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Finite amplitude Holmboe waves. (English) Zbl 0661.76033
We investigate the evolution of a parallel shear flow which has embedded whithin it a thin, symmetrically positioned layer of stable density stratification. The primary instability of this flow may deliver either Kelvin-Helmholtz waves or Holmboe waves, depending on the strength of the stratification. We describe a sequence of numerical simulations which reveal for the first time the behavior of the Holmboe wave at finite amplitude and clarify its structural relationship to the Kelvin-Holmholtz wave.
The flows investigated have initial profiles of horizontal velocity and Brunt-Vaisala frequency given in nondimensional form by \(U=\tanh \zeta\) and \(N^ 2=J \sec h^ 2 R\zeta,\) respectively, in which \(\zeta\) is a nondimensional vertical coordinate, J is the value of the gradient Richardson number \(N^ 2/(dU/d\zeta)^ 2\) at \(\zeta =0\), and \(R=3\). Linear stability theory predicts that the flow will develop Holmboe instability when J exceeds some critical value \(J_ c\), and Kelvin- Helmholtz instability when J is less than \(J_ c\); \(J_ c\) being approximately equal to 0.25 when \(R=3\). We simulate the evolution of flows with \(J=0.9\), \(J=0.45\), and \(J=0.22\), and find that the first two simulations yield Holmboe waves while the third yields a Kelvin-Helmholtz wave, as predicted.
The Holmboe wave is a superposition of two oppositely propagating disturbances, a right-going mode whose energy is concentrated in the region above the centre of the shear layer, and a left-going mode whose energy is concentrated below the centre of the shear layer. The horizontal speed of the modes varies periodically, and the variations are most pronounced at low values of J. If \(J\leq J_ c\), the minimum horizontal speed of the modes vanishes and the modes become phase-locked, whereupon they roll up to form a Kelvin-Helmholtz wave as predicted by J. Holmboe [Geophys. Publ. 24, 67 ff. (1962)]. When J is moderately greater than \(J_ c\), the Holmboe wave ejects long, thin plumes of fluid into the regions above and below the shear layer, as has often been observed in laboratory experiments, and we examine in detail the mechanism by which this occurs.

MSC:
76E05 Parallel shear flows in hydrodynamic stability
76V05 Reaction effects in flows
76M99 Basic methods in fluid mechanics
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