Exact Solution and Instability for Geophysical Waves with Centripetal Forces and at Arbitrary Latitude

Abstract

We consider three-dimensional geophysical flows at arbitrary latitude and with constant vorticity beneath a wave train and above a flat bed in the β-plane approximation with centripetal forces. We consider the f -plane approximation as well as the β-plane approximation. For the f -plane approximation, we prove that there is no bounded solution. For the β-plane approximation, we show that the flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity. Our results reveal some essential differences from those results in the literature, due to the presence of centripetal forces. Moreover, for the case exhibiting the surface tension, we prove that there are no flows exhibiting constant vorticity. 2010 Mathematics Subject Classification. Primary 35Q31, 35J60, 76B15.Vorticity is adequate in describing the motion of both equatorial and nonequatorial flows. The nonzero vorticity serves as a tool for describing interactions of waves with non-uniform currents. From the history perspective, the mathematical theory of rotational water waves was original started by Gerstner in the beginning of the 19th century [24], in which an explicit family of periodic travelling waves with non-zero vorticity was constructed using Lagrangian coordinates. In recent works [6,17,19,20,21,29,34,38], the assumption of nonzero constant vorticity has been assumed, which is the simplest rotational setting and corresponds to a uniform current. Although such an assumption is for physical viewpoints (see the discussion in [31]), the main consideration lies on more convenient in mathematical analysis, for example, constant vorticity flows have the advantage that their velocity field consists of harmonic functions (see the modern discussions in [19,20]). The importance of the vorticity in the realistic modeling of ocean flows is highlighted in the very recent papers [1,16,32,33]. See [11,12,18,30,32,39,40] and the monograph [5]for more results on rotational water waves.Among the results on vorticity in the literature, a feature is to determine the dimensionality of the flow. From the mathematical perspective, the study of the two-dimensionality for the rotational flow was started by the work [6], in which Constantin showed that a free surface water flow of constant nonzero vorticity beneath a wave train and above a flat bed must be two-dimensional and the vorticity must have only one nonzero component which points in the horizontal direction orthogonal to the direction of wave propagation. After [6], more results along this line has been obtained in different settings. In the presence of Coriolis forces, Martin in [33] proved the two-dimensionality of the equatorial flows in the f -plane approximation, and it was found that there is a striking difference between the geophysical flows and the classical gravity flows, that is, the two-dimensionality holds even if the vorticity vector vanishes due to the presence of Coriolis forces. Martin also proved in [36] and [35] that for the equatorial and non-equat...

Some Explicit Solutions to the Three-Dimensional Nonlinear Water Wave Problem

“…By means of an essential and relevant extension of the Gerstner wave solution, Constantin has constructed explicit solutions in Lagrangian coordinates [4][5][6][7] that portray three-dimensional geophysical flows with a preferred propagation direction. The skeleton of the Gerstner wave was used to further the study of geophysical fluid dynamics [1,11,18,19,[22][23][24]28,[30][31][32].…”

Variational formulations of steady rotational equatorial waves

“…Starting with this pioneering paper, recently some essential results on equatorial water waves have been proved in the literature. See [5,6,9,10,14,19,20,32,36,45]. In the model constructed by Constantin in [9], the upper boundary of the centre layer is assumed to be at, while the lower boundary is the thermocline near which the equatorial undercurrent resides.…”

Constant vorticity atmospheric Ekman flows in the $ f- $plane approximation

“…o latitude [1,8]. In the β−plane approximation, exact nonlinear geophysical Gerstner-like solutions were constructed by Constantin [9,10,11], there have recently emerged a considerable literature on the geophysical water waves related to different physical scenarios, including the β−plane approximation [4,12,25,5,3,38], the f −plane approximation [17,26,6], and the large-scale ocean currents with distinctive and persistence patters [14,15]. Exact solutions for equatorial geophysical water waves incorporating a constant underlying mean current in the β−plane approximation were presented in [27,28], which are the first rate papers.…”