1. Introduction

Spatial scales of turbulence in the atmospheric boundary layer have been characterized for a range of conditions (Kaimal et al. 1972), but observations of turbulence scales in oceanic boundary layers have been limited to unstratified tidal flows (Soulsby 1977; Gross and Nowell 1983). Spatial scales of weak turbulence in a strong mean flow can be estimated from time series measurements at a single point by means of the frozen-turbulence hypothesis (Taylor 1938). In many oceanic environments, surface waves complicate the interpretation of point measurements (Lumley and Terray 1983), and determination of turbulence scales requires a spatial array of sensors.

In this study, observations from an array of velocity sensors are used to estimate scales of turbulence just seaward of the surf zone on an ocean beach. The array was designed to estimate a spatial structure function D_υw (Δy), defined by

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Here, x and (x, y, z) denote the position vector, with x positive onshore, y alongshore, and z positive upward and equal to zero at the bottom; Δy is the alongshore separation; j is a unit vector in the y direction; t is time; (U, V, W) is the velocity vector; (u, υ, w) is the departure of the velocity vector from its hour-averaged value; and an overbar denotes an hour average.

The analysis is based on the assumption that velocity fluctuations with timescales of less than 1 h are caused by superposed surface waves and turbulence. At the scales resolved by the array, the operation of subtraction in (1) largely removes effects of waves (Trowbridge 1998). In principle, D_υw (Δy) increases from zero to 2υ′w′, where primes denote turbulent fluctuations, as Δy increases from zero to values greater than the correlation scale of the turbulence. Thus, D_υw (Δy) describes the alongshore scales of the turbulence contributing to the Reynolds shear stress −ρ₀υ′w′, where ρ is the density and the subscript 0 denotes a fixed reference value. This component of the Reynolds stress is a dominant term in the alongshore momentum balance in shallow water (Longuet-Higgins 1970; Feddersen et al. 1998; Lentz et al. 1999).

The null hypothesis is that the dynamics of turbulence in the shallow coastal ocean, at heights above bottom that are small relative to the water depth, but larger than the thickness of the oscillatory boundary layer produced by surface waves, are consistent with local application of Monin–Obukhov scaling (Monin and Yaglom 1971). In this scaling, the statistical properties of the velocity and density depend only on z, ρ₀, the vertical buoyancy flux B = (g/ρ₀)ρ′w′, where g is gravity, and the vertical flux of streamwise momentum, here approximated by ρ₀υ′w′, because hour-averaged currents at the measurement position were predominantly shore parallel, so that U ≪ V and u′w′ ≪ υ′w′. Dimensional consistency requires that properly scaled statistics of the velocity and density fields be functions of z/L, where L = |υ′w′|^3/2/(κB) is the Monin–Obukhov length, with the empirical von Kármán constant κ ≃ 0.40. Monin–Obukhov scaling implies that

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where L_c is a length characterizing the alongshore correlation scale of the stress-carrying turbulence, ϕ is a function that can be derived from semiempirical results obtained in the atmospheric surface layer, and N² = −(g/ρ₀)∂ρ/∂z is the squared buoyancy frequency, which can be positive or negative. Here, oceanic measurements of D_υw and z are fit to a model to determine υ′w′ and L_c, and these estimates are combined with measurements of N² to test (2).

2. Methods

b. Models of D_υw, L_c/z, and N²

A representation of D_υw follows from a semiempirical expression for the cospectrum of streamwise and vertical velocities in the atmospheric surface layer, given by

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The coordinate system is defined so that the y axis is aligned with the mean velocity; Co_υw is the cospectrum of υ and w, defined so that ∫^+∞_−∞ Co_υw(k_y) dk_y = υ′w′; k_y is the y component of the radian wavenumber; and L_c is given by

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The Fourier transform of Co_υw(k_y) is E_υw(Δy), defined to be the even part of υ(x, t)w(x + jΔy, t) (Monin and Yaglom 1975). By definition, D_υw(Δy) = 2[E_υw(0) − E_υw(Δy)]. It follows that (3) is equivalent to

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where ξ is a dummy variable of integration. Equations (3) and (4) are similar to a model proposed by Kaimal et al. (1972) for neutral and stable stratification (B ≥ 0), and they are consistent with theory and measurements at large k_y for both B < 0 and B ≥ 0 (Kaimal et al. 1972; Wyngaard and Cote 1972). The integral on the right side of (5) depends only on the ratio of Δy/z to L_c/z and can be evaluated numerically. According to (5), D_υw increases monotonically with Δy and is within 10% of 2υ′w′ for Δy ≥ 3L_c.

Standard results for the atmospheric surface layer lead to an expression for the squared buoyancy frequency (Hogstrom 1988):

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Equations (4) and (6) define ϕ in (2).

c. Analysis

Structure functions were computed for each record at each of the six nonzero spatial separations by removing the hour average of alongshore and vertical velocity, computing products according to (1), and averaging over the 1-h record. Estimates of z and the mean water depth h were obtained from the altimeter and pressure measurements. Estimates of wave statistics were obtained from the pressure and velocity measurements using linear wave theory.

Estimates of ∂T/∂z, where T is the temperature, were obtained by regressing array-averaged estimates of ∂T/∂t, obtained from centered differences of the temperature time series, against array-averaged measurements of −w. This procedure is based on a linearized model of temperature fluctuations given by

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Both the variance of w and the covariance of ∂T/∂t and w were dominated by surface wave fluctuations at frequencies between approximately 0.05 and 0.5 Hz. The standard deviation of w was much larger than the bottom slope times the standard deviation of u, which suggests that estimates of ∂T/∂z were not affected significantly by terms neglected in (7), such as u∂T/∂x, or by uncertainties in the orientation of the ADVs. The squared buoyancy frequency was estimated from ∂T/∂z by assuming a linear relationship between temperature and density, with a slope of 0.24 kg (m³ °C)⁻¹, determined from previous temperature and salinity observations near the measurement site (Lerczak 2000).

Estimates of υ′w′ and L_c/z were obtained by fitting measurements of D_υw to (5), with the measured z/Δy, using iterative least squares regression. First, the 854 × 6 measurements of D_υw were regressed against (5) to produce 854 preliminary estimates of υ′w′ and a single estimate of L_c/z, representative of the entire measurement period. The measured z and N² and the preliminary estimates of υ′w′ were then used to compute κ²N²z²/|υ′w′| for each 1-h record. To exclude observations in which υ′w′ was too small to be estimated accurately, 54 1-h records for which |κ²N²z²/υ′w′| > 2 were omitted from the analysis. The remaining 800 records were sorted into four equally sized groups in order of increasing κ²N²z²/|υ′w′|. The 200 × 6 measurements of D_υw within each group were then regressed against (5) to produce 200 final estimates of υ′w′ and a single estimate of L_c/z for that group. The preliminary and final estimates of υ′w′ were not significantly different. The result of this procedure was four estimates of L_c/z, ordered by κ²N²z²/|υ′w′|, and 800 estimates of υ′w′. Error bounds for L_c/z were obtained by a standard procedure based on Gaussian statistics and linearization of the nonlinear regression about the best-fit values of the parameters (e.g., Draper and Smith 1966). In the calculation of error bounds, the six measurements of D_υw obtained during each 1-h record were assumed to have only 2 degrees of freedom, because concurrent measurements of D_υw at different spatial separations were not statistically independent (e.g., Cressie 1993).

3. Results

During the 70-day measurement period, the significant wave height ranged from approximately 0.5 to 1.5 m, the standard deviation of the cross-shore velocity ranged from approximately 0.2 to 0.5 m s⁻¹, and the hour-averaged alongshore velocity usually was less than 0.1 m s⁻¹, with maximum values of about 0.2 m s⁻¹ (Fig. 2). Hour-averaged cross-shore velocities were much smaller than hour-averaged alongshore velocities. Wave incidence angles were within 10° of shore normal. The mean and standard deviation of ∂T/∂z were −0.01° and 0.03°C, respectively. Stable stratification usually occurred during southward alongshore flows (V < 0), and unstable stratification usually occurred during northward alongshore flows (V > 0). The hour-averaged water depth varied between approximately 3.0 and 5.5 m, primarily owing to tidal fluctuations. The height of the velocity measurements above the bottom ranged between 0.75 and 0.93 m as the seafloor eroded and accreted. The measurement height z was larger than the thickness of the oscillatory boundary layer produced by surface waves, estimated at a few centimeters (Grant and Madsen 1979) but was small in comparison with the hour-averaged water depth h (0.15 < z/h < 0.30). Approximately 60% of the variance of the alongshore current was at frequencies below 2 cycles per day, approximately 20% was at semidiurnal frequencies, and the remaining 20% was at frequencies greater than semidiurnal.

Visual observations obtained nearly daily by divers suggest that the seafloor was smooth, with no signs of wave-orbital or larger-scale ripples or bedforms. The seafloor elevation varied by centimeters on timescales of hours, but the differences between the two altimeters indicate there were no bedforms with horizontal scales smaller than the altimeter spacing and vertical scales larger than the altimeter accuracy of about 0.01 m (Gallagher et al. 1998).

The qualitative dependence of the measured structure functions on spatial separation and stratification is consistent with expectations (Fig. 3). The magnitude of D_υw is smaller than 2υ′w′ at small Δy, and it asymptotically approaches 2υ′w′ as Δy increases. The approach of D_υw to 2υ′w′ with increasing Δy is more rapid for positive υ′w′, corresponding to V < 0 and stable stratification, than it is for negative υ′w′, corresponding to V > 0 and unstable stratification. The alongshore turbulence scales suggested by Fig. 3 are roughly 1.0 m for stable stratification and are between about 1.5 and 2.5 m for unstable stratification.

Estimates of υ′w′ are approximately consistent with a quadratic drag law, given by

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with a drag coefficient c_d = (1.3 ± 0.1) × 10⁻³ at 95% confidence (Fig. 4), which is similar to previous estimates of the drag coefficient in nearshore flows under unbroken waves (Feddersen et al. 1998; Trowbridge and Elgar 2001).

The dependence of L_c/z on κ²N²z²/|υ′w′| indicates suppression of turbulence scale by stable stratification and agreement with the atmospheric results in (4) and (6) (Fig. 5). Corresponding estimates of the drag coefficient indicate no significant dependence on κ²N²z²/|υ′w′|.

4. Discussion

The observations indicate that turbulence length scales in the shallow coastal ocean, at heights above the bottom that are small in comparison with the water depth but are larger than the thickness of the wave boundary layer, are influenced by stratification in a manner consistent with local application of Monin–Obukhov scaling (Fig. 5). This result implies that the dynamics of near-boundary turbulence in the shallow coastal ocean and the atmosphere are similar, even though the environments differ. In the atmospheric surface layer, both υ′w′ and B are nearly independent of z. At water depths of a few meters seaward of the surf zone in the ocean, scaling and measurements (Feddersen et al. 1998; Lentz et al. 1999) indicate that Coriolis acceleration is negligible in the hour-averaged alongshore momentum balance and that the dominant terms in the depth-integrated balance are wind stress and bottom stress, implying that υ′w′ is nearly independent of z, as in the atmospheric surface layer. However, the seafloor usually is idealized as nonconducting for both heat and salt, so that B → 0 as z → 0, and thus B must vary with z if B ≠ 0. The present results indicate that the dependence of B on z does not have a dominant effect on the local dynamics of the turbulence.

For a fixed turbulent Reynolds shear stress, increasing stable stratification reduces the eddy viscosity and increases the vertical derivative of the hour-averaged velocity. Measurements in the atmospheric surface layer indicate that (Monin and Yaglom 1971)

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where β is an empirical constant, approximately equal to 6.0 (Hogstrom 1988). For the observations used here, z/L was as large as 0.3, which corresponds in (9) to a reduction in eddy viscosity and an increase in ∂V/∂z by a factor of 2.5. Measurements of V(z) often are fit to (9) to infer υ′w′. In the present case, this procedure would result in overestimates of υ′w′ by factors of roughly 6 during stable stratification.

The measurements indicate a 50% reduction in length scale during stable stratification (Fig. 5) but no significant dependence of the drag coefficient on stratification (section 3). This result is consistent with local application of Monin–Obukhov scaling. A simple model for near-bottom flow in the ocean, where B → 0 as z → 0, is υ′w′ = constant and B = z∂B/∂z, where ∂B/∂z = constant. For this case, integration of (9) with respect to z gives

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where z₀ is the bottom roughness length, which includes the effect of the bottom boundary layer produced by surface waves (Grant and Madsen 1979). For the data discussed here, the second term in square brackets in (10) is substantially smaller than the first, so that stratification has a small effect on the local relationship between the hour-averaged velocity and the turbulent Reynolds shear stress, even though it has an important effect on the vertical gradient of the hour-averaged velocity in (9).

Divergence from Monin–Obukhov scaling at small k_y is observed in atmospheric measurements during unstable conditions (B < 0) and is attributed to large-scale, convective circulations driven by the unstable density structure (Kaimal et al. 1972), which presumably depend on scales in addition to B, υ′w′, and z. Here, the manifestation of similar processes would be L_c/z larger than predicted by (4) and (6) at large, negative κ²N²z²/|υ′w′|. This behavior is suggested by the measurements (Fig. 5), but the uncertainty is sufficiently large that the observations are not necessarily inconsistent with (4) and (6).

The alongshore turbulence scale during unstable stratification was the same order of magnitude as the water depth h (Figs. 3 and 5), but h is unlikely to have placed an important constraint on turbulence scale at the measurement height. The quantity z/h, a dimensionless measure of the importance of h, was small, and the measurements indicate no systematic dependence of L_c/z on z/h. This result is consistent with laboratory measurements in open channels indicating that the water depth affects the turbulence scales only for z/h > 0.5 (Nezu and Nakagawa 1993). Also, much of the energy appearing at long spatial scales in turbulence statistics referenced to a single coordinate direction, such as D_υw(Δy) and Co_υw(k_y), is aliased from shorter scales, because wavelengths of individual Fourier components appear longer if the wavenumber is not aligned with the coordinate direction of interest. Thus, the actual scale of the stress-carrying turbulence was smaller than 3L_c, the value of Δy for which D_υw is within 10% of 2υ′w′.

The wave-induced strain rate (of order kσ_u, where k is the wavenumber of the surface waves and σ_u is the standard deviation of u), was larger than the strain rates of the hour-averaged velocity field and the stress-carrying turbulence (both of order |υ′w′|^1/2/z), but straining by the wave-induced velocity field is unlikely to have had an important effect on the turbulence at the measurement position. The measurements indicate no systematic dependence of L_c/z on kσ_uz/|υ′w′|^1/2, the ratio of wave-induced to turbulent strain rates. The wave-induced strain rate is oscillatory and therefore is unlikely to have had an important effect when averaged over many wave periods. Also, an order-of-magnitude argument adapted from Phillips (1961) indicates that wave-induced straining of turbulence had a negligible effect on the hour-averaged budget for the turbulent vorticity variance. Although it does not directly address stress-carrying scales, the argument of Phillips (1961) suggests that wave-induced straining had a minor effect on the turbulence.

5. Conclusions

Measurements from an alongshore array of near-bottom velocity sensors located approximately 0.8 m above the seafloor in 4-m water depth on an ocean beach were used to estimate the alongshore scales of the turbulent Reynolds stress just seaward of the surf zone. The analysis is based on a spatial structure function that rejects contributions from waves. Estimates of Reynolds stress are consistent with a quadratic drag law, with a drag coefficient similar to previous observational estimates. Turbulent length scales were approximately 2 m during unstable stratification and were reduced by a factor of roughly 2 during stable stratification, in a manner consistent with atmospheric results based on Monin–Obukhov scaling. Effects of finite water depth and straining of turbulence by the wave-induced velocity field were small. The results imply that turbulence in the shallow coastal ocean, just above the oscillatory bottom boundary layer produced by surface waves, is dynamically similar to turbulence in the atmospheric surface layer.