The Sea Surface Heat Flux at a Coastal Site

L. Mahrt aNorthwest Research Associates, Corvallis, Oregon

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Erik Nilsson bDepartment of Earth Sciences, Uppsala University, Uppsala, Sweden

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Anna Rutgersson bDepartment of Earth Sciences, Uppsala University, Uppsala, Sweden

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Abstract

We analyze approximately four years of heat-flux measurements at two levels, profiles of air temperature, and multiple measurements of the water temperature collected at a coastal zone site. Our analysis considers underestimation of the sea surface flux resulting from vertical divergence of the heat flux between the surface and the lowest flux level. We examine simple relationships of the heat flux to the wind speed and stratification and the potential influence of fetch and temperature advection. The fetch ranges from about 4 to near 400 km. For a given wind-direction sector, the transfer coefficient varies only slowly with increasing instability but decreases significantly with increasing stability. The intention here is not to recommend a new parameterization but rather to establish relationships that underlie the bulk formula that could lead to assessments of uncertainty and improvement of the bulk formula.

Significance Statement

The behavior of surface heat fluxes in the coastal zone is normally more complex than over the open ocean but has a large impact on human activity. Our study examines extensive flux measurements on a tower in the Baltic Sea that allows partitioning of the fluxes according to wind direction without seriously depleting the data for a given wind-direction sector. Because some of the normal assumptions for the usual parameterization are not met, our study examines relationships behind the parameterization of the surface fluxes.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: L. Mahrt, mahrt@nwra.com

Abstract

We analyze approximately four years of heat-flux measurements at two levels, profiles of air temperature, and multiple measurements of the water temperature collected at a coastal zone site. Our analysis considers underestimation of the sea surface flux resulting from vertical divergence of the heat flux between the surface and the lowest flux level. We examine simple relationships of the heat flux to the wind speed and stratification and the potential influence of fetch and temperature advection. The fetch ranges from about 4 to near 400 km. For a given wind-direction sector, the transfer coefficient varies only slowly with increasing instability but decreases significantly with increasing stability. The intention here is not to recommend a new parameterization but rather to establish relationships that underlie the bulk formula that could lead to assessments of uncertainty and improvement of the bulk formula.

Significance Statement

The behavior of surface heat fluxes in the coastal zone is normally more complex than over the open ocean but has a large impact on human activity. Our study examines extensive flux measurements on a tower in the Baltic Sea that allows partitioning of the fluxes according to wind direction without seriously depleting the data for a given wind-direction sector. Because some of the normal assumptions for the usual parameterization are not met, our study examines relationships behind the parameterization of the surface fluxes.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: L. Mahrt, mahrt@nwra.com

1. Introduction

In general, the momentum flux over the sea has been examined more extensively than the heat flux, including the comprehensive study of (e.g., Edson et al. 2013). Our study examines the behavior of the surface heat flux. While the increase of transfer coefficient for heat Ch from stable to unstable conditions is generally found to be significant, a constant transfer coefficient can form a simple credible approximation (Blanc 1985; Brunke et al. 2002). Friehe and Schmitt (1976) surveyed a number of papers and proposed transfer coefficients of 0.86 × 10−3 for stable conditions, 0.97 × 10−3 for unstable conditions, and 1.46 × 10−3 for very unstable conditions. A number of studies have formulated Ch in terms of dependence on stability; for example, see Rutgersson et al. (2001) and citations therein.

Liu et al. (1979) revealed a decrease of Ch at larger U caused by wave sheltering where the vertical exchange is weaker in the lee of the wave crest. Veron et al. (2008) found that the direct effect of waves normally enhances the heat flux. Smedman et al. (2007) identified an increase of Ch for U greater than about 11 m s−3. Ch can also show little dependence on U (Grachev et al. 2011). Mahrt et al. (1998) showed enhancement of Ch for very small wave age and significant reduction of Ch with shallow internal boundary layers in offshore flow. Smedman et al. (2007) noted a roughly linear increase of wθ¯/U with increasing air–sea temperature difference δθ, with similar slope for both upward and downward heat flux, where θ is the potential temperature and wθ¯ is the surface heat flux. The comparable slopes in the relationships for upward and downward heat fluxes imply that the change in Ch with the change of the sign of wθ¯/U was not large for their dataset, although the scatter was significantly greater for downward heat flux. The impact of nonstationarity of the wind has been examined primarily in terms of the surface momentum flux (Rieder 1997), which presumably feeds back upon the heat flux. Over tropical oceans with weak winds, a significant fraction of the heat flux can be carried by mesoscale motions (Sun et al. 2006).

Offshore flow leads to internal boundary layers due to sharp changes of temperature at the coast and a sudden decrease of roughness. The vertical variation of the heat flux is large within thin boundary layers, often as internal boundary layers in offshore flow (Fairall et al. 2006). Thin internal boundary layers in a complex fetch-limited environment may not obey similarity theory partly due to rapid decrease of the flux with height, particularly in stable internal boundary layers in warm air over cooler water (Garratt 1987; Garratt and Ryan 1989; Rogers et al. 1995; Vickers et al. 2001; Skyllingstad et al. 2005; Dörenkämper et al. 2015). Offshore flow of colder air over warmer water leads to rapidly deepening internal boundary layers. The influence of greater roughness over land leads to advection of turbulence over the water, which decreases with increasing distance offshore (Sun et al. 2001). Ortiz-Suslow et al. (2021) provides a general analysis of thin coastal-zone boundary layers.

Open ocean sea surface temperature fronts also induce significant horizontal temperature advection for both warm to cold surfaces and cold to warm surfaces (Friehe et al. 1991; Thum et al. 2002; Skyllingstad et al. 2007), although the horizontal temperature gradients are usually small relative to those near coastal boundaries. The edge of tropical cold pools and currents can function as fronts (Small et al. 2008; de Szoeke et al. 2017). Even larger scale mesosynoptic organization of surface temperature on horizontal scales of hundreds of kilometers cannot be generally posed in terms of fronts although tropical warm pools initiate moisture convection that may propagate away in terms of squall lines (Skyllingstad et al. 2019). Even very weak tropical warm pools can initiate converging air (de Szoeke and Maloney 2020). Samelson et al. (2020) examine the flow response to summertime cold water anchored to a coastal upwelling regime.

Our analysis contrasts the behavior of the heat flux for short-fetch offshore flow with longer-fetch flows. We follow a hierarchy of simple relationships to examine the behavior of the surface heat flux posed in terms of the near surface stratification δzθ and wind speed U. Following the usual bulk formula, the surface heat flux wθ¯ is examined as a function of zθ, which reveals several flow regimes. Our study subsequently evaluates Ch as a function of zθ, stability, and wind direction.

2. Measurements

We analyze measurements from the mast at the Östergarnsholm station beginning 17 April 2014 and ending 1 January 2022. The measurements are not continuous and are equivalent to about 3.75 years of data. We focus on the Campbell CSAT sonic anemometers located at 10 and 26 m. Maps of the site include Figs. 1–2 in Rutgersson et al. (2001), Fig. 1 in Smedman et al. (2009), and Fig. 1 in Gutiérrez-Loza et al. (2019). For the current study, see Fig. 1. The observational site is described by Smedman et al. (1999), Rutgersson et al. (2001), Sahlée et al. (2008), Högström et al. (2008), and Rutgersson et al. (2020) and citations therein. The wind-direction dependent fetch varies by two orders of magnitude from about 4 to near 400 km. The potential influence of the local bathymetry and the presence of the low flat island to the north (2 km across) were also discussed. Fluxes measured at the mast compared well to buoy measurements offshore for the open-sea wind directions (e.g., Högström et al. 2008).

Fig. 1.
Fig. 1.

Map of the local site, with inset to show its relative location, generated using Generic Mapping Tools, version 6 (Wessel et al. 2019). The aerial photograph was adapted from https://map.openseamap.org/ and the OpenStreetMap-Project (CC-BY-SA 2.0).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

We divide the wind direction into sectors based on the fetch and bathymetry (Rutgersson et al. 2020). For the northeasterly wind direction (40°–80°), the fetch averages about 220 km. For the southeasterly wind direction (80°–160°), the fetch ranges from 130 to 250 km. The southerly flow sector 160°–220° is the most common direction and the fetch is generally 300–400 km. The fetch is short for the westerly directions, 220°–295°, with values as small as 4 km. The remaining broad northerly sector contains a mixture of land and sea and is excluded.

Fast-response measurements of the fluxes are located at 10, 16, and 26 m, although the 16-m level is missing considerable data and is not included here. Slow-response air temperature measurements are located at 7, 12, 14, 20, and 29 m. The measurement of water temperature is discussed later in this section.

Definitions

The flow variables are decomposed as
ϕ=ϕ+ϕ¯,
where ϕ is the potential temperature or one of the velocity components and ϕ′ is the fluctuation. The heat flux is written as wθ¯. Perturbations are based on deviations from 30-min trends except that fluxes are based on 30-min block averaging without detrending. Fluxes are subjected to an outlier filtering procedure. The wind speed is computed as
Uu¯2+υ¯2.

Some of the analysis is posed in terms of bin averages, symbolized as [ϕ]. Bin averages are plotted only if the bin has 20 points or more. Standard errors are shown for some of the plots. However, the standard error significantly underestimates the actual uncertainty because of lack of independence between adjacent points (nonstationarity).

The vertical difference of potential temperature can be estimated in terms of a within-air temperature difference or the difference between an air temperature and a measure of the surface water temperature. The within-air temperature difference is generally much smaller and thus more erratic than the air–water temperature difference. The upper level of the air temperature may extend above the thin surface layer, as suggested by the vertical divergence of the heat flux examined below.

The current field program includes a number of different measurements of water temperature. The water temperature requires correction for the vertical temperature variation between the level of the water temperature measurement and the sea surface. HOBO sensors recorded water temperature at depths of 0.5, 1, and 4 m for a period of about 2 years. Deeper measurements at 8, 10, and 20 m are available for shorter periods. The temperature difference between the 0.5- and 4-m HOBO measurements averaged 0.26 K. As an example of extreme cases, 0.4% of the values exceed 3 K, which are generally confined to low wind speeds. The Seabird temperature at 5-m depth and the 4-m Sami temperature compared well. We estimate the sea surface temperature as the 5-m Seabird temperature plus a positive adjustment of 0.25 K to account the averaged stratification within the water indicated by the HOBO sensors. This adjustment is marginally significant in the results below.

The Seabird water temperature is located about 1 km to the southeast of the station. The water temperature measurements could be less representative for the westerly and northeasterly flow sectors because the Seabird sensor is well outside of these wind-direction sectors. This issue could be important for small air–sea temperature differences and lead to fictitious countergradient heat fluxes. Such anomalous behavior was not observed for small air–sea temperature differences for westerly flow but was observed for northeasterly flow. The northeasterly flow might be complex partly because the angle between the flow and the eastern coastline of the small flat peninsula is small (Fig. 1). Aerial photographs of the wave field shows that sometimes southerly waves do not make it to the east coast of the small flat peninsula, suggesting the influence of bathymetry.

We now examine the relationship between the heat flux and vertical temperature difference that provides insight into the bias (offset) of the temperature measurements. We first use the vertical difference of temperature between the adjusted 5-m water temperature and the slow-response atmospheric temperature at 7 m to predict the heat flux. The heat flux is predicted with no detectable offset of the heat flux at vanishing air–sea temperature difference (Fig. 2a). A bias might be expected because the vertical temperature difference is based on two different types of instruments, one in the air and one at the 5-m water depth. The lack of even a small offset could be fortuitous.

Fig. 2.
Fig. 2.

The heat flux at 10 m as a function of the vertical difference (a) of temperature between the 5-m water temperature and the slow-response atmospheric temperature at 7 m and (b) of the slow-response temperatures at 7 and 29 m. The red line represents bin averages.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

The heat flux at 10 m as a function of the vertical difference of the slow-response temperature between 7 and 29 m (Fig. 2b) indicates a small temperature offset of only 0.1 K, which is comparable to the expected instrumental error at a single level. A larger offset might have been expected because the 7–29-m layer is probably often above the surface layer, as can be inferred from the vertical variation of the heat flux in the next subsection. The heat flux appears to be more systematically dependent on δzθ based on the water–air temperature difference partly because the within-air temperature differences are much smaller and more difficult to measure.

3. Flux divergence

Because the marine boundary layer is often thin, the vertical divergence of the flux between the 10-m observational level and the sea surface can be important (Fairall et al. 2006). This generally leads to underestimation of the surface flux based on the 10-m flux. Using the 26- and 10-m observational levels, we linearly extrapolate the heat flux down to the surface (Fig. 3) by computing
δzwθ¯C(wθ2¯wθ1¯),
where here levels 1 and 2 are the 10- and 26-m observational levels, respectively; C = 10 m/16 m converts the vertical difference of the heat flux between the observation levels to an estimate for the difference across a 10-m layer using the same linear height dependence. Using such linear extrapolation, the surface heat flux can be estimated from the 10-m value as
wθsfc¯=wθ¯(10 m)Cδzwθ¯,
which corrects wθ¯ at the lowest observational level (10 m) with the estimated change of heat flux between the 10-m level and the surface, Cδzwθ¯. The relative heat-flux divergence
Cδzwθ¯wθ¯(10m)
is the error resulting from estimation of the surface heat flux based on the flux at the 10-m observational level [Eq. (3)], scaled by the 10-m heat flux. Negative values of δzwθ¯ correspond to the usual vertical convergence of the upward heat flux in the unstable boundary layer and the usual divergence of the downward heat flux in the stable boundary. Estimation of the flux divergence from shorter periods that included three levels of fluxes did not provide a clear estimate of the errors expected from downward extrapolation using only two levels of fluxes.
Fig. 3.
Fig. 3.

Estimation of the surface heat flux using downward linear extrapolation from the 10-m level for the case of upward heat flux. The red line identifies the change of flux between the 10- and 26-m observational levels, i.e., δzwθ¯. The red curly brace identifies the flux correction Cδzwθ¯ needed to estimate the surface flux from the 10-m flux.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

We examine the dependence of the flux divergence on the measure of the forcing of the heat flux, defined as zθ. For the long-fetch southerly flow (black line in Fig. 4), δzwθ¯ is positive for positive stratification (positive zθ), corresponding to decreasing downward heat flux with height. The δzwθ¯ is negative for unstable negative stratification corresponding to decreasing upward heat flux with height. Similar results occur for southeasterly flow (magenta) and short-fetch westerly flow (cyan). However, the magnitude of the divergence of the heat flux for westerly flow is significantly larger than that for the southerly and southeasterly flow. For the short-fetch westerly flow, the boundary layer is expected to be thinner, corresponding to greater height dependence of the flux.

Fig. 4.
Fig. 4.

The dependence of δz[wθ¯] on zθ for the four wind direction sectors.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

The δzwθ¯ for northeasterly flow (red) is smaller and more complex. For most of the range of δzwθ¯, the magnitude of the heat flux increases with height and presumably reaches a maximum and then decreases with height toward the boundary layer top. This structure could be forced by height-dependent advection of temperature or other complications (section 2).

The relative heat-flux divergence predicts the percent underestimation of the surface heat flux. We compute the relative heat-flux divergence by averaging measurements for the classes of unstable and stable stratification after discarding data with abs(wθ¯)<0.002Kms1. The results were not very sensitive to the choice of this cutoff value. Because the frequency distribution of zθ weights small magnitudes, averages over positive values of zθ and over negative values of zθ can be significantly smaller than visual estimates from the figures. Averaged over the class of unstable stratification, the relative heat-flux divergence (flux error) is about 17% for the long-fetch southerly flow, 32% for the short-fetched westerly flow, 5% for the southeasterly flow, and small for the northeasterly flow. Averaged over the class of stable stratification, the relative heat-flux divergence is about 14% for the long-fetch southerly flow, 18% for the short-fetched westerly flow and 10% for the southeasterly flow. For northeasterly flow, the downward heat flux increases with height corresponding to 11% overestimation of the surface heat flux. Altogether, the heat-flux divergence and the relative heat-flux divergence are greatest for the short-fetch westerly flow, particularly for upward heat flux. This heat-flux divergence may be associated with a thin internal boundary layer with cold-air advection over the short fetch over water. Normally internal boundary layers are expected to be thinner for flow of warm air over cold water. However, the boundary layer advected from land may not always have sufficiently decayed to permit formation of the stable internal boundary layer.

In general, the correction of the heat flux due to flux divergence below the lowest observational level appears to be significant. However, the accuracy of the estimates of the flux divergence is not known with confidence, and we do not apply such adjustments to the surface flux in our study. Instead, we consider the relative flux divergence as a measure of the uncertainty of the estimation of the surface flux.

4. Bulk formula

The surface heat flux is normally formulated in terms of the bulk formula
wθ¯ChUδzθ,
where δzθ is theoretically estimated by integrating similarity theory from a level in the surface layer and the surface, such that δzθ = θ(z) − θ(0). Technically, θ(0) is the unavailable surface aerodynamic temperature, and θ(0) is generally replaced with the sea surface radiation temperature or the temperature of the water near the surface. Over the sea, θ(z) generally varies more rapidly in time than the surface temperature such that the air temperature generally controls the time dependence of δzθ. Wind speed U plays a dual role through shear generation of turbulent mixing and influencing δzθ through temperature advection.

a. Heat budget

Over the sea where diurnal variations are generally small, advection of temperature can play a major role. The heat budget of the air is expressed as
θ¯t+Vθ¯=wθ¯zuθ¯xυθ¯y+R*,
where θ is potential temperature, V is the horizontal wind vector, u is the horizontal velocity component directed toward the east, υ is the horizontal velocity component directed toward the north, and R* represents other terms such as the clear-air radiative-flux divergence and evaporation or condensation in the air, which will all be neglected. Over the open sea, all of the terms in Eq. (7) are sometimes small and difficult to measure.
Over homogenous land surfaces with strong diurnal variation of surface temperature, we can write
θ¯twθ¯z
such that local cooling or warming is caused completely by the turbulent heat-flux divergence. More appropriate to marine boundary layers, an advective balance is defined for cases with horizontal heterogeneity such that the advection is significantly larger than the local time variations, in which case
Vθwθ¯z.

For unstable conditions, the relative heat-flux divergence is generally negative because the upward heat flux decreases with height (δzwθ¯<0 and wθ¯>0). The heat-flux convergence is roughly balanced by cold-air advection. For stable conditions, the relative heat-flux divergence is also generally negative because the downward heat flux decreases with height (δzwθ¯>0 and wθ¯<0). The heat-flux divergence is roughly balanced by warm-air advection. With near stationary flow over the open sea, the temperature advection and heat-flux divergence normally approximately balance.

Integrating between the surface and some level in the boundary layer h we obtain
sfch(Vθ¯)wθh¯+wθsfc¯.
Using the bulk formula for the surface flux, we obtain
sfch(Vθ¯)wθh¯ChUδzθ,
where U is a near-surface wind speed.
Neglecting the vertical advection and the heat flux at the top of the boundary layer, and expressing the horizontal advection in terms of scaling variables,
hVδxθLxChUδzθ,
where V is the layer-averaged wind speed and δxθ is the horizontal variation of θ in the direction of the wind vector over distance Lx. The surface wind speed U is proportional to the layer-averaged wind speed such that U/V does not vary much. Noting that this is only a scaling analysis, we neglect the difference between the 10-m wind speed and the layer-averaged wind speed, in which case
δzθhδxθChLx.
Thus, δzθ becomes proportional to δxθ, subject to the assumptions above. For example, if the temperature increases along the wind direction (δxθ > 0), cold-air advection normally generates δxθ < 0. Rearranging,
ChhδxθδzθLx.
Thin boundary layers (small h) restrict large eddies, reduce the turbulent transport, and decrease Ch.

b. Dependence of δzθ on U

The relationship of δzθ to U provides a simple description of the overall flow regime based on these two contributions to the bulk formulation. Over land with clear skies, the magnitude of δzθ is generally reduced by mixing and is therefore inversely related to U. Over the sea, δzθ is often maintained by temperature advection. Then δzθ becomes related to the horizontal temperature gradient such that the dependence of δzθ on U becomes obscure, as suggested by Fig. 5 and represented by Eq. (13) above. The upper-right region of Fig. 5a corresponds to large horizontal advection where significant δzθ is evidently maintained at least partly by warm-air advection over a cooler sea surface. The lower-right region of Fig. 5b predicts large horizontal advection where significant δzθ is presumably maintained at least partly by cold-air advection over a warmer sea surface.

Fig. 5.
Fig. 5.

(a) The weak relationship between δzθ and U for (a) stable and (b) unstable stratification for large-fetch southerly flow (160°–220°). Black lines indicate constant forcing zθ (1, 10, and 20 K m s−1) discussed in section 5. Inferences on temperature advection are based on a rough balance between the parameterized vertical heat flux and horizontal advection of temperature, as in Eq. (12).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

The lower-left corner with small δzθ and small U is probably vulnerable to observational errors. The stratification for the other wind-direction groups (not shown) also shows little dependence on wind speed, presumably because temperature advection plays an important role for all of the wind-direction sectors. A curve running from “very stable” to “weakly stable” or “very unstable” to “weakly unstable” describes variation dominated by stability as normally occurs with clear skies over land.

5. Dependence of heat flux and Ch on zθ

We now consider a sequence of relationships that leads to a bulk formula with a stability-dependent transfer coefficient. The bulk formula assumes that the heat flux is related to zθ, which can be considered as the forcing for the heat flux. We now examine the relationship between the actual heat flux and zθ (Fig. 6). Although the right-hand side of this figure generally corresponds to downward heat flux, while the left-hand side corresponds to unstable stratification, zθ itself is not a measure of the stability.

Fig. 6.
Fig. 6.

The dependence of the heat flux on zθ for long-fetch southerly flow. The red line represents bin averages. The cyan line represents prediction of the heat flux based on linear regression where the discrete values are Ch = 0.90 × 10−3 for stable stratification and Ch = 1.10 × 10−3 for unstable stratification. The standard error is indicated by the vertical bars, which underestimate the actual uncertainty because adjacent samples are not independent (nonstationarity).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

Interpretation of Fig. 6 requires recognition that a little more than 90% of the data correspond to abs(zθ) < 20 K m s−1 (farthest-right black lines labeled 20 in Fig. 5) and thus dominate the regression fits in Fig. 6 (cyan lines). Therefore, the actual regression lines (cyan lines in Fig. 6) may appear to disagree with the visual interpretation of the points, particularly for large abs(zθ). The heat flux depends roughly linearly on zθ for stable conditions and also for unstable conditions but with a modestly steeper slope (Fig. 6).

The Ch is proportional to the slope of the dependence of wθ¯ on zθ and therefore in Fig. 6 is larger for unstable conditions. The cyan lines represent a prediction of wθ¯ based on linear regression. These predictions correspond to Ch = 0.90 × 10−3 for the class of stable stratification and Ch = 1.10 × 10−3 for the case of unstable stratification. We refer to this simplified representation of the stability influence as the “discrete” Ch. These discrete values of Ch based on linear regression provide an overall value that captures the trend. The discrete values of Ch are statistically more robust than estimating Ch(z/L) from bin averages because the discrete values are based on all of the data to obtain only two values. Although of uncertain practical value, the discrete values form a basis for interpreting the stability dependence of Ch in section 7. The discrete value of Ch for southerly flow is 20% larger for unstable stratification relative to that for stable stratification. Below, we examine the simple discrete Ch to explore the directional dependence.

a. Variability and large forcing

The variability in Fig. 6 is partly due to variation of stability z/L within each class but also is due to surface roughness, wave state, advection, nonstationarity, and curved upwind trajectories where the wind-direction boundaries are violated. In addition, corrections have not been made for vertical divergence of the heat flux. The scatter is much larger for stable stratification than for unstable stratification mainly due to the dependence of the flux on stratification whereas the effect of stratification for the class of unstable stratification is small (section 6). Similar impact of stability is found for the short-fetch westerly flow (Fig. 7) and the southeasterly and northeasterly wind-direction sectors (not shown).

Fig. 7.
Fig. 7.

The dependence of the wθ¯ on zθ for short-fetch westerly flow. The red line represents bin averages. The cyan line represents a regression prediction of wθ¯ corresponding to Ch = 0.66 × 10−3 for stable stratification and Ch = 1.05 × 10−3 for unstable stratification.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

For southerly flow with stable stratification and large forcing of zθ > 40 K m s−1 (Fig. 6), the standard deviation of the heat flux is particularly large. For unstable stratification and large forcing −zθ > 40 K m s−1, the small quantity of measurements corresponds to much less scatter relative to that for the stable stratification but includes a tendency toward greater upward heat flux in comparison with the regression prediction (cyan). The significance of this deviation is unknown but is due to extreme events that account for less than 1% of the data. The scatter includes countergradient heat flux where the heat flux is of opposite sign to that predicted by the air–sea temperature difference. This has been observed over the sea for events of intermittent significant upward heat flux embedded within general stable stratification and very weak downward fluxes (Mahrt et al. 2012).

b. Variation of Ch with wind direction

The dependence of the heat flux and Ch on zθ appears to vary significantly between wind-direction groups for stable conditions (Fig. 8). The error bars (not shown) indicate significant differences also on the unstable side. However, even with modest nonstationarity, error bars can underestimate the actual sampling errors by more than 100% due to lack of independence between samples (Mahrt and Thomas 2016). For Fig. 8, attempting to correct the estimation of error bars for lack of independence (Wilks 2006) flips some of the classifications from statistically significant to statistically insignificant. Because correction of error bars for lack of independence is an uncertain process in itself, we forego inclusion of error bars for this particular figure and make no formal conclusions on the statistical significance.

Fig. 8.
Fig. 8.

The bin-averaged wθ¯ as a function of zθ for longer-fetch southerly flow 160°–220° (black), short-fetch westerly flow 260°–295° (cyan), northeasterly flow 40°–80° (red), and southeasterly flow 80°–160° (magenta).

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

Alternatively, we have computed values of Ch independently for unstable stratification and for stable stratification using linear regression for each of the wind-direction sectors. The visual interpretation of Fig. 8 for the different wind-direction sectors may differ from the actual linear regression that is dominated by abs(zθ) < 40 K m s−1 where 90% or more of the points are concentrated (see also cyan lines, Fig. 5). For unstable stratification, the discrete value of Ch for short-fetch westerly flow is about the same as that for the southerly flow (1.05 × 10−3) and the northeasterly flow (1.06 × 10−3). For the southeasterly flow, Ch is larger (1.27 × 10−3). The transfer coefficients for stable stratification vary more between wind directions: 0.90 × 10−3 for southerly flow, 0.66 × 10−3 for westerly flow, 0.76 × 10−3 for southeasterly flow, and 0.51 × 10−3 for northeasterly flow. The relatively small value of Ch for stable westerly flow is consistent with the diagnostic relation Eq. (14) where shallow boundary layer depth h prevents large transporting eddies. The Ch for stable conditions for northeasterly flow is particularly small, about half of the magnitude for unstable conditions. Northeasterly flow seems to depart from flow from the other wind-direction sectors in a number of ways, including significant vertical convergence of the heat flux (section 3). A heat-flux offset with respect to vanishing zθ for northeasterly flow (Fig. 8) corresponds to mean countergradient upward heat flux for small zθ, of unknown significance.

6. Impact of within-class variation of stability

We now qualitatively assess the role of stability, z/L, in wθ¯zθ space by dividing both the class of unstable stratification and the class of stable stratification into three subclasses based on threshold values of z/L (Fig. 9). Recall that zθ is not a measure of the stability. As an aside, integrating over positive values zθ for different intervals of z/L provides an estimate of Ch as a function of z/L. Because L for individual records can be erratic, records with u* < 0.1 m s−1 are eliminated for the calculation of L where u* is based on the along-wind component of the momentum flux.

Fig. 9.
Fig. 9.

The heat flux at 10 m related to the forcing zθ for southerly flow for weakly unstable (0 > z/L > −0.2; red dashed), unstable (−1 < z/L < −0.2; black dashed), very unstable (z/L < −1; cyan dashed), very stable (z/L > 1; cyan solid), stable (z/L > 0.2; black solid), and weakly stable (z/L < 0.2; red solid); Ch is proportional to the slope of the wθ¯zθ relationship.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

For casual investigation of the stability dependence, the class of unstable stratification is divided into subclasses of weakly unstable (0 > z/L > −0.2; red dashed), unstable (−1 < z/L < −0.2; black dashed) flow, and very unstable (z/L < −1; cyan dashed). The class of stable stratification is divided into very stable (z/L > 1; cyan solid), stable (z/L > 0.2; black solid), and weakly stable (z/L < 0.2; red solid) conditions. The threshold values of 0.2 and 1.0 are somewhat arbitrary. The maximum downward heat flux occurs at about z/L = 0.05, similar to Fig. 1 in Grachev et al. (2005). More significant stability (z/L > 0.2) generally captures records with small downward heat flux. For very stable (z/L > 1), the magnitude of the downward heat flux reaches very small values (Fig. 9, cyan). A significant fraction of the observations for very stable conditions are countergradient heat flux, evident in Fig. 6.

The composited heat flux for unstable stratification (Fig. 9) is not sensitive to stability as indicated by the subclasses of weakly unstable, unstable, and very unstable flow. For unstable stratification and −zθ < 30 K m s−1, Ch ≈ 1.1 × 10−3 (the slope in Fig. 9). The limited dependence of the heat flux on z/L for the unstable class leads to much less scatter relative to the that for the class of stable stratification.

For the class of stable stratification (Fig. 9), Ch decreases significantly with increasing z/L. For weakly stable subclass (red) for zθ < 30 K m s−1, Ch is approximately constant with about the same value as for the class of unstable stratification. Evidently, the generation of the turbulence in this “extended regime” is not significantly affected by the stability and is primarily shear generated. Presumably, the heat flux does not significantly feedback on the turbulence nor Ch. This includes upward heat flux approaching 0.03 K m s−1 and downward heat flux approaching −0.02 K m s−1. In comparison with the unstable and weakly stable classes, Ch is significantly smaller for the stable class (black) and even smaller for the very stable subclass (cyan).

For stable stratification and zθ < 25 K m s−1, a fit to the bin-averaged quantities predicts Ch = 0.97 × 10−3 for the weakly stable subclass, 0.57 × 10−3for the stable subclass, and about 0.2 × 10−3 for very stable conditions. The organized dependence of wθ¯ on z/L and zθ breaks down for zθ > 25 K m s−1 where the scatter becomes large (Fig. 6) and the dependence of wθ¯ on zθ becomes more irregular. Apparently, large warm-air advection prevents δzθ from decreasing with high winds (Fig. 4).

The behavior of the heat flux for the short-fetch westerly flow (Fig. 10) is similar to that for the southerly long-fetch flow. A fit to the bin-averaged quantities predicts Ch = 0.80 × 10−3 for the weakly stable class, 0.55 × 10−3 for the stable class, and 0.27 × 10−3 for the very stable class. Again, the behavior becomes more erratic for downward heat flux and large zθ. The “constant” Ch regime for downward heat flux extends to about zθ = 15 K m s−1 for the two easterly wind sectors (not shown) where high wind speeds are less common.

Fig. 10.
Fig. 10.

As in Fig. 9, but for westerly flow.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

Relating wθ¯ to z/L involves the heat flux as a shared variable, and interpretation must be viewed with caution. For a fixed value of zθ and downward heat flux, this self-correlation corresponds to smaller z/L with smaller downward heat flux, opposite to what is observed, and thus precludes important self-correlation. For unstable stratification, the impact of z/L on the heat flux is very small, precluding large self-correlation. The analyses in wθ¯zθ space are less sensitive to these issues. Mahrt and Hristov (2017) found that the systematic dependence of Ch on the surface layer Richardson number was partly due to self-correlation resulting from the shared variables of U and δzθ.

7. Dependence of Ch on stability

Figure 11 provides an estimate of the dependence of Ch on z/L for the different wind-direction sectors. Here, the data are partitioned according to the heat flux and not according to the stratification. The heat flux and zθ are bin-averaged over intervals of z/L, and then Ch is computed for each bin. Directly compositing Ch is not as well behaved because individual values of Ch includes outliers, particularly with very small values of wind speed or δzθ. Because of this multistep process of computing Ch, the corresponding standard error is more complex and is without direct theoretical support.

Results in Fig. 11 more or less agree with values of Ch inferred from Figs. 67 and 910 except that Ch is significantly larger for upward heat flux with long-fetch southerly flow (Fig. 11, black). While all of the analyses indicate generally larger Ch for long-fetch southerly flow relative to short-fetch westerly flow, it is not known if the significantly larger values of Ch for upward heat flux in southerly flow (Fig. 11) are completely representative. Most of the data for upward heat flux with southerly flow are for small −z/L, so the visual assessment of Fig. 11 overestimates the typical value of Ch whereas the estimate of Ch based on the regression (Fig. 6) weights all of the data points equally. The results also depend on the elimination of outliers and the influence of self-correlation between shared variables. The bin averaging depends on the order of the averaging, the bin width, and the required number of points per bin. The weak dependence of Ch on z/L for upward heat flux, and more significant dependence on z/L for downward heat flux (Fig. 11), agrees with the general findings in Figs. 910.

Fig. 11.
Fig. 11.

The dependence of Ch on z/L for the long-fetch southerly flow (black; 160°–220°), the short-fetch westerly flow (cyan; 260°–295° and relatively long-fetch southeasterly flow (magenta; 80°–160°). The red T indicates a transition at z/L = 0.2.

Citation: Journal of Physical Oceanography 52, 12; 10.1175/JPO-D-22-0094.1

The larger values of Ch for southerly flow relative to westerly flow (Fig. 11) may also be due to a number of features of the westerly flow. Because the vertical divergence of the upward heat flux is large for westerly flow (Fig. 4), the heat flux and Ch might be significantly underestimated for westerly flow. The smaller values of Ch for short-fetch westerly flow relative to the long-fetch southerly flow might also be due to less developed wave state (Veron et al. 2008). In addition, large eddies are suppressed in thin internal boundary layers, which reduces Ch (Mahrt et al. 1998). The short-fetch westerly flow includes a few cases of δzθ > 10 K and δzθ < −10 K probably associated with large horizontal temperature advection. Large positive δzθ is augmented by summertime cold upwelling east of the large island of Gotland (Sproson and Sahlée 2014), which can extend eastward past the Östergarnsholm site (Fig. 1). However, in general, horizontal gradients of water temperature in the Baltic Sea appear to be relatively small, particularly away from the shorelines. The dependence of Ch on z/L is more erratic for the northeasterly flow, partly because of countergradient heat flux of uncertain reliability, and is not included in Fig. 11.

The Ch makes a transition to significantly smaller values as z/L increases to values greater than about 0.2, which can be envisioned as the transition between weakly stable conditions and very stable conditions. This transition is larger than the change of Ch across neutral stability. As z/L exceeds 0.2, the transfer coefficient begins to decrease more slowly with further increases of z/L. Increasing the resolution in z/L shifts the transition toward z/L = 0, reduces the change of Ch across z/L = 0 to a smaller value, and incurs additional noise in Ch(z/L).

We summarize the uncertainty and variability of Ch by collecting statistics from the above materials (Table 1). The relative heat-flux divergence is largest for westerly flow, particularly for unstable conditions where it reaches about 32%. The relative divergence is significant for both stability categories for southerly and westerly flow where the surface fluxes appear to be significantly underestimated. Coefficient Ch based on the regression approach varies between wind-direction groups by about 20% for unstable conditions and about 25% for stable conditions. The regression estimate of Ch is about 40% greater for unstable conditions than for stable conditions.

Table 1

Estimates of relative vertical divergence of the heat flux [Eq. (5)] and Ch for unstable conditions and stable conditions (in parentheses) based on the regression approach and based on z/L using bin averaging. The Ch are in units of 10−3.

Table 1

To assess the variation of Ch(z/L), we evaluate the approximate value of Ch for z/L = 0.4 for stable conditions and z/L = −0.4 for unstable conditions. These two values of z/L provide most of the range of Ch without relying on the largest values of z/L where the data is sparse. The resulting values of Ch vary between wind-direction groups by about 35% for unstable conditions and about 20% for stable conditions. The values of Ch(z/L) are about 50% greater for unstable conditions than for stable conditions. The dependence of Ch on z/L seems to be generally systematic even though Ch depends on the method of calculation (Table 1). The dependence on wind direction (fetch) is also evident.

8. Conclusions

In contrast to typical fair-weather boundary layers over land, the vertical temperature difference over the sea δzθ is often maintained by advection of temperature, even with high wind speeds and associated mixing. Over land with high wind speeds, significant δzθ is often eliminated by the mixing. With these differences in mind, we examined the dependence of the heat flux over the sea on wind speed, direction (fetch), δzθ or z/L, and method of analysis (sections 57).

The sea surface heat flux was found to be a strong function of the forcing, zθ, as assumed in the bulk formula, but Ch revealed some unexpected tendencies. For unstable stratification, the upward heat flux could be roughly approximated by a constant Ch for this data, corresponding to constant slope for the dependence of wθ¯ on zθ. Thus, Ch did not depend significantly on z/L, particularly for the long-fetch southerly winds. For the short-fetch westerly flow and unstable stratification, Ch depends only weakly on z/L. The Ch for stable stratification decreases significantly with increasing stability z/L for all of the wind-direction sectors. For zθ > 25 K m s−1, the dependence of the downward heat flux on the stratification is less organized, and large values of zθ are maintained for large U, presumably due to important warm-air advection.

Based on bin-averaged values, Ch decreases most rapidly with increasing z/L as z/L increases beyond about 0.2, which can be envisioned as the transition of Ch between weakly stable conditions and very stable conditions. The transition across neutral stability (z/L = 0) is smaller. The dependence of the heat flux on zθ and the behavior of Ch varied significantly between wind-direction sectors. The Ch was generally smaller for short-fetch conditions, probably due partly to suppression of large eddies by thin internal boundary layers. The interrelated fetch, surface roughness, wave state, temperature advection, and the depth of any internal boundary layers are difficult to isolate.

Our long dataset allows an estimate of vertical divergence of heat flux near the surface, here projected onto a 10-m layer. This projection predicts a significant underestimation of the surface heat flux of 15%–30%, being greatest for short-fetch unstable flow and smallest for long-fetch unstable flow. We did not apply these corrections for our default analysis, but the corrections provide an estimate of the potential underestimation of Ch.

Because bin averaging hides important physics, case studies are needed, particularly for short-fetch cases. More flux levels would improve estimation of the surface heat flux based on downward vertical extrapolation. Improved parameterization would also benefit from measurements at additional locations provided that a relatively long dataset can be obtained. Ultimately, comparisons between the measurements and boundary layer models are needed.

Acknowledgments.

The very helpful comments of the two reviewers are gratefully acknowledged. Author Larry Mahrt is funded by Grant N00014-19-1-2469 from the U.S. Office of Naval Research. The Östergarnsholm station is funded by ICOS (Swedish Research Council; Grant 2015-06020 and Uppsala University). Authors Erik Nilsson and Anna Rutgersson were partly funded by the Swedish Research Council (Grant 2015-06020). The authors are grateful for the work by the research engineers, researchers, and Ph.D. students and their technical efforts in making measurements at the Östergarnsholm site possible. Emily Moynihan at BlytheVisual, LLC, provided Fig. 3.

Data availability statement.

The data are not yet available for general use.

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  • Blanc, T. V., 1985: Variation of bulk-derived surface flux, stability and roughness results due to the use of different transfer coefficient regimes. J. Phys. Oceanogr., 15, 650669, https://doi.org/10.1175/1520-0485(1985)015<0650:VOBDSF>2.0.CO;2.

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  • Brunke, M. A., X. Zeng, and S. Anderson, 2002: Uncertainties in sea surface turbulent flux algorithms and data sets. J. Geophys. Res., 107, 3141, https://doi.org/10.1029/2001JC000992.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • de Szoeke, S. P., and E. D. Maloney, 2020: Atmospheric mixed layer convergence from observed MJO sea surface temperature anomalies. J. Climate, 33, 547558, https://doi.org/10.1175/JCLI-D-19-0351.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • de Szoeke, S. P., E. D. Skyllingstad, P. Zuidema, and A. S. Chandra, 2017: Cold pools and their influence on the tropical marine boundary layer. J. Atmos. Sci., 74, 11491168, https://doi.org/10.1175/JAS-D-16-0264.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dörenkämper, M., M. Optis, A. Monahan, and G. Steinfeld, 2015: On the offshore advection of boundary-layer structures and the influence on offshore wind conditions. Bound.-Layer Meteor., 155, 459482, https://doi.org/10.1007/s10546-015-0008-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Edson, J. B., and Coauthors, 2013: On the exchange of momentum over the open ocean. J. Phys. Oceanogr., 43, 15891610, https://doi.org/10.1175/JPO-D-12-0173.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., and Coauthors, 2006: Turbulent bulk transfer coefficients and ozone deposition velocity in the international consortium for atmospheric research into transport and transformation. J. Geophys. Res., 111, D23S20, https://doi.org/10.1029/2006JD007597.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Friehe, C. A., and K. F. Schmitt, 1976: Parameterization of air–sea interfacial fluxes of sensible heat and moisture by the bulk aerodynamic formulas. J. Phys. Oceanogr., 6, 801809, https://doi.org/10.1175/1520-0485(1976)006<0801:POASIF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Friehe, C. A., and Coauthors, 1991: Air–sea fluxes and surface layer turbulence around a sea-surface temperature front. J. Geophys. Res., 96, 85938609, https://doi.org/10.1029/90JC02062.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., 1987: The stably stratified internal boundary layer for steady and diurnally varying offshore flow. Bound.-Layer Meteor., 38, 369394, https://doi.org/10.1007/BF00120853.

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  • Fig. 1.

    Map of the local site, with inset to show its relative location, generated using Generic Mapping Tools, version 6 (Wessel et al. 2019). The aerial photograph was adapted from https://map.openseamap.org/ and the OpenStreetMap-Project (CC-BY-SA 2.0).

  • Fig. 2.

    The heat flux at 10 m as a function of the vertical difference (a) of temperature between the 5-m water temperature and the slow-response atmospheric temperature at 7 m and (b) of the slow-response temperatures at 7 and 29 m. The red line represents bin averages.

  • Fig. 3.

    Estimation of the surface heat flux using downward linear extrapolation from the 10-m level for the case of upward heat flux. The red line identifies the change of flux between the 10- and 26-m observational levels, i.e., δzwθ¯. The red curly brace identifies the flux correction Cδzwθ¯ needed to estimate the surface flux from the 10-m flux.

  • Fig. 4.

    The dependence of δz[wθ¯] on zθ for the four wind direction sectors.

  • Fig. 5.

    (a) The weak relationship between δzθ and U for (a) stable and (b) unstable stratification for large-fetch southerly flow (160°–220°). Black lines indicate constant forcing zθ (1, 10, and 20 K m s−1) discussed in section 5. Inferences on temperature advection are based on a rough balance between the parameterized vertical heat flux and horizontal advection of temperature, as in Eq. (12).

  • Fig. 6.

    The dependence of the heat flux on zθ for long-fetch southerly flow. The red line represents bin averages. The cyan line represents prediction of the heat flux based on linear regression where the discrete values are Ch = 0.90 × 10−3 for stable stratification and Ch = 1.10 × 10−3 for unstable stratification. The standard error is indicated by the vertical bars, which underestimate the actual uncertainty because adjacent samples are not independent (nonstationarity).

  • Fig. 7.

    The dependence of the wθ¯ on zθ for short-fetch westerly flow. The red line represents bin averages. The cyan line represents a regression prediction of wθ¯ corresponding to Ch = 0.66 × 10−3 for stable stratification and Ch = 1.05 × 10−3 for unstable stratification.

  • Fig. 8.

    The bin-averaged wθ¯ as a function of zθ for longer-fetch southerly flow 160°–220° (black), short-fetch westerly flow 260°–295° (cyan), northeasterly flow 40°–80° (red), and southeasterly flow 80°–160° (magenta).

  • Fig. 9.

    The heat flux at 10 m related to the forcing zθ for southerly flow for weakly unstable (0 > z/L > −0.2; red dashed), unstable (−1 < z/L < −0.2; black dashed), very unstable (z/L < −1; cyan dashed), very stable (z/L > 1; cyan solid), stable (z/L > 0.2; black solid), and weakly stable (z/L < 0.2; red solid); Ch is proportional to the slope of the wθ¯zθ relationship.

  • Fig. 10.

    As in Fig. 9, but for westerly flow.

  • Fig. 11.

    The dependence of Ch on z/L for the long-fetch southerly flow (black; 160°–220°), the short-fetch westerly flow (cyan; 260°–295° and relatively long-fetch southeasterly flow (magenta; 80°–160°). The red T indicates a transition at z/L = 0.2.

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