A simple description of turbulent transport in a stratified shear flow devoted to the simulation of thermohydrodinamics of inland waters

The paper presents three major models used to describe the thermohydrodynamics of inland water objects: the standard one-dimensional E-ε (k-ε) 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-ε 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-ε model based on experimental data are presented.


Introduction
By now, the role of inland water bodies in the processes of general circulation of the atmosphere, the ocean and land, their thermohydrodynamic and biological properties in weather forecasts, and also in climate change has been established [1][2][3][4][5][6]. The basis of physico-mathematical simulation of thermodynamics and hydrodynamics of inland waters is RANS (Reynolds-averaged Navier-Stokes equations) [7]. Since this system contains unknown quantities (turbulent flows), it is necessary to involve additional hypotheses for its closure. Practically significant is the so-called E-ε (k-ε) model based on the equations for the kinetic energy of turbulence E and its dissipation rate ε. The ratio of the coefficient of momentum eddy diffusivity to the coefficient of heat transfer eddy diffusivity (the turbulent Prandtl number, PrT) is assumed to be constant. This, in particular, limits the description of the known effect [8][9] associated with the existence of turbulence for large gradient Richardson numbers Ri. At the same time, both laboratory and field measurements demonstrate the dependence of PrT on Ri. In this connection, special attention in the simulation of the thermohydrodynamic regime of inland water bodies is paid to stratification and, in particular, to its role in the processes of turbulent mixing, thermocline dynamics, etc. Recently, modernized approaches to the description of geophysical turbulence are actively developing, taking into account stable and unstable stratification, as well as internal waves interacting with turbulence and even amplifying it. Authors propose geophysical turbulence models that take into account the two-sided transformation of the kinetic and potential energies of turbulent pulsations. This is the so-called EFB model [10] and the closed Reynolds system of equations (RANS) [11] obtained on the basis of the sequential kinetic approach. These models allowed, in particular, to explain the case of maintaining turbulence observed in laboratory and field conditions by weak velocity shears, as well as to calculate the dependence of PrT on Ri. In the present work, a parameterization of the coefficient of heat transfer eddy diffusivity for the E-ε model is proposed in order to take into account stratification when calculating the thermohydrodynamic regimes of inland waters. Verification of the proposed parameterization is presented.

E-ε (kε) model for the description of thermohydrodynamics of inland waters
In this paper, the model is used in the form proposed in [12]: where a E and a ε are dimensionless constants, P is total generation of turbulence energy through the shear of velocity and effect of density stratification, C 1 is the function of Reynolds number. The beginning of the vertical downward z-coordinate is supposed to be coupled with water surface z=h(t), so we can turn from z to the new independent variable = , where h=h(t) is a depth of water body, and t is a time.
The coefficient of momentum eddy diffusivity is as follows: where С е is dimensionless constant.
Vertical heat exchange, expressed as the coefficient of heat transfer eddy diffusivity λ, is defined as: = . (2. 3) It is used in the calculations of the temperature distribution (see below) and the parameter P in equations (2.1-2. 2) The numerical model for calculating the temperature distribution is based on solving a onedimensional heat equation: where c is heat capacity of the water, ρ -its density, λthe coefficient of heat transfer eddy diffusivity, T -temperature, = 0 -water balance at free surface, S -solar radiation flux penetrating the water body. As for the water object dynamics, in [12] the zonal u and the meridional v components of the flow velocity used to compute the parameter P and the gradient Richardson number Ri in our study, are found using the following equations: where f is the Coriolis frequency. Components of friction stress are used to calculate the boundary conditions for u and v according to the Monin-Obukhov similarity theory at the upper boundary (water -atmosphere) and according to the Chezi formulas [13] taking into account the surface roughness at the lower boundary (water -soil ).

EFB-model and parameterization of turbulent Prandtl number
This model, presented in [9][10], is based on the balance equations for the kinetic energy of turbulence E K , potential energy E P , turbulent fluxes of momentum τ i3 and potential temperature F z , as well as on the relaxation equation for the turbulent time scale t T : Here is the substantial derivative, K E , K FM , K FH , K T are the coefficients of turbulent transport, β is the buoyancy parameter, E z is the vertical component of the kinetic energy, C P , Cτ, C θ , C F , C R are constants, Θ is the average potential temperature, t TE is the equilibrium relaxation time. It should be noted that this model is rather difficult for numerical simulation, but it allows to calculate various characteristics of turbulence in the conditions from neutral stratification to extremely stable one. The model of gradient transfer under the condition of equilibrium versions of balance equations for turbulent fluxes of momentum and potential temperature follows from the system (3.1-3.5). In this case: is the coefficient of heat transfer eddy diffusivity (is equivalent to λ), is the coefficient of momentum eddy diffusivity (is equivalent to ), = , = П , П = .
Let us consider the following simple task of the evolution of homogeneous turbulence in a stationary shear flow with a given velocity profile = 0 , where 0 = . Shear flow is stably stratified and with a constant Brunt-Väisälä frequency N: 2 = Θ . The turbulent regime of such a fluid will be described by a system of equations of the first order in time: The system has two equilibrium states: an unstable one = = 0 and stable one with a non-zero value of the energy parameter of the stratification П = , defined by the relation: where П ∞ is the limiting value of the parameter equal to 0.287. So, from the definition of the turbulent Prandtl number and (3.7-3.9) we get:  Fig. 1 (see below) the dependence of the turbulent Prandtl number on the gradient Richardson number is presented. You can see that PrT(Ri) is an unboundedly increasing function, and for large ≫ 1, the Prandtl number grows linearly.

Turbulent transfer model by L. Ostrovsky and Yu. Troitskaya
On the basis of the kinetic approach, the expressions for turbulent fluxes and average values of the corresponding hydrodynamic quantities were obtained in [11]. This allowed us to close the RANS equations: where R is a parameter characterizing the anisotropy of small-scale turbulence and depending on the ratio of the vertical and horizontal scales of correlation field of density fluctuations. Values of R equal to 0.3 and 0.5 were considered, PrT (Ri) is presented in Fig. 1.

Results of calculation
In the previous work of the group of authors [14] fairly accurate results for calculating the thermodynamics of inland water bodies were obtained with the standard E-ε model. They were used to verify the results of the current study. The verification was based on experimental data obtained at the Gorky reservoir experimental site in the period 2014-2017. In the present work, the parameterization of the coefficient of heat transfer eddy diffusivity for the E-ε model is proposed in order to take into account stratification when calculating the thermohydrodynamic regimes of inland water bodies. Formula (2.3) turns into the following form: It is demonstrated that the use of parameterizations made it possible to obtain no less accurate results, but, as the analysis of the standard deviation (see Table 1) showed, even more accurate ones. Case 1 (Fig. 1a) 0.23 0.19 0.20 Case 2 (Fig. 1b) 0.63 0.60 0.60

Conclusion
In this paper, we consider three main models of turbulence, which are used to describe the thermohydrodynamics of both inland waters, and the ocean and atmosphere. 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-ε 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. Based on the previously obtained results [14], it was found that the use of parameterizations allows reducing the standard deviation from the real values Due to the fact that such parameterization allows to correctly describe the processes of turbulence in inland waters, in the future they are planned to be used for the conditions of the ocean and atmosphere, including for large values of the gradient Richardson number.