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Stability of jet flows in a rotating shallow water layer. (English. Russian original) Zbl 1354.76060

Fluid Dyn. 51, No. 5, 606-619 (2016); translation from Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza 2016, No. 5, 29-42 (2016).
Summary: The problem of the stability of an isolated jet flow and two counter-streaming jet flows in a rotating shallow-water layer is considered. These flows are described by exact solutions of the Charny-Obukhov equation with one or two discontinuities of the potential vorticity, respectively. The isolated jet flow is shown to be stable. For the system consisting of two jet flows the dependence of the characteristics of the unstable wave modes on a geometric parameter, namely, the ratio of the spacing between the jet axes to the deformation radius, is determined. On the basis of the contour dynamics method a weakly-nonlinear model of the longwave instability is developed.

MSC:

76E07 Rotation in hydrodynamic stability
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[1] J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin & New York (1987). · Zbl 0713.76005
[2] C. J. McWilliams, Fundamentals of Geophysical Fluid Dynamics, Cambridge Univ. Press, Cambridge (2011). · Zbl 1233.86003
[3] A. E. Gill, Atmosphere-Ocean Dynamics, Acad. Press, New York (1992).
[4] E. Palmén and C.W. Newton, Atmospheric Circulation Systems, Acad. Press, New York (1969).
[5] J. Masters, “The Jet Stream is Getting Weird,” Scientific Amer. 311 (6), 68 (2014).
[6] A. M. Obukhov, “On the Geostrophic Wind,” Izv. Akad. Nauk SSSR. Ser. Geogr. Geofiz. 13 (4), 281 (1949).
[7] V. V. Alekseev, S. V. Kiseleva, and S. S. Lappo, Laboratory Models of Physical Processes in the Atmosphere and the Ocean [in Russian], Nauka, Moscow (2005).
[8] F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, “Stability and Vortex Structures of Quasi-Two-Dimensional Shear Flows,” Usp. Fiz. Nauk 160 (7), 1 (1990).
[9] A. E. Gledzer, E. B. Gledzer, A. A. Khapaev, and O. G. Chkhetiani, “Experimental Discovery of the Vortex and Rossby Wave Transfer Blocking upon MHD Excitation of Quasi-Two-Dimensional Flows in a Rotating Cylindrical Reservoir,” Pisma Zh. Eksp. Teor. Fiz. 97, 359 (2013).
[10] M. E. Stern, “Lateral Wave Breaking and Shingle Formation in Large-Scale Shear Flows,” J. Phys. Oceanogr. 15, 1274 (1985).
[11] Lord Rayleigh, “On the Stability or Instability of Certain Fluid Motions,” Proc. London Math. Soc. 9, 57 (1880). · JFM 12.0711.02
[12] E. Heifetz, C. H. Bishop, and P. Alpert, “Counter-Propagating Rossby Waves in the Barotropic Rayleigh Model of the Shear Instability,” Quart. J. Roy. Met. Soc. 125, 2835 (1999).
[13] E. Heifetz and J. Methven, “Relating Optimal Growth to Counter-PropagatingRossbyWaves in Shear Instability,” Phys. Fluids 17, 064107 (2005). · Zbl 1187.76209
[14] D. I. Pullin, “Contour Dynamics Method,” Annu. Rev. Fluid Mech. 24, 89 (1992). · Zbl 0743.76021
[15] V. P. Goncharov and V. I. Pavlov, Hamiltonian Vortex and Wave Dynamics [in Russian], Geos, Moscow (2008).
[16] M. A. Sokolovsky and J. Verron, Dynamics of Vortex Structures in a Stratified Rotating Fluid, Springer, New York & London (2014). · Zbl 1384.86001
[17] L. J. Pratt and M. E. Stern, “Dynamics of Potential Vorticity Fronts and Eddy Detachment,” J. Phys. Oceanogr. 16, 1101 (1986).
[18] L. J. Pratt, “Meandering and Eddy Detachment to a Simple (Looking) Path Equation,” J. Phys. Oceanogr. 18, 1627 (1988).
[19] B. Cushman-Roisin, L. J. Pratt, and E. Ralph, “A General Theory for Equivalent Barotropic Thin Jets,” J. Atmos. Sci. 23, 91 (1993).
[20] G. Sutyrin, “Generation of Deep Eddies by a Turning Baroclinic Jet,” Deep Sea Res. 101, 1 (2015).
[21] A. V. Gruzinov, “Contour Dynamics of the Hasegawa-Mima Equation,” Pisma Zh. Eksp. Teor. Fiz. 55 (1), 75 (1992).
[22] I. S. Gradshtein and I.M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Acad. Press, New York (1980).
[23] A. H. Nayfeh, Perturbation Methods, Wiley, New York (1973). · Zbl 0265.35002
[24] P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge Univ. Press, Cambridge (2002). · Zbl 0997.76001
[25] D. G. Dritshel and J. Vanneste, “Instability of a Shallow-Water Potential- Vorticity Front,” J. Fluid Mech. 561, 237 (2006). · Zbl 1098.76031
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