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## 5. Zonal momentum balance

In this section we examine the dynamics of month-to-month variations described above through a diagnosis of the zonal momentum equation.

### a. Diagnostic equation

We can write the zonal momentum equation as

(1)

where u, v and w are zonal, meridional and vertical velocity components, p is pressure, (= 10 kg m) is density, f is the Coriolis parameter, A is vertical eddy viscosity and K is horizontal eddy viscosity. Subscripts denote differentiation and the operator is a horizontal gradient operator. We can neglect the Coriolis term on the equator since f = 0 there. Horizontal diffusion will also be neglected because even though it may be important (e.g., Hansen and Paul 1984; Bryden et al. 1986), it cannot be estimated from only equatorial data.

Theory indicates that meridional advection of momentum is likely to be important in the dynamics of the Undercurrent (see section 2 and Philander and Pacanowski 1980). We would expect therefore that the term vu will be significantly nonzero in the latitudinally sheared zone of the EUC north and south of the equator. This term can also be large near the surface where meridional winds can produce significant cross-equatorial flows (e.g., Cane 1979). However, at the depth of the EUC on the equator, vu will be locally small relative to the pressure gradient since on seasonal time scales meridional velocity variations are typically only O(1 cm s) (e.g., Fig. 3) and u 0 by symmetry (Lukas and Firing 1984). Hence, we neglect vu in (1).

The vertical diffusion term is not easily estimated from our data because 1) the eddy viscosity is highly variable with depth and precisely known (e.g., Peters et al. 1988) and 2) second derivatives of u would be very noisy computed from data at only seven depths. However, integrating from 0 m to 250 m, we get

(2)

where the overbar denotes depth integral, e.g.,

and is the zonal wind stress. For consistency with pressure and stress integrals, 10 m velocity data are first linearly extrapolated to the surface. Differences between velocity integrals to 0 m and to 10 m are only a few percent, however, so our results are insensitive to the choice of upper bound. Equation (2), which governs zonal transport variability, now requires an estimate of A at only one depth (250 m) and an estimate of zonal velocity shear instead of curvature.

We have data from discrete times and depths at the mooring locations from which to calculate terms in (2). We therefore use finite differences in time to estimate acceleration and in depth to estimate vertical shear. We also use finite differences in the zonal direction to estimate the zonal pressure gradient and the zonal advection. To compare two-point zonal gradient estimates of velocity and pressure with other terms in (2), we average zonally to get

(3)

The angle brackets denote zonal average, e.g.,

where x and x are eastern and western longitudes and x = x - x. For convenience, we will designate the three intervals of 110°-124.5°W, 124.5°-140°W and 110°-140°W as 117°, 132° and 125°W, respectively.

Averaging zonal velocity across 15° and 30° of longitude is reasonable given the high correlations noted for between 110° and 140°W in the previous section. However, there is an error involved in estimating averages in (3) across an interval (x, x) using data from just the end points (Bryden 1977). As an extreme example of how this can affect our results, consider the 60-90 day wave for which one quarter wavelength fits between 110°and 140°W. Using Bryden's formulae for a pure progressive sine wave, averages in the angle brackets would be too low by about 20% relative to the true average. For less extreme variations across the array, as for example 1/10 to 1/20 of a cycle in transport at 1 cpy, estimates of average amplitude would be reduced by 10% or less. Average gradients, i.e. ( - )/(2 x), are exact.

Zonally averaging zonal wind stress for use in (3) poses a problem because of the large data gaps in the records at each longitude. However, one can see from the solid curves in Fig. 5 that there is a high correlation between records at adjacent mooring sites. Between 110° and 124°W this correlation is 0.96 for 14 monthly mean pairs; and between 124° and 140°W it is 0.94 for 8 monthly mean pairs. Hence, we can use regression analysis based on adjacent locations to fill record gaps for improved zonal averages. These regression fills are shown in Fig. 5 by the dashed lines. Fills were made only for "nearest neighbor" sites since the very tight regression weakens over 30 degrees of longitude to 0.63 between 100° and 140°W.

The nonlinear term ½u has been calculated by first forming products of daily data, then smoothing with the 51-day Hanning filter. Vertical velocity is too weak to measure directly and we have no reliable way of estimating it from our data. We attempt calculation of time mean vertical advection in (3) though, using an estimate of w from Bryden and Brady's (1985) diagnostic model of the steady equatorial circulation in the region 110° to 150°W. This mean velocity (their Fig. 7) has a maximum upwelling of about 3 × 10 cm s at 60 m, crosses zero at about 180 m and shows downwelling at a rate of 1 × 10 cm s at 250 m. Vertical velocity profiles similar in magnitude and structure have been estimated from mooring data in the equatorial Pacific at 110°W (Halpern and Freitag 1987) and between 110°-150°W (Bryden et al. 1986).

### b. Results

Figure 11 shows examples of pressure gradient, local acceleration, and zonal advection before integration over depth for the interval centered at 125°W. Also shown in Fig. 11c is our estimated profile of time mean vertical advection. It is interesting to note that the mean profiles in Figs. 11a and 11c are quantitatively and qualitatively similar to profiles from Bryden and Brady's (1985) diagnostic model of the mean circulation between 110° and 150°W. Their results were based solely on hydrographic data in contrast to ours which are based principally on time series measurements.

Figure 11. Time series estimates of (a) zonal pressure gradient, p, relative to 250 m, (b) local acceleration, u, and (c) zonal advection, ½u for the interval centered at 125°W. Contour interval is 1.0 × 10 N m, though the ±0.5 × 10 N m contour is also drawn. Means are estimated from a 6-parameter regression fit. Dashed line in the mean panel of (c) is the vertical advection term WU.

Zonal pressure gradient fluctuations (Fig. 11a) are vertically coherent with largest amplitudes of O(10 N m) in the upper 100 m, decreasing to zero at the 250 m reference level. The gradient is always negative with seasonal minima in boreal spring of each year. However, 60-90 day waves tend to obscure seasonal variations, especially in the latter part of 1984. A distinct interannual increase in the gradient is seen between 1983-84 and 1985-86.

Fluctuations in local acceleration of O(10 N m) (Fig. 11b) are comparable to those of the pressure gradient. The 60-90 day periodicity is dominant and tends to be vertically coherent as for the pressure gradient. There is no obvious evidence of a seasonal variation in U; nor are there trends in velocity over the record length as indicated by the near zero mean of acceleration.

Zonal advection changes sign at the approximate depth of the EUC core and is comparable to or larger than the pressure gradient below about 100 m (Fig. 11c). It exhibits an annual cycle between 50-100 m which reflects vertical movement of the EUC core across these depths (Fig. 6). Also evident is variability at 60-90 day periods of O(10 N m) in the upper 250 m, as for zonal pressure gradient and local acceleration.

Mean zonal and vertical advection tend to oppose one another above and below the EUC core (Fig. 11c), consistent with the fact that the EUC shoals towards the east. The tendency for the two terms to cancel suggests that nonlinearity on the zonal plane is important in redistributing momentum in the upper 250 m. However, the cancellation is not complete since near the surface there is a large mean upward advection of eastward momentum from the depth of the EUC core. This leads to a reduction of westward flow at the surface and a large zonal transport along the equator relative to that expected from linear theory (q.v., Philander and Pacanowski 1980).

Figure 12 shows estimates of the terms in the depth integrated momentum balance (3) for the intervals centered at 117°, 125° and 132°W. For convenience we write (3) in a simplified notation given by

(3')

Record gaps in the wind stress time series indicate when zonally averaged stress could not be calculated with at least 50% observed data based on the solid curves in Fig. 5. Thus for example, we did not estimate averaged stress in late 1984 and early 1985 across the intervals centered at 117° or 125°W. Overplotted smooth lines are reconstituted time series based on the mean, trend, and 1 cpy harmonics from the regression analyses (Tables 2, 3). The 2 cpy was not included because, except for the zonal winds, this harmonic was generally insignificant.

Figure 12. Estimates (in N m) of terms in (3) for the intervals centered at (a) 117°, (b) 125° and (c) 132°W. Shown in the uppermost panel at each location is ten times the vertical shear stress at 250 m. Also shown as dotted and long dashed lines are time series reconstituted from the least squares determined trend, mean, and 1 cpy variations.

##### If your browser cannot view the following table correctly, click this link for a GIF image of Table 2.
Longitude 117 Nov 83–Nov 84 –24.1 ± 1.8 –21.9 ± 3.9 X X X 117 May 84–Sep 85 X –27.2 ± 3.4 (–0.3 ± 7.0) –5.7 ± 0.9 –13.2 125 Nov 83–Nov 84 –29.5 ± 2.1 –28.7 ± 3.1 (–2.3 ± 6.7) –7.8 ± 2.6 –16.1 125 June 85–May 86 –55.5 ± 3.6 –49.1 ± 4.3 (0.7 ± 9.8) –4.9 ± 2.0 –17.8 132 Nov 83–Nov 84 –34.4 ± 2.5 –34.3 ± 3.9 X X X 132 May 84–Sep 85 –64.3 ± 2.1 –52.8 ± 4.8 (0.5 ± 6.5) –3.2 ± 1.1 –22.5
##### If your browser cannot view the following table correctly, click this link for a GIF image of Table 3.
Longitude Dates P U U 117 Nov 83–Nov 84 Amplitude 12.0 ± 3.5 14.0 ± 9.9 X X Phase 113 ± 29 97 ± 47 X X 117 May 84–Sep 85 Amplitude X 8.8 ± 5.1 (7.8 ± 9.7) (1.1 ± 1.4) Phase X 99 ± 31 (41 ± 80) (251 ± 72) 125 Nov 83–Nov 84 Amplitude 13.7 ± 3.8 13.2 ± 8.6 (6.9 ± 23.7) (2.6 ± 5.9) Phase 115 ± 29 89 ± 36 (45 ± 85) (289 ± 178) 125 June 85–May 86 Amplitude 19.9 ± 9.7 11.6 ± 11.2 (13.9 ± 37.5) (3.2 ± 6.5) Phase 98 ± 34 100 ± 71 (50 ± 62) (262 ± 89) 132 Nov 83–Nov 84 Amplitude 15.1 ± 4.7 14.4 ± 11.9 X X Phase 120 ± 33 76 ± 37 X X 132 May 84–Sep 85 Amplitude 24.2 ± 3.0 (5.0 ± 7.2) (8.5 ± 9.1) 1.6 ± 1.5 Phase 149 ± 7 (133 ± 79) (44 ± 67) 36 ± 60

Shown in the uppermost frame at each location is ten times the averaged zonal shear stress at 250 m calculated using an eddy viscosity of A = 10 m s. This choice of A is based on recent microstructure measurements in the thermocline of the eastern equatorial Pacific (Peters et al. 1988). Though subject to considerable uncertainty and derived from data no deeper than 140 m, this value of A leads to estimates of stress at 250 m that are 100 times smaller than at the surface. Thus, we will neglect in subsequent discussion since it is unlikely to affect our conclusions.

To examine the relationship between zonal wind stress and pressure gradient, we first subsampled the time series every 31 days, then generated scatter diagrams and linear regression fits for the intervals centered at 117°, 125° and 132°W (Fig. 13). The correlation between stress and gradient is positive in all three intervals and is >0.80 at 132° and 125°W. The slope of the regression lines in these latter two intervals indicates that on a month-to-month basis, is 10%-20% larger than P. The inequality could be due to a combination of the shallow 250 m reference level for P , uncertainty in the wind stress drag coefficient, and the importance of nonlinearity. The lower correlation between P and in the interval centered at 117°W vis-a-vis 132°W may reflect the fact that the zonal wind stress weakens towards the east (e.g., Fig. 5) so that remote effects take on a larger significance there.

Figure 13. Regression analysis of P and at (a) 117°W from November 1983 to August 1985; (b) 125°W from November 1983 to April 1986; and (c) 132°W from November 1983 to September 1985. Correlation coefficients (r) and number of monthly data (n) are shown.

Harmonic analyses for simultaneous records centered at 117°, 125° and 132°W are summarized in Tables 2 and 3. Two separate analyses were done at 125°W for approximately year long records bracketing the wind data gap in 1984-85. Similarly, two analyses were done for the intervals centered at 117°W and 132°W, one for November 1983 to November 1984, and one for May 1984 to September 1985.

In the mean, P with a maximum difference of 20%; in general, though, these two terms are not different from one another to within one standard error (Table 2). Both the mean wind stress and the pressure gradient are significantly stronger in the interval centered at 132°W as compared to 117°W (consistent with Fig. 1). The nonlinear terms are smaller than either P or with zonal (vertical) advection being 10%-30% (35%-60%) of the pressure gradient. The sign of the nonlinear terms is always negative implying acceleration of flow to the east, which is consistent with the observation that the mean eastward transports along the equator are larger than expected from linear theory. Together ½U and WU tend to overcompensate for the difference between wind stress and pressure gradient by about -10 × 10 to -20 × 10 N m. This imbalance could be due to the cumulative error involved in estimating terms in (3). However, it could also be due to the neglect of lateral turbulent diffusion due to instability waves which lead to an effective zonal stress of about O(-10 × 10 N m) in the upper 200-300 m of the eastern tropical Pacific (Hansen and Paul 1984; Bryden et al. 1986).

Table 2 also indicates an interannual change in both the zonal wind stress and pressure gradient which nearly double between 110° and 140°W from 1983-84 to 1985-86. Thus the increase in thermocline depth and dynamic height over the record length (Figs. 8b, 10b) can be dynamically related to the increasing strength of the trades. The warmer SSTs in the latter half of the temperature record in the eastern Pacific (q.v., Fig. 8) may be related to the large scale pressure field adjustment to increased winds, since increasing the slope of the thermocline would remove the cold water reservoir further from the surface. This could reduce the efficiency of upwelling, leading to a rise in SST in spite of an expected cooling tendency due to stronger meridional Ekman divergence and zonal advection.

Table 3 presents data on the annual harmonic in each of the three intervals centered at 117°, 125° and 132°W. We note that zonal wind stress and pressure gradient are generally comparable in magnitude and phase if one takes into account the uncertainty of our estimates due to sampling errors. Strongest (weakest) easterlies and zonal pressure gradients occur from September to November (March to May) depending on location and specific time period. However, the zonal pressure gradient in the interval centered at 132°W does not exceed one standard error for the period May 1984 to September 1985. At this time and location, the 60-90 day waves are most strongly developed in pressure gradient and are of sufficient amplitude to obscure the annual cycle. In contrast, there is an approximate balance between P and over the first year of data at 132°W when the waves are weaker.

The local acceleration term, U, may be comparable at times to the zonal pressure gradient at 1 cpy, e.g., from June 1985 to May 1986 in the interval centered at 125°W. However, estimates of local acceleration are highly uncertain and, unlike zonal pressure gradient, never exceed one standard error. Also, the zonal advection term does not appear to be significant at 1 cpy. These results suggest an equilibrium balance between zonal wind stress and pressure gradient at the annual period.

Harmonic analysis indicates that variations at 2 cpy are generally smaller than at 1 cpy and with few exceptions (notably in wind stress) statistically insignificant. Thus, while it is likely that wind-driven 2 cpy variations occur in the eastern equatorial Pacific based on previous historical data analyses and modeling studies (e.g., Meyers 1979; Kindle 1979; Busalacchi and O'Brien 1980), we cannot critically examine the dynamics at this frequency with our data. Conversely, the residuals about our six-parameter regression equation show a consistent pattern. This is illustrated for the interval centered at 132°W during May 1984-September 1985 when 60-90 day wave variability was especially pronounced (Fig. 14); similar results apply to the intervals centered at 125° and 117°W.

Figure 14. Residuals about the mean, trend, 1 and 2 cpy regression for terms in the dynamical balance (3) at 132°W. Numbers adjacent to the time series are root-mean-square variations in N m; maximum cross correlation with U is shown in parentheses. Maxima generally occur at nonzero lag, with U leading P, , and ½U by 12, -3, an -1 days, respectively.

Figure 14 shows that residuals around the six-parameter regression fit are dominated by the 60-90 day period, with the largest amplitude fluctuations in P and U. The maximum correlation between residuals of P and U is -0.66, which is significant at the 95% level of confidence. Residual U leads P by 12 days, which could be due to a phase shift introduced by averaging U across longitude in (3). The zonal pressure gradient is weaker than the local acceleration as would be expected if the residual variability were due to low baroclinic model Kelvin waves, since the 250 m reference level is too shallow to capture deep modal pressure variations. Indeed, referencing U to 250 m brings the magnitude of these two terms into closer agreement. Zonal wind stress variations are typically weaker than variations in either P or U, and maximum correlation between residual and U is only 0.42 (not significant at the 95% level). Thus, it appears that the residual variance is remotely forced rather than locally forced. The amplitude of the nonlinear term is only about 15% the amplitude of the local acceleration term, but significantly negatively correlated (-0.66) with U. Hence, nonlinearity is significant, though of secondary importance on this time scale.

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