Analysis of the annual mean energy cycle of the Black Sea circulation for the climatic, basin-scale and eddy regimes

Abstract

This work presents an analysis of the Lorenz energy cycles derived from the simulation results of the Black Sea circulation. Three numerical experiments are carried out based on an eddy-resolving z-model with a horizontal resolution of 1.6 km and taking into account different atmospheric forcing: climatic data, 2011, and 2016. The annual mean circulation for these time intervals reflects the climatic basin-scale, basin-scale (2011) and eddy (2016) regimes. Main differences between experiments are (1) the intensity of atmospheric fluxes and (2) SST assimilation and direct consideration of shortwave radiation in the realistic forcing simulations. The Lorenz energy cycles components are considered in detail. Some common features between climatic and realistic energetics are detected. The annual mean energy conversion from mean motion to the eddy is observed for all circulation regimes. Also, it is obtained that the annual mean buoyancy work enhances the mean current for all experiments, which evidences about maintaining of isopycnal surfaces slope such that a condition for converting available potential energy into kinetic energy is realized. Qualitative difference in the energy transfers for the climatic calculation, basin-scale and eddy regimes is revealed. Conversion from the eddy kinetic energy to eddy available potential energy is observed only for climatic circulation. For the basin-scale circulation the eddy kinetic energy is increasing mainly due to the transfer from the mean current kinetic energy through barotropic instability. The growth of the eddy kinetic energy for the eddy regime is provided by conversion of the available potential energy due to baroclinic instability.

Appendix

1.1 MHI model formulation

The three-dimensional nonlinear eddy-resolving MHI model is a computing complex for simulation of the Black Sea dynamics. The equations are written in the Boussinesq, hydrostatics and seawater incompressible approximations in a Cartesian coordinate system (the x-axis is directed to the east, the y-axis is to the north, the z-axis is downward from the surface to the bottom).

$${u}_{t}-(\xi +f)v+w{u}_{z}=-g{\zeta }_{x}-\frac{1}{{\rho }_{0}}(P\mathrm{^{\prime}}+E{)}_{x}+({\nu }_{V}{u}_{z}{)}_{z}-{\nu }_{H}{\nabla }^{4}u$$

(4)

$${v}_{t}+(\xi +f)u+w{v}_{z}=-g{\zeta }_{y}-\frac{1}{{\rho }_{0}}(P\mathrm{^{\prime}}+E{)}_{y}+({\nu }_{V}{v}_{z}{)}_{z}-{\nu }_{H}{\nabla }^{4}v$$

(5)

$$P=g{\rho }_{0}\zeta +g{\int }_{0}^{z}\rho dz=g{\rho }_{0}\zeta +P\mathrm{^{\prime}}$$

(6)

$${u}_{x}+{v}_{y}+{w}_{z}=0$$

(7)

$${\zeta }_{t}+\underset{0}{\overset{H}{\int }}\left({u}_{x}+{v}_{y}\right)dz=\left(Pr-Ev\right)/{\rho }_{1}$$

(8)

$${T}_{t}+(uT{)}_{x}+(vT{)}_{y}+(wT{)}_{z}=-{\kappa }_{H}{\nabla }^{4}T+({\kappa }_{T}{T}_{z}{)}_{z}-({\rho }_{0}{c}_{p}{)}^{-1}{I}_{z}$$

(9)

$${S}_{t}+(uS{)}_{x}+(vS{)}_{y}+(wS{)}_{z}=-{\kappa }_{H}{\nabla }^{4}S+({\kappa }_{S}{S}_{z}{)}_{z}$$

(10)

$$\rho =\rho (T,S)$$

(11)

where u, v, w are the components of velocity; ζ is the sea level; f is the Coriolis parameter; g is the gravity acceleration; P is the pressure; H is the sea depth; Pr is the precipitation rate; Ev is the evaporation rate; T is the temperature; S is the salinity; I is the solar radiation; c_p is the specific heat capacity of seawater; ρ is the seawater local density; ρ₀ = 1 g cm⁻³; ρ₁ is the area-averaged surface density; ν_H and κ_H are the horizontal viscosity and diffusion coefficients; ν_V is the vertical viscosity coefficient; κ_T and κ_S are the vertical heat and salt diffusion coefficients; $\xi =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$; $E={\rho }_{0}\frac{{u}^{2}+{v}^{2}}{2}$

Equation (8) is obtained under the assumption of implementation of the linear kinematic condition on the free surface (z = 0): $w=-{\zeta }_{t}+\frac{Pr-Ev}{{\rho }_{1}}$. In Eq. (11), the density non-linearly depends on temperature and salinity (Mamaev 1975). The temperature variability due to solar radiation in Eq. (9) is determined following (Paulson and Simpson 1977):

$$I\left(z\right)=SWR\left[a\mathrm{exp}\left(-z/{b}_{1}\right)+(1-a)\mathrm{exp}(-z/{b}_{2})\right]$$

where SWR is the shortwave radiation on the sea surface; a, b₁, b₂ are the empirical constants (set according to (Jerlov, 1968) and equal to 0.77, 1.5, and 14, respectively).

The Mellor-Yamada turbulent closure model level 2.5 (Mellor and Yamada 1982) is used to describe vertical mixing. Equations (4)–(11) are supplemented with equations for determining the turbulent kinetic energy ${e}^{2}$ and the turbulence macroscale $l$ :

$$\frac{d{e}^{2}}{dt}=\frac{\partial }{\partial z}\left({\mu }_{V}\frac{\partial {e}^{2}}{\partial z}\right)+2{\nu }_{V}\left[{\left(\frac{\partial u}{\partial z}\right)}^{2}+{\left(\frac{\partial v}{\partial z}\right)}^{2}\right]+\frac{2g}{{\rho }_{0}}{\kappa }_{T,S}\frac{\partial \rho }{\partial z}-\frac{2{e}^{3}}{{B}_{1}l}-{\nu }_{e}{\nabla }^{4}{e}^{2}$$

(12)

$$\frac{d({e}^{2}l)}{dt}=\frac{\partial }{\partial z}\left({\mu }_{V}\frac{\partial ({e}^{2}l)}{\partial z}\right)+l{E}_{1}{\nu }_{V}\left[{\left(\frac{\partial u}{\partial z}\right)}^{2}+{\left(\frac{\partial v}{\partial z}\right)}^{2}\right]+\frac{l{E}_{3}g}{{\rho }_{0}}{\kappa }_{T,S}\frac{\partial \rho }{\partial z}-\frac{{e}^{3}}{{B}_{1}}{H}_{1}-{\nu }_{e}{\nabla }^{4}\left({e}^{2}l\right)$$

(13)

$${\mu }_{V}=le{S}_{e},$$

(14)

where H₁ is the empirical function; ${E}_{1},\hspace{0.33em}{E}_{3}$ are the empirical constants; ${\nu }_{e}={\nu }_{H};{S}_{e}=0.2$.

Coefficients ν_V and κ_T,S are calculated as follows:

$${\nu }_{V}=le{S}_{M},\hspace{0.33em}\hspace{1em}{\kappa }_{T,S}=le{S}_{H}$$

(15)

where ${S}_{H},\hspace{0.33em}{S}_{M}$ are the stability functions, which determined as Eqs. (16) and (17).

$${S}_{M}={A}_{1}\left[\left(1-\frac{6{A}_{1}}{{B}_{1}}-3{C}_{1}\right)+9\left(2{A}_{1}+{A}_{2}\right){S}_{H}{G}_{H}\right](1-9{A}_{1}{A}_{2}{G}_{H}{)}^{-1},$$

(16)

$${S}_{H}={A}_{2}\left(1-\frac{6{A}_{1}}{{B}_{1}}\right){\left[1-3{A}_{2}{G}_{H}(6{A}_{1}+{B}_{2})\right]}^{-1},$$

(17)

Here, A₁, A₂, B₁, B₂, C₁ are the empirical constants; ${G}_{H}={\left(\frac{Nl}{e}\right)}^{2}=\frac{{l}^{2}}{{e}^{2}}\frac{g}{{\rho }_{0}}\frac{\partial \rho }{\partial z}$; $N={\left(\frac{g}{{\rho }_{0}}\frac{\partial \rho }{\partial z}\right)}^{1/2}$ is the buoyancy frequency.

The boundary conditions on the free surface for Eqs. (4), (5), (9), and (10) are as follows:

$${\rho }_{1}{\nu }_{V}{u}_{z}=-{\tau }^{x}, {\rho }_{1}{\nu }_{V}{v}_{z}=-{\tau }^{y}, {\rho }_{1}{\kappa }_{T}{T}_{z}={Q}^{T}, {\kappa }_{S}{S}_{z}=\frac{Pr\text{ } - {\text{Ev}}}{{\rho }_{1}}{S}_{0}+\beta \left({S}_{0}-{S}^{cl}\right),$$

(18)

where τ^x and τ^y are the wind stress components; ${Q}^{T}$ is the heat flux (superposing thermal, sensible and latent heat fluxes); S₀ is the surface salinity; ${S}^{cl}$ is the climatological salinity; $\beta$ is the relaxation parameter, β = 2/1728 cm s⁻¹. If wind forcing provides a velocity W, then the wind stress are recalculated by the formula:

$${\varvec{\uptau}}={C}_{d}{\rho }_{a}{\mathbf{W}}^{2}$$

where ρ_a is the air density; C_d is the drag coefficient with magnitudes by (Uppala et al. 2005).

The boundary conditions are:

on the bottom

$$u=v=w=0, {T}_{z}=0, {S}_{z}=0$$

(19)

on the meridional solid boundary

$$u=0,\hspace{1em}{\nabla }^{2}u=0,\hspace{1em}{v}_{x}=0,\hspace{1em}{\nabla }^{2}{v}_{x}=0,\hspace{1em}{T}_{x}=0,\hspace{1em}({\nabla }^{2}T{)}_{x}=0,\hspace{1em}{S}_{x}=0,\hspace{1em}({\nabla }^{2}S{)}_{x}=0,$$

(20)

on the zonal solid boundary

$$v=0,\hspace{1em}{\nabla }^{2}v=0,\hspace{1em}{u}_{y}=0,\hspace{1em}{\nabla }^{2}{u}_{y}=0,{T}_{y}=0,\hspace{1em}({\nabla }^{2}T{)}_{y}=0,\hspace{1em}{S}_{y}=0,\hspace{1em}({\nabla }^{2}S{)}_{y}=0.$$

(21)

The Dirichlet conditions are used for liquid boundary

on the meridional one

$$u={u}^{r},\hspace{0.33em}\hspace{0.33em}{\nabla }^{2}u=0,\hspace{0.33em}\hspace{0.33em}{v}_{x}=0,\hspace{0.33em}\hspace{0.33em}{\nabla }^{2}{v}_{x}=0, T={T}^{r},\hspace{0.33em}\hspace{0.33em}S={S}^{r},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}({\nabla }^{2}T{)}_{x}=0,\hspace{0.33em}\hspace{0.33em}({\nabla }^{2}S{)}_{x}=0,$$

(22)

on the zonal one

$$v={v}^{r},\hspace{0.33em}\hspace{0.33em}{\nabla }^{2}v=0,\hspace{0.33em}\hspace{0.33em}{u}_{y}=0,\hspace{0.33em}\hspace{0.33em}{\nabla }^{2}{u}_{y}=0,$$

$$T={T}^{r},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}S={S}^{r},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}({\nabla }^{2}T{)}_{y}=0,\hspace{0.33em}\hspace{0.33em}({\nabla }^{2}S{)}_{y}=0,$$

(23)

and on the region Bosphorus Strait (upper flow) and Kerch Strait (if flow directs from Black to Azov Sea)

$$v={v}^{s},\hspace{1em}{\nabla }^{2}v=0,\hspace{0.33em}\hspace{0.33em}{u}_{y}=0,\hspace{0.33em}\hspace{0.33em}{\nabla }^{2}{u}_{y}=0,\hspace{1em}{T}_{y}=0,\hspace{1em}{S}_{y}=0,\hspace{0.33em}\hspace{1em}({\nabla }^{2}T{)}_{y}=0,\hspace{0.33em}\hspace{0.33em}({\nabla }^{2}S{)}_{y}=0.$$

(24)

The upper indexes r and s in Eqs. (22)–(24) denotes river and strait, respectively. The used model configuration takes into account the runoff of the Dnieper, Danube, Dniester, Sakarya, Kizilirmak, Yeshilirmak, Rioni rivers and exchange through the straits. The runoff of other Caucasian rivers is set at two points in the northeastern part of the sea. Salinity at river mouths is equal to 7 ‰, the temperature is set according to climatological data (Simonov and Altman 1991). The upper Bosphorus flow extends to depth of 27.5 m, the temperature and salinity here are the same as in the sea. In the lower Bosphorus flow (depth up to 68.75 m), salinity is taken equal to 35 ‰ and temperature is equal to 16 °C, that corresponds to the annual characteristics of the Marmara Sea. Seasonal variability of the river runoff and the upper Bosphorus discharge are calculated from the monthly climatological values obtained by long term data (Simonov and Altman 1991) and presented in Table 3. Negative values in Table 3 indicate that water flows out of the Black Sea. The seasonal discharge value of the lower Bosporus is estimated on the assumption that the mass of water in the Black Sea remains constant over year. So, superposition of the river runoff, discharges through the straits, precipitation, and evaporation over year is obtained zero.

Boundary and initial conditions for Eqs. (12) and (13) are as follows:

$${e}^{2}={B}_{1}^{2/3}{\left[\left({\tau }^{x}+{\tau }^{y}\right)/{\rho }_{0}^{2}\right]}^{1/2}, {e}^{2}l=0\;for\;z=0,$$

$${e}^{2}=0, {e}^{2}l=0\;for\;z=H\left(x,y\right),e={e}^{0}, l={l}^{0}\;for\;t={t}^{0}$$

(25)

The horizontal turbulent coefficients in Eqs. (4), (5), (9), and (10) are equal to

ν_H = 10¹⁶ cm⁴ s⁻¹ and κ_H = 10¹⁶cm⁴ s⁻¹.

The constants in Eqs. (12), (13), (18), and (25) are selected according to (Bagaiev and Demyshev 2011) as next: A₁ = 0.92; A₂ = 0.74; B₁ = 16.6; B₂ = 10.1; C₁ = 0.08; E₁ = 1.8; E₃ = 1.8; ${e}^{0}=$ 10 cm² s⁻²;${l}^{0}=$ 10 cm.

The sea level, temperature, salinity, and horizontal velocity are set at initial moment t⁰:$u={u}^{0}(x,y,z),\hspace{1em}v={v}^{0}(x,y,z),\hspace{1em}\varsigma ={\varsigma }^{0}(x,y),\hspace{1em}T={T}^{0}(x,y,z),\hspace{1em}S={S}^{0}(x,y,z)$

Table 3 Climatological monthly the Black Sea rivers runoff, the Upper Bosphorus and Kerch Strait discharges (m³ s⁻¹)

To increase accuracy of the simulated temperature in subsurface layer, the daily sea surface temperature (SST) are assimilated on the upper model horizon. We can use the reanalysis or satellite data, and the assimilation procedure are constructed with taking into account possible lack of satellite data. The deviation between the model surface temperature ${T}_{0}^{m}$ and the assimilated one ${T}^{as}$ is calculated, and the equation for the residual temperature ${\delta }^{T}$

$${\nabla }^{2}{\delta }^{T}=0$$

with the conditions on solid (${\delta }^{T}=0$) and liquid (${\delta }^{T}={T}_{0}^{m}-{T}^{as}$) boundaries is solved. The liquid boundary separates a region (or regions) for which observational data are available from the rest of the sea area. The model temperature is corrected according to the equation

$${T}_{*}^{m}={T}_{0}^{m}+{\delta }^{T}.$$

(26)

Then the diffusion equation is solved to filter small-scale disturbances

$${\left({T}_{*}^{m}\right)}_{t}=-{\kappa }_{H}{\nabla }^{4}{T}_{*}^{m}$$

with the boundary conditions ${\left({T}_{*}^{m}\right)}_{n}={\left({\nabla }^{2}{T}_{*}^{m}\right)}_{n}=0$, where n is the normal to the boundary.

The finite-difference approximation of the model equations, the boundary and initial conditions is implemented on the C grid. The leapfrog scheme (Roache 1998) is used for the time approximation. The advective terms in Eqs. (9) and (10) is approximated by TVD scheme (Harten 1983). Horizontal turbulent mixing in Eqs. (4), (5), (9), and (10) is described by biharmonic Laplace operator, which filters the computational noise and stabilizes the numerical solution. Discrete equations are exact consequence of the finite-difference model equations, and it allows to provide conservation of main invariants.

The model domain is constructed on uniform grid (698 × 390 points) with horizontal resolution of (1/48)° on longitude and (1/66)° on latitude. It equals about of 1.6 km, and less than the Rossby deformation radius in the Black Sea (10–30 km (Poulain et al. 2005)). The vertical resolution is 27 horizons with depth 2.5, 5, 10, 15, 20, 25, 30, 40, 50, 62.5, 75, 87.5, 100, 112.5, 150, 200 м, from 200 to 500 m through 100 m, from 700 to 2100 m through 200 m. Time step is equal 1.5 min. The atmospheric fields are linearly interpolated to each time step. The model domain bathymetry is built on the data of the European Marine Observation and Data Network (EMODnet, http://portal.emodnet-bathymetry.eu) with resolution of (1/8)'.