## 1. Introduction

Over land, the geometric roughness *k* and corresponding aerodynamic roughness *z*_{o} for surface features can be considered temporally constant. Over the open ocean, *z*_{o} 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, *C*_{d} ~ *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 *C*_{d}, owing to their tower being deployed on a sand spit. During Hurricane Ike in 2008, Zachry et al. (2013) and Powell (2008) measured elevated *C*_{d} values in the near shore compared with the open ocean. Anctil and Donelan (1996) found increased *C*_{d} values for waves shoaling from 12 m to breaking in 2-m water depth. Shabani et al. (2014, 2016) found that measured *C*_{d} 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 *C*_{d}.

*z*

_{o}is composed ofwhere

*z*

_{υ}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

*z*

_{w}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

*z*

_{f}is the aerodynamic roughness due to spray droplets and foam and is often included in

*z*

_{w}or

*z*

_{υ}. Though

*z*

_{o}can be a linear summation,

*C*

_{d}is not a linear summation (Edson et al. 2013). The terms

*z*

_{o}and

*C*

_{d}at 10 m (subscript 10) for neutral atmospheric stability (subscript

*N*) are related bywhere

*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

*z*

_{υ}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

*z*

_{w}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

*U*

_{N10}using a modified

*α*in Eq. (2). Golbraikh and Shtemler (2016) developed a

*z*

_{f}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 *C*_{d} 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 *z*_{f} generated by depth-limited wave breaking inside the surfzone also contributes to the increased *C*_{d} (Fig. 1). Within the surfzone, since surface gravity waves are decaying, the potential influence of the wave form drag *z*_{w} relative to *z*_{f} may be reduced, while at the same time *z*_{f} is increasing due to increased foam coverage by breaking waves. Using Golbraikh and Shtemler (2016), a modified *C*_{d} 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 *H*_{sig}, average wave period *T*_{avg}, 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.

*H*

_{sig}and

*T*

_{avg}ranged between 0.3 and 2 m and 6 and 13 s, respectively, associated with local storm-generated events. The wind speed

*U*

_{6}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,

*U*

_{6}> 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*, wherewhere

*T*

_{υ}is virtual temperature,

*w*′ and

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 *U*_{6} is referred to as the decoupling distance *x*_{dc}, inside of which the decreasing speed of shoaling waves may increase drag (Anctil and Dolelan 1996). For the experiment, *x*_{dc} 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.

### Footprint analysis

*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 bywhere

*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

*z*

_{u}is defined asResearchers 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 bywhere the particular footprint source region

*f*(

*x*,

*z*) is defined between two cross-shore locations (

*x*

_{1}and

*x*

_{2}). 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

*C*

_{d}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.

## 3. Results

*C*

_{dN10}are well recognized, resulting in a wide range of

*C*

_{d}, even over homogeneous terrains (Andreas et al. 2012). To avoid these uncertainties,

*C*

_{d6}is estimated first directly at

*z*= 6 m bywhere

*ρ*

_{a}is the air density,

*u*′ and

*w*′ are the turbulent horizontal and vertical velocity perturbations (as measured herein), and < > denotes time average;

*C*

_{d6}is

*O*(2 × 10

^{−3}) for the surfzone (Fig. 2a), and

*C*

_{d6}seaward of the surfzone is

*O*(1.5 × 10

^{−3}) (Fig. 2a). This suggests that

*C*

_{d6}increases over the surfzone. The

*C*

_{dN10}calculated as a function of

*U*

_{N10}using Eq. (8) collapses toward

*O*(1.5 × 10

^{−3}) (Fig. 2b). The

*U*

_{N10}for nonneutral conditions is calculated bywhere

*ψ*(

*ζ*) is the empirical function of the stratification based on stability. Observed open-ocean unstable estimates of

*C*

_{d10}are larger than

*C*

_{dN10}(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,

*C*

_{d10}(−

*ζ*) [

*C*

_{d10}(+

*ζ*)] values corrected to

*C*

_{dN10}are reduced (increased). In practice, the

*C*

_{d}per source region is dependent upon

*ζ*, which will collapse to a similar

*C*

_{dN10}. For the moment, the similarity of

*C*

_{dN10}(Fig. 2b) is suggested as unique and that the different regions (Fig. 2a) potentially represent different mechanisms for modifying

*C*

_{d}.

## 4. Surfzone foam coverage drag coefficient model

Golbraikh and Shtemler (2016) developed an empirical model for *C*_{d} as function of *U* and foam coverage *δ*_{f}; *C*_{d} linearly increases with fractional foam coverage owing to whitecapping until saturated foam coverage. Holthuijsen et al. (2012) suggests *z*_{o} 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 *z*_{o} 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.

*z*

_{o}averaged over the sea surface

*S*is described aswhere

*S*=

*S*

_{ff}+

*S*

_{f}, where

*S*

_{ff}is the foam-free surface and

*S*

_{f}is the foam surface,

*z*

_{ff}is the foam-free aerodynamic roughness,

*z*

_{f}is the foam-covered aerodynamic roughness, and

*δ*

_{f}=

*S*

_{f}/

*S*is the fractional foam coverage. For the open ocean, Holthuijsen et al. (2012) developed a

*δ*

_{f}approximation as function of a

*U*

_{10}. For the surfzone,

*δ*

_{f}is approximated for depth-limited wave breaking, as given by Sinnett and Feddersen (2016):where

*m*≅ 400 and is a fit parameter,

*h*is the water depth (Battjes 1975; Feddersen 2012a,b). The roller dissipation is given bywhere

*E*

_{r}is the roller energy density and the slope of the roller surface sin

*β*= 0.1 (Deigaard 1993; Duncan 2001). The dissipation

*E*is the wave energy density

*C*

_{g}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

*δ*

_{f}are provided in Figs. 3a and 3b for the experiment conditions.

*δ*

_{f}averaged over the surfzone from

*H*

_{sig}(max) to the beach is estimated for a range of wave heights and wave periods, resulting in a

*δ*

_{f}of 0.35–0.55 (Fig. 3a). The foam roughness is defined aswhere

*k*is the geometric roughness of foam. Applying constant

*z*

_{ff}= 2 × 10

^{−4}m (Charnock 1955) and

*z*

_{f}= 2 × 10

^{−3}m (Soloviev and Lukas 2006), the resulting

*C*

_{dN10}is

*O*(2 × 10

^{−3}) (Fig. 3b). The open-ocean estimate of

*z*

_{f}being similar to

*k*is most likely an overestimate in the surfzone, owing to the foam not moving. Reducing

*z*

_{f}by ~

*k*/3, as suggested by land relationships by Nield et al. (2013), results in a

*C*

_{dN10}

*O*(1.5 × 10

^{−3}) (Fig. 3c) similar to the observations (Fig. 2b).

*z*

_{ff}empirical relationship can be described as a function of wave age

*z*

_{o}and

*C*

_{dN10}(Figs. 4c,d). The quantity

*z*

_{ff}[Eq. (15)] increases within the surfzone, owing to decreasing

*c*, while

*H*

_{sig}is decreasing (Fig. 4a). It is also suggested that

*z*

_{ff}should decrease in the surfzone, as the waves are decreasing in amplitude, which should reduce the form drag. For low winds within the surfzone,

*z*

_{o}and

*C*

_{dN10}appear to be governed more by foam [Eq. (14); Figs. 4c,d]. As the winds increase,

*z*

_{ff}[Eq. (15)] unrealistically grows (Figs. 5a,b) because

*c*remains a depth-limited constant but

*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

*z*

_{ff}and

*z*

_{f}~

*k*/3 in Eq. (10) (black line in Figs. 5a,b) results in similar observed surfzone

*C*

_{dN10}estimates (black dots in Figs. 5a,b). This suggests that the summation of the Charnock formulation [Eq. (2)] for

*z*

_{ff}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 *z*_{o} associated with form drag decreases in the surfzone, while surface foam stress increases. Modifying a *z*_{o} foam model for the open ocean to a surfzone foam model results in predicted values similar to observed surfzone *C*_{d}.

## Acknowledgments

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.

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