×

A class of exact solutions for two-dimensional equations of geophysical hydrodynamics with two Coriolis parameters. (Russian. English summary) Zbl 1446.35211

Summary: The article proposes a class of exact solutions of the Navier-Stokes equations for a rotating viscous incompressible fluid. This class allows us to describe steady shear inhomogeneous (i.e., depending on several coordinates of the selected Cartesian system) flows. Rotation is characterized by two Coriolis parameters, which in a rotating coordinate system leads to the fact that even for shear flows the vertical velocity is nonzero. The inclusion of the second Coriolis parameter also clarifies the well-known hydrostatic condition for rotating fluid flows, used in the traditional approximation of Coriolis acceleration. The class of exact solutions allows us to generalize Ekman’s classical exact solution. It is known that the Ekman flow assumes a uniform velocity distribution and neglect of the second Coriolis parameter, which does not allow us to describe the equatorial counterflows. In this paper, this gap in theoretical research is partially filled. It was shown that the reduction of the basic system of equations, consisting of the Navier-Stokes equations and the incompressibility equation, for this class leads to an overdetermined system of differential equations. The solvability condition for this system is obtained. It is shown that the constructed nontrivial exact solutions in the general case belong to the class of quasipolynomials. However, taking into account the compatibility condition, which determines the solvability of the considered overdetermined system, leads to the fact that the spatial accelerations characterizing the inhomogeneity of the distribution of the flow velocity field turn out to be constant. The article also provides exact solutions for all components of the pressure field.

MSC:

35Q86 PDEs in connection with geophysics
86A05 Hydrology, hydrography, oceanography
35Q35 PDEs in connection with fluid mechanics
35N10 Overdetermined systems of PDEs with variable coefficients
76D05 Navier-Stokes equations for incompressible viscous fluids
76D17 Viscous vortex flows
76U60 Geophysical flows
76D50 Stratification effects in viscous fluids
35N99 Overdetermined problems for partial differential equations and systems of partial differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Aristov, S. N., Eddy currents in thin liquid layers, Optimization of boundary and distributed controls in semilinear hyperbolic systems. Dr. Sci. [Phys. -Math. ] Diss. Vladivostok, 303 (1990)
[2] Aristov, S. N.; Myasnikov, V. P., Time-dependent three-dimensional structures in the near-surface layer of the ocean, Physics. Doklady, 8, 358-360 (1996) · Zbl 1146.86301
[3] Aristov, S. N.; Knyazev, D. V.; Polyanin, A. D., Exact solutions of the Navier-Stokes Equations with the linear dependence of velocity components on two space variables, Theoretical Foundations of Chemical Engineering, 5, 642-662 (2009)
[4] Aristov, S. N.; Prosviryakov, E. Y.; Couette, Inhomogeneous, Inhomogeneous Couette flow, Russian Journal of Nonlinear Dynamics, 2, 177-182 (2014) · Zbl 1310.76074
[5] Aristov, S. N.; Prosviryakov, E. Y., Large-scale flows of viscous incompressible vortical fluid, Russian Aeronautics, 4, 413-418 (2015)
[6] Aristov, S. N.; Prosviryakov, E. Yu., A new class of exact solutions for three-dimensional thermal diffusion equations, Theoretical Foundations of Chemical Engineering, 3, 286-293 (2016)
[7] Aristov, S. N.; Frik, P. G., Dynamics of large-scale flows in thin liquid layers, Preprint No. 146,, 48 (1987), Sverdlovsk, Institute of Continuous Media Mechanics, Academy of Sciences USSR
[8] Aristov, S. N.; Frik, P. G., Nonlinear effects of the Ekman layer on the dynamics of large-scale eddies in shallow water, Journal of Applied Mechanics and Technical Physics, 2, 189-194 (1991)
[9] Aristov, S. N.; Shvartc, K. G., Vortex flows of an advective nature in a rotating fluid layer, 153 (2006), Perm, Perm State University
[10] Burmasheva, N. V.; Prosviryakov, E. Yu., Thermocapillary convection of a vertical swirling liquid, Theoretical Foundations of Chemical Engineering, 1, 230-239 (2020)
[11] Burmasheva, N. V.; Prosviryakov, E. Yu., Exact solution of Navier—Stokes equations describing spatially inhomogeneous flows of a rotating fluid, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2, 79-87 (2020)
[12] Gorshkov, A. V.; Prosviryakov, E. Y., Ekman convective layer flow of a viscous incompressible fluid, Izvestiya. Atmospheric and Oceanic Physics, 2, 189-195 (2018)
[13] Gushchin, V. A.; Rozhdestvenskaya, T. I., Numerical study of the effects occurring near a circular cylinder in stratified fluid flows with short buoyancy periods, Journal of Applied Mechanics and Technical Physics, 6, 905-911 (2011) · Zbl 1298.76074
[14] Ziryanov, V. N., Theory of steady ocean currents, 248 (1985), Leningrad, Gidrometeoizdat Publ.
[15] Kalashnik, M. V.; Chkhetiani, O. G., Optimal perturbations with zero potential vorticity in the Eady model, Izvestiya. Atmospheric and Oceanic Physics, 5, 415-422 (2018)
[16] Kalashnik, M. V.; Chkhetiani, O. G.; Chagelishvili, G. D., A new class of edge baroclinic waves and the mechanism of their generation, Izvestiya. Atmospheric and Oceanic Physics, 4, 305-312 (2018)
[17] Koprov, B. M.; Koprov, V. M.; Solenaya, O. A.; Chkhetiani, O. G.; Shishov, E. A., Technique and results of measurements of turbulent helicity in a stratified surface layer, Izvestiya. Atmospheric and Oceanic Physics, 5, 446-455 (2018)
[18] Korotaev, G. K.; Mikhaylova, E. N.; Shapiro, N. B., The theory of equatorial countercurrents in the oceans, 208 (1986), Kiev, Naukova Dumka Publ.
[19] Monin, A. S., Theoretical foundations of geophysical hydrodynamics, 424 (1988), Leningrad, Gidrometeoizdat Publ.
[20] Pedlosky, J., Geophysical fluid dynamics, 710 (1987), Springer-Verlag · Zbl 0713.76005
[21] Sidorov, A. F., Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, Journal of Applied Mechanics and Technical Physics, 2, 197-203 (1989)
[22] Chkhetiani, O. G.; Vazaeva, N. V., On algebraic perturbations in the atmospheric boundary layer, Izvestiya. Atmospheric and Oceanic Physics, 5, 432-445 (2019)
[23] Aristov, S. N.; Nycander, J., Convective flow in baroclinic vortices, Journal Physical Oceanography, 9, 1841-1849 (1994)
[24] Burmasheva, N. V.; Larina, E. A.; Prosviryakov, E. Yu., Unidirectional convective flows of a viscous incompressible fluid with slippage in a closed layer, AIP Conference Proceedings, 030023-030023 (2019)
[25] Burmasheva, N. V.; Prosviryakov, E. Yu., Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation., Journal of Samara State Technical University, 2, 341-360 (2019) · Zbl 1449.76027
[26] Couette, M., Etudes sur le frottement des liquides, Annales de chimie et de physique, 433-510 (1890) · JFM 22.0964.01
[27] Ekman, V. W., On the Influence of the Earth’s rotation on ocean currents, Arkiv for matematik, astronomi och fysik, 11, 1-52 (1905) · JFM 36.1009.04
[28] Hoff, M.; Harlander, U., Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence, Journal of Fluid Mechanics, 522-543 (2019) · Zbl 1430.76178
[29] Lin, C. C., Note on a class of exact solutions in magneto-hydrodynamics, Archive for Rational Mechanics and Analysis, 391-395 (1958) · Zbl 0083.42103
[30] Meirelles, S.; Vinzon, S. B., Field observation of wave damping by fluid mud, Marine Geology (2016)
[31] Patel, P. D.; Christman, P. G.; Gardner, J. W., Investigation of unexpectedly low field-observed fluid mobilities during some CO2 tertiary floods, SPE Reservoir Engineering, 4, 507-513 (1987)
[32] Polyanin, A. D.; Zaitsev, V. F., Handbook of exact solutions for ordinary differential equations. 2nd ed. Boca Raton, Chapman& Hal, 803 (2003), Boca Raton, Chapman& Hall/CRC · Zbl 1015.34001
[33] Precigout, J.; Prigent, C.; Palasse, L.; Pochon, A., Water pumping in mantle shear zones, Nature Communications (2017)
[34] Privalova, V. V.; Prosviryakov, E. Yu.; Simonov, M. A., Nonlinear gradient flow of a vertical vortex fluid in a thin layer, Russian Journal of Nonlinear Dynamics, 3, 271-283 (2019) · Zbl 1440.76035
[35] Smagorinsky, J., The Global Weather Experiment-Perspective on Its Implementation and Exploitation, A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program (GARP), National Academy of Science, 4-12 (1978)
[36] Smagorinsky, J., The beginnings of numerical weather prediction and general circulation modeling: Early recollections, Advances in Geophysics, 3-37 (1983)
[37] Smagorinsky, J.; Phillips, N. A., Scientific problems of the global weather experiment. The Global Weather Experiment, Perspectives on Its Implementation and Exploitation, A Report of the FGGE Advisory Panel to the U.S. Committee for the Global Atmospheric Research Program (GARP), National Academy of Science, 13-21 (1978)
[38] Stefani, F.; Gerbeth, G.; Gundrum, Th.; Szklarski, J.; Rudiger, G.; Hollerbach, R., Liquid metal experiments on the magnetorotational instability, Magnetohydrodynamics, 2, 135-144 (2009)
[39] Woumeni, R. S.; Vauclin, M., A field study of the coupled effects of aquifer stratification, fluid density, and groundwater fluctuations on dispersivity assessments, Advances in Water Resources, 7, 1037-1055 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.