×

Lie symmetry analysis and exact solutions of the quasigeostrophic two-layer problem. (English) Zbl 1315.86001

Summary: The quasigeostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras, we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Wherever possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.{
©2011 American Institute of Physics}

MSC:

86A10 Meteorology and atmospheric physics
76F45 Stratification effects in turbulence
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76B65 Rossby waves (MSC2010)
76E20 Stability and instability of geophysical and astrophysical flows
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abraham-Shrauner, B.; Govinder, K. S., Provenance of type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys., 13, 4, 612 (2006) · Zbl 1110.35321
[2] Andreev, V. K.; Kaptsov, O. V.; Pukhnachov, V. V.; Rodionov, A. A., Applications of Group-Theoretical Methods in Hydrodynamics (1998) · Zbl 0912.35001
[3] 3.Bihlo, A. and Popovych, R. O., “Symmetry justification of Lorenz’ maximum simplification,” Nonlinear Dyn.61(1-2), 101 (2009);10.1007/s11071-009-9634-5e-print arXiv:0805.4061v2. · Zbl 1204.37026
[4] 4.Bihlo, A. and Popovych, R. O., “Lie symmetries and exact solutions of the barotropic vorticity equation,” J. Math. Phys.50, 123102 (2009);10.1063/1.3269919e-print arXiv:0902.4099. · Zbl 1272.86001
[5] 5.Bihlo, A. and Popovych, R. O., “Symmetry analysis of barotropic potential vorticity equation,” Commun. Theor. Phys.52(4), 697 (2009);10.1088/0253-6102/52/4/27e-print arXiv:0811.3008v2. · Zbl 1182.76937
[6] 6.Bihlo, A. and Popovych, R. O., “Point symmetry group of the barotropic vorticity equation,” In Proceedings of 5th Workshop “Group Analysis of Differential Equations and Integrable Systems” (June 6-10, 2010, Protaras, Cyprus), pp. 15-27 (2011); e-print arXiv:1009.1523. · Zbl 1234.35014
[7] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989) · Zbl 0698.35001
[8] Carminati, J.; Vu, K., Symbolic computation and differential equations: Lie symmetries, J. Symb. Comput., 29, 1, 95 (2000) · Zbl 0958.68543
[9] 9.Fushchych, W. I. and Popovych, R. O., “Symmetry reduction and exact solutions of the Navier-Stokes equations,” J. Nonlinear Math. Phys.1(1-2), 75 (1994);10.2991/jnmp.1994.1.1.6e-print arXiv:math-ph/0207016. · Zbl 0956.35099
[10] Golovin, S. V., Applications of the differential invariants of infinite dimensional groups in hydrodynamics, Commun. Nonlinear Sci. Numer. Simul., 9, 1, 35 (2004) · Zbl 1036.58033
[11] 11.Head, A. K., “LIE, a PC program for Lie analysis of differential equations,” Comput. Phys. Commun.77(2), 241 (1993);10.1016/0010-4655(93)90007-Ysee also http://www.cmst.csiro.au/LIE/LIE.htm. · Zbl 0854.65055
[12] Hirschberg, P.; Knobloch, E., Mode interactions in large aspect ratio convection, J. Nonlinear Sci., 7, 6, 537 (1997) · Zbl 0902.76035
[13] Holton, J. R., An Introduction to Dynamic Meteorology (2004)
[14] Hydon, P. E., How to construct the discrete symmetries of partial differential equations, Eur. J. Appl. Math., 11, 5, 515 (2000) · Zbl 1035.35005
[15] Ibragimov, N. H.; Aksenov, A. V.; Baikov, V. A.; Chugunov, V. A.; Gazizov, R. K.; Meshkov, A. G.; Ibragimov, N. H., Applications in Engineering and Physical Sciences, CRC Handbook of Lie Group Analysis of Differential Equations (1995)
[16] Katkov, V. L., A class of exact solutions of the equation for the forecast of the geopotential, Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer. i Okeana, 1, 630 (1965)
[17] Katkov, V. L., Exact solutions of the geopotential forecast equation, Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer. i Okeana, 2, 1193 (1966)
[18] Landsberg, A. S.; Knobloch, E., Oscillatory bifurcation with broken translation symmetry, Phys. Rev. E, 53, 4, 3579 (1996)
[19] Meleshko, S. V., A particular class of partially invariant solutions of the Navier-Stokes equations, Nonlinear Dyn., 36, 1, 47 (2004) · Zbl 1098.76059
[20] Meleshko, S. V., Methods for Constructing Exact Solutions of Partial Differential Equations, Mathematical and Analytical Techniques with Applications to Engineering (2005) · Zbl 1081.35001
[21] Olver, P. J., Application of Lie Groups to Differential Equations (2000) · Zbl 0937.58026
[22] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982) · Zbl 0485.58002
[23] Pedlosky, J., Geophysical Fluid Dynamics (1987) · Zbl 0713.76005
[24] Phillips, N. A., Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model, Tellus, 6, 3, 273 (1954)
[25] Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists (2002) · Zbl 1027.35001
[26] Popovych, H. V., Lie, partially invariant and nonclassical submodels of the Euler equations, Proceedings of Institute of Mathematics of NAS of Ukraine, 178-183 (2002) · Zbl 1039.35088
[27] 27.Popovych, R. O. and Bihlo, A., “Symmetry preserving parameterization schemes,” 36 pp. (2010); e-print arXiv:1010.3010. · Zbl 1277.58021
[28] 28.Popovych, R. O., Boyko, V. M., Nesterenko, M. O., and Lutfullin, M. W., “Realizations of real low-dimensional Lie algebras,” J. Phys. A36(26), 7337 (2003)10.1088/0305-4470/36/26/309see also the extended and revised version of e-print arXiv:math-ph/0301029v7. · Zbl 1040.17021
[29] Popovych, R. O.; Ivanova, N. M., New results on group classification of nonlinear diffusion-convection equations, J. Phys. A, 37, 30, 7547 (2004) · Zbl 1067.35006
[30] 30.Popovych, R. O., Kunzinger, M., and Eshraghi, H., “Admissible transformations and normalized classes of nonlinear Schrödinger equations,” Acta Appl. Math.109(2), 315 (2010);10.1007/s10440-008-9321-4e-print arXiv:math-ph/0611061. · Zbl 1216.35146
[31] 31.Prokhorova, M., “The structure of the category of parabolic equations,” 24 pp. (2005); e-print arXiv:math.AP/0512094.
[32] Pukhnachov, V. V., Invariant solutions of Navier-Stokes equations describing free boundary motion, Dokl. Akad. Nauk SSSR, 202, 2, 302 (1972)
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.