Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

The aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.


Introduction
The combined forces of gravity and Coriolis (due to Earth's rotation), triggered by the wind stress, drive circulating ocean currents, called gyres, in which the ocean flow adjusts to these two major forces acting on them so that those forces balance one another (see the discussions in [12,13]). Since the Coriolis effect deflects winds, the deflection of gyres is clockwise in the Northern Hemisphere and counter-clockwise in the Southern Hemisphere. These geophysical flows are dominantly horizontal, with horizontal velocities being about a factor 10 4 larger than the vertical velocities [26]. At the same time, friction also plays a role in the generation of the currents (see the discussion in [7]). However, these currents persist after the wind stress has ceased, and in this regime one can neglect frictional effects.
Recently, in [9] a model of the general motion of a gyre in spherical coordinates as a shallow-water flow on a rotating sphere was obtained, in which the gyre motion is described in terms of a stream function by the solution of an elliptic boundary value problem, which can be transfomed into a second-order differential equation by neglecting azimuthal variation. In recent works [2][3][4][5][6], the first author used this equation to model the arctic gyres as a boundary value problem on semi-infinite interval. Such an equation with suitable boundary conditions was also used to model the Antarctic gyres in [8,15,16,[18][19][20][21][22][23][24][25]. The presence of a non-uniform current in the Southern Ocean is of great physical relevance, since this region features the largest known waves (see discussions in [10,27]). Moreover, new flow phenomena appear in wave-current interactions due to non-uniformity in currents (see [11]). Now let us recall the model of the general motion of a gyre in spherical coordinates. Let θ ∈ [0, π) be the polar angle (with θ = 0 corresponding to the North Pole) and ϕ ∈ [0, 2π) be the angle of longitude (or azimuthal angle) as shown at Fig. 1.
Let ψ(θ, ϕ) be the stream function and where ω > 0 is the non-dimensional form of the Coriolis parameter. The governing equation for the gyre flow (see the discussion in [9]) is 1 where F (Ψ − ω cos θ) is the oceanic vorticity, which is typically one order of magnitude larger than the planetary vorticity 2ω cos θ, generated by the Earth's rotation. The total vorticity of the gyre flow is the sum of the oceanic vorticity F (Ψ − ω cos θ), and of the planetary vorticity 2ω cos θ.
Using the stereographic projection of the unit sphere (centred at the origin) from the North Pole to the equatorial plane (see Fig. 2 where (r, φ) are the polar coordinates in the equatorial plane, can be transformed into the semilinear elliptic partial differential equation where Δ = ∂ 2 x + ∂ 2 y denotes the Laplace operator. Since the ACC presents a considerable uniformity in the azimuthal direction it corresponds to radially symmetric solutions ψ = ψ(r) of the problem (4). The change of variables with r = e −s/2 for and 0 < r − < r + < 1, Eq. (4) is transformed to the second-order ordinary differential equation The flow in a jet component of the ACC, between the parallels of latitude defined by an appropriate choice of r ± ∈ (0, 1) with r + /r − ∈ (1, 2), is modeled by coupling (7) with the boundary conditions which expresses the fact that the boundary of the jet is a streamline-since the flow is steady, this means that a particle there will be confined to the boundary at all times. Given 0 < s 1 < s 2 , the change of variables transforms (7) and (8) into the equivalent two-point boundary-value problem where and

Semilinear Cases
In this section, we will establish the existence results for (10) in the semilinear case, that is, the function F is L 1 -Caratheodory and there exist two positive constants α, β such that To prove the results, we shall apply the method of topological degree together with some facts on the weighted eigenvalue of the corresponding linear problems. For more details on semilinear differential equations with and without a singularity we refer to [28][29][30] and the references therein. Let us first consider the following weighted eigenvalue problem where a(t) is given by (11). It is easy to show that all eigenvalues λ of (13) are positive. Indeed, multiplication (13) where It is a proven fact that the Dirichlet eigenvalues in (15) are simple and can be ordered in an increasing sequence going to infinity (see [17]).

Theorem 1.
The eigenvalue problem (13) has a sequence of eigenvalues:

Theorem 2. Assume that the function F (u) satisfies the inequality
for some constant β > 0 and Then the problem (10) has at least one solution.
Proof. Let us deform the Eq. (10) to the following one u (t) + βu(t) = 0 and write a homotopic equation in the form: where G(t, u, τ ) = τ (−a(t)F (u) − b(t)) + (1 − τ )βu. The problem (18) is equivalent to a fixed point equation in the space with the C 1 -norm. By assumption (17), when τ = 0, the problem (18) has only a trivial solution. As the operator M 0 is odd in u ∈ X, deg(M 0 , X.0) = odd = 0, which means that the theorem will be proved, if we find a priori bounds for all solutions of (18).
By (16), there exists a constant 0 ≤ ψ < +∞ such that for all u ∈ R for a.e. t ∈ [0, 1]. Thus Let u(·) be the solution of (18) for some τ ∈ [0, 1]. On the one hand, we have and on the other hand, λ 1 (a) has the following property: Thus an estimate holds Since u(0) = u(1) = 0, there exists some constant c 1 > 0 such that By assumption, the function F (u) is L 1 -Caratheodory function, and it implies that there is some for all t, τ and all solutions u(·) of (18). Due to the Dirichlet boundary conditions in (18), there exists some t 1 ∈ (0, 1), for which u (t 1 ) = 0 and |u (t)|= Hence all solutions of the boundary value problem (18) are bounded and the theorem is proved.

Theorem 3.
Assume that the function F (u) satisfies condition (12) and there exists an integer k ≥ 2 such that then problem (10) has at least one solution.
Since γ > α, we can conclude that On the other hand, since γ < β it follows that Thus, for any n ∈ N λ n (γ) = 1. Hence, the problem (25), (14) has only a trivial solution, and therefore the nonlinear problem (10) has at least one solution.
Remark 1. Moreover, if in Theorems 2 and Theorem 3 function F (u) is assumed to be nondecreasing, it is easy to prove that the solution of the semilinear boundary value problem (10), (14) is unique (see Theorem 5.1 in [25]).

Superlinear Cases
In this section, we prove one existence result for (10) in the superlinear case at u = ∞, which means that For example, both and F (u) = e αu , α > 0, are typical superlinear functions. It is easy to see that looking for a solution of (10) is equivalent to looking for a fixed point of the following operator equation where where G(t, s) is the Green function given by and To present the main result, we will use the notations

G(t, s)α(s)ds,
and Obviously, the functions A(t) and γ(t) are nonnegative, and A * > 0, γ * > 0. The proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in [14, page 120-130]. (I) T has at least one fixed point in Ω. (II) There exists x ∈ ∂Ω and 0 < λ < 1 such that

Theorem 4. Assume that F : R → R is continuous and there exists a nondecreasing continuous function
Suppose further that there exists a positive number r > γ * such that Then Eq. (10) has at least one solution u with 0 < u < r.
Proof. Let us first consider the family of problems where λ ∈ [0, 1]. Note that a solution of (31) is just a fixed point of the operator equation where F is given as (28). We claim that for any λ ∈
Hence r ≤ g(r)A * + γ * , which is a contradiction to the condition (30).
Obviously Ω is an open subset in X with γ(t) ∈ Ω since γ * < r. Now using Lemma 1, we know that u = T u has a fixed point u in Ω, i.e., Eq. (10) has a solution u with u < r. Finally, one may readily see that u is a nontrivial solution since u(t) ≥ γ(t).