On shear instability and propagation of surface gravity waves in sea straits
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THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
On shear instability and propagation of surface gravity waves in sea straits V.R.K. Reddy1,a and M. Subbiah2 1 2
Department of Mechanical and Aerospace Engineering, IIT, Hyderabad, India Department of Mathematics, Pondicherry University, Kalapet, Puducherry, India Received: 23 February 2018 / Revised: 23 September 2018 Published online: 26 November 2018 c Societ` a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2018 Abstract. Normal mode analysis of stability of shear flows of inviscid, incompressible fluids in channels with variable cross section and with a free upper surface is undertaken. Some general analytical results are obtained for basic flows with smooth velocity profiles in channels with smooth topography function. A Kelvin-Helmholtz instability study in a channel with constant topography function is also analyzed.
1 Introduction Stability analysis of shear flows in sea straits is important in physical oceanography [1]. In the context of sea straits departures from rectangular geometry seem to be quite important as sea straits rarely have simple cross sections. For example Bab el Mandab, the strait that connects the Red Sea to the Gulf of Aden has a deep central trough, bordered by shallow flanges [1]. A continuously stratified exchange flow takes place in which dense water flows out of the Red Sea at depth and is replaced by less dense water near the surface. The outflow is largely confined to the central trough whereas the inflow is spread over the whole strait. In [1] the velocity U0 (z) of a shear flow and the stratification parameter N (z) were found from observed data and analytical approximations to the topography. Primarily to aid in interpretation of data collected in sea straits the stability equation for stability of shear flows in channels with variable cross sections was developed in [2]. Specifically they considered a channel that is unbounded in the x-direction, bounded between two lateral walls at y = yL (z), and y = yR (z) in the y-direction, and bounded between two rigid walls at z = 0 and z = D in the z-direction. The width function of the channel b(z) is given by b(z) = yR (z) − yL (z), and the topography by T (z) = (ln b(z)) , where a prime denotes differentiation with respect to z. Let (U0 (z), 0, 0) be the velocity and ρ0 (z) the density of the basic flow whose stability is being studied. For statically stable density stratification (ρ0 ≤ 0) we have N 2 (z) = (−gρ0 )/ρ0 ≥ 0, where g is the acceleration due to ik(x−ct) be the vertical component of the disturbance velocity, where k > 0 is the wave number gravity. Let w(z)e ˆ and c = cr + icr is the complex wave velocity. Here ci > 0 corresponds to an unstable mode and ci = 0 to a neutral mode. If ci = 0 and cr lies within (outside) the range of U0 (z) then we have a singular (non-singular) neutral mode. It was found in [2] that the stability equation is an extended Taylor-Goldstein (ETG) equation. As they were interested in understanding the propagation of internal
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