Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates

Abstract

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.

Surface waves over currents and uneven bottom

“…A general overview on mathematical tools employed for the study of problems in physical oceanography is given in Johnson (2018). We would also like to point out the recent emergence of solutions modelling large-scale ocean currents with distinctive persistence patterns like gyres (Constantin and Johnson, 2017a) and the Antarctic Circumpolar Current (Constantin and Johnson, 2016b). An analysis of a solution describing nonlinear surface waves propagating zonally on a zonal current in the presence of Coriolis effects and exhibiting two modes of wave motion was presented by Constantin and Monismith (2017).…”

The modeling of the equatorial undercurrent using the Navier–Stokes equations in rotating spherical coordinates

“…These considerations show that (21) specifies, at leading order, the background flow, provided that the stream function ψ(ϕ, θ, z) solves the vorticity equation (22) subject to the boundary condition (19) and to the last two constraints in (20); Equations (16)-(17) then determine the associated pressure field, taking also into account the boundary condition represented by the first constraint in (19). Let us note that if we ignore the z-dependence (and thus, implicitly, consider an inviscid setting in which the wind forcing plays no role), then (22) simplifies to the ocean gyre model derived recently in [24] as a shallow-water asymptotic solution of Euler's equation in rotating spherical coordinates (with the stipulation that θ stands in [24] for the polar angle, and not for the angle of latitude) and further investigated in [25][26][27][28][29] in the context of the Antarctic Circumpolar Current -the largest ocean current on Earth, flowing clockwise from west to east around Antarctica (see [30,32,37]) so that, due to the lack of any landmass connecting with Antarctica, it keeps the warm ocean waters from lower latitudes away from Antarctica and thus maintains the huge ice sheets encountered near the South Pole (see the discussion in [13,31]).…”

Introduction to the Special Issue on Mathematical Aspects of Physical Oceanography

“…This motivates the investigation of shallow-water largescale ocean flows in a rotating frame in spherical coordinates. Using structural properties that become apparent in the shallow-water regime, solutions can be obtained that capture essential features of large-scale ocean gyres (Constantin and Johnson, 2017b), of general nearsurface wind-induced ocean currents (as discussed in the paper by Constantin and Johnson), and of the Antarctic Circumpolar Current, as shown in the paper by Haziot and Marynets.…”