Volume 123, Issue 10 p. 7172-7184
Research Article
Open Access

The Competing Effects of Breaking Waves on Surfzone Heat Fluxes: Albedo Versus Wave Heating

Gregory Sinnett

Corresponding Author

Gregory Sinnett

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USA

Correspondence to: G. Sinnett,


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Falk Feddersen

Falk Feddersen

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USA

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First published: 17 September 2018
Citations: 6
This article was corrected on 29 MAY 2019. See the end of the full text for details.


Depth-limited wave breaking modifies the heat flux in the surfzone relative to the inner-shelf (where waves are not breaking). Surfzone wave breaking generates heat through viscous dissipation (wave heating), but also increases surface foam coverage and albedo, thereby reducing solar heating, that is, cooling relative to the inner-shelf. These two competing breaking wave effects are quantified with a yearlong experiment at the Scripps Institution of Oceanography Pier. Cross-shore averaged surfzone albedo estimates were more than three times higher than inner-shelf albedo, reducing the yearly averaged surfzone water-entering shortwave radiation by 41 W/m2 relative to the inner-shelf. Surfzone breaking wave dissipation added an additional yearly averaged 28 W/m2 relative to the inner-shelf. The albedo-induced solar heating reduction in spring, summer, and fall was usually greater than the wave heating. However, in winter, large waves and relatively weak shortwave solar radiation (due to both lower top of the atmosphere solar radiation and clouds) resulted in a nearly equal number of days of breaking wave-induced heating or cooling. These two heat flux terms are coupled via wave breaking dissipation. Averaged over the surfzone, the albedo-induced solar radiation reduction is linearly related to the downwelling solar radiation and is independent of wave height. Consequently, the albedo-induced cooling to wave heating ratio is a function of breaking wave height to the −3/2 power, allowing evaluation of the relative importance of these terms in other geographic regions.

Key Points

  • Surfzone breaking waves heat via dissipation; foam increases albedo, reducing solar radiation
  • Over a year, the albedo-induced solar heating reduction was most significant
  • The net effect depends on incident wave height, latitude, seasons, beach slope, and cloudiness

Plain Language Summary

Temperature variation in nearshore waters affects the local ecology, and is also used to study important physical processes. Wave breaking contributes to surfzone temperature variation in two ways. First, breaking waves dissipate their energy in the surfzone creating friction (heat) and foam. Surfzone foam reflects sunlight reducing solar warming of the surfzone, thus leading to cooling relative to no wave breaking. These two competing wave effects (addition of frictional heating and reduction in solar heating) are quantified with a yearlong experiment at the Scripps Institution of Oceanography pier (La Jolla, CA). On average, frictional wave heating added 28 W to each square meter of surfzone. At the same time, surface foam reduced the solar heating in each square meter of surfzone by 41 W on average. The relative contribution of these competing effects varied depending on the wave height and the available sunlight, which depended on seasons and clouds. Temperature variation caused by these two effects can be estimated at other locations if the wave height and the amount of sunlight are known.

1 Introduction

The surfzone (region of depth-limited wave breaking) and adjacent offshore shallow inner-shelf (no depth-limited wave breaking) comprise the nearshore; a physically dynamic, economically important, and biologically diverse part of the ocean. Temperature is an important physical attribute here, as temperature variation affects growth rates, recruitment rates, and egg mass production rates of various species (e.g., Broitman et al., 2005;Fischer & Thatje, 2008;Phillips, 2005) as well as pathogen ecology (e.g., Goodwin et al., 2012). Pathogen mortality is related to both temperature (Surbeck, 2009) and exposure to solar shortwave radiation (e.g., Boehm et al., 2002; Sinton et al., 1999, 2002). In the nearshore, temperature can also be a tracer for nutrient delivery (e.g., Omand et al., 2012) or surfzone to inner-shelf water mass exchange (e.g., Hally-Rosendahl et al., 2014).

Consequently, quantitatively understanding physical mechanisms affecting the inner-shelf heat budget has been an active area of recent study. Inner-shelf heat budgets include upwelling (e.g., Fewings & Lentz, 2011;Lentz, 1987), wind stress (e.g., Austin, 1999), eddies (e.g., Wilkin, 2006), internal waves (e.g., Shroyer et al., 2010), and the passage of weather systems on time scales of days to weeks (e.g., Austin & Lentz, 1999). Heat transfer between the air-sea interface occurs through radiative solar shortwave heating, net long-wave heat flux, as well as net latent and sensible heat exchange and is often parameterized (e.g., Beardsley et al., 1998; Fairall et al., 1996, 2003) when applied to observational and modeling studies (e.g., Davis et al., 2011; Etter et al., 2004; Lentz, 1987; Wilkin, 2006).

Closer to shore, rip currents (narrow wave-driven ejections from the surfzone) have been associated with strong temperature variation on the inner-shelf (Hally-Rosendahl et al., 2014; Smith & Largier, 1995), interacting with and adjusting the vertical temperature profile and influencing the inner-shelf cross-shore heat flux (Kumar & Feddersen, 2017). Thus, surfzone temperature (relative to the stratified inner-shelf) is an important determining factor for how this transport mechanism is established and evolves. Additionally, the presence of fecal indicator bacteria (FIB) near the Southern California coast varies with temperature (Boehm et al., 2004), and predictive models for pathogen transport in the surfzone include temperature and shortwave radiation (e.g., Boehm, 2003). In addition, solar radiation-induced Enterococcus (FIB) mortality contains cross-shore variation, and modeled FIB concentrations and decay rates were best predicted when cross-shore mortality gradients were included (Rippy, Franks, Feddersen, Guza, & Moore, 2013). Thus, cross-shore variation of temperature and solar radiation affects many important biological processes, motivating a more complete understanding of surfzone to inner-shelf temperature and solar radiation differences.

Many aspects of the surfzone heat budget are similar to the inner-shelf heat budget, although surfzone wave breaking modifies terms and creates a new term. The new wave heating term is generated by surfzone wave breaking, which through viscous dissipation, generates heat. Also, breaking wave-induced foam increases the surfzone albedo and thereby reduces the water-entering solar shortwave radiation relative to the inner-shelf. Further, surfzone wave breaking affects the sensible (MacMahan et al., 2018) and potentially the latent air-sea fluxes. Here the wave heating and surfzone foam albedo effects are explored.

The wave heating contribution to the surfzone heat budget results from mechanical wave energy being converted to heat (internal energy) through viscous dissipation. Waves outside the surfzone shoal and break in the shallow surfzone, generating turbulent kinetic energy. Some wave energy is reflected from the shoreline, however on shallow sloping beaches (such as in this study) the percentage of reflected wave energy is typically small (<3%; Elgar et al., 1994). Other surfzone processes driven by wave breaking are frictionally balanced with energy pathways still leading to viscous heating. For example, breaking wave-driven alongshore currents are frictionally balanced (Feddersen et al., 1998). Similarly, surfzone wave breaking can suspend sediment or inject bubbles into the water column, yet their fall or rise is also frictionally balanced. Acoustic noise energy generated by wave breaking does radiate away but noise generation is negligible (6–10 orders of magnitude smaller) relative to breaking wave dissipation (e.g., Kennedy, 1992; Klusek & Lisimenka, 2013). Additional export of mechanical energy from the surfzone (via rip currents or undertow, e.g.) has been estimated to be many orders of magnitude smaller than incident wave energy flux on similar beaches (Sinnett & Feddersen, 2014). Thus, the bulk of the incident wave energy is dissipated in the surfzone through turbulence throughout the water column, and eventually converted to heat. Wave heating heats the surfzone relative to the inner-shelf.

Solar heat flux is a major surfzone heat budget term (Sinnett & Feddersen, 2014), so changes to the albedo, and thus the amount of absorbed solar radiation, are consequential. The surfzone surface is a combination of foam-free and foam-covered areas due to the recent passage of breaking waves (e.g., Frouin et al., 1996). As foam has a higher albedo (α ≈ 0.55; Whitlock et al., 1982) than foam-free water (α ≈ 0.06; Payne, 1972), the average albedo is higher in the surfzone than in the relatively foam-free inner-shelf (Frouin et al., 1996). Deep-water albedo parameterizations have been developed for wind-generated whitecapping (e.g., Frouin et al., 1996; Jin et al., 2011; Koepke, 1984). However, surfzone foam is due to depth-limited wave breaking and does not require wind, making these parameterizations inappropriate for the surfzone. Recently, a surfzone albedo parameterization has been developed that uses offshore wave conditions, bathymetry, and a surfzone wave model (Sinnett & Feddersen, 2016).

The breaking wave-related surfzone albedo increase can be large (as much as 8× the inner-shelf albedo; Sinnett & Feddersen, 2016), and the subsequent decrease in solar radiation is significant. Thus, elevated surfzone albedo results in surfzone cooling relative to the inner-shelf. Similarly, the wave heating term can be a significant source of heat as including wave heating improved a surfzone heat budget (Sinnett & Feddersen, 2014). However, breaking wave albedo effects were not included, although a residual net surfzone cooling was inferred. Thus, the relative importance of these two competing effects is unknown, as is how parameters such as wave height, beach slope, or latitude affect relative heating or cooling.

Here surfzone parameterizations of wave heating (Sinnett & Feddersen, 2014) and wave-induced albedo increase (Sinnett & Feddersen, 2016) are applied to yearlong observations quantifying the competing wave heating and albedo affects on surfzone heat fluxes. The experiment and analysis methods are detailed in section 2. Results quantifying the competing effects of wave heating and albedo-induced solar heating reduction are described in section 3. The implications of these competing effects for different parameter space (wave height, beach slope, latitude) is discussed in section 4.1. These competing wave-related heating and cooling effects are discussed relative to a previous heat budget at the same location (Sinnett & Feddersen, 2014) in section 4.2. Section 5 is a summary.

2 Methods

2.1 Instrumentation and Data Processing

A yearlong study was conducted at the Scripps Institution of Oceanography (SIO) pier (La Jolla California, 32.867N, 117.257W) between 25 October 2014 and 25 October 2015. The SIO pier extends 322 m west-north-west (288°) from Scripps beach into water depth h ≈  7 m (Figure 1a). The roughly alongshore uniform shoreline extends 200 m north to 500 m south of the pier. Cross-shore bathymetry profiles were conducted along the pier at 0.5 to 1 month intervals as wave conditions allowed. The cross-shore profile slopes gently with yearly bathymetric changes less than 0.3 m at any location, causing slope variation of less than 5%. The average slope in depths h< 3.5 m (typically includes the surfzone) is s ≈ 0.023 (Figure 1b). A pier-end NOAA station (9410230) measured 6-min averaged tidal elevation η relative to the mean tide level. The cross-shore x coordinate is positive onshore, with the mean shoreline (x = 0) where mean tide level intersects the mean bathymetry. The alongshore coordinate y is positive toward the north, with y = 0 at the northern edge of the pier.

Details are in the caption following the image
(a) Google earth image of the Scripps Institution of Oceanography pier experiment site near midtide with the x and y coordinates indicated. Locations of the wave and tide gauges (square) and radiometer (triangle) are shown relative to the pier. The surfzone width Lsz (white dotted) extends from the offshore limit of breaking xsz to the effective shoreline xsl where h = 0.28 m depth. (b) Cross-section along the Scripps Institution of Oceanography pier depicting MTL and mean bathymetry z =− h(x) versus cross-shore coordinate x with wave gauge (square) and radiometer (triangle) locations indicated. The radiometer elevation above MTL is z = 6.5 m (not to scale in b). MTL = mean tide level.

For the 365 days beginning 25 October 2014, hourly significant wave height Hs (zeroth moment of the hourly energy spectrum) and peak period Tp (period of the highest spectral energy density) were observed at the pier-end (square, Figures 1a and 1b) by the Coastal Data Information Program station 073 pier-mounted Paros pressure sensor. When the sensor was inoperative (<7% of the time), a spectral refraction wave model with very high skill and initialized from offshore buoys was used (O'Reilly & Guza, 1991, 1998; O'Reilly et al., 2016).

Concurrently, a Campbell Scientific NR01 four-way radiometer located midpier (triangle, Figures 1a and 1b) recorded 1-min averaged downwelling urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0001 and reflected upwelling urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0002 solar shortwave radiation (wavelengths 300 to 2800 nm) as described in Sinnett and Feddersen (2016). Although the radiometer was cleaned at regular intervals, rain or very dense fog caused water to accumulate on the glass optics. Additionally, rarely occurring extremely low tides moved the shoreline seaward of the radiometer location so that the sensors viewed sand rather than water. Data during these times were flagged and removed from the record (6% of all data). For this study, radiation data were hourly averaged onto the same temporal grid as the wave observations. These wave and radiation data were used to calibrate a parameterization relating offshore wave energy to surfzone albedo as described in section 2.2.3 and detailed in Sinnett and Feddersen (2016).

2.2 Analysis

2.2.1 Wave Model

The cross-shore transformation of normally incident narrow-banded waves on alongshore uniform beaches is described by one-dimensional wave and roller transformation models (e.g., Battjes & Stive, 1985; Ruessink et al., 2001; Thornton & Guza, 1983). The wave transformation is given by
where E is the wave energy density, cg is the linear group velocity given by peak period and depth, and εb is the bulk breaking wave dissipation. The wave energy density is
where ρ is water density, g is gravity, and Hs is the significant wave height. The cross-shore wave energy flux at location x is

The model adapted here follows Church and Thornton (1993) with standard breaking parameters (B =0.9 and γ = 0.57).

Similarly, the wave roller transformation describes the dissipation along a breaking wave face with energy equation (e.g., Ruessink et al., 2001)
Here Er is the roller energy density, c is the linear phase speed, and roller dissipation εr (analogous to foam) is
with wave slope β = 0.1 (e.g., Deigaard, 1993; Walstra et al., 1996). The model boundary conditions are the pier-end yearlong hourly Hs and peak period observations.

An example cross-shore wave transformation over bathymetry is illustrated (e.g.) on 5 May 2015 at 14:00 PDT (Figure 2a). Observed offshore wave height Hs = 1.4 m slightly increases onshore before breaking (black, Figure 2b) due to the shallowing bathymetry. Wave set-up and set-down are ignored in the transformation model as these adjustments contribute to a negligibly small variation in shoreline location. As waves break, Hs decreases from the outer surfzone to the shoreline, also reducing the wave energy flux Fwave (red, Figure 2b).

Details are in the caption following the image
Example cross-shore (x) hourly averaged parameters from 5 May 2015 at 14:00 local time. (a) Bathymetry h(x) (solid) and mean water level η (dotted), (b) significant wave height Hs (black) and associated cross-shore wave energy flux Fwave from 3 (red), (c) nondimensionalized roller energy dissipation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0003 from 12 and foam fraction ζ from 13 as black and red, respectively, and (d) albedo on the inner-shelf αθ and in the surfzone αsz 14. The offshore breaking location xsz=−170 m and effective shoreline xsl=−22 m (black dashed in b, c, and d). The cross-shore averaged surfzone albedo 〈αsz〉=0.21 (black dashed) and albedo where waves are not breaking (with clear sky conditions) αθ=0.04 are indicated in (d).
The outer surfzone boundary, xsz (vertical dotted in Figures 2b–2d), is defined as where breaking wave dissipation is nonnegligible and corresponds to the maximum in Hs. Wave transformation models are not designed for shallow swash zones. Thus, an effective shoreline, xsl, is defined as the first offshore location where h > 0.28 m, where the wave roller model is still applicable. Waves in water shallower than h = 0.28 m are considered swash and this region is ignored. The effective surfzone width Lsz is

For the example in Figure 2, xsz=−170 m and xsl = −22 m, making the effective surfzone width Lsz = 148 m.

2.2.2 Wave Heating

Cross-shore wave energy flux is dissipated across the surfzone by breaking 1. Since wave reflection on shallow sloping beaches is small (Elgar et al., 1994) as is export of mechanical energy from the surfzone (Sinnett & Feddersen, 2014), the bulk of the wave energy flux is frictionally dissipated inside the surfzone, eventually as heat. Note that the wave heating estimate here is an upper bound. Assuming the surfzone is well mixed, the heating from wave energy flux dissipation occurs over the entire surfzone width. Thus, the cross-surfzone averaged additional heat flux (relative to no wave breaking on the inner-shelf) due to the dissipation of breaking waves is
where superscripts indicate the cross-shore flux location. This term Qwave is denoted wave heating. In the example, at xsz, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0015 = 7,500 W/m but at xsl urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0016 = 33 W/m (red, Figure 2b), implying that at this exampletime, there is a 7,467 W/m energy flux convergence in the surfzone (or ≈ 50 W/m2) which is largely viscously dissipated and converted to heat. Over the year, hourly Qwave is estimated from observed Hs through 7 and 3.

2.2.3 Solar Radiation

Top of the atmosphere shortwave solar radiation ( urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0017 in Figure 3) is
where S is the solar constant, θs is the solar zenith angle (sun declination angle from vertical) which varies on diurnal and seasonal time scales, and Γ is the ratio of the actual to mean earth-sun separation distance, which varies annually (e.g., Whiteman & Allwine, 1986). Atmospheric attenuation and clouds reduce urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0019 so that the downwelling radiation at the ocean surface is Qswd < Qswtop(Figure 3). The atmospheric reduction in downwelling shortwave solar radiation is defined as
and indicates atmospheric optical depth or cloudiness. The shortwave albedo (reflectance) is the ratio of the total reflected (upward) solar radiation to the downwelling solar radiation at the ocean surface,
so that the water-entering shortwave radiation (Figure 3) is
Details are in the caption following the image
Schematic depicting the shortwave solar radiation (Qsw, arrows) at different locations: the top of the atmosphere (dotted line) urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0011, downwelling to the ocean surface urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0012, water-entering urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0013, and upwelling (reflected) urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0014 from the ocean surface. The solar zenith angle is θs.

Thus, changes to either the available downwelling radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0023 or the albedo α affect the water-entering shortwave radiation and thus solar heating.

2.2.4 Inner-Shelf and Surfzone Albedo

In direct sunlight, standard nonwave breaking albedo parameterizations depend only on solar zenith angle θs (Briegleb et al., 1986; Payne, 1972; Taylor et al., 1996). In diffuse light (defined here when the ratio of atmospheric reduction in shortwave radiation to top-of-atmosphere shortwave radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0024), ocean surface albedo is near 0.06 and no longer depends on θs (Payne, 1972). Thus, here the inner-shelf albedo (where waves are not breaking) αθ is defined following Taylor et al. (1996) with specular reflection for urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0025 (direct sunlight). In diffuse light ( urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0026) α ≈ 0.06 (Payne, 1972). Latitude and local time define θs following Reda and Andreas (2008). This θs dependent parameterization works well for inner-shelf observations at this site (Sinnett & Feddersen, 2016),

Surfzone albedo is parameterized following Sinnett and Feddersen (2016). The foam fraction ζ is a function of the nondimensionalized wave roller dissipation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0027,
where nondimensionalization is denoted with urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0029. The example cross-shore urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0030 profile (black, Figure 2c) has peaks where waves are breaking over shallowing bathymetry and troughs where bathymetry is flatter or wave height is very low. Over the range of urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0031 typically observed at this location, the foam fraction ζ and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0032 are linearly related (Sinnett & Feddersen, 2016) so that
where m = 398 is a constant best-fit parameter. The example cross-shore ζ profile (red, Figure 2c) includes locations near x = −75 m and x = −40 m that are nearly continuously covered in foam, while only a few (large) waves break seaward of x = −150 m reducing ζ. Under extremely energetic wave conditions, parts of the surfzone can saturate so that the fit produces ζ > 1. When this occurs (less than 4% of the time) the foam fraction is restricted to the physical maximum ζ = 1.
The wave affected (surfzone) albedo αsz has contributions from both the foam-covered and foam-free surface, making

(Figure 2d). Here the best-fit αf=0.465 (Sinnett & Feddersen, 2016) and αθ is the θs parameterized albedo of foam-free water (Taylor et al., 1996). Onshore of the outer surfzone limit (xsz, where waves begin to break) albedo increases above αθ due to surface foam. Generally, albedo increases as the surfzone depth decreases, with variations caused by undulations in bathymetry. In the very shallow inner-surfzone, nearly all waves are breaking and the surfzone is nearly saturated in foam, so that αszαf.

The cross-shore surfzone average foam fraction is
which with 14 yields a cross-shore average surfzone albedo 〈αsz〉 (as in Figure 2d),
Here 〈.〉 indicates cross-shore averaging. From 11, the surfzone averaged albedo-induced solar heating reduction relative to the inner-shelf is then

Both the amount of available downwelling radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0038 and the albedo difference between the surfzone and inner-shelf affect urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0039. As 〈αsz〉>αθ, the surfzone has an albedo-induced cooling relative to the inner-shelf. Over the year, hourly urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0040 is estimated with Hs and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0041 via 17.

At this quartz-sand beach, this albedo parameterization does not explicitly consider the albedo of the seabed and suspended sediment, which can be important for other regions such as coral reefs (e.g., Hochberg et al., 2003) and estuaries (e.g., Fogarty et al., 2017). At small θs, the albedo of wet sand is about 0.07 (e.g., Dickinson, 1983), thus seabed reflections are weak. Furthermore, due to breaking wave-generated turbulence suspending sediment, the surfzone optical depth is typically small (e.g., Rippy, Franks, Feddersen, Guza, & Warrick, 2013) such that little light penetrates to the seabed. Surfzone suspended sediment concentrations above 5 g/L are unusual except near the seabed (e.g., Beach & Sternberg, 1996), and thus near-surface sand reflectance that contribute to albedo is also expected to be weak. Colocated instantaneous surfzone albedo and video observations clearly show that breaking wave foam drives albedo time-dependence, and when no waves are breaking, observed albedo agrees with the Taylor et al. (1996) parameterization (Sinnett & Feddersen, 2016).

3 Observations and Results

3.1 Observed urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0042, Hs, Fwave, and Albedo

The top of the atmosphere urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0043 varies with θs and Γ on diurnal and seasonal time scales, so that the daily maximum urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0044 varies seasonally (red, Figure 4a). At the water surface available downwelling solar radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0045 primarily varied diurnally, but also varied at synoptic to seasonal time scales (black, Figure 4a). On clear days, atmospheric attenuation resulted in urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0046. Clouds decreased the available urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0047 further (Figure 4b). In winter, cloudy periods usually lasted a few days (jagged peaks, Figure 4b) and were frequently accompanied by rain causing short urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0048 data gaps. In the very late spring and early summer, coastal fog persisted for longer periods causing urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0049 to remain elevated (Figure 4b). Early spring, late summer, and early fall were typically less cloudy.

Details are in the caption following the image
Yearlong time series of (a) daily maximum solar radiation at the top-of-atmosphere max urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0050 (red) and hourly averaged downwelling shortwave solar radiation to the ocean surface urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0051 (black), (b) daily percent reduction of downwelling solar shortwave radiation due to cloud cover urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0052, (c) pier-end significant wave height Hs, and (d) surfzone width Lsz 6. Seasons denoted in (a) are 91 days long, centered on each solstice and equinox. Data in (a) are removed when rain obscured the radiometer.

Pier-end significant wave height Hs typically varied synoptically between 0.5 and 1.5 m, with generally larger waves in winter and spring, and smaller waves in summer and fall (Figure 4c). Pier-end peak wave period was usually between 7 and 13 s (not shown). The mixed barotropic tide typically varied ±1 m (not shown) inducing a roughly ±43 m variation in xsl. Wave and tide conditions, together with the evolving bathymetry, affected the surfzone width Lsz (Figure 4d). Average Lsz = 84 m, but was at times above 150 m during strong wave events and as small as 4 m when waves were small. Time periods were excluded from analysis when waves were very small and xsz was in less than 0.5 m depth (i.e.,Lsz<10, less than 0.2% of all data).

At the outer surfzone boundary, wave energy flux mean and standard deviation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0053 = 2,149 ± 1,826 W/m driven primarily by variable Hs through 3 on synoptic time scales (Figure 5a). Large wave events have an outsized contribution to Fwave due to the quadratic relationship between Fwave and Hs 3. Seasonal Hs variability generally elevated Fwave in wintertime and reduced Fwave in summertime. The cross-shore average surfzone albedo mean and standard deviation 〈αsz〉=0.28 ± 0.07 (Figure 5b) and was more than three times the mean inner-shelf albedo. Surfzone albedo 〈αsz〉 varied on tidal, diurnal, and seasonal time scales, and usually much more rapidly than Fwave.

Details are in the caption following the image
Hourly (a) pier-end wave energy flux Fwave 3 and (b) cross-shore averaged surfzone albedo 〈αsz16 versus time of year.

The daylight variation of 〈αsz〉 and αθ is examined with ensemble averages. Albedo estimates are removed when solar zenith angle is large (|θs|> 80°) to remove near-horizon effects. For each day, the daylight albedo estimates are normalized onto a standard 12 hr time-period removing seasonal daylight variations. These are subsequently binned over all the days in the year, allowing interday surfzone and inner-shelf albedo comparison. Daily ensemble averaged αθ (blue line, Figure 6) has strong solar zenith angle θs dependence, with elevated albedo at low sun angles near sunrise and sunset. Seasonal variation in θs and cloud cover variation account for the relatively small αθ deviation from the mean (blue shaded). As the surfzone has fractional foam coverage, 〈αsz〉 retains some θs dependance, although weaker than αθ, with elevated 〈αsz〉 at larger |θs| (red line, Figure 6). However, surfzone foam elevates 〈αsz〉 above αθ, with midday ensemble averaged 〈αsz〉 elevated by 0.19 over αθ. Wave, tide, and bathymetry variability influence 〈ζ〉 and thus contribute to the relatively large 〈αsz〉 variability (red shaded).

Details are in the caption following the image
Daily ensemble averaged albedo with no wave breaking αθ (blue) and daily ensemble averaged cross-shore averaged surfzone albedo 〈αsz〉 (red) versus normalized time of day. Shading is ± one standard deviation from the mean.

3.2 Competing Wave Effects: urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0054 and Qwave

Breaking wave energy dissipation leads to surfzone wave heating Qwave 7. Wave breaking also increases albedo, thereby reducing the water-entering shortwave solar radiation relative to the inner-shelf by an amount urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0055 17. Here these two competing effects are examined. Variability in Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0056 occur on seasonal, synoptic, diurnal, and semidiurnal time scales through variation in Hs, θs, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0057, and Lsz. Here Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0058 are daily (24 hr) averaged to examine their relative effects on synoptic and seasonal time scales. Henceforth, all Q variables will be daily averaged.

Breaking wave-related heat flux contributions varied over the year (Figure 7) with Qwave always increasing (positive) surfzone heat flux and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0059 always reducing (negative) surfzone heat flux relative to the inner-shelf. Over the year, the mean and standard deviation of the daily averaged Qwave = 28 ± 11 W/m2 (red) and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0060 = −41 ± 16 W/m2 (blue). Thus, at this location, the combined effect of Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0061 typically reduced the surfzone heat flux relative to the inner-shelf. Both daily averaged Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0062 varied on synoptic to seasonal time scales. However, daily averaged Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0063 were uncorrelated (r2 = 0.04) as Qwave depends on incident Hs (Figure 4c) whereas urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0064 depends also on clouds and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0065. Throughout most of summer, clouds reduced urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0066 and waves were small (Figures 4a–4c). Thus, the yearly maximum urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0067 occurred in April when waves were larger and cloudiness lower, rather than at the summer solstice (21 June) when urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0068 is maximum.

Details are in the caption following the image
Yearlong time series of daily (24 hr) averaged wave heating Qwave 7 and albedo-induced solar heating reduction urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0084 17 as indicated in the legend.

The relative effects of Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0069 have a seasonal dependance (Figure 8). In winter, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0070 is low and cloudiness urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0071 can be high reducing urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0072. Wintertime waves are also relatively large with Qwave> 40 W/m2 about 20% of the time. The combined effect in winter heats the surfzone (to the right of the 1:1 line) relative to the inner-shelf 47% of the time (Figure 8a). In contrast, summertime waves were relatively small with Qwave> W/m2 only 5% of the time. The combined effect in summer cools the surfzone relative to the inner-shelf 96% of the time (Figure 8c).

Details are in the caption following the image
Daily averaged albedo-induced solar heating reduction urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0090 versus the daily averaged wave heating Qwave for each season. Symbols are shaded according to urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0091 representing bulk cloudiness (Figure 4b). Daily averaged values fall to the right or left of the 1:1 line (solid black) contributing to net heat flux increase or decrease, respectively, relative to the inner-shelf.

Spring is characterized by a wide range of both Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0073 (Figure 8b). Spring had few clouds, with urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0074 > 40% only a quarter of the time (compared to over half the time in summer). Spring also contained some of the largest Hs, resulting in the daily averaged Qwave> 50 W/m2 11% of the time. The fall urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0075 distribution is slightly lower than in summer (Figure 8d). Fall urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0076 is smaller than in summer (red, Figure 4a), yet fall skies were clearer (lower urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0077) relative to summer such that mean urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0078 was reduced by only 5%. Occasional large wave events in late fall (more typical of winter conditions) widened the fall Qwave distribution compared to summer. The seasonal variation in the Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0079 relationship demonstrates the effect of parameters such as the incident Hs, cloudiness, and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0080.

3.3 Surfzone Adiabatic Temperature Change

As temperature is relevant to circulation dynamics, cross-shore exchange, and ecology, relating Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0081 to an adiabatic temperature change is useful for understanding their relative effects. Relative to the inner-shelf, the daily averaged combined surfzone heat flux Qnet is
with positive Qnet implying surfzone warming relative to the inner-shelf. For a planar beach slope, the surfzone daily adiabatic temperature change ΔT induced by Qnet is
where tday = 86,400 s is the duration of a day and hsz is the outer surfzone boundary depth. Here the surfzone is assumed adiabatic (insulated) with no other breaking wave-induced heat fluxes (e.g., surfzone to inner-shelf exchange or air-sea fluxes).

Over the year, the daily adiabatic ΔT (18) was negative 75% of the time (black dots, Figure 9), with a mean and standard deviation of ΔT = −0.5 ± 0.6 °C. The 30-day ΔT mean and standard deviation also varied seasonally (red dots and red lines, Figure 9). Wintertime mean and standard deviation ΔT = 0.0 ± 0.4 °C as wintertime Qnet is near zero. Beginning in early spring, ΔT typically becomes negative, with mean and standard deviation ΔT = −0.7 ± 0.5 °C between March and September. In late summer and early fall with low clouds and small waves, ΔT can be as low as −1.9 °C. Daily ΔT variability was largest in spring and late summer when urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0085 was high, but intermittent clouds or coastal fog caused large changes in urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0086. The late fall urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0087 reduction and overall Hs increase (Figures 4a and 4c) prompted a return to winter conditions. In the adiabatic limit, net surfzone heat flux changes induced by Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0088 are substantial and can induce significant ( urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0089(1 °C)) temperature changes.

Details are in the caption following the image
Yearly time series of daily adiabatic surfzone temperature change ΔT (black dots) as in (18) due to the competing wave effects of wave heating Qwave and albedo-induced solar heating reduction urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0092. 30-day averages (red dots) and ± standard deviation (red lines), along with the ΔT = 0 (dashed black), are highlighted for reference.

4 Discussion

4.1 Scaling for an Idealized Surfzone

Parameters affecting surfzone averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0093 and Qwave are explored with scalings for a constant slope surfzone, lending insight to potential application at other sites with variable urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0094, clouds, incident waves, and beach slope. Although urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0095 and Qwave are uncorrelated, both depend on incident wave conditions, so a relationship exists between the two with added variability from the other nonwave factors such as bathymetric slope s and downwelling solar radiation at the water surface urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0096. For an idealized surfzone of constant bathymetric slope s and constant γ, the surfzone averaged foam fraction 〈ζ15 can be related to the nondimensionalized roller dissipation through 13 by surfzone averaging both the numerator and denominator in 12. The surfzone averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0097 is simply Qwave=Fwave/Lsz, and for a planar slope the representative (surfzone averaged) h3/2 becomes urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0100. Thus, bulk surfzone nondimensional roller dissipation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0101 can be scaled as
where the outer surfzone boundary depth hsz=Hsb/γ, Hsb is the significant wave height at breaking, and γ = 0.57 is the breaking parameter. The surfzone averaged foam fraction 〈ζ〉 is found applying 18 to 13 so that
where m = 398. The surfzone averaged 〈ζ〉 is independent of Hs, yet is linearly related to bathymetric slope s. Thus, the ratio of daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0104 and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0105 is expected to be
where urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0107 is the constant daily averaged albedo of the inner-shelf. For constant s and γ and daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0108, the daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0109 and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0110 is expected to be linearly related.

The linear relationship between daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0111 and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0112 (Figure 10, has squared correlation r2 = 0.48 (p < 0.01)) with best-fit slope −0.19 (red line). This implies that the daily averaged surfzone albedo is on average 0.19 larger than the inner-shelf. With an idealized (constant) bathymetric slope s = 0.023, daily averaged clear-sky inner-shelf albedo urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0113 (e.g., Payne, 1972), and foam albedo αf=0.465 as in section 2.2.4, the surfzone averaged foam fraction 20 applied to 21 yields a theoretical slope urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0114 (dashed black line, Figure 10) which is less than 1% different from the best-fit slope to observations. Deviations from the scaling 21 are potentially due to tidal and incident Hs variation together with the realistic and variable nonplanar bathymetry. The linear relationship correlation between urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0115 and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0116 (Figure 10) that matches the scaling 21 demonstrate the suitability of 20 and 21 to effectively scale urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0117 on gently sloping and alongshore uniform beaches.

Details are in the caption following the image
Daily averaged albedo-induced solar heating reduction urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0098 17 versus observed daily averaged downwelling solar radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0099. Symbols are identified (color and shape) by season. The best-fit slope of −0.19 (red line) deviates less than 1% from the idealized slope (dashed black line) from 21. The squared correlation r2 = 0.48.
Next, for a planar slope using 21, the ratio of surfzone daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0118 magnitude to Qwave is
where the bracketed quantity is a constant and is independent of bathymetric slope. Thus, the ratio urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0120 largely depends on urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0121. The downwelling solar radiation urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0122 depends on cloudiness and top of the atmosphere urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0123. Daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0124 is found from 8, and cloudiness (atmospheric attenuation or optical depth) may be estimated from terrestrial or satellite products (e.g., CERES, 2018). The wave height at the breakpoint Hsb can be well modeled (e.g., Ruessink et al., 2003) given incident wave conditions.

The urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0125 scaling for an idealized surfzone 22 is compared with observations (Figure 11), illustrating how clouds, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0126, and Hsb affect the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0127 ratio. The observed urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0128 ratio is largest for small Hsb and decreases for larger Hsb consistent with the scaling. For Hsb> 1.5 m, the observations and scaling have urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0129 (relative heating) at this location. For a clear sky (no clouds or constant atmospheric attenuation), urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0130 in 22 depends only on urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0131, varying only by season and latitude. At a latitude of 33°N (near the SIO pier) for the clear-sky summer solar maximum, the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0132 scaling 22 bounds the upper limit on the observed urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0133 for a particular Hsb (Figure 11, solid black). For the 33°N clear-sky winter solar minimum, the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0134 scaling intersects the observations (Figure 11, black dashed). Without clouds, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0135 observations are expected to fall between the black solid and dashed curves. However, the presence of clouds lower the observed urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0136 (and subsequently the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0137) for a particular Hsb. Thus, the scaling 22 sets an upper bound.

Details are in the caption following the image
Normalized joint PDF (gray shaded) of daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0138 and significant wave height at breaking Hsb. The urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0139 line is highlighted, delineating relative cooling (>1) and heating (<1). Daily averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0140 observations at various For clear skies (25% atmospheric attenuation), the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0141 ratio versus Hsb is found from 22 and plotted for the summer (solar maximum, solid) and winter (solar maximum, dotted) at the equator (0°N), 33° and 66° latitude (colored). Summer (solid) 33° and 66° latitude (black and blue) curves are nearly on top of each other.

The scaling for urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0142 22 can be used to estimate the relative importance of urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0143 to Qwave at other locations with variable latitude and seasonal top of the atmosphere urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0144, cloud, beach slope, and wave conditions. At the equator (0°N) seasonal variation in urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0145 is very small, resulting in a similar clear-sky urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0146 and Hsb relationship year-round (red solid and dashed curves in Figure 11). At high latitudes, the seasonal difference in urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0147 is large, expanding the summer to winter difference. At 66°N, the summer clear-sky urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0148 to Hsb relationship (Figure 11, blue solid ) is nearly the same as at 33°N (the experiment site). However, for wintertime clear skies, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0149 for any Hsb > 0.4 m (Figure 11, blue dashed) indicating wave heating nearly always dominates at any beach exposed to the open ocean (not iced in). In contrast at 33°N, wintertime clear-sky urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0150 only for Hsb> 1 m. These significant latitude and seasonal differences in clear-sky urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0151 will have implications for surfzone heat budgets from equator to Arctic. Note that carbonate sands or coral reef surfzones, which often have high optical clarity, may have additional albedo affects due to seabed reflections (e.g., Hochberg et al., 2003).

4.2 Wave heating Qwave and Albedo-Induced Solar Radiation Reduction urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0152 in Context

At the La Jolla, CA, experiment site, the parameters Hs, h(x), urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0153, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0154, and cloudiness ( urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0155) contribute to the breaking wave-induced positive or negative surfzone heat flux relative to the inner-shelf. Here the two terms Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0156 are placed in the context of a previous surfzone heat budget. Including wave heating (Qwave) but not urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0157 improved a summertime binned mean surfzone heat budget on diurnal and longer time scales (Sinnett & Feddersen, 2014). However, here the summertime urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0158 was usually greater than |Qwave| (Figure 8c). However, Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0159 are uncorrelated (r2 < 0.04). Thus, including Qwave but not urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0160 still improved the binned mean heat budget slope by reducing the unexplained variance. Sinnett and Feddersen (2014) also inferred a net surfzone cooling of ≈ 5,200 W/m (or ≈ 90 W/m2 over the average Lsz for the same period) required to balance the surfzone heat budget. Here the summer-averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0161 = 44 W/m2 (compare to the yearly averaged urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0162 = 41 W/m2) may account for nearly half the Sinnett and Feddersen (2014) inferred required net cooling. Advective processes, such as transient rip currents (e.g., Hally-Rosendahl et al., 2015) or nonlinear internal wave run up (e.g., Sinnett et al., 2018) may also contribute to the required relative surfzone cooling.

Breaking wave-induced changes to the surfzone latent or sensible heat flux are also modified by wave breaking due to surfzone spray and aerosol generation, which may also contribute to the surfzone heat budget. Parameterized (COARE) surfzone sensible heat flux estimations required an additional spray contribution when compared to surfzone covariance measurements (MacMahan et al., 2018). For the average wave dissipation observed at this site, the additional sensible heat flux due to breaking wave spray is ≈ 5 W/m2, relatively small compared to Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0163 at Scripps Beach. Spray droplets produced by breaking are typically large (Andreas, 2016) and quickly fall back to the surface before exchanging latent heat (MacMahan et al., 2018; Veron, 2015). However, the enthalpy exchange coefficient may be larger for a foamy sea surface than a foam-free surface (Chickadel, 2018), potentially enhancing surfzone latent heat flux. Examination of all surfzone heat flux terms is warranted to properly understand all the ways that breaking waves can affect the surfzone heat budget.

5 Summary

Nearshore heat and solar radiation budgets typically overlook breaking wave effects, and the relative importance of this adjustment is unknown. Here the relative effects of wave heating due to viscous dissipation of breaking waves Qwave and albedo-induced solar heating reduction relative to the inner-shelf are studied with yearlong observations at the SIO (La Jolla, CA) pier. Wave energy flux at the outer surfzone boundary urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0164 = 2,149 ± 1,826 W/m, which dissipated over Lsz yielding a daily averaged wave heating contribution Qwave = 28 ± 11 W/m2. Breaking waves partially covered the surfzone in foam, increasing albedo on average by a factor of 3 relative to the inner-shelf. The increased surfzone albedo subsequently created a solar heating reduction relative to the inner-shelf of urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0165 = 41 ± 16 W/m2. Usually at this location, the net effect (QwaveQsww) together act to cool the surfzone relative to the inner-shelf. However, the combined (Qwave and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0166) effect had seasonal dependence, with a net heating roughly half the time in winter, but only 4% of the time in summer.

On a beach of constant slope, the average surfzone foam fraction can be scaled as a function of beach slope, resulting in a surfzone averaged albedo 〈αsz〉 that is independent of Hs. At the experiment site, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0167 and urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0168 are linearly related and are in good agreement with the scaling. Scalings also are developed to the relative breaking wave surfzone heat flux contribution. The amount of additional surfzone cooling or heating relative to the inner-shelf is related to the ratio of urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0169 to urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0170 at the outer surfzone boundary. Clouds, urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0171, and Hsb affect the urn:x-wiley:jgrc:media:jgrc23132:jgrc23132-math-0172 ratio and thus the relative surfzone cooling or heating. This scaling can be applied at other locations to determine the relative heating or cooling effects of surfzone breaking waves.


This publication was prepared under NOAA Grant NA14OAR4170075/ECKMAN, California Sea Grant College Program Project R/HCME-26, through NOAAS National Sea Grant College Program, U.S. Dept. of Commerce. Additional support for this work was provided in part by ONR (N00014-15-1-2117) and NSF (OCE-1558695). The radiometer boom arm was designed and manufactured by B. Woodward, B. Boyd, K. Smith, and R. Grenzeback. Deployment was aided by C. McDonald, R. Walsh, S. Mumma, G. Boyd, L. Parry, and many San Diego lifeguards. Wave data from the Coastal Data Information Program (CDIP) were provided with assistance from W. O'Reilly and C. Olfe. Atmospheric conditions and tide data were from NOAA station 9410230 and Earth Networks. J. Norris, R. Clemesha, S. Giddings, A. Suanda, D. Grimes, G. Pawlak, K. Davis, and two anonymous reviewers provided helpful feedback. We sincerely thank these people and institutions. Data are available at iodlabs.ucsd.edu/falk/szheat


    In the originally published version of this article, a labeling error in Figure 11 switched “warming” and “cooling” above and below the y = 1 line. This error has been corrected, and this may be considered the authoritative version of record.