Observations of the heat flux over the open sea and in the coastal zone are analyzed to reexamine the relation of the heat flux to the air–sea temperature difference and wind speed. The study begins by examining problems with different methods for estimating the air–sea temperature difference. The difference between the air and within-water temperature is found to be most suitable for one dataset, while the difference between the air and the radiometrically measured sea surface temperature must be used for the second dataset. On average, the heat flux is linearly proportional to the product of the air–sea temperature difference and wind speed corresponding to approximately constant transfer coefficient for heat CH. Deviations of the heat flux from this simple relationship were generally weakly related or unrelated to the surface bulk Richardson number Rb, even for the coastal zone site. Similar results are also found for the moisture flux. In contrast to the general success of constant transfer coefficient for heat, CH plotted directly as a function of Rb is systematically related to the square root of Rb. The role of the wind speed as a shared variable between CH and Rb also predicts such a square root dependence, suggesting that the relationship could be largely due to self-correlation, which is also supported by a nominal study of self-correlation. However, confident isolation of the influences of stability, self-correlation, uncertainties in the air–sea temperature difference, and other physics requires more extensive data.
The literature has largely excluded measurements of small values of the air–sea temperature difference for prediction of the surface heat flux because of suspected important observational errors and perceived ill-defined behavior in the relationship between the surface heat flux and small values of the air–sea temperature difference. Exceptions include Smedman et al. (2007), who were able to determine the air–sea temperature difference down to small values on the order of 0.5 K or less by using air temperature sensors with an accuracy of 0.02 K located within a vertically mixed ocean surface layer.
Sometimes, upward heat flux is observed with small positive difference between the air temperature and the radiometrically measured sea surface temperature, implying either countergradient transfer or measurement errors. Application of the bulk formula requires gradient transfer. Mahrt et al. (2012) inferentially attributed at least some of the observed countergradient transfer to variations of the sea surface temperature within the averaging area for aircraft data. For small air–sea temperature differences, the area-averaged air–sea temperature difference may be dominated by stably stratified parts of the averaging area, while the heat flux may be dominated by unstable parts of the averaging area where the turbulence is a little stronger. Analogous countergradient transfer can result from variations within the time-averaging window applied to measurements from fixed site such as offshore towers. Microscale variations of surface temperature can be induced by patches of foam that are observed to be cooler than the surrounding water surface (Marmorino and Smith 2005). Microscale wave breaking at moderate and strong wind speeds can disrupt the cool skin layer and produce micropatches of relatively warm surface water (Zappa et al. 2004). Sea spray may also lead to countergradient transfer of sensible heat for stronger wind speeds (Andreas 2011).
Previous studies of the sea surface sensible heat flux have generally excluded countergradient heat fluxes by 1) eliminating them as apparent observational errors (e.g., Zeng et al. 1998; Kalogiros and Wang 2011), 2) imposing conditions on the minimum magnitude of the air–sea temperature difference such as 1 K or greater (e.g., Larsén et al. 2004), or 3) adjusting the radiatively measured surface temperature to reduce the frequency of countergradient cases (e.g., Vickers and Mahrt 2006). Removal of countergradient cases or removal of small air–sea temperature differences may be questionable because any contribution of biases in those measurements will also affect the data not removed. The choice of data removal method influences the behavior of the transfer coefficient for heat in the transition between stable and unstable conditions. This problem is of interest because much of the atmospheric surface layer over the ocean is near neutral.
While the increase of transfer coefficient for heat CH from stable to unstable conditions is thought to be significant, a constant transfer coefficient might 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. Smedman et al. (2007) found a roughly linear increase of /V with increasing air–sea temperature difference δθ with similar slope for both upward and downward heat flux, where θ is the potential temperature, is the surface heat flux, and V is the wind speed. The comparable slopes in the relationships for upward and downward heat fluxes imply that the change in CH with change of the sign of the stability was not large for their dataset although the scatter was significantly greater for stable conditions.
Liu et al. (1979) found a decrease of CH at larger V due to wave sheltering where the vertical exchange is weaker in the lee of the wave crest. In contrast, Smedman et al. (2007) found an increase of CH for V greater than about 11 m s−3. Because large V can be associated with small air–sea temperature difference, the accuracy of the measurement of air–sea temperature difference becomes critical. Veron et al. (2008) found that the direct effect of waves normally enhances the heat flux. This effect is expected to contribute to the dependence of CH on V.
For Monin–Obukhov similarity theory, CH is closely related to the stability z/L, suggesting a large influence of stability. However, L is a function of itself so that a significant relationship between and z/L can result from the shared variable . Our goal here is to relate the turbulence exclusively to the mean flow rather than relating turbulence quantities to other turbulence quantities, as in Monin–Obukhov similarity theory. The heat flux can be related to the mean flow alone by expressing the stability in terms of the surface bulk Richardson number (Louis 1979) instead of relating the heat flux to a stability function that is a function of the heat flux. Such an approach is easier to interpret because it relates the turbulent flux exclusively to the mean flow. However, relating CH directly to the Richardson number is still influenced by shared variables, which are examined further in our study.
A significant influence of stability on the turbulence over the sea is most likely with the flow of cold air from land over warmer water or flow of warmer air from land over cooler water. In these cases, stratification is maintained by horizontal advection. Isolation of the impact of stability on the fluxes is sometimes difficult due to the impact of growing, steep, young waves, shoaling waves, nonequilibrium turbulence, and thin internal boundary layers in a complex fetch-limited environment where similarity theory may or may not be valid (e.g., Garratt 1987; Garratt and Ryan 1989; Rogers et al. 1995; Vickers et al. 2001; Sun et al. 2001; Dörenkämper et al. 2015). With flow of warm air over cooler water, the stable boundary layer can be sufficiently thin that the surface layer, if fully developed, lies below the lowest observational level (Fairall et al. 2006; Mahrt et al. 2016). Open-ocean sea surface temperature fronts also induce significant horizontal temperature advection and air–sea temperature differences (Friehe et al. 1991; Thum et al. 2002; Mahrt et al. 2004; Small et al. 2008), although the horizontal sea surface temperature contrasts and associated atmospheric stability or instability do not generally reach the values found at coastal boundaries.
Counter to the ever increasing complexity of formulations of surface fluxes over the sea, Andreas et al. (2012) and Vickers et al. (2015) found that the friction velocity could be estimated from a function of wind speed alone without significant loss in parameterization skill. Mahrt et al. (2016) found that even in the coastal zone was not obviously dependent on the air–sea temperature difference but was instead augmented by small nonstationary submeso motions. Rieder and Smith (1998) revealed the potential impact of nonstationarity of the wind on the drag coefficient. It is not known if these findings carry over to the heat transfer.
Our study investigates the potential impact of stability on the surface heat fluxes over the sea by first studying the basic relationship between the heat flux and the product Vδθ without assuming any similarity theory. The analysis will be organized by the framework in Fig. 1. Important uncertainties in different estimates of δθ are discussed in detail in section 3. Relating to Vδθ and the subsequent assessment of the dependence on stability is examined in sections 4 and 5.The importance of advection and impact of limited fetch are briefly addressed in section 5. The role of self-correlation due to shared variables is assessed in section 6.
The Roughness and Radiation Duct experiment (RED) provides the primary measurements for our study (Anderson et al. 2004; Högström et al. 2013). The observations were taken between 2 and 15 September 2001 in the trade wind regime approximately 10 km upwind from the coast of Oahu, Hawaii, United States. Although this short dataset is limited primarily to unstable conditions with moderate winds speeds, the Floating Instrument Platform (FLIP) profiles of temperature during RED allow analyses not typically possible over the open ocean. Our study analyzes 50-Hz measurements of the velocity components from Campbell Scientific CSAT3 sonic anemometers deployed at 5.1, 6.9, 9.9, and 13.8 m. We primarily use 10-min-averaged values computed from the 9.9-m sonic anemometer, although the heat flux, averaged over the field program, varied by less than 3% between 6.9- and 13.8-m levels. Mean temperature was measured with Hart thermistors at 6.9, 9.9, 13.8, 16.8, and 18.8 m. Specific humidity was computed from dewpoint temperatures measured at 6.9, 9.9, 13.8, and 16.8 m. The water surface elevation was measured with four wave-staffs [Scripps Institution of Oceanography (SIO) WaveWire]. The wind speed at 10 m was typically 6–8 m s−1, while the wind direction was typically northeasterly to easterly. A Hart model 1529 thermistor measured and recorded water temperature at approximately 1.5-m depth. The moisture flux was computed from moisture fluctuations measured by the Licor 7500 at 9.9 m.
All but 11 of the 1670 10-min values of are upward. The downward values are retained even though they may be counter to the vertical gradient of . Excluding such values as unphysical may convert random error to systematic error in that it would eliminate random variations of one sign but not the other sign. Even though such values are included in the determination of relationships, they are off scale in some plots.
Observational errors such as solar radiational warming of the sensors due to incomplete shielding and distortion of temperature and wind profiles by the FLIP superstructure are both thought to be small for the temperature levels used here. However, their significant influence cannot be ruled out. Most of the data that were removed by simple quality control procedures were obviously erroneous values that occurred during the only day when precipitation occurred.
b. ASIT and CBLAST low wind
We also analyze data from the Air–Sea Interaction Tower (ASIT) collected during the CBLAST low-wind experiment in late summer of 2003 (Edson et al. 2007). The offshore tower is located 3 km south of Martha’s Vineyard in 15 m of water. We analyze 20-Hz turbulence measurements collected using a CSAT3 sonic anemometer at approximately 6 m above the mean sea surface to calculate heat fluxes. We use the lowest level of sonic anemometer measurements because of thin boundary layer depths for stable conditions. Slow-response measurements of temperature (Vaisala HMP 35A sensors) at 7 m are used for the mean temperature. The sea surface temperature (SST) was measured by a Heitronics KT15 radiometer. An Eppley precision infrared radiometer (PIR) infrared flux sensor was used to correct for sky reflections. In addition to the nominal quality control that eliminated obviously impossible values, data with wind direction between 0° and 120° were eliminated to reduce the effects of flow distortion by the tower. We do not convert the measurements to 10-m values because we wish to avoid assuming the validity of similarity relationships.
The flow is partitioned as
where ϕ is potential temperature or specific humidity, is the average over the averaging time τ, and ϕ′ is the deviation from such an average. The vertical heat flux for a given averaging window is then computed as . For RED, turbulent fluctuations are defined as deviations from 10-min averages after linearly detrending each 10-min window. Because the ASIT data include significantly stable cases, a smaller averaging window of 1 min is used to reduce contamination of the computed fluctuations by nonturbulent motions. The 1-min averaging begins to exclude detectable flux for stronger winds where transporting eddies become larger and more elongated in the wind direction. However, based on longer averaging times such as 1 h, this flux loss did not systematically affect the relationship between , V, and δθ. Wind speed was computed from the wind components averaged over the averaging windows. The term θ of the air is defined with respect to the sea surface using the dry adiabatic lapse such that θ = T + 0.01 K m−1 × Z, where Z is the height of the sensor above the mean sea surface.
Because the scatter can be large with offscale outliers, the measurements of the heat flux are also bin averaged for intervals of some independent variable such as the air–sea temperature difference. Here, bin averaging requires at least 20 data points for each interval that is included. Standard errors are reported although they are likely an underestimation of the uncertainty because adjacent flux calculations are not completely independent. Correction for such dependence (Mahrt and Thomas 2016) is not attempted.
3. The vertical difference of potential temperature, δθ
a. Measurement of the surface temperature
Formally, Monin–Obukhov similarity theory and the subsequently derived similarity theory based on the surface bulk Richardson number (Louis 1979) relate the surface heat flux to the difference between the air temperature in the surface layer and the surface aerodynamic temperature. The latter is generally not directly available and often is replaced with SST, which requires recalibration of the similarity theory. The SST is generally radiometrically measured θSST. While the radiometrically measured surface temperature over land often varies erratically on microscales and is not very well posed, the surface radiation temperature over the water is better defined. Even then, wave breaking and foam can lead to significant local variation of the surface temperature. Variations of emissivity due to wave breaking, foam, and general roughness of the sea surface (Katsaros 1980) and upward reflection of the downward longwave radiation can also contaminate the radiometrically measured surface temperature. Finally, calibration errors of the radiometer itself may be significant and drift with time.
Measurement of the within-water temperature avoids these complications but may be different from the surface temperature due to stratification between the surface and the level of the water temperature measurement, particularly with weaker wind conditions. This stratification may include concentrated vertical temperature gradients at the surface (Fairall et al. 1996; cool skin effects). In addition, within-water stable stratification is often generated by warm-layer effects associated with absorption of solar radiation in the near-surface water (Fairall et al. 1996). Donlon et al. (2002) document significant differences between the radiometrically measured surface temperature and the in-depth water temperature as a function of wind speed and time of day. Nonetheless, the within-water temperature can become a valuable estimate of the true sea surface temperature (Smedman et al. 2007).
b. Three versions of δθ
For the RED measurements, the water temperature varies diurnally with an amplitude of typically 0.3 K (not shown). The diurnal variation of the air temperature is a little greater but not as well defined. The diurnal variation of the radiometrically measured sea surface temperature is the smallest and least well defined. For the RED measurements, we define three measures of the vertical difference of potential temperature δθ for prediction of the sensible heat flux, each with its own source of errors. The air–sea temperature difference is estimated from the radiometrically measured sea surface temperature θSST as
A bulk measure is defined as
where θwater is the temperature measured within the water used as an estimate of the true SST.
The vertical difference of the potential temperature within the air at the RED site is computed from the measurements at 18.9 and 6.9 m as
Although use of the surface water temperature in the bulk formula is physically more ambiguous than use of the within-air vertical temperature difference for prediction of the surface heat flux (Sun and French 2016), nearly all applications of the bulk formula over the sea use the air–sea temperature difference.
For analogous examination of the moisture flux, δqbulk is computed from the RED measurements by using the specific humidity at 9.9 m and using the saturation specific humidity at the surface computed from the sea surface temperature. For the ASIT measurements, only δθSST is computed, which uses the 9-m potential temperature and the radiometrically measured sea surface temperature.
c. Thermal structure and dependence on V
For the RED measurements, the air temperature θair increases with increasing V (Fig. 2a) possibly due to greater heat flux from the sea surface and/or greater downward mixing of warm air from the boundary layer top. The radiometrically measured sea surface temperature θSST is cooler than the water temperature θwater, implying measurement errors or unstable stratification of the water.
The differences between θ(9.9), θSST, and θwater all decrease with increasing V (Fig. 2). The |δθair| [Eq. (4)] increases by about 30% as V increases from 3.5 to 9.5 m s−1 (Fig. 2b), possibly due to the influence of increasing cold air advection by the northeasterly trade winds over warmer water with increasing V. A completely self-consistent physical explanation of Fig. 2 cannot be constructed. We now examine the relationship between the heat flux and the three different measures of the vertical temperature difference.
d. The – relationship
Although tends to vary linearly with all three measures of δθ (Fig. 3), appears to remain upward with vanishing δθ for all three measures of δθ. This suggests that the measured δθ might include instrumental offsets and might be affected by physical processes that can lead to heat flux of the sign opposite to that predicted by δθ (countergradient flux; see introduction).
The bulk formula requires that vanishes with vanishing δθ, by definition. Without adjustment of δθ, the transfer coefficient for becomes negative for small δθ. We apply adjustments to δθ. The relationship of to δθ depends on V because generally increases with increasing V, as is evident by comparing the black points with the red points in Fig. 3. However, attempts to formulate the offset correction as a function of V might inadvertently capture some of the influence of stability on the heat flux.
The relationship of to δθbulk (Fig. 3b) has less offset and more regular behavior for small |δθbulk| compared to the use of δθSST based on the radiometrically measured surface temperature. The water is the reservoir of heat, whereas the skin temperature has little heat capacity and is more erratic. Extrapolating the least squares fit to the bin averages of for intervals of δθbulk using all of the data yields an offset of δθbulk at zero of 0.87 K. For subsequent analyses, δθbulk is reduced by 0.87 K.
Because the needed adjustment for δθbulk is smaller than that for δθSST and the relationship of to the unadjusted δθbulk includes no countergradient examples, our investigations below will use δθbulk for examination of the dependence of on Vδθ in the context of the bulk formula. That is, we place more confidence in the water temperatures compared to the complex radiometrically measured surface temperature. Although similar qualitative conclusions on the dependence of on Rb are reached with use of δθSST, numerical differences are not negligible. Different methods for determining the offset also causes nonnegligible differences in the dependence of on Rb. The estimation of δθ includes a degree of uncertainty that is revealed only with extensive exploration of different approaches for determining the offset.
Offsets in the relationship between and δθ were estimated by first bin averaging the data for intervals of δθ to reduce the role of outliers and then applying a least squares fit (green line, Fig. 3a). Extrapolating the fit to zero heat flux indicates that vanishes for δθSST ≈ 1.2 K. For the subsequent analyses, δθSST is adjusted by subtracting 1.2 K.
e. Within-air δθair
The within-air δθair is more coupled to variability of the turbulence compared to δθ based on the SST (Sun and French 2016). Because δθair is not generally used in the bulk formula, no adjustments will be made to δθair. Approximating in terms of a linear dependence on δθair (green line in Fig. 3c) underestimates the upward heat flux for the relatively small number of points corresponding to the smallest magnitudes of δθair. As with the other two unadjusted measures of δθ (Figs. 3a,b), upward heat flux is implied with vanishing δθ in the absence of any adjustment of δθ.
Because is reasonably well-correlated to even very small within-air vertical temperature differences (Fig. 3c), we conclude that the measurements of air temperature are quite accurate compared to the radiometrically measured SST and the measurement of the within-water temperature. If so, the above adjustments of δθSST and δθbulk are probably adjusting primarily for issues with the water temperatures, not air temperature.
Even though our study examines the relationship between the sensible heat flux and the vertical difference of the potential temperature, stability must be expressed in terms of the difference of virtual potential temperature to include the contribution of moisture to the buoyancy. For the RED data, the moisture flux contributes to about 30% of the buoyancy flux, which in turn generates turbulence for unstable conditions. At the same time, the contribution of the moisture content to the stability was small, but we include it for completeness. Based on the virtual potential temperature difference δθυ, we define a bulk Richardson number
where δθυ is computed from δθbulk for the RED measurements and δθSST for the ASIT measurements, and the specific humidity at the water surface is assumed to be at the saturation value without corrections for salinity or surfactants.
Because Rb can take on quite large values with large positive skewness, plots have often been made in terms of the log of Rb. We use , which serves a similar purpose, and the analysis of observations (section 4) indicate a roughly linear dependence of the heat flux and CH on . For unstable cases where Rb < 0, we compute , which retains the negative sign indicating instability.
4. Heat flux dependence
The heat flux based on the RED measurements tends to be linearly proportional to Vδθbulk (Fig. 4a). The green line in Fig. 4a is a reference line corresponding to CH = 10−3, which underestimates heat flux by about 5% except for the smallest magnitudes of Vδθbulk, where the underestimation is larger. It is not known if the underestimation is significant compared to the uncertainties in estimating the air–sea temperature difference.
a. Dependence on stability
The scatter in Fig. 4 is substantial and is undoubtedly partly related to random variation of . Is any of the scatter related to the stability of the mean flow? We now examine the possible dependence of the deviation heat flux
on the bulk Richardson number (Fig. 5a), where
The reference heat flux is a simple plausible approximation that neglects stability. Zero corresponds to CH = 10−3. The bin-averaged varies systematically with Rb by about 0.003 K m s−1, which is 23% of the mean value. This variation appears to be significant compared to the error bars, although such standard errors underestimate the uncertainty and the role of errors in δθ is not known.
The value of CH increases roughly linearly with increasing (Fig. 5b) by about 15%. Because , V, and δθbulk are independently bin averaged, the estimation of the standard error is complex and not attempted here. This modest dependence of CH on Rb seems consistent with Fig. 5a. The role of self-correlation due to shared variables between CH and Rb and the role of the limited range of stability are discussed in section 6.
b. Relation to within-air vertical gradients
Vertical mixing can feed back on the vertical gradients of temperature within the air more quickly compared to vertical differences that involve the slowly changing water temperature (Sun and French 2016). Although the vertical difference of θ within the air is quite small for the RED measurements, the relatively good relationship between and δθair (section 3) and the relatively good relationship between and Vδθair (Fig. 4b) suggest that the vertical differences of air temperature are adequately measured. The coefficient for δθair is much larger than that for δθbulk because δθair is small. The offset for the heat flux is inherited from the offset in the relationship of to Vδθair (Fig. 3c), discussed above.
The relation of to δθair seems systematic and suggests the possibility of estimating with observations of V and δθair alone where the latter requires two levels of accurate temperature measurements. This estimate avoids the sometimes problematic measurement of the water temperature or could serve as a supplementary estimate of in addition to using the bulk method. We do not pursue parameterization of in terms of δθair but encourage more extensive careful measurements of δθair.
c. Moisture flux
The moisture flux for the RED measurements is now related to δq = q(air) − qsat(Twater), where Twater was augmented by 0.87 K (section 3). The saturation surface specific humidity qsat(Twater) depends nonlinearly on the water temperature and is thus sensitive to the adjustment of the surface temperature. Nonetheless, the dependence of the moisture flux on V δq does not produce countergradient cases.
The transfer coefficient for moisture is defined in terms of
Based on the bin-averaged values (red, Fig. 6), Cq ≈ 1.0 × 10−3. Fitting all of the points and forcing the fit through the origin (green, Fig. 6) corresponds to Cq ≈ 1.1 × 10−3. The significance of the 10% difference between the two values of Cq is unknown. The scatter in Fig. 6 is larger than in the relationship of to Vδθbulk (Fig. 5) possibly because the moisture flux measurements are more difficult than heat flux measurements due partly to the separation between the sensor for w′ and the sensor for q′. The deviation of the moisture flux from Eq. (8) shows no dependence on Rb (not shown).
5. Coastal zone
The open-ocean conditions for the RED program are now contrasted with the ASIT field program in the coastal zone that includes numerous fetch-limited cases with advection of warmer air from the heated land surface located to the north and west of the site. Southerly flow is not fetch limited in the strict sense but includes advection of warm air from warmer water to the south of the site. The majority of the measurements represent positive air–sea temperature difference as a result of the advection of warmer air over the cooler water (Mahrt et al. 2016). Because δθSST is generated primarily by advection, δθSST increases with increasing V. For the measurements from the RED site where the absolute value of δθSST tends to be smaller and the airflow is not fetch-limited, temperature advection probably has a smaller influence compared to the ASIT site and δθSST decreases with increasing V.
Suitable measurements of the water temperature below the surface were not available for the ASIT measurements. We use instead the adjusted radiometrically measured sea surface temperature. Following procedures in section 3, δθSST was reduced by 1.8 K. The dependence of on VδθSST at the ASIT site (Fig. 7) is characterized by larger scatter than that for the RED data (Fig. 4). The larger scatter is partly due to a complex wind-directional dependence (Mahrt et al. 2016) and the need for smaller averaging time for stable conditions (1 min in ASIT compared to 10 min during RED). The 1-min averaging was used to filter out nonturbulent motions for stable conditions even though this smaller averaging length can increase the random error for the individual flux values.
The dependence of the bin-averaged heat flux on VδθSST at the ASIT site (red, Fig. 7) agrees well with 10−3VδθSST for both stable and unstable stratification. Small systematic deviations occur where the sample size is less and the scatter is relatively large.
The relationship 10−3VδθSST also performed well for the RED data (Fig. 4). The agreement between the two datasets could be fortuitous because the analysis of the RED data uses the within-water temperature and the atmospheric observations at approximately 10 m, whereas the analysis of the ASIT measurements uses the surface radiation temperature and atmospheric observations at 6 m. The ASIT data include 1 day with strong northerly flow and large cold air advection (unstable conditions) characterized by large magnitudes of VδθSST (−50 to −80 K m s−1) that is off scale from Fig. 7a. The northerly flow corresponds to a fetch of only about 5 km; CH = 10−3 still predicts the mean value of the heat flux reasonably well for this case, although the scatter in the observations is large, and conclusions are not possible based on only 1 day.
The deviation of from the linear model shows no detectable dependence on (Fig. 8a). The heat-flux deviations become small for vanishing Rb, where the mean heat flux also tends to vanish, as predicted by vanishing air–sea temperature difference. The CH increases significantly with increasing for unstable conditions (Fig. 8b) and decreases with increasing for stable conditions.
The dependence of CH on is almost linear for both stable and unstable conditions except for the most stable conditions. For these most stable conditions, CH no longer decreases with further increase of partly because CH is constrained to be positive no matter how large Rb. The boundary layer depth can become quite thin, less than 20 m (Mahrt et al. 2016) with the most stable conditions where the fetch is small. For these conditions the measurements even at 6 m can significantly underestimate the surface flux. The thinness of the internal boundary layer may also directly restrict vertical transfer through suppression of large eddies (Vickers and Mahrt 1999). Distortion of the profiles of the temperature, wind, and fluxes by advection could also alter the estimation of CH (Rb).
Systematic dependence of CH on Rb seems to contradict both the relatively good performance of the simple model CH = 10−3 (Fig. 7) and the relative insensitivity of deviations from such a model to Rb (Fig. 8a). An important role of shared variables is suspected (section 6).
The potential dependence of CH on stability is further investigated by restricting the analysis to the most unstable conditions and averaging the heat flux, δθSST, and V over such data. This analysis still leads to CH of approximately 10−3, and such results are not sensitive to the choice of “most unstable.” The corresponding CH for the most stable conditions is significantly smaller than 10−3 but does depend on the threshold for defining “most stable.” The short-fetch boundary layer depth is thin, such that surface fluxes are probably underestimated. In addition, the sample size for both the most unstable and the most stable conditions at the ASIT site is too small to make definite conclusions.
6. The small range of stability and the role of self-correlation
The deviation of the heat flux from the simple prediction based on CH = 10−3 is not related to Rb in the ASIT data, despite some significantly stable conditions, yet CH varied systematically with Rb. The relationship between CH and Rb could be related to the shared variable V because CH and Rb are both inversely related to V. The influence of the shared variable δθbulk acts to produce an inverse relationship between CH and Rb, which is not observed. Self-correlation due to the shared variable V predicts the observed square root dependence of CH on Rb (Figs. 5b, 8b). The square root dependencies disagree with the standard formulation of CH (Rb; Louis 1979) even for the near-neutral asymptotic behavior approaching from both the stable and unstable sides.
A brief examination of self-correlation for the RED and ASIT measurements is conducted by randomizing V and δθbulk and then recomputing CH and Rb (Klipp and Mahrt 2004). Linearly regressing CH on Rb for the RED measurements corresponds to an R-squared value of only 0.11 for the original data and 0.19 for the randomized data. If there was no self-correlation, the R-squared value for the randomized data would be zero. The small R-squared values are partly due to the greater random variability of ratios. The increase of the variance explained by randomization might be at least partly due to the larger ranges of CH on Rb based on the randomized data. We do not pursue more sophisticated approaches for examination of self-correlation. Randomizing the values of V and δθSST in ASIT suggests that the relationship between Rb and CH is mainly due to self-correlation for unstable conditions, although the much greater range of Rb and CH for the randomized data again prevents confident conclusions. Analysis of the role of self-correlation for the stable case is inconclusive.
While Figs. 5b and 8b might also imply that the range of stability is generally too small to significantly impact CH, independent verification is surprisingly difficult. Our attempts to compare the range of the stability in our two datasets with the range of stability over land in fair weather conditions where stability is known to be important, confirmed the general expectation that the range of stability is much less over the sea. However, this comparison had to contend with the complexity of the surface radiation temperature over land and the difficulty of measuring the small values of δθair over the sea and measuring air temperature sufficiently close to the sea surface. The ASIT data include advection of warm air from land over cooler water that appears to lead to some values of stability that would be quite important over land. However, these advective cases are difficult to interpret partly because of the small amount of data, short-fetch wave fields, and boundary layer depths less than 20 m, where even 6 m is too high to estimate the surface fluxes. In summary, the weak dependence of the heat flux on Rb is difficult to isolate probably because of the small range of stability, the role of self-correlation, and the importance of other physical influences.
The dependence of the heat flux on stability depends significantly on the instruments used to estimate the vertical temperature difference. Use of the within-water temperature during RED is more reliable than use of the radiometrically measured sea surface temperature because the radiometric measurements are physically more difficult to interpret and vary more erratically. These data did not include very weak winds where stratification within the water might be significant. However, adjustments of δθ are required for application to the bulk formula where must vanish with vanishing δθ by definition. The different approaches in the literature used to estimate the air–sea temperature difference may partly explain the difference between the results of different studies.
For the RED and ASIT measurements, , on average, is approximately linearly proportional to Vδθ (approximately constant CH). Deviations of from this linear relationship are only weakly dependent on Rb for the RED measurements and show no detectable dependence on Rb at the more complex ASIT coastal site, even though Rb exhibits a larger range of values compared to the RED measurements. The moisture flux for RED also follows a linear relationship to Vδqbulk. The deviation of the moisture flux from this linear relationship does not depend on Rb.
In contrast to the above conclusions, directly plotting CH as a function of Rb roughly indicates a systematic square root dependence on Rb for both datasets. A brief study of self-correlation indicates that the relationship between CH and Rb based on randomized values of V and δθ was as strong as with the unrandomized data. In addition, CH seems to follow a square root dependence on Rb, which is predicted by the influence of the shared variable V in CH and Rb. As a result of these issues, the weak dependence of the heat flux on Rb cannot be confidently formulated. A simple formulation based on constant CH might be useful and easier to amend for inclusion of additional physics such as wave state. We note that use of Monin–Obukhov similarity relates the heat flux to a stability parameter based on the heat flux and can directly create an influence of stability even over the sea that does not necessarily exist when expressing the stability exclusively in terms of the mean flow (Mahrt et al. 2016).
The smallness of the impact of stability and its agreement with that predicted by self-correlation may be due to the limited range of stability compared to that over land, issues with measurement of the small air–sea temperature difference, and other physics such as wave state. The most stable conditions occur with limited fetch and very thin boundary layers for offshore flow. Our future work will incorporate additional datasets to investigate the generality of these findings, conduct a more extensive comparison of the range of stability over the sea with that over land, and more explicitly consider the implications for the bulk aerodynamic relationship.
We gratefully acknowledge the extensive valuable comments of Simon de Szoeke and one anonymous reviewer and the helpful discussions with Jielun Sun. We thank Jim Edson for the ASIT data. This work was supported by the U.S. Office of Naval Research through Awards N00014-16-1-2600 and N00014-11-1-0073.