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On formal stability of stratified shear flows. (English) Zbl 1328.76030

Summary: A novel linear stability criterion is established for the equilibria of general three-dimensional (3D) rotating flows of an ideal gas satisfying Boyle-Charles’ law by a newly refined energy-Casimir convexity (ECC) method that can exploit a larger class of Casimir invariants. As a conventional ECC method cannot be applied directly to stratified shear flows, in our new approach, rathern than checking the local convexity of a Lyapunov functional \(L\equiv E+C_{E}\) defined as a sum of the total energy and a ceratin Casimir, we seek the condition for non-existence of unstable manifolds: orbits (physically realisable flow in phase space) on the leaves of invariants including \(L\) as well as other Casimirs connecting a given equilibrium point \(O\) and other points in the neighbourhood of it. We argue that the separatrices of the second variation of \(L\;(\delta^{2}L=0)\) generally consist of such unstable manifolds as well as pseudo unstable ones for which either the total energy or Casimirs actually serves as a barrier for escaping orbits. The significance of new method lies in the fact that it eliminates the latter so as to derive a condition that \(O\) being an isolated equilibrium point in terms of orbital connections.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
37B25 Stability of topological dynamical systems
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[1] H. D. I. Abarbanel, D. D. Holm, J. E. Marsden and T. S. Ratiu, Richardson number criterion for the nonlinear stability of three-dimensional stratified flow, Phys. Rev. Lett. 52 (1984), 2352-2355.
[2] , Nonlinear stability analysis of stratified fluid equilibria, Philos. Trans. Roy. Soc. London Ser. A 318 (1986), 349-409. · Zbl 0637.76119 · doi:10.1098/rsta.1986.0078
[3] V. I. Arnol’d, Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR 162 (1965), 975-978 (in Russian); English transl.: Soviet Math. Dokl. 6 (1965), 773-777. · Zbl 0141.43901
[4] , On an a priori estimate in the theory of hydrodynamical stability, Izv. Vyssh. Uchebn. Zaved. Mat. 54 (1966), 3-5 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 79 (1969), 267-269. · Zbl 0191.56303
[5] J. M. Ball and J. E. Marsden, Quasiconvexity at the boundary, positivity of the second vari- ation and elastic stability, Arch. Ration. Mech. Anal. 86 (1984) 251-277. · Zbl 0552.73006 · doi:10.1007/BF00281558
[6] J. G. Charney, Planetary fluid dynamics, Reidel, 1973. · Zbl 0297.76036
[7] J. G. Charney and M. E. Stern, On the stability of internal baroclinic jets in a rotating atmosphere, J. Atmospheric Sci. 19 (1962), 159-172.
[8] F. V. Dolzhansky, Fundamentals of geophysical hydrodynamics, Springer, 2013. · Zbl 1286.86001 · doi:10.1007/978-3-642-31034-8
[9] D. D. Holm, B. A. Kuperschmidt and C. D. Levermore, Canonical maps between Poisson brackets in Eulerian and Lagrangian descriptions of continuum mechanics, Phys. Lett. A 98 (1983), 389-395.
[10] D. D. Holm, J. E. Marsden, T. S. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep. 123 (1985), 1-116. · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6
[11] J. R. Holton, An introduction to dynamic meteorology, 4th ed., Elsevier, 2004.
[12] L. N. Howard, Note on a paper of John W. Miles, J. Fluid Mech. 10 (1961), 509-512. · Zbl 0104.20704 · doi:10.1017/S0022112061000317
[13] Y. Kuroda, Symmetries and Casimir invariants for perfect fluid, Fluid Dynam. Res. 5 (1990), 273-287.
[14] R. G. Littlejohn, Singular Poisson tensors, in Mathematical methods in hydrodynamics and integrability in dynamical systems (La Jolla, CA, 1981), AIP Conf. Proc. 88, Amer. Inst. Phys., 1982, 47-66. · Zbl 0537.70014
[15] A. Mahalov, M. Moustaoui and B. Nicolaenko, Three-dimensional instabilities in non- parallel shear stratified flows, Kinetic Related Models 2 (2009), 215-229. · Zbl 1372.76051
[16] J. E. Marsden and P. J. Morrison, Noncanonical Hamiltonian field theory and reduced MHD. in Fluids and plasmas: geometry and dynamics (Boulder, CO, 1983), Contemp. Math. 28, Amer. Math. Soc., 1984, 133-150. · Zbl 0546.58024 · doi:10.1090/conm/028/751979
[17] J. W. Miles, On the stability of heterogeneous shear flow, J. Fluid Mech. 10 (1961), 496-508. · Zbl 0101.43002 · doi:10.1017/S0022112061000305
[18] P. J. Morrison and J. M. Greene, Noncanonical Hamiltonian density formulation of hy- drodynamics and ideal magnetohydrodynamics, Phys. Rev. Lett. 45 (1980), 790-794.
[19] , Noncanonical Hamiltonian density formulation of hydrodynamics and ideal mag- netohydrodynamics addition, Phys. Rev. Lett. 48 (1982), 569.
[20] P. J. Olver, Applications of Lie groups to differential equations, Springer, 1986. · Zbl 0588.22001
[21] H. Sakuma and M. Ghil, Stability of propagating modons for small-amplitude perturbations, Phys. Fluids A 3 (1991), 408-414. · Zbl 0850.76239 · doi:10.1063/1.858096
[22] R. Salmon, Lectures on geophysical fluid dynamics, Oxford Univ. Press, 1998.
[23] E. C. G. Sudarshan and N. Mukunda, Classical dynamics: A modern perspective, Inter- science, New York, 1974. · Zbl 0329.70001
[24] G. K. Vallis, G. F. Carnevale and W. Young, Extremal energy properties and construction of stable solutions of the Euler equations, J. Fluid Mech. 207 (1989), 133-152. · Zbl 0681.76007 · doi:10.1017/S0022112089002533
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