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# The spherical coordinate system, where θ ∈ [0, π ] is the polar angle, ϕ ∈ [0, 2π ) is the azimuthal angle and r is the distance from the origin: (π/2 − θ ) is the conventional angle of latitude, so that the North Pole corresponds to θ = 0, the South Pole to θ = π , and the Equator to θ = π/2.

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Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter:...

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We consider the system of relativistic fermions in the presence of rotation. The rotation is set up as an enhancement of the angular momentum. In this approach the angular velocity for the angular momentum plays the same role as the chemical potential for density. We calculate the axial current using the direct solutions of the Dirac equation with...

## Citations

... The problem at hand is in particular fully nonlinear. This method was originally devised as a means to describing ocean gyres [8,12], but has been since applied to other oceanic flows [11,17,18], including the Antarctic Circumpolar Current, as well as atmospheric flows (via a slightly different scaling-see [6,9,10]). Nevertheless, as a consequence of the thin-shell approximation, the motion is essentially two-dimensional, as the vertical motion vanishes at leading order. ...

... As mentioned in the Introduction, the equations that we are going to consider in this paper have been derived as an asymptotic model from the Euler equations in spherical coordinates, in a frame of reference that is rotating with the Earth. Since the derivation of the model can be found at several places in the literature (see, for instance, [8,12,22]), we will omit it here. Instead, we will merely write down the equations for the leading-order dynamics: these are, essentially, the non-dimensional incompressible Euler equations on the surface of a rotating sphere, written in the spherical coordinates (2.1), given by ...

We consider a model for the Antarctic Circumpolar Current in rotating spherical coordinates. After establishing global-in-time existence and uniqueness of classical solutions, we turn our attention to the issue of stability of a class of steady zonal solutions (i.e., time-independent solutions that vary only with latitude). By identifying suitable conserved quantities and combining them to construct a Lyapunov function, we prove a stability result.

... The mathematical research on the system of geophysical fluid dynamics nonlinear model established by Constantin and Johnson has attracted widespread attention and discussion [6][7][8][9][10][11][12]. In this trend, some results related to geophysical fluid dynamics have been obtained, such as exact solutions, stability, instability, existence and uniqueness, and so forth. ...

... where ω > 0 is the non-dimensional form of Coriolis parameter. F( − ω cos θ) and 2ω cos θ are the oceanic vorticity and the planetary vorticity, respectively [12]. By applying the stereographic projection of the unit sphere centered at the origin from the North Pole to the equatorial plane, the model (1) in spherical coordinates can be transformed into an equivalent plane elliptic partial differential equation, defined by ...

... Meanwhile, linking (11), (12), (17)- (19) and defining the operator ...

In this paper, we study the existence and uniqueness of solutions of a second-order differential equation arising from the Antarctic Circumpolar Current (ACC) and the Ulam–Hyers stability under different ocean vorticity cases. Firstly, we present the expression of general solution for constant vorticity. Secondly, by constructing a new Green’s function corresponding to a second-order ordinary differential equation, which is related to the boundary conditions. Finally, we demonstrate the existence and uniqueness of solution in cases of linear vorticity and nonlinear vorticity.

... Following the discussion in [8] and [10], we make use of the small aspect ratio of the ocean (that is, the ratio between the depth of the ocean and the radius of the Earth) to derive a nondimensional asymptotic model-the thin-shell approximation-from the Euler equations in spherical coordinates on the rotating Earth. However, in contrast to these two works, we do not assume the density to be constant, but we merely require incompressibility (a reasonable assumption in the ocean). ...

... This, alongside mass conservation, will enable us to introduce a pseudo stream function ψ on which the density ρ will be functionally dependent (similarly to [4,22]). Thence we rewrite the equations of motion as a single vorticity equation for ψ and ρ(ψ), whose structure can be exploited to obtain an elliptic equation for ψ that extends the one derived in [8] and [10] (see also [16,17]). ...

... Please note that an analogous derivation was published by the first author of the present paper in [12]; nevertheless, as the present manuscript was supposed to precede [12], we will present the full derivation at this place as well, adding some important details that were omitted in [12]. Following the discussion in [8] and [10], the dynamics of gyres is modelled via an asymptotic shallow-water approximation of the water wave equations above. To set this up, we define the non-dimensional variables t, z, u, v, w, ρ and P by t = R U t, r = ...

In the setting of the thin-shell approximation of the Euler equations in spherical coordinates for oceanic flows with variable density on the spinning Earth, we study a vorticity equation for a pseudo stream function $\psi$ ψ , whereby the assumption of incompressibility allows us to express the density as a function of $\psi$ ψ . Via an elliptic comparison argument, we show that, under certain assumptions, the (explicit) solution in the case of zero rate of rotation (i.e., on a fixed sphere) in a bounded region with smooth boundary contained either in the Northern or in the Southern Hemisphere is an approximation, in a suitable sense, of the corresponding solution of the equation with positive rate of rotation in the same region. This provides new insight into the dynamics of ocean gyres.

... Constantin and Johnson [6] introduced an approach that employed Euler's equation in conjunction with the integration of the equation of mass conservation and associated boundary conditions. This methodology resulted in a solution capable of accurately depicting the fundamental characteristics of gyres on the Earth's surface, regardless of their size. ...

... Subsequent to their work, Hus and Martin [16] delved into an exploration of solutions and the function pertaining to the ACC. Building upon these contributions, Marynets [20] reexamined the governing equation from Constantin and Johnson's study [6], transforming it into a second-order two-point boundary value problem (BVP) suitable for analyzing ocean flows devoid of azimuthal variations (1 + e s ) 3 , and the Coriolis parameter is denoted by the dimensionless symbol ω > 0. The author employs an approach based on the topological transversality theorem to prove the existence of solutions for a class of oceanic vorticities. Chu and Marynets [5] further extended the work of [20] and studied the same secondorder two-point BVP. ...

This article explores the Hilfer fractional derivative within the context of fractional differential equations and investigates a mathematical model formulated as a three-point boundary value problem (BVP). The primary focus is on the application of these models to analyze the jet flow of the Antarctic Circumpolar Current. The study establishes the existence of stream functions using Schaefer’s fixed point theorem under the assumption of the continuity of the vorticity function Φ$\Phi$. Furthermore, the article delves into the existence and uniqueness results of the stream functions by employing the Banach fixed point theorem. This analysis is conducted under the condition that the vorticity function Φ$\Phi$ is Lipschitz continuous with respect to the stream function. Additionally, the stability of the stream functions of the BVP is explored through Ulam–Hyers and generalized Ulam–Hyers stability analyses. In contrast to the foundational results presented for the three-point BVP, the article includes illustrative examples aimed at validating the findings.

... where φ 0 ∈ R is some constant and F represents the total vorticity of the flow (spin vorticity, due to the Earth's rotation, plus oceanic vorticity). The above works are mainly based on [14], in which Constantin and Johnson established the Euler's equation and boundary conditions in spherical coordinates for large gyres and also obtained that the elliptic boundary value problem in terms of a stream function. See [10,12,[15][16][17]19] for more results on Arctic gyres. ...

This paper is concerned with the bounded solutions for a nonlinear second-order differential equation with asymptotic conditions and boundary condition which arise from the study of Arctic gyres. In the case of Lipschitz continuous nonlinearities, we prove the existence, uniqueness and stability of the bounded solution. An existence result for the general nonlinear vorticity term is also obtained.

... Of course the study of gyres in the arctic region has a significant importance. Based on a recent model describing as an elliptic boundary value problem in terms of a stream function [20] for the general motion of ocean currents in the setting of spherical coordinates, Chu derived a nonlinear model of an infinite-interval boundary-value problem for a second-order ordinary differential equation to describe the motion of arctic gryes. See the discussions in [5,8]. ...

... Specifically, the geographic coordinate system used in this paper exhibits a singularity at the North Pole, and this issue has been extensively discussed in the recent paper [17]. However, this singularity is not a problem since there are no gyres near the North Pole; instead, the ocean flow takes the form of the transpolar drift current (see in [20]). Let us consider the inviscid Euler equations in spherical coordinates (ϕ, θ, r ) with ...

... where R ≈ 6378 km is the mean radius of the Earth, P s is the surface pressure, h is the free surface of the ocean, and d is the bottom topography of the ocean. See the discussions in [20] for the above governing equations. Now let us reformulate the above problem by using an asymptotic shallow-water approximation. ...

In this paper, we derive a nonlinear model for stratified arctic gyres, and prove several results on the existence, uniqueness and stability of solutions to such a model, by assuming suitable conditions for the vorticity function and the density function. The approach consists of deriving a suitable integral formulation for the problem and using fixed-point techniques.

... We also assume that the viscosity is constant -while our formulation does allow it to vary, the analytical details become too intricate, making it virtually impossible to find an explicit flow pattern that captures the essence of what is observed. (For the interested reader, a mathematical description of large gyres is given in [54].) With m = ρ = 1, (3.29)-(3.32) ...

... In the case of the EI and SG system, there are no geographic breaks, and the mean surface currents are <6 cm s −1 between both islands (Chaigneau and Pizarro, 2005;Meerhoff et al., 2018), which considering an approximate distance between the island of 400 km yields an average transit time of ∼77 days at such speed (6 cm s −1 ). EI and SG are located in the east-central South Pacific gyre (von Dassow and Collado-Fabbri, 2014), a typical gyre that circles around large areas of essentially stationary, calm water (Constantin and Johnson, 2017). Hence, the mean water flow between the islands is relatively weak compared to other zones in the south-oriental Pacific Ocean. ...

Population connectivity has a fundamental role in metapopulation dynamics with important implications for population persistence in space and time. Oceanic islands, such as Easter Island (EI) and the Salas & Gómez Island (SG), are ideal for the study of population connect-ivity because they are separated by 415 km and isolated from other islands in the Pacific Ocean by >2000 km. Considering that the dispersal process could play a critical role in the persistence of their populations, we evaluated the connectivity pattern of the endemic gastro-pod Monetaria caputdraconis between EI and SG using population genetics and biophysical modelling. Eleven microsatellite loci did not show differences in the allelic frequency of individuals located in EI and SG, suggesting the presence of one genetic population. Historical reciprocal migration implies that 0.49% of the recruits in EI come from SG and 0.37% in SG come from EI. Considering year-round larval release and a larval development of 2 weeks in the plankton, a Lagrangian experiment based on a regional oceanic simulation indicated a weak population connectivity with a high rate of self-recruitment. Interestingly, self-recruitment showed both monthly and interannual variation ranging from 1 to 45% of returned larvae, with lower values estimated in SG compared to EI. The results suggest that few larvae/individuals arrive at each other's island, possibly due to stochastic events, such as rafting. Overall, our results indicate that both islands maintain population connectivity despite their distance; these findings have implications for designing conservation strategies in this region.

... In recent several years, there are some interesting results [4][5][6][7] on the existence, uniqueness, monotonicity and the stability for equation (1.4) with (1.2)-(1.3) starting with the works [3,16]. We point out that the specific form of asymptotic conditions (1.3) is due to physically relevant considerations: the unbounded interval originates in a suitable change of coordinates (starting with a stereographic projection from spherical coordinates) and the fact that it is semi-infinite is because gyres never cross the equator, and the boundary condition (1.3) express the fact that the center of the gyre is a stagnant point. ...

... We point out that the specific form of asymptotic conditions (1.3) is due to physically relevant considerations: the unbounded interval originates in a suitable change of coordinates (starting with a stereographic projection from spherical coordinates) and the fact that it is semi-infinite is because gyres never cross the equator, and the boundary condition (1.3) express the fact that the center of the gyre is a stagnant point. See the discussions in [16,18]. We also refer to [14,17,26] for the descriptions of the special features of arctic ocean flows and [23,24,34] for recent results on arctic gyres. ...

We prove a Nagumo-type uniqueness result for a nonlinear differential equation on a semi-infinite interval. A typical example of such boundary value problems is a recently derived nonlinear model for the ocean flow in arctic gyres. Successive approximations are constructed to converge uniformly to the unique solution. A stability result is also proved.

... Recently, a classical model of gyres is obtained in [8], which is a model of gyres in spherical coordinates as shallow-water flow on a rotating sphere. By means of stereographic projection and neglecting azimuthal variation, the elliptic equation which describes the model of gyres can be reduced to a second-order differential equation (see [4,5,15,16,[20][21][22]). ...

... Recently, Constantin and Johnson [8] derive the classical model of geophysics gyres which acts as shallow-water flow on a rotating sphere and the governing equation for gyres is given by ...

... To end this section, we present the explicit solutions for zero vorticity and constant vorticity in the next two examples, respectively. In fact, the importance about the assumption of the zero vorticity and constant vorticity in modeling the ocean flows is not only because the mathematical analysis, but because the physical perspective as well (see [8,16,17]). (9) with F = γ via (12), by direct computation, we obtain ...

In this paper, we derive a new integral equation model for the Antarctic circumpolar current (ACC) by considering the radial solutions for a semi-linear elliptic equation model of gyres and applying Green’s function.We give the representation of solutions
for constant vorticity and linear vorticity and show the existence and uniqueness of solutions for nonlinear vorticity. Finally, we present the Ulam–Hyers stability for the ACC involving Lipschitz-type nonlinear vorticity.