1. Introduction
Exact solutions in fluid mechanics are rare and must be treated with some caution and some respect. On the one hand, these solutions are likely to be very limited in their application to a particular physical phenomenon with all its inherent complexity. On the other hand, they are precise and clear in their validity, detail, and structure. Of course, one of the skills of the theoretician interested in using a mathematical framework to describe and understand physical processes is to learn how to combine the exact with the approximate in order to make progress. Here, we present one aspect of this process: the construction of an exact solution that relates to a flow observed in the oceans on Earth. Ultimately, we would wish to describe the motions, for example, that include surface waves and their interaction with variable currents: a complex flow. In addition, the topography of the ocean bottom and the existing landmasses around the globe add further complications to the flows observed on Earth. All these difficulties are set aside for the purposes of this discussion. Here, in particular, we will focus on the problem of providing an exact description of the flow that moves completely around the globe on a circular path. This, we propose, can be regarded as a model for the Antarctic Circumpolar Current (ACC). We note that nonlinear exact solutions for oceanic east–west wave propagation are available in the research literature, but these have been derived using the f-plane approximation close to a fixed latitude (as in Pollard 1970; Mollo-Christensen 1979) or, for equatorial flows, the β-plane approximation (see Constantin 2012, 2014; Henry 2013; Hsu 2014; Ionescu-Kruse 2015). Recently, however, exact nonlinear solutions in spherical coordinates (so with a more precise geometry) were presented in Constantin and Johnson (2016) for equatorial flows. The present paper builds on this work by showing not only that on a rotating, spherical Earth we can accommodate azimuthal flows that follow a small circle but also that a meridionally localized jet structure, which decays monotonically with depth, can be described.
The ACC is arguably the most significant current in our oceans and the only current that completely encircles the polar axis; almost unobstructed by landmasses, being constricted only in the region of the Drake Passage (about 800 km wide), it flows around Antarctica. This primarily wind-driven flow moves eastward through the southern regions of the Atlantic, Indian, and Pacific Oceans (see Fig. 1). Although the mean current speed is relatively low (4–25 cm s−1, about 3% of the ambient wind speed; the southern latitudes have some of the strongest westerly winds on Earth), the ACC transports vast volumes of water; indeed, its effects are felt from the surface down to depths as much as 4–5 km, it is about 23 000 km long, and in places its width extends over 2000 km. Unlike other major oceanic currents, the ACC is not a single flow, being composed of a number of high-speed, vertically coherent, seafloor-reaching jets (with speeds commonly exceeding 1 m s−1 and typically 40–50 km wide) that run largely parallel to the ocean ridge system that surrounds Antarctica, separated by zones of low-speed flow. [Further information can be found, e.g., in Ivchenko and Richards (1996), Rintoul et al. (2001), Firing et al. (2011), and Olbers et al. (2004).] Here, the essential, simplified, geometric model for this configuration is to consider flow in the neighborhood and following the path of a small circle at some given latitude around the polar axis of a spherical Earth. We will formulate the problem in spherical coordinates [as outlined in Constantin and Johnson (2016)] and combine this with a suitable flow structure for the ACC.
Sketch of the ACC. It resides between two circumpolar fronts: the inner Polar Front and the outer Subantarctic Front (red curves). Landmasses are drawn in black and oceanic regions in white, with depths shallower than 3.5 km shaded blue.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-16-0121.1
As with almost all exact solutions of fluid systems, we must expect that some elements of the physics cannot be accommodated in any form; the current exercise is no exception. Although we can incorporate an arbitrary flow that follows a small circle on a rotating, spherical Earth, it is altogether unrealistic to expect that a lot of the observed detail can be incorporated. But we do demonstrate that an exact solution is available that allows for any chosen flow profile (both at the surface and varying with depth), although unchanging as it moves in the azimuthal direction on a circular path. The modeling of fairly realistic profiles—that is, profiles that are observed in some average sense—is impossible using the conventional Euler equation but accessible if a nonconservative force field is introduced. [One of the essential difficulties of working with a conservative system comes about as a consequence of the Taylor–Proudman theorem; see Taylor (1917) and Proudman (1916), as we briefly discuss in section 2.] There can be no doubt that many of the properties, both observed and assumed, in these oceanic flows, come about by the action of nonconservative processes. Because the governing Euler equation (which we take as providing our overarching theoretical principle) can admit a general body force, we take advantage of this in our development. The upshot is that we can describe, via an exact solution, a broad family of flows, which may extend over a wide latitudinal arc and behave in any desired manner below the surface, allowing for a decrease in speed with depth but not necessarily dropping to zero on the bottom (even though this is a natural choice). So we can, for example, model data for the ACC, which indicates that there is a current even at considerable depths [e.g., 5 cm s−1 at about 2 km, with 15 cm s−1 at the surface in some regions; see Ivchenko and Richards (1996) and Firing et al. (2011)]. Certainly we can accommodate profiles that enable the appropriate volume of water to be transported by the ACC (see Johnson and Bryden 1989), and for this the details of the vertical structure are less important. Therefore, the model that we shall work with is that of an essential core of fluid, of arbitrary velocity profile, that constitutes the ACC moving eastward around Earth with unchanging form in that direction.
The plan in this paper is to outline the problem in spherical coordinates [and more details of this and associated geometries can be found in Constantin and Johnson (2016)] and the special choices that underpin the construction of the exact solution that is appropriate for the ACC. We will explain that this involves the need to use a nonconservative force field in order to produce a realistic flow but that this automatically caters for many of the ad hoc mechanisms invoked in other models. Indeed, we advocate the use of exact solutions, the more complete the better, before there is any attempt to model other physical processes. The way in which we can impose the shape of the profile, and the pressure condition at the free surface, will be described and an example presented. We conclude with a brief indication of the possible ways in which this exact solution can be used as part of a wider theoretical study of the ACC and its properties.
2. Formulation of the problem and of the equations
The fundamental assumption that predicates our analysis is that we work with an incompressible, inviscid fluid. [The density of water changes by about 0.025% when there is a change of 500 kPa in pressure, and it is generally accepted that the Reynolds number is extremely large for these oceanic flows; see Maslowe (1986).] We introduce a set of (right handed) spherical coordinates (r, θ, φ): r is the distance (radius) from the center of the sphere, θ (with 0 ≤ θ ≤ π) is the polar angle (and then π/2 − θ is conventionally the angle of latitude), and φ (with 0 ≤ φ ≤ 2π) is the azimuthal angle, that is, the angle of longitude. The North and South Poles are at θ = 0, π, respectively, and the equator is on θ = π/2; the Antarctic Circumpolar Current sits at about θ = 3π/4. The unit vectors in this (r, θ, φ) system are (er, eθ, eφ), respectively, and the corresponding velocity components are (u, υ, w); eφ points from west to east, and eθ from north to south (see Fig. 2).
The spherical coordinate system: θ is the polar angle (with π/2 − θ being the angle of latitude), φ is the azimuthal angle (the angle of longitude), and r is the distance from the origin.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-16-0121.1










3. Modeling the ACC

a. Analysis of the balances and linearization


b. Prescription of the velocity profile at the free surface






Sketch of the region of validity of the velocity profile for a conservative body force; the center of the spherical Earth is O and the South Pole is S. The free surface of the ocean is in blue, and E marks the position of the equator; the bottom of the ocean is in green, the velocity profile is imposed on the thick red circular arc, and the vertical, black lines (and dotted line) associated with the red arc indicate lines of constant
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-16-0121.1


A surface representing the speed of the flow in an example of an ACC velocity profile. (left) The general orientation is consistent with that used in Fig. 3; the speed increases to the left and the depth increases upward. The nearer edges correspond to zero speeds at the meridional extremes of the current and also along the bottom; the maximum of the profile at the surface is away from the viewer. (right) The same profile viewed from below, with the current at the surface (the lowest edge of the plot) clearly evident.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-16-0121.1
An important consequence of our approach to this modeling problem, that is, by introducing a nonconservative body force, which is used to mimic the complicated processes that produce (it is believed) the observed type of ACC profiles, is that we need it only where the ACC exists. Outside the ACC, we take the flow to be stationary—the only motion in our model is that of the ACC—and the stationary state requires simply the conventional conservative body force to maintain the existing pressure distribution. Thus, we define G [see Eqs. (4) and (9)] so that it is zero in the regions where w = 0, which, in our example, is for |θ − Θ| > θ0; we then see that the pressure determined from Eqs. (14) to (16), with K = 0, is that consistent with only gravity, centripetal and Coriolis contributing. Furthermore, it is instructive to observe that the body force is also conservative if there is no dependence on r, that is, N(r) = constant in our chosen form for
c. Matching to descriptions from field data
In our general discussion above, we have presented a theory that does not select a specific form of nonconservative forcing that is driven by any particular physical principles that might underpin the flow. We now indicate how this can be done by referring to the vertical structure descriptions of the ACC in the Drake Passage from direct velocity observations, presented in Firing et al. (2011).
The first type of mean velocity profile, advocated by Ferrari and Nikurashin (2010), is linear, with
4. Discussion and conclusions
This analysis has demonstrated how it is possible to produce an exact model (“exact” in the sense of an exact solution of the full set of governing inviscid equations written in spherical coordinates) for the ACC. We have shown that the conventional conservative system has some significant shortcomings but that the essential difficulties are overcome if we invoke a nonconservative body force to maintain the ACC. This, we have argued, is a suitable mathematical maneuver that successfully replaces the ad hoc modeling of the many nonconservative processes that certainly contribute to this flow. Indeed, our approach enables considerable freedom in the choice of velocity profile, surface distortion, and surface pressure distribution.
The ocean–atmosphere interactions in the circumpolar belt where the ACC resides are significant and varied. Satellite data and observation by vessels show consistently large eastward wind speeds (of average 8–12 m s−1) with little variation in wind direction. The wind regime is dominated by frequent storms and has strong meridional variations, with the maximal wind speed near 50°S, and gale force winds occur frequently (as much as 20% of the time; see Tomczak and Godfrey 1994). The winds blowing over the ocean surface drive a near-surface flow, and the latitudinal variations in the wind velocities create areas of convergence and divergence at the surface that are balanced by upwelling and downwelling flows. Importantly, from our point of view, the wind regime generates a predominantly zonal flow in a fragmented system of more or less intense circumpolar jet streams; observations indicate that about 75% of the total flow in the oceanic region of the ACC occurs in zonal jets, which occupy about 19% of the cross-sectional area (see Tomczak and Godfrey 1994). Indeed, it has been observed that these zonal jets are not restricted to a near-surface layer but extend to great depths (in most places to the ocean floor). Drag due to topography (form drag, i.e., drag that arises from the pressure distribution generated by the shape of an object) slows down the current, preventing the zonal flow from accelerating indefinitely, despite being continuously fed with momentum by the westerly wind stress along an unobstructed band of latitude: the eastward flow establishes a pressure difference across the four major north–south submarine ridges that are encountered (and crossed at gaps, a process that enhances the filamentary structure of the ACC; see Klinck and Nowlin 2001). The associated jets that contribute to the ACC can be included within our exact solution simply by combining a number of solutions of the type that we have described, that is, a maximum at the center, dropping to zero on either side, and this repeated a number of times, the junction between each pair of jets being along the zero-speed line at a suitable latitude. All this is possible, we believe, because the Reynolds stresses and viscous terms are very small in these flows (see Olbers et al. 2004), and this permits us to model the flow dynamics by the inviscid Euler equations but with meridional nonconservative forcing. Furthermore, the wind-driven forcing, by which energy is transferred to the system, can be captured in our model by allowing pressure changes at the surface; the classical theory of surface gravity waves, on the other hand, takes the pressure on the surface to be constant, that is, atmospheric pressure. [The mechanism by which the wind imparts momentum on the water, using various models, is discussed in Johnson (2012) and Walsh et al. (2013).] We regard the current as the eventual by-product of wind blowing over the water, but we remain agnostic as to how exactly that generation takes place, and, importantly, such details are not required in our model.
The fact that we have described the construction of an exact solution should be regarded as no more than the starting point for a careful and systematic theoretical investigation of these flows; a number of possibilities spring to mind. Our results presented here allow for an underlying flow to be modeled as accurately as we wish, at least in principle, by choosing an appropriate body force; this is clearly an avenue that needs further investigation, presumably based on available data and physical models for the flow. With a suitable background flow in place, it is natural, we believe, to examine the effects on wave propagation. Wave–current interactions are of considerable interest, and our exact solution makes this type of investigation readily accessible. The procedure is to perturb the solution described here, regarded as a prescribed background state, by adding surface waves and examining their general properties, evolution, and so on. The wave–current interaction then becomes a problem that can be studied as an irrotational wave perturbation of a background pure current flow; see Constantin and Johnson (2015), where this type of approach has been successfully applied to equatorial wave–current interactions. Another avenue would be to allow the flows of the type developed here to evolve slowly in the azimuthal direction; this will enable slow depth changes, for example, to be incorporated. Of course, both these give rise to approximate—no longer exact—solutions, for which suitable asymptotic techniques are likely to be relevant.
Finally, we make a brief general observation about the interpretation of our modeling assumptions in the light of available oceanographic data. We take the view that the proficiency of comprehensive and complicated numerical models that simulate the complex dynamics of the Southern Ocean actually spur the need for an understanding, at a basic level, of the analytical structure of these problems, helping to elucidate what is possible and what is allowed. The practical benefit of an analytically tractable model is that it offers a global perspective, albeit of a simplified version of the system as a whole, but with the inherent ability to grasp the dominant features and the fundamental processes of the overall dynamics. The interactive exploration of analytical ideas and numerical simulations is a very powerful tool.
In conclusion, we submit that this new, exact solution, which provides a model for the Antarctic Circumpolar Current, can be the starting point for further, and more directly relevant, theoretical and numerical investigations of this important oceanic flow.
Acknowledgments
The authors are grateful for helpful comments from the referees.
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