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

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 physical expectations. A regularity result concerning the interface is also derived.


expectations. A regularity result concerning the interface is also derived. INTRODUCTION
Undoubtedly, the Antarctic Circumpolar Current (ACC) is one of the most significant currents in the Earth's oceans. To name one of its many unique features, we say that it is the only current that completely encircles the polar axis, occupying a tremendous area: indeed, ACC flows eastward through the southern regions of Atlantic, Indian, and Pacific Oceans along 23 000 km, and extends in places over 2000 km in width, cf. [1][2][3][4] . Its massiveness is also reflected by the huge volumes of water it transports estimated to be between 165 million and 182 million cubic meters of water every second, cf. 5 , which represents more than 100 times the flow of all the rivers on Earth.
Among the many factors that shape the complex behavior of ACC is the presence of stratification that accommodates observed sharp changes in water density (due to variations in temperature and salinity, cf. [6][7][8][9][10][11] ). So-called fronts or jets arise due to stratification in the meridional direction, cf. 12 ; the two main fronts of the ACC are the Subantarctic Front to the north and the Polar Front further south, see Figure 1. Stratification in the vertical direction can cause internal waves; for the underpinning mechanics of internal wave generation, we refer the reader to Refs. 13, 14, and 15. However, allowing for (discontinuous) stratification complicates the analysis of an already very demanding analytical problem. Certain progress for two-dimensional stratified flows was made quite recently, cf. 7,[16][17][18][19][20][21][22][23][24][25][26][27] . On a related note, we would like to point out the very recent results by Escher et al. 28 on stratified water flows with singular density gradients.
Here we address, from a mathematical perspective, the topic of stratified geophysical water flows exhibiting vertical structure, internal waves (arising as a result of the discontinuous stratification), and a preferred propagation direction. In doing so, we derive and, subsequently analyze an exact, partially explicit solution to the geophysical water wave equations written in spherical coordinates and in a rotating frame. This solution portrays an incompressible, inviscid, stratified, steady flow moving purely in the azimuthal direction, that is, the velocity profile and the pressure are described below and up to the free surface as a function of depth and the angle of latitude. As such, this solution is suitable for a depiction of ACC. Our contribution here follows a line of work initiated by Constantin 6,[29][30][31]8,32,33 on the derivation of explicit and exact solutions to the governing equations of geophysical fluid dynamics (GFD) that describe surface waves and their interactions with the underlying currents that are ubiquitous in Earth's ocean basins. For a selective list of recent references, in this direction, we point the reader to Refs. 34, 35, 36, 37, 38, 39, 40, 41, 42, and 43. We would like to emphasize that the availability of exact and/or explicit solutions in fluid mechanics is special and quite rare-due to the complexity of the governing equations. However, once available, they provide new avenues of investigation of physically realistic flows, by means of asymptotic 44  The new aspect of our investigation-compared to the existing mathematical literature on ACC (e.g., Refs. 33,45,46,47,48,49,50, and 51)-is the presence of a discontinuous density stratification. More precisely, we allow a vertical layering of the flow, with two layers of different, nonconstant densities, where the denser layer sits below the less dense one (stable stratification): thus, an interface, which plays the role of an internal wave, arises. Although, in general, the interface is given implicitly, we are able to devise a scenario where it arises as the solution of an explicitly solvable differential equation. As an added bonus we prove interesting regularity properties for the interface. Besides stratification, our solution accounts for forcing terms being necessary for the dynamical balance of the ACC, cf. 33,52 . In line with these ideas, we point to Refs. 53 and 54 for the relevance of stratification to maintain the equilibrium of ACC: baroclinic instability (arising from stratification) generates eddy-induced cells (acting to flatten the isopycnals) that counterbalance the wind-driven Ekman cell (acting to steepen isopycnals).
The layout of the paper is as follows: we introduce in Section 2 the governing equations (in spherical coordinates) and their boundary conditions for geophysical flows. Thereafter, we derive in Section 3 explicit solutions for the velocity field and the corresponding pressure function in the two layers of the fluid domain. From the dynamic boundary condition, we find an implicit relation between the imposed pressure and the resulting surface distortion. The interface defining function also appears implicitly as a condition expressing the balance of forces at the interface. The two implicit relations are then subjected to the implicit function theorem; this way we are able to prove that any small enough perturbation of the pressure required to preserve an undisturbed free surface (following the Earth's curvature) triggers unique functions, describing the surface and the interface, respectively. The last section of the paper is devoted to proving that the solution we derived displays expected physical properties: a decay of the surface height occurs as soon as the pressure along the free surface increases. A regularity property of the interface defining function is also proved, followed by a particular explicit example.

PHYSICAL PROBLEM AND GOVERNING EQUATIONS
In this section, we provide the governing equations for geophysical flows written in spherical coordinates to accommodate the shape of the Earth, together with the boundary conditions for the free surface and a rigid bed. We will work in a system of right-handed coordinates ( , , ) where denotes the distance to the center of the ball, ∈ [0, ] is the polar angle (the convention being that ∕2 − is the angle of latitude), and ∈ [0, 2 ] is the azimuthal angle (angle of longitude). Although in this coordinate system, the North and South poles are located at = 0 and = , respectively, the Equator sits on = ∕2, the ACC is situated around = 3 ∕4. The unit vectors in this system are = (sin cos , sin sin , cos ), = (cos cos , cos sin , − sin ), = (− sin , cos , 0), with pointing from West to East and from North to South, cf. Figure 2. Throughout this paper, we make the following simplifying assumption on ACC's location. The angle of latitude is assumed to lie in the compact interval : We are guided in our study by the observations made in Maslowe, 55 asserting that the Reynolds number is, in general, extremely large for oceanic flows. Accordingly, we will consider incompressible and inviscid flows. For 0 < 2 < 1 ≪ and ∶= + , = 1, 2, we consider the two fluid layers separated by an interface and bounded by the bottom and a free surface, which are described by the graphs of the functions ℎ, , and , respectively, cf. Figure 3: The spherical coordinate system: is the polar angle, is the azimuthal angle (angle of longitude), and represents the distance to the origin F I G U R E 3 A schematic illustration of the fluid domain in the flattened ( , )-plane being bounded by the prescribed ocean bottom at = and the free surface at = 1 + . The fluid layers 1 and 2 are separated by the internal wave at = 2 + ℎ. The boundaries of the intervals ( 2 + ℎ − , 2 + ℎ + ) and ( 1 + − , 1 + + ), which contain the interface and the free surface, respectively, as well as the deepest depth − are indicated by dashed lines We associate with the Earth's radius. The given function ≈ describes the bottom topography, whereas ℎ and describe the unknown deviations of the interface and the free surface from their unperturbed locations at 2 and 1 , respectively. We assume that ℎ is restricted to some interval (ℎ − , ℎ + ) for all ( , ) ∈ × [0, 2 ]; likewise, ∈ ( − , + ) by assumption. To rule out mixing of the two layers, we furthermore assume that < 2 + ℎ − and 2 + ℎ + < 1 + − . The prescribed density is assumed to satisfy ( , , ) = ( ) = + ( ) in for positive constants 1 < 2 , and a slight depth depending smooth variation ∶ ( − , 1 + + ) → ℝ, which satisfies ≡ 0 in ( 2 + ℎ − , 2 + ℎ + ); here − ∶= min ( , )∈ ×[0,2 ] ( , ). In particular, is a discontinuous depth-dependent function with a jump of height 2 − 1 at the interface 2 + ℎ. The function | ( )| ≪ 2 − 1 accounts for comparably small density gradients away from the interface due to, for example, slight changes in salinity or temperature. By writing ( ), we indicate that is taken from the layer , = 1, 2, whereas always refers to the before mentioned constants. Let denote the velocity field within the fluid layer .
which incorporate both Coriolis effects and centripetal acceleration (Ω ≈ 7.29 × 10 −5 rad s −1 refers to the constant rotation speed of the Earth), cf. 32 . Here, ( , , ) denotes the pressure field and = ( , , ) is the prescribed body-force vector. Additionally to (8), the equation of mass conservation is supposed to be satisfied: The GFD governing equations (8) and (9) are supplemented with the following boundary conditions. At the free surface = 1 + ( , ), we require the dynamic boundary condition (for a prescribed function 1 ) and the kinematic boundary condition to be satisfied. At the interface = 2 + ℎ( , ), we require the normal components of the velocity fields to be equal: Moreover, to ensure a balance of forces, we require that At the rigid ocean bottom = ( , ), it holds that

EXACT EXPLICIT AND IMPLICIT SOLUTIONS
We seek for a steady flow governed by (8) which can be rewritten as Thus, can be eliminated, and the flow (0, 0, ) satisfies in , = 1, 2. This equation can be explicitly solved by employing the method of characteristics (cf. 48 ): Plugging (18) Integrating (20) in the lower layer ( = 2) with respect to yields that where for = 1, 2, we set which by (19) satisfies , ( , ) = csc , ( csc , ), and therefore, 2 is given by In the upper layer, we obtain 1 (ℎ, , ) = ∫ sin ( 2 +ℎ( )) sin for the function 1 (ℎ, ) ∶= At this point, we note that without loss of generality, it can be assumed that Letting 0 1 denote the surface pressure related to the undisturbed interface and free surface (i.e., ℎ = ≡ 0), Equation (30) wherẽ= + − 1 2 2 Ω 2 sin 2 absorbs the centripetal and gravitational acceleration; cf. the discussion in Constantin and Johnson. 33 As 1̃i s relatively small, we infer from (35) in combination with Assumption 1 that

Implicit description of interface and surface
In the next step, we employ dimensionless variables to obtain an implicit formulation for the interface. Let and consider the nonlinear operator  on the space of dimensionless interfaces h defined by Noting that 1 (ℎ, ) = 1 ((1 + h( )) 2 , ) by (27) and (28), where (ℎ, ) is expressed in terms of h according to (37), we find that To infer an implicit formulation for the free surface, we set and define where 1 (ℎ, ⋅, ⋅) is expressed by means of h via (37). Then and by setting  0 1 ∶= −1 atm 0 1 , we infer from (31) and (41) that Generally, the dimensionless interface h and surface k are implicitly given by the abstract operator equation We aim to solve (44) locally around the undisturbed state with the help of the implicit function theorem. For this purpose, we calculate the derivatives of  and  at 0 and (0, 0,  0 1 ), respectively, with respect to both h and k. As  does not depend on k, it holds that k (h) = 0 for arbitrary h. Furthermore, a direct calculation using (24) and (29) hence, ( h (0)h)( ) = −1 atm ( 2 ( 2 sin ) − 1 ( 2 sin ) − 2 ( 2 − 1 ) +[ 2 ( 2 sin , ) − 1 ( 2 sin , )] 2 sin )h( ).

QUALITATIVE ANALYSIS OF SOLUTIONS
This final section presents qualitative results for the interface and the free surface, and a specific example allowing for an explicit description of the interface. Proof. By Theorem (1), it holds that ((h))( ) = 0 for all ∈ . Hence, via differentiation with respect to , we infer that ( ) ∶= ((1 + h( )) 2 sin , )(1 + h( )) 2 sin .