Quantifying the length-scale dependence of surf zone advection

Authors


Abstract

[1] We investigate the momentum balance in the surf zone, in a setting which is weakly varying in the alongshore direction. Our focus is on the role of nonlinear advective terms. Using numerical experiments, we find that advection tends to counteract alongshore variations in momentum flux, resulting in more uniform kinematics. Additionally, advection causes a shifting of the kinematic response in the direction of flow. These effects are strongest at short alongshore length scales, and/or strong alongshore-mean velocity. The length-scale dependence is investigated using spectral analysis, where the effect of advective terms is treated as a transfer function applied to the solution to the linear (advection-free) equations of motion. The transfer function is then shown to be governed by a nondimensional parameter which quantifies the relative scales of advection and bottom stress, analogous to a Reynolds Number. Hence, this parameter can be used to quantify the length scales at which advective terms, and the resulting effects described above, are important. We also introduce an approximate functional form for the transfer function, which is valid asymptotically within a restricted range of length scales.

1. Introduction

[2] Depth- and time-averaged surf zone currents, initiated by the transfer of momentum due to wave breaking, are often classified in the literature as alongshore-uniform (1DH) or nonuniform (2DH). 1DH flow corresponds to an alongshore-uniform shore-parallel jet [Bowen 1969a; Longuet-Higgins, 1970a; Thornton and Guza, 1986; many others], and occurs on long straight beaches with oblique wave angles. 2DH flows contain across-shore-directed currents, with an extreme case being a rip current system [MacMahan et al., 2006 and references therein]; these typically occur on alongshore-nonuniform beaches with small (nearly shore-normal) wave angles.

[3] Between these two extremes, there is a continuum of flows typified by meandering or separation of the alongshore current. In these cases, an alongshore current is propagating over spatially varying bathymetry or through a spatially varying wave field, and hence exhibits somewhat-2DH behavior, although perhaps less pronounced than a rip current. These, too, have been well documented [Sonu, 1972; Slinn et al., 2000], but have received less theoretical treatment than the extreme cases described above. One exception is the numerical study by Wu and Liu [1984], who simulated steady 2DH flows on undulating bathymetry with oblique wave angles, and found that the results did not correspond to a linear superposition of simple 1DH and 2DH end-members. Instead, nonlinear advection was shown to play a leading-order role in determining the magnitude and alongshore position of flow undulations. When the strength of the background alongshore current was increased, flow undulations became increasingly weaker, and were pushed downstream, illustrating the fact that advection acts to bypass minor variations in forcing. This effect can be important for the prediction of surf zone circulation in the field. For example, Long and Özkan-Haller [2005] present a realistic case where a linearized model predicted the presence of a rip current extending past the surf zone, whereas a nonlinear model did not. They explain that the inertia of the background flow was acting to “sweep away” smaller rips in the nonlinear model. Hence, an understanding of what constitutes a “strong” background flow, or a “minor/short” variation in forcing, is important for bridging the gap between 1DH and 2DH regimes.

[4] Our approach is to study the role of nonlinear advective terms in a weakly nonlinear setting, using a combination of numerical experiments and linearized analysis. In section 2, we investigate advection by selectively including/excluding it in a numerical model (equations (1)-(3), in which the advective terms are in square brackets). The difference between the alongshore current with and without advection is interpreted in terms of an effective transfer function, analyzed using spectral methods, which provides a quantified description of the length-dependent shifting/dampening of flow perturbations described above. In other words, we hypothesize that advection can be viewed as a spatial filter, which we seek to characterize using a transfer function. Then, in section 3, we analyze simplified equations of motion to better understand the numerical results. We find that the numerical results can be largely explained using a single nondimensional parameter, which quantifies the strength of the advective terms. We also derive an asymptotic approximation for the transfer function. This approximation is only valid under specific simplifying conditions, which we describe, but may represent useful progress toward a more general understanding.

2. Advection Transfer Function: Numerical Experiments

2.1. Basic Approach

[5] Our approach to studying advection in surf zone flows will be to analyze numerical model results in terms of a transfer function representing the effect of the advective terms on the alongshore current. In this section, we define the model used for surf zone dynamics, the hypothesized transfer function, and the method of extracting the transfer function from numerical results.

[6] We will limit our analysis to cases with weak alongshore variability, such that leading order properties of the flow can be obtained from an alongshore average (i.e., the ratio between alongshore perturbations and alongshore averages is assumed small). We also focus on the region just shoreward of the maximum velocity of alongshore current, where the alongshore current is relatively strong, and hence where we expect the strongest influence from advective terms. These assumptions will be reflected in the analysis of section 3. The main reason for the restriction of weak variability is to avoid nonlinear interactions between different length scales; hence the advection transfer function can be computed for a range of length scales from a single model simulation (or one ensemble, see section 2.4). Treatment of large-amplitude variability, though potentially important for real-world flows, would require more detailed numerical experiments and would lead to a more difficult theoretical analysis. This should be the subject of subsequent study.

2.2. Numerical Model

[7] The numerical model used here assumes steady-state depth-averaged and wave-averaged equations of motion [Mei, 1989],

display math(1)
display math(2)
display math(3)

where inline image are the across-shore and alongshore currents, h is the still water depth, η is the mean water surface elevation, inline image are constant forces due to radiation stress gradients [Longuet-Higgins and Stewart, 1964], M represents momentum mixing, and inline image is the bottom shear stress. Subscripts denote x and y components. The basic physics captured by this model include the transfer of momentum from waves (represented by radiation stress gradients) and the response of the free surface and 2DH currents to generate pressure gradients and bottom stresses. These physics are represented in most contemporary models for 2DH surf zone dynamics [following, e.g., Bowen, 1969b]. The differences among such models largely lie in the choice of physical parameterizations, used to obtain quantitative accuracy for field and laboratory prediction [e.g., Thornton and Guza, 1986; Reniers and Battjes, 1997; Ruessink et al., 2001; Haas et al., 2003, many others]. In order to study surf zone advection in as simple a setting as possible, we choose to employ relatively simple parameterizations, described next.

[8] Momentum mixing is parameterized by

display math(4)

where the eddy viscosity inline image is computed as in Haas et al. [2003].

[9] Bottom stress is modeled using a linear parameterization,

display math(5)

where μ is a constant drag coefficient. Although this parameterization can be inaccurate for quantitative prediction in natural settings [Feddersen et al., 2000], it is often used to facilitate analytical interpretations of numerical results (e.g., in studies of nonlinear shear instabilities; Dodd et al. [1992]; Allen et al. [1996]; Slinn et al. [1998]; Feddersen [1998]; Slinn et al. [2000]; or in the present case, section 3]). As a reality check, we will also include one test case where the bottom stress is modeled using a more-realistic nonlinear parameterization [Svendsen and Putrevu, 1990].

[10] To compute radiation stress gradients, and other wave properties required for computing inline image, we use the numerical code SWAN, which solves the spectral action-balance equation [Mei, 1989; Booij et al., 1999]. This equation governs the energy flux of waves propagating from the deep ocean into the surf zone, where they are modified by the presence of the shallow bottom, and ultimately break (wave breaking is parameterized in SWAN using the model of Battjes and Janssen [1978]). These processes in turn determine the transfer of momentum represented by the radiation stress gradient terms F, which are an output of SWAN. The effect of the mean flow on the waves (i.e., wave-current interaction) is neglected, hence F is considered a fixed forcing represented as a body force in equations (1)-(3).

[11] Numerical solutions of equations (1)-(3) are calculated using a modified version of the numerical code Shorecirc (version 2.0; Svendsen et al. [2000]). This code solves equations (1)-(3) in time-dependent form. To obtain the desired steady-state solution, radiation stress gradients are smoothly ramped up from zero over a period of 2500 s, then are kept fixed for 10 h. No-flow boundary conditions are applied at the 10 cm depth contour, and a 2-D radiation boundary condition is applied at the offshore boundary [van Dongeren and Svendsen, 1997].

2.3. Hypothesized Transfer Function

[12] To reiterate, based on previous studies [Sonu, 1972; Wu and Liu, 1984; Long and Özkan-Haller, 2005] we assume that the presence of advective terms causes changes in the alongshore current v, which are dependent on the alongshore length scale. Specifically we expect that, at short length scales, the inertia of the flow will cause changes in the position and magnitude of alongshore-variability in v. To quantify this, we hypothesize that advection can be approximated in terms of a transfer function,

display math(6)

where vl is the current which would result if advective terms were not present. The hat operator in equation (6) represents a Fourier Transform in y.

[13] The hypothesized advection transfer function (equation (6)) includes a real-valued gain factor, I, which represents the reduction of flow variability (i.e., smoothing) when advection is included; it also includes a real-valued phase shift, inline image, which represents the shifting of flow features downstream. Note we allow I and inline image to be dependent on x (across-shore); this will be investigated, although we will initially focus our attention on the region immediately shoreward of the maximum of alongshore current, where the mean alongshore current is strong, i.e., where the effects of advection are likely to be most pronounced.

2.4. Method for Estimating Transfer Function

[14] The equations governing v and vl are defined and solved as in section 2.2, with the exception that, when solving for vl, the terms in square brackets in equations (2) and (3) are dropped. Once these solutions are obtained, we estimate the advection transfer function inline image from equation (6) as follows.

[15] We adopt the usual notation for the cross spectrum of two functions inline image and inline image,

display math(7)

where the asterisk denotes complex conjugation. The hypothesized cross spectrum between v and vl is, using (6),

display math(8)

so that the gain and phase shift due to nonlinear advection can be calculated from

display math(9)
display math(10)

[16] In practice, these estimates are computed from discretized numerical model outputs, using the discrete cross-spectral density,

display math(11)

where the summation is over the N model gridpoints (with grid spacing inline image). Another practical consideration is the fact that equations (9) and (10) can be unreliable at spectral bands for which inline image is small: assuming equation (6) is not exact, calculations at such low-energy bands would be sensitive to small errors. In practice we found that, for a given realization of h (see later), this problem almost always occurred for one or more bands. Hence, we opt to compute our estimates using a weighted ensemble average in equations (9) and (10), using 20 realizations of bathymetry/flow (section 2.5), where the weights are chosen equal to inline image. We also discard bands below a noise floor, inline image m3/s2, which occurred only at high wave numbers. In addition to having low signal, such high wave number bands are likely to be dominated by momentum mixing, which is not of primary interest for the present work. Also, we note that the wave model used here is unlikely to produce meaningful outputs for length scales less than a typical wavelength, which puts an upper bound of inline image m−1 for the cases we are studying.

2.5. Definition of Test Cases

[17] In the present tests, the numerical model grid has dimensions of 512 gridpoints in the alongshore direction, and 180 gridpoints across-shore, having grid spacings of 10 and 5 m, respectively. The domain is taken to be alongshore-periodic.

[18] We will test two bathymetries, which we refer to as the “barred” beach and the “troughed” beach. These bathymetries are chosen to represent possible field conditions, although they are not meant to describe any one beach (nor do they describe all beaches in general). Both bathymetries are defined using the formula

display math(12)

where inline image is a 2-D random field having unit standard deviation. To compute n, we apply a Gaussian low-pass filter to white noise; the filter has standard deviation of 15 m in the across-shore and 30 m in the alongshore direction, and the filter footprint is inline image gridpoints (see sample results in Figures 1 and 2). As stated above, 20 realizations of n are computed for each test case.

Figure 1.

Nonlinear model outputs for troughed beach, inline image m/s, plotting 1 km in y for a single realization (complete domain is 5 km wide, not shown). From left to right: (a) bathymetry perturbation inline image, in meters, where h0 is the alongshore-averaged bathymetry; (b) velocity deviations from the alongshore-average (red scale arrow in upper right indicates 10 cm/s); (c) momentum balance at transect x = 100 m (indicated by black dashed lines in (a) and (b); solid black line indicates location of maximum v0, for reference), in cm2/s2. The momentum balance terms in Figure 1c are: blue, inline image; red, inline image; magenta, Fy; green, inline image; black, inline image. Dashed lines in Figure 1c are the result when advection is not included in the model.

Figure 2.

As in Figure 1, but for barred beach, inline image m/s, x = 275 m.

[19] The first model bathymetry, which we call the troughed beach, is defined by the parameters inline image (beach slope), W = 75 m (across-shore width of trough), inline image m (offshore location of trough), inline image m (depth of trough), inline image (amplitude of perturbations within trough). With these parameters, the bathymetry is planar (alongshore-uniform) in the surf zone, but contains a depression, or trough, which in our case will be outside the surf zone. Alongshore variability within the trough causes modification of the incoming wave field, hence the surf zone wave-induced forcing is alongshore-nonuniform. This is reminiscent of the beach studied by Long and Özkan-Haller [2005], where bathymetric undulations in an offshore canyon (well outside the surf zone) were shown to produce surf zone rip currents.

[20] The second model bathymetry, which we call the barred beach, consists of a plane beach profile superimposed with a bar which in our case will be located in the surf zone: inline image, W = 75 m, inline image m, inline image m, and inline image. The slight alongshore-variability in the bar shape causes alongshore-nonuniform wave-induced forcing, as in the troughed beach, however in this case the surf zone bathymetry is also alongshore-varying. The smaller value of D for the barred beach was chosen such that the magnitude of variability in alongshore current was similar to that of the troughed beach.

[21] For both beaches, the wave model was initialized at the offshore boundary using a JONSWAP wave spectral shape. Wave conditions were chosen on a trial-and-error basis in order to satisfy three constraints. First, we required that waves were breaking onshore of the trough in the case of the troughed beach, and over the bar in the case of the barred beach. Second, we required that the two beaches have similar alongshore currents, in terms of the width and maximum velocity of the alongshore current jet. And third, we required that the background alongshore current be linearly stable, such that shear instability did not occur (this being outside the scope of our intended work). To that end, the troughed beach uses significant wave height inline image m, peak period T = 10 s, and mean wave incidence inline image (measured from shore normal). The barred beach uses inline image m, inline image s, and inline image.

[22] Finally, to judge the effect of changing the alongshore-averaged v, while keeping wave forcing constant, we perform tests for each beach using two different drag coefficients: inline image m/s and 0.004 m/s. In total, then, we are testing four different model setups (two bathymetries, and two values of μ).

2.6. Results

[23] The leading order (alongshore-averaged) steady fields for the troughed and barred beaches are shown in Figure 3. Figures 1 and 2 depict the bathymetry perturbations, the nonlinear model perturbation velocity fields, and a transect of the alongshore momentum balance, for inline image m/s, taking a single realization on a representative 1 km alongshore stretch. For both beaches, the full 2-D circulation takes the form of a meandering alongshore current.

Figure 3.

Ensemble of alongshore-mean model outputs for troughed beach (red) and barred beach (blue): Topography ( inline image), significant wave height (H0), and alongshore current (v0). Solid lines, inline image m/s; thin dashed lines, inline image m/s. Thick vertical dashed lines correspond to validation transect locations.

[24] In all cases, the bottom stress contribution to the nonlinear model momentum balance (solid green lines in Figures 1 and 2) was more alongshore-uniform than the other terms. This appears to be due to a tendency for the advective terms to balance with the combined alongshore-varying part of radiation stress and pressure gradients. In other words, advection acted to balance much of the alongshore-variable part of the dynamics; when advection was not included (dashed lines in Figures 1 and 2), this variability was instead largely balanced by bottom stress.

[25] The kinematic implications of the above dynamics are that, when advection is included, the flow is more alongshore-uniform. This is shown in Figure 4, which compares v from the linear and nonlinear models, for an alongshore transect in the surf zone. Closer inspection also shows that advection causes major flow features to be shifted in the downstream direction (toward smaller y). It should be mentioned that these changes in the flow also resulted in adjustment in η. This effect was subtle, however: alongshore variations in η were typically of order 2–3 mm, and changes in η due to advection were typically less than 0.5 mm.

Figure 4.

Comparison between numerical model velocity with advection (red), and without advection (blue); and predicted velocity inline image from equation (57) (black; see section 3). Two tests are shown, for a single realization of the (top) troughed beach and (bottom) barred beach, at x = 100 m and x = 275 m, respectively, using inline image m/s.

[26] Next, we consider the length scale dependence of the above advective effects. Figure 5 shows the advection transfer function, calculated using the methods of section 2.4. Results were similar for other transects, although further discussion of the across-shore dependence of the transfer function is deferred to later sections. Overall, the various model runs reveal a consistent picture of the effects of advection. At the largest length scales, advection has a minimal effect ( inline image, inline image); for smaller length scales, advection acts to reduce alongshore-variability in v ( inline image), and the variability which does survive is phase-shifted in the downstream direction ( inline image).

Figure 5.

Numerical model estimates of transfer function factors I (top four plots) and inline image (bottom four plots), using weighted ensemble-averaged calculations from equations (9) and (10). Four tests are shown: beach type and drag coefficient are listed above each plot. Data represent transect x = 100 m for troughed beach, x = 275 m for barred beach.

[27] As an aside, recall that spectral estimates with very low energy in vl have not been included; this leads to high wave number bands being suppressed in Figure 5. If this energy constraint was relaxed, the result tended toward I = 1 and inline image (i.e., inline image) at the largest wave numbers. The shift toward this regime is visible in some high wave number bands of Figure 5. One caveat is there is nearly no energy in those bands. Further, a logical explanation is that mixing dominates the dynamics at these scales, hence the effect of advection becomes negligible.

3. Analytical Approximation

[28] The above results show numerical experiments designed to illustrate the effects of nonlinear advection in a surf zone model. We have interpreted the advective terms as a transfer function, which causes scale-dependent shifting and dampening of variability in the alongshore current. Estimates of this transfer function indicate generally consistent behavior among four experimental setups. In this section, we attempt to reconcile this behavior with the equations of motion. We introduce an asymptotic approximation to explain the present results, and discuss when this approximation is valid.

3.1. Perturbation Equations

[29] We begin by restating the equations of motion (1)–(3), but making the simplifying approximations of rigid-lid dynamics, i.e., inline image, and negligible momentum mixing, inline image. Under these assumptions, the equations read

display math(13)
display math(14)
display math(15)

[30] Some justification for neglecting momentum mixing is provided by Thornton and Guza [1986] who found alongshore current predictions were insensitive to inclusion of mixing, as long as the effects of a random wave field were included. However, the assumption of negligible momentum mixing might still be invalid for some cases, and/or at small enough length scales; this will be discussed further in section 3.5. The assumption of rigid-lid dynamics, on the other hand, is more generally valid. This can be formalized by defining a scaling inline image, where inline image defines the velocity scale. The rigid-lid approximation inline image is then seen to be equivalent to assuming small Froude Number, inline image, which is very often valid based on field data [although some laboratory data may conflict; e.g., see Liu and Dalrymple 1978]. For instance, using the 4 month data set collected during the SandyDuck '97 experiment (Duck, NC, 1997; Elgar et al. [2001]), looking at the most-shoreward alongshore transect of sensors (having nominal depths of 1–2 m), and excluding calm periods (measured inline image cm/s) for which Froude Number is trivially small, we found the Froude Number was below 0.25 for 90% of the data set, its mean value was 0.15, and its maximum value was 0.58.

[31] Next, we decompose the problem into a background part, which is independent of the alongshore coordinate y, and small (i.e., second-order) perturbations:

display math(16)
display math(17)
display math(18)
display math(19)
display math(20)

[32] In other words, we assume the amplitude of inline image is much less than one, and similar for the other variables listed. Note we make no assumption regarding length scales at this stage, except to say that the background fields have inline image.

[33] Inserting this decomposition into the governing equations and neglecting second-order terms yields

display math(21)
display math(22)
display math(23)

[34] Hence, to leading order, the background solution satisfies the well-known equations for alongshore-uniform flow [e.g., Longuet-Higgins, 1970b; Bowen, 1969a]. The across-shore velocity u0 is zero, the across-shore-directed force inline image is balanced by a free surface gradient (wave-induced setup), and the alongshore-directed force inline image is balanced by bottom stress.

[35] Next, we have the leading-order equations governing the perturbations:

display math(24)
display math(25)
display math(26)

[36] These equations include the effect of advection on the alongshore-varying part of the flow, i.e., the effect we have analyzed numerically above. To isolate this effect, consider the solution to the corresponding equations if the advective terms were suppressed:

display math(27)
display math(28)
display math(29)

[37] Here the subscript l denotes “linear model”. Note we have used the fact that the background equations (equations (21)-(23)) do not include advective terms, hence inline image, inline image, and inline image. Now, subtracting from the original equations (24)-(26), we get

display math(30)
display math(31)
display math(32)

where we have defined

display math(33)
display math(34)
display math(35)

[38] These primed variables represent the changes in the flow field caused by the presence of advective terms, i.e., the changes analyzed numerically in the preceding sections.

[39] Next, we manipulate equations (30)-(32) to obtain the following form,

display math(36)
display math(37)
display math(38)

where

display math(39)
display math(40)
display math(41)

[40] This form of the equations highlights the natural nondimensional variables and parameters in the problem. Three equations are to be solved for the three unknown nondimensional variables inline image, inline image, and inline image, given two “forcing” terms inline image and inline image. The length scale Λ represents the length at which advective and frictional terms are of similar order in equations (25) and (26), which we will refer as the “advective length scale”. The parameters S and B represent the magnitude of background shear and beach slope, respectively. All three of these parameters are functions of the background fields, hence are functions of x.

[41] The fact that derivatives in equations (36)-(38) always appear premultiplied by Λ suggests the definition of another nondimensional parameter,

display math(42)

where inline image is a wave number corresponding to a particular flow length scale under consideration. We choose to refer to this parameter as the “shallow water Reynolds Number”, because it represents (like the Reynolds Number) a ratio of advective and viscous (drag) scales, but for shallow water equations [also cf., Tomczak 1988]. When R is large, advective terms are important to the flow (and vice versa). It will be shown, below, that this is a key factor for addressing the questions posed in the Introduction regarding quantification of “short” alongshore length scales and “strong” alongshore currents. Also note in general R is a function of both x (via h0 and v0) and k, and can only be treated as a constant when analyzing a specific across-shore location (x) and flow-perturbation length scale ( inline image). In the present work, however, we mainly focus on individual alongshore transects, so that R only varies with length scale. Much of our discussion will focus on such an interpretation, equating small R with large length scales and vice versa.

3.2. Advection Transfer Function

[42] We now outline a range of parameters for which (36)–(38) yields an approximation for the advection transfer function. In so doing, we will introduce several scaling assumptions, which effectively restrict the length scales for which the resulting approximation is valid. The results will therefore be valid only within a finite range (hopefully nonzero) of length scales; this will be summarized and discussed in section 3.3.

[43] We begin by defining local scales for shear and beach slope, considering a specific across-shore location,

display math(43)
display math(44)

[44] Here inline image and inline image are constants representing the local background shear and beach slope. In this section, we will assume weak shear and mild slope, inline image, which will simplify the analysis (later, in section 3.3, we explore the accuracy of the result in cases where this assumption is violated). This assumption effectively sets a maximum length scale for which the subsequent analysis will be valid. For example, the specific case of a linear background velocity profile implies we are restricted to length scales much smaller than the surf zone width; in cases where the background shear is not constant in x, as in Figure 3, some across-shore locations will have smaller ks and the range of allowable length scales will be increased at those locations (e.g., this is usually the case just onshore of the maximum of v0). Similar reasoning applies for kb. Finally, note the assumption inline image becomes increasingly restrictive near the shoreline where v and h become small.

[45] Next, we assume length scales are similar in the x and y directions, such that

display math(45)

[46] Equivalently, using the continuity equation and recalling we have already assumed inline image above, this implies we are considering a range of wave numbers for which u1 and v1 have similar scales, i.e., an isotropic regime. Note this assumption is being applied only to the derivatives of perturbation fields inline image, not the background fields inline image (for which inline image, hence inline image and inline image). In the present numerical experiments, perturbation flow features do tend to have inline image (equivalently, inline image), as shown in Figures 1 and 2.

[47] Given the above orderings, we now rewrite equations (36)-(38), neglecting higher-order terms,

display math(46)
display math(47)
display math(48)

[48] Or, taking the Fourier Transform in y,

display math(49)
display math(50)
display math(51)

[49] Next, moving toward a simple transfer-function-like solution for inline image, we introduce a final assumption which further simplifies equations (50) and (51). First, we recognize that solutions should scale as functions of R, this being the only relevant parameter in the system (given previous scaling assumptions). We will assume R is large, and subsequently restrict attention to cases where inline image. Some justification for this can be found by considering the linear model y-momentum equation (29); following the same steps as above, it gives at leading order

display math(52)

[50] Note the bottom stress term ( inline image) and the forcing term ( inline image) are expected to be of the same order of magnitude (this can be seen after eliminating pressure via cross differentiation with equation (28)). This, in turn, suggests the pressure term ( inline image) must be also of the same order or smaller, i.e., it is at most of order inline image. Based on this, we similarly assume inline image is at most of order inline image, reasoning that pressure would not differ by an order of magnitude as a result of advection. Hence, we expect inline image for large R, as stated above. Using this in equations (50) and (51) then gives asymptotic equations

display math(53)
display math(54)

which implies the following form for the transfer function,

display math(55)
display math(56)

[51] As expected, this transfer function predicts that flow perturbations are reduced in amplitude and shifted downstream (relative to the linear model solution), as R is increased.

3.3. Range of Validity

[52] The range of wave numbers for which the approximate transfer function of equations (55) and (56) is valid depends on the various assumptions made above. Specifically, we have introduced the following four constraints:

[53] (a) The flow length scales are of the same order, i.e., inline image

[54] (b) The length scales associated with shear, inline image, and beach slope, inline image, are large compared to inline image

[55] (c) the advective length scale Λ is large compared to inline image (i.e., R is large); and

[56] (d) the length scale associated with mixing is small compared to inline image.

[57] If all of these constraints are met simultaneously, then we may apply (55), (56).

[58] To illustrate this, the first plot in Figure 6 shows cross spectra quantifying the relative scale of variables in equations (36)-(38), as a function of wave number k, for the troughed beach test case with inline image m/s. We note that a range of wave numbers exists for which inline image, approximately inline image m−1, satisfying (a) (similarly inline image and inline image in this same range). The background shear wave number (estimated locally at this transect) was inline image m−1, and the beach slope wave number was inline image m−1, so that we must take inline image m−1 to satisfy (b) (assuming “large” in this context corresponds roughly to a factor of five). The background inline image m/s and inline image m at this transect result in inline image m, hence large R corresponds roughly to inline image, to satisfy (c). Finally, we can confirm that within this range the scale of the pressure term inline image is bounded from above by inline image, and hence inline image for large R, consistent with the assumed ordering.

Figure 6.

Numerical model cross spectra (estimated using methods of section 2.4, with uniform-weighted averaging of 20 realizations) quantifying relative scale of terms in equations (36)-(38), for transect x = 100 m of troughed beach test case with inline image m. Subscripts x and y in legends denote partial derivatives, and the notation inline image has been adopted for brevity. Top plot shows relative scale of kinematic terms; orderings introduced in text are verified here within a range of k, by comparing to unity and to 1/R (black solid curve). Middle plot shows terms in x-momentum balance, and bottom plot shows terms in y-momentum balance; note the balance predicted by equations (53) and (54) corresponds to the blue curve being equal to one.

[59] Next, the lower two plots of Figure 6 show the magnitude of terms in the momentum balance equations (37) and (38), relative to inline image and inline image, for the same transect as above. In these plots, the approximate balance predicted by equations (53) and (54) is valid when the blue line is equal to one. Indeed, there is a range of wave numbers where this occurs, roughly inline image m−1. The contribution from the neglected mixing term is also shown (green lines in lower plots of Figure 6); breakdown of equations (53) and (54) at larger wave numbers may be partly due to an increasing role of momentum mixing, i.e., constraint (d) above. The breakdown at smaller wave numbers apparently corresponds to a regime where all terms in equations (37) and (38) become of similar order, and hence no simplified balance exists.

[60] In the preceding example, comparing the actual range of validity to the expected range of validity for constraints (a)–(c), outlined above, we find that (a) and (c) are the most important factors for accuracy. Constraint (b), on the other hand, produces a too-conservative requirement of inline image m−1. This suggests the restriction of weak shear and mild slope may not be as important as previously suggested. The reason this assumption was introduced in the analysis was twofold: to ensure solutions were cast in terms of R only, and to ensure the term inline image did not enter into the asymptotic balances of equations (53) and (54). In other words, this assumption was primarily intended to simplify the formal analysis, not necessarily to ensure an accurate solution. It appears it may not be a strong factor in the final accuracy of the result.

[61] The effect of nonnegligible momentum mixing is also apparent in Figure 6, and may help to guide intuition as to the role of momentum mixing in our results. The figure shows that at sufficiently small length scales, differences in mixing (represented by inline image) between the full model and its counterpart with no advection eventually become of similar order to inline image and inline image. The reason that mixing differs between the two models (i.e., inline image) is because it is acting on two different velocity fields, one of which is modified due to advection. The resulting momentum difference must be absorbed by the other terms in equations (37) and (38), eventually interfering with the assumptions leading to (53) and (54).

[62] Similar results are found when repeating the above analysis using the barred beach transect, Figure 7. In that case, constraint (a) is satisfied for roughly inline image m−1, constraint (b) is satisfied for inline image m−1 (in this case, inline image m−1, and inline image m−1), and constraint (c) is satisfied for inline image m−1 (in this case, inline image m/s, and inline image m, hence inline image m). Hence, if we include the constraint of weak shear and mild slope we would expect no range of validity for equations (53) and (54). However, Figure 7 shows the approximation does appear to have accuracy in a narrow range, perhaps inline image m−1, with the upper limit of validity being associated with increasing momentum mixing. Hence, again it appears the most important factors for accuracy are constraint (a) of isotropic perturbation velocities, constraint (c) of large R, and constraint (d) of weak mixing.

Figure 7.

As in Figure 6, but for barred beach, x = 275 m, inline image m/s.

3.4. Application to Numerical Experiments

[63] Finally, we apply the above results to the numerical experiments from section 2.6. Figure 8 shows the estimated transfer functions for all four numerical experiments, plotted versus. R. This demonstrates that R is the appropriate nondimensional parameter to collapse the present numerical results. Note the collapse of data is quite general across a broad range of wave numbers; regardless of the exact functional form of the transfer function, it clearly is a function of R. We also overlay the transfer function predicted by equation (55) and (56). It appears these equations are a close approximation to the numerically estimated transfer function in the present experiments. Given the verification of the ordering of terms in section 3.3, this is not particularly surprising. One unexpected result is that equations (55) and (56) continue to predict the transfer function for small R all the way down to R = 0. This may be fortuitous, a result of interpolating through a fairly smooth regime to join with the expected I = 1 and inline image at the largest length scales.

Figure 8.

Collapse of data in Figure 5 when plotted versus the parameter R, and comparison to equations (55) and (56) (black lines), for various model experiments. Four tests are shown: troughed beach at x = 100 m, with inline image m/s (red) and inline image m/s (green); and barred beach at x = 275 m, with inline image m/s (blue) and inline image m/s (magenta).

[64] Figure 4 presents the results in a different way, applying equations (55) and (56) as a “filter”. That is, we compare the nonlinear model v to the following approximation:

display math(57)

where inline image denotes the inverse Fourier Transform. Example results are shown for single realizations from the troughed and barred beaches, with inline image m/s.

3.5. Accuracy at Different Across-Shore Locations

[65] The above results indicate equations (55) and (56) may be fairly accurate for predicting the spectral properties of advection, based on an analysis of a particular across-shore position in our numerical experiments. Next we seek to quantify that accuracy over a range of across-shore positions, using the root-mean-square error (RMSE),

display math(58)

where inline image is given by equation (57), and recall v1 is defined by equation (17).

[66] Figure 9 shows RMSE as a function of x, for each of the four experiments from section 2.6. To judge whether a given value of RMSE represents positive skill, we consider a comparison to two baseline models: the first baseline model assumes advection completely attenuates all flow variability, so that RMSE is equal to inline image (blue line in Figure 9); the second baseline model assumes advection has no effect on the flow, so that RMSE is equal to inline image (green line). If the RMSE from equation (58) is less than the RMSE for both these baseline models, we say equation (57) has positive skill. Comparing Figure 9 with Figure 3, we find positive skill occurs mainly at locations somewhat shoreward of the maximum in v0. Accuracy degrades at locations close to the shoreline, likely due to the fact h0 and v0 become small there (e.g., the assumption inline image will be violated near the shoreline; similar for the assumption inline image).

Figure 9.

Root-mean-square error of equation (57) as a function of across-shore position, for all ensemble members, compared to error for two baseline models. Red: RMSE, equation (58); blue: inline image (a baseline model); green: inline image (another baseline model). Four tests are shown, with 20 realizations for each test (dashed lines; mean of realizations plotted as solid line). Equation (57) is judged to have positive skill at locations where the red line is smaller than both the green and blue lines. Black dashed lines indicate the alongshore transects presented in previous figures (e.g., Figure 4), and black solid lines indicate the location of maximum v0. Also see Figure 3 for additional context in terms of alongshore-averaged variables.

3.6. More-Complex Flow and Physics

[67] The above results were obtained using relatively simple physical parameterizations, and for relatively simple flows. We now explore how the results apply under less-ideal conditions. Figure 10 shows the estimated filter transfer function obtained when certain simplifying assumptions are relaxed, discussed next. For these tests we use the barred beach, which already is the more complex of the existing two test cases. Similar results were obtained for the troughed beach, not shown.

Figure 10.

As in Figure 8, but for cases which violate the assumptions leading to equations (55) and (56) (black lines). Three tests are shown, each using the barred beach: red, x = 315 m, where momentum mixing is relatively strong; green, x = 275 m, using nonlinear spatially varying bottom stress; blue, x = 275 m, using inline image m/s, and allowing for time dependence in governing equations, which leads to shear instabilities (subsequently removed by time-averaging before analysis).

[68] First we consider the effect of momentum mixing, which was assumed negligible in equations (55) and (56). A notable result in Figure 9 is equation (57) becomes relatively less accurate at locations near to and offshore of the peak of the alongshore current v0, where mixing is more important. To investigate this, the red lines in Figure 10 show the result of the analysis of Figure 8, but using a transect at the peak of v0 (located at x = 315 m). At this location, inline image was at its maximum, on average 6.8 times larger than at the transect x = 275 m (analyzed previously). The resulting advection transfer function is biased toward slightly larger I, and smaller inline image, compared to equations (55) and (56), which leads to decreased quantitative skill for equation (57). Hence, a correction would be required for cases with large mixing, which would be a useful extension to the present work.

[69] Similarly, the present results have so far been restricted to cases where the background current is linearly stable, i.e., shear instability of the alongshore current [Bowen and Holman, 1989] did not occur. This allowed us to investigate the perturbation flow without having to consider random fluctuations due to shear waves. Another approach to this problem would be to use time averaging to attempt to remove those random fluctuations, which results in the blue lines in Figure 10. This result was obtained by using inline image m/s for the barred beach, and allowing time variability in the governing equations (adding the appropriate terms to equations (1)-(3)), which resulted in an unstable current. The instabilities were allowed to develop for 10 h, and the flow was then averaged over 10 h to remove the random component.

[70] As an aside, it is worth noting that inline image for the linearly unstable case. The reason is that shear waves in the nonlinear model cause significant across-shore mixing which is not accounted for in the linear model. This had only a small effect on the results in Figure 10, however it did affect the predictions of equation (57); in fact, equation (57) lacked skill at nearly all across-shore locations, using the RMSE-based skill criteria defined in section 3.5. If the error due to inline image was taken out prior to computing RMSE, equation (57) was skillful in the range inline image m (again, by the criteria defined in section 3.5).

[71] Finally, we note that while the present tests and analysis have assumed a simple Rayleigh bottom stress formulation, similar results are found when using a more realistic nonlinear spatially varying bottom stress. To test this, we repeated the barred beach test case using the bottom stress formulation of Svendsen and Putrevu [1990], with drag coefficient 0.007. The result is shown by the green lines in Figure 10. When computing R for this case, we noted that the alongshore-averaged value of inline image was equal to 0.0024 m/s on the analyzed transect (x = 275 m), and this value was used to represent μ. Repeating the analysis at different across-shore transects (not shown), we found the across-shore dependence of accuracy was similar to the result using Rayleigh bottom stress shown in Figure 9.

[72] In summary, each of the above tests violate assumptions leading to the simplified theory of equations (55) and (56), hence the quantitative accuracy of that theory is reduced. The results are encouraging, however, in that the qualitative behavior of advection is not affected. We believe this point to the possibility of extending the present work to include more complex cases.

4. Discussion

[73] The above numerical experiments investigate the previously documented phenomenon where background alongshore currents can overcome small undulations in forcing, owing to advection [Wu and Liu, 1984]. The strength of this effect depends on the relative scale of the advective terms compared to the bottom stress, a ratio which is quantified by the shallow water Reynolds Number, R. As a first approximation for a restricted class of flows and flow length scales, we have introduced equations (55) and (56); they predict that values of inline image correspond to a reduction of over 75% in the amplitude of variability of v1, due to advection. For nominal values inline image m/s, inline image m, inline image m/s, this corresponds to alongshore length scales of less than 400 m, scales that are relevant for some applications.

[74] The present work is naturally limited in scope, owing to the reliance on a linearized regime (small perturbations), scaling assumptions (rigid-lid dynamics, weak mixing), assumptions about flow dynamics (linearly stable, steady state), and simplified parameterizations (depth-uniformity, Rayleigh bottom stress). These assumptions were made in an attempt to study advection in the simplest context possible. Hence, extensions are likely necessary for the more complex flows and physics that can occur in the natural surf zone. For example, we have seen that the quantitative accuracy of equations (55) and (56) is degraded as momentum mixing is increased, if shear instability occurs, or if bottom stress is parameterized differently; this would need to be quantified and accounted for in a more general theory.

[75] That said, we believe the basic qualitative results shown here are likely to hold even in more complex cases. The result that advection causes dampening and shifting of flow perturbations is a natural outcome of a balance among wave forcing (assumed constant), bottom stress (related to velocity) and advection (related to derivatives of velocity). Such an effect should depend mainly on the relative importance of advection compared to bottom stress, i.e., should be scaled by the shallow water Reynolds Number. Below, we discuss how this conceptual framework can be used to help understand the role of advection in field studies of surf zone flows. We will attempt to highlight how characterizing the length scale dependent effects of advection using R can distinguish cases where advection should be more important, and hence guide our intuition on how it affects the flow.

4.1. The Case of Large R

[76] At large R (small alongshore length scales), we have seen advection causes smoothing of the alongshore-varying part of the velocity field. In the extreme case, the kinematic response will appear nearly alongshore-uniform (alongshore-variability may still exist, but only at wavelengths which are longer than the scale of interest). Dynamically, this occurs because the advective terms counteract the combined alongshore-nonuniform part of the radiation stress and pressure gradients (as in the momentum balances shown in Figures 1 and 2). The remaining balance, then, is largely between the bottom stress and the alongshore-averaged forcing. Note this is the same balance which results from an a priori assumption of inline image (i.e., the 1DH assumption). Hence, large R corresponds to near closure of the 1DH balance, but only because of self-cancellation of 2DH terms.

[77] This fact has implications for field applications. 1DH models are often used in field settings, usually for reasons of efficiency. Hence, there is a need to determine when their outputs can be trusted. Ruessink et al. [2001] have suggested a metric inline image for this purpose, which quantifies alongshore variability in bathymetry: in practice, if inline image is below a threshold value, the beach can be considered approximately 1DH. The present work suggests this threshold could be increased when R is large, assuming the only goal is an efficient prediction of v. Likewise, if the threshold value of inline image is to be determined empirically (by measuring 1DH model accuracy as a function of inline image), a consideration of R may help to guide this process.

[78] Similarly, consideration of R may be important for studies which attempt to experimentally verify the 1DH momentum balance. A prominent example is Feddersen and Guza [2003], who present a comparison of surf zone-averaged terms in the 1DH momentum balance, calculated from observations over a four month experiment. They found a strong correlation between the alongshore-directed wave forcing (calculated using offshore wave measurements, and assumed constant in the alongshore direction) and bottom stress, and concluded that the flow was consistent with 1DH dynamics. We note that such a correlation would be dominated by times when the wave forcing was strong (leading to large v0, large R), and hence when 2DH terms would likely be self-cancelling based on the above interpretation. In light of the present results, then, correlation of terms in the 1DH momentum balance does not necessarily imply that other terms are negligible. In fact, Feddersen and Guza [2003] did attempt to estimate the advective contribution to the momentum budget, and found it to be of the same order as the total forcing. However, those calculations also had a characteristic which could be reproduced by instrument noise alone, and were therefore rejected.

[79] Finally, we reiterate that even when the overall alongshore length scale of the surf zone is large, R can still be large if one is considering localized features, for example a narrow rip channel. This is illustrated by the work of Sancho [1998], who studied a simplified set of equations assuming inline image. Their model predicts a dominant balance between wave-induced forces, pressure gradient forces, and bottom stress, while across-shore advection was assumed to play a role similar to mixing (their approach follows Putrevu et al., [1995], with a correction to ensure consistent ordering of terms). They found this simplified model became inaccurate near rip channels, when compared to a full model including all terms. The reason was traced to the neglected advection terms, which became of similar order to pressure gradient terms. The result was that the flow predicted by the simplified model was overly nonuniform in those locations. In other words, the scale-dependent smoothing effects of advection became important at smaller length scales, consistent with the present work.

4.2. The Case of Small R

[80] Small R corresponds to flows at long length scales, e.g., over a long stretch of coastline. At these scales, we expect the alongshore current to respond linearly to the forcing, with advection having a negligible effect, similar to the model studied by Sancho [1998]. This can be seen by assuming inline image, and inline image (i.e., R small) in equations (24)-(26), in which case the advective terms are found to be negligible (note this result is separate from that of equations (55) and (56), which is for the case of large R).

[81] As an example, consider the observational and modeling study conducted by Apotsos et al. [2008] (hereafter A08). Their focus was on flows onshore of a submarine canyon, where alongshore variability was strong due to offshore focusing of waves [Long and Özkan-Haller, 2005], hereafter L05). In their analysis, A08 neglected advection a priori, and found inclusion of the pressure gradient term was necessary to balance the observed bottom stress. This suggests advection played a relatively minor role in the overall dynamics. L05, meanwhile, using the same bathymetry and offshore wave conditions, had an apparently contradictory conclusion: advection was found to play a significant role in their model simulations of rip currents. This contradiction can be resolved by considering the different scales of the two analyses. The alongshore scales analyzed by A08 were large: alongshore gradients in A08 were computed by centered differencing of observational transects, which were spaced (nominally) 250 m apart. That is, the differencing used to calculate alongshore gradients spanned 500 m, and hence the resolved wavelengths of variability were of order 1 km and above. Also, the observations were in shallow water, inline image m. This points to a large value of R, hence it is perhaps not surprising that advection was relatively small at the scale considered by A08, compared to other terms in the momentum budget. The model results of L05, on the other hand, focused on much smaller alongshore length scales (order 100–200 m), suggesting R differed between the two studies by a factor of 5–10, which may explain their differing conclusions regarding the role of advection.

5. Conclusions

[82] We have presented numerical model experiments exploring the role of advection in alongshore-nonuniform surf zone circulation. We focus on the case where alongshore variability is weak compared to the alongshore-averaged current. These flows are typically described as meandering alongshore currents [Sonu, 1972], and are known to be strongly affected by advection [Wu and Liu, 1984; Long and Özkan-Haller, 2005].

[83] In our experiments, we find advection acts to dampen and shift short-scale alongshore variability in the dynamics. The remaining momentum flux at these scales, balanced by bottom stress, is more alongshore-uniform than it would be if advective terms were not present. In other words, at these short scales, terms which would be neglected when assuming a 1DH balance ( inline image) are naturally self-cancelling. For that reason, an understanding of the scale-dependence of advection is important for the correct interpretation of surf zone dynamics in terms of 1DH versus 2DH flows.

[84] For the present numerical tests, the effects of advection could be approximated in wave number space as a function of the parameter R, a ratio between advective and frictional scales which we refer to as the shallow water Reynolds Number. The estimated transfer functions from the numerical model were shown to collapse when plotted as a function of R. We also showed how the transfer function can be approximated analytically as a function of R, under certain simplifying conditions. This approximation was found to be a good match to our numerical experiments, and its derivation was shown to be consistent with the dynamics of those experiments. The extension of this theoretical analysis to include more complex flows and physics is an important subject for future work.

Acknowledgments

[85] We wish to thank the many people who provided comments, suggestions, and interesting discussion. This work was supported by Office of Naval Research Grants N00014-02-1-0198, N00014-07-1-0852, and N00014-11-10393. This paper made use of a data set collected for the SandyDuck '97 experiment by S. Elgar, R.T. Guza, H.T.C. Herbers, and T.C. O'Reilly.

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