Over the years, the problem of dissipation rate of turbulent kinetic energy (TKE) in stable stratification remained unclear because of the practical impossibility to directly measure the process of dissipation that takes place at the smallest scales of turbulent motion. Poor representation of dissipation causes intolerable uncertainties in turbulence-closure theory and thus in modelling stably stratified turbulent flows. We obtain a theoretical solution to this problem for the whole range of stratifications from neutral to limiting stable; and validate it via (i) direct numerical simulation (DNS) immediately detecting the dissipation rate and (ii) indirect estimates of dissipation rate retrieved via the TKE budget equation from atmospheric measurements of other components of the TKE budget. The proposed formulation of dissipation rate will be of use in any turbulence-closure models employing the TKE budget equation and in problems requiring precise knowledge of the high-frequency part of turbulence spectra in atmospheric chemistry, aerosol science, and microphysics of clouds.

Until the
present, the dependence of dissipation rate,

For certainty, we consider the dissipation rate of TKE in terms of dry atmosphere, where
fluctuation of buoyancy,

Stratification involves the Obukhov length scale:

We consider horizontally homogeneous stationary boundary-layer flow in semi-space

Then, substituting

To comprehensively validate the above analyses, we performed DNS of stably stratified

Usual height,

In our DNS total (turbulent

Following Obukhov (1942), we distinguish between “absolute geometry” characterized by
the usual height over the surface,

In semi-space, the “internal geometry” coincides with “absolute geometry”:

In Figs. 2–4 we show our DNS data together with atmospheric data on the
dissipation rate retrieved from observations in the surface layers
indirectly:

via the Kolmogorov

via the steady-state TKE budget Eq. (6) from the measured
turbulent fluxes of momentum,

Flux Richardson number,

Figure 2 shows flux Richardson number,

Dimensionless dissipation rate in stable stratification,

The energy Richardson number,

Figure 3 shows dimensionless dissipation rate,

In Fig. 2 we consider the flux Richardson number:

The concept of the TKE dissipation rate directly relates to the definition of the
turbulent timescale,

We emphasize that

The above analyses are done for the simplest surface-layer (or Couette) flow, where
dimensionless height

Expressing the dissipation rates of TKE and TPE in the steady state through the
dissipation timescale,

There is an essential advantage to

The dissipation rate of TKE,

By providential coincidence, the formulations happen to be precisely the same in the
asymptotic limit

Universal analytical formulation of

The data can be accessed at

SZ conceived the concept, did theoretical derivations, and led the research. EM (author of INM-RAS DNS code) and OD (author of IAP-RAS DNS code) carried out independent numerical simulation of turbulent Couette flow aiming at verification and validation of the theory. IR collected and analysed complementary data from observations in the atmospheric surface layer over various types of the Earth surface. EK utilized all these data, carried out the entire work on verification and validation, and produced all figures. YT and AG contributed to theoretical analyses. All authors participated in writing the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Pan-Eurasian Experiment (PEEX)”. It is not associated with a conference.

The authors acknowledge support from the Academy of Finland project ClimEco no. 314 798/799; Russian Foundation for Basic Research (project nos. 18-05-60299, 16-05-01094 A, 18-55-11005). Analysis of data from atmospheric observation was supported from the Russian Science Foundation grant no. 17-17-01210. Post-processing of numerical data from IAP was supported by the Russian Science Foundation grant no. 15-17-20009 and Russian Foundation for Basic Research grants nos. 17-05-00703 and 18-05-00292. Edited by: Imre Salma Reviewed by: three anonymous referees