Physical flow properties for Pollard-like internal water waves

Abstract

We present a study of the physical flow properties for a recently derived three-dimensional nonlinear geophysical internal wave solution. The Pollard-like internal wave solution is explicit in terms of Lagrangian labelling variables, enabling us to examine the mean flow velocities and mass flux in the three-dimensional setting. We show that the Pollard-like internal water wave does not have a net wave transport.

Mass transport for Pollard waves

“…Numerical evaluation of (13) shows this decrease in mean Eulerian velocity, as depicted in Figure 4. There are no Eulerian mean velocities in the vertical or meridional direction, as v E and w E lead to an integral akin to (12), albeit with an odd, L-periodic integrand.…”

“…This was the first of many -and in recent years increasingly complex -exact solutions for geophysical waves (see [6,9,10] and references therein). Pollard's zonally propagating wave has since been studied by a number of authors -its stability was investigated by Ionescu-Kruse [8], who also considered the restriction to the equatorial f -plane [7], and it was recently extended to internal waves by Kluczek [13,12]. A thoroughgoing investigation of Pollard's solution, including the effects of an underlying current, was performed by Constantin & Monismith [3].…”

Exact solution and instability for geophysical waves at arbitrary latitude

“…The approach pioneered by Gerstner [20] (for modern detailed descriptions see [3,22]) of finding explicit exact solutions for gravity fluid flows within the Lagrangian framework [2], was extended to geophysical flows too (see the survey [27]). Gerstner-like three-dimensional solutions were obtained in the f -plane approximation at an arbitrary latitude in [45] and very recently in [14,17], the internal wave in [38,39], in the β-plane approximation in [5,7,8,23] or in a modified β-plane approximation [25,26]; for other studies, we refer the reader to [24,30,31,32,43,44]. We mention that the above exact solutions fail to capture, for example, strong depth variations of the flows.…”

“…where is a small parameter, A, A are vector functions and Φ, B, B are scalar functions. Plugging (37) and (38) into the linearized equations (35)- (36) and equating the terms of the same order in , we get for the leading order terms the relations A • ∇Φ = 0 for all t ≥ 0, B = 0, and the following coupled system, consisting of the eikonal equation for the wave phase and the transport equation for the wave amplitude of the velocity,…”

“…Since the next-order terms A and B depend only on A and ∇Φ, and the remainder terms u rem , p rem are bounded, in an appropriate norm, at any time t by functions that can depend on t but are independent of , then, in (37), (38) these terms can be made as small as we want by multiplication with and they are dominated by the growth of the leading order terms. See, for example, [33] for details.…”