University of South Florida Scholar Commons
Marine Science Faculty Publications College of Marine Science
6-1997
The Zonal Momentum Balance of the Equatorial Undercurrent in the Central Pacific
L. Qiao University of South Florida
Robert H. Weisberg University of South Florida, [email protected]
Follow this and additional works at: https://scholarcommons.usf.edu/msc_facpub
Part of the Marine Biology Commons
Scholar Commons Citation Qiao, L. and Weisberg, Robert H., "The Zonal Momentum Balance of the Equatorial Undercurrent in the Central Pacific" (1997). Marine Science Faculty Publications. 147. https://scholarcommons.usf.edu/msc_facpub/147
This Article is brought to you for free and open access by the College of Marine Science at Scholar Commons. It has been accepted for inclusion in Marine Science Faculty Publications by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. 1094 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
The Zonal Momentum Balance of the Equatorial Undercurrent in the Central Paci®c
L. QIAO AND R. H. WEISBERG Department of Marine Science, University of South Florida, St. Petersburg, Florida (Manuscript received 29 February 1996, in ®nal form 25 November 1996)
ABSTRACT Current velocity data from an array of subsurface moorings deployed during the Tropical Instability Wave Experiment from May 1990 to June 1991 are used to diagnose the upper-ocean zonal momentum balance at 0Њ, 140ЊW. The ¯ow ®eld and associated zonal momentum ¯ux divergence are fully three-dimensional over the upper 250 m, consistent with the earliest descriptions and theoretical ideas of the Equatorial Undercurrent (EUC). Estimates of the vertical stress divergence show dynamical ¯ow regimes that change between the surface and the base of the EUC, being essentially linear (modi®ed by nonlinearity) near the surface, weakly nonlinear at the EUC core, and fully nonlinear below the core. The vertical stress divergence is much larger over the lower portion of the EUC than previously reported, but this is consistent with the observed downstream deceleration of the EUC and the idea that vertical mixing is important in maintaining the thermostad. Nonlinearity becomes increasingly important with decreasing frequency, but tends to cancel upon vertical integration.
1. Introduction Global Atmosphere (TOGA) Tropical Instability Wave Experiment (TIWE). The array provides estimates of the The discovery by Cromwell et al. (1954) of a very vertical circulation (Weisberg and Qiao 1996, unpub- swift, subsurface current, ¯owing eastward on the equa- lished manuscript) and thus a three-dimensional view of tor in opposition to the winds, initiated an ongoing di- the ¯ow ®eld. The present paper uses these data to es- alog on the dynamics of this remarkable Equatorial Un- timate the zonal momentum ¯ux divergence and, com- dercurrent (EUC). Early descriptive studies by Knauss bined with other data from the TOGA-Tropical Atmo- (1960, 1966) showed the EUC to be continuous along, sphere Ocean (TOGA-TAO) array, to diagnose the upper- symmetric about, and tightly con®ned to the equator ocean zonal momentum balance. Section 2 reviews pre- with transports comparable to other major ocean cur- vious work on this topic. Section 3 describes the data rents. Early theoretical studies, beginning with the non- and methods. Section 4 attempts a quantitative diagnosis linear, inertial jet arguments of Fofonoff and Montgom- of the record-length averaged, depth-dependent zonal ery (1955) and the linear, frictional arguments of Arthur momentum balance and provides a description of how (1960) and Stommel (1960), were followed by numer- the dynamics change between the surface and the base ous articles combining these arguments into more com- of the EUC. Section 5 offers a more qualitative (owing plete theories. Central to all of these is the three-di- to data limitations) view of the time-dependent variations. mensionality of the ¯ow ®eld, driven by a depth-de- These ®ndings are then summarized and discussed in pendent zonal pressure gradient (ZPG) whose vertical section 6. integral tends to balance a westward surface wind stress. This three-dimensionality is what makes the circulation so important to contemporary climate-related studies, 2. Background because it largely determines the equatorial sea surface temperature distribution. But, it is also what makes a Knauss (1960, 1966) gives a comprehensive descrip- quantitative understanding of the EUC so dif®cult since tion of the EUC along with dynamical inferences. The it requires resolving the circulation's divergence. ¯ow is three-dimensional with a meridional conver- In May 1990, an array of ®ve subsurface acoustic gence upon the EUC core compensated by a vertical Doppler current pro®ling moorings was deployed about divergence away from the core. Intense vertical mixing 0Њ, 140ЊW for 13 months as part of the Tropical Oceans is surmised on the equator to account for both the ob- served material property distributions in the meridional plane and the approximate geostrophic balance (meri- dionally) found for the near-equator zonal currents. Me- Corresponding author address: Dr. Robert H. Weisberg, Depart- ment of Marine Science, University of South Florida, 140 Seventh ridional convergence upon the EUC core is the essential Avenue South, St. Petersburg, FL 33701-5016. element in Fofonoff and Montgomery (1955) where the E-mail: [email protected] speed of the EUC is accounted for by conservation of
᭧1997 American Meteorological Society JUNE 1997 QIAO AND WEISBERG 1095 absolute vorticity. This convergence is attributed to an of the ZPG and the estimated nonlinear accelerations eastward directed ZPG force owing to a westward wind were very similar to those of Bryden and Brady (1985), stress over a bounded basin. Recognizing that the effects as was the imbalance in the integrated ZPG and the of the wind-induced surface stress may extend vertically surface stress. The mean accelerations tended to oppose over the same region for which the ZPG is dynamically each other, suggesting that nonlinearity redistributes signi®cant, Arthur (1960) calculated a velocity pro®le momentum vertically in the zonal plane within the upper on the equator from the balance between the vertical 250 m. Assuming a small, constant vertical eddy vis- Ϫ4 2 Ϫ1 stress divergence and the ZPG. Thus, the EUC core cosity coef®cient Av ϭ 1 ϫ 10 m s (motivated by occurs where the stress is zero and the stress divergence the microstructure measurements of Peters et al. 1988), crosses zero together with the ZPG. Charney (1960) and the vertical stress at 250 m was calculated to be 100 Charney and Spiegel (1971) combined these inertial and times smaller than the surface stress. Thus, the vertically viscous effects in a constant density EUC model and integrated imbalance was not resolved. A time-depen- found that the relative importance of these terms greatly dent analysis showed that on intraseasonal timescales affects the resulting three-dimensional ¯ow ®eld. the vertically integrated ZPG varied with the surface Theory and observations con®rm that the zonal mo- stress to within about the same imbalance as the mean. mentum balance on the equator must entail conver- In an attempt to resolve the role of turbulent stress gences of momentum ¯ux and stress along with a ZPG, divergence, Hebert et al. (1991) used shipboard micro- but further advances have been hampered by data lim- structure and moored measurements collected between itations. Simultaneous data have been unavailable for 140ЊW and 110ЊW in spring 1987. With zonal advection estimating these constituents, leaving uncertainty in the being the only calculable nonlinear acceleration term, zonal momentum balance for both analytical and nu- correspondences were not achieved between the esti- merical model results. mated ZPG, acceleration, and turbulent stress diver- In a diagnostic study, Bryden and Brady (1985) used gence. It was suggested that annual averages are nec- historical hydrographic data for a box bounded by 5ЊS essary for comparing estimates of turbulent stress di- and 5ЊN, 150ЊW and 110ЊW, and 500 db and the surface. vergence with the diagnostic calculation of Bryden and The mean horizontal pressure gradient was referenced Brady (1985). to 500 db; the horizontal velocity components were es- Turbulent stress divergence occurs over synoptic as timated from the horizontal pressure gradient compo- well as microstructure and intermediate scales. At syn- nents and the climatological wind stress using geo- optic scales, the tropical instability waves are particu- strophic and Ekman assumptions, and the vertical ve- larly important. For example, the horizontal Reynolds locity component (w) was then calculated by mass con- stress divergence on the equator in the central Paci®c servation. The ZPG decreased monotonically to zero is a signi®cant fraction of the wind-stress-induced body between the surface and the lower portion of the EUC force (Hansen and Paul 1984), and similarly for the and was slightly westward below. Its vertical integral Atlantic (Weisberg and Weingartner 1988). Additional balanced the surface wind stress to within about 80%. supporting evidence is found in Lukas (1987), Wilson The vertical integral of the nonlinear accelerations over- and Leetmaa (1988), Bryden and Brady (1989), and compensated the surface wind stress/ZPG imbalance, Luther and Johnson (1990). However, even when av- implying a signi®cant stress (and stress divergence) at eraging over a record length suf®cient to include the least to the base of the EUC. On the equator upwelling synoptic scale (Johnson and Luther 1994), the role of was found above 180 db with smaller downwelling be- the turbulent stress divergence on the equator has re- low, and the ¯ow was described as being primarily along mained an unresolved issue. isopycnals. In summary, the available datasets show an approx- The w pro®le of Bryden and Brady (1985) has been imate balance on the equator between the vertically in- used in subsequent studies for estimating the vertical tegrated ZPG force and the surface stress. Estimates of advection of eastward momentum. An example is given the nonlinear acceleration terms suggest that these tend by Wilson and Leetmaa (1988), employing data from to cancel, but not completely. Since the nonlinear ac- several shipboard velocity pro®le and hydrographic sur- celerations may be important in adjusting the ¯ow ®eld veys on the equator roughly between 150ЊW and 90ЊW. to external forcing, their resolution is necessary for de- The time-dependent variations for the estimated terms termining the vertical distribution of stress divergence were found to be as large as their means. Upon vertical and hence an improved understanding of the equatorial integration the ZPG closely balanced the wind stress, currents' zonal momentum balance. but the lack of data on individual terms precluded anal- yses on the vertical pro®les. However, their estimate of 3. Data and methods vertical eddy viscosity showed large values below the EUC core. Another example is that of McPhaden and The TIWE equatorial moorings, designated TIW1± Taft (1988), wherein the zonal momentum balance is TIW5, were deployed in a diamond shaped array cen- studied between 140ЊW and 110ЊW using TOGA-TAO tered upon 0Њ, 140ЊW (Fig. 1). Hourly velocity pro®les array moored current meter data. The vertical pro®les were sampled by RD-Instruments 150-kHz acoustic 1096 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
FIG. 1. The location of the TIWE equatorial array in relation to the tropical Paci®c Ocean's climatological SST distribution for September (Courtesy of M. McCarty and M. McPhaden, NOAA/PMEL).
Doppler current pro®lers (ADCP) with 20Њ transducer ®ltered to exclude oscillations at timescales shorter than con®guration. Mooring performance and data editing 10 days. The u-component isotachs are representative procedures are given in Weisberg et al. (1991). With of the zonally oriented equatorial currents reported on ambient sound speed correction, the instruments nom- by numerous precedent studies. Observed is a highly inally sampled at 9-m vertical intervals. Hourly velocity variable, near-surface-con®ned South Equatorial Cur- components resampled at 10-m intervals between 250 rent (SEC) overriding the EUC whose high speed core m and the surface are used herein. Resampling was by is located within the thermocline (Fig. 6). The ®ve lo- linear interpolation between 250 and 30 m (the last bin cations show an EUC that is maximum on, and nearly unbiased by surface re¯ection) and by linear extrapo- symmetric about, the equator and a SEC that is maxi- lation over the upper 20 m using the 30±40 m shear. mum to the north of the equator. The primary variations The analysis period is 12 May 1990 to 18 June 1991 in the EUC are both annual and intraseasonal. Annually, and the mooring locations and nominal instrument the depth of the EUC's high speed core varies with the depths are listed in Table 1. thermocline with maximum EUC speed observed in July The time and depth variations of the zonal (u) and 1990 and April 1991 when the core was relatively shal- meridional () velocity components at the ®ve mooring low. In contrast, an intraseasonal maximum is observed locations are shown in Figs. 2 and 3, where the data in December 1990 when the core was relatively deep. here and in subsequent time series plots are low-pass In agreement with previous studies the SEC is most developed when the EUC is deepest, except when in- TABLE 1. TIWE equatorial array mooring positions and nominal tervened upon by large intraseasonal events such as one instrument depths. in December l990. When the EUC is shallow, the SEC Mooring Position Instrument depth is weak, and westward ¯ow was generally absent on the name (lat/long) (m) equator from April to June 1991. TIW1 0Њ 01.4ЈN 273.6 In contrast to u, the component consists of season- 141Њ 50.6ЈW ally modulated, higher frequency oscillations. In par- TIW2 0Њ 57.8ЈS 280.5 ticular, a series of regular, large amplitude oscillations 139Њ 57.5ЈW are observed at all of the sample locations from August TIW3 0Њ 02.4ЈN 281.5 137Њ 57.7ЈW to December 1990. These are the tropical instability TIW4 0Њ 03.2ЈS 276.5 waves and a description of their kinematics during this 140Њ 08.4ЈW time is given by Qiao and Weisberg (1995). The - TIW5 1Њ 01.5ЈN 266.4 component oscillations are largest on the equator within 139Њ 57.4ЈW the westward-¯owing SEC, with amplitudes decreasing JUNE 1997 QIAO AND WEISBERG 1097 and all time series have been low-pass 1 Ϫ component, with westward ¯ow denoted by light stippling and eastward ¯ow u W: (a) Њ component, with northward ¯ow denoted by stippling. The contour intervals are 0.2 m s v denoted by dark stippling and (b) 1 Ϫ . 2. Horizontal velocity components as a function of depth and time for moorings along 140 IG F greater than 0.8 m s ®ltered to exclude ¯uctuations at timescales shorter than 10 days. 1098 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 . 3. As in Fig. 2 but along the equator. IG F JUNE 1997 QIAO AND WEISBERG 1099
FIG. 4. Record-length-mean vertical pro®les for the u and v components at the ®ve mooring locations. The solid lines denote the three equator moorings (the thick one being 140ЊW) and the dashed and dotted lines denote moorings north and south of the equator, respectively. precipitously across the thermocline to relatively small and meridional derivatives followed by vertical inte- values at the EUC core. gration using trapezoidal rule such that Mean horizontal velocity component vertical pro®les 0 are obtained by averaging over the 13-month record u3152Ϫ u Ϫ w(z)ϭw(0) ϩϩ⌬z, (2) length (Fig. 4). For the u component the three equatorial zx31Ϫxy 52Ϫy locations are similar, except that the speed at the EUC core increases downstream as the core depth shoals to where the subscripts denote the station locations and ⌬z the east. The core speeds at the two off-equator locations is 10 m. A rigid-lid approximation is used for w(0), Ϫ2 are nearly symmetric and about 60% of the values on which is correct to a factor of about 10 for the synoptic the equator. The SEC is shallow at all locations but or longer timescales considered. penetrates slightly deeper off the equator. For the com- The resulting (low-pass ®ltered) w at 0Њ, 140ЊWis ponent, the off-equator locations indicate antisymmetric shown as a function of time and depth in Fig. 5 and the behavior consistent with a surface Ekman divergence corresponding record-length-averaged vertical pro®les and a subsurface geostrophic convergence. The three for u, , and w are given in Fig. 6. The low-frequency Ϫ5 Ϫ4 Ϫ1 locations on the equator all show similar vertical pro®les variations in w are of order 10 to 10 ms . The intermediate between the off-equator ones so the mean mean w pro®le shows maximum upwelling of about 2.3 Ϫ5 Ϫ1 meridional divergence is nearly symmetric about the ϫ 10 ms at 60-m depth and a zero crossing at 140 equator. At 250 m, or the base of the EUC, all of the m just below the EUC core. The lower portion of the component pro®les approach zero. EUC is thus a region of downwelling on average. The Vertically integrating the continuity equation between ¯uctuations in w are a factor of 5±10 larger than the the surface and depth z provides an estimate w: mean and these may be of the same sign over the entire region of the water column sampled. A speci®c example is the intraseasonal event in December 1990, whenץ uץ 0 w(z) ϭ w(0) ϩϩdz, (1) downwelling is observed at the leading edge of an east- yץxץ ͵ z ward momentum pulse followed by upwelling at the where x, y, and z are positive to the east, north, and up, trailing edge, consistent with the passage of an equa- respectively. Central differences are used for the zonal torial Kelvin wave. A more detailed discussion of w, 1100 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
FIG.5.Thewcomponent as a function of depth and time estimated at 0Њ, 140ЊW. Upwelling is denoted by stippling and the contour interval is 5 ϫ 10Ϫ5 msϪ1. The time series have been low-pass ®ltered to exclude ¯uctuations at timescales shorter than 10 days. including error estimates, comparisons with previous that the potential for systematic errors owing to compass estimates, and implications regarding the upper-ocean calibration when vertically integrated to 250 m are an heat balance are in preparation (Weisberg and Qiao order of magnitude less than the estimated mean w there, 1996, unpublished manuscript). Here we note that the and that kinematic, dynamic, and thermodynamic con- mean w pro®le is robust with regard to random errors, sistencies argue against large ®nite differencing errors. This is important since the downwelling estimated be- low the EUC core (with magnitude similar to that of the upwelling above the core) is an essential element in the momentum arguments that follow. Temperature and wind data used for estimating the ZPG and the surface wind stress, respectively, are from the TOGA-TAO array. Figure 7 shows (low-pass ®l- tered) temperature as a function of time and depth on the equator at 170ЊW, 140ЊW, and 125ЊW. Consistent with previous observations, the thermocline slopes up to the east giving rise to the depth-dependent ZPG. Su- perimposed on the isotherms at 170Њ and 140ЊW is the EUC core depth, which coincides and shoals with the thermocline. Also, the depth of penetration for the west- ward SEC decreases eastward as the mixed layer be- comes shallower.
4. The mean zonal momentum balance Beginning with the zonal momentum balance in the form uץץ Pץ u 1ץ uץ uץ uץ ϩuϩϩwϪfϭϪ ϩ zץzץ xץz ץ yץ xץ tץ (١u), (3)´ϩ١
FIG. 6. The record-length-mean vertical pro®les of the u (thin sol- where f is the Coriolis parameter, p is pressure, is is a horizontal gradient ١ id), (dashed), and w (thick solid) components at 0Њ, 140ЊW. density, is viscosity, and JUNE 1997 QIAO AND WEISBERG 1101
FIG. 7. Isotherm depths as a function of time from moorings at 0Њ, 125ЊW; 0Њ,140ЊW; and 0Њ,170ЊW (from the TOGA- TAO array courtesy of M. McPhaden, NOAA/PMEL). Superimposed at 0Њ,170ЊWand0Њ,140ЊW are the EUC core .z ϭ 0 and the westward-¯owing SEC regions denoted by stipplingץ/uץ depths de®ned by 1102 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 operator, and then averaging over the record length, sional nature of the nonlinear zonal momentum ¯ux. gives the mean equation This same symmetry, however, biases the ¯ux diver- gence scheme relative to the advective scheme. For Uthese reasons the ¯ux divergence scheme results areץץ Pץ u 1ץ uץ uץ u ϩ ϩw ϪfVϭϪ ϩ A z presented ®rst to develop the three-dimensionality ofץzvץ xץz Όץ yץ xץΌΌΌ the zonal momentum ¯ux. This is followed by the ad- -Ah١U), (4) vective scheme results, and the bias for the ¯ux diver)´ϩ١ where the angle brackets denote the record-length av- gence scheme is accounted for in appendix B. Esti- erage, capitalized variables are the record-length means, mation is then made of the ZPG and the zonal wind x and Av and Ah are vertical and horizontal eddy viscosity stress (0 ) and the results are combined into vertically coef®cients dependent upon the averaging interval. integrated and pointwise momentum balances from Each of the nonlinear acceleration terms may be de- which the vertical stress divergence follows as a resid- composed into mean circulation and Reynolds stress ual. Integrating the vertical stress divergence then re- terms as sults in vertical pro®les of stress and Av. uЈץ Uץ uץ u ϭ U ϩ uЈ a. Material acceleration by ¯ux divergence x formulationץx Όץ xץΌ uЈ Vertical pro®les of the nonlinear terms of (5) areץ Uץ uץ ϭV ϩ Ј ,y shown in Fig. 8. They are arranged such that the leftץy Όץ yץΌ central, and right columns provide the divergence of uЈ means, the divergence of the Reynolds ¯uxes, and theץ Uץ uץ w ϭW ϩ wЈ , (5) divergence of their sums, respectively, and the rows z ץz Όץ zץΌ from top to bottom provide the zonal, meridional, and where the primes denote (hourly) ¯uctuations about the the vertical derivatives comprising these mean diver- record-length means. gences. Standard deviations due to random variations The nonlinear acceleration terms in the above for- (derived in appendix A) are indicated by dashed lines. mulation are expressed using an advective scheme. Al- Relative to these, all of the estimated nonlinear terms ternatively, using the continuity equation, these may be are signi®cantly different from zero at most depths (ex- y) and the shapes and zero-crossings ofץ/uЈЈ͗͘ץ expressed using a ¯ux divergence scheme, wherein their cepting record-length averages are the vertical pro®les are robust. x is nearly zeroץ/UUץ The mean zonal derivative UWץ UVץ UUץuw͗͘ץu͗͘ץuu͗͘ץ ϩϩ ϭϩϩ above 40 m, positive from 40 m to 140 m, and negative z below 140 m. The zero-crossing at 140 m (about 30 mץyץxץzץyץxץ uЈЈ͘ below the EUC core) is consistent with the fact that the͗ץ uЈuЈ͗͘ץ ϩϩ ¯ow within the EUC core accelerates as the core shoals x y downstream (Fig. 4) at this location. This ®nding of a ץ ץ uЈwЈ͘ zero-crossing below the core differs from the Bryden͗ץ ϩ . (6) -z and Brady (1985) and McPhaden and Taft (1988) ®ndץ ings of zero-crossings either at or above the core. Zonal In either case, central differences are used to estimate ®nite differences for these previous studies, however, the horizontal derivatives; thus, for the advective and were over 30Њ of longitude, a distance over which the ¯ux divergence schemes the acceleration terms become vertical position of the core changed on a scale com- parable to the measurement's vertical resolution. With pq p 33 q Ϫ pq 11ץqq3Ϫq 1ץ x is observedץ/UUץ p ϭp4 or ϭ higher zonal and vertical resolution xxx xxx -31Ϫ to be maximum and positive at the EUC core and neg ץ 31Ϫ ץ ative below the EUC core. pq p 55 q Ϫ pq 22ץqq5Ϫq 2ץ p ϭp4 or ϭ , The vertical pro®le of the mean meridional derivative yyϪyץ yyϪyץ -y is negative with largest magnitudes at the surץ/UVץ 52 52 where p and q represent the appropriate horizontal ve- face and at the EUC core. At the surface, this is due to locity components and subscripts denote horizontal lo- the Ekman divergence of westward momentum away cations (Table 1). The vertical derivatives are calculated from equator and at the EUC core, this is due to the by forward differencing consistent with the w estima- geostrophic convergence of eastward momentum onto tion. the equator. The minimum at 40 m marks the transition Given the mean circulation's symmetry about the between Ekman divergence and geostrophic conver- y then approaches zeroץ/UVץ y͘ being gence dominance, andץ/uץ͗ equator (that results in both ͗͘ and appproximately zero on the equator) the ¯ux divergence again below 220 m as the geostrophic convergence goes scheme is necessary to reveal the fully three-dimen- to zero at the base of the EUC. JUNE 1997 QIAO AND WEISBERG 1103
FIG. 8. Vertical pro®les of the individual constituents making up the record-length-mean zonal momentum ¯ux divergence estimated about 0Њ, 140ЊW. The dashed lines represent standard deviations by random ¯uctuations (appendix A). 1104 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
The vertical pro®le of the mean vertical derivative Fig. 9. The divergence of the mean circulation zonal V U is positive and maximum within ´ ١ y with rel- momentum ¯uxץ/UVץ z varies in opposition to that ofץ/UWץ ative maxima and minima of opposite sign occurring at the EUC core and negative near the surface and below the same depths. In particular, within the EUC core, the the EUC core. The divergence of the Reynolds mo- vЈuЈ͘ is positive and maximum near the͗´١ divergence of eastward momentum by the vertical cir- mentum ¯ux culation nearly cancels the convergence of eastward mo- surface, negative at the EUC core, and relatively small mentum by the meridional circulation. Moreover, all of but negative below the EUC core. Upon summation, the these mean derivative terms are of the same magnitude, total divergence of the mean zonal momentum ¯ux -vu͘ is small near the surface, large and positive with͗´١ showing that the mean divergence about the equator of the mean eastward momentum is fully three-dimen- in the EUC core, and equally large but negative below sional. the EUC core. This development shows that the record- The mean divergences of the zonal momentum ¯ux length-averaged divergence of the zonal momentum ¯ux owing to the Reynolds ¯uxes (the three central panels at 0Њ, 140ЊW is fully three-dimensional. The contribu- in Fig. 8) are generally smaller than, and tend to oppose, tions by the mean circulation are the largest, but the their corresponding mean circulation terms (the three contributions by the Reynolds ¯uxes cannot be ignored, left panels of Fig. 8). Intuitively, the Reynolds ¯uxes particularly between the EUC core and the surface. The vu͘ pro®le, of order 3 ϫ 10Ϫ7 m͗´١ act to smooth the mean gradients. An exception is the magnitude of the UU/ sϪ2, is comparable to that of the ZPG. Bias, due to theץ x, which adds toץ/uЈuЈ͗͘ץ near-surface behavior of uЈuЈ͘/ curvature in U is largest at the EUC core. However, this͗ץ ,x between 30 and 90 m and below 150 m. Alsoץ x above 50 m. can be quanti®ed (appendix B) and corrected, givingץ/UUץ x is larger thanץ Because the Reynolds ¯ux divergences are relatively ¯ux divergence scheme results that are nearly identical small, when added to the mean circulation momentum to those from the advective scheme presented next. ¯ux divergence terms, the total mean momentum ¯ux divergence terms (the three right panels of Fig. 8) retain b. Material acceleration by advective formulation their general shapes, but with small changes in mag- nitude and extrema depths. With respect to the EUC, Using the same format as Fig. 8, Fig. 10 shows the zonal momentum converges meridionally upon the core individual terms calculated by the advective scheme. and divergences vertically away from the core. Zonal Term by term the advective and ¯ux divergence scheme momentum diverges zonally within and above the core results are different. The exception is the top row where where the EUC accelerates downstream and converges the zonal derivative terms by the advective scheme are zonally below the core where the EUC decelerates roughly half the magnitude of those by the ¯ux diver- downstream. Adding these terms together gives the ma- gence scheme as expected. Zonally, within and above terial rate of change of zonal momentum, as shown in the EUC core ¯uid accelerates downstream, while below
FIG. 9. Vertical pro®les of the record-length-mean zonal momentum ¯ux divergence by the mean circulation, the resolvable Reynolds stresses, and their sum. The dashed lines represent standard deviations by random ¯uctuations (appendix A). JUNE 1997 QIAO AND WEISBERG 1105
FIG. 10. Vertical pro®les of the individual constituents making up the record-length mean advection of zonal momentum estimated at 0Њ, 140ЊW. The dashed lines represent standard deviations by random ¯uctuations (appendix A). 1106 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
FIG. 11. Vertical pro®les of the record-length-mean advection of zonal momentum by the mean circulation, the resolvable ¯uctuations, and their sum. The dashed lines represent standard deviations by random ¯uctuations (appendix A). the core ¯uid decelerates downstream. Vertically, ¯uid shown in Fig. 11. When compared with the ¯ux diver- decelerates as it diverges away from the core. However, gence scheme results, the Reynolds stress contributions it is not meaningful to compare the individual terms for are found to be nearly identical. The mean circulation these schemes since it is only the summation of all terms contribution, on the other hand, is smaller within the that constitutes the material rate of change of zonal mo- EUC core since that is where the velocity component mentum. The sum of the three component terms are thus curvature is largest. Elsewhere, the two forms are very similar. Upon correcting the ¯ux divergence bias (see appendix B) the discrepancy at the EUC core is ac- counted for, so consistent results are obtained from the two very different formulations. From this comparison between zonal-momentum-¯ux diagnostic schemes it is clear that considering individual terms without the context of their total summation makes little sense because of the three-dimensionality in the material rate of change. It is inappropriate to mix formulations between terms because uninterpretable bias errors result and regardless of formulation the ver- tical circulation is a critical factor.
c. Zonal pressure gradient and wind stress The TOGA-TAO array temperature data on the equa- tor at 125ЊW, 140ЊW, and 170ЊW are used for estimating the ZPG. Salinity is held constant at 35 ppt for lack of data, and using constant salinity versus a historical T/S relationship does not result in signi®cant error (e.g., Weisberg and Weingartner 1986). When referenced to 250 m, the ZPG calculated for the three station pairs are shown in Fig. 12. For the 125Њ±170ЊW and 140Њ± 170ЊW pairs the mean ZPG force is eastward every- where above 250 m, while for the 125Њ±140ЊW pair it reverses to very small westward values below 180 m. FIG. 12. Vertical pro®les of mean zonal pressure gradients estimated between 170Њ and 140ЊW (dotted), 140Њ and 125ЊW (dashed), and The vertical pro®les of the ZPG are consistent with 170Њ and 125ЊW (solid) using a reference level of 250 m. previous measurements (e.g., Mangum and Hayes 1984; JUNE 1997 QIAO AND WEISBERG 1107
FIG. 13. Mean vertical pro®les for vertically integrated constituents of the zonal momentum balance at ∫∫∫000 ;(xdz(long dashedץ/uץtdz(thin solid); uץ/uץ ;(0Њ, 140ЊW. The left panel includes zzzdu/dt dz (thick solid ∫∫00 ∫ 0 xdzestimated betweenץ/pץzdz(dotted). The right panel includes zϪץ/uץydz(short dashed); and wץ/uץzz 170Њ±140ЊW (dotted), 140Њ±125ЊW (dashed), and 170Њ±125ЊW (solid).
-z, a vertically integrated analץ/xץ Bryden and Brady 1985; Weisberg and Weingartner problem of unknown 1986; McPhaden and Taft 1988). From Mangum and ysis between the surface and any depth z is considered Hayes (1984), which presents eight individual pro®les ®rst. The diagnostic equation is of the ZPG estimated between 150Њ and 110ЊW refer- P͗x͘ץenced to 1000 db over the 2-yr period 1979±81, it is 0001 ١u͘dz Ϫ fVdzϭϪ dz ϩ 0´v͗ observed that choosing a reference level anywhere be- Όx 0 ץ͵͵͵ tween 200 and 1000 db would affect the mean ZPG zzz U0ץ ,pro®le by less than 0.2 ϫ 10Ϫ4 NmϪ3. Consequently ,١U) dz A)´ϪAϩ١ z Η ͵ hץ choosing the reference level to be 250 m herein (con- v zz sistent with Bryden and Brady 1985 and McPhaden and (7) Taft 1988) should not be the limiting factor for the dy- x namical inferences to be drawn. where ͗͘0 is the mean zonal wind stress and 0 is the Following Large and Pond (1981), and in the same mean surface density. The Coriolis term is zero on the manner as McPhaden and Taft (1988), the zonal com- equator and the horizontal stress divergence is assumed ponent of wind stress is calculated after transforming small relative to the vertical stress divergence. It is noted the buoy winds measured at 4 m to winds at the standard that the horizontal stress divergence is explicitly con- 10-m height (under neutrally stable conditions) using a tained within the estimated Reynolds stress divergence drag coef®cient of 1.2 ϫ 10Ϫ3 and an air density of 1.2 for scales of motion that are resolved. Thus, neglecting kg mϪ3. The resulting record-length-averaged zonal the horizontal stress divergence in (7) only neglects ef- wind stress component at 0Њ, 140ЊWis6.1ϫ10Ϫ2 N fects from unresolved scales. Since the tropical insta- mϪ2. bility waves are the major source of Reynolds stress at all of the ®ve mooring locations, it follows that ne- glecting the unresolved horizontal stress divergence d. Vertically integrated zonal momentum balance term here and in subsequent analyses should not be a The large, vertically sheared currents on the equator signi®cant source of error. The integrated ͗Du/Dt͘, the suggest a balance between the ZPG and the vertical individual terms forming it, and the integrated ZPG are z), with any imbalance resulting shown as functions of depth on Fig. 13. The integratedץ/xץ) stress divergence in a material acceleration ͗Du/Dt͘. To circumvent the ZPG increases monotonically with depth, reaching the 1108 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
z͘ isץ/xץ͗ FIG. 14. Mean vertical pro®les at 0Њ, 140ЊW of (left panel) the three terms making up the zonal momentum balance where x obtained as the residual of the ZPG and ͗Du/Dt͘; (center panel) ͗ ͘; and (right panel) Av. The dotted, dashed, and solid lines represent results using the ZPG estimated between 170Њ±140ЊW, 140Њ±125ЊW, and 170Њ±125ЊW, respectively.
x estimated value of ͗͘0 /0within about 10 m of the EUC an independent physical consistency check on the es- core (depending upon station pair). Below the core the timation of ͗Du/Dt͘. Along with quanti®able random integrated ZPG attains a nearly constant value. The zon- errors (appendix A), it may now be argued that the ®nite- al and meridional terms constituting ͗Du/Dt͘ tend to differencing errors are not overwhelming either. cancel with the vertical term, so upon integration the acceleration remains small until below the core where e. The implied vertical stress distribution it reaches a relatively large negative value. Across the core, roughly between 80 to 160 m, the magnitude of Assuming a balance between the ZPG force, ͗Du/ z͘, the latter term may be estimated as aץ/xץ͗ the integrated ͗Du/Dt͘ is relatively constant at less than Dt͘, and x 10% the magnitude of ͗͘0/0. residual as shown in Fig. 14. Near the surface the bal- Two important points follow. First, when integrated ance is primarily between the eastward directed ZPG -z͘. This is modץ/xץ͗ vertically to the vicinity of the EUC core (located in force and the westward directed the thermocline) the momentum balance is linear to i®ed by ͗Du/Dt͘ shaping the near-surface stress diver- within 10%. This explains why linear, reduced-gravity gence. Thus, at 50 m where ͗Du/Dt͘ has a westward models of the equatorial thermocline's variability in re- directed maximum because ¯uid particles decelerate as x sponse to0 perform well (e.g., Busalacchi and O'Brien they move upward away from the EUC core, there must 1981; Weisberg and Tang 1990) despite the presence of be a corresponding westward directed maximum in ver- nonlinear currents. Second, the relatively small, constant tical friction to account for this deceleration. The in- magnitude of the integrated ͗Du/Dt͘ near the EUC core crease in ͗Du/Dt͘ from 50 m to the surface is due to is dynamically consistent with the fact that ͗x͘/ ϭ both a decrease in the vertical velocity and positive y͘). By 20-m depthץ/uЈץz ϭ 0 at the core. From (7), if the integrated ZPG Reynolds stress (in particular ͗Јץ/UץAv x equals Ϫ͗0 ͘/0 at the vicinity of the core, as observed, (the mean westward ¯owing SEC region) the Reynolds then ͗x͘/ must equal the integrated ͗Du/Dt͘ (neglecting stress is suf®cient to reverse the sign of ͗Du/Dt͘, thereby the horizontal Reynolds stress divergence due to un- accelerating the ¯ow (see Fig. 11) making it less west- z͘ pro®leץ/xץ͗ resolved scales, as argued previously). With ͗x͘/ ϭ ward downstream. A similar near-surface
-z ϭ 0 at the core, it follows that the integrated was reported by Wacongne (1989) in a numerical cirץ/UץAv ͗Du/Dt͘ should also be zero. Since this occurs within culation model analysis applied to the equatorial Atlan- x 10 m of the core, or to within 10% of ͗͘0 /0, we have tic Ocean and such may be inferred from the Bryden JUNE 1997 QIAO AND WEISBERG 1109 and Brady (1985) results. This behavior, resulting from 5. Temporal evolution of the zonal momentum nonlinearity, is now understood in terms of the speci®c balance processes that give rise to the observed ͗Du/Dt͘, the underlying cause being vertical advection, but equally Since the relative magnitudes and spatial structures important in setting the magnitude and causing a change of the SEC and EUC are functions of time, the relative in direction near the surface is the Reynolds stress. importance of the various terms making up the zonal momentum balance may also be time dependent. Of Below 50 m, as the ZPG force decreases and ͗Du/ particular interest are the timescales over which non- Dt͘ increases with depth toward the EUC core, these linearity is important. Figure 15 shows the terms making two terms tend to offset resulting in a minimum west- x up Du/Dt as a function of time and depth, computed by -z͘ at the core. Thus, the balance advective formulation using raw hourly data. For preץ/ ץ͗ ward directed within the EUC core, where the ¯ow accelerates down- sentation, these time series along with others in this stream, is weakly nonlinear with the sum of similar section (except the ZPG) have been low-pass ®ltered to x .z͘ bal- exclude ¯uctuations at timescales shorter than 10 daysץ/ ץ͗ magnitude eastward ͗Du/Dt͘ and westward ancing the eastward directed ZPG force. Below the EUC The same calculation using low-pass ®ltered time series core, where the ZPG approaches zero, the balance be- gave nearly identical results, showing that the essential -z͘. It is within nonlinearities over the spatial scale of the TIWE equaץ/xץ͗ comes one between ͗Du/Dt͘ and this regime that the EUC decelerates downstream re- torial array occur over timescales longer than 10 days quiring a frictional retarding force in the absence of a and that higher-frequency motions do not bias the non- suf®ciently large westward directed ZPG force. linear calculations. -١u is vertically inhom´To summarize, the dynamical ¯ow regimes change in The nonlinear acceleration v going from the surface through the base of the EUC. ogenous and time dependent. It tends to be positive with The zonal momentum balance goes from one in which largest variability within the EUC core (which migrated the pressure gradient force balances the wind-induced between 80 and 150 m) and negative above and below t has largest magnitudeץ/uץ ,١u´frictional force near the surface (essentially a linear re- the core. Compared to v gime modi®ed by nonlinearity), to a weakly nonlinear near the surface and relatively uniform sign with depth. regime at the EUC core where the pressure gradient A notable event is the December 1990 eastward mo- force drives a nonlinear acceleration equal in magnitude mentum pulse. While the local acceleration is well de- to the frictional retarding force, to a fully nonlinear ®ned and large, the nonlinear acceleration is as large at regime below the core in which a relative maximum in the EUC core. Such pulses are generally identi®ed as the frictional retarding force decelerates the EUC down- intraseasonal, linear, equatorial Kelvin waves (e.g., stream. Knox and Halpern 1982; McPhaden and Taft 1988) with -z͘ gives an phase lines traceable across the equatorial Paci®c (Kesץ/xץ͗ Integrating the residual-determined estimate of the ͗x͘ pro®le, as shown in Fig. 14. Con- sler et al. 1995). While linearity reasonably describes ١u may´sistent with the summary above, ͗x͘ is large, negative, the vertical integral (Fig. 18), it is found that v .t at individual depthsץ/uץ and monotonically increasing from the surface to the be as large as Of the terms making up v´ u, w u/ z has the largest ץ ץ ١ EUC core. A zero-crossing occurs within 10 m of the ¯uctuations that tend to reverse sign across the EUC core, below which ͗x͘ is positive (as higher momentum yץ/uץx tends to be largest in the core, and ץ/uץcore, u ¯uid above is rubbing against lower momentum ¯uid tends to be largest near the surface. These are all com- below) and monotonically increasing. This physically -t. As with the recordץ/uץ parable in magnitude with required change in sign at the core occurs independent ١u is also fully´length average, the time dependence of v of the ͗Du/Dt͘ estimation, providing some con®dence three-dimensional. that the errors, both random and systematic, are not The time dependence of Du/Dt should re¯ect changes x z provides an esti- in external forcing. Figure 16 shows the ZPG referencedץ/Uץcontrolling. Dividing ͗ ͘ by x ±z͘ is to 250-m depth estimated between 170Њ±140ЊW, 140Њץ/ ץ͗ mate of Av (except near the EUC core where 2 2 .z , since the shear is zero), as shown 125ЊW, and 170Њ±125ЊW as functions of time and depthץ/U ץdivided by in Fig. 14. This results in values of 40±50 (ϫ 10Ϫ4 m2 The large data gaps and inadequate zonal resolution sϪ1) in the near-surface mixed layer decreasing to values preclude analyses as performed for the record-length around 3 ϫ 10Ϫ4 m2 sϪ1 within the EUC core and then mean, but some qualitative comparisons can be made. increasing again below the core. Over the range 150± The two relative maxima and intervening minimum in x z͘ are maximum, Av is the ZPG from July through October 1990 correspondץ/ ץ͗ m, where ͗Du/Dt͘ and 200 estimated between 10±20 (ϫ 10Ϫ4 m2 sϪ1). It is noted to the relative maxima and intervening minimum in the that while the station pairs used for the estimation of EUC core speed shown in Fig. 2, and the ZPG maximum the ZPG affects the magnitude of the stress divergence, in December 1990 corresponds to the Kelvin wave pulse the stress, and the eddy coef®cient estimates of Fig. 14, at that time. Thus, the zonal momentum within the EUC the different station pairs do not affect the pro®le shapes responds to changes in the ZPG as expected since an z adjusts to strike aץ/xץ or the implications that follow from these. acceleration must occur until 1110 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
t), and individual constituentsץ/uץ) ١u), the local acceleration´FIG. 15. The nonlinear acceleration (v z) as functions of depth and time. Positiveץ/uץy, and wץ/uץx, vץ/uץmaking up the nonlinear acceleration (u values are stippled, the contour interval is 5 ϫ 10Ϫ7 msϪ2, and all time series have been low-pass ®ltered to exclude ¯uctuations on timescales shorter than 10 days. new balance. However, with a complicated superposi- Kelvin waves and a three-dimensional mean background tion of local (wind stressÐsee Fig. 19) and remote circulation ®eld increases with decreasing frequency. ١u tend to´pressure gradient) forcing, the time-dependent adjust- As with the means, the time-dependent v) .z cannot be inferred from the present da- cancel upon vertical integration. This is shown in Figץ/xץ ments of .١u´t, and vץ/uץ ,taset, primarily due to inadequate ZPG resolution. 18, comparing the integrated Du/Dt ١u is small above the EUC core and´Over what frequency range does nonlinearity become The integrated v important? To address this question, variance density then increases in magnitude to maximum values just .١u are shown as a function of below the core (at about 150 m) before decreasing again´t and vץ/uץ spectra of frequency and depth in Fig. 17. At frequencies higher Figure 19 provides a qualitative comparison between /uץ ,t spectral densities are at least the vertically integrated (surface to 250 m) Du/Dtץ/uץ than those shown, the an order of magnitude larger than those of v´ u. The /uץ t, ZPG, andx . The vertically integrated Du/Dt andץ ١ ١u) spectra decrease (increase) with decreasing 0´t (vץ/uץ t are similar and there is considerable correspondence ץ frequency. Comparable magnitudes begin to occur in between these accelerations and thex ¯uctuations. Lim- the EUC for frequencies between 0.01 cph and the in- 0 stability wave frequencies of about 0.002 cph. At lower ited correspondence with the ZPG is also evident; par- t, es- ticularly for an event in December 1990 attributable toץ/uץ ١u spectra exceed those of´frequencies the v pecially in the EUC core. The EUC thus becomes in- an intraseasonal Kelvin wave that propagated through creasingly nonlinear approaching the mean. This is com- the array, but since the TOGA-TAO array does not re- patible with the ®nding of Johnson and McPhaden solve synoptic-scale ¯uctuations in the ZPG, as ob- (1993) that the nonlinear interaction between equatorial served in the wind and acceleration, the time-varying JUNE 1997 QIAO AND WEISBERG 1111
FIG.15.(Continued) zonal momentum balance at these scales cannot be di- length (13 months) means and ¯uctuations. In either agnosed. case the ¯ow is three-dimensional, tending to converge meridionally upon, and diverge vertically away from, the EUC core, consistent with the ®rst comprehensive 6. Summary and discussion descriptions of the equatorial circulation by Knauss Current velocity data from the TIWE equatorial array (1960, 1966) and the ®rst theory for the EUC by Fo- have been used to diagnose the upper-ocean zonal mo- fonoff and Montgomery (1955). The ensuing vertical mentum balance at 0Њ, 140ЊW. The array resolves the circulation is found to be a critical element of the zonal large-scale divergence of the upper-ocean currents, momentum balance. The record-length average, dynam- thereby permitting an estimation of the zonal momen- ical ¯ow regime changes between the surface and the tum ¯ux divergence, the unknown factor in previous base of the EUC. At the surface it is essentially linear, empirical studies. Analyses are made of the record- with the ZPG force balancing the wind-induced fric- 1112 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
FIG. 16. The ZPG estimated between 170±140ЊW, 140Њ±125ЊW, and 170Њ±125ЊW as functions of depth and time. The small region of westward-directed pressure gradient force is stippled and the contour interval is 10Ϫ4 NmϪ3. tional force modi®ed by nonlinearity. At the EUC core with the visco±inertial theory of Charney and Spiegel it is weakly nonlinear, with the ZPG force driving a (1971) in which downward vertical advection is nec- nonlinear acceleration equal in magnitude to the fric- essary to balance friction in the absence of a suf®ciently tional retarding force. Below the core (where the ZPG large ZPG. is nominally small or slightly westward) it is fully non- As with the ¯ow ®eld, the nonlinearities are three- linear, with the frictional retarding force accounting for dimensional, requiring all three components to char- the downstream deceleration of the EUC, consistent acterize the zonal momentum ¯ux divergence. The mean JUNE 1997 QIAO AND WEISBERG 1113
١u as functions of depth and frequency. The´t and vץ/uץ FIG. 17. Log variance densities for contour interval is 0.5. Light stippling highlights densities greater than 10Ϫ0.5 (m sϪ2)2 cphϪ1. The spectra were averaged over a 0.92 ϫ 10Ϫ3 cph bandwidth for approximately 18 degrees of freedom.
/xץ͗ͦ zonal momentum ¯ux divergence also contains impor- nitude of this acceleration. Below the EUC core zͦ͘ increases again to a relative maximum necessary toץ -tant Reynolds stress divergences found to occur pri marily over the tropical instability wave scales. Their account for the downstream deceleration of the EUC effect in the mean zonal momentum ¯ux divergence is there. The implied vertical pro®le of Av has near-surface to decelerate downstream both the surface SEC (making values of 40±50 (ϫ 10Ϫ4 m2 sϪ1), decreasing to around it less westward) and the subsurface EUC (making it 3 ϫ 10Ϫ4 m2 sϪ1 at the EUC core and increasing again less eastward), with the horizontal (vertical) Reynolds to 10±20 (ϫ 10Ϫ4 m2 sϪ1) within the deceleration region ¯ux working against the SEC (EUC). below the core. Wilson and Leetmaa (1988) reported a z͘ pro®le is largely affected by the mean similar pro®le. The minimum at the core is consistentץ/xץ͗ The zonal momentum ¯ux divergence. The mean vertical with the thermocline inhibiting vertical mixing and the advection of eastward momentum combines with the estimate agrees with that of Wyrtki and Bennett (1963) divergence of the horizontal Reynolds ¯uxes to produce arrived at from simple energy considerations using a .zͦ͘ maximum at 50-m depth. Without these non- similar momentum balanceץ/xץ͗ͦ a linear effects, particularly the mean vertical advection Quantitatively, the near-surface results for Av agree -zͦ͘ would be a maximum reasonably well with results from numerous microstrucץ/xץ͗ͦ ,of eastward momentum at the surface as recently found at midlatitude by Cher- ture investigations (e.g., Gregg 1987; Peters et al. 1988; .(zͦ͘ is a minimum Dillon et al. 1989; Hebert et al. 1991; Lien et al. 1995ץ/xץ͗ͦ eskin (1995). At the EUC core owing to the downstream acceleration of the EUC, and However, they differ greatly at and below the EUC core it would even be smaller were it not for the resolvable where microstructure measurements suggest values at vertical Reynolds ¯ux divergence that reduces the mag- least an order of magnitude smaller. With the observed 1114 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
∫∫∫000 ١u´tdz) and nonlinear( vץ/uץ ) FIG. 18. The vertically integrated zonal acceleration ( zzzDu/Dt dz) and its local dz) constituents as functions of lower limit of integration and time. Positive values are stippled and the contour interval is 5 ϫ 10Ϫ5 m2 sϪ2. downstream deceleration of the EUC below the core be inconsistent with the observed meridional mass con- demanding a suf®ciently large westward directed force, vergence that does not approach zero until the base of three possible explanations are offered: 1) there exists the EUC below 230 m. There is no basis herein for on average a large (Ͼ2 ϫ 10Ϫ4 NmϪ3), westward di- discussion of the second, but the third may have merit. rected ZPG force below the EUC core; 2) microstructure Friction ultimately occurs on molecular scale. Eddy fric- measurements within the EUC are incorrect; or 3) mi- tion is just a parameterization for unresolved scales. crostructure measurements do not resolve the scales of When resolved, these scales result in quanti®able Reyn- z͘ across the EUC. In the olds ¯uxes, which for the present measurements occurץ/xץ͗ motion that produce absense of observational evidence (e.g., references cited on synoptic scales. So, it is possible that the gap between in section 4c), the ®rst seems unlikely, and it would also the synoptic and the microstructure scales may contain JUNE 1997 QIAO AND WEISBERG 1115
FIG. 19. Time series of the vertically integrated (0±250 m) ZPG, local acceleration, total acceleration, and surface zonal wind stress. other physical processes causing eddy momentum ¯ux the ͗Du/Dt͘ estimate, relatively small errors in the in- x divergence. Regardless of such conjecture, something dependently estimated ZPG or0 do not alter the in- is necessary to decelerate the ¯ow and to account for ferences drawn. the mixing of heat (the thermostad) and all other ma- The dynamical inferences drawn for the region below terial properties below the EUC core (e.g., Jones 1973; the EUC core are different from the numerical model Wyrtki and Kilonsky 1984; although other opinions on results of Wacongne (1989) and Yin and Sarachik thermostad formation have also been offered, e.g., Lu- (1993). In these model cases, however, the magnitude kas 1986; McPhaden 1984). In this regard, it is noted of the frictional force below the EUC core and its par- that the TIWE dataset gives consistent results for the tition between horizontal and vertical stress divergences mixing of heat below the EUC core (Weisberg and Qiao are essentially preset by the subgrid-scale turbulence 1996, unpublished manuscript). Also, by not altering parameterizations (a constant horizontal eddy diffusivity the shapes of the curves in Fig. 14, which derive from and a Richardson-number-dependent vertical eddy dif- 1116 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 fusivity). The a priori results of such parameterizations In summary, the TIWE equatorial array descriptions are relatively small (when compared with our results) of the three-dimensional circulation and zonal momen- frictional forces below the EUC core and hence no need tum ¯ux divergence are consistent with the earliest de- of a momentum ¯ux divergence by downward vertical scriptions and theoretical ideas of the EUC. Estimates advection. This may help explain why the vertical dis- of the vertical stress divergence show dynamical ¯ow tribution of the vertical velocity component in the mod- regimes that change between the surface and the base els is upwelling everywhere within the EUC, as con- of the EUC. The vertical stress divergence is much larg- trasted with the change from upwelling above and with- er over the lower portion of the EUC than previously in the EUC core to downwelling below the core cal- reported. Nonlinearity becomes increasingly important culated with the TIWE equatorial array data. Charney with decreasing frequency and in the mean the EUC and Spiegel (1971) provide a physical argument for this circulation is nonlinear. Nonlinearities tend to cancel difference. In their model, with the ZPG and the vertical upon vertical integration consistent with reduced-grav- eddy diffusivity independent of depth, as the eddy dif- ity models being able to account for equatorial ther- fusivity decreases, the vertical shear and hence the fric- mocline variations. Future process experiments on the tional force below the EUC core increases, beyond equatorial circulation and heat balance will require high- which it can no longer be balanced by the ZPG force. er resolution for the temperature and pressure gradients At this point a downward transport of momentum must than provided by the present TOGA-TAO array. develop to balance the frictional force. These ®ndings bring into question the relative roles of the fully three- dimensional circulation versus horizontal and vertical Acknowledgments. Support was provided by the eddy diffusivity in maintaining momentum, heat, and Ocean Sciences Division, National Science Foundation, other water property balances near the equator. Model Grants OCE-8813378 and OCE-9302811. R. Cole and studies using smaller horizontal eddy diffusivities and J. Donovan assisted with the ®eld work and analyses, larger background state vertical eddy diffusivities would and sea going operations were facilitated by the of®cers seem warranted, along with additional empirical studies. and crew of the R/V Wecoma and Alpha Helix. The The analyses presented for the time-dependent zonal TOGA-TAO array wind stress and temperature data momentum balance were more qualitative. Ironically, were kindly provided by M. McPhaden and L. Mangum the ZPG (the best determined of the terms in prior stud- of the NOAA/PMEL. ies) is the limiting factor for the time-dependent analysis here. Designed for large-scale monitoring, the TOGA- TAO array does not resolve the spatial scale of the large APPENDIX A amplitude synoptic variability. Consequently, there is a Mean Zonal Momentum Flux Divergence Random large mismatch in the ¯uctuations observed in either x Errors 0 or Du/Dt with the ¯uctuations in the ZPG. TIWE lacked the resources to augment the TOGA-TAO array It is assumed that the measured velocity components with closely spaced moorings for temperature and sa- are stationary, Gaussian-distributed random variables linity. This was a mistake that future process experi- satisfying the ergodic hypothesis. If their variations re- ments should avoid. Despite this shortcoming, the large sult from the superposition of several different physical x and Du/Dt ¯uctuations showed considerable corre- 0 processes and the record is long enough (has a suf®cient spondence. Current nonlinearity is largest within the number of degrees of freedom) to sample these, then EUC core region, but upon vertical integration the non- linear terms tend to cancel. Thus, for intraseasonal Kel- the Gaussian assumption is supported by the central vin waves, as an example, the local acceleration exceeds limit theorem. the nonlinear effects, especially upon vertical integra- The w component is estimated by linear operation tion. But, nonlinearity increases with decreasing fre- upon the measured u and components, so its distri- quency, becoming very important in the mean. This bution is also Gaussian. The zonal momentum ¯ux di- y, andץ/uץ ,xץ/uuץ) vu) and its constituents´١) agrees with the observation that the mean meridional vergence z) are estimated by nonlinear operation, so theirץ/uwץ scale of the EUC is much less than an equatorial Rossby radius of deformation (the standard deviation of a me- distributions cannot be speci®ed a priori. However, after ridional Gaussian distribution). For example, from Fig. forming time series of the velocity component products, 4 an equivalent meridional scale of about 1Њ latitude is their distributions follow the same argument given for calculated, which is smaller by a factor of 3 than an the individual velocity components. equatorial Rossby radius of deformation [3Њ latitude for The standard deviation for a stationary, Gaussian ran- a Kelvin wave speed of 2.5 m sϪ1, as typically reported dom variable x is de®ned as the positive square root of (e.g., Kessler et al. 1995)]. Thus, while the intraseasonal its variance. Let the record-length-averaged estimate of ¯uctuations may be described as nearly linear, the mean the mean value of x be denoted by X and let the true EUC is nonlinear. A similar conclusion was found for mean value of x determined by the expectation operator the equatorial Atlantic by Tang and Weisberg (1993). E{x}bex. The variance of X is thus JUNE 1997 QIAO AND WEISBERG 1117
2 TABLE A1. Covariance of mean velocity pairs used in the standard 1 T var{X} ϭ E{[X Ϫ ]}2 ϭE (x(t)Ϫ)dt , deviation estimates. x T͵ x Ά·[]0 U V W U V W i i i j j j which for large enough T [such that the autocovariance Ui 1 0 0 0.8 0 0 V 0 1 0 0 of x, Cx(), tends to zero] reduces to (e.g., Bendat and i W 0 1 0 0.8 Piersol 1972) i 1 T var{X} ഠ Cx() d. T ͵ E{xii} ϭ ϪT
Upon drawing the equivalence between this variance of var{xii} ϭ the mean estimate and the variance of the mean estimate for band-limited noise, this becomes xiijjϪxϪ E ϭCij, Άij· 0 var{X} ϭ Cx(0), T respectively. The expected values of products of the xi may then be written as where E{xxij}ϭC ijij ϩ ij T C () d E{xxxijk}ϭ kijC ijϩ jikC ikϩ ijkC jkϩ ijk ͵ x ϪT E{xxxx}ϭ[CC ϩCCϩCC] 0 ϭ ijklijklijklikjliljk Cx(0) ϩklijC ijϩ jlikC ikϩ jkilC il is the integral timescale and NϭT/0 is the equivalent number of degrees of freedom (e.g., Tennekes and Lum- ϩiljkC jkϩ ikjlC jlϩ ijklC kl ley 1972 or Davis 1977). The signi®cance of the integral ϩ. timescale is that statistically meaningful estimates of X ijkl may be obtained if T is a suf®ciently large multiple of These allow the variances of the ®nite difference terms
0, and for T ϭ N0 (N Ͼ 1) the time series may be to be expressed as considered as having N equivalent independent samples. 22 var{xx12Ϫxx 34}ϭE{[xx 12Ϫxx 34]}ϪE{[xx 12Ϫxx 34]} Herein, 0 was estimated using the smoothed spectrum T ∫ 22 2 22 at zero frequency [since S(0) ϭ ϪTCx()d]. It is noted ϭ1 2[C 12ϩ1] ϩ 2 1 2 1 2C 12ϩ 1 2 that suf®ciently large N supports the Gaussian assump- 22 22 2 tion via the central limit theorem. In the present dataset ϩ21ϩ 34[C 34ϩ1] only u near the EUC core had N as small as 15. For all ϩ2C ϩ22ϩ 22 other variables, linear or nonlinear, N was generally larg- 3434 34 34 43 er than 30. Ϫ21234[CC 1324ϩCC 1423] The analysis of the momentum ¯ux divergence is split into two parts: Ϫ22413C 13Ϫ2 2314C 14
.vЈuЈ͘. (A1) Ϫ21423C 23Ϫ2 1324C 24͗´VUϩ١´ vu͘ϭ١͗´١ (A3) The total and the Reynolds ¯ux terms were analyzed using the above time series approach. With N larger than The standard deviation estimates for ⌬UU/⌬x, ⌬UV/⌬y, 30 for all of the nonlinear terms and their sums the and ⌬UW/⌬z then follow by replacing the variables xi Gaussian assumption appears to be satisfactory. Esti- in (A3) with the appropriate U, V, and W and then 2 2 2 mating the standard deviation for the ®rst term on the multiplying by a factor 1/⌬x ,1/⌬y ,or1/⌬z. right-hand side of (A1) is more complex. In ®nite dif- The covariance values Cij used in the estimations are ference form this term may be written as shown in Table A1. These were obtained from estimates of coherence at zero frequency (by averaging over the ⌬UU ⌬UV ⌬UW lowest-frequency portion of the spectrum) with the un- (VU ϭϩϩ. (A2´١ ⌬x ⌬y ⌬z derstanding that the superposition of variations having mixed symmetry properties about the equator will de- Since this term and its constituents are not time series, crease covariances between component pairs at different their random errors must be calculated from the prop- locations. Thus, high covariance between U pairs and erties of (assumed Gaussian) random variables. For the W pairs are obtained, compared with lower coherence set of Gaussian random variables xi (i ϭ 1, ´´´, n) let between off-equator V pairs. With no consistent set of the mean, the variance, and the covariance be denoted values to choose for lower coherence pairs, these were by set to zero in the standard deviation estimation. Since 1118 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 the small, omitted terms have different signs in (A3), ⌬uu ⌬u ⌬uw ⌬u ⌬u ⌬u ϩϩ ഠuϩϩw they tend to cancel so the net effect of this omission is ⌬x⌬y⌬z⌬x⌬y⌬z relatively small. With this technique the standard de- viations for each of the individual terms on the right- uϩuu Ϫu ϩ3131 hand side of (A2) were estimated. The standard devi- 2xϪx ation of the sum was not. As a divergence, the errors 31 should tend to cancel like the terms making up the sum, uϩuϪ ϩ5252 but there is no way of objectively determining this. Sim- 2yϪy ply adding the variances of each term together would 52 provide a meaningless overestimate of the standard de- uϩuw Ϫw ϩ u 1 u 1 . viation for the divergence. 2 ⌬z A similar development using the expected values of Thus, an equivalency is achieved between the ¯ux di- products of the xi was employed for the standard de- viation analysis of the advective scheme terms. vergence and the advective schemes if the term in pa- rentheses on the right-hand side is zero. This occurs by continuity if APPENDIX B u ϩ uuϩuuϩu 31,u152 , and Advective and Flux Divergence Scheme 22 2 Comparison are equal. The mean velocity component pro®les of Fig. The central ®nite differences for the individual terms 4 show that (u3ϩu1)/2 and (uuϩu1)/2 are both approx- in advective and ¯ux divergence schemes, respectively, imately equal to u4; however, (u5ϩu2)/2 is generally less are than u4. This discrepancy is largest within the EUC core where (u5ϩu2)/2 is about u4/2. Thus, near the core uu31Ϫuץ uu u31315252ϩ uu ϪuuϩuϪuuϩuw1uϪw1 ഠ4 ϩϩ xx31Ϫxץ 2x31Ϫx2y 52Ϫy2⌬z uu52Ϫuץ ഠ4 u3152ϪuϪwuϪwu1452Ϫ yy52Ϫy ഠu4 ϩϩ Ϫץ x31Ϫxy 52Ϫy ⌬z 2y 52Ϫy uuuϪu1ץ wഠwÅ uϪ ,z ⌬z ϭϪ 45 2ץ 2y52Ϫy and where the continuity equation has been applied. There- fore, the ¯ux divergence scheme is biased relative to uu u331131313131 u Ϫ uu u ϩuu Ϫuuϩuu Ϫuץ ഠ ϭϩ the advective scheme, with the bias xx31Ϫx 2x 31Ϫx 2x 31Ϫxץ u45 Ϫ 2 , uv u55 Ϫ u 22 5ϩ 25uϪuu 2 5ϩu 25Ϫ 2 Ϫץ ഠ ϭϩ 2y52Ϫy yyϪy 2yϪy 2yϪyץ 52 52 52being due to the meridional curvature of the u com-
-uw u w Ϫ uw u Ϫuuϩuw Ϫw ponent causing (u5ϩu2)/2 to be different from u4. Physץ ഠuu 11ϭwÅu 1ϩu 1 u 1, z ⌬z ⌬z 2 ⌬z ically, the ¯ux divergence formulation fails to conserveץ mass; but, a posteriori, this is a quanti®able, correctable where the subscripts are either station numbers or the error so that inferences drawn from the ¯ux divergence upper and lower layers, respectively; wÅ is the average scheme remain valuable. For the TIWE equatorial array w between upper and lower layers and ⌬z is the 10-m data such bias applies primarily to the mean circulation. layer spacing. In differential form, these two schemes For the Reynolds ¯uxes, in contrast to the mean, the are equivalent via continuity. Is this true of the ®nite- results from the two different schemes are nearly iden- difference forms, or is one biased relative to the other? tical, since the meridional curvature for the ¯uctuations Note that with small acceptable error the central values is small relative to the mean and u4 and 4 may be approximated by averages of their uЈϩuЈ horizontally adjacent values 52ഠuЈ. 24 u ϩ u ϩ uഠ31,ഠ 52. 4422 REFERENCES This approximation allows the ¯ux divergence form to Arthur, R. S., 1960: A review of the calculation of ocean currents at be rewritten as the equator. Deep-Sea Res., 6, 287±297. JUNE 1997 QIAO AND WEISBERG 1119
Bendat, J. S., and A. G. Piersol, 1972: Random Data. Wiley-Inter- Turbulence variability at the equator in the central Paci®c at the science, 407 pp. begining of the 1991±1993 El NinÄo.J. Geophys. Res., 100, 6881± Bryden, H. L., and E. C. Brady, 1985: Diagnostic model of three- 6898. dimensional circulation in the upper equatorial Paci®c Ocean. J. Lukas, R., 1986: The termination of the Equatorial Undercurrent in Phys. Oceanogr., 15, 1255±1273. the eastern Paci®c. Progress in Oceanography, Vol. 16, Per- , and , 1989: Eddy momentum and heat ¯uxes and their gamon Press, 63±90. effects on the circulation of the equatorial Paci®c Ocean. J. Mar. , 1987: Horizontal Reynolds stresses in the central equatorial Res., 47, 55±79. Paci®c. J. Geophys. Res., 92, 9453±9463. Busalacchi, A. T., and J. J. O'Brien, 1981: Interannual variability of Luther, D. S., and E. S. Johnson, 1990: Eddy energetics in the upper the equatorial Paci®c in the 1960s. J. Geophys. Res., 86, 10 901± equatorial Paci®c during the Hawaii-to-Tahiti shuttle experiment. 10 907. J. Phys. Oceanogr., 20, 913±944. Charney, J. G., 1960: Non-linear theory of a wind-driven homoge- Mangum, L. J., and S. P. Hayes, 1984: The vertical structure of the neous layer near the equator. Deep-Sea Res., 6, 303±310. zonal pressure gradient in the eastern equatorial Paci®c. J. Geo- , and S. Spiegel, 1971: Structure of wind driven equatorial cur- phys. Res., 89, 10 441±10 450. rents in homogeneous oceans. J. Phys. Oceanogr., 1, 149±160. McPhaden, M. J., 1984: On the dynamics of equatorial subsurface Chereskin, T. K., 1995: Direct evidence for an Ekman balance in the countercurrents. J. Phys. Oceanogr., 14, 1216±1225. California Current. J. Geophys. Res., 100, 18 261±18 269. , and B. A. Taft, 1988: Dynamics of seasonal and intraseasonal Cromwell, T., R. B. Montgomery, and E. D. Stroup, 1954: Equatorial variability in the eastern equatorial Paci®c. J. Phys. Oceanogr., under-current in Paci®c Ocean revealed by new methods. Sci- 18, 1713±1732. ence, 119, 648±649. Peters, H., M. C. Gregg, and J. M. Toole, 1988: On the parameter- Davis, R. E., 1977: Techniques for statistical analysis and prediction ization of equatorial turbulence. J. Geophys. Res., 93, 1199± of geophysical ¯uid systems. Geophys. Astrophys. Fluid Dyn., 1218. 8, 245±277. Qiao, L., and R. H. Weisberg, 1995: Tropical instability wave kin- Dillon, T. M., J. N. Moum, T. K. Chereskin, and D. R. Caldwell, ematics: Observations from the Tropical Instability Wave Ex- 1989: Zonal momentum balance at the Equator. J. Phys. Ocean- periment. J. Geophys. Res., 100, 8677±8693. ogr., 19, 561±570. Stommel, H., 1960: Wind-drift near the Equator. Deep-Sea Res., 6, Fofonoff, N. P., and R. B. Montgomery, 1955: The Equatorial Un- 298±302. dercurrent in the light of the vorticity equation. Tellus, 7, 518± Tang, T. Y., and R. H. Weisberg, 1993: Seasonal variations in equa- 521. torial Atlantic Ocean zonal volume transport at 28ЊW. J. Geo- Gregg, M. C., 1987: Diapycnal mixing in the thermocline: A review. phys. Res., 98, 10 145±10 153. J. Geophys. Res., 92, 5249±5286. Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence. Hansen, D., and C. Paul, 1984: Genesis and effects of long waves The MIT Press, 300 pp. in the equatorial Paci®c. J. Geophys. Res., 89, 10 431±10 440. Wacongne, S., 1989: Dynamical regimes of a fully nonlinear strati®ed Hebert, D., J. N. Moum, C. A. Paulson, D. R. Caldwell, T. K. Cher- model of the Atlantic Equatiorial Undercurrent. J. Geophys. Res., 94, 4801±4815. eskin, and M. J. McPhaden, 1991: The role of the turbulent stress Weisberg, R. H., and T. J. Weingartner, 1986: On the baroclinic re- divergence in the equatorial Paci®c zonal momentum balance. sponse of the zonal pressure gradient in the equatorial Atlantic J. Geophys. Res., 96, 7127±7136. Ocean. J. Geophys. Res., 91, 11 717±11 725. Johnson, E. S., and M. J. McPhaden, 1993: Structure of intraseasonal ,and , 1988: Instability waves in the equatorial Atlantic Kelvin waves in the equatorial Paci®c Ocean. J. Phys. Ocean- Ocean. J. Phys. Oceanogr., 18, 1641±1657. ogr., 23, 608±625. , and T. Y. Tang, 1990: A linear analysis of equatorial Atlantic , and D. S. Luther, 1994: Mean zonal momentum balance in the Ocean thermocline variability. J. Phys. Oceanogr., 20, 1813± upper and central equatorial Paci®c Ocean. J. Geophys. Res., 99, 1825. 7689±7705. , J. C. Donovan, and R. D. Cole, 1991: The Tropical Instability Jones, J. H., 1973: Vertical mixing in the Equatorial Undercurrent. Wave Experiment (TIWE) Equatorial Array: A report on data J. Phys. Oceanogr., 3, 286±296. collected using subsurface moored acoustic Doppler current pro- Kessler, W. S., M. J. McPhaden, and K. M. Weickmann, 1995: Forcing ®lers, May 1990-June 1991. Tech. Rep., 84 pp. [Available from of intraseasonal Kelvin waves in the equatorial Paci®c. J. Geo- Dept. of Marine Science, University of South Florida, 140 Sev- phys. Res., 100, 10 613±10 632. enth Ave., St. Petersburg, FL 33701.] Knauss, J. A., 1960: Measurements of the Cromwell Current. Deep- Wilson, D., and A. Leetmaa, 1988: Acoustic Doppler current pro®ling Sea Res., 6, 265±286. in the equatorial Paci®c in 1984. J. Geophys. Res., 93, 13 947± , 1966: Further measurements and observations of the Cromwell 13 966. Current. J. Mar. Res., 24, 205±240. Wyrtki, K., and E. B. Bennett, 1963: Vertical eddy viscosity in the Knox, R. A., and D. Halpern, 1982: Long range Kelvin wave prop- Paci®c Equatorial Undercurrent. Deep-Sea Res., 10, 449±455. agation of transport variations in Paci®c Ocean equatorial cur- , and B. Kilonsky, 1984: Mean water and current structure during rents. J. Mar. Res., 40(Suppl.), 329±339. the Hawaii-to-Tahiti Shuttle Experiment. J. Phys. Oceanogr., 14, Large, W. G., and S. Pond, 1981: Open ocean momentum ¯ux mea- 242±254. surements in moderate to strong winds. J. Phys. Oceanogr., 11, Yin, F. L., and E. S. Sarachik, 1993: Dynamics and heat balance of 324±336. steady equatorial undercurrents. J. Phys. Oceanogr., 23, 1647± Lien, R.-C., D. R. Caldwell, M. C. Greg, and J. N. Moum, 1995: 1669.