Exact solution and instability for geophysical waves at arbitrary latitude

This paper is concerned with some analytical aspects pertaining to the Antarctic Circumpolar Current. We use spherical coordinates in a rotating frame to derive a new exact and partially explicit solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible azimuthal flow with a discontinuous density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows, that is, flows that are observed in an averaged sense in the ocean. The discontinuous density triggers the appearance of an interface that plays the role of an internal wave. Although the velocity and the pressure are determined explicitly, we use functional analytical techniques that uniquely render the surface and interface defining functions in an implicit way as soon as a small enough pressure is applied on the free surface. Additionally, we consider a particular example, where the interface can be determined explicitly. We conclude our discussion by setting out monotonicity relations between the surface pressure and its distortion that concur with the physicalThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…𝑟𝜌 p𝜃 is relatively small, we infer from (35) in combination with Assumption 1 that 𝐺(𝑟, 𝜃) ≈ −2Ω𝑢 0 cos 𝜃 ≥ 0.…”

Section: Exact Explicit and Implicit Solutionsmentioning

confidence: 96%

Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates

Martin

Quirchmayr

2021

Stud Appl Math

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“…An alleviation of this aspect is represented by a Bernoulli-type relation between the imposed pressure at the surface and the resulting surface distorsion; an implicit formula for the interface defining function being given by the balance of forces at the interface (5d). For a selective list of papers presenting exact solutions pertaining to geophysical fluid dynamics we refer the reader to [1,4,6,7,[11][12][13][14]16,23,24,[31][32][33]39,[42][43][44][45]48].…”

Section: Explicit and Exact Solutionsmentioning

confidence: 99%

Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

Martin

2021

Physics of Fluids

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We are concerned here with an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates which incorporates discontinuous fluid stratification. This solution represents a steady, purelyazimuthal equatorial flow with an associated free-surface and an interface separating two fluid regions, each of which having its own continuous distribution of density. However, the two density functions do not match along the interface. Following the derivation of the solution we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. Interestingly, it turns out that the interface defining function has infinite regularity.

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“…Besides the work [3] mentioned above, an extension of the exact solution [27] for equatorial waves in the f -plane approximation to the cases at arbitrary latitude and in the presence of a constant underlying background current was presented in [23]. A β-plane approximation at arbitrary latitude in the presence of an underlying current and a Gerstner-like solution to this problem was very recently provided in [2].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we focus on geophysical ocean waves in which both Coriolis and centripetal effects of the Earth's rotation play a significant role. In recent years, the mathematical analysis of geophysical flows [22,37] has attracted much attention for their wide applications (see the references [1,7,9,10,13,14,15,25,28,29,40] for the flows in the equatorial region and [2,3] for the flows at arbitrary latitude). However, in most existed results, centripetal forces are typically neglected because they are relatively much smaller than Coriolis forces.…”

Section: Introductionmentioning

confidence: 99%

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

Chu

Ionescu-Kruse

Yang

2019

J. Math. Fluid Mech.

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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...

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Exact solution and instability for geophysical waves at arbitrary latitude

Cited by 31 publications

References 45 publications

Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates

Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates

Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

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

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