A convergent instability wave front in the central tropical Pacific

Deep-Sm Rewarrh II. Vol. 43. No. 4-6. p_n.753-778, 1996 Copyright c 1996 Elsevier Science Ltd Printed m Great Britain. All rights reserved 09674X45/96 515.00 + 0.00

Pergamon

PII: So967-0645(96)00034-3

A convergent instability wave front in the central tropical Pacific ERIC

S. JOHNSON*

(Received3 April 1995; in revisedform 3 February 1996; accepted 13 March 1996)

Abstract-A narrow front encountered at 2. ION, 14O”W was characterized by high productivity and a 2°C temperature drop. Satellite sea-surface temperature (SST) imagery showed this front to extend over 400 km from just north of the equator to 5”N along the western (leading) edge of a tropical instability wave cold cusp. During 3days of observations the front propagated westward at while oriented SW-NE. Water velocities and densities around the front were measured 64cms-’ using a shipboard acoustic Doppler current profiler and hydrographic stations. These data showed warm, fresh northern water moving southward at speeds of up to 50 cm s-’ to encounter cold, salty equatorial water moving northward at up to 40 cm s-‘; both slabs were embedded in the South Equatorial Current, here a uniform 100 m deep westward flow of 90 cm s-‘. At their meeting a net convergence of 30 cm s- ’across the l-km wide front drove intense downwelling of up to 0.9 cm s- ‘. Both T-S relations and velocity fields indicate that cold water subducted beneath the warmer water and continued northward to beyond 3”N. deepening to 120 m. For analysis the data were gridded in a co-ordinate system centered on the front and moving with it. The dynamics near the front consisted of a balance between pressure gradient and non-linear advection; the Coriolis force was smaller and associated mostly with the broader scale flow. Thus, the front itself was not geostrophic, but rather the leading edge of a density-driven flow that can be accurately modeled as a dissipative lock-exchange. The subduction released potential energy at the to previous estimates of large-scale energy rate of 5290 $- 230 W m-’ of front length, comparable conversion. Nevertheless, the kinetic energy gain by the larger-scale instability was only 84Ok 31 W rn-‘% implying that the rest of the energy was lost to the large-scale flow. Turbulent dissipation as derived from the rate of mixing accounted for only 500+300 W m-’ of this loss. On the full 100 km-scale of the layered flow, the Coriolis force became important: the orientation of the surface front and the angle of the interface’s northward deepening produced a pressure gradient that balanced the meridional flow geostrophically. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION In August and September 1992 the JGOFS fall survey cruise sampled a section from 12”N to 12”s along 14O”W in the tropical Pacific. At 2.1”N the ship crossed a surprisingly intense section of the equatorial front. Satellite sea-surface temperature (SST) images (Yoder et al., 1994) indicated that the front was oriented SW-NE along the leading edge of a westward propagating, tropical instability wave cold cusp, and that the ship crossed the front about mid-way between its southern and northern extremes near the equator and 5”N. The intensity of this front’s circulation and its associated biological activity (Yoder et al., 1994) interested both the physical and biological oceanographic communities. Although previous observations of such features are sparse, instability waves themselves are ubiquitous

*Joint Institute for the Study of the Atmosphere WA 981954235. U.S.A.

and Oceans,

753

Box 354235. University

of Washington,

Seattle,

754

E. S. Johnson

features of the tropical circulation and are known to have been energetic during this period (Kessler and McPhaden, 1995). Thus, the front’s association with tropical instability waves raises the possibility that such fronts might be a relatively common, and perhaps important, component of the near-equatorial ocean physical and biological environment. The instability wave itself was not resolved by this cruise, and the dynamics of such waves in general are not yet well understood (e.g. Qiao and Weisberg, 1994). Nevertheless, the smaller region within tens of kilometers of the front’s midpoint was intensively sampled during repeated crossings over a period of several days. The present study focuses on the local structure and dynamics of the front itself, neglecting the larger-scale structures from which it arose. Obviously, further studies will be needed both to refine our understanding of the larger-scale instability wave dynamics and to integrate the present work into it.

2. DATA The data used here consist of three sets: velocity profiles from an acoustic Doppler current profiler (ADCP); near-surface temperatures from the same instrument; and hydrographic data from CTD stations.

2.1. ADCP

velocities

Water velocity relative to the ship was sampled continuously by a 150 kHz shipboard ADCP, which returned measurements between 19 m and about 400 m depth. Both the bin width and the pulse width of the instrument were set to 8 m depth, giving a vertical resolution of about 16 m. The ADCP configuration was further tailored for accuracy in high-shear regions as suggested by Chereskin et al. (1989). The returned velocities were averaged into I-min blocks to reduce both noise and data volume. Earth referenced currents in a layer between 139 and 195 m depth were subsequently calculated by subtracting a 20-s filtered, GPS-derived, time-series of the ship’s velocity from the profiler’s relative velocities. The resulting reference-layer velocities had a marked spectral gap at 2 h period, presumably separating real water velocities at long periods from shorter-period navigational noise. Low-pass filtering removed the variability with periods shorter than 2 h, thus restricting the reference-layer velocities to timescales greater than 20 min (corresponding to space scales greater than 8 km when the ship is underway). For each profile, velocities below and above the reference layer were built up using the observed vertical shears: thus, shallow features are available at full resolution to the extent that their variability is independent of the deeper reference layer.

2.2. ADCP

temperatures

In order to facilitate speed-of-sound corrections to the recorded velocities, the ADCP instrument measured ambient water temperatures at the transducer head. This temperature was recorded with each ADCP ensemble, forming a continuous, 1-min sampled time-series of temperature at 7 m depth. Post-cruise calibration relative to the CTD data indicates that the thermistor read about 0.4”C high. This is sufficient here, since the ADCP temperatures are used only in a qualitative sense to identify the location of the front’s sharp temperature gradients.

Convergent

instability

wave front in the tropical

Pacific

755

2.3. CTD measurements

Numerous CTD stations were occupied in the course of the survey along 14O”Wand also in the region of the front itself. Casts were conducted to 200 m depth, sufficient to sample the variability of the thermocline and the region above it. Details of the data preparation are given by Murray et al. (1995). 3. LARGE-SCALE

FLOW AND FRONT

OVERVIEW

The observed large-scale zonal flows (Fig. 1) are typical of the tropical mid-Pacific. The Equatorial Undercurrent (EUC) is centered on the equator and at about average strength. Above and poleward of this current are the westward flows of the South Equatorial Current (SEC). It is unusualy strong, reaching speeds of over 60 cm s-l. The North Equatorial Countercurrent (NECC) flows eastward between 4” and 11“N, and appears to have a weak counterpart in the southern hemisphere. In meridional velocity the expected mean circulation of wind-driven upwelling near the equator and poleward surface flow (e.g. Wyrtki and Kilonsky, 1984; Luther and Johnson, 1990) is obscured by the much stronger flows associated with tropical instability waves: these waves were energetic during the study period (Kessler and McPhaden, 1995) and are known to produce meridional velocities comparable to those observed. The resulting flow field has a strong surface divergence about

Zonol Velocity.

Meridional

-15

-10

-5

Velocity,

0 Latitude

cm/s

cm/s

5

10

15

Fig. 1. Sections of zonal and meridional velocity along 14O”W from the JGOFS fall survey cruise. Velocities are positive eastward and northward, and have been smoothed to suppress space scales less than 100 km.

756

E. S. Johnson

the equator, which appears to be coupled to strong convergences around 5”s and 2”N (see Fig. 2). The latter convergence is the focus of this paper. An expanded view of the northern convergence, spanning the region between the EUC to the south and the NECC to the north, shows the southward flow apparently overriding the northward flow from the equator to form a layered flow field. This layered meridional flow occurs in a region of strong westward flow reaching speeds of 100 cm s- ‘. Zonal flow, however. has no comparable layering; rather it is relatively uniform above the strong thermocline. The region of southward surface flow is associated with warm temperatures and low salinities (Fig. 3) while the northward flow comprises low temperatures and high salinities: where the two meet the cold, saline water evidently subducts and continues northward at SO-100m depths to form a high-salinity tongue, consistent with the meridional velocity field. In temperature-salinity diagrams this intrusion forms a saline cusp, beginning at the front with salinities close to that of the northward flow but slowly eroding as it penetrates northward beyond 3”N (Fig. 4). By the next available hydrographic station at 5”N the intrusion no longer can be identified. Meridional velocity shows the subsurface advection continuing northward only to about 3.2”N (Fig. 2), making that the likely limit of the subducted tongue. Thus, the data present a picture of cold surface water flowing from the

Zonal

Meridional

r

1

I

I

Velocity,

cm/s

Velocity,

cm/s

I 2

3 Latitude,

4

“N

Fig. 2. As in Fig. 1 but for the region between 1” and 4”N. Southward velocities (dashed contours) appear to override northward velocities to produce a strong convergence with surface expression at 2.1 “N. The gap at 2”N represents a time discontinuity of 2.7 days; the almost vertical zero contour just south of the gap is a repeat sampling of the front. Data have been smoothed to suppress space scales less than 10 km.

Convergent

instability

wave front in the tropical

Temperature.

2 Latitude,

Fig. 3. of Fig.

Pacific

“C

3 degrees

As in Fig. 2 but for temperature and salinity. The strong southward (northward) velocities 2 are associated with warm-fresh (cold-salty) water, respectively. Data locations are represented by small crosses.

T-S

Dlot

c~cross

front,

10-l

50

m

deaths

25

---

300krn

N.

100km

.o

.____._

s ‘;

N

7krn

N.

Skm

5.

100km

5.

$20 E 2

/ 15 -

34.5

34.6

34 7

34.8 Salinity

34.9

35.0

35

Fig. 4. A temperature-salinity plot of five CTD stations from 10 to 150m depths, labeled by meridional distance from the front. The top of the thermocline is at about 23°C. The cold, salty surface water south of the front intrudes into the northern water column as a subsurface cusp of high salinity.

757

758

E. S. Johnson

equator to 2”N, where it subducts and continues northward beneath warmer surface water flowing southward from the NECC region. In a further expansion of the frontal region itself (Fig. 5) the southern surface water appears to subduct beneath the northern water within the space of a few kilometers, directly beneath the intense surface convergence and its implied downwelling. On the other hand, zonal velocity shows no comparable front-related features, a remarkable observation considering that the front itself is angled about 45” horizontally. Concurrent measurements of 7-m temperatures barely resolve the front even at 280m resolution (Fig. 6): a single sampling interval spans most of the front’s temperature contrast. This confirms the visual observation of Yoder et al. (1994) of the front’s extreme narrowness and intensity.

4. FRONTAL

CIRCULATION

4.1. Frontal position The front moved large distances during the period of observation. The ship entered the region from the northwest en route to a scheduled station at 2”N, 14O”W, then meandered westward with the moving front for 2.7 days, finally departing southeastward from 2”N, 141 .S’W (Fig. 7). During this period it made 11 crossings of the front (defined here as the

Meridional

2 00

2.05

Velocity,

2.10 Latitude.

cm/s

2.15 ON

2.20

2.25

Fig. 5. As in Fig. 2 but for the region between 1.98”Nand 2.25”N at 1km resolution, the highest resolution practicable due to noise constraints. The southern surface water subducts beneath the northern water within the space of a few kilometers. Zonal velocity shows no comparable frontrelated features.

Convergent

instability

ADCP

27

wave front in the tropical

7

m

Temperature

-

,o

+

^ z 3 26 z b F :

759

Pacific

-

+

25

-

i ,d’,

,

2.00

2.05

Latitude, Fig. 6.

,

,

2.10

,

,

,

,

2.15

,

,

,

,

,

, 2

2.20

!5

“N

As in Fig. 5 but for temperature at 7m depth. The sampling interval corresponding to about 280 m along the ship’s track.

is

1min of time,

25.5”C isotherm), providing an extensive sampling of its location in space and time. Ten of these crossings occurred within the first 2 days and yielded velocity sections of remarkable uniformity; the last occurred 1 day later when the compact structure of the front appeared to be breaking up. In order to resolve the front as a physical feature the data must be transformed into a coordinate system moving with it. Motivated by satellite imagery (Yoder et al., 1994), the surface front is assumed to be linear in form and moving steadily westward. Then its orientation and zonal phase speed can be found using a linear least-squares fit to the 10 earliest positions. This simple fit succeeds in explaining 99.3% of the front’s position

Ship,

Front,

and

2.4,

CTD

Positions

+x-,*++++x+ + I

++ ++ +

++ +++

+ ++

1.8

X ++

++

g s ’ c g 141.4

+.++,, 141.2

Ship Front

? ?CTD

Position

‘“s,

Crossing Station

, , , , , , , , , , , , , , , , , , , , , , , 141.0

140.8

Longitude,

140.6

140.4

140.2

1,4(1.0

W

Fig. 7. The spatial locations of CTD stations and every twentieth I-min ADCP profile. Positions where the ship crossed the front are also shown: the front is oriented NE-SW approximately along one limb of the crosses.

760

E. S. Johnson

variance, leaving a root-mean 2, and phase speed, c, are

square

(rms) residual

of 4.2 km. The fitted front orientation,

(II = 44.4 f 4.5”trigonometric c = 64.1 f 2.4 cm s-‘. angle corresponds

The trigonometric position is (6 - X())l.ll

to a compass

cm x 105= c(t - t0)86,400 -s- + cot(a)b oLong day

for -x0= - 140.7099”; :,=2.2196”; to= 1618 GMT, longitude in “E, y is latitude in “N and t is time in days.

4.2. Composite

heading

of 45.6”. The fitted front

- vO) 1.111 x 1O5= “Lat

8 August

1992 and

where

x is

velocity sections

The fit is so successful at reproducing the front’s position according to their distance from the front. The perpendicular D, = sin(cr)[x,

that it can be used to bin data distance to point p is

- x0+,, tp)]

where the function xCy,t) is the frontal position above. By assuming that the front is uniform along its length (about 40 km of which was sampled), one can assemble even scattered CTD data into useful sections. Further, the six usable ADCP sections can be gridded according to D, and cornposited into an average section. The concurrent 7-m temperatures were used to closely align the individual sections during this process, removing even the few kilometers of residual position error. The resulting composite velocity field (Fig. 8) is virtually indistinguishable from the individual sections due to their uniformity. The variance of the individual sections give standard errors for the composite velocity fields ranging from 2 to 4cm s-‘.

4.3. Cross-jkont-vertical

circulation

The co-ordinates relevant to the front’s local dynamics are those perpendicular and parallel to the front itself. Thus, the velocity components of Fig. 8 are rotated into those of Fig. 9, where along-front velocity is now parallel to the front and cross-front velocity is perpendicular to it. The cross-front component of frontal propagation [i.e. c sin(a)] is removed from cross-front velocity to give motion relative to the front itself; along-front velocity is left relative to the fixed Earth to facilitate calculation of the Coriolis force in the next section. Along-front velocity shows some shear across the front: the warm water north of the front has a component of motion southwestward along it. Note, however, that the thermal wind balance cannot be expected to hold at these small spatial scales: vertical shears are not necessarily associated with horizontal temperature gradients, but with the boundaries of water masses. Cross-front velocity shows the subduction of colder, equatorial water as before. It is important to realize that despite the front’s rapid westward propagation, it is moving easflvard relative to the even faster background flow of the SEC: thus, cold water

Convergent

instability

Zonol

’ ,’ -__-,’

Velocity

wave front in the tropical Composite.

! .-.’

. t

761

Pacific

cm/s

.I\ t-- ----_____ 11 \.__-. I,I. ..’:

50 ,’ \,

,I.\

w :..:’ 1 : ; 5 ‘ t.: e : cl ,oo.,.::3 _________-----___------_ _8a,-~~~~~~~~~~~~~~~~~ ---______,-‘ --____~---_~ ---______---____e--- _50-_..-_---________-. ------______--. --______---______----r-_______-________--__40----.___--__ .___--* L-. .-_r _ L I I _--. 301_---__ .-. 1 I -5

-10

Meridional

Velocity

tI.-“‘.““““““~1 -10

Fig. 8.

A composite

-5 Cross-front

10

5

0

Composite.

Cm/s

10

5 km

0 distance.

of six ADCP velocity sections resolved in terms of distance to the front. Standard errors range from 2 to 4 cm s-‘.

1.

__

-’

,__--__,

:_-*_-::I..

.____________---------____

perpendicular

----__*

-60‘,,.:2.

_------______-30.--. -_____----_____--__P----__--. ---___ _40_-__

_____~

.---

-__----.____e- --~

8, -10

,

.____,-20_~------‘.___ I

,

I

I.-.

I

Velocity

I

-j

-I--

10

5

0

-5

Cross-front

*

Composite.

cm/s IO 20

a

C&s-front

Odistonce.

km

Fig. 9. As in Fig. 8 but for velocity components rotated into the front’s orientation: velocity is positive northeastward, while cross-front velocity is positive northwestward. front component of the front’s propagation speed has been subtracted from cross-front give velocity relative to the front itself.

along-front The crossvelocity to

762

E. S. Johnson

moves west and north to overtake the front from behind and subduct beneath it. Note also that within the shallow, warm slab the cross-front velocity goes to zero at about 20 m depth. This is consistent with the expectation that the front is to first order a material surface admitting no through-flow. If comparable shears continue toward the surface then water above 20m depth must flow toward the front while water below 20m flows away. This implies a shallow, overturning circulation within the warm slab itself, possibly driven by mixing (see Section 5.1). Having cross-front velocity one can calculate vertical velocity from the continuity equation if the along-front component of convergence is negligable. Satellite SST imagery (e.g. Yoder et al., 1994) indicates that the linear segment of the front continues for 300 km: assuming along-front velocities of 100 cm s-l gives along-front convergence scales of about 0.3 x 10K5 SK’. In contrast, the cross-front convergence is at least 20cm s-’ over 5 km, or 4 x 1OK5 s- ‘, an order of magnitude larger. Thus, one may safely calculate averaged vertical velocity from averaged cross-front convergence. Individual crossings, however, show substantial variability in convergence across the front, which probably can be attributed to small-scale, along-front structure (D. Archer, personal communication, 1995). Since this structure is unresolved it will appear as stochastic noise in the vertical velocity composite. Velocities are extended to the surface using natural cubic splines to allow for a sheared surface layer. This is consistent with the strong stratification found as shallow as 10 m depth on the warm side of the front. The result shows strong downwelling within 1 km of the front reaching a maximum of 0.92 cm s- ’at 45 m depth (Fig. 10). Since the standard error here is about 0.2 cm s-l these values are highly significant. Below this, downwelling diminishes again, reaching nonsignificant speeds at the top of the thermocline. This picture accords nicely with our expectations: the downwelling is largest at the level of the bottom of the warm slab (where the greatest amount of fluid is moving downward) and nears zero again at the top of the thermocline, which should act as a floor bounding the circulation of the less dense waters above. Finally, the non-divergent fields of vertical and cross-front velocity can be transformed into a stream function of the circulation around the front (Fig. 11). The resulting streamlines closely parallel contours of water properties (Fig. 12), indicating that the flow fields conserve both heat and salt to first order. This provides confidence in the observed flow field.

4.4. Composite

CTD section

Fields of temperature, salinity and (T()are gridded from CTD stations scattered in time but resolved according to perpendicular distance from the front (Fig. 12). As noted above they are consistent with the observed flow fields. Several caveats, however, are in order. First, the horizontal resolution of the CTD stations is much poorer than that of the continuously sampled ADCP data. While reasonable contouring assumptions (vertical and horizontal mapping scales of 5 m and 1.5 km, respectively) produce reasonable-looking plotted fields, the actual structure between CTD stations was unmeasured: for example, the strong temperature gradients of the front (e.g. Fig. 6) are in Fig. 12 spread out between the closest bracketing CTD stations. No attempt is made to “remanufacture” the front beyond what was actually sampled. Secondly, the CTD data do not represent a synoptic section, but contain aliased signals from internal waves, tides and, possibly, longer period variability. While the water masses

Convergent

I

,

,

I

instability

wave front in the tropical

763

Pacific

I

-10

-5

10

5

0

Cross-front

Distance,

km

Fig. 10. Vertical velocity derived from the convergence of cross-front velocity, in cm s-l. Standard errors are about 0.2cms-‘: thus, the intense downwelling associated with the front is highly significant. The zero contour has been suppressed to reduce clutter, and extra contours at +O.l added.

adjoining the front appeared relatively stable during the 2.7-day sampling period, the depth of the main thermocline was not. Individual CTD stations show a rms variability of about 5 m in this depth. In addition, the thermocline appeared to be about 10 m shallower on the warm side of the front than on the cold side (Fig. 13). While this displacement is statistically significant, it probably does not represent real spatial variability. Concurrent TAO ATLAS mooring temperatures transitioned from warm to cold surface water at some time between 23 and 24 August, but no clear depression of the thermocline accompanied the front’s passage (Fig. 14). Rather it rose over the next day or two and then fell again. Since all the cold-side CTD stations were taken during 24 and 25 August, and the warm-side stations during 26 August, the observed thermocline “tilt” is much more likely to be aliased time

0

Stream ,.I..,..,.

Function

-5

Cross-front

11. The stream function Streamlines

Front

(m‘s_‘)

I ’ ’ ’ ’ I

-10 Fig.

around

roughly

0

5

Distance,

of verticakross-front parallel the contours

10

km

circulation calculated from Figs 9 and of cross-front velocity in Fig. 9.

10.

764

E. S. Johnson

-10

-5

Cross-front

Fig. 12. Sections of temperature, as small crosses. The streamlines

0 diSto”Ce,

5 km

10

salinity and be comparable with Fig. 9: data locations are shown of Fig. 11 also parallel these contours, implying conservation of these quantities.

variability than spatial structure. Thus, while Fig. 12 offers a reasonably of the front, no firm conclusions can be made regarding the thermocline’s 5. FRONTAL

consistent picture spatial structure.

DYNAMICS

Having determined the circulation and density structure across the front, the dynamics of the flow now can be addressed quantitatively. The first step is to check the volume and mass fluxes of the flow: if these are not reasonably balanced, it is futile to advance to the more sensitive calculations involving momentum and energy. 5.1. Continuity

and mass fluxes

Neglecting along-front variations reduces continuity to a balance between cross-front and vertical convergences. Since the thermocline acts as a bottom boundary to the circulation the vertical convergence can be eliminated by integrating vertically from the boundary to the surface. A depth of 110 m is chosen for the boundary depth, though the choice makes little difference. In anticipation of the mass-balance calculation below, continuity also is integrated across the front to regions where the flow is horizontally

Convergent

instability

wave front in the tropical

Mean lsoo~cncl

Fig.

13.

Isopycnal

depths

765

Pacific

Depths

averaged over nine cold-side and six warm-side 13 km of the front, with 95% confidence limits.

CTD stations

within

uniform (and hence where the density profiles can be reasonably estimated from nearby CTD data). Thus, the volume flux through a profile south of the front is compared to the equivalent flux through the north. The southern or cold-side velocity profile is defined as the average of cross-front velocity (Fig. 9) between - 11 and - 5 km: similarly the northern (or warm-side) profile is the average between + 5 and + 11 km. The vertical averages of these respectively. The discrepancy between them is small profiles are 29.5 and 24.7cms-r, 2”N, 14O”W Temperature,

_

“C

8Or_

Fig. 14. Thermal structure at 2”N, 14O”W over a period spanning in situ observations of the front. Individual data points, shown as dots, are I-day binned averages of several scattered observations. Data courtesy of the TOGA-TAO mooring group.

766

E. S. Johnson

relative to the overall transport, and is disposed of here by simply adjusting each profile by half that amount. This will be important in the energy flux calculation (see the Appendix). The volume flux as a function of density is constructed from these adjusted velocity profiles and the respective density profiles (Fig. 13). Cold-side transport toward the front is restricted to a very narrow density range, consistent with the well-mixed nature of that flow (Fig. 15). Warm-side transport away from the front shows a similar but smaller flow, augmented by a “shoulder” of less-dense flow above and an extended region of denser flow below. The shoulder can be attributed to mixing between the cold water subducting beneath the front and the less-dense water moving toward it above. The structure of this mixed shoulder is quite robust; in its middle at ao=23.5 kgme3 the water velocity is 20 cm s-’ away from the front (e.g. Figs 9 and 12). Since the standard error of cross-front velocity is about 4.2 cm s-l, the error for transport in the mixed shoulder is around 20%. This mixing drives a shallow overturning circulation that also is observed in the streamlines (Fig. 11): warm water flows toward the front above 20m depth, subducts on its warm side and recirculates poleward mixed with the underlying cold flow. The warm-side flow below rso = 24.0 kg m -’ is likely spurious. Recall from Section 4.4 that the elevated warm-side thermocline is an artifact of aliased time variability, so that the observed flow is erroneously perceived as having penetrated into the thermocline. This spurious density increase also results in the creation of spurious potential energy. For the purposes of the energy calculations in Section 6.1, the warm-side thermocline is adjusted downward to match that of the cold-side, with linear interpolation joining the adjusted warm-side thermocline below 99 m depth to the observed densities above 71 m depth. In effect the density variability above 7 1 m is accepted as being true spatial structure, while that below is rejected as being due to thermocline heave. The resulting adjusted profile gives warm-side mass fluxes in better agreement with the cold-side fluxes (Fig. 15).

Fig. 15. Transport

The cross-front transport of water on the cold and warm sides as a function of potential density. per unit density is proportional to the area under the curves. The warm-side density profile is adjusted for subsequent potential energy calculations to better conserve mass: see text.

Convergent

instability

wave front in the tropical

Pacific

767

5.2. Momentumfluxes In determining momentum balance it is important to distinguish between the forces maintaining the large-scale flow and the more intense, localized forces associated with the front itself. The former have broad structure and are expected to be of order 10m4 cm s-* or less (e.g. Bryden and Brady, 1985; Johnson and Luther, 1994). The latter have highly localized structures and are shown to be substantially stronger. Thus, by focusing the analysis closely on the length scales of the front itself one can neglect the smaller forces of the background flow. At the same time the available resolution is limited by the spacing of CTD stations (Fig. 12). Therefore, forces here will be averaged over the 6 km centered on the front, matching space scales with the CTD data and at the same time suppressing small-scale noise in the ADCP data. The equation for cross-front momentum can be written

where V,, U, and Ware the cross-front, along-front and vertical velocities; P is pressure, f is the Coriolis parameter and y here is the cross-front direction. On the scale of the front, two terms can be neglected. The time tendency is small due to the remarkable stationarity of the flow: changes of several tens of cm s- ’over the 2-day sampling period yield accelerations of only lop4 cm s-*. Vertical stresses also are neglected although this cannot be justified strictly on physical grounds: turbulent forces are not directly measured, and their parameterization for this relatively unique flow is beyond the present effort. Thus, our simplified momentum balance is

Advective forces are calculated using simple finite differences of the composite fields, with standard errors determined from their variances among the six usable sections forming the composites. The first term on the left of the balance above, the force due to the meridional advection of momentum, is positive above 45 m depth and negative below, returning to zero around 110 m (Fig. 16). This corresponds to the deceleration of surface water as it nears the front and its subsequent northward acceleration after subducting beneath it. The second term, the force due to vertical advection, is zero at the surface where vertical velocity vanishes. Below, it is negative to about 60m depth, consistent with the overturning circulation within the warm slab: some downwelling occurs on the warm-side of the front where vertical shears are strongly negative, rather than only on the cold-side where shears are positive. From 60 to 90 m the force turns positive, consistent with the subduction of cold flow beneath the front. All of these observed features are statistically significant at 95% confidence. The vertical advective term also has substantial positive values extending well below 100 m, even though the thermocline should act as a lower bound to the flow. This artifact arises from spurious downwelling through the thermocline (Fig. 10): unfortunately the intense shear there produces a large force from this smaller, statistically insignificant, downwelling. Discounting this artifact, the sum of the advective terms yields positive forces above 25 m and negative forces below down to the thermocline, consistent with the sinuous path of the flow.

768

E. S. Johnson Advective

Acceleration.

Fig. 16.

Advective

components 6 km centered

Forces

lo-‘cm

se2

of the cross-front momentum balance. horizontally on the front. Error bars are one standard error.

averaged

over

The Coriolis force is relatively small and quite well observed (Fig. 17). It has no structure in the upper 100 m, having more to do with the broader-scale background flow than the front itself. Thus, the front is not in any sense geostrophic. The cross-front pressure gradient is built up from CTD dynamic heights. The reference level is chosen at 100 m depth, thus focusing on pressure gradients produced by the density structure of the front above. (The thermocline below also exerts pressure forces, but as

Summed

0

E

60

r x 2

60

Forces

100

140 -2

Acceleration, -1 0

1 OJcm

5.’

Fig. 17. The sum of the terms of Fig. 16 along with the remaining momentum terms, averaged Fig. 16. Above 100 m depth the advective terms roughly balance the pressure gradient.

as in

Convergent

instability

wave front in the tropical

Pacific

769

mentioned in Section 4.4 these are not well observed here: presumably on larger space and time scales they geostrophically balance the Coriolis force of the SEC.) The pressure gradient errors are determined from repeat CTD stations, and are due mostly to time variability at the top of the thermocline: the vertical structure above is more precisely known. The front produces strongly sheared pressure gradient forces, negative above 30 m and positive below (Fig. 17), consistent with the tendency of the front to rotate under the force of gravity. The pressure gradient torque roughly opposes that produced by the summed advective forces. Their total, together with the smaller Coriolis force, gives a residual near zero above 100 m. Though noisy, this residual contains no evidence of the front’s two-layer structure: only the spurious vertical advective signal below 100m and the unknown large-scale pressure gradient remain. Therefore the front’s primary small-scale dynamical balance is between advective and pressure gradient forces. This is in marked contrast with the longterm background flow, which has zonal pressure gradients balanced primarily by Coriolis and wind stress forces (e.g. Johnson and Luther, 1994) and meridional gradients by Coriolis forces alone (Lukas and Firing, 1984). 5.3. Modeledflow?: the lock-exchange problem The relative insignificance of Coriolis forces at the front allow it to rotate freely under the force of gravity, a process known as the lock exchange in reference to the subduction of salt water beneath fresh following the opening of a lock gate. By observation the lighter, upper layer takes on a streamlined “nose” profile as it propagates into the denser fluid at speed Ca (Fig. 18). In the reference frame of the advancing front this nose breasts a current that accelerates beneath it to become a layer of uniform thickness at sufficient distance. The relevant theory for the free problem (i.e. having unlimited fluid supplies) was developed by Benjamin (1968). He found a unique solution for inviscid, energy conserving flow in which the final layer thicknesses each equal half the total fluid depth d (Fig. 18). At this point the denser fluid passes beneath the lighter with relative velocity (c2)2 = gd-, P2

where pi and p2 are densities for the upper and lower layers, respectively. By continuity C0 then equals half C2. This flow is supercritical with Froude number 1.4. Comparisons between this model and the data must be in a co-ordinate system aligned with the layered flow itself: this is evidently the north-south direction, since no shear appears in zonal velocity (Fig. 8). The relative density difference between the two layers (Fig.

J Fig. 18.

A schematic

of the lock-exchange

J+ problem

from the reference

frame of the moving front.

770

E. S. Johnson

13) is approximately 8 x 10W4, with d set to 110 m by the relatively rigid thermocline. Then the calculated C2 is 93 cm s-i, with Ce equal to 46.5 cm s-i. Comparable observed speeds relative to the front can be found by averaging cross-front velocity (Fig. 9) over depth and rotating back into N-S co-ordinates: the resulting Cc (averaged O-100 m) is 44 cm s-l, very close to expectations. The midpoint between the two layers occurs near 35 m depth (Figs 13 and 15) giving an /z2 of 75 m, or 0.68d, somewhat larger than expected. The observed C2 less than predicted. Finally, the Froude (averaged 40-100 m) is 55 cm s-l, substantially number is

Thus, the lower layer flow is less intense than energy conserving lock-exchange theory expects. Benjamin (1968) also showed, however, that the supercritical sheared flow downstream of the nose is unstable to small perturbations, and, in fact, can convert to a slower, subcritical flow via a dissipative hydraulic jump. After such a jump the flow should have hZ=0.78d, C2 = 60 cm s-i and Fr = 0.72. These values reproduce the observed flow much more closely, indicating that the observed flow closely approximates a dissipative lock-exchange. Since no distinct jump is observed in the layered flow it must be incorporated into the front itself. More elaborate theories incorporate the effects of friction and entrainment between the fluid layers. These effects are important for thin layers produced by the restricted outflow of rivers, but, for the present, unrestricted flow the effects are small and the solutions of Benjamin remain valid (Britter and Simpson, 1978).

6. FRONTAL

ENERGETICS

6.1. Energpf1u.ue.y and conversion The subducting transport amounts to about 3 Sverdrups per 100 km of front length. This subduction of relatively dense water converts potential energy into kinetic energy, which then may be incorporated into the larger instability wave or dissipated. The magnitude of the conversion can be estimated from the net flux of energy into the front. The total mechanical energy of a water parcel is: TE=$“-pgz+P. V is taken here as the cross-front velocity, since the assumptions of small Coriolis force and invariance in the along-front direction reduces along-front velocity to a passive tracer. (Indeed, inclusion of the along-front velocity in the calculation of kinetic energy makes no significant difference.) Then the flux of energy into the frontal region is the difference between fluxes through profiles bracketing the front: 0 [

s4

V2TE2 - I’,TEi]dz

=

$o(I$ P[-d

- I$) - gz (V 2~2

-

J,'IPI)+(V~~'~

-

~'IPI)

1 dz.

Through continuity a constant may be removed from p, while z and P may be taken relative to any fixed point; here height and pressure are taken relative to the cold-side

Convergent

instability

wave front in the tropical

Pacific

771

surface, and a mean value of 1023.5 kg m-’ is removed from density. Continuity also can reduce the kinetic energy term: a fuller treatment of the calculations and their errors is given in the Appendix. The net fluxes of energy are calculated using the adjusted profiles of velocity and density from Section 5.1. Pressure profiles are produced as for density, with the cross-front pressure gradient set by a surface boundary condition (Benjamin, 1968): since the surface represents a streamline, application of Bernoulli’s theorem along it requires the pressure difference to equal the kinetic energy difference. This removes any pressure gradient due to the thermocline tilt, which presumably is balanced by large-scale Coriolis forces. Kinetic energy converges on the front above 50 m depth, where the flow itself is convergent, and diverges below (Fig. 19). Though the two amounts roughly balance, there is a significant export of kinetic energy: the vertical integral above 110 m is - 840 f 3 1 W m-‘, where the uncertainty represents one standard error. This power goes to accelerate the subducted cold flow. The combined pressure and potential energy convergence is less readily interpretable since its vertical structure depends on the choice of reference level and mean density. It is strongly positive, reflecting the subduction of dense water and has large errors below 100 m depth due to the thermocline’s large variability. Its vertical integral is 5290 + 230 W m- ‘, a gain far greater than the kinetic energy export. Thus, the front has a net gain of 4450 k 230 W m- ’that somehow must be lost from the flow. This result confirms that the observed flow is not energy conserving. The Benjamin (1968) theory predicts that the equivalent two-layer lock-exchange flow scaled to the observed volume flux would convert 2310 W m-l of potential and pressure energy into 1830 W m- ’ of kinetic energy, with only 480 W m-i lost to dissipation. Thus, the observed flow has approximately twice the potential energy gain of the purely two-layer problem and half its kinetic energy export. Both these effects are due to the surfaceintensified nature of the observed cold-side flow: faster velocities higher in the water column Energy

Fig. 19. The net convergence to 0 m depth on the cold-side.

Convergence

with Standord

Errors

of energy on the front, with pressure and potential energy referenced The standard errors shown are for each term’s contribution to the vertical integral (see the Appendix).

772

E. S. Johnson

increase both the kinetic and potential energy of the inflow. Since the observed outflow roughly matches two-layer theory, the net result is an approximate order-of-magnitude increase in the amount of energy that must be lost through dissipation. 6.2. Dissipation

and mixing

The signs of dissipative energy loss in the flow are unmistakable. For example, the cold flow starts with positive vertical shear in cross-front velocity (Fig. 9) but immediately acquires negative shear after subducting beneath the front, indicating that momentum mixes between the two layers. Density mixes at the front as well, since it generates water at densities intermediate between the cold and warm inflows (Fig. 15). There is additional mixing in the downstream flow: the gradient Richardson number dips to 0.2 around 58 m depth (Fig. 21) indicating active deepening of the mixing region. In general this mixing must continue to the north since the surface layer cools and increases in salinity as it approaches the front. At 7 km from the front it is already a mixture of about 20% equatorial water relative to its characteristics 100 km to the north (Fig. 4). Nevertheless, the bulk of the mixing must occur at the front itself since within a few kilometers of it the flow stabilizes into layers matching two-layer dissipative theory. The dissipation rate typically is estimated using intensive microstructure measurements (e.g. Dillon et al., 1989). Here the few available CTD profiles probably are not sufficient to determine this rate with any reliability. Alternatively, one can deduce dissipation from the resultant mixing. By assuming a constant ratio between the total turbulent energy generated and that absorbed by mixing, one can estimate the net dissipation to heat from the potential energy of the mixed water. Examples are Henyey and Hoering (submitted) (using data) and

60

-

cI~~~~‘*~*.‘~* 22.5

23.0 Ue,

23.5 1 O-‘g/cm’

Fig. 20. The warm-side averaged density (solid) and velocity (dashed) profiles for the depth range in which mixing has occurred. Shown in lighter lines are the two-layer densities and velocities proposed for a hypothetical “unmixed” profile (see text).

Convergent

instability

wave front in the tropical

Pacific

773

Bogucki and Garrett (1993) (using a model). Osborn (1980) calculated this ratio, or “mixing efficiency”, for both Reynolds stresses in time-varying shear flows and the collapse of Kelvin-Helmholtz instabilities: the result was a relatively uniform value of 0.15. While the present flow conforms to neither of these models, Osborn’s mixing efficiency can still be used to construct a straw-man estimate of its dissipation. The difficulty lies in distinguishing the increase of potential energy due to mixing from the much larger decrease due to subduction. Observations cannot distinguish them since the two processes are intermingled at the front. Therefore, a hypothetical profile must be constructed showing the water column as it would appear after subduction but before mixing. There are a number of possibilities: the upper layer could remain stratified (as for a heated surface layer) or it could be set to constant density (as for a well-mixed layer); the velocity profile could remain unchanged or set to a two-layer flow conserving mass and/or volume relative to the mixed flow. Various combinations were tried, with results agreeing within a factor of 2. The calculation detailed here is for two layers well mixed in both velocity and density (Fig. 20). The upper layer’s initial density is set to that observed at 10 m depth (A), and its velocity is zero since prior to mixing there can be no flow toward the front. The lower layer’s velocity and density are matched to the fluid directly beneath (B), and the interface depth (C) is set to conserve the overall flux of cold fluid. This “unmixed” profile is then assumed to mix and produce the observed profile. The resulting increase in potential energy flux is 88 W m- ‘. This number’s uncertainty arises mostly from the choice of unmixed profile and is about a factor of 2: the reported case is at the high end of that range. Now by the definition of Osborn’s mixing efficiency, CI,

l -AE ct

A&eat = AEtota~- AE,o, = -

pot.

For c(=O.15 the total dissipation to heat becomes 500 W m-‘. This approximates the theoretical two-layer value of 330 W m-l , but is far less than the observed large-scale energy gain of 4450 f 230 W m- ‘. Observational errors cannot possibly account for this difference; therefore mixing between the two layers cannot absorb the front’s large energy gain. Mixing is also possible between the cold flow and the top of the thermocline. Unfortunately this is difficult to quantify since the warm-side thermocline was displaced upward by aliased time variability (Section 4.4); as a result the calculated Richardson number (Fig. 21) indicates strong stability near the top of the warm-side thermocline (90 m depth) rather than the marginal stability found on the cold side. Thus, the raised warm-side thermocline is not a product of mixing. What would indicate mixing is a rounding of the density profile corner between the thermocline and the weakly stratified cold flow above: this corner is only slightly more rounded on the warm side than on the cold side (Fig. 13), and the resulting increase in potential energy is negligible. Thus, the bulk of the front’s energy loss must occur though some means other than turbulent dissipation. 7. DISCUSSION 7.1. Internal waves as an energy sink Internal waves could absorb some of the front’s excess energy production. Indeed for weak hydraulic jumps much of the generated energy may be radiated away by a downstream

774

E. S. Johnson Inverse

Richardson

Number

“m

Ri“

Fig. 2 1. The inverse gradient Richardson number calculated from averaged potential density (Fig. 13) and velocity profiles averaged between 5 and 11 km of the front on both sides (Fig. 9).

train of short, standing waves (Benjamin, 1966). As the jump’s Froude number increases, however, dissipation grows more important until, for Fr > 1.25, turbulance suppresses the waves and most of the energy is lost to dissipation (Benjamin, 1966). Since the lockexchange flow would have Fr = 1.4 in the (hypothetical) region preceeding the hydraulic jump, significant radiation of energy through internal waves is not expected, though the present data cannot clearly rule it out.

7.2. Zonal geostrophic balance The lack of geostrophic balance across the front is a necessary consequence of its short time scales. Although the local inertial time scale of 2.2 days is shorter than the instability wave scale of 3.2 days (20-day periods), nevertheless the front has shorter scales yet: crossfront speeds of 30 cm s-t move particles through the 22-km sections shown in less than 1 day. Thus, the front is a non-geostrophic flow embedded in a larger, more geostrophic, instability. Yet the cold water subducted at the front continues northward at least 100 km beneath the upper layer (Fig. 2). Given the observed meridional velocities this requires a time scale of several days, comparable to the inertial time scale. Thus, on scales larger than the front itself the Coriolis force must be important in the flow’s zonal balance of forces. The layered flow’s zonal structure was not sampled: only one meridional section is available, showing the interface steadily deepening from 2.2” to 3.2”N (Fig. 2). Nevertheless, by assuming the layers are locally uniform in the along-front direction, one can approximate

Convergent

instability

wave front in the tropical

Pacific

775

their interface with a plane. This plane intersects they-z plane at an angle a of -0.035” from the horizontal, and intersects the sea surface at an angle u of 44.4” from eastward (Section 4.1). Thus, the two-layer thermal wind equation in the zonal direction becomes Vi - V, =T (tana! tar@)P2

and the resulting geostrophic shear is -74cm s-i. This is quite close to the observed meridional velocity shears (Fig. 2), indicating that the layered flow is close to geostrophically balanced. 7.3. Tropical instability

wave energetics

The front’s conversion of mean-flow potential energy into local kinetic energy is no doubt an energy source for the parent instability wave. Previous measurements of such sources rely on time- and volume-averaged eddy heat fluxes. The present synoptic measurement can be converted to an equivalent volume average by distributing its energy release over an active instability region 4” latitude wide and 50 m deep. Then the observed energy conversion of 5290 W m- ’gives a volume-averaged power of 79 + 3.5 ,uW m-’ after assuming that the net length of the subducting front is one-third the instability’s zonal wavelength. The comparable area-averaged heat flux is 78 W m-z. [These values are probably underestimates: the subducted water continues to deepen to 100m depths as it moves northward, possibly doubling the energy release. Also Harrison’s (1996) model instabilities have subducting fronts fully a wavelength long.] The observed energy production is less than the 200-300 PW m-’ found at 2”N, 1lo”--13O”W by Hansen and Paul (1984), but comparable to the 60 PW m-’ found by Luther and Johnson (1990) at 150”-158”W during a similar period of active baroclinic instability on the equatorial front. Note, however, that only 16% of the potential energy released at the front is exported as kinetic energy; the rest is evidently lost to the large-scale flow. Since calculations of meanflow to eddy potential energy conversion generally make no allowance for such losses, they can seriously overestimate the energy actually captured by the instability wave. 7.4. Meridional

orientation

of the subducting flow

Although the structure of the larger instability wave is beyond the reach of the present data, the orientation of the subducting flow offers an intriguing hint at larger-scale dynamics. Why should the flow be oriented precisely north-south? Is it a consequence of the most unstable instability mode maximizing down-gradient heat flux into the cold tongue? Clearly a great deal remains to be learned about the dynamics which engendered and constrained the observed lock-exchange flow. 8. SUMMARY

Shipboard ADCP velocity profiles and hydrographic stations were used to study an intensely convergent manifestation of the equatorial front formed at the western (leading) edge of a tropical instability wave cold cusp. During 2 days of observations the front’s propagation and associated flow fields were sufficiently steady that the data could be composited into a single, representative section. The front was formed by the convergence of

776

E. S. Johnson

warm, fresh NECC water with colder, saltier, equatorial water in a region of rapid westward background flow between the EUC and the NECC. The equatorial water subducted beneath the front at vertical speeds reaching 0.9kO.2 cm s-i, and continued northward beneath the NECC water as a saline plume visible in both velocity and hydrographic data, deepening to 120 m depths as it passed 3”N. The forces across the front consisted of a balance between pressure gradients and nonlinear advective forces; the Coriolis force was smaller and associated mostly with the broader scale flow. Thus, the front itself was not geostrophic. Instead, it resembled a surge of warm water freely overriding the colder water, with relative velocities oriented precisely along north-south lines even though the front itself angled southwest-northeast. Simple, non-rotating, gravity-flow theory developed for the lock-exchange problem allows two distinct solutions for such flows having unconstrained fluid sources: an energy-conserving, super-critical solution and a dissipative, sub-critical one. The latter solution accurately predicts the observed layer depths, speeds and Froude number, indicating that the subducting front was dissipative. This is consistent with mass fluxes across the front: water approaching the front was of two distinct densities, warm and cold, while flow away included only the cold water and water of intermediate density. This mixing drove a secondary, overturning circulation within the warm water mass; surface water moved south toward the front, subducted and returned northward mixed with the deeper flow. Thus, water converged toward the front from both directions, resulting in a dense concentration of floating organisms (Welling et cd., 1996). The subduction of cold water released potential and pressure energy at the rate of 5290+230 W m-’ of front length, a rate comparable to previous large-scale estimates of baroclinic energy conversion by tropical instability waves. Nevertheless the flow produced kinetic energy at only 840 & 3 1 W m- ‘, about 16% of the total potential energy conversion. Thus, the subducting front was far less efficient at converting potential to kinetic energy than expected from two-layer theory. Estimates of the total turbulent dissipation derived from the potential energy of the mixed water via a mixing efficiency (Osborn, 1980) gave a total dissipation of 500 f 300 W m- ‘. Thus, most of the front’s energy loss must have been through some other mechanism; radiating internal waves are possible, although theory makes it unlikely. It is concluded that estimates of baroclinic energy conversion derived from large-scale heat fluxes and density fields (e.g. Hansen and Paul, 1984; Luther and Johnson, 1990) can substantially overestimate the energy actually captured by instability waves, since they make no allowance for inefficient conversion. The layered flow extends north more than 100 km, a scale over which Coriolis force becomes important. On this scale the orientation of the layer interface produces a pressure gradient that just balances the meridional shear of the flow. Thus, in the north-south direction the flow is a gravity-driven lock-exchange, while east-west it is geostrophic. The relation of this flow to the dynamics and energetics of the parent instability wave should prove a challenging and interesting problem for both observationalists and meso-scale modelers. Aclino~~~/~~~/~er~l~nr.v-DrJames

Murray’s

help has been crucial

throughout

this work, from securing

funding

to

providing CTD data to shepherding the manuscript along as editor. I am grateful to Patricia Pullen and Charles Eriksen for advice on configuring the ADCP. and to Michael Relander for collecting the raw ADCP data. Timothy Boyd, LuAnne Thompson and Meghan Cronin contributed helpful discussions. while David Archer and John Milliman commented on the manuscripts. Frank Henyey and an anonymous reviewer inspired great improvements in the analysis. The ATLAS temperature data of Fig. 14 was provided by the TOGA-TAO Project Office, Dr

Convergent

instability

wave front in the tropical

Pacific

Michael J. McPhaden, Director. This work was supported by NSF grant 9024379 and by NOAA’s Environmental Laboratory. JISAO contribution 324. JGOFS contribution 214.

777 Pacific Marine

REFERENCES Benjamin T. B. (1966)Internal waves of finite amplitude and permanent form. Journal of Fluid Mechanics, 25, 241-270. Benjamin T. B. (1968) Gravity currents and related phenomena. Journa/ of Fluid Mechunics, 31. 209-248. Bogucki Darek and C. Garrett (1993) A simple model for the shear-induced decay of an internal solitary wave. Jownd of Ph~:rical Oceanograph!, 23. 1767-l 776. Britter R. E. and J. E. Simpson (1978) Experiments on the dynamics of a gravity current head. Journnl of F/d Mechanics. 88. 223-240. Bryden H. L. and E. C. Brady (1985) Diagnostic model of the three-dimensional circulation in the upper equatorial Pacific. Journa/ of Ph~:sicn/ Oceanography,, 15, 1255-I 273. Chereskin T. K.. E. Firing and J. A. Cast (1989) Identifying and screening filter skew and noise bias in acoustic Doppler current profiler measurements. Journal of Atmospheric and Oceanic Technology, 6. 10401054. Dillon T. M.. J. N. Mourn, T. K. Chereskin and D. R. Caldwell (1989) Zonal momentum balance at the equator. Journul of Ph!:ricul Oceunograpl~~. 19. 561-570. Hansen D. V. and C. A. Paul (1984) Genesis and effects of long waves in the equatorial Pacific. Jozrrnul of Geoph~~.vico/ Resenrch, 89, 10.43 I - 10,440. Harrison D. E. (1996) Vertical velocity in the central tropical Pacific: a circulation model perspective for JGOFS. Deep-Sen Rcreurrh II. 43. 687-705. Henyey F. S. and A. Hoering (submitted) Energetics of borelike internal waves. Journa/qfGeop/~~.~icu/ Research. Johnson E. S. and D. S. Luther (I 994) Mean zonal momentum balance in the upper, central, equatorial Pacilic Ocean. Jownnl of Geoph.vsicnl Research. 99, 7689-7705. Kessler W. S. and M. J. McPhaden (1995) The 1991-1993 El Niiio in thecentral Pacific. Deep-Sea Research II. 42, 295-333. Lukas R. and E. Firing (1984) The geostrophic balance of the Pacific Equatorial Undercurrent. Deep-Sen Reserrrch. 31. 61-66. Luther D. S. and E. S. Johnson (1990) Eddy energetics in the upper equatorial Pacific during the Hawaii-to-Tahiti Shuttle Experiment. Jownul of Physical Oceunogruphy, 20, 9 13-944. Murray J. W.. E. S. Johnson and C. Garside (1995) A U.S. JGOFS process study in the equatorial Pacific (EqPac): Introduction. Deep-Se0 Resenrch II, 42, 275-293. Osborn T. R. (1980) Estimates of the local rate of vertical diffusion from dissipation measurements. Journd of Ph~~sical Oceortogroph~~. 10. 83-89. Qiao L. and R. H. Weisberg (1994) Tropical instability wave kinematics: observations from the Tropical Instability Wave Experiment (TIWE). Jowml qf Geophysicd Research, 100. 8677-8693. Welling L. A.. N. G. Pisias. E. S. Johnson and J. R. White (I 996) Distribution of polycystine radiolaria and their relation to the physical environment during the 1992 El Nifio and following cold event. Deep-Se0 Rexrrrd~ Il. 43. 1413-1434. Wyrtki K. and B. Kilonsky (1984) Mean water and current structure during the Hawaii-to-Tahiti Shuttle Experiment. Jowmd of Ph.wicul 0ceanograph.v. 14, 242-254. Yoder J. A., S. G. Ackleson. R. T. Barber and P. Flament (1994) A line in the sea. Narure. 371. 689-692.

APPENDIX ERROR

CALCULATIONS

Continuity provides a powerful constraint parcel’s energy that passively advects through

FOR

ENERGY

FLUXES

on the net energy fluxes, allowing one to neglect that part of a water the region. The most straightforward applications are in the choice of

778

E. S. Johnson

local reference level and the removal of mean density. This focuses the calculation of pressure and potential energy on their local changes. Similarly for kinetic energy one can decompose velocity into its vertically averaged and residual, vertically varying parts. By continuity the vertical means of the profiles are equal (assuming the flow is indeed constrained below by the thermocline and uniform along the front). Then the net kinetic energy flux becomes

;po[V; s’ -
V;] dr =

s -d

;,%[3ML(s~

- VI) + 3M(v; - I$) + (v; - v;)] d-_,

where M is the vertical mean velocity and the rs are residuals of profiles bracketing the front. Terms proportional to M’ have already been canceled on the right-hand side. Similarly the M’ term is zero in the vertical integral: thus it figures only in the net energy flux at a given depth. Since the vertically integrated flux is of most concern here, the subsequent analysis will focus on the last two terms, which are clearly smaller than the two canceled terms for any region dominated by through-flow (i.e. for M> 11)~ Thus, adjusting the velocity profiles to ensure exact continuity is important: any net flow into the region would throw the large canceled terms out of balance, possibly swamping the smaller. physically interesting terms. Assuming the variables are joint normally distributed and the residual velocities of the profiles are not correlated with each other or with errors of the mean velocity, the mean-squared-errors (MSEs) contributed by one profile can be found as:

where GA, and O, are the standard deviations of the mean and residual velocities, respectively. Mean-squared errors for the net flux are the sum of errors for both profiles. Similarly, errors for the combined pressure and potential energy flux contribution of one profile are

where. again. the mutual correlations between velocity. density and pressure have been neglected. Mean-squared errors for the vertical integral of energy flux are then the vertical integral of the errors above. divided by an assumed four degrees of freedom in the vertical.