The theory of stratified turbulent flow developed earlier by the authors is applied to data from different areas of the ocean. It is shown that turbulence can be amplified and supported even at large gradient Richardson numbers. The cause of that is the exchange between kinetic and potential energies of turbulence. Using the profiles of Brunt–Väisälä frequency and vertical current shear given in

At present, it is well established that the processes in the upper mixed layer of the ocean and inland waters play a significant role in both the development of global climate models and the creation of regional weather forecast models (e.g.

In

In this paper, the kinetic model of turbulence is used to describe the evolution and structure of the upper turbulent layer with the parameters taken from in situ observations. Particular attention is paid to the cases of the large Richardson number and the role of turbulent potential energy in explaining the observation data. As an example, we use some data from the paper

Profiles of the Richardson number for three cruises calculated from Fig.

Temporal variation of kinetic

The general equations obtained in

Before analyzing the full system (2), we note that some significant conclusions can be made from a reduced, local ODE system following from Eq. (2) after neglecting the last, diffusive terms in these equations:

Profiles of kinetic (blue) and potential (green) energies for JC29 at

Profiles of the turbulent kinetic energy dissipation rate for JC29. Green – interpolated data of

The coordinate

In what follows, we solve systems (2) and (3) using the Wolfram Mathematica 13 program and compare them with each other and the data of in situ measurements.

Profiles of kinetic (blue) and potential (green) energies for D306 at

Profiles of the turbulent kinetic energy dissipation rate for D306. Green – interpolated data of

As mentioned, here we apply the theory to the data of the three cruises described in

Note that, for a water density of 1000 kg m

Profiles of kinetic (blue) and potential (green) energies at

Profiles of the turbulent kinetic energy dissipation rate for D321. Green – interpolated data of

Now, using the interpolation of digitized data for

Cruise JC29: comparison of profiles of kinetic

Cruise D306: comparison of profiles of kinetic

Cruise D321: comparison of profiles of kinetic

Here, the constant levels of energy are established in several hours. It is natural to use the asymptotic values for comparison with the measurement data. In the subsequent plots we use the log–lin presentation, following

Using this solution, we calculate the turbulent dissipation rate (Eq. 6) and compare it with the data of

To save space, for another two cruises, we show only the asymptotic profiles of the corresponding values at large times. For cruise D306, the values of kinetic and potential energies are of the same order as for cruise JC29 (Fig.

Figure

Here again, the difference between the theory and data of

The corresponding dependencies for cruise D321 are given in Figs.

Here again, one can see good agreement between the theory and the mean measured profile.

The above results were obtained neglecting vertical turbulence diffusion. To verify this approximation, we solved the full system (2) with the same parameters and initial conditions, adding boundary conditions for fluxes of kinetic and potential energy:

In all three cases, the local and full models are practically identical. Evidently, this means the closeness of data for the dissipation rate that is a function of

In this paper we demonstrated that including the potential energy of turbulence (associated with density fluctuations in the presence of stratification) in the semi-empirical Reynolds-type equations of a turbulent flow allows us to explain the existence and evaluate the parameters of small-scale turbulence at large Richardson numbers. Application of these equations to the results of

For further progress, more specific experimental data sets are desirable. Indeed, in the above calculations, we used the average data for buoyancy frequency, velocity shear, and rate of kinetic energy dissipation plotted in red in Fig. 2 of

Maximal (black) and minimal (gray) profiles of the turbulent kinetic energy dissipation rate for D306 calculated from the maximal possible scatter of data for N2 and S given in

Note that the data scattering for this value shown in

Note in conclusion that the kinetic approach used in the equations used above allows us to naturally include the potential energy under consideration. Considering a large variation of empirical parameters given in different sources

Here we briefly outline the general system of equations for a turbulent stratified flow obtained in

Equation (A2) for the mass flux includes the summand

As a result, one obtains the equations for the mean values of velocity, density, turbulent kinetic energy

All the materials in the text and figures were created by the authors using standard mathematical and numerical analysis using Wolfram Mathematics software tools. Implementations of specific expressions given in the work can be provided upon request by email to the contact author.

The work did not use any external data other than those explicitly given in

The concept of the article was proposed by LO. Data curation was performed by IS and DG. The formal analysis was carried out by LO and DG. The study was supervised by LO and YT. The original draft was prepared by LO, IS and DG. LO and DG worked on the review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The work was supported by RSF project no. 23-27-00002.

This research has been supported by the RSF (project no. 23-27-00002). Publisher's note: the article processing charges for this publication were not paid by a Russian or Belarusian institution.

This paper was edited by Harindra Joseph Fernando and reviewed by two anonymous referees.