Cross-Equatorial Flow Through an Abyssal Channel Under The

Home , Antarctic Bottom Water, Continental rise, Weddell Plain

Cross-equatorial ﬂow through an abyssal channel under the complete Coriolis force: two-dimensional solutions

A. L. Stewart and P. J. Dellar OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB

Abstract The component of the Coriolis force due to the locally horizontal component of the Earth’s rotation vector is commonly neglected, under the so-called traditional approximation. We investigate the role of this “non-traditional” component of the Coriolis force in cross-equatorial flow of abyssal ocean currents. We focus on the Antarctic Bottom Water (AABW), which crosses from the southern to the northern hemisphere through the Ceara abyssal plain in the western Atlantic ocean. The bathymetry in this region resembles a northwestward channel, connecting the Brazil Basin in the south to the Guyana Basin in the north. South of the equator, the AABW leans against the western continental rise, consistent with a northward flow in approximately geostrophic balance. The AABW then crosses to the other side of the abyssal channel as it crosses the equator, and flows into the northern hemisphere leaning towards the east against the Mid-Atlantic Ridge. The non-traditional component of the Coriolis force is strongest close to the equator. The traditional component vanishes at the equator, being proportional to the locally vertical component of the Earth’s rotation vector. The weak stratification of the abyssal ocean, and subsequent small internal deformation radius, defined a relatively short characteristic horizontal lengthscale that tends to make non-traditional effects more prominent. Additionally, the steep gradients of the channel bathymetry induce large vertical velocities, which are linked to zonal accelerations by the non-traditional component of the Coriolis force. We therefore expect non-traditional effects to play a substantial role in cross-equatorial transport of the AABW. We present steady asymptotic solutions for shallow water flow through an idealised abyssal channel, oriented at an oblique angle to the equator. The current enters from the south, leaning up against the western side of the channel in approximate geostrophic balance, and crosses the channel as it crosses the equator. The “non-traditional” contribution to the planetary angular momentum must be balanced by stronger westward flow in the channel, which leads to an increased transport in a northwestward channel, and a reduced transport in a northeastward channel. Our results suggest that as much as 10–30% of the cross-equatorial flow of the AABW may be attributed to the non-traditional component of the Coriolis force. Keywords: Antarctic Bottom Water, abyssal currents, equatorial currents, complete Coriolis force, traditional approximation, shallow water equations

1. Introduction tion to the roleˆ of the so-called “non-traditional” part of the Coriolis force, the part due to the locally horizontal component Abyssal ocean currents, being denser than their surround- of the Earth’s rotation vector that is neglected under the tradi- ing fluid, are largely driven by gravity and steered by tional approximation. their local bathymetry. Over large scales they are also We focus on the Antarctic Bottom Water (AABW), an strongly influenced by the Coriolis force, which tends to abyssal current that originates in the Weddell Sea near Antarc- push northward/southward-flowing currents against the west- tica, where meltwater flows down the Antarctic continental ern/eastern boundary in the southern hemisphere, and vice rise and northward into the Atlantic ocean. As the AABW versa in the northern hemisphere. The dynamics of these flows northward toward the equator, it mixes with the overly- currents becomes complicated close to the equator, where ing Lower North Atlantic Deep Water (LNADW), so its north- geostrophic balance breaks down as the locally vertical com- ward transport decreases with latitude. McCartney and Curry ponent of the Earth’s rotation vector changes sign. Direct mea- (1993) reported a northward transport of 7.0 Sv at 30◦S (1 Sv = 6 3 1 surement of abyssal equatorial currents requires flowmeters to 10 m s− ), dropping to 5.5 Sv at 11◦S. They give a geostrophic be set up at depths of thousands of metres, and years of mea- estimate of 4.3 Sv for the cross-equatorial transport, but later surement may be necessary to capture the slow variation of the measurements by Hall et al. (1994), Hall et al. (1997), and flow (e.g. Hall et al., 1997). In this paper we investigate how Rhein et al. (1998) place this figure closer to 2 Sv. In Figure 1 an obliquely-orientated channel might enhance or inhibit cross- we plot the large-scale bathymetry around the Ceara abyssal equatorial flow of an abyssal current. We pay particular atten- plain, where the AABW crosses the equator. We highlight

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Figure 1: Contours of the bathymetry in the region where the AABW crosses the equator. The data has been taken from Amante and Eakins (2009), interpolated onto a grid of 331 by 181 points, and smoothed via six applications of a nine-gridpoint average (c.f Choboter and Swaters, 2004) to remove small-scale features and highlight the large-scale structure. The dotted lines highlight the northwesterly cross-equatorial channel, whose average proﬁle is shown in Figure 2.

the west-northwesterly channel formed between the continental ing vertical wall to the east, by switching from the western to rise to the southwest and the Mid-Atlantic Ridge to the north- the eastern wall of the channel as it crossed the equator. The east. In Figure 2 we plot the averaged profile of this channel, numerical results of Nof and Borisov (1998) corroborate this illustrating its characteristically steep sides and relatively flat finding, showing that a portion of the current should flow back middle. The AABW approaches the equator as a deep western across the equator when the channel is wider than the defor- boundary current, but a large portion of it follows the bathymet- mation radius. Stephens and Marshall (2000) represented the ric contours eastward, exiting the region shown in Figure 1 at AABW using the frictional-geostrophic shallow water equa- the eastern edge between 1◦S and 2◦S. This explains the sub- tions, which they integrated numerically over almost the en- stantial drop in transport between 11◦S and 0◦. tire Atlantic ocean. Despite neglecting the advection terms that Nof and Borisov (1998) found to be necessary for an accurate From the perspective of inviscid, ideal fluid dynamics, it is representation of the AABW’s equatorial dynamics, they ob- surprising that any current is able to travel far across the equa- tained a reasonable estimate for the cross-equatorial transport in tor. As the fluid crosses the equator, the locally vertical com- the western Atlantic. Choboter and Swaters (2000, 2004) component of the rotation vector changes sign. In order to conserve pared the frictional-geostrophic and shallow water equations in potential vorticity, the fluid must acquire an increasingly large numerical simulations of the AABW through idealised and re- relative vorticity as it moves further north. Killworth (1991) alistic topography. They found that it was necessary to include showed that this generation of relative vorticity prevents fluid advection to reproduce the cross-equatorial transport measure- moving more than approximately two deformation radii beyond ments of Hall et al. (1997). the equator via geostrophic adjustment. The potential vorticity must therefore be modified by dissipative processes to permit All of the studies mentioned above employ the traditional ap- cross-equatorial flow (e.g. Edwards and Pedlosky, 1998). proximation that neglects contributions from the Coriolis force However, conservation of potential vorticity remains a strong due to the locally horizontal component of the Earth’s rotation constraint on a cross-equatorial current, and a series of in- vector. We refer to these neglected contributions as the “non- vestigations have attempted to explain why such a large por- traditional” component of the Coriolis force. We expect that tion of the AABW is able to reach the northern hemisphere. non-traditional effects associated with this component should Nof (1990) first highlighted the importance of the channel- be particularly pronounced at the equator, because the horizon- like equatorial bathymetry using a reduced-gravity shallow wa- tal component of the rotation vector is Ω cos φ at latitude φ, and ter model, showing that a northward-flowing western bound- so largest at the equator. Conversely, the vertical component ary current with an unrestricted eastern front would be blocked Ω sin φ of the rotation vector vanishes at the equator. This is by the equator. Nof and Olson (1993) then showed that the compounded by the weak stratification of the abyssal ocean, so current could cross the equator in the presence of an oppos- the Brunt–V¨ ais¨ al¨ a¨ or buoyancy frequency N becomes closer to 2 the inertial frequency Ω, thus circumventing the argument that Ω the traditional approximation is valid when N. For exam- −3900 ple, van Haren and Millot (2004) highlighted dramatic≪ changes in the near-inertial oscillation of the deep Mediterranean, where −4000 the inertial and buoyancy frequencies are comparable. Addi- tionally, the large variations in the abyssal bathymetry, shown in −4100 Figures 1 and 2, may induce large vertical velocities, which in turn induce zonal accelerations via the non-traditional component of the Coriolis force. We therefore expect non-traditional Depth (m) −4200 effects to play a prominent role in the dynamics of the AABW close to the equator. −4300 Gerkema et al. (2008) recently reviewed the role of the non- traditional component of the Coriolis force in geophysical and −4400 astrophysical fluid dynamics. Most of the previous literature focuses on linear phenomena, especially internal waves, with 0 50 100 150 200 250 300 Cross−channel distance (km) little attention paid to cross-equatorial flows. An exception is Raymond (2000), who showed that the non-traditional com- Figure 2: Profile of the west-northwesterly bathymetric channel highlighted in ponent of the Coriolis force substantially affects atmospheric Figure 1, averaged from northwest to southeast. The average has been taken cross-equatorial flow described by the two-dimensional, linear, over the full bathymetric data of Amante and Eakins (2009), rather than the inviscid governing equations on an equatorial β-plane. Hua smoothed bathymetry plotted in Figure 1. et al. (1997) showed that the potential vorticity of equator- crossing fluid must be zero, accounting for the complete Cori- olis force, or it will be subject to inertial instability. Measure- equator may only be realised when zonal variations are permit- ments by Firing (1987) indicate that deep water tends to adhere ted. In this work we investigate the cross-equatorial flow of an to a state of zero potential vorticity within approximately 3◦ of abyssal current using asymptotic solutions of the shallow water the equator, and Hua et al. (1997) conclude that this should fa- equations in an idealised channel, orientated at an oblique angle cilitate angular momentum exchanges between hemispheres. to the equator. The structure of this paper is as follows. In 2 Dellar and Salmon (2005) derived shallow water equations we briefly discuss the shallow water equations with a complet§e that include both the traditional and non-traditional components representation of the Coriolis force. In 3 we present steady of the Coriolis force, and Stewart and Dellar (2010) extended asymptotic solutions for steady flow of a§ long, narrow current these equations to include multiple layers. These equations will through a flat-bottomed channel with vertical walls, directed be discussed in detail in 2, and serve as the basis for all of § at an arbitrary angle to the equator. These solutions provide a the analysis in this paper. Stewart and Dellar (2011) used these useful description of cross-equatorial flow through an abyssal equations to study cross-equatorial flow of a steady, zonally- channel, but their validity is dubious in the case of an almost- symmetric abyssal current across a zonal channel. They dis- westward channel like the one shown in Figure 1. In 4 we cussed the contribution of the non-traditional component of modify our asymptotic scaling for the case of an almost-zona§ l the Coriolis force to the shallow water potential vorticity, and channel. This modified scaling permits an arbitrary channel showed that in a zonal equatorial channel this contribution may profile, improving on the idealised square channel of 3. Fi- partially balance the change in sign of the traditional Coriolis nally, in 5 we discuss our results and the outlook for§ future parameter, f , as fluid crosses the equator. Equivalently, fluid research in§ this area. crossing the equator through a zonal channel requires a smaller change in its zonal angular momentum, as including the complete Coriolis force accounts for changes in the angular mo- 2. Shallow water equations on a “non-traditional” β-plane mentum with the depth of the fluid. Analytical and numerical solutions for zonally symmetric flow confirm this analysis, and We model the cross-equatorial flow of abyssal currents using show that a larger cross-equatorial transport is possible when the shallow water equations, which describe the behaviour of a the complete Coriolis force is included. Including dissipation, homogeneous layer of fluid whose depth is small compared to which allows modification of the potential vorticity, further re- its breadth. The assumption of columnar motion that underlies duces the relative vorticity that the fluid must acquire to cross the shallow water equations permits a two-dimensional treat- the equator. Stewart and Dellar (2011) also considered a one- ment of three-dimensional dynamics, so they serve as a use- dimensional unsteady cross-equatorial geostrophic adjustment ful conceptual description of complicated oceanographic phe- problem in the absence of topography, and showed that non- nomena. This has motivated the use of shallow water equa- traditional effects increased both the mass flux across the equations in several previous analytical (Nof, 1990; Nof and Ol- tor and the furthest distance reached. son, 1993) and numerical (Nof and Borisov, 1998; Stephens The work of Stewart and Dellar (2011) is restricted to and Marshall, 2000; Choboter and Swaters, 2000, 2004) inves- zonally-symmetric solutions, yet the dynamical consequences tigations of cross-equatorial currents. 1 of the fluid acquiring a large relative vorticity as it crosses the In Figure 3 we present a schematic of the “1 2 -layer” shallow 3 water equations, which we use to model the flow of an abyssal most of the Earth’s surface. The exception is close to the equa- current. An active lower layer with depth-averaged horizon- tor, where the axis of rotation is almost perpendicular to the tal velocity u(x, y, t), thickness h(x, y, t) and constant density locally vertical axis. We will assume that the fluid moves in ρ, flows beneath a quiescent upper layer with a much smaller columns aligned with the locally vertical axis, even at the equa- depth-averaged horizontal velocity u u, much greater thick- tor, as this is consistent with the assumption of a small aspect 0 ≪ ness h0 h, and slightly smaller density ρ ∆ρ. The upper ratio, and because this description compares more favourably layer is≫ assumed to react passively to motions− of the internal with measurements of the AABW (e.g. Hall et al., 1997). Ad- surface z = η(x, y, t), such that the lower layer is governed ditionally, previous studies of abyssal equator crossing currents approximately by the one-layer shallow water equations with (e.g. Nof and Borisov, 1998; Choboter and Swaters, 2004) sug- the gravitational acceleration g replaced by the reduced gravity gest that inertia is an essential element in this process, so one g′∆ρ/ρ. The active layer flows directly over a prescribed bot- would not expect the fluid to move in Taylor columns. tom topography z = hb(x, y). Throughout this work we refer to Dellar and Salmon (2005) derived a set of non-traditional the actual measured height of the ocean bed as ‘bathymetry’, shallow water equations that include a complete representation and to the model ocean bed as ‘topography’. of the Coriolis force, both traditional and non-traditional parts. The coordinates x and y are defined relative to the standard Stewart and Dellar (2010) extended this derivation to the case axes X, Y, Z on an equatorial β-plane, in which X is directed of arbitrarily many layers. These non-traditional shallow water eastwards, Y northwards, and Z upwards. The “non-traditional” equations may be obtained by averaging the three-dimensional equatorial β-plane (Grimshaw, 1975) is characterised by ΩX = Euler equations over each layer, or by averaging the action 1 0, ΩY = Ω (constant), and ΩZ = 2 βY, where β = 2Ω/RE, and in Hamilton’s variational principle formulation of mechanics. RE is the Earth’s radius. These expressions correspond to first- Both derivations rely on the assumptions of a small aspect ra- order Taylor expansions of ΩY = Ω cos φ and ΩZ = Ω sin φ tio, and of columnar motion aligned with the locally vertical 1 for small φ. Dellar (2011) re-derived the non-traditional equa- axis, as detailed above. The 1 2 -layer shallow water equations torial β-plane equations by making approximations in Hamil- on the non-traditional equatorial β-plane may be written as ton’s variational principle that describes ideal fluid motions in ∂u spherical geometry. This derivation from Hamilton’s princi- Ω 1 + (u )u + βY 2 H hb + 2 h zˆ u ple ensures that the approximate equations inherit the expected ∂t ∇ − ∇ × conservation laws for energy, angular momentum, and potential ∂h + ΩH zˆ + g′(hb + h) + h(Ωxv Ωyu) = 0, (1) vorticity. × ∂t ∇ − We will use the 1 1 -layer shallow water equations on the non- ∂h 2 + (hu) = 0. (2) traditional equatorial β-plane to describe the flow of an abyssal ∂t ∇ current through an obliquely-oriented channel, so it is useful to ∂/∂ ,∂/∂ allow for arbitrary orientations of the horizontal axes. We de- Here ( x y) is the two-dimensional gradient oper- ∇ ≡ ˆ fine θ as the positive angle from North to the y-axis, so θ = 0 ator, g′ is the reduced gravity, and z is the unit vector in the corresponds to conventional GFD axes. Throughout this pa- z-direction. The non-traditional terms are proportional to the Ω = Ω , Ω per we choose θ [ π/2, π/2], so the x-axis points across the horizontal components H ( x y) of the rotation vector. channel and the y∈-axis− points along the channel. The vertical component of the rotation vector on an equatorial β-plane is Ω = 1 βY, where Y = y cos θ + x sin θ is the merid- At the core of shallow water theory is the assumption that z 2 the fluid layer moves in columns aligned with the locally ver- ional distance from the equator, as expressed in arbitrarily ori- tical axis (e.g Pedlosky, 1987), justified by the assumption that ented x and y axes. The non-traditional component of the Cori- the vertical lengthscale is small compared to the correspond- olis force connects the vertical and zonal velocities in the full three-dimensional equations. The Ω zˆ ∂ h term corresponds ing horizontal length scale. The pressure is then predominantly H × t hydrostatic, and the horizontal pressure gradient predominantly to the non-traditional zonal acceleration due to vertical veloc- independent of depth. We say “predominantly” because we re- ity, and the non-traditional contribution to the pressure gradient lax both these assumptions to derive the non-traditional shallow corresponds to non-traditional vertical acceleration due to zonal water equations given below. velocity. The inclusion of this additional term in the vertical The Taylor–Proudman theorem states that at low Rossby momentum equation defines what White and Bromley (1995) number, an inviscid fluid should form columns aligned with the call quasi-hydrostatic balance, the non-traditional extension of axis of rotation. This axis coincides with the locally vertical the hydrostatic balance between the vertical pressure gradient axis under the traditional approximation. If the complete Cori- and buoyancy. olis force is accounted for, the axis of rotation only coincides The most important non-traditional term in (1) is the correc- β with the locally vertical axis at the poles, so away from the poles tion to the traditional planetary vorticity Y. The same correc- there is a discrepancy between the alignments of the columns as tion appears in the modified potential vorticity described by shallow water theory and by the Taylor-Proudman 1 Ω 1 ∂v ∂u theorem. However, shallow water theory only requires that the q = βY 2 H hb + h + (3) h − ∇ 2 ∂x − ∂y layer depth be much smaller than the horizontal lengthscale, which is typically smaller than the planetary radius, so there is that is materially conserved (Dq/Dt = 0) by the non-traditional geometrically little difference between the two alignments over shallow water equations. This expression for q may be obtained 4 vorticity may partially balance the βY term, and so may facilitate cross-equatorial flow. A cross-equatorial current would require almost no change in ζ, irrespective of its path across the equator, if the planetary vorticity were everywhere constant, ζp = ζp0. Substituting this into (4) yields

2 ζp0 Y ζp = 2Ω (z z) = ζp0 z = z0(X, t) Y + , (7) ∇ − ⇐⇒ − 2Ω 2RE so the mid-surface of the layer follows a meridional parabola. 1 Figure 3: A schematic of the 1 2 -layer shallow water model for an active fluid In the simplest case with ζp0 = 0 and z0 = constant, the left- layer flowing beneath a deep, quiescent upper layer of less dense fluid. hand equality in (7) reads 2Ω (z z) = 0, which states that the mid-surface of the layer is everywhere∇ − parallel to the rotation vector. The right hand equality may then be rewritten as R = by replacing the traditional planetary vorticity with the com- RE + z0, which states that the midsurface remains at a constant plete planetary vorticity distance from the rotation axis. Stewart and Dellar (2011) observed that the shape of the Ω 1 Ω ζp = βY 2 H (hb + 2 h) = 2 (z z) , (4) equatorial channel shown in Figure 1 is remarkably similar to − ∇ ∇ − the meridional parabola described by the right hand side of (7), 1 where z(x, y, t) = hb + 2 h is the mid-surface of the layer, or the and concluded that the non-traditional component of the Cori- average of z across the layer. The complete planetary vorticity olis force should thus facilitate the flow of the AABW across ζp is thus the component of 2Ω normal to the mid-surface. This this channel. They showed that including the complete Cori- is more easily understood in terms of angular momentum u˜, olis force increases the transport of a zonally-symmetric cur- with components rent crossing a steep-sided zonally-symmetric equatorial channel, even when potential vorticity modification is introduced u˜ = u + 2ΩR cos θ,v ˜ = v 2ΩR sin θ, (5) − via bottom friction. However, this analysis omits zonal variations in both the idealised model topography and the fluid mo- from which we obtain q = ( u˜)/h. Here R is the mean tion. In this paper we redress this shortcoming by considering perpendicular distance of the fluid∇ × column from the axis of ro- two-dimensional shallow water solutions for an abyssal current tation. This distance is (R +z ) cos φ in spherical geometry, and E traversing an obliquely-oriented cross-equatorial channel. the consistent equatorial β-plane approximation is

2 1 Y 3. Flow through an equatorial channel with vertical walls R = z + RE 1 2 , (6) − RE   In this section we obtain asymptotic solutions for steady flow     of a current, represented by the 1 1 -layer shallow water equa- where again β = 2Ω/RE, and φ = Y/RE is the latitude. This 2 4 2 4 3 tions, through a channel that crosses the equator. We approxi- approximation is accurate to (φ ,φ z/RE) (φ ,φ z/Y), O ≡ O mate the bathymetry shown in Figure 2 by a channel with paral- where z Y RE for typical values of z and Y. The approxi- ≪ ≪ lel vertical walls and a flat bottom, directed at an angle θ to the mation is thus consistent in the distinguished limit z/Y Y/RE, as shown formally by Dellar (2011). Moreover, the potential∼ equator. A schematic of this channel is shown in Figure 4. The vorticity obtained from (5) and (6) coincides with the materi- method of solution is similar to that of Nof and Olson (1993) ally conserved potential vorticity for the non-traditional shal- for a meridional channel, but it is complicated by the inclusion low water equations, as obtained from the particle relabelling of the complete Coriolis force and the arbitrary orientation of symmetry (Dellar and Salmon, 2005). Making the traditional the channel. We show that the current tends to cross the chan- approximation corresponds to neglecting the dependence of R nel as it crosses the equator, and that the transport through a on z, thereby neglecting contributions to the angular momentum westward channel is larger when the complete Coriolis force is due to changes in the average perpendicular distance between accounted for. fluid parcels in a column and the rotation axis. As discussed in 1, inviscid fluid crossing the equator must 3.1. Governing equations conserve the potential§ vorticity given in equation (3). The As illustrated in Figure 4, we align the y-axis along the chan- relative vorticity ζ = ∂xv ∂yu is typically smaller in mag- nel and the x-axis across the channel, such that the equator is nitude than the vertical component− of the planetary vorticity described by the line y cos θ + x sin θ = 0. Without loss of gen- βY, so as βY changes sign over the equator, the fluid can only erality we position the walls of the channel at x = W, and ± conserve potential vorticity by acquiring a large relative vor- let the current enter from the south at y = yu, the prescribed ticity ζ. This prevents fluid penetrating far into the opposite upstream position. Here it is bounded to the− west by the wall hemisphere (Killworth, 1991) unless dissipative processes are of the channel at x = W, and has an eastern front described present to modify the potential vorticity (Edwards and Ped- by x = R(y). We need− only prescribe the upstream width of losky, 1998). The non-traditional contribution to the potential the current, R( y ) = R , and the upstream depth at the wall, − u u 5 We seek a solution of the form shown in Figure 4, requiring that there be no flow normal to the boundary x = W, and that the fluid depth vanishes at the front x = R(y), −

h = 0 at x = R(y), u = 0 at x = W. (11) −

At the upstream edge, y = yu, we prescribe the position of the front and the depth of the ﬂuid− at the wall,

R = R at y = y , h = H at x = W, y = y . (12) u − u u − − u As we will show in 3.3, the boundary conditions (11) and the upstream conditions§ (12) completely define the asymptotic solution. The governing equations (8a)–(8b) and boundary conditions (11)–(12) are difficult to solve in their present form. However, (a) we can make analytical progress by assuming that the potential vorticity is zero throughout the fluid. Hua et al. (1997) showed that fluid is subject to inertial instability if its potential vorticity and vertical planetary vorticity have opposite signs, QY < 0, where Q is the three-dimensional potential vorticity with complete Coriolis force and Y is the meridional coordinate. Thus the potential vorticity tends to homogenise to zero within around 3◦ of the equator. The measurements of Fir- ing (1987) corroborate this finding when the complete Coriolis force is accounted for. It is therefore quite plausible that an abyssal cross-equatorial current should approach a state of zero potential vorticity close to the equator. Setting q = 0 immediately yields

∂h ∂h ∂v ∂u (b) β(y cos θ+x sin θ) Ω sin θ + cos θ + = 0, (13) − ∂x ∂y ∂x− ∂y Figure 4: Schematic of the asymptotic channel-crossing solution presented in 3, viewed (a) from above, and (b) along the channel. In a typical solution the and substituting q = 0 into (8a) leads to a Bernoulli-type equa- §current enters at the southern edge leaning up against the western wall, crosses tion the channel as it crosses the equator, and exits at the northern edge leaning against the eastern wall. 1 2 1 2 u + v + g′h + Ωh (v sin θ u cos θ) = 0. (14) ∇ 2 2 − Finally we rewrite the mass conservation equation (8b) as h( W, y ) = H , to define the solution completely, as we will − − u u ∂ ∂ show in 3.3. In the northern hemisphere we assume that the (hu) + (hv) = 0, (15) current has§ a western front at x = L(y), and is bounded to the ∂x ∂y east by the wall at x = W. This is motivated by the form of the such that (13)–(15) forms our set of exact two-dimensional gov- solution in y < 0, which crosses the channel as y 0. erning equations. 1 → We seek a steady solution of the 1 2 -layer shallow water equations (1) and (2) with flat bottom topography, hb 0. This 3.2. Asymptotic expansion allows us to rewrite the governing equations in the form≡ We solve (13)–(15) via an asymptotic expansion in a small hq zˆ u + Φ = 0, (8a) parameter ε, to be defined. We first define the fundamental ve- × ∇ locity and length scales of the problem as the internal gravity (hu) = 0, (8b) ∇ wave speed and internal radius of deformation, respectively, where q is the potential vorticity, c c = g′Hu, Rd = . (16) 1 ∂v ∂u β q = β(y cos θ + x sin θ) Ω h + , (9) h − ∇ ∂x − ∂y Using the equatorial planetary radius, RE 6400 km, and 5 ≈ 1 and the Montgomery potential Φ is given by Earth’s rate of rotation Ω 7.3 10− rads− , we obtain 11 1 1≈ × β 2.3 10− rad m− s− . Then using typical values of 1 2 1 2 ≈ 3× 2 1 Φ = u + v + g′h + h(Ω v Ω u). (10) g = 10 ms and H = 500 m, we obtain c 0.71ms 2 2 x − y ′ − − u ≈ − 6 and Rd 180 km. We assume that the appropriate dynamical consider only the leading order solution, which is equivalent to lengthscale≈ in the y-direction is the upstream distance y , and simply neglecting all terms of (ε) in (19) and (20). u O that this is much larger than Rd. We define the small parameter We will henceforth drop the hat ˆ notation for dimensionless ε as quantities, and instead use a star ⋆ to denote dimensional quan- 2 Rd tities. At leading order in ε, the governing equations (19), (20), ε = 1. (17) y ≪ and (15) reduce to u The channel indicated by the dashed lines in Figure 1 has an ∂h ∂v approximate total length of 1130 km, and lies more south of the y cos θ δ sin θ + = 0, (22a) − ∂x ∂x equator than north, so here yu 980 km and ε 0.03. How- ≈ ≈ 1 v2 + h + δ h v sin θ = 0, (22b) ever, the plotted channel may not be the optimal representation ∇ 2 of the bathymetric channel, so perhaps a less conservative es- ∂ ∂ (hu) + (hv) = 0, (22c) timate for yu is the half-length of the channel, yu 560 km, ∂x ∂y which yields ε 0.1. In the analysis below we neglect≈ (ε) ≈ O contributions to equations (13) and (14), even though these con- subject to the following dimensionless boundary conditions, tributions may only be an order of magnitude smaller than the leading-order contributions in the AABW. However, we will u = 0 at x = W, (23a) − show in 3.8 that the solution does not change dramatically h = 0 at x = R(y), (23b) when we§ include (ε) terms. O 1/2 h = 1 at x = W, y = 1, (23c) We scale the y-coordinate by yu = ε− Rd, equivalent to as- − − R = R at y = 1, (23d) suming slow variations of the solution in the y direction. We u − simultaneously assume rapid variations in the x direction, scal- where W = W⋆/εR and R = R⋆/εR are the dimensionless ing the x-coordinate by ε1/2R = R2/y . We scale the along- d d d d u positions of the wall and the front respectively. channel velocity by the internal gravity wave speed c, so we must scale the across-channel velocity by εc to ensure that the mass conservation equation (15) balances at leading order in ε. 3.3. Solution in the southern hemisphere Finally, we scale h by the upstream depth at the wall, Hu. Our The across-channel velocity u appears in the governing equa- complete non-dimensionalisation may be written as tions only in (22c). We may therefore eliminate u by integrating (22c) from x = W to x = R(y) and simplifying using (23a) and 1/2 1/2 ˆ − x = ε Rd xˆ, y = ε− Rd yˆ, u = εcuˆ, v = cvˆ, h = Hu h, (23b), which yields (18) where the hat ˆ denotes an (1) dimensionless variable. These R(y) O hv dx = T, (24) scales describes a flow that varies slowly along the channel and W rapidly across the channel, and which has very small cross- − channel velocities. where T is the constant dimensionless along-channel transport. Substituting (18) into the zero-potential vorticity equation We will solve (22a) and (22b) for h and v, subject to (24). In (13) yields principle, u may then be calculated, once h and v have been determined, by integrating (22c) with respect to x, ˆ ˆ ∂h ∂h ∂vˆ 2 ∂uˆ yˆ cos θ+ε xˆ sin θ δ sin θ εδ cos θ+ ε = 0, (19) 1 ∂ x − ∂xˆ − ∂yˆ ∂xˆ − ∂yˆ u(x, y) = h x′, y v x′, y dx′. (25) −h(x, y) ∂y W − and the Bernoulli equation (14) becomes Equation (22a) is an exact derivative with respect to x, and ˆ 1 ε2uˆ2 + 1 vˆ2 + hˆ + δ hˆ vˆ sin θ εδ hˆ uˆ cos θ = 0, (20) (22b) is an exact gradient, so we integrate to obtain ∇ 2 2 − where xy cos θ δ h sin θ + v = A(y), (26a) H − = Ω u 1 2 δ (21) 2 v + h + δ h v sin θ = B, (26b) g′ measures the strength of non-traditional effects. The parame- where A is an arbitrary function of y, and B is the Bernoulli con- ter δ is the ratio of the non-traditional velocity scale ΩH, the stant. Equations (26a) and (26b) form an algebraic system that change in zonal velocity experienced by a fluid parcel that rises completely determine h and v in terms of B and A(y). We first a distance H while conserving angular momentum (White and find B by substituting the upstream conditions (23c) and (23d) Bromley, 1995), to the internal wave speed c. For the approx- into the algebraic equations (26a) and (26b). This yields four imate values of Ω, Hu and g′ listed above, δ 0.05 is actu- algebraic equations for four unknowns, v( W, 1), v(Ru, 1), ≈ A( 1) and B, which may be solved to obtain− − − ally comparable to ε, but we will see that this measure under- − estimates its effect on the solutions in 3.9. We may formally § 1 2 2 3 2 2 2 pose an asymptotic expansion of (19) and (20) in ε, of the form 1 + xu cos θ + δ sin θ δxu sin(2θ) B = 1 2 2 − , (27) hˆ = hˆ (0) + εhˆ (1) + etc. However, in this section we will 2 x cos θ 2δ sin θ  u   −  7     where xu = Ru + W is the upstream width of the current. With speed. We therefore discount solutions corresponding to the B known, A(y) may be determined by substituting the boundary limit (32) as unphysical. condition (23b) that holds at the front x = R(y) into (26a) and Thus the only physically reasonable limit in (31) is h 0 → (26b), and eliminating v x=R, as y 0. To conserve mass, the northward transport T must | remain→ constant as y 0 in (24). The velocity at the front A(y) = Ry cos θ + √2B. (28) → is v x=R = √2B, which excludes the possibility of v as y | 0, so instead the width of the current must approach→ ∞ in- We choose the positive root for v to ensure northward flow. finity,→ R(y) as y 0. From this behaviour Nof (1990) We may now determine v and h as functions of x, y and R(y) concluded that,→ ∞ in the absence→ of an opposing channel wall, a by substituting (27) and (28) into (26a) and (26b), and then northward-flowing western boundary current should be blocked solving algebraically. Under the traditional approximation this by the equator and turn eastward. In our formulation the flow is is straightforward and leads to confined by an opposing wall at x = W, and so the front must make contact with the wall at some point y = yc < 0, where vtrad = √2B (x R)y cos θ, (29a) − − − R( yc) = W. This is illustrated in Figure 4, and is also visi- 1 2 2 2 − htrad = √2B (x R)y cos θ (x R) y cos θ. (29b) ble in the specimen solution presented in Figure 5. At y = y − − 2 − c the solution varies rapidly with y, so the assumptions underly-− We omit the corresponding expressions when the complete ing our asymptotic solution are no longer valid. Following Nof Coriolis force is included, as they are considerably more com- and Olson (1993), we consider the solution only in the range plicated and difficult to interpret directly. To complete the solu- 1 y < yc, and connect the flow in the southern hemisphere tion in y < 0, we require the shape of the eastern front x = R(y). y− <≤0 to the− northern hemisphere y > 0 by requiring that the We first calculate the constant transport T by integrating (24) at transport T and the Bernoulli constant B are conserved across y = 1, y = 0. Despite the breakdown of the asymptotic expansion − Ru T = h(x, 1) v(x, 1) dx, (30) close to y = 0, conservation of transport T and Bernoulli con- W − − stant B must be preserved because they are a direct result of the − where h(x, 1) and v(x, 1) are determined exactly by substi- exact mass conservation equation (8b) and the assumption of tuting R = −R and y = −1 into (26a) and (26b). In principle zero potential vorticity in (8a). u − we may then find R(y) for arbitrary y by substituting the known We now show that the solution in the northern hemisphere expressions for v(x, y, R(y)), h(x, y, R(y)), and T into (24), and (y > 0) may be chosen to be symmetric to the solution in integrating with respect to x. However, it is typically more prac- the southern hemisphere (y < 0) under a rotation of π radi- tical simply to use (24) to compute R for any required value of ans about the origin. We denote quantities associated with the y. A typical solution is presented in Figure 5. northern hemisphere using a prime ′, so the dependent variables are v′(x′, y′), h′(x′, y′), and the current has a western front 3.4. Solution in the northern hemisphere where

As the current approaches the equator, its width, if unre- h′ = 0 at x′ = L(y′). (34) stricted by the opposing wall at x = W, would approach infinity. This is because the depth h approaches 0 as y 0. In the tradi- The leading-order governing equations, (26a) and (26b), may → tional solution (29b) it is straightforward to show that htrad 0 be rewritten for the northern hemisphere as as y 0 unless R(y) as y 0. More generally, taking→ the limit→ y 0 in (26a),→ (26b), ∞ and→ (28), and eliminating v and (x′ L(y′)) y′ cos θ δ h′ sin θ + v′ = √2B′, (35a) A yields → − − 1 2 2 v′ + h′ + δ h′ v′ sin θ = B′, (35b) √ 3 2 2 h 1 + 2 2B δ sin θ + 2 δ h sin θ 0 as y 0, (31) → → where we have used (34) to determine the function A (y ) = ′ ′ assuming yR(y) 0 as y 0. Thus either h 0 or √ + → → → 2B′ L(y′) y′ cos θ, analogous to A(y) in (26a). Let v(x, y), h(x, y), R(y), y < 0 describe the solution in the southern hemi- 1 + 2 √2B δ sin θ h (32) sphere, satisfying (26a), (26b), (23b)–(23d) and (24). Now let → − 3 δ sin2 θ x = x and y = y < 0 and suppose that the northern solu- 2 − ′ − ′ tion is related to the southern solution via v′(x′, y′) = v(x, y), as y 0. We require h > 0 for a physical solution, so the h (x , y ) = h(x, y) and L(y ) = R(y). Substituting these defini- → √ ′ ′ ′ ′ limit (32) is only valid for θ < 0, and only then if 2B > 1/2δ. tions into (35a) and (35b) yields− Substituting (23b) into (26b) yields v x=R = √2B, so condition (32) is a valid limit for h as y 0 only| if → (x R(y)) y cos θ δ h(x, y)sin θ + v(x, y) = √2B , (36a) − − ′ 1 1 2 v = > . (33) v (x, y) + h(x, y) + δ h(x, y) v(x, y)sin θ = B′. (36b) |x R 2δ 2 Typically δ 1, so this condition states that the along-channel Comparison of (35a) and (35b) with (26a) and (26b) shows that ≪ velocity at the front must be much larger than the gravity wave these equations are satisfied only if B′ = B. The northern flow 8 for all points x, y satisfying 1 y < y and W x R(y). − ≤ − c − ≤ ≤ For a fixed value of y, the minimum of vtrad lies at x = W, so in fact (38) may be simplified to −

v ( W, y) = √2B + (R(y) + W)y cos θ > 0, (39) trad − for all y [ 1, yc). It may be shown, using (29a), (29b) and (24), that∈ − − v ( W, y) √2B as y 0. (40) trad − → − → so if yc is sufficiently close to 0, there will be backflow at the western edge of the channel at y = yc. However, yc only approaches 0 in the limit of very large− channel width (W ), which violates the asymptotic assumption of a long narrow→ ∞ channel. Additionally, the assumption of a solution that varies slowly with y is no longer valid close to y = yc, so we may attribute any backflow that arises here to the breakdown of the asymptotic ordering. The minimum of vtrad( W, y) typically lies at y = 1, so the no-backflow condition (39)− becomes − √2 v ( W, 1) = √2B x cos θ > 0 or x < . (41) trad − − − u u cos θ

There is thus no backflow provided the upstream width xu is not too large. This requirement is most restrictive at θ = 0, corresponding to a northward channel, and least restrictive as θ π/2, corresponding to an almost-eastward or almost- → ± westward channel. Even if the minimum of vtrad does not lie at ( W, 1), condition (41) must still be satisfied to avoid back- Figure 5: Plot of a typical solution for cross-equatorial flow in a square channel. flow− at the− upstream edge. We show the fronts at the eastern (x = R(y)) and western (x = L(y)) edges of the current, under the traditional approximation (dashed line) and with the complete In the non-traditional case (δ , 0), we may obtain an equa- Coriolis force (solid line). The thick solid lines show the edges of the channel. tion for v as a function of x, y and R(y) by eliminating A and h ⋆ We have chosen the dimensions (yu = 500 km, W = 150 km, Hu = 500 m) to between (26a), (26b) and (28). It may be shown that v(x, y) > 0 match approximately the channel shown in Figure 1, though the orientation is 5 1 if less extreme, θ = π/4. We have used earth-like values of Ω = 7.3 10− rad s− 3 2 × √ + + and R = 6400 km, with a reduced gravity of g′ = 10− m s− , corresponding to 2B (R(y) x)y cos θ Bδ sin θ > 0. (42) 4 − ∆ρ/ρ = 10− . Again, for fixed y the minimum of (42) lies at x = W, so there is no backflow in the solution if − √2B + (R(y) + W)y cos θ + Bδ sin θ > 0, (43) must also conserve total along-channel transport T ′, where

W W R(y) for all y [ 1, yc). The term proportional to δ in (43) com- − ∈ − − T ′ = h′v′ dx′ = h′v′ d( x′) = hv dx = T. plicates the analysis of the backflow condition. Even at y = 1, − − − L(y′) L(y′) W where R = R , (43) and (27) together yield a quartic polynomial − − (37) u in x that must be solved to obtain a condition analogous to (41). Thus the symmetric solution guarantees conservation of B and u This means that for each solution considered in this work, we T across y = 0. The symmetry of the problem allows us to must check that (43) is satisfied via direct computation. analyse all possible solutions by considering only the flow in y < 0. 3.6. Dependence on xu, θ, and δ Under our asymptotic scaling (18), the governing equations 3.5. Condition for unidirectional flow and boundary conditions depend on only four dimensionless The solution described in 3.3 is only physically plausible if parameters: δ, θ, Ru and W. Moreover, the solution does not the flow of the current is unidirectional,§ i.e. if v > 0 everywhere. depend on Ru and W independently, but only on the upstream This is typically the most restrictive constraint on the solution. width of the current, xu = Ru + W. To see this, consider the We will first discuss the no-backflow condition under the tradi- boundary condition (23b) at the eastern front front of the cur- tional approximation (δ = 0), and then generalise to the more rent. Substituting this into (26a) and (26b) yields complicated non-traditional case (δ , 0). 1 2 2 3 2 2 The along-channel velocity is positive everywhere if 1 + 2 xu cos θ + 2 δ sin θ δxu sin(2θ) v x=R(y) = √2B = − , | xu cos θ 2δ sin θ v (x, y) = √2B + (R(y) x)y cos θ > 0, (38) − (44) trad − 9 for all y [ 1, y ). Both v = and h = thus depend only if ∆R(y) =∆W for all y. Therefore, changing the channel width ∈ − − c |x R |x R on δ, θ, and xu. In the following we fix δ and θ, and consider from W to W +∆W does not produce any relative change in only the dependence of the solution on xu. We rewrite equations the solution if the upstream width xu is fixed. The entire solu- (26a) and (26b), which determine v and h as functions of x, y, tion is simply shifted by a distance ∆W in the x-direction. In and R(y), in the form our asymptotic expansion we have neglected the x-variation of the vertical component of the rotation vector, and our equato- (x R)y cos θ δ h sin θ + v = √2B, (45a) rial β-plane approximation neglects all variation of the horizon- − − tal components, so we should expect the solution to be invari- 1 v2 + h + δ h v sin θ = B. (45b) 2 ant under translations in the x-direction. However, the position y = y , at which the front meets the eastern wall at x = W, We have used (28) to eliminate A(y). This rewriting shows that − c v and h depend on x only through the distance (x R) from the does vary with W. front. We may therefore write the solution as a function− of x R and y, so v = v(x R, y) and h = h(x R, y). We may calculate− 3.7. Effect of including the complete Coriolis force the along-channel− transport T by solving− (45a) and (45b) for h As outlined in 1, we wish to determine how the non- § and v at y = 1. Changing the integration variable in (30) from traditional component of the Coriolis force might enhance or − x to ξ = x Ru yields hinder cross-equatorial flow of an abyssal current. We are − therefore principally interested in the cross-equatorial transport Ru 0 T = T(x ,θ,δ). In Figures 6(a) and 6(a) we illustrate the de- T = h(x, 1)v(x, 1) dx = h(ξ, 1)v(ξ, 1) dξ. (46) u pendence of T on xu and θ respectively, for typical parameter W − − xu − − − − values. Including the complete Coriolis force leads to a larger From (27) we know that B = B(xu,θ,δ), so (45a) and (45b) cross-equatorial transport in a northwestward channel, and a show that v and h also depend only on xu, θ and δ, in addition smaller transport in a northeastward channel. The difference to x R(y) and y. Integrating hv from ξ = xu to ξ = 0 in is more pronounced when the current is narrower upstream. In − − (46) can only introduce an additional dependence on xu, so the Figure 6(a) we extend the plot only as far as xu = 2, as condi- transport must also be depend only on xu, T = T(xu,θ,δ). This tion (41) states that a larger width will lead to backflow in the may be shown directly by carrying out the integration in (46) current for θ = π/4. to obtain an algebraic expression for T. Under the traditional The most striking feature of these plots is that there are points approximation (δ = 0), the transport does not depend on xu at at which T approaches infinity. These asymptotes correspond to all, zeros in the denominator of the Bernoulli constant B, i.e. where 1 T = . (47) trad 2 cos θ x cos θ 2δ sin θ = 0 (49) u − When the complete Coriolis force is included (δ , 0), T de- in (27). These zeros correspond to a breakdown of cross- pends strongly on x , as shown in Figure 6, but the algebraic u channel geostrophic balance. Equations (22a) and (22b) to- expression for T is much too large to interpret directly. gether imply a geostrophic equation for the along-channel ve- We still have to show that the shape of the front, x = R(y), locity, depends only on xu, and remains unchanged for variations of W and Ru that leave xu unchanged. We consider a solution v(x, y), ∂h ∂ h(x, y) with a western wall at x = W and an eastern front y cos θ δ sin θ v = (h + δhv sin θ). (50) − ∂x ∂x at x = R(y). We then shift the western− wall to x = W +∆W, − keeping xu, δ and θ fixed, and denote the corresponding solution Setting δ = 0 in (50) immediately yields the familiar, traditional as v′(x, y), h′(x, y), with an eastern front at x = R(y) +∆R(y). geostrophic relationship between v and ∂h/∂x, though the x- For convenience we define V = hv and V′ = h′v′, the mass dependent part of the Coriolis parameter has been neglected as fluxes in the y-direction. From (45a) and (45b) we know that it appears at (ε) in (19). The non-traditional (δ) terms in V (x, y) = V(x ∆R, y), because V is a function of x R, and V (50) are proportionalO to v and ∂v/∂x, so we obtainO an explicit ′ − − ′ is the same function of x (R +∆R). As xu is identical in each expression for v by substituting (22a) back into (50) to eliminate solution, the transport T must− also be identical, so from (24), ∂v/∂x,

R R+∆R ∂h 1 + δ2h sin2 θ δyh cos θ sin θ V(x, y) dx = T = V′(x, y) dx ∂x − W W+∆W v = . (51) − − ∂h R+∆R R y cos θ 2δ sin θ = V(x ∆R, y) dx = V(x, y) dx − ∂x W+∆W − W+∆W ∆R − − − We obtain the condition (49) from the denominator of (51) by R W − applying the upstream condition y = 1, and replacing ∂h/∂x = V(x, y) dx + V(x, y) dx. (48) − W W+∆W ∆R by its average, − − − In a solution with no backflow, V(x, y) > 0 everywhere, so ∂h 1 Ru ∂h 1 1 = dx = h h = . (52) the leftmost and rightmost expressions in (48) are equal only x=Ru x= W ∂x xu W ∂x xu | − | − − xu − 10 dimensional equations. However, the solution breaks down if the orientation of the channel is very close to westward (θ π/2), or if the current is very narrow in a northwestward channel→ (θ > 0, x 0). Beyond the asymptotes marked in u → Figure 6, where xu cot θ < 2δ, the solution no longer satisfies the boundary condition (23c) at the western wall. We will return to the case the case of an almost-westward or almost-eastward channel in 4, using a modified asymptotic scaling. § 3.8. Extended solution with δ = (ε) O For the physical parameters discussed in 3.2, which are relevant to the AABW, we find ε δ 0.1. Terms§ that are (ε) should thus be as important as≈ terms≈ that are (δ). WeO now consider the distinguished asymptotic limit in whichO δ = (ε). This is consistent for a layer of vanishing depth with otherO pa- 1/2 (a) rameters fixed, because δ, ε Hu as Hu 0. The natural approach is to∼ pose an asymptotic→ expansion of (19) and (20) in ε, e.g. hˆ = hˆ (0) + εhˆ (1) + . However, the leading-order solution to (19) and (20) in the distinguished limit with δ = (ε) is simply the traditional solution (ε = δ = 0) describedO in 3.3–3.6. The asymptotes shown in Figure 6, a crucial feature§ of the solution, are fixed at θ = π/2, and are not modified by higher-order corrections to the± leading-order solution. It may be possible to correct this non-uniform conver- gence of the asymptotic expansion using the method of strained coordinates (e.g. Lighthill, 1949; Hinch, 1991). We return to this point in 5.2. A more practical,§ if less rigorous, alternative is simply to truncate our asymptotic expansion of (19) and (20), retaining terms up to (ε). This yields O ∂h ∂v y cos θ + εx sin θ δ sin θ + = 0, (53a) (b) − ∂x ∂x 1 v2 + h + δ h v sin θ = 0, (53b) Figure 6: Plots illustrating the change in the along-channel transport T when ∇ 2 the complete Coriolis force is included. In (a) we plot T against xu for δ = 0 Note that (53a) and (54a) differ from (22a) and (22b) only in (dashed line), and for δ = 0.1 with θ = 45 (solid line) and θ = 45 (dash- ◦ − ◦ the addition of the εx sin θ term in (53a) that accounts for the x- dotted line). In (b) we plot T against θ (measured in degrees) for xu = 0.5 with δ = 0 (dashed line) and δ = 0.1 (solid line). The dotted lines mark the dependence of the Coriolis parameter f . Integrating (53a) and asymptotes when δ = 0.1. (54a) yields a set of algebraic equations analogous to (26a) and (26b), 1 2 2 √ (x R)y cos θ + 2 ε(x R )sin θ δh sin θ + v = 2B, If the mean gradient ∂h/∂x of h approaches (1/2δ) cot θ, then − − − (54a) the gradient of h at some point x [ W, R ] must− also approach u 1 v2 + h + δ h v sin θ = B, (54b) (1/2δ) cot θ, by the mean value∈ theorem,− so the along-channel 2 − velocity tends to infinity by (51). In the traditional case (δ = 0), where again B is the Bernoulli constant, and we have deter- geostrophic balance breaks down only as y cos θ 0, i.e. as mined the function of integration in (54a) using the boundary → θ π/2. condition (23b) at the edge of the current. The method of solu- → ± Including the complete Coriolis force thus leads to a larger tion is identical to that described in 3.3, but the additional term transport in a northwestward channel, because the smaller de- in (54a) leads to several qualitative changes§ in the results, as denominator in (51) that expresses cross-channel geostrophic bal- scribed below. The solution retains the antisymmetry property, ance produces a stronger flow for a given pressure gradient. i.e. v(x, y) = v( x, y), h(x, y) = h( x, y), L(y) = R( y) for Similarly, a weaker flow is required to maintain cross-channel y > 0, which may− be− shown via a procedure− − similar− to that− de- geostrophic balance in a northeastward channel. These effects scribed in 3.4. As in 3.5, we must ensure that the flow is uni- are more pronounced when the channel is close to eastward or directional§ for the solution§ to be valid, but a simple condition westward (θ close to π/2). The zonal component of the along- analogous to (41) appears to be unavailable even in the tradi- channel velocity is then± large, and the non-traditional Corio- tional case (δ = 0). We have therefore checked this condition lis terms connect zonal and vertical motions in the full three- via direct computation for each solution presented here. 11 (θ = π/4) and northeastward (θ = π/4) channels. In the northwestward channel, the current begins− to turn as it approaches the equator, but it is only possible to compute the position of the front x = R(y) to the extent of the solid lines. The dotted lines in the left panel of Figure 5 suggest how the current might widen as it approaches equator, giving an impression of what the complete solution might look like. In the northeastward channel the front turns approximately parallel to the equator, and the position of the front may be computed beyond y = 0. However, close to y = 0 the solutions in the northern and southern hemispheres are overlapping, so this section of the front is again plotted simply to give a visual impression. The breakdown of our asymptotic solution close to the equator means that we can only speculate about the nature of the solution in the region that joins the inflow in the southern hemisphere to the outflow in the northern hemisphere. However, in general we expect the current to turn as it approaches the equator, flow along the equator as it crosses the channel, and then turn again to exit the channel. The more northwestward the orientation of the channel, the sharper these turns must be, and so the steady solution may be less stable. The explicit x-dependence in (53a) also means that the solution is no longer determined completely by δ, θ and xu, as shown in 3.6. To illustrate this, consider the cross-channel Figure 7: Typical plots of the eastern and western fronts (solid lines) of the § current when the x-dependence of the Coriolis parameter f is retained. The left geostrophic balance implied by (53a) and (53b), figure corresponds to a northwestward channel, θ = π/4, and the right figure ∂h to a northeastward channel, θ = π/4. In each case the dashed line marks the 1 + δ2h sin2 θ δh sin θ (y cos θ + ε x sin θ) equator. Both solutions break down− close to y = 0, but the dotted lines give an v = ∂x − . (56) impression of how the θ = π/4 solution might approach the equator. ∂h y cos θ + ε x sin θ 2δ sin θ − ∂x The asymptotes in the along-channel transport T occur where In 3.3, our neglect of the εx sin θ term in (53a) places the § the across-channel average of the denominator in (56) at y = 1 dynamic equator in the leading-order problem at y = 0. Under approaches zero, as in 3.7. Thus T approaches infinity where− the traditional approximation (δ = 0) cross-channel geostrophic § 1 balance (50) requires that ∂h/∂x 0 as the current approaches xu cos θ ε sin θxu xm 2δ sin θ = 0, xm = 2 (Ru W), (57) the equator (as y 0). As h = 0→ at the eastern front, by (23b), − − − → the current’s height much become vanishing small everywhere. where xm is the midpoint of the current at the upstream edge of Conversely, to conserve the along-channel transport, the cur- the channel. This condition gives rent’s width must become infinitely large as it approaches the xu equator. This broadening of the current is eventually blocked tan θ = . (58) 2δ + εxu xm by the opposing channel wall at x = +W. The same broadening also occurs in the non-traditional case, as shown in 3.4, but the The channel geometry requires that xm 0, so the asymptote § ≤ 1 mechanism is less intuitive. where T lies in θ (0, π/2) if δ > 2 εxu xm , and in θ → ∞ 1 ∈ | | ∈ The εx sin θ term in (53a) accounts for the fact that the equa- ( π/2, 0) if δ < εxu xm . These results may all be interpreted − 2 | | tor actually lies along the line y = εx tan θ. We determine in terms of the breakdown of cross-channel geostrophic balance where the current is blocked by setting−h = 0 in (54a) and (54b), (56), which now depends not only on the upstream width of which yields v = √2B and the current xu, but also on its absolute upstream position xm. Otherwise, the physically relevant results for ε , 0 are broadly (x R) y cos θ + 1 ε(x + R)sin θ = 0. (55) similar to those described in 3.7. − 2 § The height of the current must thus approach zero as x R(y), 3.9. Implications for the AABW which corresponds to the edge of the current, and as→ y We now apply this channel-crossing solution to the AABW. 1 ε(x + R(y)) tan θ, which is almost identical to the line mark-→ The channel highlighted in Figure 1 makes an angle of approx- − 2 ing the equator. This suggests that the equator should still act as imately 8◦ with the equator, and the flow is close to westwards. a barrier to cross-equatorial flow, in the absence of an opposing This corresponds to θ 1.43 rad 82◦ in our convention of wall at x = +W. In Figure 7 we plot the current profiles for measuring θ from the≈ eastward direction,≈ so the y cos θ and the same parameters used in Figure 5, for both northwestward εx sin θ terms in (53a) are of similar magnitude. We therefore 12 orientation close to westward (θ π/2) the validity of this solution becomes questionable, as→ the leading-order contribution y cos θ to the Coriolis parameter in (53a) vanishes. In 4 we ad- dress this shortcoming using an alternative asymptotic§ scaling in the limit θ π/2. → ±

4. Flow through an almost-zonal channel with arbitrary topography

The orientation of the channel outlined in Figure 1 makes an angle of approximately 8◦ or 0.14 radians with the equator, corresponding to θ 1.43 rad 82◦ when we measure the angle from the northward≈ direction.≈ The leading order contribution y cos θ to the traditional Coriolis parameter thus becomes com- 1/2 parable with the lengthscale ratio Rd/yu = ε 0.26 in the solutions described in 3.9. This motivates an alternative≈ asymp- § Figure 8: Plot of the dimensional along-channel transport against the upstream totic scaling of the zero-potential vorticity governing equations width of the current, under the traditional approximation (dashed line) and in- (13)–(15), in which the orientation of the channel is asymptot- cluding the complete Coriolis force (solid line). The dotted line marks a vertical ically close to eastward/westward. This scaling also allows us asymptote. The channel has similar dimensions to that plotted in Figures 1 and ⋆ to retain arbitrary variations of the bottom topography in the 2: yu = 500 km, W = 150 km, Hu = 500 m, and θ = 1.43. We use a small 4 2 cross-channel direction. reduced gravity g′ = 3 10− m s− to avoid unrealistically large velocities. × Figure 9 illustrates this problem schematically. We consider a channel that is uniform in the y-direction, and with arbitrary variations in the x-direction, crossing the equator at an angle retain the (ε) x-dependent contribution to the Coriolis param- θ with North. The current enters from the south, leaning up eter in ourO solutions, as described in 3.8. Figures 1 and 2 sug- against the western side of the channel, and exits to the north gest that the bathymetry approximately§ forms a channel with leaning against the eastern side, consistent with cross-channel ⋆ a half-length of yu = 500 km, a half-width of W = 150 km, geostrophic balance. This differs from the situation presented and a height of Hu = 500 m. We use approximate Earth-like in Figure 4 in that the current now has two fronts instead of one, values for the planetary radius, RE = 6400 km and the rotation and the positions of both fronts may vary along the channel. 5 1 rate, Ω = 7.3 10− rads− . We choose a somewhat small re- × 4 2 duced gravity, g′ = 3 10− ms− , to avoid unrealistically large 4.1. Scaling for an almost-zonal channel velocities in the current.× Even with g this small, the along- ′ Following 3.1, we seek a steady solution of the 1 1 -layer channel velocity and transport are an order of magnitude larger 2 shallow water§ equations under the assumption of zero potential than observations suggest (see Hall et al., 1997). This may be vorticity. Equations (13)–(15) become attributed to the absence of dissipation in our idealised model. The only remaining unspecified parameter is the width x⋆ of ∂v ∂u u β(y cos θ + x sin θ) + the current upstream, so in Figure 8 we plot the dimensional ∂x − ∂y along-channel transport T ⋆ as a function of x⋆. The plot is u ∂ 1 ∂ 1 ⋆ ⋆ 2Ω sin θ hb + h 2Ω cos θ hb + h = 0, (59a) reminiscent of Figure 6(a), but T varies with xu even when − ∂x 2 − ∂y 2 δ = 0 because we have retained the (ε) x-dependent term O 1 2 1 2 in (53a). Including the complete Coriolis force increases the u + v + g′(hb + h) + Ωh (v sin θ u cos θ) = 0, (59b) ∇ 2 2 − cross-equatorial transport, regardless of the upstream width of ∂ ∂ the current. For x⋆ & 120 km the current flows southward (hu) + (hv) = 0, (59c) u ∂x ∂y (v < 0) at the western wall, so the solution is no longer valid. We therefore consider our solution as representing the fast- in dimensional variables u, v, h, x, y, and with arbitrary bottom flowing core of the AABW, rather than giving a complete de- topography at z = hb(x, y), Equation (59b) contains the usual scription of the flow in the whole channel. In the non-traditional topographic term in the pressure, since the horizontal pressure case (δ , 0), cross-channel geostrophic balance (56) breaks gradient is proportional to the gradient of the free surface as down for x⋆ . 20 km, as discussed in 3.7. However, this z = h +h. The topography also alters the mean distance of fluid u § b width is unrealistically small, even for the core of the AABW parcels in a column from the rotation axis, and hence alters the (see Rhein et al., 1998). angular momentum of the column. This explains the origin of In summary, this solution suggests that the complete Coriolis the hb term in the condition (59a) for zero potential vorticity. force may play a substantial role in the cross-equatorial flow of In 3.2 the upstream depth Hu served as a natural vertical the AABW, though the Figure 8 suggests that the transport is lengthscale,§ but in the situation shown in Figure 9 the depth ev- very sensitive to inflow conditions. Additionally, for a channel erywhere is determined by the positions of the fronts at x = L(y) 13 Our asymptotic assumptions (17) and (61) together constrain the northward distance that the current can travel. The northward coordinate is Y = y cos θ + x sin θ, so the meridional position of the channel centre (x = 0) at the upstream boundary (y = y ) is − u Y(0, y ) = Y = y cos θ y ε1/2Θ = R Θ. (63) − u u − u ≈ − u − d The current can thus only cover a northward distance comparable to the deformation radius Rd. Given that Rd is typically around 150 km, this limitation is appropriate for the channel shown in Figure 1. We scale (59a)–(59c) on the assumption that yu is an appropriate lengthscale in the y-direction, so the solution is slowly- (a) varying along the channel. However, we now allow for (1) variations the x-direction, and need only assume that the cross-O channel velocity u is small to balance the mass conservation equation (59c) at leading order. Our scaled dimensionless variables are thus

1/2 1/2 x = Rd xˆ, y = ε− Rd yˆ, u = ε cuˆ, v = cvˆ, h = H hˆ. (64)

As we impose no asymptotic ordering on x, we may allow for arbitrary variations of the topography in the cross-channel direction, hb = Hhˆ b(ˆx). The scaling (64) leaves (59c) unchanged in dimensionless variables, whilst (59a) and (59b) become

∂vˆ ∂uˆ yˆ Θ 1 εΘ3 xˆ 1 1 εΘ2 + ε − 6 ± − 2 ∂xˆ − ∂yˆ (b) ∂ ∂hˆ δ 1 1 ε Θ2 2hˆ + hˆ δεΘ = ε2 , (65a) ∓ − 2 ∂xˆ b − ∂yˆ O Figure 9: Top-down (a) and along-channel (b) schematics of the solutions presented in 4. The solution has a similar form to that shown in Figure 4, except 1 2 1 2 § ˆ εuˆ + vˆ + hˆ + hˆ we allow for arbitrary variations of the topography in the x-direction, and the ∇ 2 2 b channel is oriented arbitrarily close to eastward or westward. δhˆvˆ 1 1 ε Θ2 δεΘhûˆ = ε2 . (65b) ± − 2 − O and x = R(y). We therefore use an arbitrary vertical lengthscale Under this scaling, the dependence of the Coriolis parameter H to define the gravity wave speed and deformation radius, f = βY on both x and y is retained at (1). All terms involving the cross-channel velocityu ˆ appear at O(ε), so the leading-order c solution is determined entirely in termsO ofv ˆ and hˆ. We introduce c = g H, Rd = . (60) ′ β a non-traditional parameter δ analogous to that in (21), We again prescribe the inflow of the current at y = yu, and − H assume that this distance is large compared with the radius of = Ω 2 δ , (66) deformation Rd, defining the small parameter ε = (Rd/yu) 1 g′ as in (17). We alter the scaling described in 3.2 by assuming≪ that the orientation of the channel is arbitrarily§ close to west- which is again the ratio of the non-traditional velocity scale ΩH ward/eastward, to the internal wave speed c. π θ = ε1/2Θ , (61) We expand the dimensionless variables as asymptotic series ± 2 − in ε, e.g. hˆ = hˆ (0) + εhˆ (1) + , and solve (65a) and (65b) at each where Θ = (1). The ‘+’ corresponds to an almost-westward order in ε. However, here we will consider only the leading- O channel, and the ‘ ’ to an almost-eastward channel. Using ε order solution, so it is sufficient simply to neglect terms of (ε) − ≪ 1, we may expand the trigonometric terms in (59a)–(59c) using and above, O Taylor series, ∂ ∂vˆ 1 2 2 ˆ ˆ sin θ = 1 ε Θ + ε , (62a) Θyˆ xˆ δ 2hb + h + = 0, (67a) ± − 2 O ± ∓ ∂xˆ ∂xˆ cos θ = ε1/2Θ 1 ε3/2Θ3 + ε5/2 . (62b) ˆ 1 vˆ2 + hˆ + hˆ δhˆvˆ = 0. (67b) − 6 O ∇ 2 b ± 14 Equations (67a), (67b) and (59c) must be solved subject to the The algebraic manipulations described here are prohibitively thickness of the layer vanishing at the frontsx ˆ = Lˆ (yˆ) andx ˆ = complicated in the most general case, and the exact formulae for Rˆ (yˆ), where the upstream positions of the fronts are prescribed, vˆ (xˆ, yˆ) and hˆ (xˆ, yˆ) would be much too large to interpret directly. However, the solutions presented in 4.3 have been obtained via ˆ h = 0 atx ˆ = Lˆ (yˆ) , (68a) exactly the procedure outlined above,§ but for specific cases of hˆ = 0 atx ˆ = Rˆ (yˆ) , (68b) the parameters Θ, δ, Lˆ u and Rû. A useful result that may be obtained directly is the equation for cross-channel geostrophic Lˆ = Lˆ aty ˆ = 1, (68c) u − balance implied by (67a) and (67b), Rˆ = Rû aty ˆ = 1. (68d) − ∂ ∂ hˆ + hˆ + δhˆ Θyˆ + xˆ δ 2hˆ + hˆ We will show that prescribing the upstream positions of the ∂xˆ b ± − ∂xˆ b vˆ = . (72) fronts in (68a)–(68d) determines the entire leading-order solu- ∂ tion. Θyˆ xˆ 2δ hˆ + hˆ ± ∓ ∂xˆ b 4.2. Method of solution Following 3.7 and 3.8, we expect the transport Tˆ(Θ, δ, Lˆ u, Rû) We now describe an analytical solution to the leading-order to have asymptotes§ § when the cross-channel average of the de- equations and boundary conditions in 4.1. The solution is pos- nominator in (72) is zero, i.e. where sible due to the assumption of zero potential§ vorticity, but the Θ Rˆ Lˆ 1 Lˆ 2 Rˆ2 2δ hˆ Lˆ hˆ Rˆ = 0. (73) algebra still rapidly becomes unmanageably complicated. We u − u ± 2 u − u ∓ b u − b u therefore outline the steps that could be taken, in principle, to determine the complete solution for arbitrary values of the pa- Algebraic manipulation of (70) shows that (73) is exactly the ˆ rameters. condition for the denominator of the Bernoulli constant B to vanish. In 4.3 we consider solutions in an almost-westward Equations (67a) and (67b) are exact derivatives, and so may § 2 2 be integrated immediately to obtain channel, and we always have Ru Lu 0 and Lu Ru 0, so typically the transport does not approach− ≥ infinity. However− ≥ , we 1 2 expect that equivalent solutions in an almost-eastward channel Θxˆyˆ xˆ δ 2hˆ b + hˆ + vˆ = Aˆ (yˆ) , (69a) ± 2 ∓ should be strongly affected by the asymptotes in the transport. 1 vˆ2 +hˆ + hˆ δhˆvˆ = Bˆ, (69b) 2 b ± 4.3. Solutions in an AABW-like channel where A is an arbitrary function ofy ˆ and Bˆ is the Bernoulli In this section we specialise the previous very general ap- constant. Substituting the boundary conditions (68a) and (68b) proach, capable of accommodating an arbitrary topographic into (69a) and (69b) yields two equations for Aˆ (yˆ) in terms of profiles, to solutions that are most relevant to the AABW. We the unknowns Lˆ (yˆ), Rˆ (yˆ) and Bˆ, take the channel orientation to be almost westward, and the channel shape to be ˆ ˆ 1 ˆ 2 ˆ ˆ ˆ ˆ ˆ A (yˆ) = ΘLyˆ 2 L 2δhb L + 2 B hb L ± ∓ − m x ˆ 1 ˆ2 ˆ ˆ ˆ ˆ ˆ hb(x) = H . (74) = ΘRyˆ 2 R 2δhb R + 2 B hb R . (70) x ± ∓ − 0 These may be solved simultaneously for Bˆ in terms of the pre- Here H is the maximum channel height, which serves as the scribed parameters Θ, δ, Lˆ u and Rû using (68c) and (68d). Equa- vertical lengthscale, whilst x0 is the channel half-width. We use tion (70) then determines Aˆ (yˆ) intermsofy ˆ, Lˆ (yˆ) and Rˆ (yˆ), and x0 = 150 km, H = 500 m, and m = 4, which yields a shape also directly relates Lˆ (yˆ) and Rˆ (yˆ). We may then solve (69a) that agrees qualitatively well with the averaged bathymetry and (69b) algebraically to determinev ˆ and hˆ as functions ofx ˆ, shown in Figure 2. For symmetric topography such as (74), ˆ ( ) ˆ ( ) an antisymmetric solution with v(x, y) = v( x, y), h(x, y) = yˆ, L yˆ and R yˆ . Thus the solution is completely determined − − ˆ ( ) ˆ ( ) h( x, y), and R(y) = L( y), satisfies the governing equations in terms of the unknown positions of the fronts, L yˆ and R yˆ . − − − − To complete the solution, we integrate the mass conservation (69a), (69b), boundary conditions (68a)–(68d), and transport equation (59c) across the current to obtain a condition analo- condition (71). This may be shown via a procedure similar to that described in 3.4. We therefore compute the solution only gous to (24), § Rˆ(yˆ) in one half of the channel (y < 0) and reflect it to obtain the hˆvˆ dˆx = Tˆ, (71) solution in y > 0. ˆ L(yˆ) In Figure 10 we plot the positions of the fronts in a typical where Tˆ is the constant along-channel transport. We may de- solution with AABW-like parameters. We have obtained this termine Tˆ in terms of the prescribed parameters Θ, δ, Lˆ u and solution under the traditional approximation, as this simplifies Rû using the upstream conditions (68c) and (68d). Thus (70) the computation greatly, yet the fronts do not differ dramatically and (71) constitute two equations relating Lˆ (yˆ) and Rˆ (yˆ), which when the complete Coriolis force is included. The behaviour of may be solved simultaneously to determine the positions of the the current is broadly similar to the solutions of 3, in that it fronts everywhere. This in turn determinesv ˆ and hˆ everywhere crosses from the western wall to the eastern wall§ of the chan- via (69a) and (69b). nel as it crosses the equator. However, in 3 the solution in the § 15 mum depth shifts towards the centre of the channel, following the equator, until it reaches the state shown in Figure 11(d) at y = 0. The reverse takes place as the current flows towards the northern edge of the channel. Rapid changes in the positions of the fronts are inconsistent with out assumption(64) about the solution varying slowly with y, so the validity of our asymptotic expansion is dubious at these points. If the upstream position Ru is brought closer to Lu, the boundary R(y) downstream makes a discontinuous jump from a state resembling Figure 11(b) to one resembling Figure 11(c), which certainly invalidates (64). If Ru is closer to the equator, the front will cross smoothly to the eastern wall of the channel, and if the current straddles the equator at the channel entrance (Lˆ u < Θ < Rû) the fronts need not cross the equator at all. Though the region close to the equator may require a more sophisticated analysis, this solution is a useful description of how the core of the AABW might flow steadily across the equator.

4.4. Effect of including the complete Coriolis force We are principally interested in the role of the non-traditional component of the Coriolis force in transporting fluid across the equator. The along-channel transport T is determined by the conditions at the southern entrance to the channel, where the Figure 10: Plots of the western and eastern fronts, x = L(y) and x = R(y) in assumption of a slowly-varying solution holds well. For values an AABW-like solution under the traditional approximation. The solid lines of the reduced gravity g′ that are realistic for the abyssal ocean, mark the positions of the fronts, and the dashed line marks the equator. In this e.g. g = 10 3 ms 2, the along-channel velocity and transport in 3 2 ′ − − solution Lu = 150 km, Ru = 20 km, θ = 1.43, yu = 500 km, g′ = 10− m s− , our solutions tend to be around an order of magnitude larger the and the channel− topography is− described by (74). measured values (e.g. Hall et al., 1997). This may be attributed to the absence of friction in this solution. Smaller values of g′ yield more realistic velocities, but cause the transport to behave southern (northern) hemisphere was constrained by the require- unexpectedly. ment that it must have non-zero depth at the western (eastern) We illustrate this in Figure 12 using the upstream solu- wall, and a front to the east (west). This precluded a formal tion for the same parameters as in Figure 10, but with g′ = 4 2 matching of these previous solutions at the equator. In the cur- 3 10− ms− . Including the complete Coriolis force leads to rent solution both fronts are free to vary with y, so we are able a substantially× shallower, faster-flowing current, but the trans- to compute the solution along the entire length of the channel. ports are almost identical. This may be understood qualitatively The most interesting feature of the solution shown in Fig- by redimensionalising (69a) and (69b) for an almost-westward ure 10 is the rapid transition to cross-channel flow at the equa- channel, tor, which we illustrate in Figure 11 with plots of the depth pro- 2 file at various points on the y-axis. To understand this, consider 1 β x + y 1 π θ 2Ω h + 1 h + v = A (y) , (76a) 2 2 − − b 2 the equation for cross-channel geostrophic balance (72) under 1v2 + g (h + h) + Ωhv = B. (76b) the traditional approximation (δ = 0) in an almost-westward 2 ′ b channel, We have added an extra y-dependent term to both sides of (76a), ∂ηˆ (Θyˆ + xˆ) vˆ = , η = hˆ b + hˆ. (75) absorbing it into A(y) on the right hand side. The left hand side ∂xˆ is then more easily recognisable as the angular momentumv ˜ As the current approaches the equator, the scaled Coriolis pa- from (5) in an almost-westward channel, so (76a) states that v˜ rameter Θyˆ + xˆ approaches zero. By conservation of along- is constant across the channel. This accounts for the fact that channel transport (71), the along-channel velocity may not ap- the fluid carries a planetary angular momentum that varies with proach infinity, so the gradient of the surface must approach the average vertical position of the fluid parcels in a column, as zero instead. Thus, in Figure 10 the current leans against west- discussed in 2. Fluid that lies higher up on the channel slope ern wall, as in Figure 11 panels (a) and (b), until it can not pro- thus experiences§ a larger planetary angular momentum, and re- ceed any further along the channel without crossing the equa- quires a larger along-channel velocity to balance it. Meanwhile, tor. Around y 250 km the solution makes a rapid transition the large westward velocity induces a downward vertical accel- to a state resembling≈ − Figure 11(c), leaning against both sides eration via the non-traditional component of the Coriolis force of the channel with a minimum depth (∂η/∂ˆ xˆ = 0) at the equa- in the three-dimensional equations. This leads to a positive con- tor. As the current moves further along the channel, the mini- tribution to the pressure in the Bernoulli equation (76b) via the 16 (a) (a)

(b)

Figure 12: Plots of (a) the along-channel velocity and (b) the layer height at the upstream edge of the channel, both under the traditional approximation (dashed line) and with the complete Coriolis force (solid line). In (b) the thick solid line marks the bottom topography. All parameters match the solution shown 4 2 in Figure 10, except g′ = 3 10− m s− . The traditional and non-traditional along-channel transports T are× 6.9 Sv and 6.8 Sv respectively.

Ωhv term. The gravitational pressure g(hb + h) decreases to compensate, so the current becomes shallower. These interpre- (c) tations for Figure 12 are not robust, as it is possible to find solutions that are counter-examples, but they provide an intuitive explanation for the effects of including the complete Coriolis force in typical solutions. A consequence of these counteracting effects on the transport is that the non-traditional component of the Coriolis force mod- ifies T more when g′ is larger. This is counter-intuitive, because the non-traditional parameter δ defined by (66) is larger when g′ is smaller, so we expect non-traditional effects to be more pronounced when the stratification is weaker. Figure 13 shows the percentage change in the along-channel transport over a range of values of g′, with all other parameters equal to those used in Figure 10. For g & 10 3 ms 2, including the complete Coriolis (d) ′ − − force leads to a clear increase in the transport. This is because Figure 11: Along-channel profiles of the solution shown in Figure 10, at (a) g′ is sufficiently large that the non-traditional contribution to y = 500 km = yu, (b) y = 250 km, (c) y = 200km, and (d) y = 0 km. the pressure is negligible compared to the gravitational terms − − − − in (76b), so the thickness h is almost identical in the traditional and non-traditional cases. Meanwhile the relative sizes of the terms in (76a) are independent of g′, so there is still a substan- 17 15

% change in transport 0

−5 × −4 −3 × −3 −2 3 10 10 ′ 3 10 10 g (m s−2)

Figure 13: Percentage increase in the along-channel transport T when the com- Figure 14: Plots of the along-channel transport T against the orientation of plete Coriolis force is included, against the reduced gravity g′. In all solutions the channel θ, under the traditional approximation (dashed line) and including Lu = 150 km, Ru = 20 km, yu = 500 km, and θ = 1.43rad 82◦. the complete Coriolis force (solid line). In all solutions Lu = 150 km, Ru = − − ≈ 3 2 − 20 km, yu = 500 km, and g = 10 m s . − ′ − − tial increase in the along-channel velocity v when the complete 3 2 Coriolis force is included. For g′ . 10− ms− the percent- 4 2 age change decreases rapidly, and, for g′ . 3.5 10− ms− the transport is actually slightly larger under the traditi× onal approximation. This is because the non-traditional term in (76b) has become comparable in magnitude to the gravitational term, forcing the layer to become shallower (c.f. Figure 12) and re- 4 2 ducing the transport. For g . 2 10− ms− the solution fails to satisfy the boundary conditions× (68a)–(68d).

Given this unusual dependence of the solution on g′, it seems reasonable to analyse the effect of including the complete Coriolis force using a realistic reduced gravity for a weakly- 3 2 stratified ocean, g′ = 10− ms− , and attribute the resulting unrealistically large velocities and transports to the absence of friction in the model. In Figure 14 we plot the transport T against the orientation of the channel θ for typical AABW-like Figure 15: Plot of the average along-channel mass flux, T/xu, against the up- parameters. The transport is quite sensitive to changes in θ, stream width, xu = Ru Lu, under the traditional approximation (dashed line) − particularly when the complete Coriolis force is included. This and including the complete Coriolis force (solid line). A straightforward plot of T against xu obscures the detail in this plot. In all solutions Lu = 150 km, is relevant to the AABW because the channel sketched in Fig- 3 2 − yu = 500 km, θ = 1.43rad 82◦, and g′ = 10− m s− . ures 1 and 2, with θ 1.43 rad 82◦, is not an exact rep- ≈ resentation of the equatorial≈ bathymetry.≈ In general the cross- equatorial transport is larger when the complete Coriolis force is included, by almost 35% as θ 90 . For θ . 70 the independent of xu under the traditional approximation. Again → ◦ ◦ transports are approximately equal, and for θ . 65◦ the solu- the cross-equatorial transport is typically larger when the com- tion breaks down, as the boundary conditions (68a)–(68d) are plete Coriolis force is included, and may be as much as 30–40% no longer satisfied. larger in a narrow current. As xu approaches the width of the In Figure 15 we plot the average along-channel transport channel (300 km), the solution fails to satisfy the boundary conditions (68a)–(68d). T/xu against the upstream width xu = Ru Lu. This is perhaps the most difficult parameter to determine from− observational ev- idence (e.g. Hall et al., 1997), because our solution represents 5. Discussion only a swift core of the AABW, rather than the entire AABW water mass. In this solution T is typically proportional to xu, We have investigated the role of the “non-traditional” com- except close to xu = 0, so simply plotting T against xu obscures ponent of the Coriolis force in cross-equatorial flow of abyssal some of the detail in Figure 15. This differs substantially from currents through channel-like bathymetry. Our focus, particu- the solution of 3.6, in which the along-channel transport is larly in 4, has been on the AABW, as an unexpectedly large § § 18 portion of this current crosses the equator in the western At- and cross the channel, so the breakdown of the solution is in- 1 lantic ocean. We have used the steady 1 2 -layer shallow water evitable. equations as an idealised model of a current, or of the swift core In both solutions the positions of the fronts are minimally af- of a larger water mass. Our analysis has been greatly simplified fected by the inclusion of the complete Coriolis force, whilst by the assumption the current adheres to a state of zero potential the along-channel velocity v, and therefore the along-channel vorticity to avoid symmetric instability as it crosses the equator transport, is much more strongly affected. As discussed in 3.7, (see Hua et al., 1997). Measurements of the deep ocean suggest this is a direct consequence of cross-channel geostrophic§ bal- that this condition is approximately satisfied within 3◦ of the ance, and leads to a larger transport in westward channels, and a equator (Firing, 1987). smaller transport in eastward channels. The difference becomes more pronounced if the current is narrower, which corresponds 5.1. Asymptotic Results to larger gradients in the surface height. These larger gradients are associated with larger vertical velocities, and therefore In 3 we obtained asymptotic solutions for flow through a with larger zonal accelerations due to the non-traditional com- § flat-bottomed channel with vertical walls, crudely approximat- ponent of the Coriolis force. An alternative interpretation is ing the along-channel average of the real equatorial bathymetry, that equations (26a), (54a) and (69a) are statements of cross- as shown in Figure 2. Following Nof and Olson (1993), our fun- channel conservation of along-channel angular momentumv ˜, damental asymptotic assumption (17) was that the deformation as defined in (5). When the complete Coriolis force is included, radius Rd was small compared with the upstream position yu, the fluid acquires an additional westward velocity to balance 1/2 i.e. ε = Rd/yu 1. Furthermore, we assumed in (18) that the additional planetary angular momentum it acquires when ≪ 1 yu was the relevant along-channel lengthscale, and that εRd was its half-layer depth z = hb + 2 h is larger. This increases the the relevant cross-channel lengthscale. Applying these scalings, transport through a westward channel, and reduces the trans- which describe a a long, narrow current, yields leading-order port through an eastward channel. The results of 4.3 confirm equations (22a)–(22b) that may be integrated exactly, and that that the effect of including the complete Coriolis force§ is most are independent of the cross-channel velocity u. The solution pronounced when the channel is almost westward, as shown in may be calculated explicitly, at least in principle, everywhere Figure 14. The effect is also more pronounced when the current except close to y = 0, where the asymptotic assumptions are vi- is narrower, though this is difficult to discern from Figure 15. olated. We therefore connected the solutions in y < 0 and y > 0 via conservation of mass and Bernoulli function. The resulting 5.2. Non-traditional geostrophic balance solution is characterised by Figure 5, which shows the change in the positions of the fronts along the channel. The solutions presented in 3 and 4 suggest that the tradi- The validity of the solutions obtained in 3 becomes ques- tional approximation underestimates§ the§ transport of westward- tionable when the orientation of the channel§ is close to west- flowing current, and overestimates the transport of an eastward- ward or eastward. When θ is sufficiently close to π/2 that flowing current. This may be understood as consequence of the cos θ becomes (ε), the dimensionless traditional Coriolis± pa- combination of cross-channel geostrophic balance and cross- rameter fˆ = yÔcos θ + εxˆ sin θ becomes asymptotically small. channel conservation of angular momentum. These together Maintaining the leading-order cross-channel geostrophic bal- yield an explicit expression for the along-channel velocity, 1 ance (51) thus requires an (ε− ) along-channel velocity v. This given by (51), (56) and (72) for the solutions we have con- motivates the alternativeO scaling in 4, in which the channel sidered. However, the interpretation of these expressions is is almost-eastward or almost-westward,§ and in which thex ˆ- not straightforward, so here we elucidate the role of the non- dependent part of fˆ is retained at leading order. This scaling traditional component of the Coriolis force in geostrophically- permits arbitrary cross-channel variations of the bottom topog- balanced flow over relatively steep topography. raphy z = hb(x), allowing us to bring our idealised topography Consider a current flowing steadily westward over a zonally into better agreement with the averaged bathymetry shown in symmetric topographic slope, as illustrated in Figure 16. We Figure 2. The method of solution is similar to that described in align the y-axis with the direction of the flow because this is 3, though the arbitrary topography quickly leads to algebraic consistent with the solutions discussed earlier in this paper. Un- § expressions that are too complicated to interpret directly. der the assumption of a steady, zonally-symmetric solution with 1 The most obvious difference between the results of 3 and 4 no flow in the x-direction, the 1 2 -layer shallow water equations is that the solutions in an almost-westward channel no§ longe§ r (1)–(2) reduce to break down close to y = 0, as shown in Figures 10 and 11. In- stead, they undergo a rapid transition to cross-channel flow as ∂z ∂ 1 f f˜ v(x) = g′η + f˜ hv . (77) they cross the equator. This transition is smooth when the cur- − ∂x ∂x 2 rent is sufficiently wide, but forms a discontinuity if the current 1 is too narrow upstream. A complete description of the channel- Here z = hb + 2 h is the half-layer depth, as in (4), f = crossing flow is possible because the positions of the fronts, 2Ω sin φ is the traditional Coriolis parameter, f˜ = 2Ω cos φ x = L(y) and x = R(y), are unconstrained. In 3 the current is the non-traditional Coriolis parameter, and φ is the latitude. enters from the south with a finite depth at the§ western wall, The components of the rotation vector in these coordinates are and no mechanism is provided for it to detach from the wall 2Ω = ( f˜, 0, f ). We approximate both f and f˜ by constants for 19 where the approximation follows from neglecting terms of (δ2) in the denominator. The denominator of (80) may then Obe rewritten as

2Ω (z h ) 2Ω nˆ = 2Ω sin(φ α). (82) ∇ − b ≈ − Under the approximation that neglects (ǫδ) terms in (77) we thus recover the standard expression forO geostrophic balance,

g η v = ′ x , (83) 2Ω sin φe

but with the geometrical latitude φ replaced by φe = φ α. This “effective latitude” is the angle between the rotation− vec- Figure 16: A schematic of non-traditional westward geostrophic flow over relatively steep topography in the northern hemisphere. The solution is uniform tor and the topography, rather than the angle between the ro- along the y-axis, which points into the page. Under the complete Coriolis force tation vector and the horizontal plane (as it is under the tradi- the latitude φ is replaced by an “effective latitude” φe = φ α, where α(x) is tional approximation). The effective latitude may be treated as a − the elevation of the topographic slope. “strained coordinate” (e.g. Lighthill, 1949; Hinch, 1991) whose introduction circumvents the non-uniformity of the expansion of (79) in δ that is due to the vanishing of the leading-order simplicity, giving what Gerkema and Shrira (2005) named the denominator sin φ at the equator. “non-traditional f -plane”. The latitude is typically larger in magnitude than the to- Equation (77) resembles traditional geostrophic balance, but pographic slope, so φ and φe have the same sign. In with additional non-traditional terms proportional to f˜ on both topographically-dominated flow ηx and α have the same sign sides. To clarify how these non-traditional terms affect the because ηx hb,x = tan α. The flow is then westward if φ along-channel transport, we further simplify (77) by assuming and α have the≈ same sign, and the geostrophic velocity is larger that the flow is dominated by the bottom topography, like the than it would be under the traditional approximation because solutions considered previously in this paper. We nondimen- φe < φ . Conversely, the flow is eastward if φ and α have op- sionalise (77) using different vertical lengthscales for the fluid posite| | | signs,| and its velocity is smaller than it would be under thickness, h = Hhˆ, and the bottom topography, h = H hˆ , and the traditional approximation because φ > φ . Very close to b b b | e| | | assume that their ratio ǫ = H/Hb 1 is small. We introduce the equator, the magnitude of the topographic slope may ex- an arbitrary horizontal lengthscale,≪x = Lxˆ, and use traditional ceed the latitude, α > φ . A geostrophic flow that would be geostrophic balance to determine an appropriate velocity scale, westward under the| traditional| | | approximation then becomes an v = (g′Hb/2ΩL)ˆv. The dimensionless equation for geostrophic eastward flow under the complete Coriolis force, because φ and balance in the x-direction is thus φe have opposite signs. ∂ ∂ More generally the y-axis, and thus the direction of the flow, sin φ δ cos φ hˆ + 1 ǫhˆ vˆ = hˆ + ǫhˆ + 1 ǫδhˆvˆ cos φ . may be inclined at an angle θ from northwards. The compo- − ∂xˆ b 2 ∂xˆ b 2 nents of the rotation vector are then 2Ω = ( f˜sin θ, f˜cos θ, f ), (78) so westward flow corresponds to θ = π/2. Making the same The aspect ratio δ = Hb/L 1, as required for the validity of the shallow water equations.≪ We neglect terms that are (ǫδ), approximations as above, the geostrophic along-slope velocity but retain terms that are (ǫ) or (δ) to obtain the approxima-O is O O tion ∂ ˆ ˆ g η + 1 f˜ hv sin θ ∂hb ∂h ′ 2 g η + ǫ = ∂x = ′ x ∂xˆ ∂xˆ v Ω (84) vˆ = . (79) f f˜zx sin θ 2 x,z sin(β α) ∂hˆ − | | − sin φ δ cos φ b − ∂xˆ where 2Ωx,z = ( f˜sin θ, 0, f ) is the component of the rotation In dimensional form this becomes vector that lies in the x-z plane. The elevation of Ωx,z above the = 1 g η g η x-axis is β cos− (cos φ sin θ). = ′ x = ′ x v Ω (80) f f˜ hb,x 2 (z hb) − ∇ − 5.3. Conclusion where the subscript x indicates differentiation with respect to 1 The solutions obtained in this paper suggest that the non- x. We now define α = tan− (∂hb/∂x) to be the angle between the x-axis and the topographic slope, as sketched in Figure 16. traditional component of the Coriolis force may account for 10– The upward unit normal nˆ to the topographic slope is 30% of the cross-equatorial transport of the AABW. We might expect to see a similar increase in the cross-equatorial trans- (z hb) port in more sophisticated models, or even in the real ocean, as nˆ = ∇ − (z hb) (81) 2 1 + (hb,x) ≈ ∇ − long as the current remains in near-geostrophic balance close 20 to the equator. However, our idealised model omits many po- Lighthill, M. J., 1949. A technique for rendering approximate solutions to phys- tentially important features of the abyssal dynamics. Our so- ical problems uniformly valid. Philosophical Magazine Series 7 40, 1179– lutions over-predict the along-channel velocity and transport, 1201. McCartney, M., Curry, R., 1993. Transequatorial flow of Antarctic Bottom Wa- due to the absence of frictional effects. These may be expected ter in the western Atlantic Ocean: Abyssal geostrophy at the equator. Journal to be particularly important for an abyssal current flowing over of Physical Oceanography 23, 1264–1276. highly variable bathymetry. We also do not account for mix- Nof, D., 1990. Why are some boundary currents blocked by the equator? Deep Sea Research 37, 853–873. ing of the AABW with the less dense overlying layers. Possi- Nof, D., Borisov, S., 1998. Inter-hemispheric oceanic exchange. Quarterly Jour- bly the most important omission is time dependence, as there nal of the Royal Meteorological Society 124, 2829–2866. is no guarantee that the steady states calculated here would be Nof, D., Olson, D., Feb. 1993. How do western abyssal currents cross the equa- stable in a time-dependent computation. For example, the nu- tor? Deep Sea Research 40, 235–255. Pedlosky, J., 1987. Geophysical Fluid Dynamics. Springer. merical solutions of Nof and Borisov (1998) and Choboter and Raymond, W. H., 2000. Equatorial meridional flows: Rotationally induced cir- Swaters (2000) for shallow water flow in an idealised channel culations. Pure and Applied Geophysics 157, 1767–1779. suggest that unsteady solutions could completely diverge from Rhein, M., Stramma, L., Krahmann, G., 1998. The spreading of Antarctic bot- those presented here. In future work we will use unsteady so- tom water in the tropical Atlantic. Deep Sea Research 45, 507–528. Stephens, J. C., Marshall, D. P., 2000. Dynamical pathways of Antarctic Bottom lutions of the non-traditional shallow water equations (1)–(2) Water in the Atlantic. Journal of Physical Oceanography 30, 622–640. to study the role of the complete Coriolis force in transporting Stewart, A. L., Dellar, P. J., 2010. Multilayer shallow water equations with fluid through idealised cross-equatorial channels and realistic complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. equatorial bathymetry for the AABW. Journal of Fluid Mechanics 651, 387–413. Stewart, A. L., Dellar, P. J., 2011. 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