A nonlinear , three-dimensional model for ocean flows , motivated by some observations of the Pacific Equatorial Undercurrent and thermocline

Using the salient properties of the flow observed in the equatorial Pacific as a guide, an asymptotic procedure is applied to the Euler equation written in a suitable rotating frame. Starting from the single overarching assumption of slow variations in the azimuthal direction in a two-layer, steady flow that is symmetric about the equator, a tractable, fully nonlinear, and three-dimensional system of model equations is derived, with the Coriolis terms consistent with the β-plane approximation retained. It is shown that this asymptotic system of equations can be solved exactly. The ability of this dynamical model to capture simultaneously fundamental oceanic phenomena, which are closely inter-related (such as upwelling/downwelling, zonal depth-dependent currents with flow reversal, and poleward divergence along the equator), is a novel and compelling feature that has hitherto been elusive. While details are presented for the equatorial flow in the Pacific, the analysis demonstrates that other flow configur...


I. INTRODUCTION
An aspect of oceanic flows which, over the last few decades, has received a lot of attention is the interaction of equatorial flows-especially undercurrents-and thermoclines (and then the way this relates to the El Niño phenomenon).Great strides have been made in our understanding of much of what we observe, enhanced by the significant increase in the availability of reliable data.However, although there is a general appreciation of many of the processes involved, based mainly on a combination of interpretation of observations and theoretical studies within the framework of linear theory, the importance of nonlinear processes is also recognised; see Ref. 12.These are manifest in the emergence and persistence of large coherent structures that are indicative of fundamental nonlinear laws of self-organisation, rather than of linear, reductive paradigms suitable for small perturbations of an explicit closed-form solution that describe a relatively simple state of the system.Indeed, we believe that a general mathematical approach, which retains some nonlinearity and elements of three-dimensionality, will produce more useful, wide-ranging, and reliable results than any ad hoc modeling.In this article, we develop this idea in the context of oceanic flows, using the Pacific equatorial flow as the basic example.Although we will develop the details as they pertain to this particular flow, we will make the generality of the approach and technique clear.a) Electronic mail: adrian.constantin@univie.ac.at b) Electronic mail: robin.johnson@newcastle.ac.uk In order to set the scene, we first describe the particular phenomenon-one of many in the rich variety of ocean dynamics-that prompted this investigation.Although we were aiming, initially, to look in depth at this flow (and some of our earlier attempts were reasonably successful; see Refs.7 and 8), the work took us in a slightly different, but a more general and, we believe, a more exciting direction.Thus we must emphasise, at the outset, that there is much that we cannot include in our approach to these problems but, on the other hand, we believe that we have here a new and useful method for analysing such flows.So let us describe the specific example that motivated this investigation; indeed, success could be measured by producing a model that recovers much that is observed in this case.
The dynamics of the ocean near the equator present some unique features.First, there is a pronounced stratification in equatorial regions, greater than anywhere else in our oceans; see Ref. 13.Second, while in mid-latitudes the Coriolis terms lead to a frequency-scale separation between slow potential vorticity dynamics and the fast oscillatory effects induced by gravity, this is no longer the case at the equator, where the meridional component of the Coriolis force vanishes.In particular, the widely used mid-latitude geostrophic balance (between the Coriolis force and the meridional pressure gradient) breaks down in equatorial regions (see Ref. 30).Third, there is a redeeming feature: due to the vanishing of the meridional component of the Coriolis force there, the equator works as a (fictitious) natural boundary that facilitates azimuthal-flow propagation (see Ref. 13).More precisely, in the Pacific Ocean, within a band about 150 km on either side of the equator, there is a pronounced stratification in the near-surface region where a pycnocline/thermocline resides; see Ref. 13.This separates a layer of relatively warm water (and so less dense) from a deeper layer of denser, colder water; the difference in density across this line is about 0.5%.It is usual-and we follow the same route here-to model the thermocline (pycnocline) as a line of discontinuity in density: above it we have a density ρ 0 and below it a density (1 + r)ρ 0 , where r is a small, positive constant (see Refs. 28 and 29).The thermocline is situated at depths of about 50-150 m below the surface.This variability is not a measure of uncertainty, rather, the thermocline drops a short distance away from the western edge of the Pacific and then rises gradually towards the east at a rate of about 1 m over one degree of longitude at the equator. 13It should be noted that although the density change across the thermocline plays a crucial role in the coupling between waves here and on the surface, it does not contribute in any significant way to the structure of the velocity profile below the surface-and we will not be discussing waves in this analysis.All this, however, turns out to be a relatively minor complication of the flow in this region.
The flow and its dynamics are dominated by the presence of underlying non-uniform currents; indeed, in the oceans generally, these come in many forms, including upwelling and downwelling, leading to complex flow patterns.In the neighbourhood of the Pacific equator, the currents are quite significant (and well known, e.g., see Ref. 32).In particular, near the surface and extending no more than about 100 m down is a westward drift that is driven by the prevailing trade winds.
[Of course, these are generated and maintained by the stresses induced by wind action at the surface of the ocean, see Refs.7 and 9; we will take this motion as prescribed.The main equatorial oceanic phenomena occur on length scales much larger than those directly affected by molecular viscosity or diffusivity: it is accepted that the Reynolds number is extremely large for these flows (see Ref. 31), and so we make the obvious choice to model the fluid as inviscid and laminar, but rotational.Thus our mathematical model will be based on the Euler equation.]Returning to the specific equatorial flow of interest to us in this initial phase of the investigation, we note that just below this near-surface layer is the Equatorial Undercurrent (EUC); this is a relatively high-speed jet flowing eastwards, with its core more-or-less aligned with the thermocline (and so, typically, this also rises from west to east).Below the EUC, starting at depths of about 500-1800 m is the abyssal region where the water is essentially stagnant (stationary).In addition, as we have already mentioned, there are regions of upwelling and downwelling superimposed on this flow, which convert an essentially one-dimensional flow into a three-dimensional one.For the ocean in the neighbourhood of the Pacific equator, the three-dimensionality results in a flow that rises to the surface, all along the equator, and moves away from it close to the surface (although this upwelling may be restricted only to regions fairly near the surface).The consequence of this is clearly observed: the flow at the surface moves predominantly to the west (wind-driven), with a small component of the velocity to the north or to the south but, lower down, the flow structure is not known in any detail; 24,25,27 the basic flow configuration is depicted in Fig. 1.
The challenge, in the first instance, is to develop a consistent and systematic mathematical approach (necessarily asymptotic) that will capture the essential properties of the type of flow observed in the neighbourhood of the Pacific EUC.However, we cannot hope to produce an accurate model for this particular flow, which also includes all the elements that lead to its complexity; rather, our aim is to produce a sufficiently general and robust approach that might elucidate some of the fundamental mechanisms that underpin this and many other similar oceanic flows.Above all, we will show that a nonlinear, three-dimensional theory is possible, i.e., there is no need to oversimplify by resorting to linear models.We have discovered that a nonlinear approach based on Euler's equation, coupled with one overarching assumption, namely, that the two-layer flow in the equatorial direction evolves slowly, is sufficient for our purpose.This slow evolution provides us with our fundamental parameter-there are no other parameters essential to the development of the solution-and so the FIG. 1.A sketch of the structure of the equatorial ocean flow in the Pacific.The figure on the left depicts the main features of the flow along the surface and along the thermocline, indicating the equatorial upwelling and downwelling, and the zonal and meridional tilting of the thermocline.The figure on the right depicts the vertical profile of the underlying currents along the equator, with a near-surface westward wind-drift above the eastward EUC, whose maximum flow-speed region is more-or-less aligned with the thermocline, and below the EUC there is an abyssal region of essentially still water.The eastward net flow occurring within a band restricted to about 1 • latitude either side of the equator is compensated by a return flow at higher latitudes-a situation that departs from what one can observe in wave tanks.
problem will still retain contributions from the rotation of the Earth and the density change across the thermocline.However, the jump in density is small (see above) and contributes in only a superficial way to the solution of the particular problem that we describe here, as we will explain.(The role of this jump is, on the other hand, very important in the description of the various modes of wave propagation because in this case the jump contributes to the coupling of the pressure changes across the two oscillating surfaces; see Ref. 7.) In the context of the overall structure of a velocity profile (describing the flow in the azimuthal direction), the jump in density is altogether unimportant-the thermocline simply provides kinematic and pressure-continuity conditions that are appropriate across any surface in the flow field-and so, if it is useful, we may invoke the smallness of the density change.Indeed, the upshot is that we can proceed with a rather more straightforward and transparent presentation, without affecting the final result to any significant extent.Our reduction of the problem still retains both nonlinearity and three-dimensionality, together with the effects of rotation (provided by the approximation to the Coriolis contribution via the β-plane approximation but, as we show, we must be careful how this approximation is interpreted geometrically).The retention of the rotation effects provides a novel feature here, even though in a strictly numerical sense they are associated with a small parameter; we will return to this issue in the discussion at the end of Sec.III, when we describe the relevant research literature and specific quantitative features of the equatorial Pacific flow.However, this is not the parameter that drives the fundamental asymptotic approximation that we develop here.Nevertheless, there are opportunities to invoke its smallness and thereby simplify the calculation without losing the essential character of the flow.This simplification, coupled with that of small density change across the thermocline, is used, where appropriate, to make the underlying solution more transparent.
In summary, the plan is to describe the slow development, along the equator-other circles could be chosen, but the flow in the neighbourhood of the Pacific EUC is the obvious one to start with-and then in a region that is limited in width (to be consistent with the β-plane approximation).Thus, of course, the analysis cannot produce an accurate representation of the behaviour of the flow to the north and to the south, linking it to the flows in the surrounding ocean away from the equator.Necessarily, we are restricted to regions fairly close to the line of the equator.We hope, however, to capture the detailed behaviour of the evolution of the flow along the equator and the flow in the ocean in the neighbourhood of the equator.As implied above, we retain all the terms associated with the Coriolis effect, as represented by the β-plane approximation, even though this turns out to be associated with a fairly small parameter in our development.We do retain the essential ingredients, in this context, that contribute to both the nonlinearity and the three-dimensionality of the flow, but the model implies, not surprisingly, some important simplifications by virtue of the slow evolution.Some previous attempts have been made to describe nonlinear flows in this same context (see Refs. 4-6, 18-20, and 23).However, the models described by these authors, although leading to exact solutions, retain only weak dependence in the meridional direction and, further, there is no meridional flow and no upwelling/downwelling process.
The paper is organized as follows.Using the general properties of the flow observed in the equatorial Pacific as a guide, an asymptotic procedure is developed which is based on slow variations in the azimuthal direction; this is applied to the governing equations which are presented in Sec.II.The result is a new system of model equations which incorporate both the thermocline and rotational effects, and is fully nonlinear and three-dimensional (see Sec. III).The equations, with the Coriolis terms consistent with the β-plane approximation retained, and the boundary conditions at the free surface and at the thermocline included, can be solved exactly; furthermore, the background flow describing the velocity profile from the surface downwards may be arbitrarily assigned.A simplified model is introduced in Sec.V and the method of solution is discussed in Secs.VI and VII, by taking advantage of the small change in density across the thermocline and the relatively slow rate of rotation (as described in Sec.IV).The simplification also includes a geometrical adjustment, implying that all deviations are measured relative to planes parallel to the local tangent plane (but all consistent with the underlying curvilinear coordinate system).We choose, in Secs.VIII-X, to model profiles that describe a westward flow near the surface and a faster eastern flowing current lower down (typical of the Pacific Equatorial Undercurrent).The flow properties for two velocity profiles are obtained; the example in Sec.IX includes a region of uniform flow (around the maximum of the profile), and the one in Sec.X uses a quadratic-quartic profile to represent the whole profile down to a region of stationary flow; the thermocline is embedded close to the region of maximum flow speed.In addition, some streamline patterns for the flow in the plane normal to the equator are presented in Sec.XI; these examples show that one, two, or three cells are possible, the resulting flows representing upwelling, downwelling, and flow towards/away from the equator.Finally, in Sec.XII, we discuss our results and their implications for future studies.As the analysis demonstrates, the approach is quite general and so it will be relevant to other similar, but different, flows because there is sufficient freedom in the analysis to accommodate, for example, arbitrary velocity profiles below the surface and any suitable path for the maximum flow speed of this profile.In particular, the equatorial flows encountered in the Atlantic Ocean and the Indian Ocean are of definite interest.We also point out the possibility of accommodating more general flows, as a part of future work, by removing some of the additional simplifying assumptions that we choose to incorporate in this initial investigation.

II. GOVERNING EQUATIONS
In the light of our general comments above, we present the ideas as they pertain to the Pacific EUC, although it will become clear that the methods are more generally applicable.
The model that underpins the approach that we adopt is that of an incompressible, inviscid fluid; the jump in density is accommodated by allowing the presence of two regions, each Phys.Fluids 29, 056604 (2017)

FIG. 2. (a)
The rotating frame of reference based on the tangent plane, with the x-axis chosen horizontally due east, the ȳ-axis horizontally due north, and the z-axis vertically upward.(b) Schematic depiction, at fixed x on the equator, illustrating the fact that the curved surface of the Earth drops below the tangent plane, whose cross section is the line P 0 P (at the equator) and the line PP 0 (at P).Here P(x, y, z) is a point on the surface of the ocean, and P 0 is located on the equator at the same longitude.Since the circular arc P 0 P has length y and the distance of P 0 and P to the centre of the Earth, O, is R + z, the angle of latitude θ of P equals y/(R + z), while tan θ = y/(R + z).Since θ = tan θ in the β-plane approximation, then y = y at leading order.Note that the curved surface of the Earth drops below the (x, y)-plane by the distance between P and P : exhibiting constant density (see Ref. 17).We choose a coordinate system that rotates with the Earth, with x pointing due east, y due north, and z vertically upwards; see Fig. 2. (We use overbars to denote physical variables; these will disappear when we introduce our non-dimensional scheme.)This coordinate system is the familiar one associated with the tangent plane (and therefore consistent with the β-plane approximation); however, we must be circumspect in its interpretation.The β-plane approximation adopts a local "flat" Cartesian coordinate system, while still capturing the effects of the curvature of the Earth's surface.[Readers who are altogether familiar with the standard development that leads to the β-plane approximation may wish to go directly to Eqs. ( 9)- (14), which are the relevant equations and boundary conditions, written in this approximation, expressed in non-dimensional form.]The derivation of these coordinates from the original spherical coordinate system requires two manoeuvres: first, the great circle (the equator here) is straightened to become the generator of a cylinder, its circular cross section now representing the interior of the sphere.The coordinate perpendicular to this, on the surface, must then be expressed as the arc length along the circular rim of the cylinder, following the circumference at a mean radius of the Earth; it is this definition which appears in the conventional version of the governing equations (details can be found in Refs.8 and 15).The coordinate y in Fig. 2 is in the direction of the tangent at a point on the equator-and this is not the curvilinear coordinate that appears in the equations; we write the latter as y.In typical Cartesian coordinates, the z-axis points vertically upwards from the centre of the Earth, for all points on the surface; we denote the corresponding cylindrical (radial) coordinate by z.With this in mind, the two governing equations (Euler and mass conservation) may therefore be written as where u = (u, v, w) is the velocity of the fluid [corresponding to x = (x, y, z)], t is the time (and D/Dt the associated material derivative), while p and ρ are the pressure and the (constant) density, respectively.The body force is represented by F (which incorporates constant gravitational acceleration only) and Ω = Ω (0, cos θ, sin θ) is the angular-velocity vector describing the rotation of the Earth (with |Ω| = Ω ≈ 7.29 × 10 −5 rad s 1 and θ the angle of latitude).
In order to make the problem reasonably accessible and tractable, we invoke a suitable approximation of the Coriolis term close to the equator; the same general procedure can be adopted in the neighbourhood of any small circle around the polar axis.Thus, for small values of y/R, where R is an average radius of the Earth, we write the term 2Ω × u as 2Ω (w − y v/R, y u/R, −u), since sin θ ≈ θ and cos θ ≈ 1 for |θ| 1; this constitutes the equatorial " β-plane" approximation, which is regarded as altogether adequate within about 2 • of the equator (see Ref. 17).We therefore have x measured, equivalently, along the curvature of the equator (eastwards) and y measured along a line of longitude (northwards).Note that, with this same approximation, the curved surface of the Earth drops below the (x, y)-tangent-plane by the amount y 2 /2R, approximately; indeed, consistent with, and a consequence of, the β-plane approximation, we have z = z + 1 2 y 2 /R and y = y [see Fig. 2(b)].With this interpretation of the chosen coordinate system, the distance from the centre of the Earth is R+z, so that the gravitational potential is z g and thus the gravitational field (its negative gradient) becomes (0, 0, −g): the gravity term appears only in the z-component equation.The governing equations for a steady flow therefore become (where we write y for y) with g ≈ 9.8 m s 2 the constant acceleration of gravity, R≈ 6371 km an average radius of the Earth, and ρ 0 ≈ 1027 kg m 3 an average constant density of the ocean.These equations have been expressed within the limitations of the β-plane approximation and, precisely as written, applied to the region above the thermocline; below it we must insert (1 + r)ρ 0 in place of ρ 0 , where r ≈ 5 × 10 −3 is a constant (and here we quote the value appropriate for the EUC).
The boundary conditions that we require involve the pressure at the surface (z = η(x, y))-there are no other stresses in our model-and a kinematic condition that describes the motion of this surface; these are, respectively, where P s (x, y) is the pressure prescribed at the surface (which will be discussed later).There is a corresponding kinematic condition at the thermocline (z = T (x, y)), across which we must also impose the continuity of pressure; this condition and ( 7) are redundant if no thermocline is present.Finally, we assume that the bed of the ocean [z = −d(x, y)] is a stationary, impermeable surface, so This last boundary condition will turn out to be superfluous in the solution that we develop here: below a known level, we will consider the flow to be stagnant (stationary), and therefore on the bed we will automatically have u = v = w = 0. Of course, we retain the boundary condition (8) for more general flows; this is one of the aspects of the problem that can be examined in the future, particularly if suitable field data are available to give some guidance as to the nature of the three-dimensional flows at greater depths.

III. NON-DIMENSIONALISATION AND THE SLOW SCALE
The first stage in expressing our governing equations in a useful form is to introduce a suitable non-dimensionalisation of the variables; to this end, we write where the length scales are (L, l, h) and U is an appropriate speed scale.(The omission of the overbars now indicates that we are using the non-dimensional version of the corresponding variable.)The velocity components have been nondimensionalised in a form that guarantees the existence of a stream function, via a balance of terms of equal size, no matter whether the flow is in the (x, z) or (y, z) frame.Equations ( 2)-( 5) therefore become with the boundary conditions 13) where η = hη and T = hT .In view of the comment after (8), we have dispensed with the bottom boundary condition altogether.
It is convenient, at this stage, to redefine the pressure so that we work hereafter with the pressure as measured relative to the ambient hydrostatic pressure: we set To proceed, we must now decide on a specific limiting process associated with the various parameters that have appeared in Eqs. ( 9)-( 14).This is to be selected so that we capture the essential mathematical problem that we wish to investigate here.In order to construct a reasonably sensible underlying (formal) approximation, we will take the sizes of these parameters as guided by the properties of the flow associated with the Pacific equator.Thus we have a length of about 13 × 10 3 km and the typical depth to the bottom of the EUC is about 200 m; the total width of the EUC is about 300 km.With these choices as an initial guide for L, h, and l, respectively, and with U = 0.5 m s 1 (a typical speed at the surface), we In the context of this type of problem, a more convenient choice, given (U, L, h), is which is the natural choice, consistent with the β-plane approximation, because it follows the curvature of the ocean away from the line of the equator.Indeed, R h is proportional to the width over which the surface drops by h below the tangent plane due to the local curvature.[For the EUC, this is about 36 km, and so the EUC extends, approximately, over −4 < y < 4 in these variables; we then have (l/L) 2 ≈ 7 × 10 −6 .]With this choice for l, we invoke the limiting pro- → 0 and all other parameters held fixed.The governing equations are therefore taken as with the boundary conditions P(x, y, z) = P 0 (x, y) + κ η(x, y) w = uη x + vη y on z = η(x, y) , ( 18) where ω = Ω h/U and κ = g h/U 2 (plus the continuity of pressure across the thermocline).The governing equations that correspond to ( 16) and ( 17), but valid below the thermocline, are written with P replaced by P/(1 + r), which we write-see below-in a more convenient form later.We will interpret our choice of scales so that, in the flow direction, we have 0 ≤ x ≤ 1 (and in the context of the EUC, we could associate the western end, away from any shorelines, by x = 0 and correspondingly x = 1 at the eastern end); the width must be restricted (by virtue of the β-plane approximation), which we represent by −y 0 < y < y 0 , for suitable (finite) y 0 .Note that the "traditional β-plane approximation," used in the overwhelming majority of studies of equatorial flows, 38  neglects the terms 2Ωw and −2Ωu in ( 2) and ( 4), respectively, which are associated with the horizontal component of the Coriolis force involving the vertical velocity and the vertical component of the Coriolis force.While technical convenience, such as the possibility of separation into normal modes, 17 drives the widespread use of the "traditional approximation," some studies have revealed that the neglected non-traditional components of the Coriolis force have significant effects on the equatorial ocean dynamics.For example, a vorticity analysis and an angular momentum analysis on the sphere 2,3 show that, if the ocean flow varies on a horizontal scale comparable to H R, where H is the vertical scale of the motion, then the effect of the horizontal component of Earth's rotation must be taken into account.In the equatorial Pacific, due to the presence of deep zonal jets in the subthermocline flow, 21 situated at about 1800 m beneath the surface, we have H R ≈ 100 km, so that the ratio of vertical to horizontal length scales, measuring the importance of the non-traditional components of the Coriolis force, is about 1.8 × 10 −2 , one order of magnitude larger than the parameter typical 35 for the traditional approximation.The derivation of model equations with the retention of the non-traditional terms has been pursued in the recent research literature by the application of variational methods which neglect vertical accelerations, either outright 36 or as a by-product of vertically averaging across homogeneous layers. 10,11,34The resulting shallow-water equations, owing to their simplified two-dimensional description of three-dimensional flows with large horizontal scales, represent a promising starting point for in-depth studies of the effects arising from non-traditional rotation.However, their intrinsic two-dimensional form does not capture the observed three-dimensional equatorial flow patterns generated by the interaction of zonal depth-dependent currents with upwelling/downwelling processes.Our proposed description of this problem, ( 16)- (19), is a reduction based on just one simplifying assumption that there is a slow variation of the flow properties along the equator.This is the most satisfactory form of approximation of the mathematical problem: we take just one limit (which we interpret here as h/L → 0) and retain all the other parameters in the system of equations.Furthermore, an important consequence is that these equations contain all the Coriolis contributions associated with the rotation of the Earth (even though the corresponding parameter, ω, is quite small); it is the contribution of these terms in ω which led to the discovery of an exact solution of the original system (2)-( 8), described in Ref. 8. In the rest of this paper, we present the details of the solution to system ( 16)- (19) as they relate to the flows in the neighbourhood of the Pacific equator, this being the obvious example in this context.We shall see that our approach enables us to capture the detailed structure of the near-surface ocean dynamics achieving, in particular, a depiction of the three-dimensional structure typical for these flows.

IV. PRESSURE ACROSS THE THERMOCLINE AND AT THE SURFACE
Before we are able to study the problem defined by ( 16)- (19) in any detail, we must first examine the form taken by the pressure across the thermocline; for this purpose, we consider the last two equations in (16) and their counterparts below the thermocline.Above the thermocline, we have precisely the last two equations in (16); for the flow below, it is natural to write [cf.(15)] and so we have the pair of equations 2ω y û = − Py , 2ω û = Pz , which correspond precisely with the appropriate pair valid in the region above the thermocline so, both above and below the thermocline, we have the same structure for u(x, y, z), namely, u = u(x, z − 1 2 y 2 ); let us introduce and we observe that ζ = z/h is the (non-dimensional) coordinate measured from the tangent plane; we then have where A(x) and Â(x) are arbitrary functions.At this stage, we allow the velocity profile to take a different form below the thermocline, so there could be a discontinuity across this line; we denote the dependent variables below the thermocline with the circumflex.Once u has been prescribed throughout the depth of the ocean, both above and below the thermocline, we may determine φ and φ [and it is convenient to define φ(x, 0) = 0 and φ(x, 0) = 0].The requirement that u takes the form described in (20), and the corresponding relation for û below the thermocline, is the first indication of the role played by the rotation terms, at least in the case of a slow evolution along the length of the equator, consistent with the β-plane approximation: such an imposed structure is absent if these terms are ignored.As we will see below, this also has ramifications for the way in which we interpret the boundary conditions and how these are linked to the choice of the coordinate system.Note that the dependence on (z − 1 2 y 2 ) corresponds in dimensional variables to a dependence on (z − 1 2 R ȳ2 ), which defines the ratio of the vertical and horizontal widths of the flow: if the vertical width is ∆z, then, taking into account the symmetry with respect to the equator, the total horizontal width is In particular, for R ≈ 6400 km and ∆z ≈ 4 km, (22) yields ∆z ≈ 160 km, which is reasonable for the observed width of the EUC.At the free surface [z = η(x, y)], the pressure boundary condition now becomes and at the thermocline [z = T (x, y)], the continuity of pressure requires −κ T (x, y) + P(x, y, T (x, y)) which can be written, upon using ( 21), as The form of these two boundary conditions becomes clearer when, first, we describe the free surface relative to the tangent plane.We set and then we have the pressure at the surface given by which connects the pressure prescribed at the surface to the distortion of the surface.(This important idea is developed in Ref. 8 in the context of some exact solutions.)Second, we may write the condition at the thermocline, (23), in the form This completes our formulation of the problem that is relevant to the flow along the Pacific equator, in the β-plane approximation.This most general problem, under the umbrella of a slow variation in the azimuthal direction, is therefore described by Eqs. ( 16) and ( 17), with the surface pressure condition, (25), and the pressure condition across the thermocline, (26), together with the two kinematic conditions [in (18) and (19)].The boundary condition on the bottom has been suppressed here, by virtue of our assumption of stagnant flow at greater depths.It is evident that this system, although a significant reduction of the original, contains a number of complicating features.So, as a first attempt to investigate if we have a problem which captures some important and relevant features of these flows, we proceed by considering a simplified version; as will be shown, this still retains both threedimensionality and rotation and enables the recovery of some realistic flow patterns.However, it is clear that a more comprehensive study is required, based on the system described above; this is for the future.

V. A SIMPLIFIED THREE-DIMENSIONAL NONLINEAR MODEL
In order to make more transparent the problem posed in Sec.IV, we seek simplifications that produce a technically simpler-but still worthwhile-problem.This involves two simplifying assumptions: we take the density change across the thermocline to be small and we work with a simplified geometry.First, for sufficiently small r, we see that (26) gives A solution of the equation obtained by ignoring the error O(r) is and then from (21) we see that the expressions for the pressure become identical above and below the thermocline.This is what we would recover if the thermocline was absent, although we observe that it is still present in this model through the kinematic condition, (19), and its presence can be guaranteed by prescribing its position at one end.In this formulation of the problem, we suppose that u is given, as a bounded function (and we may as well allow it to be continuous) and so φ will be likewise.In this case, for a given thermocline [z = T (x, y)], the terms in r can be regarded as providing a (uniformly) small correction to φ (to give φ), and hence to u (to give û), for the flow below the thermocline.Second, the free surface described in (24) incorporates the curvature in the y-direction as measured relative to the tangent plane, and the simplest choice is to allow no further dependence on y, i.e., h = h(x) only.Further, in the context of the EUC, it is a property of the Pacific that the level of the ocean from east to west (relative to the curvature of the Earth) rises by about 0.5 m because of the action of the trade winds. 14 This variation could be embodied within our function h(x), but it is so small and unimportant, we believe, to the overall flow pattern that we are investigating here, that we elect to prescribe that the free surface is flat in the x-direction [i.e., h(x) = 0] in our model.The free surface is therefore the (local) undisturbed surface of the sphere.From (25), the pressure at the surface now becomes and we will invoke the simplest version of this condition: constant pressure along y = 0, so A(x) = constant.This leaves the pressure at the surface as that required to maintain the spherical surface in the β-plane approximation.The thermocline is most simply treated in the same manner: it sits in a plane parallel to the tangent plane (and so follows the curvature in the y-direction), but we allow it to evolve in the x-direction.Thus we choose to set and this will sit somewhere in the neighbourhood of the maximum speed below the surface.Thus the special problem that we discuss here is the one which follows the geometry of the curved surface (in the y-direction, locally) and has a (local) flat free surface associated with a constant atmospheric pressure at the surface along the equator and a thermocline at a level which may evolve in the equatorial direction.This is described by the set of equations where the prime denotes the derivative with respect to x, and u is now prescribed throughout the entire flow, from the free surface downwards.The pressure no longer appears explicitly in this formulation-it is now associated with the term in φand u may be arbitrarily assigned as a function of depth along y = 0, at some x.We expect that we are able to fix levels of the maximum speed, say, and of the thermocline, at one end, e.g., at x = 0 (the western end for the EUC); these will evolve in x-and determining this behaviour has become one of the aims of this analysis.The solution that we obtain will be defined from the surface down to the zero-speed level (and we impose stagnant conditions below this) and for 0 ≤ x ≤ 1 and −y 0 < y < y 0 (for some suitable y 0 consistent with the flow configuration and the β-plane approximation).
To summarise, we have developed the description of the problem with just one overarching assumption: slow variation in the azimuthal direction; this gives the problem described by Eqs. ( 16)-( 19), but with the pressure conditions replaced by (25) and (26).This is the system that, eventually, we hope will provide some avenues for further investigation.However, in this initial phase of the work, we have a rather limited objective: to simplify but retain, as far as possible, the essential character of the nonlinear, three-dimensionality of the flow field, particularly as it might apply to the neighbourhood of the Pacific equator.To this end, we have made choices (all of which are allowed within our specification of the problem) as follows: sitting on planes parallel to the tangent plane; • constant pressure on the surface along y = 0.
It is hoped that, in future work, some or all of these restrictions can be relaxed and more elaborate solutions can be constructed.

VI. METHOD OF SOLUTION
We will now show that, for the reduced problems ( 27) and ( 28), the knowledge of the azimuthal velocity component along the equator, u(x, z), specifies the entire flow, which is at the heart of the novel approach that we have developed here.
At the non-dimensional longitude x, let us denote by ζ 0 (x) < 0 the depth at which u first vanishes below the surface; note that field data indicates that u(x, 0) < 0 all along the Pacific equator at the surface (see the discussion in Ref. 7), while u vanishes in the abyssal layer where there is no motion.Since φ(x, 0) = 0, from (20) we infer that Noting these preliminary considerations, the first stage in the construction of the general solution of the system described in Eqs. ( 27) and ( 28) is to combine the equations in (27), by eliminating u x , to give (29) which can be re-expressed as where Note that the restriction ζ > ζ 0 (x) avoids singularities in expression (31) but, as we show below, these apparent complications are irrelevant once we have developed the final form of the solution.Equation ( 30) is best treated by introducing a quasi-stream-function by setting to obtain (by integration with respect to the z-variable) where (34) Expressions for v and w, given ψ, are provided by (32) and (33), We have elected to define Î so that Î(x, y, 1 2 y 2 ) = 0; the addition of an arbitrary function of (x, y) in the form chosen for Î is unnecessary: any such term vanishes identically when the expressions for v and w are constructed.The equation for ψ is determined from the first equation in (27), when rewritten as It is clear from (36) that we will need to avoid the singularity where 2ω + u ζ = 0.In the non-rotation case (ω = 0), this is situated at any vertical face of the underlying velocity profile; in practice-for realistic profiles of interest-this will occur only at the maximum speed (which will be at some point below the surface).With rotation included, this point is shifted slightly (because ω is small) and so it is close to the maximum-speed position.Because such a point will exist for our profiles, the only way forward is to ensure that the velocity components generated by this expression (see below) remain finite as any zero of 2ω + u ζ is approached; we will discuss this aspect of the problem in Sec.VII.We now seek the solution of (36) in the form ψ(x, y, z) = Ψ x, y, z − 1 2 y 2 , which gives where The choice that we have made here ensures that v = 0 on y = 0 (as required by symmetry about the equator).We now introduce the expression for Ψ, from ( 37) and ( 38), into the two velocity components given in (35); we find that The previous considerations, with the underlying working hypothesis that the singularity where 2ω + u ζ = 0 is avoided, yield a solution for the velocity field in the fluid region In the fluid region ζ < ζ 0 (x), which includes the motionless abyssal layer, we cannot repeat the considerations that led to (39) and (40) since these rely on the assumption that u 0, but we can check directly that ( 39) and (40) solve both equations in (27); this is all that we require in order to proceed.It is clear, at this stage, that we have retained a significant element of nonlinearity in our reduced model: the two velocity components, v and w, are highly nonlinear functionals of the azimuthal velocity profile u(x, ζ).We note, however, that the y-dependence takes a surprisingly simple form here but, as we shall see, this is sufficient to produce some realistic three-dimensional flows.

VII. SOME GENERAL PROPERTIES OF THE SOLUTION
Our primary intention in the study of our reduced problem is to produce some detailed information about the structure of the velocity field described by ( 39) and (40), which satisfies the kinematic boundary conditions, (28), for a suitable choice for u and which also ensures the removal of the singularity where 2ω + u ζ = 0 (and, for future reference, we note that the root of this equation is neither at the surface nor, for ω 0, at the maximum of the u-component).First, the kinematic condition at the surface, (28), becomes simply = 0; thus, using (20), with φ(x, 0) = 0, we obtain and so the speed at the surface is constant, This is a general property of these flows and is independent of the rotational effects; however, as we shall see shortly, some related and similar properties of the flow field do depend on ω.
We now turn to the corresponding problem at the thermocline, given in (28); this can be written as Using (20), this becomes (u 2 + 4ω φ) = constant on the thermocline ζ = −t(x), conveniently expressed as since φ(x, 0) = 0 and (20) holds.This reduces to u(x, t(x)) = constant [cf.(41)] in the absence of any rotational contribution (i.e., ω = 0): the speed of the flow at the thermocline is then also a constant.Finally, we must remove the singularity that exists where 2ω + u ζ = 0; as we mentioned earlier, we will interpret this as requiring that the velocity components v and w be defined as 2ω + u ζ → 0. From ( 39) and (40), we see that we certainly must have Let the position of the singularity, i.e., the solution of 2ω + u ζ = 0, be at ζ = ζ(x); we therefore require the existence of two limits The second limit does not introduce an additional constraint if Indeed, (20) and L'Hospital's rule ensure that Phys.Fluids 29, 056604 (2017) The numerator cancels out the zero factor in the denominator; both required limits exist.Thus conditions (43) and ( 44) are sufficient to prevent the appearance of singularities.
There is one further observation that we make, which will enable us to obtain directly some important information about particular profiles.Consider the expression From Eq. ( 20), with (42) and ( 43), this can be written as where the prime denotes the derivative with respect to x; this is identically zero, and so Immediately we see that, in the case of ω = 0, the speed on the profile, at the point that moves along the line of the singularity, is constant [cf.(41) and the comment after (42)].We also note that because the position of the singularity is given by the solution of 2ω + u ζ = 0, the choice of ω = 0 puts this at the maximum of the velocity profile (as we mentioned earlier).Thus, if we neglect the effects of rotation (ω = 0), the speeds in the flow at the free surface, at the thermocline, and at the maximum of the profile are all constant in the equatorial direction.However, the terms associated with the Coriolis effect are important in some circumstances, as we now explain.We construct the difference of Eqs. ( 42) and (45), which gives First, this result embodies an important general property: there is an exact solution in which the thermocline sits precisely along the path of the singularity, for all ω, i.e., t(x) = − ζ(x).Of course, this can be the solution only if the initial datum-the position at the western end, for example-puts the thermocline at this level; this will rarely be the case.Further, for small ω and u 2 (x, −t(x)) − u 2 (x, ζ(x)) = O(1) as ω → 0, we see that the dominant behaviour for the position of the thermocline is independent of ω: again, the rotational effects are weak.Importantly, however, we shall demonstrate that some useful model profiles possess the property that u 2 (x, −t(x)) − u 2 (x, ζ(x)) = O(ω) as ω → 0, and then the position of the thermocline is controlled solely by the terms generated by the rotation of the Earth.Therefore we conclude that, in many cases, the rotational terms in the governing equations have little effect upon the properties of the flow in the azimuthal direction nor on its associated thermocline.There are situations, however, where the rotation is important in the determination of the evolution of the flow.

VIII. OVERVIEW OF THE EXAMPLES
The plan, now, is to construct some specific solutions, based on our observations above, by choosing appropriate velocity profiles for the flow below the surface.For this, we will use the general form of the profile of the Pacific EUC as a guide but, as we have already seen, any realistic velocity profile could be used here.The aim will be to consider choices that we can handle mathematically and which are consistent, by-and-large, with the types of flows that are observed.We will discuss two model profiles which are quite closely related, but we also make some additional comments about more general profiles, as appropriate.Both profiles have the same behaviour in the upper regions of the flow, i.e., at and above the maximum speed (which could be interpreted as the core of the EUC).In this upper region, the flow profile is represented by two polynomial terms (of prescribed degrees, chosen to be appropriate to this problem), but we will comment on the consequences of electing to use just one term in the polynomial representation.In the lower regions of the profile, below the maximum speed, we make two different choices.In one, the maximum of the profile is continued as a uniform flow down to a prescribed depth; what happens even further down is irrelevant at this stage (and we will not use this example for the construction of streamlines), but conditions will necessarily be stagnant at even greater depths in our model.The second example extends the polynomial form, describing the whole profile down to a zero in speed, and then stagnant (stationary) conditions below that.Not surprisingly, the first is technically far simpler, although physically less convincing; it does, however, possess some interesting features that will allow us to expand on the comments made in Sec.VII.The second is much closer to what we might expect to use as a model for flows such as the EUC, being both physically and mathematically more convincing.Of course, these can only be examples; nevertheless, they are intended to help us explain much of the underlying flow structure.We aim to show what is possible and then, in the future, different (and perhaps more complicated) choices for the background flow could be made: the solution that we have developed allows any choice of the underlying velocity profile.The calculations that we present below come in two stages.First, we introduce the u-profile and the consequences for the general properties of the flow field; this provides the basic description for the slow evolution of the profile and of the thermocline in the azimuthal direction.The second stage in the calculation takes the full polynomial profile, and its evolution, and uses this to generate some streamlines for the flow fairly close to the equator; these exhibit the three-dimensional nature of these flows as well as retaining elements of the inherent nonlinearity of the system.

IX. A POLYNOMIAL/UNIFORM PROFILE
The first example that we consider, and which will provide much of the detail for the second example (and so save on unnecessary repetition), is the flow which mimics that observed in the EUC, represented by Phys.Fluids 29, 056604 (2017) where we have used the simplest pair of rational powers (quadratic and quartic) which generate the type of profile that is relevant in this context.The "etc." refers to any appropriate flow below the uniform state, which is indicated in Fig. 3 by the heavy dashed line (and we do not need any detail for the current calculation).We assume that U > 0, γ > 0, δ > 0, and µ > λ > 0; the uniform maximum azimuthal flow below the surface sits in −λ(x) > ζ ≥ −µ(x); the singularity to be removed is necessarily above this, and we assume a model in which the thermocline is situated somewhere within this maximum azimuthal-flow region (and not on its boundary).
The surface kinematic condition, written in form (41), gives directly where V 0 > 0 is the constant (westward) speed at the surface.Correspondingly, at the thermocline [from ( 42)], we have The singularity that is to be removed [see ( 43)] occurs where 2ω + u ζ = 0, which gives the cubic equation It is sufficient for our purposes to use the unique real solution obtained by constructing an asymptotic representation in the form which confirms that the singularity is positioned just above the point of the maximum azimuthal speed and so sits in the quadratic-quartic part of the profile.(The expansions generated here based on ω → 0 are, with the physical conditions required for a realistic solution, all uniformly valid; we use these approximations merely as a device to simplify, and to make more transparent, the calculations.It is certainly possible, if deemed expedient, to use the exact solution by solving the cubic.)The constraints ( 43) and ( 44) that ensure the removal of the singularity hold since with as ω → 0, due to (49), while (50) and (51) yield u ζ ζ ∼ −2γ, where 2ω + u ζ = 0. Note that (46) then becomes (52) In summary, with the constants in ( 49) and ( 52) evaluated in terms of conditions at some station, e.g., at the western end of the equatorial Pacific, and labelled with the circumflex, we have (48) and where the first is exact and the second is an asymptotic approximation for small ω.We now combine Eqs. ( 53) and (54 which shows explicitly in this example the controlling influence played by the (small) rotation terms; this condition is absent if we set ω = 0 in the above equations: they are then identical (with U = Û).
The most straightforward and natural way to proceed is to seek asymptotic expansions in ω for each of the unknown functions; this we do and elect to retain the first two terms only in each of the expansions for U, γ, and δ. (Of course, we could extend this asymptotic approximation to as many terms as was thought appropriate.)Thus we write with Û = Û0 , γ = γ0 , δ = δ0 , and t = t0 given at, say, the westward end, which fixes the arbitrary constants.There are various ways to proceed; the simplest is to specify the path of the EUC (if that is what we wish to model), represented by the function λ(x), and so we write chosen independently of ω, with λ = λ at the westward end.Finally, the leading-order representation of the path of the thermocline, obtained from (55), is described by This result shows that the thermocline necessarily follows the evolution of the EUC, running parallel to ζ = −λ(x) and at a level determined by the initial datum.Further, we note that this property of the flow uncouples from the rest of the calculation (for the velocity field, streamlines, etc.) and requires only, as we know, the start conditions and the path of the maximum of the EUC (but this configuration is special to this type of velocity profile, as we will see later).The interpretation that we opt for here is prescribe λ(x), choose δ(x) ∼ δ 0 (x) + ω δ 1 (x), and then determine γ 0 (x) and γ 1 (x).However, we are not at liberty to assign δ 0 (x) and δ 1 (x) arbitrarily; although we clearly have families of solutions, there are constraints that must be satisfied.This involves a careful consideration of the v-component of the velocity field: for a physically acceptable solution, this has to correspond to the flow away from the equator near the surface-we are still guided by the properties of the EUC-and this leads to the construction of the relevant constraints.The details of this routine calculation, coupled with our choice of expansion in ω, are given in Appendix A; this produces the constraint The procedure is therefore to make choices that satisfy (58) and then derive corresponding expressions for γ 0 and δ 0 .However, we leave the selection of F(x) and k [in (58) and see Appendix A] to one side, for the moment; we shall return to this when we present more detailed examples, together with the associated streamlines.

X. A QUADRATIC-QUARTIC PROFILE
We now examine a more physically acceptable profile, based on the one described in Sec.IX: we allow the quadraticquartic structure to extend down to the second zero of this expression.(There is another zero, of course, nearer to the surface, where the switch from westwards to eastwards flow occurs; see Fig. 3.) Thus we write where Much of the calculation that we have already performed (in Sec.IX) is directly applicable here, so we may quote it.Indeed, the previous calculation was done, to some extent, in order to lay the foundations for this one, which we might regard as a more satisfactory choice of the profile that leads naturally to stagnant conditions deeper down.
The kinematic condition at the surface, (48), is unchanged, as is condition (52) for the removal of the singularity which is situated at the same position, (51).The condition at the thermocline, however, is rather more involved because the azimuthal flow is no longer uniform at its maximum; so, with θ = λ − t, from (42) we obtain Expanding in ω as before, and again regarding λ = λ(x) as prescribed, we find that (A2)-(A4), from Appendix A, hold.Finally, the leading-order representation (for small ω) of the path of the thermocline is obtained from (61) using (A3), which gives (for real solutions) (62) since U 0 = Û0 .The two possible solutions here relate to the thermocline being below/above the line of maximum azimuthal speed, ζ = − λ, although most data indicate that it is above (so our solution admits the alternative as an option).The conditions that define the observed type of flow at the surface (away from the equator) are also specified by (58) because there has been no change in the form of the solution near the surface.

XI. THE STREAMLINES
We now turn to an examination of the flow pattern generated according to the mathematical description given above; this requires the construction of the streamlines.The streamlines for the flow projected onto the (y, z)-plane-equivalently, the (y, ζ)-plane when viewed relative to the tangent plane-are defined by from ( 39) and ( 40).[The complete description of the threedimensional flows requires, in addition, the solution of (dx/dt)/(dz/dt) = u/w; however, this equation and (63) uncouple.We give a representation of the full 3D flow pattern in Fig. 12.]This differential equation, (63), is briefly discussed in Appendix B, where the case for taking ω → 0 is presented and some properties of the relevant solution are discussed; the result is that, for our purposes in this initial study of the flow patterns, it is sufficient to express the streamlines in the (y, ζ)-plane in the form In order to proceed, and apply the ideas and results that we have developed thus far, we will have to make some choices.We have already seen that the underlying velocity profile may be arbitrarily assigned; now, in addition, the solution allows the specification of the path of either the line of maximum azimuthal speed or that of the thermocline.We elect, not unreasonably, to follow a fairly simple route (but it is evident that many other-and more involved-choices are possible; these are left for subsequent investigation).First, we prescribe the path taken (eastwards) by the line of maximum speed below the surface; this is represented by the function λ(x).A simple choice that is in accord, more-or-less, with the observed properties of the EUC, for example, is to take this to be a straight line that rises to the east.Further, we define the scale already subsumed into the non-dimensionalisation so that the eastern end (in the equatorial direction) is represented by x = 1; also, we use a depth of this level as defining (most appropriately at the western end) the scale used in the z-direction.Thus we set where 0 ≤ Λ < 1 is a constant; and so the line of maximum speed rises from z = 1 to z = −(1 − Λ).This specification, (65), on the face of it, is a simple and reasonable choice but, if data suggest otherwise, this can be readily incorporated.[As an alternative, we also take λ(x) = 1−Λx 2 ; although the details are not pursued here, we will present one example with this choice, for comparison, later; see Fig. 7.] Finally, our solution permits some freedom in the choice of the constant k and the function F(x) that appear in the constraint, (58); see the comments in Appendix B.
First, we set to one side the question of the position of the thermocline (although we will include its behaviour in the full description of the solutions that we shall present).From (62), we note that the path followed by the thermocline, in the direction of the equator, uncouples from the determination of the streamlines.We may place it at any level in the flow field at the western end, and then its evolution is completely determined by the functions λ(x), γ 0 (x), and δ 0 (x).In addition, the line of the thermocline can, in exceptional circumstances, uncouple in a very dramatic fashion: there is a solution t(x) = − ζ(x), for all ω; see (46).
The primary issue that has driven this preliminary investigation is the following: what form does the streamline pattern take?It is the answer to this that will show if our approach has captured some important properties of these flows.Even a fairly superficial investigation of the properties of (64) [and see (B5)], as all the various parameters are changed, shows that the possible flow patterns are many and varied.With this in mind, we give a brief overview of what can happen and then provide a few explicit examples.In one sense, the picture is not too involved: depending on the choice of parameters, the flow exhibits either a simple downwelling to the zero state [which is on and below the line ζ = −µ(x); see (60)], and up and away from the equator near the surface, or there are three cells between the bottom and the top.The existence of a line on which, and below which, the flow is stationary means that we may have flows that either decelerate towards this level (downwelling) or accelerate away from it (upwelling).[On the other hand, the special choice given in (B10) leads to solutions which exhibit upwelling from this zero state and contain two cells, with downwelling immediately below the upper zero in the velocity profile.]In the case of three cells, the flow is downwelling in the lowest cell, as in the one-cell solution, but then there is a circulating flow away from, and returning to, the equator in the middle cell; in the cell at the surface, the flow is away from the equator near the surface (exactly as before).In all these solutions, the flow is always away from the zerospeed line near the surface.Indeed, when we neglect the terms in ω (which is the case that dominates our discussion in this context), from (39) and (40) we see that v = w = 0 wherever u = 0, but the streamlines are smooth across this line; this indicates some deeper structure.This surprising feature of the flow field is an example of a special dynamical behaviour: a slow centre manifold, which is briefly described in Appendix C.
So the solutions can have one or two or three cells, and no doubt more with different choices for F(x).However, there is an important-and intriguing-addition to this picture: for a considerable range of parameter values, the flow can transform from one to the other as it evolves in the azimuthal direction.In the cases that we have investigated, this transition, if it occurs, is always from the three-cell state near the western end to the simple, single-cell downwelling profile as the flow evolves eastwards: the middle cell collapses and eventually disappears altogether, typically after a short distance.Nevertheless, there are solutions which exhibit the three cells throughout the length of the equator, but for these the change in the depth of the maximum speed (and, correspondingly, of the thermocline) is far less than that observed (but still rising towards the east).In the case of two cells, we have found that these exist Phys.Fluids 29, 056604 (2017) FIG. 4. Example 1: (a) The functions γ 0 (x) (red) and δ 0 (x) (blue) against the distance in the azimuthal direction, from west (x = 0) to the east (x = 1).(b) The path of the maximum azimuthal speed at the core of the flow (green) and the path of the thermocline (magenta); the free surface is the x-axis, plotted from west (x = 0) to the east (x = 1).throughout the length of the flow (0 ≤ x ≤ 1) in the azimuthal direction; we will provide an example of a two-cell solution.
We undertook a numerical investigation of (64), i.e., of (B5), with all the parameters included; the details of the various choices that we made are given in Appendix D, and the references there are to the examples that we describe here.We now present some of the detailed flow patterns that can be obtained from our analysis.
The simplest type of flow that we found (example 1) is of a downwelling towards the stagnant state throughout the length of the flow.The (y, ζ)-streamlines change shape slightly as the flow evolves from x = 0 to x = 1, but all have the same overall structure.The flow properties of the quadratic-quartic velocity profile, (59), are shown in Figs.4(a) and 4(b).These two figures show some interesting and relevant properties of the flow.First, in Fig. 4(a), we see that both the quadratic and quartic terms have reasonable, and equitable, coefficients at the western end but, as the flow evolves, so the quadratic term dominates; indeed, the coefficient of the quartic term drops almost to zero at the eastern end.In Fig. 4(b), we show the evolution of the thermocline, given that the line of maximum speed evolves according to a linear rule.In this example, the thermocline moves down slightly, relative to the maximum line (but, of course, it still rises, from west to east, quite significantly).The corresponding streamlines, generated by (64) and presented for the mid-point position (x = 0.5), are shown in Fig. 5 (plotted relative to the tangent plane, and so the free surface is flat: ζ = 0).This pattern of streamlines and associated directions show the flow away from the equator near the surface, and the downwelling towards the stagnant conditions that exist on and below the lower zero-line.The full threedimensionality of the flow field, which is not clear via this plot, is evident when the complete velocity field is examined, i.e., the flow field as shown in the (y, ζ)-plane (Fig. 5) is combined with the flow associated with the underlying velocity profile, (59).We will provide a depiction of this three-dimensional flow later.
For our second example (Example 2), we present a flow that exhibits three cells near the western end of the equator; the flow properties for this example are shown in Figs.6(a) and 6(b).In this case, we also see that the quartic term becomes less important as the flow nears the eastern end (x = 1), but its presence is still appreciable.The interesting observation concerning the path of the thermocline is that there is a noticeable drop at a short distance from the western end, and thereafter it rises, more-or-less in line, with the evolution of the maximum azimuthal speed line.This property of the thermocline has been observed for the flow in the Pacific equatorial region; see Refs. 13 and 26.For comparison with Fig. 6(b), we produce the corresponding result with the choice λ(x) = 1 − Λx 2 and all parameter values for Û0 , V 0 , Λ the same; see Fig. 7(a) where the drop in the thermocline level for this case is more pronounced.The streamline pattern associated with Fig. 6, at the western end, is shown in Fig. 8; this exhibits a threecell structure.For the same flow conditions, and only a short distance along the equator (at about x = 0.02), a single-cell structure appears, which exists at the eastern end; the flow pattern at x = 0.5 is shown in Fig. 9.The apparent existence of the three-cell solution for only a short distance is rather misleading: the critical measure is the (local) depth of the maximum-speed line [z = −λ(x)].For example, if the maximum speed-line in the azimuthal direction evolves from almost uniform (flat) conditions-these being maintained for some considerable distance-before gradually rising to shallower Phys.Fluids 29, 056604 (2017) FIG. 6. Example 2: (a) The functions γ 0 (x) (red) and δ 0 (x) (blue) against the distance in the azimuthal direction, from west (x = 0) to the east (x = 1).(b) The path of the maximum azimuthal speed at the core of the flow (green) and the path of the thermocline (magenta); the free surface is the x-axis, plotted from west (x = 0) to the east (x = 1).

FIG. 7. (a)
The parabolic path of the maximum azimuthal velocity at the core of the flow (green) and the associated path of the thermocline (magenta), plotted from west (x = 0) to the east (x = 1).(b) Example 3: The path of the maximum azimuthal velocity at the of the flow (green) and the path of the thermocline (magenta) for the two-cell example; the free surface is the x-axis, plotted from west (x = 0) to the east (x = 1).depths towards the eastern end, then a three-cell streamline pattern can persist to considerable distances.
Finally, we present a version of one of our examples that depicts the three-dimensional nature of the flow field.The flows that we describe here are a combination of the velocity field in the (y, ζ)-plane, represented by the projection of the streamlines on this plane, and the u-component which is at right angles to this, flowing in the azimuthal direction.In Fig. 10, we show these two components of the flow FIG. 8. Example 2 at the western end (x = 0); the free surface in the meridional direction is in blue, the zero-speed lines in black, the vertical black line is equator, and the green line is the line of maximum azimuthal speed at the equator; the streamlines are in red and the arrows indicate the direction of the flow.
field side-by-side for the most complicated pattern that we have discussed here-three cells as shown in Fig. 8. (An alternative view of the three-dimensionality, focussing on the flow near the surface, will be presented later; see Fig. 12.) We complete our description of some solutions by presenting a two-cell version (example 3) based on the choice given in (B10).The resulting solution is depicted in Figs.  the marked drop in the level of the thermocline, before it rises to the east.Finally, this configuration-the two-cell structure-is maintained throughout the length of the flow in the azimuthal direction, although it slightly distorts from west to east (essentially because of the changing levels of the zero and maximum speed lines); the figure we present (Fig. 11) is flow pattern at the mid-point (x = 0.5).

XII. CONCLUSIONS AND DISCUSSION
This work has shown how a fairly simple asymptotic procedure can be applied to a quite complicated flow configuration, producing a set of accessible results.The approach that we have adopted has made use of just one overarching assumption, namely, that the flow in the azimuthal direction evolves slowly; the associated small parameter then provides the basis for the construction of the leading-order problem.(In the asymptotic expansions that underpin our analysis, there is no suggestion of any non-uniformities arising in the higherorder-neglected-terms, for solutions that correspond to the requirements of physically realisable flows: one-term asymptotic representations are therefore sufficient for this initial investigation.)Rather than a continuous density stratification, we assume the existence of a thermocline as an abrupt (but weak) transition of the density values from the near-surface layer to the abyssal layer; the essence of this phenomenon is dynamical, relying on the fact that the ocean motion has shorter time scales than the diffusion time scales that would ensure a smoother and more gradual transition. 33Nevertheless, there is a link to thermocline theory (recently surveyed by Huang 22 and Vallis 39 ) in that our approach includes the dynamical properties that the presence of the thermocline implies for the three-dimensional near-surface flow pattern.The most relevant features of this flow are the reversal of azimuthal currents (with a westward wind-driven surface current above the strong eastward EUC, whose core is more-or-less aligned with the thermocline), upwelling/downwelling processes coupled with a meridional convergence towards the EUC core, and a poleward divergence at the surface; see Figs. 1 and 11.Since the vanishing of the meridional component of the Coriolis force at the equator leads to the breakdown of the mid-latitude geostrophic balance between the Coriolis force and the meridional pressure gradient near the equator, we propose a new balance between the inertial terms.In our original formulation, the density change across the thermocline and the rotation terms generated by the β-plane approximation were both retained.However, we took the jump in density to be negligibly small since the effects on our solution are minute (although the same does not apply to the problem of waves propagating on this background system 7 ).Thus the development of the details took advantage of this to make the process more manageable, although the thermocline was retained in the analysis by virtue of its associated kinematic condition and the prescription of its position at one end.In addition, where it was appropriate, the effects of the rotation terms (when expressed using the relevant parameter) were also neglected, although the analysis that we have presented makes clear that this is not an essential manoeuvre; it is, however, certainly very useful.
The problem that we have described is thoroughly nonlinear, and three-dimensional (as Fig. 10 makes clear, and see Fig. 12), although the velocity field has a weak coupling in the sense that we may arbitrarily prescribe the u-component and then, separately, find the (v, w) components.It is this structure of the problem (which has arisen by virtue of the slow evolution in the x-direction) which is fundamental to the success of our approach.The problem is reduced to a manageable level, yet it contains considerable intrinsic complexity and, we submit, it also admits solutions that correspond, in their general form, to the types of flows that are observed.Further, one of the strengths of this formulation is the considerable freedom that it still affords us: there are many choices of velocity profile, parameter values, and evolution in the equatorial direction, Phys.Fluids 29, 056604 (2017) FIG.12. Three-dimensional sketch of the flow whose streamline projections (on the vertical plane at a fixed longitude) were drawn in Fig. 11.Top: vectorarrow representation of the velocity field near the surface on one side of the equator only, showing the flow reversal and the downwelling; the plane of the equator is the nearest surface of the box, on the left.(The scales on the axes have no specific physical interpretation.)Bottom: depiction of four representative particle paths in the northern hemisphere (above the zero-velocity surface, between the zero-velocity surface and the thermocline, along the thermocline, and below the thermocline), with a local coordinate system of half-axes indicating, at the starting point, the octant in which the subsequent particle displacement takes place.(The vertical flow profile along the equator and the flow along the free surface are illustrated in Fig. 1.) and these can be adjusted with the aim of producing something close to what is observed (in the Pacific, or other similar, but different, ocean flows).The few choices that we have made show that flows with one, two, or three cells are possible; we have not found solutions with more cells, but we have not undertaken a comprehensive hunt for the conditions that lead to a specific number of cells-this is left for the future: we are confident that there are choices that will lead to even more complex flow patterns.Nevertheless, the flow directions consistent with these solutions indicate that, with an even number of cells, the flow is necessarily upwelling from the zero state; on the other hand, an odd number of cells implies a downwelling towards the zero state.The conclusion is that any appropriate flow field, as indicated by the data for any particular EUC-like flow, can be modelled.
In more detail, this analysis has produced a number of intriguing properties of these flows, and this is just for the EUC model; it is to be expected that other choices will lead to different flow configurations.Let us deal first with the effects of rotation (and the role of the density change, in the further simplified model that we have examined, is relatively unimportant-at least, in the absence of waves).For any u-component profile that possesses just one point of maximum speed (which is the core of an EUC-like flow)-and, admittedly, this is likely to be the type of profile of most practical interest-the small parameter associated with the Coriolis terms can be used to simplify the calculation: rotational effects are negligible.However, if the profile has a region of uniform flow at the maximum speed, and the thermocline sits in this same region, then the path of the thermocline is determined by the (small) rotational terms; this, we believe, is an altogether new observation.Indeed, as a part of this aspect of the work, we also found that there is an exact solution, for all speeds of rotation, in which the thermocline sits at the point on the profile-any profile of the type described-which corresponds to the removal of the singularity in the expressions for the (v, w) velocity field.Such a point is necessarily above, and at a speed slightly less than, the maximum; the path of the thermocline is then everywhere parallel to the line of maximum speed and above it.The observations, generally, put the thermocline a little above the maximum, but being parallel excludes the possibility of a drop in the level of the thermocline near the western end; this is also observed (in the EUC, for example) and recovered in some of our other solutions.We must conclude, therefore, that this special solution is not likely to be relevant to the physically realisable flows, even though it is an attractive solution of our system.
The profiles of most interest are those that model a smooth behaviour from the westward flow near the surface to a faster eastward flow below the surface (a representative model of the EUC, for example) and then decrease to a zero state at deeper levels.We chose to work with a quadratic-quartic profileothers could have been selected but this was the simplest that satisfied all the requirements-and we confirmed (Appendix A) that a single-term polynomial will never accommodate the flow conditions at the surface; also, we must restrict the terms to those of even powers in order to model the profiles that interest us.The analysis that we have presented has given us the opportunity to make a number of transparent choices.Thus we can readily choose a path for the maximum-speed linethis can certainly model the data more accurately, if that is required-and then consider choices for F(x); the possibilities here, we submit, are certainly worthy of a more extensive study than is appropriate in this introduction to these ideas.One Phys.Fluids 29, 056604 (2017) consequence of our analysis is that there is necessarily a limit to what depth the maximum-speed line can rise to the east.From (B6) we may write 1 < λ(x) κ − x 0 f (x ) dx < 2 , then with λ(0) = 1 and writing λ(1) = λ (the value at the western end), we obtain and so for f > 0 we must have λ > 1/2: the maximum-speed line can never rise as far as z = 1/2.
We elected to examine the simplest case: F(x) = constant, but many others are possible; indeed, a case which involves a very complicated F(x) [given in (B10)] produces a simple expression for γ 0 (x).It is clear, however, that the procedure for finding the flow properties (that describe the evolution of the u-component), and then the projection of the streamlines on the (y, ζ)-plane, is surprisingly straightforward for such a complex, nonlinear, three-dimensional flow.Furthermore, the analysis has also highlighted the appearance of an intriguing and unlooked addition: the role, in the flow near the surface, of a slow centre manifold (an important concept in dynamical systems).This produces a flow which is necessarily away from the zero-velocity line, which sits a little below the surface, a behaviour that is evident in the streamline patterns for the flow near the surface, shown in Figs. 5 and 8-11.In order to show this property more clearly, we have produced a vector-arrow schematic of a typical velocity field near the surface; see the top of Fig. 12.We can see the strongly threedimensional nature of the flow in our model and its behaviour near the flow-reversal surface.In particular, this shows the change in the direction across the zero-velocity surface-the neighbourhood of the slow centre manifold-and the significant downwelling that is present (Fig. 12, top).The lower part of Fig. 12 is a sketch of the flow whose meridional and vertical components are depicted in Fig. 11.There are five specific regions of the near-surface flow, delimited by the free surface, by the tilted zero-velocity surface, and by the tilted thermocline, in which the particle paths present the following characteristics: 1. above the zero-velocity surface, a particle moves westwards and polewards, rising slightly (unless it is on the surface, in which case it remains on it); 2. all particles on the zero-velocity surface are stagnant; 3. a particle below the zero-velocity surface and above the thermocline moves eastwards, with its path inclined slightly upwards and polewards; 4. a particle on the thermocline is confined to this interface, moving eastwards along a path which rises slightly polewards; 5. a particle below the thermocline moves eastwards, in an upward direction, away from the equator-a motion that dies out with depth, so that in abyssal regions the water is stationary.
In summary, we have presented a procedure for the investigation of a class of three-dimensional flow structures, describing the details for the Pacific EUC.The analysis has shown that choices are available for the velocity profile in the vertical direction and for the path of the maximum flow speed (for example); the solution then provides all the other details of the flow, both in the azimuthal direction (the slow variation) and in a narrow meridional band.Thus similar flows, such as those encountered in the equatorial regions of the Atlantic Ocean and the Indian Ocean, can be analysed using these techniques.We have presented, we believe, an approach and a method that will shed some light on the mathematical structure, with significant physical consequences, on these types of nonlinear, three-dimensional flows that appear in our oceans.Of course, we must hope that future studies will examine our underlying, original system ( 16)-( 19), but avoiding the simplifications that we have invoked in this first analysis; even this can tell only a part of the story.What happens well below the surface, and at some distance from the equator, cannot be incorporated in our model.In the region below our lower zero-line, where we have assumed stagnant conditions, we can incorporate any flow (consistent with zero velocity on this line), such as a mix of upwelling and downwelling.On the other hand, the flow at reasonable distances from the equator is not so easily accommodated.The theoretical model that we have used implies that the streamlines away from the equator asymptote to surfaces parallel to the free surface: we have no mechanism for closing the cells in this direction-the validity is, after all, only in a neighbourhood of the equator.To do so would require a quite different analysis of the equations that will-we must expect-lead to a region where there is some transition from the EUC-type flow to the more general oceanic flows.At the moment, this appears to be quite beyond a systematic, asymptotic study of the governing equations.
In this paper, we have explored the influence of the Coriolis force on the dynamics of the equatorial background flow.Due to the intractability of the governing equations, we derived a simpler nonlinear three-dimensional model that is able to capture simultaneously the main observed features of the equatorial flow in the mid-Pacific (depth-dependent currents with flow reversal, poleward flow, upwelling, and downwelling).Since solutions of the model are qualitatively consistent with observations, there is no compelling justification for invoking more complex hypotheses than the underlying assumption of slow variations in the azimuthal direction.The availability of a suitable background flow opens up promising perspectives for the investigation of the effects on wave propagation.In particular, the retention of non-traditional parts of the Coriolis force will modify the characteristics of the internal waves, thus conditioning the instabilities of the background flows. 2 The study of wave-current interactions in the context of non-traditional corrections is technically challenging (e.g., due to the non-separability of the linear barotropic stability problem 16 ) but represents a worthwhile endeavour: first steps, undertaken recently for long-wave zonal perturbations in the f -plane approximation, 37 reveal a strikingly different structure of the unstable modes, as compared to those within the framework of the traditional approximation.All of the above indicate that there are many avenues for future investigation.
are three cells at the western end (x = 0), but these disappear at about x = 0.12, and thereafter the flow is a single-cell downwelling; for most reasonable parameter ranges that allow three cells, the collapse to one cell occurs quite early in the evolution in the azimuthal direction.
Example 1: Û0 = 2 and V 0 = 1, with Λ = 0.4, a = 3, and b = 0.1; the thermocline is set at z = 0.85 at the western end (x = 0), i.e., a little above the maximum azimuthal speed of the flow (which is at z = 1 at x = 0).

FIG. 3 .
FIG. 3. (a)Sketch of a quadraticquartic profile from the surface down to the maximum, with a uniform flow below that; the return to zero conditions need not be specified (and is indicated here by the heavy dashed line).(b) An example of a quadratic-quartic profile, with zero conditions deeper down.

FIG. 5 .
FIG.5.Streamlines for example 1 at the mid-point, x = 0.5; the free surface in the meridional direction is in blue, the zero-speed lines in black, the vertical black line is the equator, and the line of maximum speed on the velocity profile at the equator is marked in green; the streamlines are in red and the arrows indicate the direction of the flow.Note that the flow is three-dimensional, i.e., the streamlines are composed of what we draw in the figure together with the flow in the azimuthal direction: the streamlines of the flow field are generated by the two flows when combined; this figure depicts the projection of the streamlines onto the (y, ζ ) plane.
7(b) and 11 (both of which may be compared with the corresponding examples shown earlier).
FIG. 9. Example 2: Streamlines at the mid-point (x = 0.5); the free surface in the meridional direction is in blue, the zero-speed lines in black, the vertical black line is the equator, and the green line is the line of maximum azimuthal speed at the equator; the streamlines are in red and the arrows indicate the direction of the flow.