# A steady stratified purely azimuthal flow representing the Antarctic Circumpolar Current

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## Abstract

We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.

## Keywords

Antarctic Circumpolar Current Variable density Azimuthal flows Eddy viscosity Geophysical fluid dynamics## Mathematics Subject Classification

Primary 35Q31 35Q35 Secondary 35Q86## 1 Introduction

*f*-plane at the 45th parallel south, enhanced with an eddy viscosity term and a forcing term, and equipped with appropriate boundary conditions. We propose this specific flow for representing the gross dynamics of the Antarctic Circumpolar Current (ACC)—the World’s longest and strongest Ocean current.

ACC has no continental barriers: it encircles Antarctica along a 23,000 km path around the polar axis towards East at latitudes between \(40^{\circ }\) and \(60^{\circ }\), see Fig. 1. It thereby links the Atlantic, Pacific and Indian Oceans making it the most important oceanic current in the Earth’s climate system. The structure of ACC is rich and complicated. Many factors contribute to its complex behavior—the most important driver being strong westerly winds in the Southern Ocean region. In addition to that there exist mesoscale eddies of a size up to 100 km, which transport the wind-induced surface stress to the bottom and also enable meridional mass transport; there are sharp changes in water density due to variations in temperature and salinity—known as fronts or jets—located at ACC’s boundaries (see Fig. 1); ACC is strongly constrained by the bottom topography; there are observed variations in time such as the Antarctic Circumpolar Wave; etc. We refer to [13, 15, 16, 23, 24, 25, 33, 34, 36, 37] for further information about the geophysical aspects and modeling as well as observational data and simulations for ACC.

From an analytical perspective one is forced to largely, yet reasonably, simplify the geophysical scenario to obtain a tractable model, which—in the ideal case—exhibits exact and explicit solutions opening the path for an in-depth analysis. Thus we do not account for all of the before mentioned phenomena, but assume a steady flow in purely azimuthal direction, which is vertically bounded by a flat bottom and a flat ocean surface. By considering Euler’s equation of motion in the *f*-plane, we obtain a valid approximation of the Coriolis effects close to the 45th parallel south; in this way the Earth’s curvature is neglected and no boundaries in the meridional direction are assumed. Even though the \(\beta \)-plane appears to be more accurate than the *f*-plane (especially for larger deviations in latitudinal direction), the \(\beta \)-plane approximation leads to inconsistencies when applied to non-equatorial regions; see the discussions in [9, 14]. To account for the transportation effects of mesoscale eddies we equip the system with an eddy viscosity term; furthermore we include a forcing term to ensure the dynamical balance of the flow. Both pressure and wind stress are prescribed on the ocean surface; a no-slip boundary condition is assumed for the ocean bed.

A similar setting has recently been considered in [31], where an explicit solution in terms of a given viscosity function was presented. In this note at hand we extend these results to stratified flows, i.e. we do account for variations of the water density (with depth and latitude). The established explicit solution is an analytic function of both the viscosity function and the density distribution. While the present paper aims for an explicit description of certain currents beneath a fixed surface, we point to [11, 30] for relevant studies of exact solutions for free surface waves in the *f*-plane approximation at mid-latitudes. A collection of numerous recent analytical studies concerning the dynamics of ACC can be found in [7, 17, 18, 22, 26, 27, 28, 31] and the references therein.

## 2 Model under study

*x*,

*y*,

*z*), where

*x*denotes the direction of increasing longitude,

*y*is the direction of increasing latitude and

*z*represents the local vertical, respectively. Denoting with

*t*the time variable, and with (

*u*(

*x*,

*y*,

*z*,

*t*),

*v*(

*x*,

*y*,

*z*,

*t*),

*w*(

*x*,

*y*,

*z*,

*t*)) the velocity field, the governing equations for inviscid and incompressible geophysical ocean flows at latitude \(\phi \) are (cf. [3, 29, 35]) the Euler equations

*d*is the constant water depth) and above the flat surface situated at \(z=0\).

Our aim is to derive exact formulas for purely azimuthal flows (i.e. \(v=w=0\)) in the region of the ACC (by fixing \(\phi = -\pi /4\) resulting in the *f*-plane approximation of (2.1) at the 45th parallel south). Furthermore we incorporate the transfer of the wind-generated surface stress to the bottom, which is due to the presence of mesoscale eddies, by adding a viscosity term of the form \((\nu u_z)_z\) to the right hand side of the first equation in (2.1); the coefficient \(\nu =\nu (z)\) is a smooth function of depth being strictly greater than some positive constant, see [23]. The classical model of uniform eddy viscosity is due to [32]. We follow the more realistic approach with a depth dependent viscosity function as it was introduced in [12]. Finally, we include a forcing term \(F=F(y,z)\) to guarantee non-trivial solutions, cf. [7, 21, 31].

*wind stress*; i.e. we assume a no-slip bottom and constant pressure as well as wind stress at the surface.

## 3 Explicit solution

### Theorem 3.1

*u*,

*P*) of system (2.3)–(2.6) with boundary conditions (2.7)–(2.9) is given by

*F*can be recovered from (2.4) by means of (3.1) and (3.2).

### Proof

*x*in (2.3)–(2.5) and obtain that

*y*and for all \(z\in [-d,0]\). Thus, by means of the bottom boundary condition (2.7) we infer that

*u*satisfies (3.1).

*z*in (2.5) we obtain that

*C*(

*y*) is determined by (2.8) and satisfies

*P*satisfies (3.2) for all \((y,z)\in {\mathbb {R}}\times [-d,0]\). \(\square \)

### Remark 3.2

One immediate consequence of (3.1) is that the vorticity vector associated with the flow (3.1), given by \((0,u_z, - u_y)\), has a non-vanishing second and third component. This represents a marked difference, if compared with the case of homogeneous flows (considered in [31]) where only the middle component \(u_z\) survives, the first and the third being zero because of the lack of *y* dependence of \(\rho \). Thus, allowing for significant variations in density leads to solutions that exhibit substantial shear not only in the vertical direction but also in the latitudinal direction, as well.

We finally point out that three-dimensional effects in ocean waves were recently captured in the papers [3, 4, 10, 19, 20] within the nonlinear setting for equatorial flows.

## Notes

### Acknowledgements

Open access funding provided by Royal Institute of Technology. The authors are grateful for helpful comments and suggestions from the referee. C. I. Martin would like to acknowledge the support of the Austrian Science Fund (FWF) under research Grant P 30878-N32. R. Quirchmayr acknowledges the support of FWF, Erwin Schrödinger fellowship J 4339-N32, and thanks his host institution—the Department of Mathematics at KTH Royal Institute of Technology.

## References

- 1.Constantin, A.: On the modelling of equatorial waves. Geophys. Res. Lett.
**39**, L05602 (2012)CrossRefGoogle Scholar - 2.Constantin, A.: An exact solution for equatorially trapped waves. J. Geophys. Res. Oceans
**117**, C05029 (2012)CrossRefGoogle Scholar - 3.Constantin, A.: Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr.
**43**, 165–175 (2013)CrossRefGoogle Scholar - 4.Constantin, A.: Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr.
**44**(2), 781–789 (2014)CrossRefGoogle Scholar - 5.Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn.
**109**(4), 311–358 (2015)MathSciNetCrossRefGoogle Scholar - 6.Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr.
**46**(6), 1935–1945 (2016)CrossRefGoogle Scholar - 7.Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current. J. Phys. Oceanogr.
**46**(12), 3585–3594 (2016)CrossRefGoogle Scholar - 8.Constantin, A., Johnson, R.S.: A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific equatorial undercurrent and thermocline. Phys. Fluids
**29**, 056604 (2017)CrossRefGoogle Scholar - 9.Constantin, A., Johnson, R.S.: Steady large-scale ocean flows in spherical coordinates. Oceanography
**31**, 42–50 (2018)CrossRefGoogle Scholar - 10.Constantin, A., Johnson, R.S.: On the nonlinear, three-dimensional structure of equatorial oceanic flows. J. Phys. Oceanogr.
**49**, 2029–2042 (2019)CrossRefGoogle Scholar - 11.Constantin, A., Monismith, S.G.: Gerstner waves in the presence of mean currents and rotation. J. Fluid Mech.
**820**, 511–528 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Cronin, M.F., Kessler, W.S.: Near-surface shear flow in the tropical Pacific cold tongue front. J. Phys. Oceanogr.
**39**, 1200–1215 (2009)CrossRefGoogle Scholar - 13.Danabasoglu, G., McWilliams, J.C., Gent, P.R.: The role of mesoscale tracer transport in the global ocean circulation. Science
**264**, 1123–1126 (1994)CrossRefGoogle Scholar - 14.Dellar, P.J.: Variations on a beta-plane: derivation of non-traditional betaplane equations from Hamilton’s principle on a sphere. J. Fluid Mech.
**674**, 174–195 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Farneti, R., Delworth, T.L., Rosati, A., Griffies, S.M., Zeng, F.: The role of mesoscale eddies in the rectification of the Southern Ocean response to climate change. J. Phys. Oceanogr.
**40**(7), 1539–1557 (2010)CrossRefGoogle Scholar - 16.Firing, Y.L., Chereskin, T.K., Mazloff, M.R.: Vertical structure and transport of the Antarctic Circumpolar Current in Drake Passage from direct velocity observations. J. Geophys. Res.
**116**, C08015 (2004)Google Scholar - 17.Haziot, S.V., Marynets, K.: Applying the stereographic projection to modeling of the flow of the antarctic circumpolar current. Oceanography
**31**(3), 68–75 (2018)CrossRefGoogle Scholar - 18.Haziot, S.V.: Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete Contin. Dyn. Syst.
**39**(8), 4415–4427 (2019)CrossRefzbMATHGoogle Scholar - 19.Henry, D.: An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. B Fluids
**38**, 18–21 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Henry, D.: Equatorially trapped nonlinear water waves in a beta-plane approximation with centripetal forces. J. Fluid Mech.
**804**, R1 (2016)CrossRefzbMATHGoogle Scholar - 21.Howard, E., Hogg, A.M., Waterman, S., Marshall, D.P.: The injection of zonal momentum by buoyancy forcing in a Southern Ocean model. J. Phys. Oceanogr.
**45**, 259–271 (2015)CrossRefGoogle Scholar - 22.Hsu, H.-C., Martin, C.I.: On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current. Nonlinear Anal.
**155**, 285–293 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Ivchenko, V.O., Richards, K.J.: The dynamics of the Antarctic Circumpolar Current. J. Phys. Oceanogr.
**26**, 753–774 (2012)CrossRefGoogle Scholar - 24.Klinck, J., Nowlin, W.D.: Antarctic Circumpolar Current. In: Steele, J.H., Thorpe, S.A., Turekian, K.K. (eds.) Encyclopedia of Ocean Sciences, pp. 151–159. Academic Press, Cambridge (2001)CrossRefGoogle Scholar
- 25.Marshall, D.P., Munday, D.R., Allison, L.C., Hay, R.J., Johnson, H.L.: Gill’s model of the Antarctic Circumpolar Current, revisited: the role of latitudinal variations in wind stress. Ocean Model.
**97**, 37–51 (2016)CrossRefGoogle Scholar - 26.Marynets, K.: The Antarctic Circumpolar Current as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Deep Sea Res. Part
**II**(160), 58–62 (2019)CrossRefGoogle Scholar - 27.Marynets, K.: Stuart-type vortices modeling the Antarctic Circumpolar Current. Monatsh. Math. (2019). https://doi.org/10.1007/s00605-019-01319-0 MathSciNetzbMATHGoogle Scholar
- 28.Marynets, K.: Study of a nonlinear boundary-value problem of geophysical relevance. Discrete Contin. Dyn. Syst.
**39**(8), 4771–4781 (2019)CrossRefzbMATHGoogle Scholar - 29.Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
- 30.Pollard, R.T.: Surface waves with rotation: an exact solution. J. Geophys. Res.
**75**, 5895–5898 (1970)CrossRefzbMATHGoogle Scholar - 31.Quirchmayr, R.: A steady, purely azimuthal flow model for the Antarctic Circumpolar Current. Monatsh. Math.
**187**(3), 565–572 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Stommel, H.: Wind drift near the Equator. Deep Sea Res.
**6**, 298–302 (1960)CrossRefGoogle Scholar - 33.Tomczak, M., Godfrey, J.S.: Regional Oceanography: An Introdution. Pergamon Press, Oxford (1994)Google Scholar
- 34.Thompson, A.F.: The atmospheric ocean: eddies and jets in the Antarctic Circumpolar Current. Philos. Trans. R. Soc. A
**366**, 4529–4541 (2008)MathSciNetCrossRefGoogle Scholar - 35.Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
- 36.White, W.B., Peterson, R.G.: An Antarctic circumpolar wave in surface pressure, wind, temperature and sea-ice extent. Nature
**380**, 699–702 (1996)CrossRefGoogle Scholar - 37.Wolff, J.O.: Modelling the Antarctic Circumpolar Current: eddy-dynamics and their parametrization. Environ. Model. Softw.
**14**, 317–326 (1999)CrossRefGoogle Scholar

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