Semyon A. Grodsky, and James A. Carton
 
 

Journal of Geophysical Research, 106, 16,673-16,684, 2001

Department of Meteorology, Computer & Space Sciences Bldg., University of Maryland College Park, MD 20742
 

Abstract.

   Tropical Pacific Ocean near surface currents and their momentum and temperature balances are investigated using several data sets including drifting buoy velocities and TOPEX/POSEIDON altimetry. The data sets are combined to produce surface current estimates on a uniform grid monthly for the six-year period 1993-1998 using multivariate optimal interpolation.

    The analysis shows dramatic changes in current during December 1996 - August 1998 in response to the recent ENSO event. Eastward current anomalies of ~1 m/s were recorded in December 1996 in the western Pacific generated by sporadic westerly wind bursts. These equatorial anomalies reached the eastern boundary by April 1997, and in the summer of 1997 a band of strong eastward flow formed across the basin. This circulation pattern persisted until the beginning of 1998 when a westward equatorial jet appeared in January-April. Interestingly, the reversal of flow occurred prior to the restoration of the trade winds.

    The timing of these events as well as results of previous dynamical studies raise questions about the relative importance of terms in the zonal momentum and temperature balances. We address the first of these questions by examining the applicability of a three-term linear zonal momentum balance on the equator. In all cases we focus on anomalies from the time-mean. Our results show that local acceleration is reasonably well balanced by the difference between zonal pressure gradient anomaly and wind-induced momentum flux. This three-term balance displays significant basin-wide variations and is consistent with the conclusion that the equatorial Pacific is not in equilibrium with local wind forcing due to the presence of propagating waves.

    Examination of the temperature balance shows that at the beginning of El Niño the warming in the central Pacific is mainly supported by horizontal temperature transport, while the vertical heat exchange and transport become important in the eastern half of the equatorial Pacific during the transition from El Niño to La Niña.

1.    Introduction

    Although many models of air/sea interaction in the tropical Pacific assume that mixed layer temperature is controlled by diabatic mixing at its base [e.g. Suarez and Schopf, 1988]. Wyrtki [1985] proposed that the El Niño - Southern Oscillation (ENSO) cycling is governed by the mass and heat dynamics in the western Pacific warm pool with zonal transport playing a key role. Picaut et al. [1996] found that much of the monthly shift in the position of the pool of warm surface water that normally resides in the western basin can be explained by zonal advection of water within the mixed layer. The boundary between the warm pool of water to the west and the cooler water to the east defines a region of zonal convergence and subduction. The potential role of such advective processes in controlling sea surface temperature (SST) complicates our understanding of the air/sea feedback cycles leading to ENSO [Picaut et al., 1997; An et al., 1999; Jin and An, 1999]. The purpose of this paper is to reexamine the contribution of horizontal advection in momentum and heat balances in the tropical Pacific during the extreme events of 1997-8, making use of the extensive surface drifter and sea level data sets now available.

    There is strong independent experimental evidence [see e.g. Johnson and Luther, 1994; Qiao and Weisberg, 1997] that the momentum balance on seasonal and interannual time scales is generally linear and geostrophic away from the equator. Along the equator Yu and McPhaden [1999] have examined Atlas mooring data at four mooring locations and found a three-term balance in the zonal momentum equation between local acceleration, pressure gradient force, and wind stress whose relative strength must change as wind intensity and direction shifts.
 

    Recently, a suite of studies has revisited the ocean's role in regulating SST on seasonal and interannual time scales [e.g. Köberle and Philander 1994; Picaut et al. 1996; Delcroix and Picaut 1998; Moisan and Niiler, 1998; Kessler et al. 1998; DeWitt andSchneider 1999; Swenson and Hansen 1999], however the conclusions have not been uniform. Modeling studies by Köberle and Philander [1994] and DeWitt and Schneider [1999] suggest that the annual cycle of SST in the eastern basin results from entrainment of cool subthermocline water into the mixed layer due to meridional divergence of surface waters, while Swenson and Hansen [1999] argue that local storage is also important. In contrast, Kessler et al. [1998] conclude that three-dimensional temperature advection terms tend to cancel each other and, to first order, the seasonal variation of SST can be described as simply following the variation of net surface heat flux. It is still debatable whether or not the same terms dominate for interannual processes such as ENSO.
 

    In the central and western basin the story of the heat budget is complicated as well. As mentioned above, Picaut et al. [1996] and Delcroix and Picaut [1998] argue that on interannual timescales changes of mixed layer temperature largely reflect changes in the anomalous zonal advection of water acting on the mean westward temperature gradient, while Wang and McPhaden [1999] argue additionally for the importance of surface heat flux.

    The massive El Niño of 1997-8 provides an exciting opportunity to reexamine the momentum and heat balances of the tropical Pacific under extreme conditions. The story of this El Niño begins with a series of westerly wind bursts in the western basin in late 1996 [Yu and Rienecker, 1998; McPhaden, 1999; Wang and Weisberg, 2000]. These westerly wind bursts, an enhancement of the normal Madden Julian oscillation, were followed by a relaxation of the trade winds, warming of SST near the dateline in January-February 1997 and in the east by March. Instantaneous zonal velocity transects from Acoustic Doppler Current Profiler meridional-vertical sections taken during October 1996 November 1998 by Johnson et al. [2000] show a strong eastward surface current anomaly on the equator in response to the relaxing trade winds with peak velocities approaching 1m/s. As warm conditions were rapidly replaced by cold La Niña conditions and the trade winds accelerated [see McPhaden, 1999 for a discussion], the current anomalies also reversed direction, forming a westward jet with its maximum speed (>1m/s) north of the equator. Although, spatial discreetness and time sampling of this data complicate assessment of time and spatial derivatives needed to estimate the momentum and heat budgets.

    Fortunately for the cause of trying to understand momentum and heat balances during these extreme events, several new data sets have become available in recent years. Extensive deployments of drogued surface drifters of the WOCE Global Data, Surface Velocity Programme [Niiler et al., 1987; Hansen and Poulain, 1996] now provide ~3000 - 4000 buoy-days of direct currents observations per month in the tropical Pacific [Acero-Schertzer et al., 1997]. Much of the tropical ocean is in near-geostrophic balance with surface pressure. Thus, satellite-based altimetry from the joint US/French TOPEX/POSEIDON mission provides a powerful additional constraint on near-surface velocity. Indeed Yu et al. [1995] have shown that the velocity estimates from these two data sets are remarkably consistent. The difference between the two we believe is due to (ageostrophic) wind-driven effects, which Ralph and Niiler [1999] show can be explained to a considerable degree by Ekman-like dynamics except close to the equator.
 

    The velocity analysis we use for this study we construct by combining drifter velocity and sea level data using multivariate optimal interpolation [Carton and Hackert, 1989; Daley, 1991]. This basic algorithm has been extended to the equatorial -plane based on the Kelvin Wave scaling of Picaut et al. [1989] and Menkes et al. [1995]. To account for the wind-driven portion of currents a simple linear friction is included to allow an Ekman-like balance on the equator [following Lagerloef et al., 1999]. Our presentation of results begins with a kinematic description of currents during 1996-1998 followed by a discussion of the corresponding zonal momentum and thermal balances.
 

2.    Data and analysis

    This study is based on four data sets, altimeter sea level, drifter velocity, SST, and NCEP wind stress and surface heat fluxes. The TOPEX/POSEIDON altimeter sea level is obtained from the Pathfinder version 2.1 archive [Koblinsky et al., 1997]. This data is available with a 9.92-day repeat cycle and 1.42-degree equatorial track spacing spanning the time period from late-September 1992 through the end of our analysis period. After the usual corrections for geophysical effects the sea level estimates have been averaged into 1⁰ latitude segments. The nominal accuracy of these estimates when monthly averaged is 2cm [Cheney et al., 1994] while observed sea level anomalies are at least four times that.
 

    The drouged surface drifter currents are obtained from the WOCE/TOGA archive at the Atlantic Ocean Marine Laboratory/NOAA. The data spans the period 1979-November, 1998. After the processing described by Hansen and Poulain [1996] we averaged currents into 2⁰x3⁰x1-month bins. This data forms the basic velocity data set for this study. For velocity comparisons additional moored velocity time series were obtained from Pacific Marine Environmental Laboratory/NOAA [McPhaden et al., 1998].
 

    The surface wind stress and the net heat flux through the ocean surface are obtained from monthly mean NCEP/NCAR reanalysis product [Kalnay et al., 1996]. SST is the combined satellite AVHRR/in situ analysis of Reynolds and Smith [1994]. For the purpose of this study the wind stress, the heat flux, and the SST fields have been averaged into 2⁰x3⁰x1-month bins.
 

    Our analysis methodology is a form of multivariate optimal interpolation [see Daley, 1991], which is briefly summarized here. Let , , define the analysis grid, while the spatial vectors , , and , , are the observation station locations for sea level, , and mixed layer velocity, . The analysis velocity, , may be written as a linear combination of the weighted differences between observations and a background estimate of sea level and velocity:
 

                                                        (1)

    There are several possible choices for use as the background velocity field , . Lagerloef et al. [1999] derive flow estimates using a combination of spatially smoothed altimetry and a wind-drift current. A disadvantage of this approach is the introduction of bias into the background error. Here we have explored the use of the long-term average surface drifter field to define , . The background estimate of sea level is similarly the long-term average.

,                                                                                                 (2)

whereis the matrix of covariances between background velocity and sea level error components evaluated at the analysis and observation station locations. Similarly, and are the matrices containing background and observation error covariances evaluated at different station locations.

,                                                                    (3)

      We calculate C, B, and O based on a variety of simplifying assumptions. First we suggest that O is diagonal, meaning that the observation errors are uncorrelated. Normalizing O by the analysis variance (assumed homogeneous) the elements along the diagonal of O are simply the ratio of noise-to-signal variances. This ratio we assume to be 4 for 1-second averaged altimetry following Carton et al. [1996]. We do not yet have good estimates of the noise-to-signal ratio for velocity observations because of our lack of knowledge of unresolved physical processes. Here we assume a ratio of 0.2 following Carton and Hackert [1989]. Finally we set the background error covariances noise-to-signal ratio to 1.
 

    To complete our statistics we begin by defining sea level error covariance and derive all other background error covariances from this. In order for all derived error covariances to have appropriate properties (i.e. positive definiteness, compactness, continuous derivatives), it is most simple if we choose to have a very simple analytic form. Here we assume a homogeneous Gaussian function

                                            (4)

450 km, km,

where is the sea level error variance, assumed to be spatially homogeneous following Carton and Hackert [1989], and where are the zonal and meridional displacements of the vector . Our estimates of horizontal length-scales  follow Carton et al. [2000].

    In order to derive the remaining covariance submatrices it is helpful and reasonable to assume a dynamical relationship between sea level and velocity. We begin by assuming that analysis, observation, and background velocity can be decomposed into a geostrophic and a wind-driven component (). Following Lagerloef et al. [1999], we assume that  satisfies an Ekman-like equation balancing the difference between wind-induced momentum flux () and linear friction by the Coriolos force:

                                                                            (5)

    Some assumption must be made regarding H, the depth-scale over which wind-induced momentum is distributed. We use the thickness of the ocean mixed layer provided by the ocean reanalysis of Carton et al. [2000] as a proxy for H, and estimate the friction coefficient, r, by assuming that the mean meridional velocity near the equator is mainly wind-driven. Based on this assumption we estimate r=2x10^-4 1/s, a value similar to that proposed by Lagerloef et al. [1999]. Then we subtract from each term (, , and ) before evaluating (2). Thus (2) becomes an equation for the geostrophic component of and the error covariances may be derived from (4) by applying geostrophy, as presented in the Appendix A.

    Within a radius of deformation of the equator the Coriolis term can no longer be considered constant and indeed the geostrophic approximation may be invalid. However, if we accept Kelvin Wave scaling following Menkes et al. [1995] then we can assume that background zonal geostrophic velocity errors () on the equator are related to background sea level errors () as  or, following Carton and Hackert (1989) and Lagerloef et al. [1999] and allowing a smooth transition to equatorial dynamics:

,                                    (6)

,                                                                            (7)

,

where decreases smoothly as latitude approaches zero. The proper assumption to make for the meridional velocity component error () is less clear. Following the Kelvin Wave analogy we have assumed that background errors of sea level and meridional velocity decorrelate near the equator as shown in (7). For a positive sea level error on the equator, this assumption together with (6) implies convergence to the east of the anomaly and divergence to the west.

    The analysis results are illustrated for March 1998 in  Fig. 1. This analysis takes place during the late stage of the intense 1997-1998 Niño when equatorial currents are already westward at speeds of up to 1 m/s in the west. The drifter data coverage during this month is not dense along the equator due to the dispersing effects of the strong currents. The lack of data is reflected in the relatively high normalized analysis velocity error shown in the lower two panels of  Fig. 1.

    It is possible for us to make a more accurate assessment of analysis error by comparing our results with the three independent 10 m velocity time series from the Atlas moorings at 165⁰ E, 140⁰ W, and 110⁰ W (see Fig. 2). At these three locations, the mean analysis (and observed) zonal velocities were: 22 cm/s (10 cm/s), -20 cm/s (-19 cm/s), and -19 cm/s (-14 cm/s) based on the same time period during which buoy data is available. The largest bias of 12 cm/s at 165⁰ E may simply reflect statistical uncertainty (the buoy at this location was only in operation for 3.5 years). The mean analysis (and observed) meridional components were: -0.5 cm/s (1 cm/s), -3 cm/s (-1.5 cm/s), and 10 cm/s (6 cm/s). Standard deviation of analysis currents from observed velocities were: 29 cm/s, 38 cm/s, and 31 cm/s for the zonal component, and 16 cm/s, 12 cm/s, and 10 cm/s for the meridional component at 165⁰ E, 140⁰ W, and 110⁰ W, respectively.

3.    Results

    We begin our presentation of results by examining the changes in surface current during the dramatic events of 1997-1998. Throughout the following discussion, the time mean 1993-1998 fields () are removed. This has an effect of focusing attention on the behavior of anomalies from the time mean. We choose not to remove the seasonal cycle because of the close connection between seasonal cycle and ENSO.

    In late-1996 a strong eastward current developed in response to westerly wind bursts in the western Pacific basin (Fig. 3). By March-April 1997 the ocean responded to these disturbances in wind and current with a simultaneous increase in SST and sea level in the central and eastern basin, while the eastward current became concentrated in the equatorial zone (see the April 1997 panel of Fig. 4). During this season the strong eastward equatorial current was down the slope of sea level, while off the equator the current was weaker and variable. However, as 1997 progressed the region of westerly wind anomalies shifted eastward and the equatorial current in the west weakened (Fig. 5, December, 1997 panel). For much of 1996 and early 1997 SST changes in phase with zonal velocity along the equator (see Fig. 3). This suggests an important role for advection in regulating SST.

    In midbasin the current remained eastward even though the pressure gradient reversed sign so that already by August the eastward current was flowing up the slope of sea level. This condition persisted until the boreal spring of 1998 when the current along the equator reversed direction west of 150⁰W again allowing the current to flow downhill (see April 1998 panel of Fig. 4). By the fall of 1998 the zonal pressure gradient relaxed although the current continued to flow westward. During this later stage zonal advection and SST do not seem closely linked (see also Fig. 5). The equatorial currents swung to the west between December and January, but SST remained high through April 1998. This suggests that advection plays a lesser role in regulating SST during this stage.

    In this kinematic discussion, it has been evident that changes in the zonal pressure gradient follow shifts in the direction of the equatorial currents. The correspondence between variations of zonal velocity and SST varies significantly with time and becomes smaller at late stage when the restoration of the tongue of cold waters in the eastern Pacific occurred with 4-months delay after reversal of currents. To obtain a more quantitative understanding of the relationship between these terms we now evaluate the zonal momentum balance and temperature budget.

    There is strong experimental evidence [see e.g. Johnson and Luther, 1994] that the momentum balance is linear and geostrophic to within 5⁰ of the equator. Closer to the equator Qiao and Weisberg [1997] show that the momentum balance in the mixed layer remains essentially linear near the surface. This balance was investigated at four ATLAS mooring locations by Yu and McPhaden [1999]. On seasonal cycle and interannual time scales they found a three-term balance between local acceleration, (), zonal pressure gradient force, (), and wind stress, ():

 

                                                                                (8)

 

 
    At interannual time scales, acceleration is small, and the pressure gradient force balances wind stress. Zonal currents are in phase with these two terms, as was explicitly shown by Johnston and Merrifield [2000] who found that interannual geostrophic current anomalies derived from tide gauge network data vary in phase with both, zonal wind stress and sea level in the near-equatorial western Pacific Ocean. At shorter seasonal periods, acceleration becomes important so that the development of an anomalous zonal pressure gradient lags behind changes in wind stress and local acceleration. Although the mooring data provides good sampling in time and depth, their spatial coverage was very limited. Here we apply similar methodology to examine applicability of this three-term balance throughout the basin during the extreme events of 1997-8.

    Results are shown in Fig. 6 for three months during the early, middle and late stages of the El Niño. For this comparison, acceleration and spatial derivatives have been estimated using central differences. Error estimates (see Appendix B) are included in lower panels to indicate the uncertainties in the various data sets. We select April 1997 to represent the early stage of El Niño, when the currents reversed and SST is increasing (see Fig. 3). We choose August 1997 to characterize the conditions during the middle stage, a time when the currents and the SST remain steady. We choose May 1998 to represent the late stage when the SST begins to decrease in the eastern basin.

    In April 1997 when the easterly winds begin to reverse, they can no longer balance the negative pressure gradient associated with sea level. As a result, the currents across the basin accelerate eastward. The close balance between the three terms in (8) suggests that friction and non-linearity must play small roles during this onset phase in the surface layer. The fact that this eastward acceleration exceeds the difference between wind stress and pressure gradient by up to 2x10^-7 m/s² offers several possible explanations including a possible 0.2 dyn/cm² overestimate of the wind stress variation during this month. This value is comparable to the difference between the two zonal wind stress estimates from the NCEP/NCAR reanalysis and NCEP operational wind product.

    By the middle stage (August 1997), the positive trade wind anomalies have extended further eastward nearly in balance with the pressure gradient force. As a result, the zonal acceleration is close to zero along the equator. By the late stage (May, 1998), the positive trade wind anomalies have shifted far to the east. But an intense negative pressure gradient force causes a strong westward acceleration of up to 4 cm/s per day there. In summary (see Fig. 7), the three-term balance applies for much of the basin (correlation > 0.4). The exception is a 20⁰ band of longitudes (140⁰W-120⁰W).  In this band of longitudes, which is a region of rather strong zonal gradient of the thermocline depth and cool surface temperatures, acceleration and the sum of wind stress and pressure gradient force become uncorrelated. The missing dynamics in this region are nonlinearity and mixing processes. We can anticipate similar changes in the heat budget here.

    Now we examine the heat balance during the same three stages as for momentum focusing on spatial variations along the equator. Written for anomalies from the long-term mean and averaged over the upper ocean mixed layer of depth (H), this balance becomes following Moisan and Niiler [1998]:

 

,                                        (9)

where is specific heat capacity of water, and and F_w are downward anomaly heat fluxes across the air-sea interface and at the bottom of the mixed layer, respectively. Following Wang and McPhaden [1999], we subtract short wave radiation that penetrates through the mixed layer from the surface heat flux, so that F₀ includes the net surface radiative flux actually absorbed by the mixed layer.

   Calculations of the terms of anomaly temperature balance (9) are presented in Fig. 8. During the early stage, accelerating eastward currents transport the waters of the western warm pool toward the east. This transport is mainly due to anomaly current advection acting on the mean SST gradient, -(), and gives warming east of 180⁰W. Net surface flux is small except east of 120⁰W, where it acts to cool the ocean. This drop results from two factors, a drop in incoming short wave radiation resulting from increased cloudiness, and secondarily by an increase of latent heat loss. Because of the agreement between surface flux, advection, and heat storage rate west of 110⁰W, to within our error estimates, heat flux due to other processes such as entrainment is negligible. Hence, during this early stage the increase of SST is balanced by eastward advection of warm pool waters by the zonal current, as was suggested by Picaut et al. [1996] and Delcroix and Picaut [1998]. East of 110⁰W, the left and right hand sides of (9) balance each other to within the errors of our data. Nevertheless,  exceeds the sum of fluxes. Examination of the ocean reanalysis shows that the mixed layer thickness averaged 120⁰W - 80⁰W increased to 30 - 35 m from 15 - 20 m observed in late-1996. This deepening of the thermocline acts to reduce heat flux across the bottom of the mixed layer.
 

    During the middle stage (see August panel of Fig. 8) SST is steady and so . Heat gain by horizontal advection is small and compensated for by surface heat loss over already warm waters west of 120⁰W. East of 120⁰W surface heat loss increases while horizontal advection remains small and additional heat gain to the mixed layer is needed to complete the balance. Examination of the ocean reanalysis shows that the mixed layer depth (120⁰W - 80⁰W) increases by 25m again acting to reduce heat loss.

    By the late stage (May 1998) SST drops in the eastern half of the basin at a rate much more rapidly than can be explained by advection or surface heat flux and seems clearly to be related to the shallowing thermocline (H averaged over 160⁰W - 120⁰W decreases to 30 - 35 m from its normal value of ~50 m). Consequently, the entrainment and subthermocline mixing increase leading to increasing of . Note, that in contrast to the early stage where SST increases were supported by mean temperature gradient transport by anomaly current, in the middle and late stages all advective terms contribute comparably.

    In summary (see Fig. 7), as we found in the momentum balances, terms in the heat equation (9) become decorellated in the band of longitudes 140⁰W - 120⁰W (and perhaps further eastward).Generally, correlation is higher in the western half of the basin that indicates on important role of advection and surface fluxes in establishing the anomaly heat balance there, while exchange processes at the bottom of the mixed layer play (probably) a minor role. East of 140⁰W the correlation decreases approaching the confidence level of zero correlation. Hence, in the eastern basin the horizontal advection plays comparable (or even minor) role with respect to other processes contributing to heat balance of the mixed layer.

4.     Summary
 

    Near-surface currents play a key role in seasonal to interannual variability in the tropical Pacific Ocean. The recent availability of extensive drogued surface drifter observations has allowed us to explore this role through direct observations. During the dramatic events of 1997-8 we find massive changes in heat distribution and currents. Describing these changes and interpreting their impact on transports of momentum and heat as well as introducing a new data set is the goal of this study. Here we focus on the balances for three specific months, April 1997 representing the early stage, August 1997, representing the middle stage, and May 1998 representing the late stage of the 1997-8 Niño.

    Our story begins in late-1996 and early-1997 with a 1 m/s eastward current anomaly equatorially trapped in the western basin produced in response to a succession of westerly wind bursts. This eastward current spread toward the east throughout the summer of 1997, supported by a pressure gradient force also acting in the eastward direction. The accelerating eastward currents transported the warm waters of the western warm pool east of 180⁰W, causing a rapid warming in mixed layer temperature as the anomalous currents acted on the mean temperature gradient as was suggested by Picaut et al. [1996] and Delcroix and Picaut [1998]. Net surface flux was small at this time except east of 120⁰W, where it acted to cool the ocean as a result of increased cloudiness and latent heat loss.

    By the middle stage in August the positive trade wind anomalies had extended further eastward nearly in balance with the pressure gradient force now acting to decelerate the flow. As a result, the zonal acceleration was close to zero along the equator. SST during this period was steady. Heat gain by horizontal advection was small because of an approximate balance between the cooling effects of the mean currents acting on the anomalous temperature gradient and the warming effects of the anomalous currents acting on the mean temperature gradient. West of 120⁰W advective heat gain was balanced by surface heat loss over already warm waters. East of 120⁰W surface heat loss was larger while horizontal advection remained small. In this far eastern region additional anomalous heat gain likely came from the anomalous deepening of the thermocline by 25m and consequent reduction in cooling due to mixing and transport across the base of this layer.

    By the later stage in May 1998, the positive trade wind anomalies shifted far to the east. But an intense negative pressure gradient force caused a strong westward acceleration. SST decreased rapidly in the eastern half of the basin. This decrease was larger than could be balanced by advection or surface heat flux and seems clearly to have been related to the loss of heat associated with the shallowing thermocline (which rose by 25m between 160⁰W - 120⁰W).

    From the oceanic prospective the changes in the mixed layer observed during 1997-8 resulted from a large climate anomaly superimposed upon an even larger annual cycle. Our results indicate that the balance of terms shifts substantially through the early, middle, and late stages of El Niño. These shifting balances represent a clear requirement for realistic models of ENSO.

    Acknowledgements. This work has been supported by the National Science Foundation (OCE9530220 and OCE9812404). We gratefully acknowledge Dr. Mark Swenson of the NOAA Atlantic Oceanographic Marine Laboratory for providing the surface drifter data set. Dr. Chet Koblinsky of NASA Goddard Space Flight Center provided the altimeter data.

 
Appendix A.

    After some algebra (see Daley [1981], page 164) the covariance functions take the form:

 

                        (A1)

 

                                                                (A2)

 

                (A3)

 

(A4)

                                        (A5)

 

 
Appendix B: Error estimates

    Let denote the standard error of any variable x. Our estimate of the accuracy of the zonal velocity analysis is based on posteriori comparison with Atlas mooring measurements. Here we assume =0.3 m/s. Then the error of the acceleration term is 10^-7 m/s² for central differences taken at =2 months. If we assume the sea level error 2 cm for 1⁰x1⁰x1-month binned data [following Cheney et al., 1994], and add a smoothing factor n=6 for data averaged into 2⁰x3⁰x1-month grid then the standard error of the sea level slope becomes  2x10^-7 m/s². The wind stress error estimate depends on wind speed error, , assuming a simple bulk formula based on a drag coefficient,, as assuming =1 m/s and 1.4x10^-3.

    Error estimates in the temperature equations follow similar reasoning. We assume that the SST observations are subject to a random error of =0.1 ⁰C. Thus the standard error = 0.3x10^-70C/s. Errors in the advection term derive primarily from errors in velocity assuming 0.1 m²/s², and 0.01 m²/s², where these numbers were estimated from the mooring-analysis comparison. Finally, we accept the optimistic =5 W/m² error estimate of Kistler et al., [1999] for the surface flux data.

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Figure captures

Figure 1. Velocity for March 1998: (a) Drifter velocity observations; (b) analysis velocity; (c) normalized zonal velocity error; (d) normalized meridional velocity error. The expected analysis error variance for each velocity component calculated following Daley [1991] was normalized by the background error variance.

Figure 2. Velocity comparisons at three moorings locations. Three estimates are compared, 10 m moored currents (ATLAS), surface drifter observations (Drifter), and analysis velocity.

Figure 3. Time-longitude equatorial cross-section of zonal wind stress, surface velocity and SST during 1996-1998. The time average, shown in the lower panels, has been removed from each field. Horizontal lines mark the times for which the estimates of the upper ocean momentum and temperature balances are shown in Figures 6 and 8.

Figure 4. Monthly sea level and velocity anomalies from the climatological mean every four months from December 1996 through August 1998. Contour interval is 5 cm. Sea level anomalies exceeding |10cm| are shaded. Only currents greater than 10cm/s are shown.

Figure 5. Monthly mean currents and SST during December 1997 - April 1998. Contour interval is 1⁰C. The areas with SST exceeding 28⁰C are shaded. Only currents greater than 10cm/s are shown.

Figure 6. Longitudinal sections along the equator of the three terms in the linear zonal momentum balance during the early, middle, and late stages of El Niño. Upper panels show acceleration (), zonal wind stress (), and pressure gradient force () separately. Lower panels show the wind stress and pressure gradient terms combined (). A vertical bar indicates standard errors for acceleration, while upper and lower thin bounding lines indicate standard errors for .

Figure 7. Longitudinal sections along the equator of the correlation over time of the left and right hand sides in the zonal momentum and heat budget equations. The correlation between zonal acceleration and the difference between anomaly zonal wind stress and anomaly zonal pressure gradient is shown in a bold line, while the correlation between SST rate of change and the difference between net heat flux and temperature advection is shown in a thin line.
 

Figure 8. Longitudinal sections along the equator of the terms in the temperature balance equation during the same three months as in Fig. 6. Upper panels show the rate of change of SST (), zonal  () and meridional () advection, and surface heat flux () separately. Lower panels show heat advection and heat flux terms combined. Standard errors are shown with vertical bar for SST rate of change, and with upper and lower thin bounding lines for the term combining the effects of advection and surface heat flux. Note, that an SST rate of change of 10^{-7 0}C/s corresponds to heat flux of 28 W/m² and 10 W/m² for the mixed layer thickness of H=70 m and H=25 m, typical of the central and eastern parts of the basin, respectively.