## Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates.(English)Zbl 1404.86015

Summary: 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: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.

### MSC:

 86A05 Hydrology, hydrography, oceanography 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing 35Q31 Euler equations

### Keywords:

Euler equations; spherical coordinates; gyres
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### References:

 [1] Zhong W, Zhao J, Shi J, Cao Y. (2015) The Beaufort Gyre variation and its impacts on the Canada Basin in 2003-2012. Acta Oceanol. Sin. 34, 19-31. (doi:10.1007/s13131-015-0657-0) [2] Sandor B, Szabo KG. (2013)Simple vortex models and integrals of two dimensional gyres. In Proc. Second Conf. Junior Res. Civil Eng., pp. 284-291. Budapest, Hungary. See https://www.me.bme.hu/doktisk/konf2013/papers/284-291.pdf. [3] Paldor N. (2015) Shallow water waves on the rotating Earth. Cham, Switzerland: Springer. [4] Dijkstra HA. (2005) Nonlinear physical oceanography: a dynamical systems approach to the large scale ocean circulation and El Ninõ. Dordrecht, The Netherlands: Springer. [5] Viudez A, Dritschel DG. (2015) Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199-223. (doi:10.1017/S0022112003004191) · Zbl 1137.86304 [6] Herbei R, McKeague I, Speer KG. (2009) Gyres and jets: inversion of tracer data for ocean circulation structure. J. Phys. Oceanogr. 39, 1180-1202. (doi:10.1175/2007JPO3835.1) [7] Aksenov Yet al.(2016) Arctic pathways of Pacific water: Arctic Ocean model intercomparison experiments. J. Geophys. Res. Oceans 121, 27-59. (doi:10.1002/2015JC011299) [8] Abraham R, Marsden JE, Ratiu T. (1988) Manifolds, tensor analysis, and applications. New York, NY: Springer. · Zbl 0875.58002 [9] Daners D. (2012) The Mercator and stereographic projections, and many in between. Am. Math. Monthly 119, 199-210. (doi:10.4169/amer.math.monthly.119.03.199) · Zbl 1258.86009 [10] Andreev VK, Kaptsov OV, Pukhnachov VV, Rodionov AA. (1998) Applications of group-theoretical methods in hydrodynamics. Dordrecht, The Netherlands: Kluwer Academic Publishers. [11] Haller G. (2015) Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137-162. (doi:10.1146/annurev-fluid-010313-141322) [12] Ambrosetti A, Arcoya D. (2011) An introduction to nonlinear functional analysis and elliptic problems. Berlin, Germany: Birkhäuser. · Zbl 1228.46001 [13] Gilbarg D, Trudinger NS. (2001) Elliptic partial differential equations of second order. Berlin, Germany: Springer. [14] Berestycki H, Nirenberg L, Varadhan SRS. (1994) The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Commun. Pure Appl. Math. 47, 47-92. (doi:10.1002/cpa.3160470105) · Zbl 0806.35129 [15] Tadmor E. (2012) A review of numerical methods for nonlinear partial differential equations. Bull. · Zbl 1258.65073
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