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. ). Small-scale and mesoscale processes effectively interact with each other and provide an energy sink for currents and waves of larger scales. Such processes as wind wave breaking and surface and subsurface shear flows can cause turbulent mixing and the resulting fine-structure formation with areas of sharp gradients of temperature and salinity. Whereas the mechanisms of generation of small-scale turbulence are understood reasonably well, the problem of its interaction with other types of motions and long-time support is less clear. Earlier works were based on Miles' instability condition Ri<1/4, where Ri is the gradient Richardson number. However, in many observations, turbulence exists in a quasi-stationary regime at much larger Ri, up to 10 and more. In some works, this was explained by the presence of a fine structure of currents, with thin layers of strong shear . A more general description is based on the semi-empirical K-ε equations showing that the developed turbulence can be amplified and supported under a softer condition Ri<1 . Similar equations were used in more specific models of formation of the upper turbulent layer and of the action of internal waves on turbulence . However, even that is insufficient to explain many observations in the ocean and atmosphere, where the turbulence is observed at significantly larger Ri . New theoretical models have also appeared for describing nonstationary turbulent processes in the atmosphere and ocean. They are based on a spectral approach, confirming, in particular, the absence of a critical Richardson number for describing turbulent-wave processes in a stratified fluid .

In , within the framework of a kinetic approach, a closed nonstationary model of turbulence interacting with a variable current was suggested. It includes mutual transformation between kinetic and potential energies of turbulence. The latter is associated with the density fluctuations occurring in stratified turbulence . The theory suggested in is based on the equation for the variable probability distribution function of fluid velocity and density. This approach reduces the uncertainty of standard semi-empirical K-ε schemes and naturally includes the potential energy of turbulence in the model. As a result, small-scale turbulence can be supported at a non-zero level by the average shear at any finite values of the gradient Richardson number, without a threshold. Later this approach was further developed in , , and in application to atmospheric turbulence, where the energy and flux budget (EFB) model was added to the theory. The correspondence between this theory and the K-ε model as well as the proper parametrization using the turbulent Prandtl number are discussed in recent works .

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 that provided a relatively detailed set of measurements for three cruises taken in 2006–2009 in different areas of the world ocean: North Atlantic (cruise D3406, June–July 2006, and cruise D321, July–August 2007) and Southern Ocean (cruise JC29, November–December 2008). These experiments aimed to study turbulent mixing in the presence of a stratified shear flow associated with mesoscale motions such as eddies and fronts. With the given profiles of current shear and buoyancy frequency taken from , the theory developed in yields the results that satisfactorily agree with the measurements of the turbulent dissipation rate given in . The details of the measurements can be found in and references therein.

The general equations obtained in (see also ), are shown in the Appendix. Here they will be used for the particular case of known profiles of horizontal current shear ∂〈ux〉/∂z=Vz, where 〈ux〉=V(z) is the ensemble-average horizontal velocity, and the average density is 〈ρ(z)〉=R(z). Here, z is the vertical coordinate. As a result, we have a system of two equations for the kinetic energy of turbulence K(t,z) and potential energy P(t,z) per unit volume. The latter is related to the density fluctuations 〈ρ′2〉: 1P=〈ρ′2〉g22N2R2. Here, g is the gravity acceleration, R(z) is defined above, and N2(z) is the squared Brunt–Väisala frequency. Note that here the z direction is chosen downwards. Under the above conditions, the general Eq. (18) of and Eq. (6) of are reduced to two equations for K: 2∂K∂t=Vz2LK-N2LK1-3PK(1-G)-GK3/2L+53∂∂zLb∂K∂z,∂P∂t=N2LK1-3PK(1-G)-DK1/2LP+∂∂zLKN2R2∂(N2R2P)∂z. Here, L is the outer scale of turbulence and G is the anisotropy parameter tending to 1 for strongly anisotropic turbulence with a small vertical scale compared to the horizontal scale. For details, see . Here we consider the model of locally isotropic turbulence, for which G∼0.5. The parameter L can be taken from empirical data or found from the turbulence spectrum . C and D are empirical constants. The terms CK3/2/L and DK1/2P/L in Eq. (2) define the dissipation rates of kinetic and potential energy, respectively.

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: 3∂K∂t=Vz2LK-N2LK1-3PK(1-G)-CK3/2L,∂P∂t=N2LK1-3PK(1-G)-DK1/2LP.

The coordinate z is now a parameter in these equations. In particular, they define a stationary distribution (a stable equilibrium point on the phase plane of variables K and P): 4Kst(z)=Vz2L22Cf(Ri),Pst(z)=Vz2L2D-Kst, where Ri=Vz2/N2 is the Richardson number, and 5f(Ri)=1-(4-3G)Ri+1+Ri2(4-3G)2+Ri(4-6G)1/2. It is noteworthy that, at Ri→∞, f has a non-zero limit f∞=6(1-G)/(4-3G)>0 (it is 1.2 for G=0.5). Hence, the turbulent energy remains finite at large Richardson numbers. General features of this solution and the turbulent Prandtl number following from it are discussed in and . In what follows, the applicability of these simple solutions will be verified by comparison with the solutions of the full Eq. (2). In the experiments, the dissipation rate of kinetic turbulent energy ε is commonly measured as a characteristic of turbulence. Within the semi-empirical approach, it is defined as 6ε=CK3/2L. It is necessary to choose the empirical constants in the above equations and in Eq. (6). There exists a broad literature discussing these values for different laboratory and on-site conditions. In the semi-empirical models they are commonly defined by scaling. We choose the outer turbulence scale based on the results of the spectral approach , in which the minimal wave number for the energy-carrying spectrum approximated by empirical functions is about 2 cpm. Here we take L=0.58 m. The range of the constant C is also wide in the literature. Since we are mainly interested in the quasi-equilibrium regime when the shear source is balanced with dissipation of turbulent energy, we use the data of to take C=D=0.09. Anyway, we are mainly concerned with the order of the obtained values; indeed, in the data of considered below, the spread of the data is up to an order of magnitude.

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.

As mentioned, here we apply the theory to the data of the three cruises described in . First, we digitized the red curves in Fig.  of that paper (as mentioned, there is a large dispersion of real data, but we naturally use the mean profiles). Then, using the given profiles of Vz and N2, we calculated the Richardson number as shown in Fig. .

Note that, for a water density of 1000 kg m-3, the static pressure (dbar) coincides with depth in meters, so that in our calculations we use the depth, neglecting small differences in density. As seen from Fig. , the Richardson number exceeds 1 within all ranges of the available data, and its maximum lies in the range of 10–23.

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: 7FK=K∂K∂z,FP=K∂P∂z. They are given to be compatible with the initial conditions at initial points z0 from which the plots start in Fig.  of and tend to zero values at the deepest points. Then, the solutions for K and P were compared with those of the local system (3). The figures below show such a comparison for the asymptotic values. In what follows, the above local solutions are shown in blue and the solutions of Eq.  with diffusion in orange.

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 K. Hence, for the vertical scales of average values, vertical diffusion can be neglected, and one can use the simplified local Eq. (3).

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 , where the measurements of profiles of the buoyancy frequency, current shear and dissipation rate of turbulent energy are shown together for three different areas of the ocean, provide not only qualitative but also reasonable quantitative agreement between the theoretical and experimental data. We have also shown that the contribution of turbulent diffusion to the level of turbulent pulsations is insignificant in the above case.

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 . However, the same figure shows significant, up to 1 order of magnitude, data scatter from each cruise, obtained in different locations and on different days. The authors do not specify the confidence intervals of the data, and no correlation between the different curves for the shear and buoyancy frequency curves is known. Still, it is possible to evaluate the maximal span of the results based on the extreme curves in each plot. For that, we digitized the leftmost and rightmost curves for the shear and buoyancy, interpolated them, and used the resulting functions to calculate the limits of the theoretical dissipation rate using Eqs. () and (). For cruise 306, this is shown in Fig. 12.

Note that the data scattering for this value shown in lies within the maximal theoretical limits, which implies that, knowing the real data for a specific location and time of measurement, we would reasonably predict the corresponding depth dependence for the turbulent kinetic energy dissipation rate. Similar results take place for other cruises.

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 , for further study it seems important to find more experimental data allowing us to apply the theory. We also plan to extend the present approach to description of time-varying turbulence in the field of internal waves (e.g., ).