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Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 10
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Articles

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

Pages 2069-2077 | Received 10 Aug 2019, Accepted 21 Sep 2019, Published online: 30 Sep 2019

ABSTRACT

Using the Navier–Stokes equations in rotating spherical coordinates, we discuss a mathematical model for the equatorial current across the Pacific Ocean. Symmetry properties of the governing equations that hold in equatorial regions permit us to gain detailed insight into the structure of the ocean flow dynamics.

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

1. Introduction

The equatorial ocean flows present special challenges with respect to flows at mid-latitudes since in this context neither geostrophic theory nor Ekman theory apply (see the discussions in [Citation1,Citation2] and [Citation3], respectively) and, moreover, nonlinear effects are noticeable (see the discussion in [Citation4,Citation5]). Of all equatorial regions, that of the Pacific is the most important due to its extent – it is more than 12,000 km across the ocean, from South America to South Asia. Several recent papers are devoted to equatorial flows (see the discussions in [Citation33Citation36]), and address the issue of underlying currents, issue that was mostly ignored in the earlier research literature (see the survey [Citation6]). The papers [Citation7–11] find explicit wave solutions for the governing equations in Lagrangian coordinates in the β-plane approximation, in the presence of relatively weak underlying currents. Two-dimensional flows (which ignore the meridional velocity component and use the f-plane approximation) are investigated in [Citation1,Citation12] (see also [Citation13]) and inroads towards three-dimensional flows can be found in [Citation14–17] (see [Citation18] for a detailed description of the intricate ongoing dynamics). One issue of great current interest is the derivation of realistic models for the underlying current field. The main features of the equatorial current are that the current is mainly azimuthal and presents a strong depth-variation in the upper 500 m of the ocean (of about 4 km mean depth), while at depths larger than 500 m there is practically no motion. The most significant features of the depth-variation consist of a westward near-surface current (induced by the trade winds that blow towards the west) and a stronger eastward flow – the Equatorial Undercurrent (EUC) – that dominates the overall ocean flow, reaching speeds in excess of 1 m/s. Note the unusual feature that the EUC flows against the prevailing trade winds, as the directly wind-driven flow vanishes below 100 m. Recently, an approach was proposed to explain how such a current with flow-reversal might be induced by the action of the wind (see [Citation19]). However, the model in [Citation19] does not capture the westward Equatorial Intermediate Current (EIC), found directly below the EUC across the Pacific (see [Citation20]). Our aim in this paper is to show that the method used in [Citation19], to investigate the Navier–Stokes equations in rotating spherical coordinates by taking advantage of symmetry properties of the leading-order terms in the governing equations for equatorial ocean flows to gain insight into the structure of the velocity field, can be further refined to provide a model of higher accuracy. In particular, we are able to include the westward EIC as a component of the background state of the ocean.

2. Preliminaries

For large-scale ocean flows it is appropriate to consider a spherical Earth, rotating at a constant rate about its polar axis (see [Citation21,Citation22]). We introduce near the Equator spherical coordinates (ϕ,θ,r), where r is the radius, θ(π/2,π/2) is the angle of latitude and ϕ[0,2π) is the angle of longitude. The corresponding unit vectors are (eϕ,eθ,er), respectively, with eϕ pointing from West to East, eθ from South to North and er upwards. We denote the ocean flow velocity components by (u,v,w). We use primes for physical (dimensional) variables, and we will remove them in the non-dimensionalisation process.

In this rotating reference system the Navier–Stokes equation and the equation of mass conservation are (see [Citation19]) (1) t+urcosθϕ+vrθ+wr(u,v,w)+1ruvtanθ+uw,u2tanθ+vw,u2v2+2Ω(vsinθ+wcosθ,usinθ,ucosθ)+rΩ2(0,sinθcosθ,cos2θ)=1ρ1rcosθpϕ,1rpθ,pr+(0,0,g)+ν12r2+2rr(u,v,w)+ν2r21cos2θ2ϕ2tanθθ+2θ2(u,v,w),(1) and (2) 1rcosθuϕ+1rcosθθ(vcosθ)+1r2r(r2w)=0,(2) respectively, where p(ϕ,θ,r,t) is the pressure in the fluid, ρ the constant density, Ω7.29×105rads1 is the constant rate of rotation of the Earth, and g=constant 9.81ms2 is the gravitational acceleration. The coefficients of the viscous terms are taken to be constant, with ν1 the vertical kinematic eddy viscosity and ν2 the horizontal kinematic eddy viscosity.

Let us now describe the associated boundary conditions. At the ocean's surface, r=R+h(ϕ,θ,t), where R6378km is the radius of the spherical Earth, we impose the dynamic and the kinematic boundary condition: (3) p=Ps(ϕ,θ,t)onr=R+h(ϕ,θ,t),(3) and (4) w=ht+urcosθhϕ+vrhθonr=R+h(ϕ,θ,t),(4) respectively, where Ps is the surface pressure. The wind-stress at the ocean's surface is considered to be known, and can be expressed in the form (5) τ1(ϕ,θ,t)=ρν1ur,τ2(ϕ,θ,t)=ρν1vr,onr=R+h(ϕ,θ,t),(5) the surface wind stress (τ1,τ2) being related to the vertical eddy viscosity by ν1=σ|(τ1,τ2)| on the surface, where σ is a (dimensional) constant and (τ1,τ2)=cDρair Uwind|Uwind|, ρair being the density of air (about 1.2 kg/m3) and cD0.0013 being a (dimensionless) drag coefficient. At the impermeable, solid, stationary bottom of the ocean, r=R+d(ϕ,θ), we have the corresponding boundary condition for viscous flow: (6) u=v=w=0onr=R+d(ϕ,θ),(6) Setting r=R+z and (7) p=ρgr+12r2Ω2cos2θ+P(ϕ,θ,r,t),(7) one can non-dimensionalise the problem by performing the change of variables (8) z=Dz,(u,v,w)=U(u,v,kw),P=ρU2P,(8) where D200m is the average depth of the near-surface layer to which the wind effects are confined), U0.1ms1 is the typical speed of mid-latitude ocean currents at the surface, while the scaling factor k for the vertical velocity is less than 104; see the discussion in [Citation19]. In terms of the the shallow-water parameter ε=D/R, with a typical value of the order 105, the governing equations for steady flow are transformed into the non-dimensional form (9) u(1+εz)cosθϕ+v1+εzθ+kεwz(u,v,kw)+11+εzuvtanθ+kuw,u2tanθ+kvw,u2v2+2ω(vsinθ+kwcosθ,usinθ,ucosθ)=1(1+εz)cosθPϕ,11+εzPθ,1εPz+1Re11ε22z2+2(1+εz)1εz(u,v,kw)+1Re2(1+εz)21cos2θ2ϕ2tanθθ+2θ2(u,v,kw),(9) and (10) 1(1+εz)cosθuϕ+θ(vcosθ)+k/ε(1+εz)2z(1+εz)2w=0,(10) where ω=ΩR/U=O(1) and Rei=UR/νj (j = 1, 2) are the inverse Rossby number and the pair of Reynolds numbers, respectively. With (h,d)=D(h,d), the non-dimensional boundary conditions are (11) P=P¯s(ϕ,θ),uz=τ1(ϕ,,θ),onz=h(ϕ,θ),vz=τ2(ϕ,θ),(11) (12) kεw=u(1+εh)cosθhϕ+v1+εhhθonz=h(ϕ,θ),(12) (13) (u,v)decays rapidly belowz=h(ϕ,θ).(13)

For ocean flows in the equatorial Pacific it is adequate (see the discussion in [Citation19]) to consider ω=O(1),1Re2=ε2μwithμ=ν2ν1, Multiplying the third component of (Equation9) throughout by ϵ, we obtain the non-dimensional Navier–Stokes system with Coriolis effects (14) u(1+εz)cosθϕ+v1+εzθ+kεwz(u,v,εkw)+11+εzuvtanθ+kuw,u2tanθ+kvw,εu2εv2+2ω(vsinθ+kwcosθ,usinθ,εucosθ)=1(1+εz)cosθPϕ,11+εzPθ,Pz+2z2+2ε(1+εz)z(u,v,εkw)+ε2μ(1+εz)21cos2θ2ϕ2tanθθ+2θ2(u,v,εkw).(14) Correspondingly, the equation of mass conservation, (Equation10), becomes (15) 1(1+εz)cosθuϕ+θ(vcosθ)+k/ε(1+εz)2z(1+εz)2w=0;(15) In the non-dimensional system of equations (Equation14)–(Equation15), the parameters (ω,μ,k,ε) are held fixed, as no approximations were performed.

The shallow-water limit corresponds to the limiting process ε0 and k/ε0 (see the discussion in [Citation19]). This regime ignores the vertical velocity component and, neglecting wave perturbations by setting h = 0, the flow dynamics being governed by the horizontal flow components u and v, subject to the nonlinear system (16) ucosθϕ+vθuuvtanθ2ωvsinθ=1cosθPϕ+2uz2,(16) (17) ucosθϕ+vθv+u2tanθ+2ωusinθ=Pθ+2vz2,(17) (18) uϕ+θ(vcosθ)=0,(18) which features Coriolis terms, viscous terms and the horizontal pressure gradients, while the boundary conditions (Equation11)–(Equation13) simplify to (19) P=P¯s(ϕ,θ),uz=τ1(ϕ,,θ),onz=0,vz=τ2(ϕ,θ),(19) (20) (u,v)decays rapidly belowz=0.(20) since setting w = 0 and h = 0 makes (Equation12) irrelevant. The system (Equation16)–(Equation20) captures the physical idea that the perturbation P¯s of the hydrostatic pressure and the shear stresses τ1 and τ2, at the ocean's surface z = 0, are needed to produce a consistent solution for the wind-drift horizontal current (u,v). To gain insight into the structure of this system, let us note that (Equation18) ensures (see the discussion in [Citation23]) the existence of a stream function in spherical coordinates, ψ(ϕ,θ,z), with (21) u=ψθ,v=1cosθψϕ;(21) see [Citation23] for a proof of the fact that the particle paths for steady flow on the surface of a sphere are the level sets ψ=constant. Taking advantage of (Equation21), we can eliminate the pressure between Equations (Equation16) and (Equation17) to derive (see [Citation19]) the vorticity equation (22) ψϕθψθϕ1cos2θψϕϕψθtanθ+ψθθ+2ωsinθ=cosθ1cos2θψϕϕψθtanθ+ψθθzz.(22) These considerations show that (Equation21) specifies, at leading order, the background flow, provided that the stream function ψ(ϕ,θ,z) solves the vorticity equation (Equation22) subject to the boundary condition (Equation19) and to the last two constraints in (Equation20); Equations (Equation16)–(Equation17) then determine the associated pressure field, taking also into account the boundary condition represented by the first constraint in (Equation19). 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 (Equation22) simplifies to the ocean gyre model derived recently in [Citation24] as a shallow-water asymptotic solution of Euler's equation in rotating spherical coordinates (with the stipulation that θ stands in [Citation24] for the polar angle, and not for the angle of latitude) and further investigated in [Citation25–29] in the context of the Antarctic Circumpolar Current – the largest ocean current on Earth, flowing clockwise from west to east around Antarctica (see [Citation30,Citation32,Citation37]) 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 [Citation13,Citation31]).

3. Main results

Finding the general solution of the equation (Equation22) with the boundary conditions (23) uz=τ1(ϕ,,θ),vz=τ2(ϕ,θ),onz=0,(23) (24) (u,v)decays rapidly belowz=0.(24) where the horizontal velocity field (u,v) is specified by means of (Equation21), is not a realistic task. Instead, we try to take advantage of some special features that are typical for the equatorial region in the Pacific. We restrict our attention to an equatorial zonal band that is symmetric about the Equator, with the angle of latitude θ close to 0 and the angle of longitude ϕ near ϕ0, with ϕ0(8π/9,14π/9) corresponding to the region between 160E and 80W. For the twice differentiable functions α(z) and β(z), we look for longitude-independent meridional velocity components and we seek a linear dependence of the azimuthal velocity component on the longitude. These features lead (see [Citation19]) to the stream function (25) ψ(ϕ,θ,z)=ϕα(z)+β(z)lncosθ1sinθωsinθ+lncosθ1sinθ,(25) solution of the governing equation (Equation22), valid for |ϕϕ0|<ϕˆ and |θ|<θˆ with some fixed (and small) values ϕˆ>0 and θˆ>0. Due to (Equation21), this corresponds to the horizontal velocity field (26) u(ϕ,θ,z)=ϕα(z)+β(z)cosθ+ωsin2θcosθ,v(ϕ,θ,z)=α(z)cosθlncosθ1sinθ,(26) while the boundary conditions (Equation23) are induced by a wind stress with components (27) τ1(θ,ϕ)=ϕα(0)+β(0)cosθ,τ2(θ,ϕ)=α(0)cosθlncosθ1sinθ.(27) Rather than imposing (Equation24), we pursue the considerations in [Citation4]) and interpret the solution (Equation26) as being valid in the near-surface ocean region above the thermocline z = −T, where we require the no-stress boundary condition (28) uz=vz=0onz=T.(28)

The fundamental characteristic of the trade winds in the equatorial Pacific is that, in each hemisphere, they are oriented towards the Equator, blowing westwards with a more stronger westward direction as the Equator (θ=0) is approached. This property holds for the wind stress specified in (Equation27) if (29) α(0)<0<ϕα(0)+β(0)(29) for all relevant values of the longitude ϕ, that is, for ϕˆ<ϕϕ0<ϕˆ. Note that this orientation of the wind, in combination with the fact that the vanishing of the meridional component of the Coriolis force, ωsinθ, at the Equator prevents it from inducing a deflection from the wind direction (as is typical in Ekman theory at mid-latitudes), leads to a near-surface current that moving westward. This is ensured for the solution (Equation26) if (30) ϕα(0)+β(0)>0.(30) Moreover, if (31) α(0)>0,(31) then the meridional flow described by (Equation26) is poleward near the surface. Note that the main features of the near-surface flow in the equatorial Pacific are a westward motion at the surface and a poleward meridional flow (see the discussion in [Citation4]). One can see that a linear or quadratic dependence on the z-variable of the azimuthal velocity profile u(ϕ,θ,z) of type (Equation26) can not accommodate the constraints (Equation29)–(Equation31) and (Equation28). A solution featuring a cubic dependence was provided recently in [Citation19], namely (32) α(z)=az2+2TzT23,β(z)=aϕ01Tz33Tz,Tz0,(32) for some constant a<0. This provides an azimuthal flow that is westward near the surface and features a faster eastward jet along the thermocline, such that at any fixed longitude ϕ(ϕ0,1312ϕ0), the zonal velocity u strictly decreases from a positive value at the thermocline z = −T (corresponding to an eastward flow) to a negative value at the surface z = 0 (corresponding to a westward flow), and vanishes once above z=(T/3), having just one inflexion point, to be found between the depth levels z=(T/3) and z=(2T/3). The solution (Equation32) captures the main features of the ocean current in the equatorial Pacific: a westward wind-drift overlying the stronger eastward Equatorial Undercurrent (EUC). We now show that we can refine this result, by accommodating also a weaker eastward flow beneath the EUC (see Figure ). For this, it suffices to choose suitable quintic polynomial expressions for the azimuthal velocity profile.

Figure 1. Depiction of the vertical profile of the current in the upper 200–400 m of the Pacific Ocean along the Equator: the westward wind-drift current is near the surface z = 0, below it is the eastward EUC that dominates the subsurface flows, with the weaker westward EIC found directly below the EUC, while at great depths there is practically no motion. These are the main features across the Pacific, over 12,000 km (see the data provided in [Citation20]).

Figure 1. Depiction of the vertical profile of the current in the upper 200–400 m of the Pacific Ocean along the Equator: the westward wind-drift current is near the surface z = 0, below it is the eastward EUC that dominates the subsurface flows, with the weaker westward EIC found directly below the EUC, while at great depths there is practically no motion. These are the main features across the Pacific, over 12,000 km (see the data provided in [Citation20]).

Theorem 3.1

The quintic polynomial expressions α(z)=1442125(195T22264)z5T5+36245(195T22264)z4T412z2Tz+1,β(z)=ϕ0182125(2285T224982)z4T4+14250141180T21487411T3z3+33401260T211531T2+1z2+2Tz accommodate an equatorial current profile that features an eastward jet at mid-depth of the near-surface layer above the thermocline z=T, a weak eastward jet just above the thermocline and a stronger westward jet near the surface z = 0.

Proof.

The corresponding azimuthal velocity u(ϕ,θ,z), obtained from (Equation26), vanishes at z=(T/8), z=(2T/3) and z=(5T/6), giving the profile depicted in the figure. Moreover, since α(0)=1>0,ϕ0α(0)+β(0)=1>0,α(0)=T<0,ϕ0α(0)+β(0)=φ0T>0,α(T)=0,β(T)=0, all the constraints (Equation29)–(Equation31) and (Equation28) are verified. The considerations preceding the statement yield now the desired result.

Remark 3.2

The theorem shows that it is possible to capture within our modeling framework not only the presence of the eastward EUC and of the westward wind-drift current, but also the presence of the weaker westward EIC, found directly below the EUC (see the figure).

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The support of the WWTF [grant number MA16-009] is acknowledged.

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