The nonlinear stability of a weakly supercritical shear flow with vertical temperature (density) stratification is investigated. It is shown that the usual Lin's rule of ‘indenting’ a singularity at the point of wave-flow resonance (the so-called critical layer, CL) is inapplicable for evaluating the nonlinear effects. To this end, a consistent procedure for deriving a nonlinear evolution equation is suggested and realized for the viscous critical-layer regime. The procedure takes into account the interaction of the fundamental harmonic with the second harmonic as well as with the zeroth one (i.e. with the mean-flow distortion). It is shown that the nonlinear factors both act in the same manner - at Prandtl number η ≤ 1 they limit the instability but at η > 1 they enhance it and convey a ‘burst-like’ character to it.It is found that CL is the region of strongest interactions between the harmonics. Hence the nonlinear contribution does not actually depend on the type of original flow model chosen. A simple physical interpretation is given to illustrate the mechanism governing the nonlinearity effects on the stability in the viscous critical-layer regime.

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In a previous paper (Churilov & Shukhman 1987a) we investigated the nonlinear development of disturbances to a weakly supercritical, stratified shear flow; we now report a continuation of that study. The degree of supercriticality of the flow is assumed not too small so that — unlike Paper 1 — the critical layer that appears in the region of resonance of the wave with the flow is an unsteady rather than viscous one. The evolution equation with cubic and quintic nonlinearity has been derived. The nonlinear term is non-local in time, i.e. depends on the entire preceding development of the disturbance. This equation has been used in the analysis of the evolution of an initially small disturbance. It is shown that where wave amplitude A is small enough (A [Lt ] ν½, ν is the inverse of the Reynolds number), cubic nonlinearity dominates. In this case, as in Paper 1, the character of the evolution essentially depends on the sign of the quantity (η − 1), where η is the Prandtl number. However, independently of this sign the disturbance reaches — as it increases — the level A ∼ O(ν½) and then quintic nonlinearity becomes dominant. At this stage an ‘explosive’ regime occurs and amplitude grows as $A \sim (t_0 - t)^{-\frac{7}{4}}$. The results obtained, together with the findings of Paper 1, provide a full description of the development of small disturbances at a large (but finite) Reynolds number in different regimes which are determined by the degree of flow's supercriticality.

We analytically study a scattering of long linear surface waves on stationary currents in a duct (canal) of constant depth and variable width. It is assumed that the background velocity linearly increases or decreases with the longitudinal coordinate due to the gradual variation of duct width. Such a model admits an analytical solution of the problem in hand, and we calculate the scattering coefficients as functions of incident wave frequency for all possible cases of sub-, super-, and transcritical currents. For completeness we study both cocurrent and countercurrent wave propagation in accelerating and decelerating currents. The results obtained are analyzed in application to recent analog gravity experiments and shed light on the problem of hydrodynamic modeling of Hawking radiation.

In the linear approximation we study long wave scattering on an axially symmetric flow in a shallow water basin with a drain in the center. Besides of academic interest, this problem is applicable to the interpretation of recent laboratory experiments with draining bathtub vortices, description of wave scattering in natural basins, and also can be considered as the hydrodynamic analogue of scalar wave scattering on a rotating black hole in general relativity. The analytic solutions are derived in the lowfrequency limit to describe both pure potential perturbations (surface gravity waves) and perturbations with nonzero potential vorticity. For the moderate frequencies the solutions are obtained numerically and illustrated graphically. It is shown that there are two processes governing the dynamics of surface perturbations, the scattering of incident gravity water waves by a central vortex, and emission of gravity water waves stimulated by a potential vorticity. Some aspects of their synergetic actions are discussed.