A Discrete Transport Equation with Nonlinear Invariants: Application to the Black Sea Circulation Modelling

S. G. Demyshev✉, O. A. Dymova

Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation

✉ e-mail: demyshev@gmail.com

Abstract

Purpose. The purpose of this study is to investigate a new approximation scheme for nonlinear terms in the heat and salt transport equations. It ensures the conservation of temperature in the first and Kth (K > 2) degree, salinity in the first and Lth (L > 2) degree and density as a polynomial in temperature and salinity. To evaluate the efficacy of the new scheme for modelling sea circulation under realistic boundary conditions.

Methods and Results. The new approximation is tested on the reconstruction of the Black Sea circulation using the MHI-model and ERA5 atmospheric forcing for 2016. The results show that the Cold Intermediate Layer becomes sharper and its depth decreases; similarly, the depth of the upper boundary of the permanent halocline is reduced. The modelling results are compared with in-situ data and the hydrophysical fields obtained by using a traditional approximation scheme. Validation of the results shows that the modelling error of the salinity fields decreases in the upper 100-m layer when the new scheme is used.

Conclusions. An approximation scheme for the heat and salt advection equations that preserves nonlinear invariants was developed and generalized to the case of a functional depending on two or more functions. The decrease in modelling errors is related to the refinement of seawater density gradients, vertical mixing coefficients and upwelling/downwelling velocities. Spectral analysis of the current kinetic energy and the vertical velocity demonstrates a more accurate redistribution of energy along the motion spectrum compared to data from the earlier version of the MHI-model.

Keywords

finite-difference approximation, nonlinear invariants, conservation laws, Black Sea, numerical modelling, thermohaline circulation, energy spectrum

Acknowledgements

This work was supported by the state assignment of the FSBSI FRC MHI, project No. FNNN-2024-0001.

About the authors

Sergey G. Demyshev, Head of Wave Theory Department, Senior Researcher, Marine Hydrophysical Institute of RAS (2 Kapitanskaya Str., Sevastopol, 299011, Russian Federation), DSc. (Phys.-Math.), Scopus Author ID: 6603919865, SPIN-code: 1848-2350, ResearcherID: C-1729-2016, ORCID ID: 0000-0002-5405-2282, demyshev@gmail.com

Olga A. Dymova, Leading Researcher, Marine Hydrophysical Institute of RAS (2 Kapitanskaya Str., Sevastopol, 299011, Russian Federation), CSc. (Phys.-Math.), Scopus Author ID: 6508381809, ResearcherID: P-9669-2015, ORCID ID: 0000-0003-4036-2447, olgdymova@mhi-ras.ru

For citation

Demyshev, S.G. and Dymova, O.A., 2026. A Discrete Transport Equation with Nonlinear Invariants: Application to the Black Sea Circulation Modelling. Physical Oceanography, 33(1), pp. 156-184.

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