## 1. Introduction

The problem of determining the meridional overturning circulation (MOC) and the associated northward heat flux has been concerning climate scientists for half a century. The predominant difficulty in reconstructing the MOC from limited data, usually hydrographic, is the distribution of the barotropic circulation. The baroclinic circulation, by contrast, is usually sufficiently accurately determined by geostrophy, although subtle questions remain as to levels of no motion assumed for the calculation (e.g., Bryden et al. 2005; Hirschi and Marotzke 2007). Various authors have defined different decompositions of the meridional flow; Sime et al. (2006) give one that includes the possibility of external influences on the section. No matter which decomposition is employed, one of the main difficulties has been to establish how the surface Ekman flux is returned at depth.^{1} This return flow is to some extent an artifact of the usual methodology, since the response of the ocean to a wind stress cannot simply be defined as just an Ekman response since winds will trigger Rossby and Kelvin waves (e.g., Johnson and Marshall 2002), which cause the MOC to vary (Hirschi et al. 2007) by several Sverdrups (1 Sv ≡ 10^{6} m^{3} s^{−1}) as well as exhibit a baroclinic response.

The Ekman flux forms an important part of the total MOC [Jayne and Marotzke (2001) give an excellent review of the subject] and hence of the northward heat flux. While the value of the Ekman flux itself is well established given adequate atmospheric data, how it is returned below the mixed layer is still unclear.^{2} Jayne and Marotzke (2001) devote much of their paper to a numerical examination of just this point, concluding that for variable winds with periods up to about a year, the basin-averaged return flow is essentially depth independent.

However, this result suggests, but does not prove, that the return flow is both horizontally and vertically uniform across an ocean section such as the RAPID monitoring array at 26°N (Srokosz 2003; www.noc.soton.ac.uk/rapidmoc). The distribution of this return flow becomes important if temperatures vary across the section, because the return flow and the ocean temperature, if even slightly correlated, will produce an error in the estimated northward heat flux.

How is a time-varying Ekman flux returned at depth? For sufficiently fast variation, only barotropic modes would be triggered, and Jayne and Marotzke’s (2001) thought experiment applies. However, surface wind stress has spatial and temporal components at all values (Killworth and Blundell 2007) and the slower of these components will permit or trigger Rossby waves across the basin. This suggests that the return flow may be nonconstant both laterally and vertically. This article examines this question.

Section 2 derives a simple modal model of the wind-driven part of the MOC assuming a flat bottom and no mean flow. The model is therefore far from realistic but contains sufficient dynamics to show how the Ekman return flow is distributed. Section 3 shows some results from this model, and section 4 shows a selection of results for flows driven by a northward wind stress (which generates coastal upwelling on the east and west boundaries of the model ocean).

## 2. The linear model

*x*,

*y*,

*z*,

*t*denote distance east, north, up, and time, respectively;

*u*,

*υ*,

*p*are the eastward and northward velocities and the pressure;

*ρ*

_{0}is the mean density;

*g*is the gravitational acceleration; and

*η*are the amplitudes of the modes. The vertical eigenmodes

_{n}*p̂*(

_{n}*z*) satisfy

*c*is an eigenvalue,

_{n}*N*

^{2}is the buoyancy frequency, and

*p̂*satisfy suitable boundary conditions. The normalization used is

_{n}*H*is the (constant) depth.

*f*is the Coriolis parameter and (

*X*,

_{n}*Y*) are the decomposition of the wind stress in the vertical, given by

_{n}*h*from its surface value to zero in what follows;

*H*is the equivalent depth given by

_{n}*c*

^{2}

_{n}=

*gH*; and

_{n}*λ*is a Stommel-like linear drag to close the vorticity balance. Other closures would almost certainly give similar results; we use the simplest for convenience. Note, however, that the closure is not scale-selective, although it is indirectly mode-selective.

*ily*−

*iωt*), where

*l*is a north–south wavenumber and

*ω*is the frequency. (This is not fully consistent with the frozen beta-plane assumption which is employed here, as is well known. However, this assumption is in line with the simple model used here.) For definiteness

*l*≈ 2

*π*/3000 km

^{−1}≈ 2 × 10

^{−6}m

^{−1}. Then, postponing the substitution in the

*y*direction, we have

*u*= 0,

_{n}*x*= 0, and

*L*, where

*L*is the width of the ocean basin. If

*ω*≪ |

*f*|, then to leading order

*ω*/

*f*

^{2}are everywhere small, and we assume that the

*λ*term is only relevant in the definition of

*u*(its effects may be confined to a western boundary layer or may act as a damping over the entire ocean).

_{n}^{3}Substituting into the continuity equation, letting

*β*denote

*df*/

*dy*,

*x*and inserting the

*y*-oscillatory behavior gives to leading order

*n*= 0, the quantity

*gH*

_{n}l^{2}/

*f*

^{2}is small save near the equator.

^{4}This equation has boundary conditions

^{5}by using the divergence from Eq. (6):

*βc*/|2

_{n}*f*|(Gill 1982), so that forcing with a frequency much higher than this will tend to produce only a local response to the wind stress, approximately −

*lH*/

_{n}X_{n}*ωf*, together with some boundary layers at the east and west sides of the ocean. Frequencies much lower than this will permit damped Rossby waves, and the response will then depend on how the time for a long Rossby wave to cross the ocean,

*Lf*

^{2}/(

*βc*

^{2}

_{n}), compares with the forcing period. If the wave has sufficient time to cross the basin, an interior wavelike solution can be expected. When the frequency becomes very small, the response is Sverdrup–Stommel in nature, and from the above the frequency has very small net northward transport for this mode.

If all responses were local and proportional to *X _{n}*, this reconstitution would simply reproduce the Ekman flux in the mixed layer and a vertically constant compensating return flow below, with a maximum value for the depth-integrated flow (the model MOC) of

*LX*/

*f*, where

*X*is the surface wind stress divided by the ocean density.

Thus the picture that emerges is that for high-frequency forcing, the interior response is local and proportional to wind stress, and this will dominate the basinwide integral of northward velocity. If all modes responded in this manner, their reconstitution would be the Ekman flux in the surface layer and a uniform return flow beneath, as assumed in MOC calculations and discussed by Jayne and Marotzke (2001). However, there are two exceptions to this simple picture. First, the barotropic mode will always be able to propagate and so while the return flow may be depth-independent, it will not be laterally independent. Second, for lower frequencies the picture may change: some modes will have wavelike solutions and some will be essentially steady Stommel modes. The reconstitution of this system will not necessarily behave in the manner assumed in MOC calculations. The appendix gives a brief analysis of relevant parameter ranges.

## 3. Results

*σ*

_{+}and

*σ*

_{−}are roots of

*σ*plays the role of

*ik*, where

*k*is a complex wavenumber) and

*A*and

_{n}*B*. We write the solution in this manner to avoid excessively large numbers in the evaluation of the exponentials in the above: the

_{n}*σ*

_{−}root corresponds roughly to a western boundary layer, and the

*σ*

_{+}root to a possible Rossby wave.

*O*(

*ω*

^{−1}) to the essentially Stommel zero-frequency solution. The boundary between these is presumably at a frequency of order

*β*(

*gH*)

_{n}^{1/2}/

*f*, which is the cutoff frequency for that mode and latitude in the absence of damping. The numerical solutions below, in fact, do not tend to an MOC varying proportionally to frequency until frequencies are below about 10

^{−10}s

^{−1}. This is a surprisingly small value, suggesting that even centennial forcing gives a nonzero contribution to MOC variability.

For numerical purposes we arbitrarily choose

*l*= 2 × 10^{−6}m^{−1}, corresponding to a 30°N–S wavelength;*L*= 4000 km;*H*= 4000 m;*h*= 50 m;*λ*= 10^{−6}s^{−1}, corresponding to a Stommel layer thickness of about 100 km; and*X*= 5 × 10^{−6}m^{2}s^{−2},*Y*= 0, corresponding to an Ekman flux, integrated across the basin, of 0.2 Sv at about 40° latitude.

This latter value is small; estimates of integrated Ekman flux from the RAPID array show a variability of order 4 Sv. Jayne and Marotzke’s (2001) parallel ocean climate model (POCM) runs used an *X* of order 50 times the value used here for seasonal calculations. The problem is, of course, linear and so it scales. Of these parameters, the two that are most subjective are *l* and *λ*. Varying these by a factor of 2 makes only small changes to the solutions.

A vertical modal structure is required. We use a constant *N* ^{2}, so that the *p̂ _{n}* are cosines except for

*p̂*

_{0}, which is a constant. [Calculations have also been made with the more realistic density profile of Gill (1984), which has an analytic solution and includes an explicit mixed layer. Apart from minor quantitative changes—for example, the response is closer to the surface than for the model discussed below—the results are essentially identical, so that they may be taken as reasonably representative.]

The barotropic wave speed *c*_{0} = (*gH*)^{1/2}. The first baroclinic wave speed *c*_{1} is taken to be 3 m s^{−1}, and the higher values *c _{n}* =

*c*

_{1}/

*n*. The frequency

*ω*is varied between 2 × 10

^{−5}and 2 × 10

^{−8}s

^{−1}, corresponding to periods between a few days (2 × 10

^{−5}), a month (2 × 10

^{−6}), a year (2 × 10

^{−7}), and a decade (2 × 10

^{−8}). Two thousand modes are used in the vertical to minimize Gibbs phenomena at the base of the mixed layer. Of course, for any realistic ocean, the higher modes would be heavily damped by a variety of mechanisms, especially for longer periods. For this simple model, such effects are ignored, as are any diabatic and nonlinear effects.

Several calculations can be made. The most immediate is to compute the maximum MOC. To do this we reconstitute ∫^{L}_{0} *υ dx* as a function of depth and locate the maximum of its absolute value. This is invariably at the base of the mixed layer (i.e., *z* = −*h*). This value is contoured in Fig. 1. With the exception of the three high-frequency values that appear to produce a kind of resonance (which we shall not explore here except to note that their frequencies are far too small for the barotropic basin mode and far too large for any baroclinic basin mode), the maximum MOC is everywhere close to the natural value *XL*/*f*, but slightly smaller.^{6}

However, this close agreement masks a degree of both horizontal and vertical variability. If we first consider the vertical structure of the MOC, it is clear that while there is an almost uniform response in the mixed layer, the return flow beneath is not necessarily uniform, nor indeed completely in phase with the wind (Figs. 2 and 3). At lower frequencies, the deviations from a constant return flow become more marked, although these effects decrease with latitude; north of 35° the diagrams are close to overlaying and are not shown. This confirms the numerical results at 30° cited by Jayne and Marotzke (2001).

Figures 4 –9 show the first 10 modal northward velocities as a function of *x*. Since western boundary current values are typically high, only solutions east of 400 km are shown. The barotropic mode is fairly uniform except for high frequencies. The first baroclinic mode propagates across much of the basin for low frequencies but becomes increasingly damped for high-frequency forcing. Higher modes are only evident outside boundary layers for decadal forcing. It is noteworthy that the imaginary parts of the solution are as big as, or larger than, the real parts, showing that much of the response is not in phase with the forcing.

The full *υ* velocity is shown as a function of *x* and depth in Figs. 10 –12, for frequencies corresponding to decadal, annual, and monthly forcing, respectively. For the decadal frequency, the northward velocity is highly baroclinic below the mixed layer and is variable over the entire width of the ocean. For the annual frequency, there is a similar baroclinicity confined to the first baroclinic mode, but there is damping toward the west of the basin. For the monthly frequency, the response is barotropic, but there is still variability across the basin.

## 4. Forcing by north–south wind stress

The previous section has shown the linear response of the ocean to a time-varying east–west wind stress. However, a north–south wind stress can also produce a wind-driven response through coastal upwelling (e.g., Köhl 2005), and we briefly discuss the response here.

Figure 13 shows the dependence of the maximum MOC produced by *X* = 0, *Y* = 5 × 10^{−6} m^{2} s^{−2}. There is little response save at the three apparent resonances, so that coastal upwelling drives very little of the amplitude of the MOC compared with east–west wind stress.

Figure 14 shows the vertical structure of the MOC produced by this wind stress. Although only driven at the boundaries, about 0.5 Sv is produced with a mode 1 vertical structure at the three apparent resonances, but much smaller values are found at other frequencies. Figures 15 and 16 show the east–west modal structure of the northward velocity for decadal forcing at 26° latitude. Although the response is smaller than for eastward wind stress, there is nonetheless a response across most of the basin. Finally, Figs. 17 –19 show the *υ* field as a function of *x* and *z* for decadal, annual, and monthly forcing. While the solutions are dominated by the coastal up- and downwelling, elsewhere the northward velocities are far smaller than those produced by east–west winds.

These results are in contrast to those of Köhl (2005), who found a larger response from varying north–south winds than from east–west winds. The relative sizes of these winds are not clear since Köhl (2005) does not show the north–south wind field. Solving the north–south problem for a single boundary with no *β* or *λ* gives a similar solution to that shown, so the details of the dynamics are probably not responsible for this difference. The most likely candidate is the north–south length scale: from (8) the solution will tend to scale with *l*^{−1}.

## 5. Discussion

The simple calculations above demonstrate clearly that the return flow under the Ekman layer is both variable in depth as well as laterally. Does this matter? Whether it does depends on how the return flow is handled in calculations from observations. If the MOC is required, then at low latitudes and low frequencies even the east–west integral of the northward flow is not independent of depth. At the latitude of the RAPID array (26°), Fig. 3 shows that there can be noticeable error in assuming a depth-independent return flow on average across the basin.

*Q*is the heat flux,

*c*is the specific heat,

_{p}*T*

_{Ek}is the mixed layer temperature, and 〈

*θ*〉 is the section-averaged potential temperature. Clearly this formula assumes both a vertical and a horizontal lack of variation of the Ekman return flow.

^{7}We have shown above that this is frequently not the case, even for oscillations as rapid as monthly. For the annual cycle, the variation across the section is large enough to make a quantitative change to the heat flux calculations, although the calculations are usually more sensitive to the vertical variation. While the simplicity of the model precludes any accurate estimation of the error in

*Q*, a simple estimate can be made. If we suppose that the basin temperature is 0.5° warmer in the western side compared with the eastern side, and that the Ekman return flow is also asymmetrical, then if

*T*

_{Ek}− 〈

*θ*〉 is of order 20°, the error in the horizontal may be about 3%. As noted, the error due to nonuniform return flow in the vertical will be larger, but a more complete model is necessary for this estimation.

## Acknowledgments

This paper could not have been written without the patient help of Naomi Greene and Diane Robinson. Comments from Joel Hirschi, Jeff Blundell, and the reviewers are appreciated. This work contributes to the NERC program “Oceans 2025,” themes 1 (climate) and 9 (model development).

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## APPENDIX

### Simple Asymptotics

*L*and times on

*T*=

*L*/

*C*, the transit time for a Rossby wave. Then with

*T*becomes larger as the mode number increases (since the transit time becomes ever larger); thus, both

*ω*′ and

*λ*′ become larger with mode number, their ratio staying constant. Typically

*λ*′ is large save for the barotropic mode, which we ignore here;

*ω*′ can be large or

*O*(1);

*ω*′ can be larger or smaller than

*λ*′;

*l*′ is

*O*(1);

*β*′ is

*O*(1); and

*μ*is small except for the barotropic mode. We examine the relevant parameter ranges below.

#### Case ω′ ≫ λ′ ≫ 1

*ωμ*≫ 1 (almost always the case), the first and last term balance, giving a Kelvin wave where

*x*≈

*μ*corresponds dimensionally to

*x*≈

*a*. Thus the boundary contributions are confined near the boundary.

#### Case λ′ ≫ ω′ ≈ 1

*η*term enters the main balance instead, giving a wavelike structure of

_{nx}*λ*′ may or may not be present depending on the size of

*λ*′

*μ*, which becomes smaller with increasing mode number; we assume a small value here) together with a complementary function satisfying

#### Case λ′ ≫ ω′ ≫ 1

*δ*=

*μ*(

*λ*/

*ω*)

^{1/2}, which is almost always very small. There is a boundary layer at each side.

So in all cases of interest (except for the barotropic mode) the interior solution is either pure Ekman or a forced Rossby wave. The contribution to the MOC can come, therefore, from the interior Ekman, Rossby waves (perhaps damped) and the boundary layers.

^{1}

For low period forcing (up to 200 days), Willebrand et al. (1980) show that the ocean response to wind forcing is concentrated near the surface, although the presence of mean flow modifies this conclusion (Killworth and Blundell 2007).

^{2}

Indeed, Sime et al. (2006) note (p. 2256), “We anticipate that the Ekman-compensating return flow is poorly represented as being uniform in the along-sectional dimension.”

^{3}

Including a drag term on *u _{n}* makes no discernible difference to the results.

^{4}

There is also a term *β*/*fl* times this, but this factor is small poleward of 10°.

^{5}

The alternative approach of integrating (4) to give to leading order ∫^{L}_{0} *υ dx* = (*g*/*f* )*η _{n}*|

^{L}

_{0}− (

*LX*/

_{n}*f*) predicts a larger integrated barotropic flow than the method used above. This is due to deficiencies in the frozen beta-plane assumption.

^{6}

As noted above, in these frequency ranges there is no sign of a strong reduction in MOC.

^{7}

This is an oversimplification of what is actually calculated for MOC estimates, since these do not rule out the possibility that the thermal wind part of the calculation itself includes a wind-driven component.