Signatures of Air–Wave Interactions Over a Large Lake

Abstract

The air–water exchange of momentum and scalars (temperature and water vapour) is investigated using the Lake-Atmosphere Turbulent EXchange (LATEX) dataset. The wind waves and swell are found to affect the coupling between the water surface and the air differently. The surface-stress vector aligns with the wind velocity in the presence of wind waves, but a wide range of stress–wind misalignment angles is observed during swell. The momentum transport efficiency decreases when significant stress–wind misalignment is present, suggesting a strong influence of surface wave properties on surface drag. Based on this improved understanding of the role of wave–wind misalignment, a new relative wind speed for surface-layer similarity formulations is proposed and tested using the data. The new expression yields a value of the von Kármán constant ($\kappa$) of 0.38, compared to 0.36 when using the absolute wind speed, as well as reduced data fitting errors. Finally, the ratios of aerodynamic to scalar roughness lengths are computed and various existing models in the literature are tested using least-square fitting to the observed ratios. The tests are able to discriminate between the performance of various models; however, they also indicate that more investigations are required to understand the physics of scalar exchanges over waves.

Appendices

Appendix A

We use the formulation of MOST for differences between two heights above the surface (yielding 6 different measurements from the four levels of LATEX), as given by Eq. 10. The functional form of $\varPsi$ is taken from Eq. 2.58 and Eq. 2.63 in Brutsaert (2005). Note that even if $U_\mathrm{c}$ is taken as $U_\mathrm{r}$, the wave phase speed ccancels out from Eq. 10 since it is also related to the ‘surface state’, which is identical for the four different heights, but the cos($\gamma )$ multiplier will remain.

To test whether the modified MOST for water surfaces is better than the classic formulation, we can write Eq. 9 or  10 with $U_{c~}= \quad U$ or $U_\mathrm{c}=U_\mathrm{r}$, and then check which form better yields (i) the correct slope which should be given by $\kappa ^{-1}$ (ii) a better fit to the data with lower fitting residuals. To that end, a least-square error linear fitting can be applied to Eq. 10 for each 15-min period (t) independently, which we can write in a general form

$$\begin{aligned} \Delta U_{j} (t)=B_{1}(t)\Delta Z_{j} (t), \end{aligned}$$

(12)

where the left-hand side is either $\Delta U_{j} =\frac{U(z_{k1})-U(z_{k2})}{u_{*}}$ or $\Delta U_{j} =\frac{\left| {\cos (\gamma )U(z_{k1})} \right| -\left| {\cos (\gamma )U(z_{k2})} \right| }{u_{*}}$; j is 1–6 depending on which combination of two levels ($z_{k1}$ and $z_{k2})$ are chosen to take the differences; t is an indicator for the time period; $\Delta Z_{j}$ is $\left( \ln \left( {z_{k1} } \right) -\varPsi \left( {z_{k1} /L} \right) -\ln \left( {z_{k2} } \right) +\varPsi \left( {z_{k2} /L} \right) \right)$ and $B_{1}$ is the slope to be deduced from the least-square fitting (a value is obtained for every time period) and will yield a value of $\kappa ^{-1}$. Thus, what we are minimizing to get the optimal fit is the sum of the squared errors over the six two-level differences, for each time period independently.

To ensure the robustness of this statistical test and the validity of the constant-stress layer assumption, we impose an additional criterion commonly used for the evaluation of the von Kármán constant from atmospheric surface-layer data (Andreas et al. 2006). Specifically, we limit the data used to periods in which the median of the measured $u_{*}(z)$, used in Eq. 10, only deviates by less than 10% from the surface value $u_{*0}$ obtained from a linear fit of the friction velocity measurements at the four height following: $u_{*} = az+u_{*0}$ (a is a constant obtained from least-square fitting). In addition, following the procedure in Andreas et al. (2006), we use the median of $u_{*} (z)$ in Eq. 12 for all level combinations. The same criterion to choose runs that satisfy the random error limit as described previously is also applied. This limits the number of periods to 161. A least-square error minimization with respect to $B_{1}$ in Eq. 12 over these 15-min periods is carried out using the two velocity scales U and $U_\mathrm{r}$.

Appendix B

To calculate the scalar roughness lengths in Eqs. 2 and 3, air temperature and water vapour concentration are obtained from the Rotronic sensor; the friction velocity $u_{*}$, kinematic sensible heat flux $H/\rho C_\mathrm{p}$ and kinematic water vapour flux $E/\rho$ are the averages over the four measurement heights. The lake surface is assumed to be saturated, and thus $q_\mathrm{s}$ is calculated using the Clausius–Clapeyron relation based on lake surface temperature obtained from the Bowen ratio method. This approach using the Bowen ratio has already been used by Bintanja and Broeke (1995) to compute the scalar roughness lengths.

According to Vercauteren et al. (2009), the surface temperature was measured by two independent sensors, which are prone to different sources of errors. One is a thermocouple, which was kept at tens of mm below the surface; it had errors due to its (lack of) immersion or radiative heating. The other, an infrared sensor, had errors related to its sensor body temperature correction and infrared transmissivity of the lake water. There were periods of missing data for the thermocouple because of the immersion. The large discrepancy among three methods shown in Fig. 13b can be attributed to these errors. Note that for small discrepancy in surface temperature, $z_{0h}$is also quite consistent among the different methods as expected (See Fig. 13a). Nevertheless, the extremely high and low values in $z_{0h}$ are likely to be unphysical. Furthermore, $z_{0h}/z_{0v}$ in Fig. 13c shows that although we assumed equal turbulent transfer efficiency to infer the surface temperature as described in Sect. 5.1, this does not necessarily imply that $z_{0h}/z_{0v}=1$. It is worth noting that the three methods for computing the surface temperature (thermocouple measurement, infrared sensor measurement and Bowen ratio method) give a ratio of the scalar roughness lengths that is very close to 1 when the measured temperatures agree, suggesting that these are the periods where the results are most robust. The Bowen ratio method results in $z_{0h}/z_{0v}$ ratios that departs the least from 1, again indicating it is the preferred method.

Fig. 13

Comparisons between different methods of obtaining surface temperature ($\theta _{BR}$: Bowen Ratio method; $\theta _{IR}$: infrared gun measurement; $\theta _{TC}$: thermal couple measurement). $z_{0s}$ is either $z_{0h}$ or $z_{0v}$. a $z_{0h}$ computed from three different methods. b Lake surface temperature obtained from the three different methods. c Ratio between $z_{0h}$ and $z_{0v}$ obtained from the three different methods