## Discrete Equation for the Available Potential Energy as an Exact Consequence of the Numerical Model Equations

*S. G. Demyshev*

Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation

*e-mail: demyshev@gmail.com*

### Abstract

*Purpose*. The work is aimed at obtaining a discrete equation for the rate of the available potential energy change in strict accordance with the finite-difference formulation that ensures adequate reproduction of discrete energy, and at analyzing its terms based on the results of a numerical experiment with realistic atmospheric forcing.

*Methods and Results*. On the basis of the well-known methods of computational mathematics (method of indeterminate coefficients and imitation modeling), a finite-difference equation for the available potential energy, which corresponded to its differential form, was obtained. In the equation structure, an additional term, which was conditioned by transition to a discrete problem and had a diffusion form, appeared. Energy analysis for the hydrological winter of 2011 in the Black Sea showed that the highest values of available potential energy in the upper layer were observed in the central region of the sea. Below 100 m, the available potential energy increased towards the coast where intense mesoscale variability was observed. At the depths exceeding 200 meters, the largest stock of this energy was concentrated in the Sevastopol and Batumi anticyclones. Action of the main forces, namely the forces of buoyancy, advection and horizontal diffusion, takes place in the coastal areas of the sea.

*Conclusions*. The resulting difference equation for the rate of the available potential energy change exactly corresponds to the discrete formulation and, therefore, accurately reflects the energy of the discrete problem. Analysis of the equation permitted to show that in winter, the rate of the available potential energy change is influenced predominantly by eddy activity at the depth slope.

### Keywords

numerical simulation, available potential energy, potential energy, Black Sea, cyclonic circulation, anticyclonic eddies, discrete energy equation, difference equation

### Acknowledgements

The investigation was carried out within the framework of the state assignment on theme No. 0555-2021-0004 “Fundamental studies of oceanological processes determining state and evolution of marine environment under the influence of natural and anthropogenic factors, based on the observation and modeling methods”.

### Original russian text

Original Russian Text © S. G. Demyshev, 2022, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 38, Iss. 3, pp. 239-255 (2022)

### For citation

Demyshev, S.G., 2022. Discrete Equation for the Available Potential Energy as an Exact Consequence of the Numerical Model Equations. *Physical Oceanography*. 29(3), pp. 221-236. doi:10.22449/1573-160X-2022-3-221-236

### DOI

10.22449/1573-160X-2022-3-221-236

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