Small-scale and large-scale intermittency in the nocturnal boundary layer and the residual layer

Abstract

In high Reynolds-number turbulence, local scalar turbulence structure parameters,$( C_{\theta }^{2}) _{r}$, local scalar variance dissipation rates, $\chi _{r}$, and local energy dissipation rates, $\varepsilon _{r}$, vary randomly in time and space. This variability, commonly referred to as intermittency, is known to increase with decreasing $r$, where $r$ is the linear dimension of the local averaging volume. Statistical relationships between $\chi _{r}$, $\varepsilon _{r}$, and $( C_{\theta }^{2}) _{r}$ are of practical interest, for example, in optical and radar remote sensing. Some of these relationships are studied here, both theoretically and on the basis of recent observations. Two models for the conditionally averaged local temperature structure parameter, $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle $, are derived. The first model assumes that the joint probability density function (j.p.d.f.) of $\chi _{r}$ and $\varepsilon _{r}$ is bivariate lognormal and that the Obukhov–Corrsin relationship, $( C_{\theta }^{2}) _{r}=\gamma\varepsilon _{r}^{-1/3}\chi _{r}$, where $\gamma\,{=}\,1.6$, is locally valid. In the second model, small-scale intermittency is ignored and $C_{\theta }^{2}$ and $\varepsilon $ are treated traditionally, that is, as averages over many outer scale lengths, such that $C_{\theta }^{2}$ and $\varepsilon $ change only as a result of large-scale intermittency. Both models lead to power-law relationships of the form $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle \,{=}\,c\hspace{0.03in}\varepsilon _{r}^{\delta}$, where $c$ is a constant. Both models make predictions for the value of the power-law exponent $\delta$. The first model leads to $\delta\,{=}\,\rho _{xy}\sigma _{y}/\sigma _{x}-1/3$, where $\sigma _{x}$ and $\sigma _{y}$ are the standard deviations of the {logarithms} of $\varepsilon _{r}$ and $\chi _{r}$, respectively, and $\rho _{xy}$ is the correlation coefficient of the logarithms of $\chi _{r}$ and $\varepsilon _{r}$. This model leads to $\delta\,{=}\,1/3$ if $\rho _{xy}\,{=}\,2/3$ and if $\sigma _{x}\,{=}\,\sigma _{y}$. The second model predicts $\delta\,{=}\,2/3$, regardless of whether (i) static stability and shear are statistically independent, or (ii) they are connected through a Richardson-number criterion. These theoretical predictions are compared to fine-wire measurements that were taken during the night of 20/21 October 1999, at altitudes of up to 500 m in the nocturnal boundary layer and the overlying residual layer above Kansas. The fine-wire sensors were moved up and down with the University of Colorado's Tethered Lifting System (TLS). The data were obtained during the Cooperative Atmosphere-Surface Exchange Study 1999 (CASES-99). An interesting side result is that the observed frequency spectra of the logarithms of $\varepsilon _{r}$ and $\chi _{r}$ are described well by an $f^{-1}$ law. A simple theoretical explanation is offered.