On the Nonlinear Stability of Inviscid Homogeneous Shear Flows in Sea Straits of Arbitrary Cross Sections
In this paper we study the nonlinear stability of steady flows of inviscid homogeneous fluids in sea straits of arbitrary cross sections. We use the method of Arnol'd  to obtain two general stability theorems for steady basic flows with respect to finite amplitude disturbances. For the special case of plane parallel shear flows we find a finite amplitude extension of the linear stability result of Deng et al . We also present some examples of basic flows which are stable to finite amplitude disturbances.
V I Arnol’d, On an a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Trans. Ser. 2, vol. 79, pp. 267-269, 1969.
J Deng, L Pratt, L Howard and C Jones, On stratified shear flows in sea Straits of arbitrary cross section, Studies in Appl. Math., vol. 111, pp. 409-434, 2003.
L J Pratt, H E Deese, S P Murray and W Johns, Continuous dynamical Modes in straits having arbitrary cross sections with applications to the Bab al Mandab, J. Phys. Oceanogr., vol. 30, pp. 2515-2534, 2000.
P G Drazin and W H Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, UK, 1981.
M Subbiah and V Ganesh, On the stability of homogeneous shear flows in sea straits of arbitrary cross section, Indian J. Pur Appl. Math., vol. 38, pp. 43-50, 2007.
M Subbiah and V Ganesh, On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability, Proc. Indian Acad. Sci. (Math Sci), vol. 120, pp. 387-394, 2010.
M Subbiah and V Ramakrishnareddy, On the role of topography in the stability analysis of homogeneous shear flows, Journal of Analysis (to appear).
D D Joseph, Stability of fluid motions I, II, Springer, 1976.
V I Arnol’d, Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Sov. Math. Dokl., vol. 6, pp. 773-776, 1965.
V I Arnol’d and B Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer, 1998.
C Marchioro and M Pulvirenti, Mathematical theory of incompressible non-viscous fluids, Applied Mathematical Sciences, vol. 95, Springer, 1995.
D D Holm, J E Marsden and T Ratiu, Nonlinear stability of the Kelvin-Stuart cat’s eyes flow, Lect. in Appl. Mathematics, AMS, vol. 23, pp. 171-186, 1986.
M E McIntyre and T G Shepherd, An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows with remark on Hamiltonian structure and on Arnol’d’s stability theorems, J. Fluid Mech., vol. 181, pp. 527-565, 1987.
M Subbiah and M Padmini, Note on the nonlinear stability of equivalent barotropic flows, Indian. J. Pure Appl. Math., vol. 30, pp. 1261-1272, 1999.
M Subbiah and V Ganesh, Bounds on the phase speed and growth rate of the extended Taylor-Goldstein problem, Fluid Dynamics Research, vol. 40, pp. 364-377, 2008.