A Surface-Layer Study of the Transport and Dissipation of Turbulent Kinetic Energy and the Variances of Temperature, Humidity and CO$_2$

Abstract

We discuss scalar similarities and dissimilarities based on analysis of the dissipation terms in the variance budget equations, considering the turbulent kinetic energy and the variances of temperature, specific humidity and specific CO$_2$ content. For this purpose, 124 high-frequency sampled segments are selected from the Boundary Layer Late Afternoon and Sunset Turbulence experiment. The consequences of dissipation similarity in the variance transport are also discussed and quantified. The results show that, for the convective atmospheric surface layer, the non-dimensional dissipation terms can be expressed in the framework of Monin–Obukhov similarity theory and are independent of whether the variable is temperature or moisture. The scalar similarity in the dissipation term implies that the characteristic scales of the atmospheric surface layer can be estimated from the respective rate of variance dissipation, the characteristic scale of temperature, and the dissipation rate of temperature variance.

Appendices

Appendix 1: Density Correction Applied for $S_c$ and $\sigma _c$

While the density correction of Webb et al. (1980) is defined only for the covariances $\overline{w'c'}$ and $\overline{w'q'}$, it should also be extended to the spectra, as mentioned by Sahlée et al. (2008) who derived an alternative direct-conversion method, which performs a unit conversion directly on the high-frequency time series.

Here, we use the equations derived by Detto and Katul (2007) for the correction to the CO$_2$ density variance, ${\overline{{\rho '_c}^2}}_{WPL}$, to correct the CO$_2$ power spectrum, $S_c$. The ${\overline{{\rho '_c}^2}}_{WPL}$ quantity can be expressed according to

$$\begin{aligned} {\overline{{\rho '_c}^2}}_{WPL}= & {} \overline{{\rho '_c}^2} + 2\mu \frac{\overline{\rho _c}}{\overline{\rho _a}} \overline{{\rho '_c}{\rho '_q}} + 2 \frac{\overline{\rho _c}\left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{T}} \overline{{\rho '_c}{T'}} + {\left( \mu \frac{\overline{\rho _c}}{\overline{\rho _a}} \right) }^2 \overline{{\rho '_q}^{2}} \nonumber \\&+{\left( \frac{\overline{\rho _c} \left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{T}} \right) }^2 \overline{{T'}^{2}} + 2 \frac{ {\overline{\rho _c}}^2 \mu \left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{\rho _a}\overline{T}} \overline{{\rho '_q}{T'}} ~\text {,} \end{aligned}$$

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where $\mu ={m_a}/{m_q}$ is the ratio between the molecular mass of dry air ($m_a$) and the water vapour ($m_q$), and T is the air temperature. From the definition of standard deviation, the correction for the scalar standard deviation is ${\sigma _c}_{WPL}=\sqrt{{\overline{{\rho '_c}^2}}_{WPL}}$.

The scalar variance can also be interpreted as the integral of $S_\chi$, and the covariance as the integral of the cospectrum $Co_{\chi _1 , \chi _2}$. Therefore, the density correction for $S_c$, for each frequency (n), is expressed as

$$\begin{aligned} {S_c(n)}_{WPL}= & {} S_c(n) + 2\mu \frac{\overline{\rho _c}}{\overline{\rho _a}} Co_{c,q}(n) + 2 \frac{\overline{\rho _c}\left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{T}} Co_{c,T}(n) + {\left( \mu \frac{\overline{\rho _c}}{\overline{\rho _a}} \right) }^2 S_q(n) \nonumber \\&+{\left( \frac{\overline{\rho _c} \left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{T}} \right) }^2 S_T(n) + 2 \frac{ {\overline{\rho _c}}^2 \mu \left( 1+\mu \frac{\overline{\rho _q}}{\overline{\rho _a}} \right) }{\overline{\rho _a}\overline{T}} Co_{q,T}(n) ~. \end{aligned}$$

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Appendix 2: Method for Detection of the Inertial Subrange

Since it is crucial for the spectral method to accurately resolve the inertial subrange, we propose an iterative method to evaluate the variations of the independent constants $\alpha$ and $\beta$ defined in Eqs. 7 and 8 by keeping constant values for the dissipation rates within a frequency interval I. From the definition of the inertial subrange, both the dissipation rates and $\alpha$, or $\beta$, are constant and, therefore, the boundaries for I can be defined by evaluating the standard deviation calculated for $\alpha$, or $\beta$, in different regions of the spectra.

The spectra calculated from 18,000 data samples collected at 20 Hz are divided into 48 frequency blocks (as shown in Online Resource 1), of which the number of estimates per block is logarithmically distributed, similar to the ones proposed by Kaimal and Gaynor (1983). For each block-centred frequency, $n^*$, the average power spectrum, $\overline{S}(n^*)$, and the dissipation rate of TKE, $\epsilon ^*(\overline{S_w}(n^*),n^*)$, are calculated using the following equation derived from the Kolmogorov power law,

$$\begin{aligned} \epsilon ^*={\left( \frac{{\overline{S_w}}(n^*)}{\alpha }\right) }^{\frac{3}{2 }}{n^*}^{\frac{5}{2}}{\left( \frac{2\pi }{{U}}\right) }~\text {.} \end{aligned}$$

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The iteration among the blocks consists of a loop starting from the centred frequency block $n^*=0.0922$ Hz (index 22 in Online Resource 1) and moving the interval I towards the higher frequencies, one block per iteration, up to $n^*=4.5244$ Hz (index 44). For this method, we suggest a window size for the interval I that includes nine consecutive blocks.

For each iteration, $\alpha$ is estimated for the nine $n^*$ within interval I, keeping $\epsilon ^*$ as a constant equal to the one corresponding to the fifth block from interval I (i.e. the block in the middle of interval I). Finally, the standard deviation of $\alpha$, $\sigma _\alpha$, is calculated for each I.

The inertial subrange is then defined as the interval I that has the smallest $\sigma _\alpha$ after completing the iteration. If the smallest $\sigma _\alpha$ is greater than 0.07 (i.e. $10\%$ of $\alpha$), we consider that the method cannot define an inertial subrange. The $\epsilon$ for the corresponding 30-min segment is then estimated according to the discussion in Sect. 2.4.

A similar procedure is carried out for the estimation of interval I in the power spectra of $\theta$, q, and c. Since $\epsilon$ is now known, the following Eq. 22 is applied in an analogous way as described before, but using $\beta$ instead of $\alpha$,

$$\begin{aligned} N_{\chi _i}^*={\left( \frac{{\overline{S_{\chi _i}}}(n^*)}{\beta }\right) }{ \epsilon }^{\frac{1}{3}}{n^*}^{\frac{5}{3}}{\left( \frac{2\pi }{{U }}\right) }^{\frac{2}{3}}~. \end{aligned}$$

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