Bihlo, Alexander; Popovych, Roman O. Lie symmetries and exact solutions of the barotropic vorticity equation. (English) Zbl 1272.86001 J. Math. Phys. 50, No. 12, 123102, 12 p. (2009). Summary: Lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation. The Lie symmetries of the barotropic vorticity equations on the \(f\)- and \(\beta\)-planes, as well as on the sphere in rotating and rest reference frames, are determined. A symmetry background for reducing the rotating reference frame to the rest frame is presented. The one- and two-dimensional inequivalent subalgebras of the Lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations. This provides large classes of exact solutions, which include both Rossby and Rossby-Haurwitz waves as special cases. We also discuss the possibility of partial invariance for the \(\beta\)-plane equation, thereby further extending the family of its exact solutions. This is done in a more systematic and complete way than previously available in literature. ©2009 American Institute of Physics Cited in 1 ReviewCited in 14 Documents MSC: 86A10 Meteorology and atmospheric physics 76B47 Vortex flows for incompressible inviscid fluids 76N20 Boundary-layer theory for compressible fluids and gas dynamics 76B65 Rossby waves (MSC2010) 76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics 22E70 Applications of Lie groups to the sciences; explicit representations Keywords:Lie algebras; Lie groups; \(P\) invariance; vortices; waves PDFBibTeX XMLCite \textit{A. Bihlo} and \textit{R. O. Popovych}, J. Math. Phys. 50, No. 12, 123102, 12 p. (2009; Zbl 1272.86001) Full Text: DOI arXiv References: [1] DOI: 10.1007/978-94-017-0745-9 [2] Andreev V. K., Differentsial’nye Uravneniya 24 pp 1577– (1988) [3] Andreev V. K., Diff. Eq. 24 pp 1041– (1989) [4] Berker R., Handbuch der Physik 2 pp 1– (1963) [5] Bihlo A., Wiener Meteorologische Schriften 6 (2007) [6] Blender R., Beitr. Phys. Atmos. 63 pp 255– (1991) [7] DOI: 10.1006/jsco.1999.0299 · Zbl 0958.68543 [8] DOI: 10.2991/jnmp.1994.1.2.3 · Zbl 01506204 [9] DOI: 10.2991/jnmp.1994.1.2.3 · Zbl 01506204 [10] DOI: 10.1088/1751-8113/41/26/265501 · Zbl 1141.76051 [11] Haurwitz B., J. Mar. Res. 3 pp 254– (1940) [12] DOI: 10.1016/0010-4655(93)90007-Y · Zbl 0854.65055 [13] DOI: 10.1023/A:1015056932520 · Zbl 1031.76041 [14] Holton J. R., An Introduction to Dynamic Meteorology (2004) [15] DOI: 10.1016/j.physleta.2003.11.056 · Zbl 02023575 [16] Ibragimov N. H., Applications in Engineering and Physical Sciences 2, in: CRC Handbook of Lie Group Analysis of Differential Equations (1995) · Zbl 0864.35002 [17] Katkov V. L., Izv., Acad. Sci., USSR, Atmos. Oceanic Phys. 1 pp 630– (1965) [18] Katkov V. L., Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 2 pp 1193– (1966) [19] DOI: 10.1023/B:NODY.0000034646.18621.73 · Zbl 1098.76059 [20] DOI: 10.1175/1520-0469(1946)003<0053:TMOHWI>2.0.CO;2 [21] DOI: 10.1007/978-1-4612-4350-2 [22] Ovsiannikov L. V., Group Analysis of Differential Equations (1982) · Zbl 0485.58002 [23] DOI: 10.1175/1520-0469(1960)017<0635:TSFOTV>2.0.CO;2 [24] Poljanin A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists (2002) [25] Popovych H. V., Proceedings of the Fourth International Conference Symmetry in Nonlinear Mathematical Physics pp 178– (2002) 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.