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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

... 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.

... Recently, the existence of solutions to geophysical fluid dynamics nonlinear governing equations, proposed by Constantin et al. [10][11][12][13][14][15][16], has been widely discussed and studied in this field. In practice, these geophysical flows have horizontal velocities with about a factor 10 4 larger than the vertical velocities [32]. ...

... In practice, these geophysical flows have horizontal velocities with about a factor 10 4 larger than the vertical velocities [32]. Therefore, on a rotating sphere, a stream function can be introduced to model gyres as shallow water flows by neglecting vertical velocities in [16]. In spherical coordinates, the model can be transformed into a planar elliptic partial differential equation under the stereographic projection. ...

... where ψ(θ , ϕ) is the stream function. See [16], recording (θ , ϕ) = ψ(θ , ϕ) + ω cos θ , (1.1) where is the stream function related to the ocean's motion. The governing equation for the gyres is 1 sin 2 θ ϕϕ + θ cot θ + θθ = F(ω cos θ ), (1.2) where ω > 0 is the non-dimensional Coriolis parameter, 2ω cos θ is the planetary vorticity generated by Earth's rotation, F(ω cos θ ) is the ocean vorticity. ...

In the article, we present multiple solutions for a second-order singular Dirichlet boundary value problem that arises when modeling the ocean flow of the Antarctic Circumpolar Current. The main tools of the proof are the Leray–Schauder nonlinear alternative principle and a well-known fixed point theorem in cones.

... Tropical gyres are near the Equator but confined strictly to the Northern or Southern hemispheres (see [9-11, 20, 21]). Subtropical gyres are situated between polar and equatorial regions, where exists a gigantic ocean areas with thousands of kilometers of diameter (see [8]). Subpolar gyres are the smallest ones on the Earth and are situated in the polar regions (see [18]). ...

... The horizontal velocity of gyres is 10 4 larger than the vertical velocity (see [23]). Ignoring the vertical velocity, a model of gyres in spherical coordinates as shallow-water flow on a rotating sphere is obtained (see [8]). In [2], this model is transformed into a plane semi-linear elliptic equation boundary value problem by stereographic projection, and then it is reduced to second-order ordinary differential equation by neglecting the change of azimuthal variations. ...

... As we know, the classical model of arctic gyres is derived by Constantin and Johnson, which is a model of gyres in spherical coordinates as shallow-water flow on a rotating sphere (see [8]). One can set θ ∈ [0, π) the polar angle and θ = 0 corresponds to the South Pole. ...

By considering the radial solutions for a semi-linear elliptic equation model of gyres and introducing exponential transformation, we derive a second-order ordinary differential equation, which acts as a new model for the ocean flow in arctic gyres. Then we investigate the solutions for constant vorticity, linear vorticity and nonlinear vorticity in this model.

... These large bodies of rotating flow, with little vertical motion and existing within a relatively thin layer at the surface of the ocean, can be modelled as a classical (inviscid) vortex; see [19]. This approach, based on the full Euler equation written in spherical coordinates but within the thin-shell approximation, was discussed in [3]. This highlighted, in particular, some nonlinear characteristics of these flows; see also [14]. ...

... These equations and boundary conditions, (1)(2)(3)(4)(5)(6)(7)(8), are now non-dimensionalised according to ...

... We use the results derived above to provide some models for gyres, all based on circular paths (at the surface) in (φ, ) coordinates, where ln cos θ 1 − sin θ , which is associated with the Mercator projection; see [3,9]. Other paths could be chosen, of course, but our choice is simple and produces easily-accessible descriptions of the gyre. ...

Starting from the general equations for a viscous, incompressible fluid, written in rotating spherical coordinates, an asymptotic theory for steady flow is developed. This uses only the thin-shell approximation, by suitably defining the variables and parameters. The result is a consistent theory which produces an Ekman-type balance, expressed in spherical coordinates, at leading order. The correction terms, which are mainly the nonlinear contribution in the equations, can be accommodated by invoking the method of multiple scales and using a strained coordinate. The resulting leading order, with slow/weak corrections, provides the basis for a study of oceanic gyres. By choosing the velocity (and noting the vorticity) at the surface, some examples are presented. Various choices are made, for closed particle paths expressed in a simple form (using a transformation based on the Mercator projection): zero velocity and vorticity at the centre and on the periphery of the gyre; non-zero speed on the periphery; finite-strength line vortex at the centre. In addition, in one case, we describe how the slow-z variation affects the solution. This treatment of the problem shows that our extended version of the Ekman balance, valid in spherical coordinates over large regions, can be used to investigate the properties of gyres. Many analytical and numerical options are available for future study.

... The Antarctic Circumpolar Current (ACC), the most important ocean circulation affecting global climate, the horizontal velocity of which is approximately 10 4 times what the vertical speed is (see [2,3]). The ACC flows eastward about 23, 000 km around Antarctica on the polar axis between 40 and 60 degrees south latitude, where there is no continental barrier. ...

... Constantin started from inviscid Euler equations and mass conservation equations, and considers coriolis forces and centripetal acceleration for flows with negligible vertical velocity relative to horizontal velocity. By selecting appropriate scale factors and shallow water parameters, Constantin introduced stream function and simulated the ocean cyclotron flow as the shallow water equation on a rotating sphere (see [2]). The theoretical framework of shallow water approximation proposed by Constantin combined the vorticity generated by the Earth rotation with the potential vorticity generated by ocean motion, which is of great significance for predicting the characteristics of large-scale natural phenomena in the ocean. ...

... However, the stereoscopic projection transformation method has the limitation that it has no such properties of isometry and conformality. Therefore, in this paper, Mercator projection technique is considered to deal with the nonlinear vorticity equation derived in literature (see [2]), which is rare in existing studies (see only [38][39][40]). Mercator projection is extensively applied to the field of navigation because of its advantages of retaining the accurate direction and angle. ...

In this paper, a mathematical model of the second order elliptic equation of the
Antarctic Circumpolar Current with Dirichlet boundary is established. By introducing
truncation function and perturbation method, the existence of infinitely many solutions
for the nonlinear elliptic equations is obtained when the nonlinear ocean vorticity is
subcritical growth and super-quadratic.

... A shallow water wave is a wave with a water depth less than the wavelength [1,2]. In 1872, the French scientist Boussinesq assumed that the water depth was constant, and the vertical velocity was distributed linearly along the water depth, and then obtained a set of non-linear equations with horizontal 1-D weak dispersion. ...

In this article, the surface wave in inviscid fluid was analyzed. Based on the Euler equation and mass conservation equation, and coupled with a set of boundary conditions, the (2+1)-dimensional sixth-order Boussinesq equation is derived for the first time. According to double-series perturbation analysis and scale transformation, the one soliton solution is obtained with (G?/G)-expansion method. Finally, the effects of amplitude parameter and shallowness parameter on the amplitude of surface wave are analyzed.

... in an exterior domain G A = x ∈ R 2 : |x| > A , where A > 0 , and f is locally Hölder continuous in G A × R. Our aim is to prove that there exists a positive solution to (1) under more general assumptions on the function f (x, y). Our method relies on the technique of super/sub-solutions, one should point out that this type of problems are of current interest in geophysics -see the discussion in the papers [8,9] for ocean flows and the papers [10,23] for atmospheric flows. ...

We consider the existence of a positive (generally unbounded) solution to the semilinear Schrödinger equation in a two-dimensional exterior domain. Our method relies on the comparison method, that is, the technique of super/sub-solutions. An application to Emden-Fowler equation is also presented.

... Constantin and Johnson gave the motion control equation and the mass conservation equation in the rotating spherical coordinate system. The thin-layer asymptotic approximation was established based on the ratio of the ocean average depth to the Earth's radius (see [5]). Martin established the exact solution of the governing equations of geophysical fluid dynamics involving discontinuous stratification in spherical coordinates (see [6,7]). ...

... It is worth mentioning that if the factor of time in Equation (7) is not taken into account, the governing Equation (7) can be regarded as a mathematical model of ocean circulation in which the vertical velocity is relatively weak compared with the horizontal velocity. This was originally developed by Constantin in [5]. After that, many scholars applied functional analysis technology and differential equation theory and found a lot of meaningful results (see [22][23][24][25][26][27][28][29][30][31][32][33]). ...

... In this paper, the existence and non-uniqueness of weak solutions of vorticity equations are obtained by using the Gronwall inequality and Cauchy-Schwartz inequality under the appropriate initial and boundary conditions. Compared with the vorticity equation established in [5], our conclusion considers the time dependence of the system. The authors of [19] studied the stability of the Rossby-Haurwitz stationary solution but did not provide the existence proof for the solution. ...

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.