hydrophysicsФундаментальная и прикладная гидрофизикаFundamental and Applied Hydrophysics2073-66732782-5221St. Petersburg Research Center of the Russian Academy of Scienceshydrophysics-996Research ArticleФУНДАМЕНТАЛЬНЫЕ ВОПРОСЫ ГИДРОФИЗИКИFUNDAMENTAL ISSUES OF HYDROPHYSICSЧисленное моделирование трехмерных потенциальных волнNumerical Modeling of Three-Dimensional Potential WavesЧаликовД. В.ChalikovD. V.

Санкт-Петербург, Мельбурн

dmitry-chalikov@yandex.ruСанкт-Петербургский филиал Института океанологии им. П.П. Ширшова, РАН; Технологический университет СвинбурнаRussian Federation20142711202271731Copyright © Чаликов Д.В., 20222022Чаликов Д.В.Chalikov D.V.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/996

Разработана простая и точная схема, предназначенная для долговременного моделирования периодических трехмерных потенциальных волн. Схема основана на следующей поверхности неортогональной криволинейной системе координат. Потенциал скорости представлен как сумма аналитической и нелинейной компонент. Трехмерное уравнение для нелинейной компоненты решается итерациями. Используется Фурье-сеточный метод, аппроксимация второго порядка для вертикальных производных на растянутой сетке и схема Рунге—Кутта четвертого порядка для интегрирования по времени. Схема проверена воспроизведением бегущей волны Стокса. Однопроцессорная версия модели позволяет моделировать эволюцию волнового поля с сотнями тысяч степеней свободы в течение сотен периодов волны пика. Модель создана для исследования нелинейных свойств поверхностных волн, генерации экстремальных волн, прямых расчетов спектра нелинейных взаимодействий. После введения в модель схем расчета притока энергии и диссипации модель будет использоваться для прямого моделирования эволюции волнового поля.

.

трехмерные нелинейные поверхностные волныволны Стоксанеустойчивость Бенджамина—Фейеранелинейные взаимодействияинтеграл ХассельманнаАвтор благодарит проф. А. С. Бабанина (Технологический университет Свинбурна, Мельбурн, Австралия) за поддержку и сотрудничество, а также анонимных рецензентов за высказанные замечания, позволившие внести много уточнений. Работа поддержана грантом Российского фонда фундаментальных исследований № 11-05-0052ReferencesTolman H. L. A third-generation model for wind waves on slowly varying, unsteady, and inhomogeneous depths and currents // J. Phys. Oceanogr. 1991. V. 21. P. 782—797Tolman H. L. A third-generation model for wind waves on slowly varying, unsteady, and inhomogeneous depths and currents // J. Phys. Oceanogr. 1991. V. 21. P. 782—797Tolman H. L. A mosaic approach to wind wave modeling // Ocean Modelling. 2008. V. 25, Issues 1—2. P. 35—47.Tolman H. L. A mosaic approach to wind wave modeling // Ocean Modelling. 2008. V. 25, Issues 1—2. P. 35—47.Tolman H., Chalikov D. On the source terms in a third-generation wind wave model // J. Phys. Oceanogr. 1996. V. 26. P. 2497—2518Tolman H., Chalikov D. On the source terms in a third-generation wind wave model // J. Phys. Oceanogr. 1996. V. 26. P. 2497—2518Craik A. D. D. The origins of water wave theory // Rev. Fluid Mech. 2004. V. 36. P. 1–28. doi: 10.1146/annurev.fluid.36.050802.122118.Craik A. D. D. The origins of water wave theory // Rev. Fluid Mech. 2004. V. 36. P. 1–28. doi: 10.1146/annurev.fluid.36.050802.122118.Ronald W. Yeung. Numerical methods in free-surface flows // Annual Review of Fluid Mechanics. 1982. V. 14. P. 395—442.Ronald W. Yeung. Numerical methods in free-surface flows // Annual Review of Fluid Mechanics. 1982. V. 14. P. 395—442.Чаликов Д. В. Статистика экстремальных ветровых волн // Фундаментальная и прикладная гидрофизика. 2009. № 3(5), С. 4—24.Чаликов Д. В. Статистика экстремальных ветровых волн // Фундаментальная и прикладная гидрофизика. 2009. № 3(5), С. 4—24.Chalikov D. Freak waves: their occurrence and probability // Phys of Fluid. 2009. V. 21. doi:10.1063/1.3175713.Chalikov D. Freak waves: their occurrence and probability // Phys of Fluid. 2009. V. 21. doi:10.1063/1.3175713.Whitney J. C. The numerical solution of unsteady free-surface flows by conformal mapping // Proc. Second Inter. Conf. on Numer. Fluid Dynamics / Ed. M. Holt. 1971. Springer-Verlag, P. 458—462.Whitney J. C. The numerical solution of unsteady free-surface flows by conformal mapping // Proc. Second Inter. Conf. on Numer. Fluid Dynamics / Ed. M. Holt. 1971. Springer-Verlag, P. 458—462.Dysthe K. B. Note on modification to the nonlinear Schrödinger equation for application to deep water waves // Proc. R. Soc. Lond. V. A369. P. 105—114.Dysthe K. B. Note on modification to the nonlinear Schrödinger equation for application to deep water waves // Proc. R. Soc. Lond. V. A369. P. 105—114.Longuet-Higgins M. S., Cokelet E. D. The deformation of steep surface waves on water. I.A numerical method of computations // Proc. R. Soc. Lond. V. A350. P. 1—26.Longuet-Higgins M. S., Cokelet E. D. The deformation of steep surface waves on water. I.A numerical method of computations // Proc. R. Soc. Lond. V. A350. P. 1—26.Tulin M. P., Waseda T. Laboratory observations of wave group evolution, including breaking effects // J. Fluid Mech. 1999. V. 378. P. 197—232.Tulin M. P., Waseda T. Laboratory observations of wave group evolution, including breaking effects // J. Fluid Mech. 1999. V. 378. P. 197—232.Chalikov D., Sheinin D. Numerical modeling of surface waves based on principal equations of potential wave dynamics // Technical Note. NOAA/NCEP/OMB. 1996. 54 p.Chalikov D., Sheinin D. Numerical modeling of surface waves based on principal equations of potential wave dynamics // Technical Note. NOAA/NCEP/OMB. 1996. 54 p.Chalikov D., Sheinin D. Modeling of Extreme Waves Based on Equations of Potential Flow with a Free Surface // J. Comp. Phys. 2005. V. 210. P. 247—273.Chalikov D., Sheinin D. Modeling of Extreme Waves Based on Equations of Potential Flow with a Free Surface // J. Comp. Phys. 2005. V. 210. P. 247—273.Chalikov D., Sheinin D. Direct Modeling of One-dimensional Nonlinear Potential Waves. Nonlinear Ocean Waves. Advances in Fluid Mechanics / Ed. W. Perrie. 1998. V. 17. P. 207—258.Chalikov D., Sheinin D. Direct Modeling of One-dimensional Nonlinear Potential Waves. Nonlinear Ocean Waves. Advances in Fluid Mechanics / Ed. W. Perrie. 1998. V. 17. P. 207—258.Chalikov D., Rainchik S. Coupled Numerical Modelling of Wind and Wavesand the Theory of the Wave Boundary Layer // Boundary-Layer Meteorol. 2011. Iss. 138. P. 1—41. doi: 10.1007/s10546-010-9543-7.Chalikov D., Rainchik S. Coupled Numerical Modelling of Wind and Wavesand the Theory of the Wave Boundary Layer // Boundary-Layer Meteorol. 2011. Iss. 138. P. 1—41. doi: 10.1007/s10546-010-9543-7.Sheinin D., Chalikov D. Hydrodynamical modeling of potential surface waves // Problems of hydrometeorology and environment on the eve of XXI century. Proc. of intern. theoretical conf. St.-Petersburg, June 24—25, 2000.Sheinin D., Chalikov D. Hydrodynamical modeling of potential surface waves // Problems of hydrometeorology and environment on the eve of XXI century. Proc. of intern. theoretical conf. St.-Petersburg, June 24—25, 2000.Zakharov V. E., Dyachenko A. I., Vasilyev O. A. New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface // Eur. J. Mech. B/Fluids. V. 21. 2002. P. 283—291.Zakharov V. E., Dyachenko A. I., Vasilyev O. A. New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface // Eur. J. Mech. B/Fluids. V. 21. 2002. P. 283—291.Clamond D., Grue J. A fast method for fully nonlinear water wave dynamics // J. Fluid Mech. 2001. V. 447. P. 337—355.Clamond D., Grue J. A fast method for fully nonlinear water wave dynamics // J. Fluid Mech. 2001. V. 447. P. 337—355.Clamond D., Fructus D., Grue J., Krisitiansen O. An efficient method for three-dimensional surface wave simulations. Part II: Generation and absorption // J. Comp. Physics. 2005. V. 205. P. 686—705.Clamond D., Fructus D., Grue J., Krisitiansen O. An efficient method for three-dimensional surface wave simulations. Part II: Generation and absorption // J. Comp. Physics. 2005. V. 205. P. 686—705.Fructus D., Clamond D., Grue J, Kristiansen O. An efficient model for three-dimensional surface wave simulations. Part I: Free space problems // J. Comp. Phys. 2005. V. 205. P. 665—685.Fructus D., Clamond D., Grue J, Kristiansen O. An efficient model for three-dimensional surface wave simulations. Part I: Free space problems // J. Comp. Phys. 2005. V. 205. P. 665—685.Grilli S., Guyenne P. and Dias F. A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom // Int. J. Num. Methods Fluids. 2001. V. 35. P. 829—867.Grilli S., Guyenne P. and Dias F. A fully nonlinear model for three-dimensional overturning waves over arbitrary bottom // Int. J. Num. Methods Fluids. 2001. V. 35. P. 829—867.Fochesato C., Dias F., Grill S. Wave energy focusing in a three-dimensional numerical wave tank // Proc. R. Soc. 2006. V. A462. P. 2715—2735.Fochesato C., Dias F., Grill S. Wave energy focusing in a three-dimensional numerical wave tank // Proc. R. Soc. 2006. V. A462. P. 2715—2735.Fochesato C., Frederick D. A fast method for nonlinear three-dimensional free-surface waves // Proc. R. Soc. A. 2006. V. 462. P. 2715—2735. doi: 10.1098/rspa.2006.1706.Fochesato C., Frederick D. A fast method for nonlinear three-dimensional free-surface waves // Proc. R. Soc. A. 2006. V. 462. P. 2715—2735. doi: 10.1098/rspa.2006.1706.Cai X., Petter H., Langtangen H. P., Nielse B. F., Tveito A. A Finite Element Method for Fully Nonlinear Water Waves // J. Comp. Phys. 1998. V. 143. P. 544—568.Cai X., Petter H., Langtangen H. P., Nielse B. F., Tveito A. A Finite Element Method for Fully Nonlinear Water Waves // J. Comp. Phys. 1998. V. 143. P. 544—568.Engsig-Karup A. P., Bingham H. B., Lindberg O. An efficient flexible-order model for 3D nonlinear water waves // J. Comp. Phys. 2009. V. 228. P. 2100—2118.Engsig-Karup A. P., Bingham H. B., Lindberg O. An efficient flexible-order model for 3D nonlinear water waves // J. Comp. Phys. 2009. V. 228. P. 2100—2118.Bingham H. B., Zhang H. On the accuracy of finite-difference solutions for nonlinear water waves // J. Eng. Math. 2007. V. 58 P. 211—228.Bingham H. B., Zhang H. On the accuracy of finite-difference solutions for nonlinear water waves // J. Eng. Math. 2007. V. 58 P. 211—228.Wu-Ting Tsai, Dick K. P. Yue. Computation of nonlinear free-surface flows // Annual Review of Fluid Mechanics. 1996. V. 28. P. 249–278.Wu-Ting Tsai, Dick K. P. Yue. Computation of nonlinear free-surface flows // Annual Review of Fluid Mechanics. 1996. V. 28. P. 249–278.Li B., Fleming A. A three-dimensional multigrid model for fully nonlinear water waves. Coast. Eng. 1997. V. 30. P. 235—258.Li B., Fleming A. A three-dimensional multigrid model for fully nonlinear water waves. Coast. Eng. 1997. V. 30. P. 235—258.Haussling H. J., Van Eseltine R. T. Finite-difference methods for transient potential flows with free surfaces // Procs. of the Intern. Conf. on Numerical Ship Hydrodynamics. Univ. Extension Publ., Berkely, 1975. P. 295— 313.Haussling H. J., Van Eseltine R. T. Finite-difference methods for transient potential flows with free surfaces // Procs. of the Intern. Conf. on Numerical Ship Hydrodynamics. Univ. Extension Publ., Berkely, 1975. P. 295— 313.Dommermuth D., Yue D. A high-order spectral method for the study of nonlinear gravity waves // J. Fluid Mech. 1987. V. 184. P. 267—288.Dommermuth D., Yue D. A high-order spectral method for the study of nonlinear gravity waves // J. Fluid Mech. 1987. V. 184. P. 267—288.West B., Brueckner K., Janda R., Milder M., Milton R. A new numerical method for surface hydrodynamics // J. Geophys. Res. 1987. V. 92, N 11. P. 803–824.West B., Brueckner K., Janda R., Milder M., Milton R. A new numerical method for surface hydrodynamics // J. Geophys. Res. 1987. V. 92, N 11. P. 803–824.Zakharov V. E. Stability of periodic waves of finite amplitude on the surface of deep fluid // J. Appl. Mech. Tech. Phys. JETF. 1968. V. 2. P. 190—194. (English translation).Zakharov V. E. Stability of periodic waves of finite amplitude on the surface of deep fluid // J. Appl. Mech. Tech. Phys. JETF. 1968. V. 2. P. 190—194. (English translation).Clamond D., Francius M., Grue J., Kharif C. Long time interaction of envelope solitons and freak wave formations // European Journal of Mechanics B/Fluids. V. 25 2006. P. 536—553.Clamond D., Francius M., Grue J., Kharif C. Long time interaction of envelope solitons and freak wave formations // European Journal of Mechanics B/Fluids. V. 25 2006. P. 536—553.Thomas L. H. Elliptic Problems in Linear Differential Equations over a Network // Watson Sci. Comput. Lab Report. New York Columbia University, 1949.Thomas L. H. Elliptic Problems in Linear Differential Equations over a Network // Watson Sci. Comput. Lab Report. New York Columbia University, 1949.Qiao F., Yeli Y., Yang Y., Zheng Q., Xia C., Ma J. Wave-induce mixing in the upper ocean: Distribution and application to a global ocean circulation model // Geophys. Res. Lett., 2004. V. 31. doi:10.1029/2004GL019824.Qiao F., Yeli Y., Yang Y., Zheng Q., Xia C., Ma J. Wave-induce mixing in the upper ocean: Distribution and application to a global ocean circulation model // Geophys. Res. Lett., 2004. V. 31. doi:10.1029/2004GL019824.Babanin A. V., B. K. Haus. On the existence of water turbulence induced by non-breaking surface waves // J. Phys. Oceanogr. 2009. V. 39. P. 2675—2679.Babanin A. V., B. K. Haus. On the existence of water turbulence induced by non-breaking surface waves // J. Phys. Oceanogr. 2009. V. 39. P. 2675—2679.Savelyev I. B., Maxeiner E., Chalikov D. Turbulence production by non-breaking waves: laboratory and numerical simulations // J. of Geophys. Res-Oceans. 2012. doi:10.1029/2012JC007928.Savelyev I. B., Maxeiner E., Chalikov D. Turbulence production by non-breaking waves: laboratory and numerical simulations // J. of Geophys. Res-Oceans. 2012. doi:10.1029/2012JC007928.Toffoli A., Babanin A. V., McConochie J. The effect of turbulence induced by nonbreaking waves on the ocean mixed layer: Field observations on the Australian North-West Shelf // J. Geophys. Res. 2012. V. 117. 8 p. doi:10.1029/2011JC007780.Toffoli A., Babanin A. V., McConochie J. The effect of turbulence induced by nonbreaking waves on the ocean mixed layer: Field observations on the Australian North-West Shelf // J. Geophys. Res. 2012. V. 117. 8 p. doi:10.1029/2011JC007780.Babanin A. V., Chalikov D. Numerical investigation of turbulence generation in nonbreaking potential waves // J. Geophys. Res. 2012. V. 117. 14 p. doi: 10.1029/2012JC007929.Babanin A. V., Chalikov D. Numerical investigation of turbulence generation in nonbreaking potential waves // J. Geophys. Res. 2012. V. 117. 14 p. doi: 10.1029/2012JC007929.Stokes G. G. On the theory of oscillatory waves // Trans. Cambridge Philos. Soc. 1847. V. 8. P. 441—445.Stokes G. G. On the theory of oscillatory waves // Trans. Cambridge Philos. Soc. 1847. V. 8. P. 441—445.Crapper G. D. An exact solution for progressive capillary waves of arbitrary amplitude // J. of Fluid Mech. 1957. V. 96. P. 417—445.Crapper G. D. An exact solution for progressive capillary waves of arbitrary amplitude // J. of Fluid Mech. 1957. V. 96. P. 417—445.Chalikov D., Rainchik S. Coupled Numerical Modelling of Wind and Wavesand the Theory of the Wave Boundary Layer // Boundary-Layer Meteorol. 2010. V. 138, N 1—41. doi: 10.1007/s10546-010-9543-7.Chalikov D., Rainchik S. Coupled Numerical Modelling of Wind and Wavesand the Theory of the Wave Boundary Layer // Boundary-Layer Meteorol. 2010. V. 138, N 1—41. doi: 10.1007/s10546-010-9543-7.Chalikov D. Simulation of Benjamin—Feir instability and its consequences // Physics of Fluid. 2007. V. 19. P. 016602-15.Chalikov D. Simulation of Benjamin—Feir instability and its consequences // Physics of Fluid. 2007. V. 19. P. 016602-15.McLean J. W. Instability of finite amplitude water waves // J. Fluid Mech. 1982. V. 114. P. 315—330.McLean J. W. Instability of finite amplitude water waves // J. Fluid Mech. 1982. V. 114. P. 315—330.Battjes J. A., Zitman T. J., Holthuijsen L. H. A reanalysis of the spectra observed in JONSWAP // J. Phys. Oceanogr. 1987. V. 17. P. 1288—1295.Battjes J. A., Zitman T. J., Holthuijsen L. H. A reanalysis of the spectra observed in JONSWAP // J. Phys. Oceanogr. 1987. V. 17. P. 1288—1295.Pierson W. J., Moscowitz L. A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii // J. of Geophys. Res. 1964. V. 69, N 24. P. 5181—5190.Pierson W. J., Moscowitz L. A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii // J. of Geophys. Res. 1964. V. 69, N 24. P. 5181—5190.Hasselmann, Barnett R. P., Bouws E. et al, Measurements of wind-wave growth and swell decay during the Joint Sea Wave Project (JONSWAP). Tsch. Hydrogh. Z. Suppl. 1973. V. A8, N 12. P. 1—95.Hasselmann, Barnett R. P., Bouws E. et al, Measurements of wind-wave growth and swell decay during the Joint Sea Wave Project (JONSWAP). Tsch. Hydrogh. Z. Suppl. 1973. V. A8, N 12. P. 1—95.Chalikov D. The Parameterization of the Wave Boundary Layer // J. Phys. Oceanogr. 1995. V. 25. P. 1335— 1349.Chalikov D. The Parameterization of the Wave Boundary Layer // J. Phys. Oceanogr. 1995. V. 25. P. 1335— 1349.Onorato М., Waseda Т., Toffoli A. et al. Statistical Properties of Directional Ocean Waves: The Role of the Modulational Instability in the Formation of Extreme Events // Phys. Rev. Lett. 2009. V. 102. P. 114502.Onorato М., Waseda Т., Toffoli A. et al. Statistical Properties of Directional Ocean Waves: The Role of the Modulational Instability in the Formation of Extreme Events // Phys. Rev. Lett. 2009. V. 102. P. 114502.Hasselmann K. Weak-interaction theory of ocean waves. Hamburg: Univ. of Hamburg, 1967. 112 p.Hasselmann K. Weak-interaction theory of ocean waves. Hamburg: Univ. of Hamburg, 1967. 112 p.Lake B. M., Yuen H. C. A new model for nonlinear wind waves. Part 1: Physical model and experimental results // J. Fluid. Mech. 1978. V. 88. P. 33—62.Lake B. M., Yuen H. C. A new model for nonlinear wind waves. Part 1: Physical model and experimental results // J. Fluid. Mech. 1978. V. 88. P. 33—62.Чаликов Д. В., Либерман Ю. М. Интегрирование примитивных уравнений для потенциальных волн // Изв. АН СССР. Физ. атм. океана. 1991. 27. P. 42—47.Чаликов Д. В., Либерман Ю. М. Интегрирование примитивных уравнений для потенциальных волн // Изв. АН СССР. Физ. атм. океана. 1991. 27. P. 42—47.Чаликов Д. В. Трансформация гармонических волн на глубокой воде. Фундаментальная и прикладная гидрофизика. 2010. № 3(9). С. 14—21.Чаликов Д. В. Трансформация гармонических волн на глубокой воде. Фундаментальная и прикладная гидрофизика. 2010. № 3(9). С. 14—21.Zakharov V. E., Korotkevich A. O., Pushkarev A. N., Dyachenko A. I. JETP Mesoscopic Wave Turbulence // JETP Letts. 2005. V. 82, N 8. P. 487—491.Zakharov V. E., Korotkevich A. O., Pushkarev A. N., Dyachenko A. I. JETP Mesoscopic Wave Turbulence // JETP Letts. 2005. V. 82, N 8. P. 487—491.Benjamin T. B., Feir J. E. The disintegration of wave trains in deep water // J. Fluid. Mech. V. 27. P. 417— 430.Benjamin T. B., Feir J. E. The disintegration of wave trains in deep water // J. Fluid. Mech. V. 27. P. 417— 430.

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