Drag coefficients Cd obtained through direct eddy covariance estimates of the wind stress were observed at four different sandy beaches with dissipative surfzones along the coastline of Monterey Bay, California. The measured surfzone Cd (~2 × 10−3) is twice as large as open-ocean estimates and consistent with recent estimates of Cd over the surfzone and shoaling region. Owing to the heterogeneous nature of the near shore consisting of nonbreaking shoaling waves and breaking surfzone waves, the surfzone wind stress source region is estimated from the footprint probability distribution derived for stable and unstable atmospheric conditions. An empirical model developed for estimating the Cd for open-ocean foam coverage dependent on wind speed is modified for foam coverage owing to depth-limited wave breaking within the surfzone. A modified empirical Cd model for surfzone foam predicts similar values as the measured Cd and provides an alternative mechanism to describe roughness.
Over land, the geometric roughness k and corresponding aerodynamic roughness zo for surface features can be considered temporally constant. Over the open ocean, zo is a function of both surface texture (associated viscous surface stresses) and the local wave field (associated form drag and flow separation). The associated stresses are dynamically coupled with the wind, can evolve together, and transition from viscous stresses to wave stresses. Nonlocal wave fields further complicate the dynamical relationship. Numerous, extensive, open-ocean field studies have investigated the various stress relationships, resulting in both consistencies and discrepancies [see Edson et al. (2013) for an overview].
Until recently, there have been limited observations of the air–ocean momentum fluxes in the nearshore region of the ocean. The nearshore region includes the surface gravity wave shoaling region (~< 30 m depth) and the dissipative surfzone (~< 2 m depth). Unlike the open ocean, surface gravity waves become decoupled from the wind-wave relationship and dependent on water depth h modifying the dynamical coupling between the wind and the waves. Furthermore, depth-limited wave breaking occurs within the surfzone, reducing the wave height.
Hsu (1970) and Vugts and Cannemeijer (1981) measured elevated drag coefficients, Cd ~ O(1 × 10−3 − 5 × 10−3), related to the surfzone and swash zone. Smith and Banke (1975) recognized that depth-limited wave breaking may have increased their measured Cd, owing to their tower being deployed on a sand spit. During Hurricane Ike in 2008, Zachry et al. (2013) and Powell (2008) measured elevated Cd values in the near shore compared with the open ocean. Anctil and Donelan (1996) found increased Cd values for waves shoaling from 12 m to breaking in 2-m water depth. Shabani et al. (2014, 2016) found that measured Cd for near-neutral, atmospheric stability over the shoaling region and surfzone were O(2) times larger than open-ocean estimates, which they ascribe to the wave celerity c and shape effects. Similar to Anctil and Donelan (1996), they suggested that as the wave shoals, wave speed slows relative to the wind speed U increasing Cd.
Total aerodynamic roughness zo is composed of
where zυ is the viscous smooth flow roughness, or tangential stress, associated with the sea surface (Charnock 1955):
where α ~ 0.011 (Charnock 1955; Smith 1988; Fairall et al. 1996), g is the gravitational acceleration, and u* is the shear velocity. The quantity zw is the wave aerodynamic roughness, owing to form drag and flow separation due to the presence of waves associated with rough flow (Donelan 1990; Banner and Peirson 1998; Reul et al. 2008; Mueller and Veron 2009). The variable zf is the aerodynamic roughness due to spray droplets and foam and is often included in zw or zυ. Though zo can be a linear summation, Cd is not a linear summation (Edson et al. 2013). The terms zo and Cd at 10 m (subscript 10) for neutral atmospheric stability (subscript N) are related by
where κ (=0.4) is the von Kármán constant. Vickers et al. (2013) found that Eq. (2) generally works well for near-neutral stable observations ignoring sea state. Andreas et al. (2012) suggests that the smooth flow formulation zυ works for U < 8 m s−1, and Donelan (1990) found that the sea becomes fully rough at 7.5 m s−1. This implies that zw becomes important for U > 8 m s−1. Andreas et al. (2012) and Edson et al. (2013) found empirical data fits that are a function of UN10 using a modified α in Eq. (2). Golbraikh and Shtemler (2016) developed a zf relationship related strictly to the percentage of open-ocean foam coverage and U. It is important to recognize that roughness is increased by an order of magnitude by the presence of foam as compared with a nonfoam water surface.
Shabani et al. (2014) indirectly posed a fundamental question: If Cd increases within the surfzone, how are the surfzone waves different from the open-ocean waves? Here, an alternative hypothesis is proposed that the surface roughness of foam zf generated by depth-limited wave breaking inside the surfzone also contributes to the increased Cd (Fig. 1). Within the surfzone, since surface gravity waves are decaying, the potential influence of the wave form drag zw relative to zf may be reduced, while at the same time zf is increasing due to increased foam coverage by breaking waves. Using Golbraikh and Shtemler (2016), a modified Cd relationship is developed for surfzone foam coverage.
2. Field experiment
Collocated sonic anemometers, temperature, and relative humidity sensors were mounted on six, 6-m-high towers and deployed simultaneously on four different sandy beaches within the surfzone and near the high-tide line located along 10 km of shoreline in Monterey Bay, California. Continuous measurements for four weeks in May–June 2016 were divided into 15-min blocks for analysis. The analysis for computing momentum fluxes and procedures for quality controlling the data is given in Aubinet et al. (2012), which is similar to that described by Shabani et al. (2014) and Ortiz-Suslow et al. (2015).
A pressure sensor and temperature string was deployed in 10-m water seaward of each beach tower. Significant wave height Hsig, average wave period Tavg, and wave setup were estimated from the pressure observations (Dean and Dalrymple 1991). The tower position and elevation and beach profile were surveyed with a GPS. The distance between the waterline and tower location, including wave setup, was estimated for each stress measurement.
The quantities Hsig and Tavg ranged between 0.3 and 2 m and 6 and 13 s, respectively, associated with local storm-generated events. The wind speed U6 measured at 6-m elevation ranged from 0 to 10 m s−1, with maxima in the late afternoon reducing to near zero at night. A diurnal cycle is observed that is occasionally modified by larger, mesoscale, atmospheric storm events. The beach air temperature ranged between 10° and 20°C. The water temperature ranged from 12° to 18°C. The difference of air and water temperatures is predominantly negative, implying the atmosphere behaved as an unstable system. Owing to the limitations of empirical formulations used in comparing results, momentum flux data are filtered to limit the range of atmospheric stabilities ζ to −2 < ζ < 0.5, U6 > 3 m s−1, and to onshore wind directions that are between ±40° relative to shore normal. Atmospheric stability is measured as ζ = z/L, where
where Tυ is virtual temperature, w′ and are the turbulent vertical velocity and turbulent virtual potential temperature perturbations, and < > denotes the time average. These limitations reduced the analyzed data to 3031 onshore records, out of which 630 records are represented by the surfzone (discussed below), representing 21% of the total data acquired.
The Monterey Bay nearshore system is composed of a relatively steep (1:10) foreshore beach flattening out to a low-tide surfzone terrace (1:100), continuing with a 1:30 offshore slope (MacMahan et al. 2010). The offshore distance for which c equals U6 is referred to as the decoupling distance xdc, inside of which the decreasing speed of shoaling waves may increase drag (Anctil and Dolelan 1996). For the experiment, xdc equals 220 ± 80 m (one standard deviation). Considering the surf width is O(100) m, the surfzone represents ~30% of the nearshore region for the experimental wind conditions.
A basic assumption for computing momentum fluxes is that the measurement environment is homogeneous. The near shore is a heterogeneous environment. The footprint represents the source location where the measured turbulence originates and is estimated by an empirical model that accounts for atmospheric stability conditions (Hsieh et al. 2000). It is important to recognize that turbulence measurements obtained on the beach represent turbulence that originates over the ocean that is advected by the wind. The footprint distance x increases with increasing stability, wind speed, and measurement elevation z and is represented by a skewed probability density function f(x, z), as described by
where D = 0.28 and P = 0.59 for unstable conditions, D = 0.97 and P = 1 for near-neutral conditions, and D = 2.44 and P = 1.33 for stable conditions (Hsieh et al. 2000). The term zu is defined as
Researchers typically use the maximum of the f(x, z) to denote the source location. Here, the relative percentage of contribution for the source region R is estimated by
where the particular footprint source region f(x, z) is defined between two cross-shore locations (x1 and x2). The data were subdivided into two categories: the surfzone and seaward of the surfzone, based on f(x, z). Data for a region are only considered when R is greater than 70% for that region. Filtering the data for −2 < ζ < 0.5 and U > 3 m s−1 eliminated all dry beach observations. It is recognized that the footprint analysis approach, particularly for a heterogeneous environment, is not absolute but is the first step in evaluating Cd for the surfzone region.
This also highlights the applicability of these results to other beaches. For the surfzone to be the primary turbulent source region, the nearshore waters need to be warmer than the associated air temperatures setting up an unstable atmospheric scenario allowing for a relatively narrow footprint to develop.
The uncertainties in using stability functions based on Monin–Obukhov similarity theory for adjusting to the stability-corrected CdN10 are well recognized, resulting in a wide range of Cd, even over homogeneous terrains (Andreas et al. 2012). To avoid these uncertainties, Cd6 is estimated first directly at z = 6 m by
where ρa is the air density, u′ and w′ are the turbulent horizontal and vertical velocity perturbations (as measured herein), and < > denotes time average; Cd6 is O(2 × 10−3) for the surfzone (Fig. 2a), and Cd6 seaward of the surfzone is O(1.5 × 10−3) (Fig. 2a). This suggests that Cd6 increases over the surfzone. The CdN10 calculated as a function of UN10 using Eq. (8) collapses toward O(1.5 × 10−3) (Fig. 2b). The UN10 for nonneutral conditions is calculated by
where ψ(ζ) is the empirical function of the stratification based on stability. Observed open-ocean unstable estimates of Cd10 are larger than CdN10 (Vickers et al. 2013). Here, it is further related to the footprint analysis, where unstable (stable) conditions result in a smaller (longer) and closer (farther) footprint. Applying Monin–Obukhov similarity theory, Cd10(−ζ) [Cd10(+ζ)] values corrected to CdN10 are reduced (increased). In practice, the Cd per source region is dependent upon ζ, which will collapse to a similar CdN10. For the moment, the similarity of CdN10 (Fig. 2b) is suggested as unique and that the different regions (Fig. 2a) potentially represent different mechanisms for modifying Cd.
4. Surfzone foam coverage drag coefficient model
Golbraikh and Shtemler (2016) developed an empirical model for Cd as function of U and foam coverage δf; Cd linearly increases with fractional foam coverage owing to whitecapping until saturated foam coverage. Holthuijsen et al. (2012) suggests zo of foam is related to the characteristic size of the foam bubbles. The sea foam bubble roughness k is 0.1–2 mm (Soloviev and Lukas 2006), resulting in a surprisingly similar zo between 0.1 and 2 mm (Powell et al. 2003). The correlation between aerodynamic and geometric roughness is believed to be related to the idea that the foam is moving in high wind (Golbraikh and Shtemler 2016). For the surfzone, the foam is assumed not to be moving, as the foam is generated by a wave roller of a self-similar bore and is left behind as the bore moves forward.
Golbraikh and Shtemler (2016) suggest zo averaged over the sea surface S is described as
where S = Sff + Sf, where Sff is the foam-free surface and Sf is the foam surface, zff is the foam-free aerodynamic roughness, zf is the foam-covered aerodynamic roughness, and δf = Sf/S is the fractional foam coverage. For the open ocean, Holthuijsen et al. (2012) developed a δf approximation as function of a U10. For the surfzone, δf is approximated for depth-limited wave breaking, as given by Sinnett and Feddersen (2016):
where Er is the roller energy density and the slope of the roller surface sinβ = 0.1 (Deigaard 1993; Duncan 2001). The dissipation is estimated from the one-dimensional wave and roller transformation models (Thornton and Guza 1983; Ruessink et al. 2001) for normally incident, narrowbanded waves. The roller energy model is defined as
where E is the wave energy density , Cg is the group velocity, and x is the cross-shore coordinate frame. The Sinnett and Feddersen (2016) surfzone foam coverage model is similar to the breaking wave intensity model, as measured by whiteness (as an indication of foam) in video images by Aarninkhof and Ruessink (2004), who also find the breaking intensity is related to the roller energy dissipation. Examples of the wave height and δf are provided in Figs. 3a and 3b for the experiment conditions.
For Monterey beach, δf averaged over the surfzone from Hsig(max) to the beach is estimated for a range of wave heights and wave periods, resulting in a δf of 0.35–0.55 (Fig. 3a). The foam roughness is defined as
where k is the geometric roughness of foam. Applying constant zff = 2 × 10−4 m (Charnock 1955) and zf = 2 × 10−3 m (Soloviev and Lukas 2006), the resulting CdN10 is O(2 × 10−3) (Fig. 3b). The open-ocean estimate of zf being similar to k is most likely an overestimate in the surfzone, owing to the foam not moving. Reducing zf by ~k/3, as suggested by land relationships by Nield et al. (2013), results in a CdN10 O(1.5 × 10−3) (Fig. 3c) similar to the observations (Fig. 2b).
The foam-free zff empirical relationship can be described as a function of wave age in the open ocean to account for wave form (Drennan et al. 2003):
with the concept that wave age represents a measure of wave height and therefore roughness in generation region. Equations (2), (10), (14), and (15) are applied across the shoaling region and surfzone to evaluate the relative contributions of zo and CdN10 (Figs. 4c,d). The quantity zff [Eq. (15)] increases within the surfzone, owing to decreasing c, while Hsig is decreasing (Fig. 4a). It is also suggested that zff should decrease in the surfzone, as the waves are decreasing in amplitude, which should reduce the form drag. For low winds within the surfzone, zo and CdN10 appear to be governed more by foam [Eq. (14); Figs. 4c,d]. As the winds increase, zff [Eq. (15)] unrealistically grows (Figs. 5a,b) because c remains a depth-limited constant but continues to increase with increasing U. This questions the validity of Eq. (15) parameterized using wave age within the surfzone, particularly for faster wind cases. Using Eq. (2) (Charnock formulation) for zff and zf ~ k/3 in Eq. (10) (black line in Figs. 5a,b) results in similar observed surfzone CdN10 estimates (black dots in Figs. 5a,b). This suggests that the summation of the Charnock formulation [Eq. (2)] for zff and the modified foam model [Eq. (14)] in Eq. (10) provides a reasonable estimate of the aerodynamic roughness and corresponding drag coefficient for the surfzone.
5. Summary and conclusions
The coupled dynamical relationship between wind and waves in the nearshore region differs from the open ocean. Unlike the open ocean, where surface foam increases as a function of wind speed and concomitant wave height, the wave heights decay while the foam generation increases within the surfzone. This suggests that aerodynamic roughness zo associated with form drag decreases in the surfzone, while surface foam stress increases. Modifying a zo foam model for the open ocean to a surfzone foam model results in predicted values similar to observed surfzone Cd.
JM was supported as part of the ONR Coastal Land Air Sea Interaction (CLASI) pilot experiment (ONR Grant N0001416WX01116). I appreciate the many fruitful conversations with Ed Thornton, Tim Stanton, and Mara Orescanin. Appreciation is extended to the NPS CLASI field team (Jessica Koscinski, Darin Keeter, Paul Jessen, Keith Wyckoff, Mathias Roth, and Tucker Freismuth) and CLASI collaborators (UM: Brian Haus, Dave Ortiz-Suslow, Hans Graber, and Neil Williams; NPS: Qing Wang, Dick Lind, and Ryan Yamaguchi; and NRL: Jim Doyle, David Flagg). Greg Sinnett provided useful feedback for his model. I thank Falk Feddersen, Baylor Fox-Kemper, and an anonymous reviewer for improving the manuscript clarity.