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Keywords:

  • Darcy's Law;
  • internal waves;
  • porous media

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

[1] It has long been known that surface gravity waves induce significant seepage through the porous layer found at the lake bottom. Away from coastal regions, however, the pressure signature of surface waves at the lake bottom is weak. We consider fully nonlinear internal gravity waves, whose long wavelength and slow motion implies a sustained and strong pressure perturbation even in the deep regions of the lake. We argue that internal waves can induce significant seepage through the sediment layer, in regions where surface gravity waves have negligible impact. The pressure profile at the fluid–porous layer interface is computed from the “exact” Dubreil-Jacotin-Long theory, giving a reliable profile even for large waves. This profile is used in conjunction with Darcy's law to compute the seepage within the porous region. We find that the geometric distribution of seepage is strongly controlled by both the ratio of porous media thickness to the horizontal length of the pressure perturbation, and the bottom topography, when it is present. Based on work on the interaction of internal solitary waves with the bottom boundary layer, we develop a model to account for the changes in permeability due to wave-induced instabilities in the bottom boundary layer and enhanced benthic turbulence. This turbulence acts to unplug the pores near the surface by lifting the detritus that clogs them. The resulting changes in permeability significantly enhance exchange between the free fluid and the porous medium on the downstream side of the wave.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

[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 reviewHelfrich 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 inda Silva and Helfrich [2008].

image

Figure 1. Diagram of the formation of an internal wave train by the steepening of a lake-scale seiche. (a) A persistent wind causes a downwind tilt of the lake surface and a corresponding larger upwind tilt of the thermocline. (b) After the wind slackens, the internal seiche steepens due to finite amplitude effects and leads to the formation of a train of finite amplitude, nonhydrostatic internal waves.

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[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.

2. Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

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

  • display math

and

  • display math

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

  • display math

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,

  • display math

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

  • display math

and the density equation,

  • display math

where inline image represents the fluid velocity, p the pressure, ρ the total density, ρ0 a constant reference density, and inline image the acceleration due to gravity. As is customary, we assume that the density can be decomposed into a background profile inline image, which depends only on the vertical coordinate, and a perturbation. The total density can be written in terms of the far upstream profile, inline image, where inline imagedefines the isopycnal displacement from its far upstream height. The buoyancy, or Brunt-Vaisala, frequencyN(z), which is defined via the relation

  • display math

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 (seeSoontiens et al. [2010] and Stastna and Lamb [2002] for a detailed derivation), to a single, scalar equation for the isopycnal displacement inline image. This is the well-known Dubreil-Jacotin-Long (DJL) equation,

  • display math

where U0 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 inline image) 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 inline image 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 inline image as inline image. The functional form of the topography was selected to be a hyperbolic secant function (though this is largely a matter of mathematical convenience)

  • display math

where A0 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.

image

Figure 2. Diagram of the basic problem layout. Note the inflow and outflow conditions as we are in a frame of reference moving with the wave. The internal wave generates a pressure distribution used to solve for the seepage within the porous lake bed (gray). Note the impermeable layer under the bed (black). Behind the wave of depression, often a region of boundary layer instability will form and this is also depicted here as a superposition of three dotted ellipses.

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image

Figure 3. Pressure profile induced by an ISW with a maximum dimensionless isopycnal displacement of 0.215 at the SWI extracted from the solution of the DJL equation in the free fluid.

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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]

  • display math

where inline image 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 (m2), 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

  • display math

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

  • display math

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 inline image so that

  • display math

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

  • display math

where

  • display math

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).

[21] The linear elliptic equation (10) for the pore pressure must be supplemented by appropriate boundary conditions. We assume no flow in the porous medium far upstream and downstream. We also assume that an impermeable layer (due to the presence of clay, for example) is found at a depth inline image. Finally, we assume that the pore pressure at the SWI, inline image, is continuous, and hence specified by the fluid pressure (which may be recovered a posteriori from the solution of the DJL). The assumption that the lake bottom is a streamline for the free fluid flow is justified since the speed of the flow into the porous medium will be many orders of magnitude lower than the typical speed in the free fluid, as discussed in the introduction. The complete boundary value problem is solved on the domain for which inline image and inline image. Mathematically, the boundary conditions can be written as

  • display math

and the situation is schematized in Figure 2.

[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 inx 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 inline image (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

  • display math
  • display math

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 (seeDiamessis 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,

  • display math

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

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

3.1. Exploratory Cases

[26] As mentioned in section 1, Webb and Theodor [1968]have demonstrated that convection is insufficient to explain the observed seepage through the porous layer found at the bottom of lakes. It is thus sensible to argue, and indeed the Introduction discusses several examples from the literature, that this seepage is, at least in part, a result of gravity wave-induced flow through the lake bottom. However, in lakes deeper than a few meters, the pressure profiles of surface waves are weak (seeequation (1)). As such, seepage in fresh water lakes that is induced by surface water waves is constrained to shallow regions near the coast [Bokuniewicz, 1992]. Moreover, even when the depth is small enough so that they could influence the bottom pressure, surface waves have a short wavelength, move quickly, and generally do not retain the identity of individual wave crests for long times. The importance of the horizontal scale of the pressure profile that forces flow in the porous medium is easy to see in the limiting case of a very deep, porous lake bed forced by a sinusoidal pressure profile. For this case, the pressure in the porous medium is given by

  • display math

The pressure, and hence the seepage, decays exponentially within the porous layer at a rate given by the wave number, . Thus we see that disturbances induced by short wavelengths (large wave number) decay quickly. Since internal waves typically exhibit length scales that are in the tens of meters, we thus see that pressure fluctuations due to internal waves will be able to penetrate into thicker lake bottoms than their surface water counterparts. The vertical component of the seepage is given from Darcy's law as

  • display math

and this implies that the magnitude of seepage at the SWI decreases with wavelength.

[27] Figure 4 demonstrates the same effect for the more realistic case of an impermeable layer at the base of the porous layer. Figures 4a, 4b, and 4c output the pressure distributions as a result of sinusoidal pressure profiles at the SWI with R = 1, 2, 4, respectively. In Figure 4d, the normalized seepage profiles taken along the line x = 0 are given. It can be seen that again, the seepage due to short wavelength perturbations does not penetrate deep into the porous layer while longer wavelengths drive significant seepage at greater depths.

image

Figure 4. Normalized pressure distributions as a result of sinusoidal pressure waves of wavelengths (a) R = 1, (b) R = 2, and (c) R = 4 at the SWI. (d) The seepage profiles plotted at x = 0. Note that longer wavelengths penetrate deeper into the porous medium.

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[28] Wörman et al. [2006]argue that in their three-dimensional simulation, seepage decays with wavelength even faster with depth than similar two-dimensional simulations. However, as internal waves tend to maintain their across-lake structure (M. Preusse, private communication, 2012) over very long length scales (certainly far longer than the wavelength), three-dimensional structure is presumed to play a rather weak role.

[29] Figure 5 considers a sinusoidal perturbation with R = 10. The pressure perturbation is shown in Figure 5a, while the magnitude of the seepage with superimposed arrows indicating the direction of flow is shown in Figure 5b. It can be seen that the regions of large magnitude of seepage are concentrated near the maximum and minimum of the pressure profile gradient. Note that over the majority of the domain, the seepage is nearly horizontal.

image

Figure 5. (a) Normalized pressure profile at the SWI, (b) magnitude of the normalized seepage through the porous medium (shaded). Arrows denote the direction of the seepage. R = 10.

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[30] Elliot and Brooks [1997]have discussed a similar phenomenon, demonstrating that the mass transfer within the porous medium is proportional to the magnitude of the pressure perturbation and inversely proportional to the wavelength of the gravity wave. Their analysis is unchanged for internal waves. Both analyses thus suggest that the large wavelength of an internal gravity wave will induce less seepage at the SWI than will a shorter wavelength wave of equal magnitude. However, the induced seepage of large wavelength internal waves will penetrate deeper into the porous medium, thereby making available nutrients previously inaccessible to the main water column. Moreover, since in water depth greater than a few meters surface gravity wave-induced pressure perturbations are negligible, internal waves greatly expand the geographic area over which significant seepage can take place.

3.2. Internal Solitary Waves

[31] In a frame of reference frame moving with a single solitary wave (or one wave in the wave train schematized in Figure 1), we can use the DJL equation to compute a pressure profile at the SWI (see Figure 3) and thereby compute the seepage induced through the bottom porous layer. While both Salehin et al. [2004] and Sawyer and Cardenas [2009]have demonstrated that the distribution of small-scale inhomogeneity induced by bed forms (e.g., ripples) can have a significant impact on the fluid residence time within the lake bed (at least on submeter scales), here we are interested primarily in motions induced by a long length-scale internal solitary wave. We thus concentrate on a flat-bottomed lake and three characteristic permeability distributions that vary with depth only. We discuss the constant permeability case first, follow this with a discussion of the influence of changing the porous layer thickness parameterR, and finally discuss the influence of permeability stratification within the domain.

3.2.1. Case 1: Constant K

[32] In Figure 6 we show the pore pressure (Figure 6a), seepage rate magnitude (Figure 6b), and transport across and along the SWI for the case of constant permeability within a porous layer with R = 25. Figures 6a and 6b show that both the pore pressure and seepage rate magnitude distributions are symmetric across the wave crest. From Figure 6cwe can see that the wave induces both along and across SWI transport. The across-SWI transport is roughly three quarters of the magnitude of the along-SWI transport and consists of a region of out flux near the wave crest and a broader region of influx on the flanks of the wave.

image

Figure 6. (a) Computed pressure, (b) seepage magnitude, and (c) seepage profiles along the SWI computed with a constant permeability throughout. Seepage profiles have been normalized by the maximum seepage within the domain and hence dimensionless values are presented here. R = 25. Here tangential seepage corresponds to along SWI transport and normal seepage corresponds to across SWI transport.

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[33] We see that the ISW is acting essentially as a “vacuum cleaner,” drawing fluid up through the SWI near the wave crest. As the porous layer in this case is shallow, we see that fluid is pulled right from the bottom of the porous layer, filtering through regions which pressure pulses due to surface gravity waves would not be able to penetrate.

3.2.2. Case 2: R Sensitivity

[34] The seepage throughout the porous domain is controlled by the dimensionless parameter R. As previously defined, R denotes the ratio of width scale of the porous layer to the height. Figure 7 compares the tangential (Figure 7a) and normal (Figure 7b) fluid velocities at the SWI and its sensitivity to the parameter R, normalized by the maximum seepage when R = 25. Notice that the tangential velocity at the SWI is invariant under the value of R, while the normal seepage grows drastically as the depth of the domain is increased (while the profile of the pressure perturbation remains fixed). This result is perhaps the clearest indication that the long wavelengths characteristic of the ISW can penetrate deep into the domain and thereby boost the fluid exchange between the porous layer and the water column.

image

Figure 7. (a) Computed tangential and (b) normal seepage rates at the SWI for varying values of R. Profiles were normalized to the maximum value of seepage of R = 25.

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[35] Further sensitivity analysis (not shown) demonstrates that with the same pressure perturbation used here, values of R lower than 2.5 yield essentially no further increase in normal seepage. Thus we see that the most substantial increase in seepage rates occurs when the R value is well above 1, and hence the typical wavelength scale of the prescribed pressure distribution is larger than porous layer depth.

3.2.3. Case 3: Permeability Stratification

[36] The porous layers on lake bottoms are often stratified into layers, thereby varying permeability in the z direction. As defined above in equations (11) and (12), we allow the permeability to double or halve over the domain. Figure 8 plots the seepage throughout the domain as a result of the permeability stratification. Both the seepage distributions (Figures 8a and 8d) and profiles along the SWI (Figures 8b and 8e) are shown.

image

Figure 8. (a and d) Computed seepage and (b and e) seepage profiles along the SWI computed (c and f) a permeability stratification centered at 50% of the depth of the porous layer. Permeability decreases with depth (Figures 8a–8c) and increases (Figures 8d–8f) by a factor of 2. Seepage profiles have been normalized by the maximum seepage within the constant permeability domain. R = 25.

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[37] What we find is that the permeability stratification effectively constrains the thickness of the porous layer. When the permeability decreases (Figures 8a–8c) by a factor of two with depth, the seepage is constrained within the upper, more permeable portion of the porous domain, and hence R is effectively doubled. The opposite effect can be noticed when the permeability is doubled with depth (Figures 8d–8f). In this case the seepage is largely found within the bottom, more permeable half of the porous layer. Notice that there is a four-fold difference in seepage at the SWI between the two profiles, with the less permeable upper layer exhibiting lower values. The shape of the normal and tangential seepage profiles at the SWI, as well as their relative proportion, remain identical. These results are consistent with those found byZlotnik et al. [2011] who demonstrated that a decreasing permeability distribution resulted in a decrease in the SWI seepage but had an increase in the seepage at greater depths.

3.3. Trapped Internal Waves and Bottom Boundary Layer Interaction

[38] The presence of a “trough” topography, for example due to a drowned creek bed, allows for a trapped wave to form. Using an isolated topography, with the mathematical form given by (7), and solving the DJL equation, we compute the pressure along the SWI and use this pressure profile to drive seepage in the porous medium. Note that the domain is nonrectangular and this requires some care in the construction of the numerical method, as described above. Figure 9 plots the resultant distributions of pressure (Figure 9a), seepage (Figure 9b), and the profiles of seepage along and across the SWI (Figure 9c). Interestingly, the profile of the across SWI seepage has become bimodal about the origin, with the seepage reduced near the center of the domain. Moreover, as the topography effectively decreases the depth of the porous domain, the ratio between normal and tangential seepage decreases near the domain center. That is, the normal seepage appears to decrease in relation to tangential seepage. However, as trapped waves can reach amplitudes much greater than propagating waves, the total seepage will be significantly larger than that presented above for propagating waves.

image

Figure 9. (a) Computed pressure, (b) seepage magnitude, and (c) seepage profiles along the SWI computed with a constant permeability throughout. Topography was chosen to be a sech function for simplicity. Seepage profiles have been normalized to the maximum seepage within the flat topography domain. Dimensionless values are presented here. R = 25.

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[39] The interaction of ISW with the bottom boundary layer has been discussed extensively in the literature from numerical [Stastna and Lamb, 2008; Bogucki et al., 2005; Diamessis and Redekopp, 2006], experimental [Carr et al., 2008], and observational [Bogucki et al., 1997] points of view. The general picture that has emerged from these studies is that while waves of elevation require the presence of an upstream current (such as the barotropic tide, or lake-scale seiche) in order to yield a global instability and enhanced benthic turbulence in the bottom boundary layer, waves of depression can yield global instabilities in the boundary layer due to the adverse pressure gradient induced by the wave. However, the instabilities due to the waves of elevation are generally more vigorous. In the context of trapped waves this propensity for instability is strengthened by the curvature of the bottom boundary that can, in itself, induce separation and boundary layer instability [Schlichting and Gersten, 1979].

[40] The presence of turbulence in the boundary layer precludes the possibility of a steady state so we have performed several time-dependent simulations in order to demonstrate the development of a highly turbulent boundary layer. The simulations were completed with a pseduo-spectral code [Subich, 2011]. The two-dimensional simulations were performed at a Reynolds number based on upstream velocity, total depth, and the viscosity of water, of roughly inline image, well into the transitional regime. In Figure 10, we show normalized vorticity at four nondimensional times, inline image. The time has been nondimensionalized by the background flow and column depth so that inline image. Within the bottom boundary layer, instability leading to turbulence develops in the downstream portion of the topography. This will, in turn, lead to pickup of the smaller particulate matter at the surface of the underlying porous layer. At later times the turbulent region has expanded well downstream of the topography. This process is elaborated further in plots of the bottom stress, Figure 11. The gray lines indicate values of ±500% of the stress induced by the current upstream. In a large portion of the downstream half of the domain, the bottom stress exceeds a factor of five times the upstream value for at least a portion of the time. The spikes in the bottom stress shown can thus lead to pickup of particulate matter, with vortex induced currents transporting material up into the water column and away from the SWI.

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Figure 10. Evolution of the normalized vorticity within the water column as a result of the currents induced by the trapped wave. The four nondimensional times are (a) 11.2, (b) 12.6, (c) 14, and (d) 15.4. Notice the generation of a turbulent region in the boundary layer in the downstream portion of the topography. Four isopycnal lines have been plotted to show the wave form.

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image

Figure 11. Evolution of the scaled bottom stress after onset of instability in the boundary layer. The vertical axis marks snapshots at different times. The curves at 1, 3, 5, and 7 correspond to Figures 10a, 10b, 10c, and 10d. The gray lines represent a deviation of ±500% from the stress induced by the current upstream of the wave.

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[41] In order to model the effect of boundary layer instability and bottom-stress-induced pore unclogging, we reconsider the problem whose solution is shown inFigure 9, but with a modified permeability distribution with the form given above (equation (13)). In Figure 12 we compare the results for the trapped internal wave modified permeability profile, with those of the constant permeability case (as shown in Figure 9). Figure 9a shows six contours of the permeability distribution, Figure 9b shows six contours of the magnitude of the seepage rate, and Figure 9ccompares across SWI transport between the modified and constant permeability cases (both are scaled by the maximum seepage rate of the constant case). It can be clearly seen that the wave-induced boundary layer instability, as expressed by the modified permeability, substantially increases the seepage through the porous domain.

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Figure 12. Comparison of turbulence modified permeability with the constant permeability case. (a) Six shaded equally spaced contours of the normalized permeability, (b) Six shaded equally spaced contours of the magnitude of the seepage rate, scaled by its largest value, (c) transport across the SWI for the wave modified permeability (solid) and constant permeability (dashed) cases. R = 25.

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4. Discussion and Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

[42] The analysis of the nutrient circulation system in oligotrophic lakes is an area of continuing research [Oliveira Ommen et al., 2011]. At the same time, field experiments that identify the incidence of high-frequency, large-amplitude internal waves continue to be carried out and to provide insight into their wave structure, amplitude, and seasonality [Preusse et al., 2012; Hutter, 2011, chapter 2]. The main point of the modeling work presented above was to demonstrate that there is a significant link between the nutrient circulation system and internal waves. To date, there exists very little work in the literature concerning this coupling.

[43] In order to demonstrate the coupling between internal waves and seepage through a porous bottom layer, we have employed solutions of the DJL equation to drive seepage in a porous medium. These waves are exact solutions of the full stratified Euler equations and thus provide the best possible representation of large amplitude internal waves. The porous medium was assumed to be governed by Darcy's law. Numerical solutions were obtained using highly accurate Fourier-Chebyshev pseudo-spectral methods. Unlike previous numerical results [Higashino et al., 2009; Meysman et al., 2007], these methods included both large-scale (tens of meters) topography and inhomogeneous permeability. While the formation of internal waves from the steepening of internal seiches is a generic phenomenon that is largely independent of lake topography, this topography is crucial in allowing the formation of the trapped waves, and in modifying the wave-induced pressure perturbations that drive flow across the SWI. We briefly mention that the importance of topography has also been mentioned in the literature, whereShum [1993] demonstrates how the concentration of nutrients within the lake bed can be substantially influenced by the bottom topography. Similarly, Wörman et al. [2006] also demonstrate how complex topography can lead to stagnant regions within the porous domain.

[44] In terms of a simplified, or “cartoon” picture, the internal wave acts essentially as a vacuum cleaner, with fluid drawn out of the porous medium under the internal wave peak where wave-induced pressure is lowest (for internal waves of depression) and returned in the broad regions found along the flanks of the solitary waves. While observed in a different context, this prediction is similar to the observations found byHuettel et al. [1996]. The extent of the seepage can be understood in terms of the parameter inline image, which gives the ratio between the horizontal scale of the pressure perturbation and the vertical extent of the porous layer. For R > 1, the typical wavelengths of the prescribed pressure profile are larger than the porous layer depth. In this parameter regime we found that the seepage through the porous domain increased with increasing R due to the fact that the long wavelengths associated with the internal waves penetrate deeper into the porous medium.

[45] Finally, we proposed, and provided numerical evidence for, a mechanism by which the finite amplitude wave increases the near-surface permeability. The basic idea is that, in agreement with theoretical and experimental studies in the literature, the wave induces instabilities in the bottom boundary layer and thus enhances benthic turbulence. This turbulence lifts the small pieces of detritus that clog the near-surface pores. An idealized permeability distribution created to encapsulate this scenario led to both an increase in across SWI transport and a shift of its maximum downstream of the wave crest. This suggests that in conjunction with the pressure perturbation induced by either traveling or trapped waves, instabilities in the bottom boundary layer can augment the seepage through the bottom porous layer and further enhance chemical exchange with the porous layer. A discussion of the effect of turbulence on the vertical solute transfer can be found inHigashino and Stefan [2012].

[46] Future work should consider several extensions of the above work. First, multiscale simulations that couple the large-scale pressure profiles due to internal solitary waves to the small-scale bed forms discussed in past literature [Salehin et al., 2004; Sawyer and Cardenas, 2009; Huettel et al., 1996] should be carried out. Second, for cases in which a less permeable layer overlies a more permeable layer, the question of possible ducting by regions of lower permeability should be explored. Finally, more complete formulations of the equations governing the flow in the porous medium, e.g., Brinkman terms, should be explored.

[47] The present paper presents the potential importance of internal waves on the nutrient circulation systems in lakes. Our hope is that this paper will encourage both experimentalists and theoreticians to consider the substantial impact that internal solitary waves can have on lake-bed dynamics and their impact on the biosphere.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References

[48] This work was funded by the Natural Sciences and Engineering Research Council of Canada. Christopher Subich provided extensive help with the numerical methods used.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion and Conclusion
  7. Acknowledgments
  8. References