Turbulent Energy Spectra and Cospectra of Momentum and Heat Fluxes in the Stable Atmospheric Surface Layer

Abstract

The turbulent energy spectra and cospectra of momentum and sensible heat fluxes are examined theoretically and experimentally with increasing flux Richardson number ($\textit{Rf}$) in the stable atmospheric surface layer. A cospectral budget model, previously used to explain the bulk relation between the turbulent Prandtl number ($Pr_\mathrm{t}$) and the gradient Richardson number ($\textit{Ri}$) as well as the relation between $\textit{Rf}$ and $\textit{Ri}$, is employed to interpret field measurements over a lake and a glacier. The shapes of the vertical velocity and temperature spectra, needed for closing the cospectral budget model, are first examined with increasing $\textit{Rf}$. In addition, the wavenumber-dependent relaxation time scales for momentum and heat fluxes are inferred from the cospectral budgets and investigated. Using experimental data and proposed extensions to the cospectral budget model, the existence of a ‘$-1$’ power-law scaling in the temperature spectra but its absence from the vertical velocity spectra is shown to reduce the magnitude of the maximum flux Richardson number ($\textit{Rf}_\mathrm{m}$), which is commonly inferred from the Rf–Ri relation when $\textit{Ri}$ becomes very large (idealized with $\textit{Ri} \rightarrow \infty$). Moreover, dissimilarity in relaxation time scales between momentum and heat fluxes, also affected by the existence of the ‘$-1$’ power-law scaling in the temperature spectra, leads to $Pr_\mathrm{t} \ne 1$ under near-neutral conditions. It is further shown that the production rate of turbulent kinetic energy decreases more rapidly than that of turbulent potential energy as $\textit{Rf}\rightarrow \textit{Rf}_\mathrm{m}$, which explains the observed disappearance of the inertial subrange in the vertical velocity spectra at a smaller $\textit{Rf}$ as compared to its counterpart in the temperature spectra. These results further demonstrate novel linkages between the scale-wise turbulent kinetic energy and potential energy distributions and macroscopic relations such as stability correction functions to the mean flow and the $Pr_\mathrm{t}$–Ri relation.

Appendix: Data and Methodology

The datasets used were collected in the stably stratified ASL over a lake and a glacier surface. These datasets include measurements of three-dimensional velocity and temperature at high frequency (=20 Hz) and at four different heights. Details about the two datasets and quality control measures can be found elsewhere (Vercauteren et al. 2008; Huwald et al. 2009; Bou-Zeid et al. 2010; Li and Bou-Zeid 2011; Li et al. 2012a). In particular, data where fluxes measured at the four heights differ by more than 10 % are excluded. Calculations of turbulent fluxes follow the standard eddy-covariance method with an averaging interval of 30 min (Li and Bou-Zeid 2011; Li et al. 2012a). Calculations of spectra and cospectra for each 30-min segment follow the standard Fourier transform method (Stull 1988), which are then smoothed using a periodic hamming window without overlap. The mean velocity and temperature vertical gradients, which are needed in the calculations of $\textit{Rf}$ and relaxation time scales, are obtained by fitting second-order polynomial functions to the mean velocity and temperature at the four measurement levels and then taking the derivatives of the fitted functions. A linear interpolation method was also used to compute the vertical gradients of mean velocity and temperature and the results were found to be insensitive to the method of evaluating the vertical gradients, which is consistent with Grachev et al. (2007). The datasets are separated into eight regimes according to $\textit{Rf}$, as can be seen from Table 1. Since the lake dataset primarily spans slightly stable to mildly stable conditions and the glacier dataset spans mildly stable to very stable conditions, the first five stability regimes in Table 1 only include data from the lake and the last three regimes only include data from the glacier set. Regime e and regime f cover roughly the same range of $\textit{Rf}$ but the averaged $\textit{Rf}$ of all segments are different: regime f has a much larger averaged $\textit{Rf}$ than regime e. The calculated spectra and relaxation time scales for each segment are further averaged over each stability regime, which are shown in Figs. 1, 2, and 3.