Abstract
The stability problem of inviscid incompressible stratified shear flows in sea straits of arbitrary cross sections is formulated without the Boussinesq approximation. Series solutions of the stability equation are found by the method of Frobenius and these solutions are used to study the existence of neutral modes when the Richardson number is larger than one quarter. The energy aspect of the problem is studied by discussing the role of the Reynolds stress and its variation in the vertical direction for both unstable modes and neutral modes adjacent to the unstable modes. The interfacial conditions at layers of discontinuity of the basic velocity, density or topography are derived and these are used in the study of instability of two examples of basic flows.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pratt LJ, Deese HE, Murray SP, Johns W (2000) Continuous dynamical modes in straits having arbitrary cross sections, with applications to the Bab al Mandab. J Phys Oceanogr 30(10):2515–2534
Deng J, Pratt L, Howard L, Jones C (2003) On stratified shear flows in sea straits of arbitrary cross section. Stud Appl Math 111(4):409–434
Subbiah M, Ganesh V (2008) Bounds on the phase speed and growth rate of the extended Taylor–Goldstein problem. Fluid Dyn Res 40(5):364–377
Ganesh V, Subbiah M (2009) On upper bounds for the growth rate in the extended Taylor–Goldstein problem of hydrodynamic stability. Proc Indian Acad Sci Math Sci 119(1):119–135
Baines PG (1995) Topographic effects in stratified flows. Cambridge University Press, Cambridge
Reddy VR, Subbiah M (2015) Stability of stratified shear flows in channels with variable cross sections. Appl Math Mech Engl 36(11):1459–1480
Miles JW (1961) On the stability of heterogeneous shear flows. J Fluid Mech 10(04):496–508
Barros R, Choi W (2011) Holmboe instability in non-Boussinesq fluids. Phys Fluids 23(12):124103
Barros R, Choi W (2014) Elementary stratified flows with stability at low Richardson number. Phys Fluids 26(12):124107
Drazin PG, Howard LN (1966) Hydrodynamic stability of parallel flow of inviscid fluid. Adv Appl Mech 9:1–89
Howard LN (1963) Neutral curves and stability boundaries in stratified flow. J Fluid Mech 16(03):333–342
Huppert HE (1973) On Howard’s techniques for perturbating neutral sorutions of the Taylor–Goldstein equation. J Fluid Mech 57(02):361–368
Engevik L, Haugan PM, Klemp S (1985) Stability analysis adjacent to neutral solutions of the Taylor–Goldstein equation when Howard’s formula breaks down. J Fluid Mech 159:347–358
Baines PG, Mitsudera H (1994) On the mechanism of shear flow instabilities. J Fluid Mech 276:327–342
Ganesh V, Subbiah M (2013) Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability. Proc Indian Acad Sci Math Sci 123(2):293–302
Smeed DA (2004) Exchange through the Bab el Mandab. Deep-Sea Res Pt II 51(4):455–474
Drazin PG, Reid WH (1981) Hydrodynamic stability. Cambridge University Press, Cambridge
Banerjee MB (1969) On the stability of nonhomogeneous shear flows. Jpn J Appl Phys 8(7):821
Lindzen RS, Barker JW (1985) Instability and wave over-reflection in stably stratified shear flow. J Fluid Mech 151:189–217
Subbiah M, Ramakrishnareddy V (2011) On the role of topography in the stability analysis of Homogeneous shear flows. J Anal 19:71–86
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Reddy, V.R.K., Subbiah, M. On Neutral and Unstable Modes and on the Role of Reynolds Stress in the Stability Problem of Density Stratified Shear Flows in Channels with Variable Bottom. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 89, 777–790 (2019). https://doi.org/10.1007/s40010-018-0536-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40010-018-0536-0