2007; *Nield and Bejan*, 2006]. However, neither modification comes with a firm range of applicability and, in the present context, the literature suggests neither is important [*Meysman et al.*, 2007]. Any modeling effort attempting to include these refinements would also need to consider a generalized boundary condition, as discussed in the classical paper by *Beavers and Joseph* [1967] and in a more recent comparative work [*Le Bars and Worster*, 2006]. In the present work, we will use Darcy's law only, and hence we can supply all the necessary boundary conditions at the SWI by invoking continuity of pore pressure with the bottom pressure in the free fluid. We assume that the seepage through the porous medium is small, consistent with literature on the damping of linear and weakly nonlinear surface gravity waves by flow over a porous region [*Flaten and Rygg*, 1991; *Packwood and Peregrine*, 1980]. This assumption allows us to solve for the pressure at the SWI based purely on the physics in the free fluid. This pressure profile subsequently drives the seepage in the porous medium. The coupling across the SWI is thus one-way, in the sense that the modification of the flow in the free fluid due to the porous medium is neglected. This one-way coupling is justified by the vast separation of scales between ISW induced velocities and velocities within the porous medium.

[5] Motivated by the findings of *Webb and Theodor* [1968], who demonstrated that convection is insufficient to explain the seepage through the porous lake bottom, the above described approach has been explored by a number of authors, generally in an attempt to study seepage due to surface wave induced flow over topography (e.g., ripples with centimeter scales), and has demonstrated noticeable seepage in the porous layer [*Shum*, 1993; *Elliot and Brooks*, 1997; *Sawyer and Cardenas*, 2009; *Huang et al.*, 2011; *Cardenas and Jiang*, 2011; *Hsieh et al.*, 2003]. We argue here that internal waves induce substantially more seepage through the porous domain than do surface gravity waves in lakes deeper than a few meters.

[6] While we are concerned here mainly with the seepage through the porous layer, we note that *Shum* [1993] has constructed a model to explicitly model the flow of solutes through the porous layer. Similarly, a recent study by *Grant et al.* [2012] argued that the mass transport through the hyporheic zone may be significantly different than previously thought. We leave the coupling of such work with internal waves for future investigations.

[7] The remainder of the article is organized as follows: We briefly discuss the comparison between surface gravity waves and internal waves and their generation. The governing equations for the free fluid and their various manipulations used to find finite amplitude internal waves are also briefly summarized. This is followed by a statement of the boundary value problem in the porous medium, and a brief discussion of numerical methods. The results are divided into three parts. First, we discuss the effect of the wavelength of the pressure profile at the SWI. Second, we analyze a single, propagating internal solitary wave and the seepage it induces. We discuss the effect of varying the depth of the porous layer and vertical variation in permeability (or permeability stratification). We subsequently move on to flows induced by trapped internal waves over topography. Finally, we propose a scenario during which the wave-induced boundary layer instabilities [*Diamessis and Redekopp*, 2006; *Stastna and Lamb*, 2008] modify the permeability distributions beneath and downstream of the wave, and hence lead to increased seepage in the porous medium and transport across the SWI. Section 4 reviews the findings and discusses future directions.