Abstract
Methods for solving shallow-water equations that describe flows in rotating annular channels are considered and the results of numerical calculations are analyzed for the possible generation of global large-scale flows, narrow jets, and numerous small-scale vortices in laboratory experiments. External effects in fluids are induced using a mass source–sink and the MHD-method of interaction of radial electric current with the magnetic field generated by the field of permanent magnets. A central–upwind scheme modified to suit the specific aspects of geophysical hydrodynamics. Initially, this method was used to solve shallow-water equations only in hydraulic problems, such as for flows in dam breaks, channels, rivers, and lakes. Geophysical hydrodynamics (in addition to free surface and topography) requires a rotation of the system as a whole, which is accompanied by the appearance of a complex system of vortices, jets, and turbulence (these should be taken into account in the formulation of the problem). Accordingly, the basic features of the central–upwind method should be changed. The modifications should ensure that the scheme is well-balanced and choose interpolation methods for desired variables. The main result of this modification is the control over numerical viscosity affecting the fluid motion variety. The active dynamics of a large number of vortices transformed into jets or generating large-scale streams is the general result of modifications suitable for geophysical hydrodynamics. Because there are technical difficulties in the creation of an appropriate laboratory setup for modeling of geophysical flows with the help of numerous source–sinks, it will be appropriate to use numerical experiments for studying the motions generated by this method. Unlike this method, the MHD-method can be rather easily used in laboratory conditions to generate a large variety of flows and vortex currents in the channel by a relatively small number of permanent magnets. Specifically, this method made it possible to obtain large-scale circular flows over the entire channel area, jets, and systems of interacting vortices. For the purpose of experiments, the distributions of source–sinks and systems of permanent magnets over the bottom of annular channels are determined.
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References
Lesieur, M., Turbulence in Fluids, Dordrecht: Academic, 1997.
Weeks, E.R., Tian, Y., Urbach, J.S., Ide, K., Swinney, H.L., and Ghil, M., Transitions between blocked and zonal flows in a rotating annulus with topography, Science, 1997, vol. 278, no. 5343, pp. 1598–1601.
Rhines, P.B., Jets and orography: idealized experiments with tip jets and Lighthill blocking, J. Atmos. Sci., 2007, vol. 64, pp. 3627–3639.
Espa, St., Lacorata, G., and di Nitto, G., Anisotropic Lagrangian dispersion in rotating flows with a ß effect, J. Phys. Oceanogr., 2014, vol. 44, pp. 632–643.
Espa, St., Bordi, I., Frisius, Th., Fraedrichs, K., Cenedese, A., and Sutera, A., Zonal jets and cyclone-anticyclone asymmetry in decaying rotating turbulence: laboratory experiments and numerical simulations, Geophys. Astrophys. Fluid Dyn., 2012, vol. 106, no. 6, pp. 557–573.
Galperin, B., Sukoriansky, S., Dikovskaya, N., Read, P., Yamazaki, Y., and Wordsworth, R., Anisotropic turbulence and zonal jets in rotating flows with a ß-effect, Nonlin. Processes Geophys., 2006, vol. 13, pp. 83–98.
Baroud, C.N., Plapp, B.B., and Swinney, H.L., Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows, Phys. Fluids, 2003, vol. 15, no. 8, pp. 2091–2104.
Gledzer, A.E., Gledzer, E.B., Khapaev, A.A., and Chernous’ko, Y.L., Barotropic blocking of the motion of vortices in laboratory experiments with a rotating circular channel, Dokl. Earth Sci., 2012, vol. 444, no. 1, pp. 647–651.
Gledzer, A.E., Gledzer, E.B., Khapaev, A.A., and Chernous’ko, Y.L., Zonal flows, Rossby waves, and vortex transport in laboratory experiments with rotating annular channel, Izv. Atmos. Ocean. Phys., 2014, vol. 50, no. 2, pp. 122–133.
Dolzhansky, F.V., Fundamentals of Geophysical Hydrodynamics, Berlin, Heidelberg: Springer, 2013.
Gledzer, A.E., Gledzer, E.B., Khapaev, A.A., and Chkhetiani, O.G., Experimental manifestation of vortices and Rossby wave blocking at the MHD excitation of quasi-two-dimensional flows in a rotating cylindrical vessel, JETP Lett., 2013, vol. 97, no. 6, pp. 316–321.
Smith, C.A. and Speer, K.G., Multiple zonal jets in a differentially heated rotating annulus, J. Phys. Oceanogr., 2014, vol. 44, pp. 2273–2291.
Xia, H., Shats, M.G., and Falkovich, G., Spectrally condensed turbulence in thin layer, Phys. Fluids, 2009, vol. 21, p. 125101.
Gledzer, A.E., Numerical model of currents generated by sources and sinks in a circular rotating channel, Izv. Atmos. Ocean. Phys., 2014, vol. 50, no. 3, pp. 292–303.
Toro, E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, Berlin, Heidelberg: Springer, 2009.
Kurganov, A. and Levy, D., A third-order semidiscrete central scheme for conservation laws and convectiondiffusion equations, SIAM J. Sci. Comput., 2000, vol. 22, no. 4, pp. 1461–1488.
Kurganov, A. and Tadmor, E., New high-resolution semi-discrete central schemes for Hamilton–Jacobi equations, J. Comput. Phys., 2000, vol. 160, no. 2, pp. 720–742.
Jiang, G.S., Levy, D., Lin, C.T., Osher, S., and Tadmor, E., High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 1998, vol. 35, no. 6, pp. 2147–2168.
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 2000, vol. 160, no. 1, pp. 241–282.
Kurganov, A., Noelle, S., and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput., 2001, vol. 23, no. 3, p. 707–740.
Harten, A., Lax, P.D., and van Leer, B., On upstream differencing and Godunov-type schmes for hyperbolic conservation laws, SIAM Rev., 1983, vol. 25, no. 1, pp. 35–61.
Kurganov, A. and Petrova, G., Central-upwind schemes for two-layer shallow water equations, SIAM J. Sci. Comput., 2009, vol. 31, no. 3, pp. 1742–1773.
Kurganov, A. and Petrova, G., Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws, Numer. Methods Part. Diff. Eq., 2005, vol. 21, no. 3, pp. 536–552.
Kurganov, A. and Petrova, G., A third-order semi-discrete genuinely multideminsional central scheme for hyperbolic conservation laws and related problems, Numer. Math., 2001, vol. 88, no. 4, pp. 683–729.
Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun. Math. Sci., 2007, vol. 5, no. 1, pp. 133–160.
Singh, J., Altinakar, M.S., and Ding, Y., Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme, Adv. Water Resour., 2011, vol. 34, no. 10, pp. 1366–1375.
Bermudez, A. and Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 1994, vol. 23, no. 8, pp. 1049–1071.
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Original Russian Text © A.E. Gledzer, 2015, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2015, Vol. 8, No. 4, pp. 408–422.
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Gledzer, A.E. Generation of large-scale structures and vortex systems in numerical experiments for rotating annular channels. J Appl Mech Tech Phy 57, 1239–1253 (2016). https://doi.org/10.1134/S0021894416070051
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DOI: https://doi.org/10.1134/S0021894416070051