## 1. Introduction

The ocean currents in the equatorial Pacific are significantly nonlinear. Contributing to this nonlinearity are eddies, such as tropical instability waves (TIWs) (e.g., Legeckis 1997; McCreary and Yu 1992; Baturin and Niiler 1997). TIWs appear as oscillations of the currents, sea level, and sea surface temperature in the eastern equatorial Pacific. These disturbances are mixed barotropic/baroclinic instabilities feeding on the kinetic and potential energy of the mean currents (Masina et al. 1999). In a model, if TIWs are not resolved, the zonal equatorial currents become too strong and must be damped by artificial means, such as by horizontal friction (Pezzi and Richards 2003; Richards and Edwards 2003). However, it is unclear how important these vigorous, small-scale space and time features are in modulating the background state of this climatically important region, and thus in the seasonal cycle of the tropical Pacific and the evolution of El Niño–Southern Oscillation.

Observational studies show that substantial volumes of thermocline water flow southward across the equator in the eastern equatorial Pacific Ocean (Wijffels 1993), influencing the water masses and strength of the current system in that region. In a fully nonlinear ocean model, Brown et al. (2007b) found that the southward cross-equatorial flow occurred at all depths east of 160°W, even at the surface and despite the northward mean wind stress. Furthermore, near the equator, the meridional flow in the surface layers was aided by the TIWs (Brown et al. 2007a). The observed and numerically modeled southward flow conflicts with Philander and Delecluse’s (1983) results from a near-equatorial linear model with northward wind stress saying that the surface flow is northward. Brown et al.’s analyses could not be applied right on the equator, however, as the Coriolis term is zero and therefore the zonal momentum equation they were studying was no longer applicable.

Beyond 6° of latitude of the equator, depth-integrated flow is almost completely described by the Sverdrup balance (Sverdrup 1947). Right at the equator the extended Sverdrup balance must be used (Kessler et al. 2003). The extended balance accounts for the nonlinear and frictional effects that drive water across the strong planetary potential vorticity gradients. Once across this potential vorticity “barrier” and out of the nonlinear and frictional region, the cross-equatorial flow must then join the flow on the other side that is driven principally by wind stress curl. As the ocean can achieve this in several ways, we first assess the relative roles of nonlinearity and friction in this balance. In particular, we show how our model run (with nonlinear effects from well-resolved TIWs and minimal horizontal friction) compares with that of Kessler et al.’s (2003) study (which had less nonlinearity from TIWs and larger applied friction).

Despite the contribution of TIWs to nonlinearity and meridional flow, not all models resolve these instabilities. Models can either overdamp these eddies through applying excessive viscosity or do not generate them because the fronts they develop are too weak. Since, without eddies, the zonal equatorial currents become too strong, more realistic mean flows are achieved through applying an enhanced friction scheme (Griffies 2004). To avoid this problem, and dampen the currents without damping the waves, Griffies and Hallberg (2000) devised a “biharmonic Smagorinsky friction,” incorporating a Smagorinsky coefficient whose magnitude depends upon the shear and strain imposed by the currents (Smagorinsky 1963, 1993). Biharmonic friction dissipates the energy only of those eddies with length scales close to the smallest that can be numerically resolved, so that it does not directly damp the large-scale flow, and allows the development of vigorous TIWs.

Biharmonic Smagorinsky friction schemes are computationally expensive. Low-resolution models with simple friction schemes are simpler to run and so it is often more convenient to parameterize the contribution of the missing TIWs using a computationally cheaper constant-coefficient Laplacian friction. In this paper we explore whether we can generate a “Laplacian friction” scheme for an ocean model that adequately replicates the damping induced by the TIWs in the mean, across seasons and for phases of ENSO.

## 2. Model

We use a global ocean general circulation model (OGCM) that is a version of the Australian Community Ocean Model (ACOM3) (Schiller 2004; Schiller and Godfrey 2003). The GCM is based on the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model 3 (MOM3) code (Pacanowski and Griffies 2000), forced with European Remote Sensing Satellite (ERS) scatterometer winds. Spatial resolution was a third of a degree in latitude and half a degree in longitude (ocean models require a mesh size of at least 1° to begin to resolve TIWs; Roberts et al. 2008). Of the 36 vertical levels, the top 10 are 10 m thick and increase to a thickness of 225 m by 1000-m depth. After an 8-yr spinup, forced by a seasonal climatology derived from the original forcing fields from the same period, our model was run from July 1992 to October 2000 with daily forcing. In the vertical, a hybrid Chen mixing scheme (Power et al. 1995) was applied, which incorporates a buoyancy-driven bulk Niiler–Kraus mixed layer component plus shear-driven gradient Richardson number mixing. Geographically, the Niiler–Krauss scheme dominates in the high latitudes, while gradient Richardson number mixing dominates near the equator (Schiller et al. 1998).

Net solar shortwave radiation and precipitation are input to an atmospheric boundary layer model (among other parameterizations). This model prognostically calculates air temperature in the boundary layer above the surface. Surface latent and sensible heat fluxes are then diagnosed by bulk formulas, as described by Schiller (2004).

In general, this model run was able to capture the main features of the equatorial flow: the Equatorial Undercurrent (EUC), the South Equatorial Current (SEC), and the North Equatorial Counter Current (NECC), with strengths close to observations (Johnson et al. 2002). The model’s performance is discussed in detail in Brown (2005) and Brown et al. (2007a,b). The nonlinear Smagorinsky biharmonic mixing scheme (Griffies and Hallberg 2000) in our model allows vigorous, realistic TIWs to form spontaneously, introducing sea level anomalies of around 10–15 cm. The “eddy fluxes” driven by these TIWs are the focus of our work.

## 3. The vorticity balance

To explore the mean dynamics at the equator, Kessler et al. (2003, hereafter KJM03) extended Sverdrup’s simple depth-integrated vorticity balance to include nonlinear advection and horizontal friction. Their technique involves recasting the nonlinear advection and friction terms as “forcing terms” in the form of a divergence of a vector stress. We further extend the KJM03 balance to include time dependence.

*βy*, where

*β*is a constant) areIn the above, (

*u*,

*υ*,

*w*) are the zonal (

*x*), meridional (

*y*), and vertical (

*z*) velocities;

*p*,

_{x}*p*are the zonal and meridional pressure gradients divided by the mean density;

_{y}**F**= (

*F*,

^{x}*F*) is a frictional body force in the ACOM3 model; and the zonal and meridional components of the vertical friction are represented by

^{y}*= (*

**τ**_{z}*τ*,

_{z}^{x}*τ*), which upon vertical integration is equivalent to the surface wind stress divided by the mean density.

_{z}^{y}*h*= 1000 m (which is taken to be a depth of no motion). At this depth of no motion, velocities

*u*,

*υ*, and

*w*are small. The free surface is neglected and the following equation is obtained:where the depth-integrated velocities are

Upon depth integration the terms (*wu*)* _{z}* and (

*wυ*)

*(which contain the vertical velocity) become small. We also neglect the meridional nonlinear terms (*

_{z}*uυ*)

*+ (*

_{x}*υυ*)

*, as these become small when we take the curl. Similarly, we find that the curl of the meridional wind stress and meridional friction is small, and that*

_{y}*V*is small compared to

_{x}*U*.

_{υ}*u*,

_{t}*υ*) in (1) and (2), this is very similar to the formulation of KJM03. Without nonlinearity, friction, and time dependency, Eq. (6) reduces to the zonal component of the standard Sverdrup equation

_{t}*V*= (1/

*β*)Curl(

*τ*).

^{x}Equation (4) and its approximate form (6) are not predictive, but are useful diagnostic tools for providing physical insight into the reasons for the existence of mean meridional flows, since each term on the right-hand side of (6) has a clear effect on the vorticity balance. Equation (6), first derived by KJM03, is another way of expressing the conservation of potential vorticity (see Pedlosky 1996, chapter 4). The curl of the nonlinear term and local time derivative together are equivalent to the change of relative vorticity following a fluid parcel.

Where the time-dependent term is negligible, we can construct a streamfunction based on (6) by calculating zonal velocity from volume conservation: *U _{x}* +

*V*= 0 (as done in KJM03). The individual contributions to

_{y}*V*of the curls of wind stress, nonlinearity, and friction can be calculated and reveal their individual contributions to the streamfunction.

### a. Eddies in the vorticity balance

*ũ*) and eddy (

*u*′) component of the currents, such that

*u*=

*ũ*+

*u*′, in a similar manner to Harrison et al. (2001). The nonlinear term in (6) expands toAveraging over many eddy time scales (denoted by the overbar), the smooth/eddy cross terms become negligible, giving the approximationWe have calculated the total term on the left-hand side of (8) and the first term on the right (smooth term), with the residual attributed to the eddy term. In the eddy term we find that ∫

*dz*dominates ∫

*dz*.

### b. The Laplacian parameterization

*A*) using the classic Reynolds stress parameterization:OGCMs often use constant values for

_{H}*A*.

_{H}In situations of primarily zonal currents (as found in our study region) the area integral of −(*u*′*υ*′)(∂*u*/∂*y*) is the energy transfer from the mean flow to eddies, which must be positive if the eddies are to grow (Gill 1982, chapter 13). This can only be true if *A _{H}* in (9) is positive in most locations. Unlike molecular viscosity, the magnitude of

*A*may vary strongly in time and space, depending on where and when the background velocity shears are dynamically unstable. Where

_{H}*A*is positive the eddies are acting as a friction on the larger-scale slower flows and only in this sense can

_{H}*A*properly be described as an “eddy viscosity.”

_{H}We found it necessary to ensure that all the derivatives used in the above relationships are accurately calculated in the native numerics of the model and at the model’s time steps. If, for example, a centered difference was used when the model had used a backward derivative, then small discrepancies are introduced and over many model time steps these errors accumulate and become large. This is particularly true if a spatial integral is performed to estimate transport. In the results we show below, terms of the vorticity balance are calculated at every model step and using the native model discretization.

## 4. Results

### a. Mean balance

We explore the mean vorticity balance by using Eq. (6) to quantify the depth-integrated meridional (and via continuity the zonal) transport driven by the curl of the wind stress, nonlinear advection, and friction (Fig. 1). In our model, we find the largest contributors to the mean depth-integrated meridional velocity are the curls of the nonlinear and wind stress terms (Figs. 1a,b). Together these two terms reproduce most of the model’s mean meridional transport (Figs. 1c,d). The explicit biharmonic Smagorinsky friction is an order of magnitude smaller (Fig. 1e). Consequently, the streamfunction generated by the combined wind stress and nonlinear curls (Fig. 2b) accounts for most of the model’s total mean, depth-integrated, zonal transport (Fig. 2c), justifying the underlying assumptions in Eq. (6).

As demonstrated by KJM03, nonlinear advection is just as important as the wind stress curl in explaining the mean, depth-integrated currents in the equatorial Pacific. The flow consists of the eastward EUC on the equator with a return SEC on either side of it, and a North Equatorial Current near 5°–6°N. Wind stress drives around 15–20 Sv of the eastward flow along the equator (Fig. 2a), but including the nonlinear term greatly increases the strength of the EUC and extends it farther eastward (Fig. 2b). The nonlinear term also dramatically influences the SEC. Without nonlinear contributions, the northern portion of the SEC is almost nonexistent and in the Southern Hemisphere it is positioned too far south (Fig. 2a versus 2b). Net cross-equatorial flow is also strongly determined by the nonlinear term in the eastern equatorial Pacific, making up over half of the total southward flow of 8.4 Sv east of 160°W (Table 1).

Another place where the nonlinear term affects the flow is the western Pacific; here the NECC is dynamically driven by both the wind stress curl and the eddy nonlinear term. The eddy nonlinear term (Fig. 3c) has significant values between 2° and 5°N from the TIWs and other eddies that form there (see Lyman et al. 2007). These nonlinear terms are responsible for strengthening the NECC west of 170°W and for moving it around 1° closer to the equator, as seen in Fig. 2. The eddies are as significant as the wind stress in driving and positioning the NECC in the western equatorial Pacific.

KJM03 and this study both use the same wind stress forcing data, but the nonlinear and friction curls differ markedly (cf. their Fig. 7 with our Fig. 1). KJM03’s model had significant contributions from the explicit horizontal friction term (their Fig. 7c) while, in contrast, ours has very little (Fig. 1e). More intriguing, however, is that the final circulation (their Fig. 7d, our Fig. 1d) is very similar in both models despite their distinct physical formulations. There are only three terms in the vorticity balance (wind stress, nonlinearity, and friction) and the wind stress is the same in both studies. Thus, much of what was ascribed to “friction” in KJM03’s simpler model is now seen to be nonlinearity.

KJM03 explored the idea that the eddy component of the nonlinear term was responsible for damping the strong zonal currents (see their Fig. 8). By recreating the nonlinear term from smoothed and high-frequency currents (see section 4a), they found the smooth term enhanced flow, while the high frequency damped it.

### b. Role of eddies in determining current strength and structure

Based on our time-scale decomposition of the flow field [Eq. (8)], we find that the smooth and eddy components of the nonlinear term have quite distinct spatial structures (Fig. 3): the magnitude of the smooth nonlinear term is strongest just south of the equator in the eastern Pacific (Fig. 3b). The magnitude of the eddy nonlinear term is also large just south of the equator (in the central and eastern Pacific) but also near 2°–3°N where the TIWs are most active (Fig. 3c).

From the perspective of the momentum equations [Eq. (1)], the smooth nonlinear term accelerates the EUC while the eddy term damps it as found by Harrison et al. (2001). KJM03, however, point out that it is not the actual value of the nonlinear term that is most important but its curl (right-hand side of Fig. 3). In our model, both the eddy and smooth nonlinear curls are roughly antisymmetric about a line that angles slightly south of eastward from the point 0°, 160°W (Figs. 3d–f). This antisymmetry line follows the position of the EUC maximum (see Brown et al. 2007b). Remarkably, the eddy and smooth nonlinear curls are of opposite sign (Figs. 3e,f) and thus largely cancel each other (Fig. 3d).

East of the date line, the smooth term contributes to a meridional flow toward the antisymmetry line, and the eddy term creates a meridional flow toward it. This view is confirmed when we derive the smooth and eddy nonlinear contributions to the streamfunction (Fig. 4): the smooth and eddy nonlinear streamfunctions grow larger toward the west, as they are zonal integrals of a zonally oriented curl. The smooth nonlinear term (Fig. 4b) enhances both the eastward flow of the EUC and the westward flow of the SEC. The eddy nonlinear term (Fig. 4c) acts in the opposite direction. When these two very large terms are combined, the resulting residual is about an order of magnitude smaller than either. The smooth term only just dominates the pattern of zonal transport, but the cross-equatorial flow is dominated by the eddy term (37.7 Sv southward driven by the eddy term compared to 33.2 Sv driven northward by the smooth term; Table 1).

In summary, we find in our model that the smooth nonlinear curl is very similar to the nonlinear curl found by KJM03, while the eddy nonlinear curl is similar to their friction. These similarities suggest that the eddy signal in the nonlinear term is acting very like the Laplacian friction used in KJM03’s model—can it be parameterized as such?

### c. Eddies as a damping

We test whether a Laplacian friction term acting on the model’s smooth currents can reproduce the strong effect of the eddy nonlinear term. First, the mean case is considered and then we explore temporal variability. We apply our analysis over the top 50 m of the ocean, which is the approximate depth of the mixed layer and region of influence of the TIWs (Baturin and Niiler 1997).

Spatially −(

*A*or

_{H}*A**

*by taking the ratio of the terms in Eq. (9) so thatand then averaging over the top 50 m (Fig. 5c). The expected large positive value for*

_{H}*A**

*appears in a band from the equator to around 4°N, coincident with the occurrence of growing TIWs, and south of the equator in the eastern Pacific. Regions where*

_{H}*A*is negative often coincide with areas where the original terms [i.e.,

_{H}*A**

*seen in Fig. 5c.*

_{H}Bryden and Brady (1989) estimated *A _{H}* from mooring observations near 152° and 110°W at the equator. They found

*A*to vary from 300 to 3000 m

_{H}^{2}s

^{−1}. In some instances, our values are much larger, with coherent regions reaching 10 000 m

^{2}s

^{−1}and above.

### d. Changes to the Laplacian parameterization over time

The equatorial Pacific climate system undergoes strong seasonal variability, driven by the north–south movement of the intertropical convergence zone and associated seasonal variations in the strength of the zonal currents (e.g., Tomczak and Godfrey 1994, chapter 1). Furthermore, the nonlinearity in the zonal momentum equation associated with TIWs varies by season (Brown et al. 2007a), with the active TIWs in the second half of the year inducing a surface southward flow that cannot be explained by linear dynamics alone. Thus, we expect that the behavior of the eddy nonlinear curl will also vary between these two time periods. For this reason, we consider the quasi-equilibrium case of each half-year, from January to June (low TIW activity) and from July to December (high TIW activity) (Figs. 6, 7), and again test whether the eddy nonlinear term can be approximately represented by a Laplacian friction term.

On these seasonal time scales, we find that the sum of the wind stress curl (Figs. 6, 7a), nonlinear curl (Figs. 6, 7b), and time-dependent terms (Figs. 6, 7c) is an effective estimate (Figs. 6, 7d) of the actual depth-integrated meridional velocity (Figs. 6, 7g).

In the western part of the basin, from 160°E to 160°W, the wind-driven flow in the first half of the year is strong and northward (Fig. 6a; Table 1), whereas in the second half, it is weakly southward. The northward flow in the first half of the year (Fig. 6b) is amplified by the nonlinear curl, while in the second half it contributes little.

In the eastern part of the basin, east of 150°W, the wind stress curl provides a net southward cross-equatorial transport for the whole year. However, the dominant forcing term is the nonlinear curl, which provides a farther southward cross-equatorial flow, particularly in the second half of the year (Table 1). As for the annual mean case, the eddy nonlinear term is very important in determining the flow at this time of year, especially the cross-equatorial flow.

On this half-yearly time scale, *A*** _{H}*, the ratio of the two expressions

*A**

*becomes stronger north of the equator in the eastern Pacific, consistent with the position and timing of TIW growth. Thus, using a constant*

_{H}*A*value year-round (Figs. 8c,f) would inadequately represent the time-variable effect of the eddies (Figs. 8b,e).

_{H}## 5. Discussion and conclusions

Through analyzing the depth-integrated vorticity balance, we demonstrated that eddy-induced nonlinearity from tropical instability waves (TIWs) greatly alters the structure and magnitude of the currents in the equatorial Pacific. Within 6° of latitude of the equator, TIWs assist in damping the Equatorial Undercurrent and South Equatorial Current. The damping introduced by TIWs has, qualitatively, a similar structure to a horizontal, constant-coefficient Laplacian scheme.

However, an accurate parameterization of the effect of TIWs by such a friction scheme would require a coefficient to double in strength when the waves are more active: temporally in the second half of the year and during La Niña, and spatially in the region where TIWs grow most actively (∼2°N, 140°–100°W).

The consequences of using a constant-coefficient Laplacian friction have been explored in a number of studies (e.g., Pezzi and Richards 2003; Roberts et al. 2009). It is understood that decreasing the friction improves the representation of TIWs, but results in an unrealistic strengthening of the Equatorial Undercurrent. Improving the resolution of TIWs also reduces the “cold tongue bias” at the equator and improves the atmospheric feedbacks (Chelton et al. 2001; Pezzi et al. 2004; Small et al. 2003). Friction schemes that are derived from lateral shears (such as a biharmonic Smagorinsky scheme) perform better in ocean models than do Laplacian schemes, although they have a higher computational cost (Richards and Edwards 2003; Roberts et al. 2009). Roberts et al. (2009) show that a Laplacian friction in a high-resolution model (⅓°) degrades the output to be roughly that of a lower-resolution model.

In addition to this understanding of friction at the equator, our study has demonstrated that the TIWs themselves act as a friction on the low-frequency, large-scale flow. By using a biharmonic Smagorinsky scheme, and allowing the TIWs to be well resolved, the waves themselves damp the strong equatorial currents. The resulting magnitude of the friction needed to damp the currents is therefore quite small. Additionally, allowing TIWs to create a natural friction means that the friction varies on spatial and temporal scales according to the instabilities generated by the strength of the currents and the active phases of the TIWs. To capture the correct timing and structure of the effective eddy friction, and hence the depth-integrated ocean currents, ocean models therefore need to resolve and represent the full effects of TIWs.

It would be possible to define the friction coefficient according to season. Such an approximation would be “ad hoc” and likely to be inadequate—the equatorial Pacific undergoes strong interannual variability in eddy activity that is as large as the seasonal cycle. For example, TIW activity is suppressed during El Niño events and enhanced during La Niña events (Yu and Liu 2003). The model’s nonlinear eddy term shows substantial variation through climatic cycles such as ENSO. To ensure that we are modeling oceanic flow patterns during ENSO events correctly, it is therefore essential that eddies be allowed to develop themselves and damp the system naturally.

Our Smagorinsky-based results should be assessed with caution. Perez and Kessler (2009) found that such a scheme, within a climatologically forced model, generated predictable TIWs that repeated exactly from one year to the next (i.e., forced waves), suggesting that the waves are not purely a response to instabilities within the ocean currents. While our findings for the damping effect of TIWs still hold, the details of how the waves impact the underlying three-dimensional flow should be assessed in other modeling studies.

A consensus is building that resolving TIWs is essential for simulating the equatorial Pacific ocean for both the mean and variability (Hansen and Paul 1984; Kessler et al. 1998; Jochum and Murtugudde 2004; Jochum et al. 2007; Pezzi et al. 2004). These waves not only change the flow structure, as in our results, but alter the heat budget of the ocean and the corresponding atmosphere and coupled feedbacks. Modeling studies that use a Laplacian friction must assess how well they simulate TIWs, with corresponding caveats on their results.

## Acknowledgments

We thank Billy Kessler for his careful reviews and advice to improve this paper. We also thank two editors, Vivienne Mawson and Angelika Hofmann, for their assistance in improving the clarity of this manuscript.

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Average cross-equatorial transports for the actual model transport and the individual contributions, as shown in Fig. 1 and described by Eq. (6). Transports are given in Sv across two sections, 160°E–160°W and 160°–80°W for the whole year and each half-year, as shown in Figs. 3, 6, and 7. Each contribution is shown for the full nonlinear curl, and then separated into its smooth and eddy components, as described by Eq. (8). Transports are given in Sv across two sections, 160°E–160°W and 160°–80°W.