Long ring waves in a stratified fluid over a shear flow

Abstract

Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this paper we study long linear and weakly nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that, despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition (different from the known decomposition in Cartesian geometry), which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant current, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first-order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shapes of the wavefronts of surface and interfacial waves propagating over the same shear flow.

Initial conditions for the cylindrical Korteweg‐de Vries equation

“…Thus, the matching condition for the sKdV equation (D.1) is ( , → 0) ∼ 0 ( ) . (D. 16) We now see that, with this matching condition, the sKdV mass is zero because…”

“…It was first derived in the context of plasma waves, 9 for surface waves, [10][11][12] and then for internal waves. [13][14][15][16][17] Like the KdV equation (2) it is integrable [18][19][20][21][22] and has "soliton" solutions. It also has two conservation laws of physical interest, analogous to (3,4),…”

Observation of surface wave patterns modified by sub-surface shear currents

“…Recent theory has predicted that ship and ring waves are affected in striking ways by sub-surface shear; for instance, ring waves such as from a pebble thrown into the water no longer form concentric rings (Johnson 1990;Ellingsen 2014b;Khusnutdinova 2016), and Kelvin's famous result that ship waves in deep water always form an angle of approximately 39 • no longer holds (Ellingsen 2014a;Li & Ellingsen 2016). The physical understanding of ship waves more generally has been the focus of much recent attention (Rabaud & Moisy 2013;Darmon et al 2014;Noblesse et al 2014;Pethiagoda et al 2014).…”

Nonlinear ring waves in a two-layer fluid

“…Importantly, the coefficient µ 5 is equal to zero both when the fluid is uniform and when there is no shear flow. In these cases the equation (2) reduces to a 1+1dimensional model [10]. The general theory was then applied to the case of a two-layer fluid with a piecewiseconstant shear flow, with an emphasis on the analytical description of the wavefronts of surface and interfacial ring waves.…”

“…In Section II, we briefly overview the derivation of the cKdV-type equation from the Euler equations with application to the two-layer fluid given in [10]. Particular examples of concentric waves generated from a localised initial condition and a 2D version of the dambreak problem are modelled in Section III.…”