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 Forryan et al. (2013), the profiles of the kinetic energy dissipation rate are calculated. The results are in reasonable agreement with the experimental data.

Lagrangian stochastic models (LSM) are widely used to model the dispersion of sea spray droplets injected from the water surface into the marine atmospheric boundary layer (MABL) and for evaluation of the spray impact on the exchange fluxes between the atmosphere and the ocean. While moving through the MABL the droplets pass through the region of high gradients of air velocity, temperature and humidity occurring in the vicinity of the air–water interface. In this case, the applicability of LSMs constructed under the assumption of weakly inhomogeneous flows is questionable. In this work, we develop a Lagrangian stochastic model taking into account the strongly inhomogeneous structure of the airflow in MABL and, in particular, the anisotropy of turbulence dissipation rate. The model constants and the diffusion matrix coefficients are calibrated by comparison of the LSM prediction for the profiles of droplet concentration and the exchange fluxes of sensible and latent heat against the results of direct numerical simulation of turbulent, droplet-laden airflow over a waved water surface.

We study the evolution of a turbulent layer in a stratified ocean layer using the theory of unsteady turbulent flows in a stratified fluid developed in [1] and subsequent works. The theory starts from a kinetic equation for turbulence parameters and results in the set of equations involving the mutual transformation of the kinetic and potential energies of turbulence that is shown to significantly affect the overall dynamics of energy exchange between small-scale turbulence and mesoscopic motions and the formation of the upper mixed layer. Besides, this approach allows an account for some important but usually neglected effects such as the dependence of vertical anisotropy of turbulence on stratification. Notably, the transformation between kinetic and potential energies eliminates the restriction on the existence of turbulence at large Richardson numbers. The results are applied to the analysis of in situ data for turbulence evolution under the action of shear flows and internal waves, obtained in different regions that are significant for climate research, including the upper equatorial ocean. The fundamental role of potential energy in the formation of a turbulent flow is demonstrated.The work was supported by RSF project No. 23-27-00002.[1] Ostrovsky L.A., Troitskaya Yu.I. (1987) A model of turbulent transfer and dynamics of turbulence in a stratified shear flow. Izvestiya, Atm. and Oceanic Phys., 23(10), 767-773 (1987).

Within the framework of the theory of unsteady turbulent flows in a stratified fluid, a new parameterization of the turbulent Prandtl number is proposed. The parameterization is included in the k-ε-closure and used within the three-dimensional model of thermohydrodynamics of an enclosed water body where density distribution includes pycnocline. This allows us to describe turbulence in a stratified shear flow without the restrictions associated with the gradient Richardson number and justify the choice of closure constants. Numerical experiments, where the downward penetration of turbulence was considered, confirm the advantage of the developed approach in describing the effects neglected in the classical closures.

The interaction of small-scale turbulence with internal and surface waves is an urgent problem of hydrology and oceanology. In particular, this issue is especially important for the properties of the upper layer of the ocean and the inland waters. Small-scale processes that exist against the background of average profiles of various hydrophysical quantities (temperature, velocity, density, and large-scale currents caused, in particular, by wind forcing) are usually nonlinear and therefore effectively interact with each other. We consider some aspects of the interaction of internal waves and turbulence in the upper layer of the ocean and inland waters within the framework of the semi-empirical theory of turbulence in a stratified fluid. The model used in this study takes into account mutual transformation of the kinetic and potential energies of turbulent fluctuations [Ostrovsky&Troitskaya, 1987; Zilitinkevich et al., 2013]. The effects of amplification and maintenance of turbulence by low-frequency and high-frequency internal waves, quasi-stationary distributions of turbulent energy in the presence of a shear caused by a low-frequency internal wave are investigated; the role of the transformation of energies on the indicated processes is analyzed. A modification of the k-epsilon mixing scheme is also proposed, which removes the limitation on the existence of turbulence at large values of the gradient Richardson number. Within the framework of the modification, the parameterization of the Prandtl number is used, which makes it possible to take into account the influence of density stratification and velocity shear on mixing processes. A numerical study of the influence of vertical mixing schemes on the transfer processes of biochemical fields in an internal reservoir was also carried out. The modified scheme was implemented into a three-dimensional model of thermo-hydrodynamics and biochemistry of an inland water body [Gladskikh et al., 2021], and a series of numerical experiments was conducted. The work was supported by the RFBR (20-05-00776; 20-05-00322; 21-05-52005), and by Moscow Center of Fundamental and Applied Mathematics (agreement with the Ministry of Science and Higher Education 075-15-2019-1621). Ostrovsky LA, Troitskaya YuI (1987) A model of turbulent transfer and dynamics of turbulence in a stratified shear flow. Izv Akad Nauk SSSR, Fiz Atmos Okeana. 3:101–104. Zilitinkevich SS, Elperin T, Kleeorin N, Rogachevskii I, Esau I (2013) A hierarchy of Energyand Flux-Budget (EFB) turbulence closure models for stably-stratified geophysical flow. Boundary-Layer Meteorol. 146:341–373 Gladskikh DS, Stepanenko VM, Mortikov EV (2021) The Effect of the Horizontal Dimensions of Inland Water Bodies on the Thickness of the Upper Mixed Layer. Water Resour 48:226–234

A parameterization of the Prandtl number as a function of the gradient Richardson number is proposed in order to correctly take into account stratification when calculating the thermohydrodynamic regime of inland water bodies. This parameterization allows the existence of turbulence at any values ​​of the Richardson number. The proposed function is used to calculate the turbulent thermal conductivity coefficient in a k-epsilon mixing scheme. Modification is implemented in the three-dimensional hydrostatic model developed at the Research Computing Center of Moscow State University. It is demonstrated that the proposed modification (in contrast to the standard scheme with a constant Prandtl number) leads to smoothing all sharp changes in vertical distributions of turbulent mixing parameters (turbulent kinetic energy, temperature and thickness of the shock layer) and imposes a Richardson number-dependent relation on the empirical constants of k-epsilon turbulent mixing scheme. The work was supported by grants of the RF President’s Grant for Young Scientists (MK-1867.2020.5) and by the RFBR (19-05-00249, 20-05-00776).

An approximate analytical description of the nonstationary evolution of cylindrical nonlinear solitary waves with a complex structure is given. A modified Gardner equation with a boundary condition in the form of a “wide” soliton close to the limiting one is analyzed. The analysis shows a qualitative difference in the behavior of converging and diverging waves, as well as a difference from the quasi-stationary dynamics of cylindrical solitons.

In the framework of the modernized RANS model of turbulent closure [1], the evolution in the pycnocline and shear flow in the upper mixed layer of the ocean is studied. For this purpose, one of the variants of the model situation is considered, which consists in studying the mutual transformation of the buoyancy frequency, shear flow, as well as the kinetic and potential turbulence energies determined at the initial time at different depths. It is shown that the kinetic energy of turbulence increases with time, and its maximum shifts to the maximum of the the horizontal shear flow. However, unlike the standard gradient scheme, in the beginning there is a mutual transformation of the kinetic and potential turbulence energies, after which they quickly reach a stationary equilibrium level (at large values of the Richardson numbers). A significant change in stratification, initially having a maximum at a certain depth, was also found in the process of establishing a stationary turbulence regime. The work was financially supported by the Russian Foundation for Basic Research (projects № 18-05-00292, 18-35-00602). References: 1. Ostrovsky, L.A., Troitskaya, Yu.I., The model of turbulent transport and the dynamics of turbulence in a stratified shear flow/ Izvestiya, Atmospheric and Oceanic Physics., 1987. v.3. pp. 1031–1040

A way to parameterize the turbulent Prandtl number is proposed based on the model of turbulent transport in a stratified fluid, which allows for the two-sided transformation of the kinetic and potential energies of turbulent fluctuations. Numerical experiments aimed at studying the influence of the proposed parameterization on characteristic features of thermohydrodynamic processes in inland water bodies have been carried out.

The paper presents three major models used to describe the thermohydrodynamics of inland water objects: the standard one-dimensional E-epsilon (k-epsilon) model, the EFB-model by S. S. Zilitinkevich, and RANS-type model of turbulent transfer by L. O. Ostrovsky and Yu. I. Troitskaya. For models that take into account the two-sided transformation of the kinetic and potential energies of turbulent pulsations, the dependences of the turbulent Prandtl number on the gradient Richardson number are provided. The obtained dependences were used in the E-epsilon model to parameterize the coefficient of heat transfer eddy diffusivity in order to take into account stratification when calculating the thermohydrodynamic regimes of inland waters. The results of verification of the modified E-epsilon model based on experimental data are presented.

We have advanced the energy and flux budget turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGWs) and the shear-free stably stratified turbulence. This theory is based on the budget equation for the total (kinetic plus potential) energy of IGWs, the budget equations for the kinetic and potential energies of fluid turbulence, and turbulent fluxes of potential temperature for waves and fluid flow. The waves emitted at a certain level propagate upward, and the losses of wave energy cause the production of turbulence energy. We demonstrate that due to the nonlinear effects more intensive waves produce more strong turbulence, and this, in turn, results in strong damping of IGWs. As a result, the penetration length of more intensive waves is shorter than that of less intensive IGWs. The anisotropy of the turbulence produced by less intensive IGWs is stronger than that caused by more intensive waves. The low-amplitude IGWs produce turbulence consisting up to 90% of turbulent potential energy. This resembles the properties of the observed high-altitude tropospheric strongly anisotropic (nearly two-dimensional) turbulence.

We investigate the effect of the sea spray on the air-sea momentum exchange during the entire "life cycle" of a droplet, torn off the crest of a steep surface wave, and its fall down to the water, in the framework of a model covering the following aspects of the phenomenon: (1) motion of heavy particle in the driving air flow (equations of motion); (2) structure of the wind field (wind velocity, wave-induced disturbances, turbulent fluctuations); (3) generation of the sea spray; and (4) statistics of droplets (size distribution, wind speed dependence). It is demonstrated that the sea spray in strong winds leads to an increase in the surface drag up to 40 % on the assumption that the velocity profile is neutral.

The self-similar turbulent density jump evolution has been studied in the scope of a turbulence closure modernized theory which takes into account the anisotropy and mutual transformation of the turbulent fluctuation kinetic and potential energy for a stably stratified fluid. The numerical calculation, performed using the equations for the average density and kinetic and potential energies of turbulent fluctuations, indicates that the vertical profiles of the buoyancy frequency, turbulence scale, and kinetic and potential energies drastically change when the turbulence anisotropy is strong. The vertical profiles of the corresponding energy and spatial discontinuity parameters, calculated at a weaker anisotropy, indicate that similar drastic changes are absent and a qualitative agreement exists with the known analytical solution, which describes the density jump evolution in a freshwater basin and was obtained previously [5, 8] in the scope of a turbulence local-similarity hypothesis applied in combination with the budget equation for the turbulent fluctuation kinetic energy.

Within the framework of an approximate approach based on the representation of the Gardnerequation solitons as compound structures (different-polarity kinks), the non-quasistationary evolution of such solitary waves, which is stipulated by the variable quadratic-nonlinearity parameter α. The structure of the composite soliton is studied in cases that are critical for the quasistationary description where the predicted increase in the solitary-wave scales becomes unbounded on finite spatio-temporal intervals. The dependence of the spatial scales of the quasisoliton-field distribution on the quadratic-nonlinearity coefficient near the critical point for the power-law time dependence α(t) is studied in detail. The obtained solution is compared with the results of direct numerical simulation of the Gardner equation with variable coefficients.