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

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.

For the 365 days beginning 25 October 2014, hourly significant wave height H_s (zeroth moment of the hourly energy spectrum) and peak period T_p (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 and reflected upwelling 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

(1)

where E is the wave energy density, c_g is the linear group velocity given by peak period and depth, and ε_b is the bulk breaking wave dissipation. The wave energy density is

(2)

where ρ is water density, g is gravity, and H_s is the significant wave height. The cross-shore wave energy flux at location x is

(3)

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)

(4)

Here E_r is the roller energy density, c is the linear phase speed, and roller dissipation ε_r (analogous to foam) is

(5)

with wave slope β = 0.1 (e.g., Deigaard, 1993; Walstra et al., 1996). The model boundary conditions are the pier-end yearlong hourly H_s 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 H_s = 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, H_s decreases from the outer surfzone to the shoreline, also reducing the wave energy flux F_wave (red, Figure 2b).

The outer surfzone boundary, x_sz (vertical dotted in Figures 2b–2d), is defined as where breaking wave dissipation is nonnegligible and corresponds to the maximum in H_s. Wave transformation models are not designed for shallow swash zones. Thus, an effective shoreline, x_sl, 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 L_sz is

(6)

For the example in Figure 2, x_sz=−170 m and x_sl = −22 m, making the effective surfzone width L_sz = 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

(7)

where superscripts indicate the cross-shore flux location. This term Q_wave is denoted wave heating. In the example, at x_sz,  = 7,500 W/m but at x_sl  = 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 Q_wave is estimated from observed H_s through 7 and 3.

2.2.3 Solar Radiation

Top of the atmosphere shortwave solar radiation ( in Figure 3) is

(8)

where S is the solar constant, θ_s 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 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

(9)

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,

(10)

so that the water-entering shortwave radiation (Figure 3) is

(11)

Thus, changes to either the available downwelling radiation 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 θ_s (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 ), ocean surface albedo is near 0.06 and no longer depends on θ_s (Payne, 1972). Thus, here the inner-shelf albedo (where waves are not breaking) α_θ is defined following Taylor et al. (1996) with specular reflection for (direct sunlight). In diffuse light ( ) α ≈ 0.06 (Payne, 1972). Latitude and local time define θ_s following Reda and Andreas (2008). This θ_s 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 ,

(12)

where nondimensionalization is denoted with . The example cross-shore 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 typically observed at this location, the foam fraction ζ and are linearly related (Sinnett & Feddersen, 2016) so that

(13)

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 α_sz has contributions from both the foam-covered and foam-free surface, making

(14)

(Figure 2d). Here the best-fit α_f=0.465 (Sinnett & Feddersen, 2016) and α_θ is the θ_s parameterized albedo of foam-free water (Taylor et al., 1996). Onshore of the outer surfzone limit (x_sz, where waves begin to break) albedo increases above α_θ 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 α_sz ≈ α_f.

The cross-shore surfzone average foam fraction is

(15)

which with 14 yields a cross-shore average surfzone albedo 〈α_sz〉 (as in Figure 2d),

(16)

Here 〈.〉 indicates cross-shore averaging. From 11, the surfzone averaged albedo-induced solar heating reduction relative to the inner-shelf is then

(17)

Both the amount of available downwelling radiation and the albedo difference between the surfzone and inner-shelf affect . As 〈α_sz〉>α_θ, the surfzone has an albedo-induced cooling relative to the inner-shelf. Over the year, hourly is estimated with H_s and 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 θ_s, 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).

3.1 Observed , H_s, F_wave, and Albedo

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

Pier-end significant wave height H_s 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 x_sl. Wave and tide conditions, together with the evolving bathymetry, affected the surfzone width L_sz (Figure 4d). Average L_sz = 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 x_sz was in less than 0.5 m depth (i.e.,L_sz<10, less than 0.2% of all data).

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

The daylight variation of 〈α_sz〉 and α_θ is examined with ensemble averages. Albedo estimates are removed when solar zenith angle is large (|θ_s|> 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 α_θ (blue line, Figure 6) has strong solar zenith angle θ_s dependence, with elevated albedo at low sun angles near sunrise and sunset. Seasonal variation in θ_s and cloud cover variation account for the relatively small α_θ deviation from the mean (blue shaded). As the surfzone has fractional foam coverage, 〈α_sz〉 retains some θ_s dependance, although weaker than α_θ, with elevated 〈α_sz〉 at larger |θ_s| (red line, Figure 6). However, surfzone foam elevates 〈α_sz〉 above α_θ, with midday ensemble averaged 〈α_sz〉 elevated by 0.19 over α_θ. Wave, tide, and bathymetry variability influence 〈ζ〉 and thus contribute to the relatively large 〈α_sz〉 variability (red shaded).

3.2 Competing Wave Effects: and Q_wave

Breaking wave energy dissipation leads to surfzone wave heating Q_wave 7. Wave breaking also increases albedo, thereby reducing the water-entering shortwave solar radiation relative to the inner-shelf by an amount 17. Here these two competing effects are examined. Variability in Q_wave and occur on seasonal, synoptic, diurnal, and semidiurnal time scales through variation in H_s, θ_s, , and L_sz. Here Q_wave and 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 Q_wave always increasing (positive) surfzone heat flux and always reducing (negative) surfzone heat flux relative to the inner-shelf. Over the year, the mean and standard deviation of the daily averaged Q_wave = 28 ± 11 W/m² (red) and  = −41 ± 16 W/m² (blue). Thus, at this location, the combined effect of Q_wave and typically reduced the surfzone heat flux relative to the inner-shelf. Both daily averaged Q_wave and varied on synoptic to seasonal time scales. However, daily averaged Q_wave and were uncorrelated (r² = 0.04) as Q_wave depends on incident H_s (Figure 4c) whereas depends also on clouds and . Throughout most of summer, clouds reduced and waves were small (Figures 4a–4c). Thus, the yearly maximum occurred in April when waves were larger and cloudiness lower, rather than at the summer solstice (21 June) when is maximum.

The relative effects of Q_wave and have a seasonal dependance (Figure 8). In winter, is low and cloudiness can be high reducing . Wintertime waves are also relatively large with Q_wave> 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 Q_wave> 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).

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

3.3 Surfzone Adiabatic Temperature Change

As temperature is relevant to circulation dynamics, cross-shore exchange, and ecology, relating Q_wave and to an adiabatic temperature change is useful for understanding their relative effects. Relative to the inner-shelf, the daily averaged combined surfzone heat flux Q_net is

with positive Q_net implying surfzone warming relative to the inner-shelf. For a planar beach slope, the surfzone daily adiabatic temperature change ΔT induced by Q_net is

(18)

where t_day = 86,400 s is the duration of a day and h_sz 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 Q_net 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 was high, but intermittent clouds or coastal fog caused large changes in . The late fall reduction and overall H_s increase (Figures 4a and 4c) prompted a return to winter conditions. In the adiabatic limit, net surfzone heat flux changes induced by Q_wave and are substantial and can induce significant ( (1 °C)) temperature changes.

4.1 Scaling for an Idealized Surfzone

Parameters affecting surfzone averaged and Q_wave are explored with scalings for a constant slope surfzone, lending insight to potential application at other sites with variable , clouds, incident waves, and beach slope. Although and Q_wave 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 . For an idealized surfzone of constant bathymetric slope s and constant γ, the surfzone averaged foam fraction 〈ζ〉 15 can be related to the nondimensionalized roller dissipation through 13 by surfzone averaging both the numerator and denominator in 12. The surfzone averaged is simply Q_wave=F_wave/L_sz, and for a planar slope the representative (surfzone averaged) h^3/2 becomes . Thus, bulk surfzone nondimensional roller dissipation can be scaled as

(19)

where the outer surfzone boundary depth h_sz=H_sb/γ, H_sb 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

(20)

where m = 398. The surfzone averaged 〈ζ〉 is independent of H_s, yet is linearly related to bathymetric slope s. Thus, the ratio of daily averaged and is expected to be

(21)

where is the constant daily averaged albedo of the inner-shelf. For constant s and γ and daily averaged , the daily averaged and is expected to be linearly related.

The linear relationship between daily averaged and (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 (e.g., Payne, 1972), and foam albedo α_f=0.465 as in section 2.2.4, the surfzone averaged foam fraction 20 applied to 21 yields a theoretical slope (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 H_s variation together with the realistic and variable nonplanar bathymetry. The linear relationship correlation between and (Figure 10) that matches the scaling 21 demonstrate the suitability of 20 and 21 to effectively scale on gently sloping and alongshore uniform beaches.

Next, for a planar slope using 21, the ratio of surfzone daily averaged magnitude to Q_wave is

(22)

where the bracketed quantity is a constant and is independent of bathymetric slope. Thus, the ratio largely depends on . The downwelling solar radiation depends on cloudiness and top of the atmosphere . Daily averaged 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 H_sb can be well modeled (e.g., Ruessink et al., 2003) given incident wave conditions.

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

The scaling for 22 can be used to estimate the relative importance of to Q_wave at other locations with variable latitude and seasonal top of the atmosphere , cloud, beach slope, and wave conditions. At the equator (0°N) seasonal variation in is very small, resulting in a similar clear-sky and H_sb relationship year-round (red solid and dashed curves in Figure 11). At high latitudes, the seasonal difference in is large, expanding the summer to winter difference. At 66°N, the summer clear-sky to H_sb relationship (Figure 11, blue solid ) is nearly the same as at 33°N (the experiment site). However, for wintertime clear skies, for any H_sb > 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 only for H_sb> 1 m. These significant latitude and seasonal differences in clear-sky 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 Q_wave and Albedo-Induced Solar Radiation Reduction in Context

At the La Jolla, CA, experiment site, the parameters H_s, h(x), , , and cloudiness ( ) contribute to the breaking wave-induced positive or negative surfzone heat flux relative to the inner-shelf. Here the two terms Q_wave and are placed in the context of a previous surfzone heat budget. Including wave heating (Q_wave) but not improved a summertime binned mean surfzone heat budget on diurnal and longer time scales (Sinnett & Feddersen, 2014). However, here the summertime was usually greater than |Q_wave| (Figure 8c). However, Q_wave and are uncorrelated (r² < 0.04). Thus, including Q_wave but not 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 L_sz for the same period) required to balance the surfzone heat budget. Here the summer-averaged  = 44 W/m² (compare to the yearly averaged  = 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 Q_wave and 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.