hydrophysicsФундаментальная и прикладная гидрофизикаFundamental and Applied Hydrophysics2073-66732782-5221St. Petersburg Research Center of the Russian Academy of Sciences10.59887/2073-6673.2025.18(3)-3MPRZSLhydrophysics-1447Research ArticleГИДРОФИЗИЧЕСКИЕ И БИОГЕОХИМИЧЕСКИЕ ПОЛЯ И ПРОЦЕССЫHYDROPHYSICAL AND BIOGEOCHEMICAL FIELDS AND PROCESSESО подходах к улучшению описания затоков североморских вод в моделях общей циркуляции Балтийского моряOn Approaches to Improving the Description of North Sea Water Inflows in the General Circulation Models of the Baltic Seahttps://orcid.org/0000-0002-3891-3396ВанкевичР. Е.VankevichR. E.

ВАНКЕВИЧ Роман Евгеньевич, старший научный сотрудник, кандидат технических наук

Scopus AuthorID: 25642198100

WoS ResearcherID: M‑3215–2013

117997, Москва, Нахимовский проспект, д.36

36 Nakhimovsky Prosp., Moscow 117997

rvankevich@mail.ruhttps://orcid.org/0000-0003-2005-4949ИсаевА. В.IsaevA. V.

ИСАЕВ Алексей Владимирович, старший научный сотрудник, кандидат географических наук

WoS ResearcherID: C‑1370–2014

Scopus AuthorID: 25641182000

117997, Москва, Нахимовский проспект, д.36

36 Nakhimovsky Prosp., Moscow 117997

isaev1975@gmail.comhttps://orcid.org/0000-0003-3909-537XРябченкоВ. А.RyabchenkoV. A.

РЯБЧЕНКО Владимир Алексеевич, главный научный сотрудник, доктор физико-математических наук

WoS ResearcherID: R‑3877–2016

Scopus AuthorID: 7005479766

117997, Москва, Нахимовский проспект, д.36

36 Nakhimovsky Prosp., Moscow 117997

vla-ryabchenko@yandex.ruИнститут океанологии им. П.П. Ширшова РАНРоссияShirshov Institute of Oceanology, Russian Academy of SciencesRussian Federation2025061020251833752Copyright © Ванкевич Р.Е., Исаев А.В., Рябченко В.А., 20252025Ванкевич Р.Е., Исаев А.В., Рябченко В.А.Vankevich R.E., Isaev A.V., Ryabchenko V.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://hydrophysics.spbrc.ru/jour/article/view/1447

Рассматриваются причины недооценки придонной солёности в моделях общей циркуляции Балтийского моря. На основе высокоэффективной модели Oceananigans выполнены расчеты распространения затока солёных североморских вод в придонном слое Балтийского моря в период январь–апрель 1993 г. Результаты этих расчетов находятся в лучшем соответствии с данными В.Т. Паки высокочастотного сканирования вертикальной структуры моря с вертикальным разрешением от 0,25 до 0,5 м и расстояниями между станциями около 500 м, чем результаты аналогичного расчёта (реанализа) для той же ситуации, выполненного по модели NEMO-Nordic. Этого удалось достигнуть за счёт существенного снижения численной диффузии путём увеличения вертикального разрешения и использования адвективных схем высокого порядка точности в модели Oceananigans без использования процедуры ассимиляции данных наблюдений. Указанные усовершенствования модели Oceananigans понизили скорость вычислений всего лишь на 30 %. Разработанная модельная конфигурация даёт определённые надежды на получение без процедуры ассимиляции адекватных оценок распространения придонного гравитационного плотностного течения в Балтийском море и его взаимодействия с осадками, включая обмен кислородом, что важно для моделирования морских экосистем.

The paper considers the causes of underestimation of bottom salinity in the general circulation models of the Baltic Sea. The highly efficient Oceananigans model is used to calculate the propagation of the inflow of saline North Sea waters in the bottom layer of the Baltic Sea in the period January-April 1993. The results of these calculations are in better agreement with the V.T. Paka high-frequency scanning data of the vertical structure of the sea with a vertical resolution of 0.25 to 0.5 m and distances between stations of about 500 m than the results of a similar calculation (reanalysis) for the same situation performed using the NEMO-Nordic model. This was achieved by significantly reducing numerical diffusion by increasing the vertical resolution and using high-order advection schemes in the Oceananigans model without using the observational data assimilation procedure. The above improvements to the Oceananigans model reduced the calculation speed by only 30 %. The developed model configuration gives certain hope for obtaining, without the assimilation procedure, adequate estimates of the propagation of the bottom gravity density current in the Baltic Sea and its interaction with sediments, including oxygen exchange, which is important for modeling marine ecosystems.

Балтийское морематематическое моделированиезаток североморских водплотностное течениепридонный слойBaltic Seamathematical modelinginflow of North Sea watersdensity currentbottom layerРабота выполнена в рамках государственного задания Минобрнауки России для ИО РАН (тема № FMWE‑2024-0028)The work was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia for the Institute of Oceanology of the Russian Academy of Sciences (topic No. FMWE‑2024-0028)ReferencesMohrholz V., Naumann M., Nausch G., Krüger S., Gräwe U. Fresh oxygen for the Baltic Sea — An exceptional saline inflow after a decade of stagnation // Journal of Marine Systems. 2015. Vol. 148. P. 152–166. https://doi.org/10.1016/j.jmarsys.2015.03.005Mohrholz V, Naumann M, Nausch G, Krüger S, Gräwe U. Fresh oxygen for the Baltic Sea — An exceptional saline inflow after a decade of stagnation. Journal of Marine Systems. 2015;148:152–166. https://doi.org/10.1016/j.jmarsys.2015.03.005Gräwe U., Naumann M., Mohrholz V., Burchard H. Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014 // Journal of Geophysical Research: Oceans. 2015. Vol. 120. P. 7676–7697. https://doi.org/10.1002/2015jc011269Gräwe U, Naumann M, Mohrholz V, Burchard H. Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014. Journal of Geophysical Research: Oceans. 2015;120:7676–7697. https://doi.org/10.1002/2015jc011269Mohrholz V. Major Baltic Inflow Statistics — Revised // Frontiers in Marine Science. 2018. Vol. 5, N 384. EDN IQMGTT. https://doi.org/10.3389/fmars.2018.00384Mohrholz V. Major Baltic Inflow Statistics — Revised. Frontiers in Marine Science. 2018;5:384. https://doi.org/10.3389/fmars.2018.00384Lehmann A. A three-dimensional baroclinic eddy-resolving model of the Baltic Sea // Tellus A: Dynamic Meteorology and Oceanography. 1995. Vol. 47. P. 1013–1031. https://doi.org/10.3402/tellusa.v47i5.11969Lehmann A. A three-dimensional baroclinic eddy-resolving model of the Baltic Sea. Tellus Series A: Dynamic Meteorology and Oceanography. 1995;47:1013–1031. https://doi.org/10.3402/tellusa.v47i5.11969Meier M., Doescher R., Coward A.C., Nycander J., Döös K. RCO — Rossby Centre regional Ocean climate model: model description (version 1.0) and first results from the hindcast period 1992/93: Tech. Rep. 26, SMHI, Norrköping, Sweden, 1999.Meier M, Doescher R, Coward AC, Nycander J, Döös K. RCO — Rossby Centre regional Ocean climate model: model description (version 1.0) and first results from the hindcast period 1992/93. Tech Rep. 1999;(26): SMHI, Norrköping, Sweden.Funkquist L., Kleine E. HIROMB: An introduction to HIROMB, an operational baroclinic model for the Baltic Sea: Reports Oceanography 37. Swedish Meteorological and Hydrological Institute, Norrköping, Sweden, 2007.Funkquist L, Kleine E. HIROMB: An introduction to HIROMB, an operational baroclinic model for the Baltic Sea. Reports Oceanogr. 2007;(37): Swedish Meteorological and Hydrological Institute, Norrköping, Sweden.Berg P., Poulsen J.W. Implementation details for HBM: Tech. Rep. Danish Meteorological Institute, Copenhagen, Denmark, 2012.Berg P, Poulsen JW. Implementation details for HBM. Tech Rep. 2012; Danish Meteorological Institute, Copenhagen, Denmark.Dietze H., Löptien U., Getzlaff K. MOMBA 1.1 — a high-resolution Baltic Sea configuration of GFDL’s Modular Ocean Model // Geoscientific Model Development. 2014. Vol. 7. P. 1713–1731. EDN VFTAZT. https://doi.org/10.5194/gmd7-1713-2014Dietze H, Löptien U, Getzlaff K. MOMBA 1.1 — a high-resolution Baltic Sea configuration of GFDL’s Modular Ocean Model. Geoscientific Model Development. 2014;7:1713–1731. https://doi.org/10.5194/gmd7-1713-2014Gräwe U., Naumann M., Mohrholz V., Burchard H. Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014 // Journal of Geophysical Research: Oceans. 2015. Vol. 120. P. 7676–7697. https://doi.org/10.1002/2015jc011269Gräwe U, Naumann M, Mohrholz V, Burchard H. Anatomizing one of the largest saltwater inflows into the Baltic Sea in December 2014. Journal of Geophysical Research: Oceans. 2015;120:7676–7697. https://doi.org/10.1002/2015jc011269Рябченко В.А., Карлин Л.Н., Исаев А.В. и др. Модельные оценки эвтрофикации Балтийского моря в современном и будущем климате // Океанология. 2016. Vol. 56, N 1. P. 41–50. EDN VIOSEJ. https://doi.org/10.7868/S0030157416010160Ryabchenko VA, Karlin LN, Isaev AV, et al. Model estimates of the eutrophication of the Baltic Sea in the contemporary and future climate. Oceanology. 2016;56(1):36–45. https://doi.org/10.7868/S0030157416010160Hordoir R., Axell L., Höglund A. et al. Nemo-Nordic 1.0: a NEMO-based ocean model for the Baltic and North seas — research and operational applications // Geoscientific Model Development. 2019. Vol. 12. P. 363–386. EDN NTIIIL. https://doi.org/10.5194/gmd-12-363-2019Hordoir R, Axell L, Höglund A, et al. Nemo-Nordic 1.0: a NEMO-based ocean model for the Baltic and North seas — research and operational applications. Geoscientific Model Development. 2019;12:363–386. https://doi.org/10.5194/gmd12-363-2019Kärnä T., Ljungemyr P., Falahat S. et al. Nemo-Nordic 2.0: operational marine forecast model for the Baltic Sea // Geoscientific Model Development. 2021. Vol. 14. P. 5731–5749. EDN: ZMZZPA. https://doi.org/10.5194/gmd14-5731-2021Kärnä T, Ljungemyr P, Falahat S, et al. Nemo-Nordic 2.0: operational marine forecast model for the Baltic Sea. Geoscientific Model Development. 2021;14:5731–5749. https://doi.org/10.5194/gmd14-5731-2021Madec G., Bourdallé-Badie R., Chanut J. et al. NEMO ocean engine // Zenodo. 2019. https://doi.org/10.5281/zenodo.3878122Madec G, Bourdallé-Badie R, Chanut J, et al. NEMO ocean engine. Zenodo. 2019. https://doi.org/10.5281/zenodo.3878122Deremble B., Hogg A.M., Berloff P., Dewar W.K. On the application of no-slip lateral boundary conditions to ‘coarsely’ resolved ocean models // Ocean Modelling. 2011. Vol. 39, N3. P. 411–415. https://doi.org/10.1016/j.ocemod.2011.05.002Deremble B, Hogg AM, Berloff P, Dewar WK. On the application of no-slip lateral boundary conditions to ‘coarsely’ resolved ocean models. Ocean Modelling. 2011;39(3):411–415. https://doi.org/10.1016/j.ocemod.2011.05.002Nasser A.A. Advancing the representation of flows along topography in z-coordinate ocean models // Ocean, Atmosphere. Sorbonne Université, 2023. English. NNT: 2023SORUS446. https://theses.hal.science/tel04399405v1Nasser AA. Advancing the representation of flows along topography in z-coordinate ocean models. Ocean, Atmosphere. Sorbonne Université; 2023. https://theses.hal.science/tel04399405v1Hill C., Ferreira D., Campin J.-M. et al. Controlling spurious diapycnal mixing in eddy-resolving height-coordinate ocean models — Insights from virtual deliberate tracer release experiments // Ocean Modelling. 2012. Vol. 45–46. P. 14–26. https://doi.org/10.1016/j.ocemod.2011.12.001Hill C, Ferreira D, Campin J-M, Marshall J, Abernathey R, Barrier N. Controlling spurious diapycnal mixing in eddy-resolving height-coordinate ocean models — Insights from virtual deliberate tracer release experiments. Ocean Modelling. 2012;45–46:14–26. https://doi.org/10.1016/j.ocemod.2011.12.001Ilicak M., Adcroft A., Griffies S., Hallberg R. Spurious dianeutral mixing and the role of momentum closure // Ocean Modelling. 2012. Vol. 45–46. P. 37–58. https://doi.org/10.1016/j.ocemod.2011.10.003Ilicak M, Adcroft A, Griffies S, Hallberg R. Spurious dianeutral mixing and the role of momentum closure. Ocean Modelling. 2012;45–46:37–58. https://doi.org/10.1016/j.ocemod.2011.10.003Ilicak M. Quantifying spatial distribution of spurious mixing in ocean models // Ocean Modelling. 2016. Vol. 108. P. 30– 38. https://doi.org/10.1016/j.ocemod.2016.11.002Ilicak M. Quantifying spatial distribution of spurious mixing in ocean models. Ocean Modelling. 2016;108:30–38. https://doi.org/10.1016/j.ocemod.2016.11.002Megann A. Estimating the numerical diapycnal mixing in an eddy-permitting ocean model // Ocean Modelling. 2018. Vol. 121. P. 19–33. EDN: YIGMNE. https://doi.org/10.1016/j.ocemod.2017.11.001Megann A. Estimating the numerical diapycnal mixing in an eddy-permitting ocean model. Ocean Modelling. 2018;121:19–33. https://doi.org/10.1016/j.ocemod.2017.11.001Campin J.-M., Goosse H. Parameterization of density-driven downsloping flow for a coarse-resolution ocean model in z-coordinate // Tellus A: Dynamic Meteorology and Oceanography. 1999. 51A. P. 412–430. EDN LPTUZJ. https://doi.org/10.3402/tellusa.v51i3.13468Campin J-M, Goosse H. Parameterization of density-driven downsloping flow for a coarse-resolution ocean model in z-coordinate. Tellus Series A: Dynamic Meteorology and Oceanography. 1999;51A:412–430. https://doi.org/10.3402/tellusa.v51i3.13468Legg S., Hallberg R.W., Girton J.B. Comparison of entrainment in overflows simulated by z-coordinate, isopycnal and non-hydrostatic models // Ocean Modelling. 2006. Vol. 11, N 1–2. P. 69–97. https://doi.org/10.1016/j.ocemod.2004.11.006Legg S, Hallberg RW, Girton JB. Comparison of entrainment in overflows simulated by z-coordinate, isopycnal and non-hydrostatic models. Ocean Modelling. 2006;11(1–2):69–97. https://doi.org/10.1016/j.ocemod.2004.11.006Riemenschneider U., Legg S. Regional simulations of the Faroe Bank Channel overflow in a level model // Ocean Modelling. 2007. Vol. 17, N 2. P. 93–122. https://doi.org/10.1016/j.ocemod.2007.01.003Riemenschneider U, Legg S. Regional simulations of the Faroe Bank Channel overflow in a level model. Ocean Modelling. 2007;17(2):93–122. https://doi.org/10.1016/j.ocemod.2007.01.003Legg S., Jackson L., Hallberg R.W. Eddy-Resolving Modeling of Overflows // Ocean Modeling in an Eddying Regime. American Geophysical Union, 2008. P. 63–81 (eprint: https://onlinelibrary.wiley.com/doi/pdf/). https://doi.org/10.1029/177GM06Legg S, Jackson L, Hallberg RW. Eddy-Resolving Modeling of Overflows. In: Ocean Modeling in an Eddying Regime. American Geophysical Union; 2008. p. 63–81. (eprint: https://onlinelibrary.wiley.com/doi/pdf/). https://doi.org/10.1029/177GM06Reckinger S.M., Petersen M.R., Reckinger S.J. A study of overflow simulations using MPAS-Ocean: Vertical grids, resolution, and viscosity // Ocean Modelling. 2015. Vol. 96. P. 291–313. https://doi.org/10.1016/j.ocemod.2015.09.006Reckinger SM, Petersen MR, Reckinger SJ. A study of overflow simulations using MPAS-Ocean: Vertical grids, resolution, and viscosity. Ocean Modelling. 2015;96:291–313. https://doi.org/10.1016/j.ocemod.2015.09.006Laanaia N., Wirth A., Molines J.M., Barnier B., Verron J. On the numerical resolution of the bottom layer in simulations of oceanic gravity currents // Ocean Science. 2010. Vol. 6, N 2. P. 563–572. https://doi.org/10.5194/os6-563-2010Laanaia N, Wirth A, Molines JM, Barnier B, Verron J. On the numerical resolution of the bottom layer in simulations of oceanic gravity currents. Ocean Science. 2010;6(2):563–572. https://doi.org/10.5194/os6-563-2010Gräwe U., Holtermann P., Klingbeil K., Burchard H. Advantages of vertically adaptive coordinates in numerical models of stratified shelf seas // Ocean Modelling. 2015. Vol. 92. https://doi.org/10.1016/j.ocemod.2015.05.008Gräwe U, Holtermann P, Klingbeil K, Burchard H. Advantages of vertically adaptive coordinates in numerical models of stratified shelf seas. Ocean Modelling. 2015;92. https://doi.org/10.1016/j.ocemod.2015.05.008Winton M., Hallberg R., Gnanadesikan A. Simulation of density-driven frictional downslope flow in z-coordinate ocean models // Journal of Physical Oceanography. 1998. Vol. 28. P. 2163–2174. https://doi.org/10.1175/1520–0485(1998)0282.0.CO;2Winton M, Hallberg RW, Gnanadesikan A. Simulation of Density-Driven Frictional Downslope Flow in Z-Coordinate Ocean Models. Journal of Physical Oceanography. 1998;28:2163–2174. https://doi.org/10.1175/1520-0485(1998)0282.0.CO;2Lemarié F., Debreu L., Shchepetkin A., McWilliams J. On the stability and accuracy of the harmonic and biharmonic isoneutral mixing operators in ocean models // Ocean Modelling. 2012. Vol. 52–53, N 27. P. 9–35. EDN XZVETY. https://doi.org/10.1016/j.ocemod.2012.04.007Lemarié F, Debreu L, Shchepetkin A, McWilliams J. On the stability and accuracy of the harmonic and biharmonic isoneutral mixing operators in ocean models. Ocean Modelling. 2012;52–53:9–35. https://doi.org/10.1016/j.ocemod.2012.04.007Wagner G.L., Silvestri S., Constantinou N.C., Ramadhan A., Campin J.M., Hill C., Chor T., Strong-Wright J., Lee X.K., Poulin F., Souza A. High-level, high-resolution ocean modeling at all scales with Oceananigans. arXiv preprint arXiv: 2502.14148, 2025.Wagner GL, Silvestri S, Constantinou NC, Ramadhan A, Campin JM, Hill C, et al. High-level, high-resolution ocean modeling at all scales with Oceananigans. arXiv preprint arXiv:2502.14148. 2025.Пака В.Т., Кушников В.В. Об использовании термохалозондов в режиме буксировки. Океанология. 1989. Т. 29, № 1. С. 160–163.Paka VT, Kushnikov VV. About using temperature halosondes in towing mode. Oceanology. 1989;29(1):160–163. (in Russian)Paka V.T. Thermohaline structure of the waters over the cross sections in the Slupsk Channel of the Baltic Sea in spring, 1993 // Oceanology. 1996. Vol. 36, N 2. P. 188–198. EDN LDSPLLPaka VT. Thermohaline structure of the waters over the cross sections in the Slupsk Channel of the Baltic Sea in spring, 1993. Oceanology. 1996;36(2):188–198.Журбас В.М., Пака В.Т. Интрузионное расслоение галоклина в Готландском бассейне, обусловленное большим затоком североморских вод в Балтику в январе 1993 г. // Океанология. 1997. Т. 33, № 4. С. 549.Zhurbas VM, Paka VM. Intrusive stratification of the halocline in the Gotland Basin caused by the large influx of North Sea waters into the Baltic in January 1993. Oceanology. 1997;33(4):549. (in Russian)Zhurbas V.M., Paka V.T. Mesoscale thermohaline variability in the Eastern Gotland Basin following the 1993 major Baltic inflow // Journal of Geophysical Research. 1997. Vol. 102, N C9. P. 20,917–20,926. EDN WNAABX. https://doi.org/10.1029/97JC00443Zhurbas VM, Paka VM. Mesoscale thermohaline variability in the Eastern Gotland Basin following the 1993 major Baltic inflow. Journal of Geophysical Research. 1997;102(C9):20917–20926. https://doi.org/10.1029/97JC00443Zhurbas V.M., Paka V.T. What drives thermohaline intrusions in the Baltic Sea? // Journal of Marine Systems. 1999. Vol. 21, Iss. 1–4. P. 229–241. EDN LFSRWV. https://doi.org/10.1016/S0924-7963(99)00016-0Zhurbas VM, Paka VT. What drives thermohaline intrusions in the Baltic Sea? Journal of Marine Systems. 1999;21(1– 4):229–241. https://doi.org/10.1016/S0924-7963(99)00016-0Baltic Sea Physics Reanalysis | Copernicus Marine Service Copernicus Marine Data Store | Copernicus Marine Service. URL: https://data.marine.copernicus.eu › product › description https://doi.org/10.48670/moi-00013 (дата обращения: 10.06.2025)Baltic Sea Physics Reanalysis | Copernicus Marine Service. Copernicus Marine Data Store. URL: https://data.marine. copernicus.eu/product/description. https://doi.org/10.48670/moi 00013 (дата обращения: 10.06.2025)Nerger L., Hiller W., Oter J. PDAF-The Parallel Data Assimilation Framework: Experiences with Kalman Filtering // World Scientific. 2005. P. 63–83. https://doi.org/10.1142/9789812701831_0006Nerger L, Hiller W, Oter J. PDAF-The Parallel Data Assimilation Framework: Experiences with Kalman Filtering. In: World Scientific; 2005. p. 63–83. https://doi.org/10.1142/9789812701831_0006International Council for the Exploration of the Sea (ICES). URL: http://www.ices.dk (дата обращения: 10.06.2025)International Council for the Exploration of the Sea (ICES). ICES. URL: http://www.ices.dk (дата обращения: 10.06.2025)Silvestri S., Wagner G., Campin J.-M., Constantinou N., Hill Ch., Souza A., Ferrari R. A new WENO-based momentum advection scheme for simulations of ocean mesoscale turbulence, 2023. https://doi.org/10.22541/essoar.170110657.76489860/v1Silvestri S, Wagner G, Campin J-M, Constantinou N, Hill C, Souza A, Ferrari R. A new WENO-based momentum advection scheme for simulations of ocean mesoscale turbulence. 2023. https://doi.org/10.22541/essoar.170110657.76489860/v1Data Assimilation System. Baltic Nest Institute, Stockholm University. URL: https://nest.su.se/das/ (дата обращения: 10.06.2025)Baltic Nest Institute, Stockholm University. Data Assimilation System. URL: https://nest.su.se/das/ (дата обращения: 10.06.2025)Umlauf L., Burchard H. A generic length-scale equation for geophysical turbulence models // Journal of Marine Research. 2003. Vol. 61. P. 235–265. https://doi.org/10.1357/002224003322005087Umlauf L, Burchard H. A generic length-scale equation for geophysical turbulence models. Journal of Marine Research. 2003;61(2):235–265. https://doi.org/10.1357/002224003322005087Wagner G., Hillier A., Constantinou N., Silvestri S., Souza A., Burns K., Hill Ch., Campin J.-M., Marshall J., Ferrari R. Formulation and calibration of CATKE, a one-equation parameterization for microscale ocean mixing // Journal of Advances in Modeling Earth Systems. 2025. Vol. 17. https://doi.org/10.1029/2024MS004522Wagner G, Hillier A, Constantinou N, Silvestri S, Souza A, Burns K, et al. Formulation and Calibration of CATKE, a One-Equation Parameterization for Microscale Ocean Mixing. Journal of Advances in Modeling Earth Systems. 2025;17. https://doi.org/10.1029/2024MS004522Beckmann A., Döscher R. A method for improved representation of dense water spreading over topography in geopotential-coordinate models // Journal of Physical Oceanography. 1997. Vol. 27. P. 581–591. https://doi.org/10.1175/1520-0485(1997)0272.0.CO;2Beckmann A, Döscher R. A method for improved representation of dense water spreading over topography in geopotential-coordinate models. Journal of Physical Oceanography. 1997;27(3):581–591. https://doi.org/10.1175/1520-0485(1997)0272.0.CO;2Zhurbas V., Elken J., Paka V., Piechura J., Väli G., Chubarenko I., Golenko N., Shchuka S. Structure of unsteady overflow in the Słupsk Furrow of the Baltic Sea // Journal of Geophysical Research. 2012. Vol. 117. C04027. EDN MENQNX. https://doi.org/10.1029/2011JC007284Zhurbas VM, Elken J, Paka VT, Piechura J, Väli G, Chubarenko I, Golenko N, Shchuka S. Structure of unsteady overflow in the Słupsk Furrow of the Baltic Sea. Journal of Geophysical Research. 2012;117(C4): C04027. https://doi.org/10.1029/2011JC007284

The authors declare that there are no conflicts of interest present.