The linear stability of a stratified shear flow of an inviscid

The linear stability of a stratified shear flow of an inviscid, incompressible fluid has been extensively studied by many authors. The standard method for obtaining stability criteria from the linearized equations for an inviscid incompressible fluid in a plane parallel flow is normal-mode analysis, which leads to the Rayleigh stability equation ( Drazin and Reid 5, Drazin and Howard 6). The normal-mode stability of plane parallel flows of an inviscid, incompressible stratified fluid has been analyzed by Taylor 16 and Goldstein 7.
Drazin 6 discussed the stability of parallel flow in a parallel magnetic field at small magnetic Reynolds number. Kent (9, 10) studied the effect of varying magnetic field on the stability of parallel flows. Agarwal and Agarwal 1 analyzed the stability of heterogeneous shear flow in the presence of parallel magnetic field. Kochar and Jain 11 and Rathy and Harikishan 14 have also discussed the same problem. Small perturbations of a parallel shear flow in an inviscid, incompressible stably stratified fluid are studied by Collyer 3.
The Kuo’s eigen value problem (Kuo 12) governs the normal mode stability of barotropic zonal flows of an inviscid, incompressible fluid on a ? – plane. Barston 2 has introduced a new method in the linear stability analysis of plane parallel flows of inviscid, incompressible homogeneous fluid. Stability of stratified shear flows in channels with variable cross section was studied by Reddy and Subbiah 15. Linear stability of inviscid, parallel and stably stratified shear flow under the assumption of smooth strictly monotonic profiles of shear flow and density is studied by Hirota and Morrison 8.
In the present study, the work of Padmini and Subbiah 13 is extended to study the effect of uniform magnetic field. The stability of stratified shear flow of an inviscid, incompressible fluid confined between two rigid planes at z=±L under the influence of uniform magnetic field is considered. The following analysis is based on the linear velocity profile with long wavelength approximations.

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