On a long series of measurements of Indian Ocean equatorial currents near Addu Atoll

Deep-Sea Research, 1976, Vol. 23, pp. 211 to 221. Pergamon Press. Printed in Great Britain.

On a long series of measurements of Indian Ocean equatorial currents near Addu Atoll R. A. KNox* (Received 14 April 1975; in revised form 4 August 1975; accepted 11 August 1975)

Abstract--Results of nearly 2 years of weekly current and temperature profiles to 300 m in the central equatorial Indian Ocean are discussed. The principal current detected is the eastward jet which appears at both the spring and fall monsoon transitions, driven by strong eastward winds, as noted by WYRTKI (Science, 181, 262-264, 1973). The jet is almost as strong at the depth of the thermocline as at the surface, and zonal currents at both depths are well correlated with the local wind stress at essentially zero lag. Since climatic atlases (NEDERLANDS METEOROLOGISCHINSTITUUT,Indische Oceaan Oceanografische en Meteorologische Gegevens. Publ. No. 135, Vol. 2, 1952) reveal a period of a month or more at the spring transition during which the winds over the western equatorial ocean remain westward while those over the central and eastern ocean change to eastward, the importance to future equatorial ocean response studies of obtaining simultaneous wind and current data from a number of longitudes is underscored. A purely local balance of forces fails to account for the observed zonal accelerations; non-local effects, mainly the time-dependent zonal pressure gradient, are important. Following each appearance of the eastward jet is a month or more of westward flow, though the winds show little or no westward stress. A similar effect, due to relaxation of water piled up at the eastern coast by the jet, is found in a timedependent numerical model of O'BmEN and HURLBURT(Science, 184, 1075-1077, 1974). An eastward equatorial undercurrent beneath a westward surface flow is found in only one of the 2 years, and this is at least consistent with certain differences in the winds of these years. 1. I N T R O D U C T I O N

AN EARLmR note (KNOX, 1974) reported on a set o f current and temperature profiles taken at locations bracketing A d d u Atoll in the Maldives (see Fig. 1). The main result was that at stations well offshore to the n o r t h or south o f the atoll 0 5 nautical miles or sot), the major features o f the profiles, in particular those o f zonal velocity, were only slightly disturbed by the presence o f the land, and so were presumably representative o f nearby open ocean conditions. With this encouraging background, a p r o g r a m o f regular weekly profiles was begun in January, 1973, in an attempt to m o n i t o r equatorial current fluctuations on time scales o f weeks to seasons and to try to relate such fluctuations to features o f the strongly seasonal winds. Since interest in carrying out a m a j o r experiment designed to observe Indian Ocean response to transient winds is n o w building ( I N D E X - - I n d i a n Ocean Dynamics Experiment),

it is felt useful to present at this time an analysis o f data gathered t h r o u g h September, 1974, discussing primarily the seasonal cycle. A second paper will appear after the mid-1975 termination o f the project (due to closure o f the British military base f r o m which the observations are carried out); it will refine the analysis o f the seasonal signal using the full data set and will investigate such additional features as signals at higher frequencies, e.g. equatorial waves. 2.

METHODS

AND MEASUREMENT

ERRORS

Once each week, weather, equipment a n d military priorities permitting, a single profile to 300 m o f temperature and current is made at one o f two sites near A d d u by men o f the British *University of California, San Diego, Scripps Institution of Oceanography, P.O. Box 1529, La Jolla, California 92037, U.S.A. t l nautical mile = 1"852 km. 211

212

R.A. KNox 75°E

74°E

nose of the unit a large sheave captures a guide wire suspended from the ship. The PCM thus slides smoothly down the wire, decoupled from wave-induced vertical motions. This decreases speed errors due to rectified pumping of the Savonius rotor, an important consideration in working from small vessels, such as the 65 ft R.A.F. pinnace used in this study, which respond to short waves. The PCM housing acts as a large 0o vane for the current meter. The Aanderaa recorder samples temperature, pressure, speed and direction once every 30 s, which typically corresponds to a few meters of descent. Profiles are made in deep water at the positions N and S in Fig. 1, alternating weekly between the two. This procedure was initiated to detect any los major differences in currents north and south of Addu. No such differences are apparent in the record, and in what follows all data are analyzed 75*E 74°E Fig. 1. Chart of Addu Atoll and vicinity, with bathy° without regard to location. metric contours in meters (courtesy of R. L. Fisher). While making the profiles the vessel drifts Contour interval 500 m, except for 100-m curve around freely or steams at a constant speed and heading atolls and islands (shaded areas). Profiles are made at N to reduce wire angle. Conversion of the measured and S. Gan (G) is the southernmost island of Addu. relative velocities to absolute velocities thus Meteorological Office and of the Royal Air depends on determining the vessel track. The only Force stationed at the R.A.F. base on Gan practical means available is to loft a radar target (Fig. 1). The instrument used is the Profiling above the vessel using a kite or helium balloon Current Meter (PCM) of DUXN6 and JOr~NSON and to follow this target using the meteorological (1972); a schematic diagram is given in Fig. 2. pilot balloon tracking radar on Gan. A typical A standard Aanderaa recording current meter, plot of fixes at 5-rain intervals is shown in Fig. 3; less fins, is mounted in a long ballasted flotation package so that the whole is slightly negatively buoyant and is trimmed to lie horizontal. At the

)

I*N

1000m

/ /// /// 7 // //'/

__

Fig. 2. Schematicdiagram of PCM as used. IV, is vertical component of PCM motion.

I

_

I

!

I

2 IO00rn

I

1

I

Fig. 3. Radar track of pinnace position at 5-rain intervals,

11 July, 1974, at Sta. S. Boxes bound errors due to radar accuracy; dashed line is mean track.

A long series of measurements of Indian Ocean equatorial currents near Addu Atoll the box at each fix encloses the errors due to range and bearing accuracy of the radar, -4- 12.5 m and 4- 0.07 °, respectively. Averaging the twelve 5-min velocity vectors of this track gives vessel motion components (mean -4- standard deviation of the mean) o f --44-9 4- 7.0 cm s-x (--35.9 41"3 cm s-x) to the east (north). For all radar tracks thus obtained, 10 cm s-~ is a conservative value for the standard deviation. In all cases only the mean vessel motion has been used, as no cases of serious unsteadiness over the profile duration (about 1 h) were found. Due to rain squalls, wind conditions and other commitments for the radar, this tracking procedure has yielded useful results for only about one-third of all profiles obtained, but it is possible to estimate absolute velocities for the remainder using the fact that the deep currents (below 150 m) are small; this layer serves as a rough reference level adequate for discussion of the much faster near-surface currents. Details of this estimation and associated error, using an objective interpolation scheme, are given in the Appendix. The expected r.m.s, error o f absolute currents thus inferred is 19 cm s-~ or less. Extensive checks of the Aanderaa meter are carried out immediately before and after each profile. Pressure and direction sensors are calibrated against a deadweight tester and a set of known headings, respectively. Rotor sensitivity and the recorded count increase for a known number of revolutions are examined. Indicated temperature is checked for gross disagreement with ambient. In addition, two separate recorders alternate in use on an irregular schedule; no jumps in data values ascribable to a change of units have occurred. It is believed that with the exception of occasional failures detected by the checkouts and subsequently remedied, the instruments have operated within the manufacturer's stated limits of accuracy: 4- 10 m in depth, 4- 0.1°C, 4- 5° in direction, and about 1% in speed. A further source of error in the current measurements stems from the fact that the PCM moves along the slanted wire. The principal effect is to introduce an unrecorded horizontal motion of the unit relative to the ship of magnitude

213

W, tan ~r, where W, is the sinking rate and cr is the wire angle (Fig. 2). Normally this error is worst near the surface, where the wire angle may be measured, and it is found to be generally less than 10 em s-1. The occasions on which it exceeds this value are those on which high currents cause large wire angles, as during the eastward jet; this may explain some o f the apparent large deep currents in Autumn, 1973, in Fig. A2. The data presented here are uncorrected for this error, as no ready means of determining its size and direction as a function of depth, e.g. by acoustic ranging on the PCM, are available. Considering the combination of radar accuracy, interpolation for profiles without radar tracks, and relative motion of ship and PCM, an error of about ± 25 cm s-1 seems a reasonable value for the current data. 3.

RESULTS

AND

DISCUSSION

Figure 4 shows a plot against time of several observed quantities. The 'surface' velocity u, is defined by a depth average: 0

1 u~ = us t + v~ j = 20--~

f u(z)oz

(1)

-- 20m

where z is the vertical coordinate, u(z) is the velocity, f and j are unit vectors toward the east and north. Similarly the 'thermocline' velocity uth is --60m

^

Uth = Uth ~ -3[- Vth j - -

1 f u(z)dz,

20m

(2)

- 80In

the depth interval being chosen to coincide with the core of the undercurrent. These averages are used instead of single depth values in order to smooth effects of small-scale variability evident on some profiles, e.g. that for 18 January, 1973, in Fig. 7. Weekly average wind stress vectors in Fig. 4, plotted at the ends of the weeks involved, are computed from the routine hourly wind observa-

214

R . A . KNOX

cm s "1

0.5

5O /N

/"X~.

dyne em"2

0.5

A

T~

J ! v ! i I:Z I i I,J 1 a I A !:S:I 0 t N.I P I J , I V I,M,I a I: M ! J :1 :J ! A :I:S:L.

i

1973 1974

Fig. 5. U p p e r : surface zonal current u, and weekly average zonal wind stress ~z. Lower: comparison o f fluctuating zonal stress ~z' and acceleration a', as discussed in connection with equation (7). ~ ' exceeds a" during eastward winds (shaded regions).

tions at Gan according to

T -- % t W z y j - -

p,,Co 168 z , , I *, I. 168 t=l

(3)

Here the wt are the wind velocities in cm s-z at 10 m height, Pa = 1"15 × 10-s g cm -a and C D = 1.5 × 10-s. In Fig. 4, and more dearly in the upper half of Fig. 5, where the zonal components rx and us are plotted against time, we see the virtually simultaneous appearance of strong eastward winds and currents in the spring and fall of each year. This is the eastward jet at the transitions between monsoons, first noted in climatic data by WYRTKI (1973). It is interesting to note that not only the overall peaks in u s and %, but also many of the shorter time scale fluctuations are nearly in phase. Table 1 gives the mean winds and currents during the three occurrences of the jet; the eastward flow at thermodine depth is a substantial fraction of that at the surface. Mean southward flows are also indicated for both levels, but one should be cautious in reading too much into these smaller values obtained so close to a large chain of atolls. For comparison, 1-year averages o f the same quantities are given in the table. Eastward winds and currents dominate these also. Note that the currents and, to a lesser extent, the eastward winds of autumn, 1973 were stronger than during

Fig. 6. Raw normalized cross-correlation estimates for selected pairs o f quantities, notation and calculation method as in equation (4). Curves are offset along ordinate by 0.4 for clarity.

the two springtime jets, while WYRTKI (1973) has reasoned from climatic data that the autumn jet is the weaker one. Figures 4 and 5 show that the time lag between winds and currents is very short; this is exposed further in Fig. 6, where we show raw crosscorrelation function estimates for several pairs of series over the entire record. The calculation performed is

l cxy ( k A r ) =

z x(t,) y(tj)

Nk

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. (4)

(ti)]}

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MAR. JUN. AUG. -L_t-__J,I-U_N+.,----ll..ll--+-I--+~_U_L+-:--+, -1-1....i _A....,~I-G_.+,-----jILl--+-_S-+~_P_T+-~ --+,-1.1_,1---+,0----lC:_,-+,_---4_ _N-t0_V_,1---+--+-_D+E_C-I'_+--I-+--+-_J-tA_N_,1----+--+-F_E+B_,-+---+---+_+---+--+-L-t-----l_+---I--4---+_+---+---tL--+----l_+----t--lLt-_-+_+---+--'l---+----l_+---+-1--+---+_+---+ , , I , ,APR. , I, I MAY , , I, , I I ,JULY , ,I , SEPT, I , ,I --+1

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1973 1974 Fig. 4. Time series of the data. Curves give mixed layer temperature and depths of mixed layer (MLD) and of two isotherms; separation of the latter two is a measure of the sharpness of the upper portion of the thermocline, Vector series of currents and wind stress are as defined by equations (1) and (2), The two circled mixed layer temperature points are linear interpolations on days when the mixed layer depth was zero, Open circles in the current series indicate unusable velocity data, solid circles in any of the vector series denote zero values, [facing p. 214)

A long series of measurements of Indian Ocean equatorial currents near Addu Atoll

215

Table 1. Mean wind stress and currents for eastward jets and for full year. Mean Value

of:

21 Mar-14dune 1973 (11 profiles) Us(m/sec)

93.6

:Vs(Cal/sec)

-10.0

Uih(Cm/sec)

74.7

Vth(Cal/sec)

-17.9

13 Sept ]973-3 dan 1974 (17 profiles)

11 April-13 dune 1974 (11 profiles)

101.6

-

-

10.7

Annual 18 dan 1973-17 dan 1974 (46 proflles)

83.8

-

51.5

-13.7

5.3

90.7

58.4

51 ;8

17.7

-13.0

-12.5

• x(d.y.ne/an2)

0.38

0.48

0.44

0.36

•y(dyne/cmz)

0.I0

0.02

0.16

0.18

Here x(O, y(ti) are the two series (means removed), the N k terms of the sum in the numerator include all data pairs with 1 A

<,,_t, < (,, +3 and each x(h ) [y(ti) ]

appears in the first (second) sum in the denominator as many times as it is used in the numerator, the total number being Nx (Ny). ti, t/ are the observation times, and Ay is the elementary time lag, 10 days for all such calculations in this paper. We see that the cross-correlations of zonal wind stress with zonal currents at both levels, eu,,, and cu,h,,, exhibit strong half-yearly periodic components and no detectable phase shifts, v, appears to lead both us and T~, by about 20 days, but the cross-correlation amplitudes are low. One is naturally led to question whether the strong zonal currents of Figs. 4 and 5 are in some sense commensurate with the local wind forcing. While observations at a single point such as these cannot fully answer this question, some progress can be made in the context of a simple model. Thus, we consider a two-layer system with the interface at the depth of the actual thermocline, and write the linearized vertically integrated zonal

momentum equation for the upper layer, assuming the velocities to be depth independent:

a U _ ~yv = __I_3P + ~x _ F. 0t p Ox pH

(5)

Here H is the mean thickness of the upper layer, F is the frictional force retarding the flow (drag at the bottom of the layer), ~x is as defined above, and the remaining notation is conventional. Denoting a time average by an overbar, the mean flow obeys:

_~yp= l ~ p +~x _F. p Ox

(6)

pH

If we further assume that fluctuations of F and ap about their time averages may be Ox neglected--an ad hoc assumption forced by lack of measurement of these quantities--the remaining fluctuations, denoted by primes, obey: -~ a' = oH \(au'

~yv'

)

,

= Tx ,

(7)

from (5) and (6). The two sides of (7), relating wind forcing to acceleration a', may be computed from

216

R.A. KNOX

the data; both quantities are plotted in the lower half of Fig. 5. The agreement is not good; in particular, ~,~' exceeds a' consistently during most of each period of strong eastward winds (shaded areas).* This is presumably due to the fact that ~P is not constant, as was assumed above; the ~x eastward winds temporarily set up the sea surface to the east causing a positive (eastward) pressure gradient fluctuation, which is of the correct sign to account for the discrepancy. WVRTKI (1973) shows from historical sea level data that this setup amounts to 0.5 dyne cm -3 in equivalent stress (pgHAh/L, where Ah is the difference in surface elevation between east and west coasts separated by a distance L); this is of the right magnitude to close the shaded gaps in Fig. 5. Were we to neglect F and ~P altogether, the dx resulting model would be (7) with total fields instead of fluctuations; this would be a strictly local balance of forces. The corresponding plot would be the lower half of Fig. 5 with the ~x' curve shifted up by ~'~, and the a' curve shifted down by -- ~y H~7 for a total shift apart of 0.34 dyne cm -2. The curves would then almost never coincide; the strictly local model is inappropriate. Non-local effects, such as the zonal pressure gradient, are significant throughout the record. In Fig. 4, each appearance of the eastward jet is seen to be followed by a month or more of westward flow in the surface and thermodine layers, as the winds become more meridional and lose strength. A similar effect is seen in the computations of O'BRIEN and HURLBURT (1974), even though in their model the wind switches on impulsively and remains constant toward the east. The westward flow is due to relaxation of water piled up at the eastern boundary. Climatic atlases (NEDERLANDSMETEOROLOGISCHINSTITUUT, 1952) show some evidence of weak surface westward flow along portions of the equator in the summer months, following the springtime jet, and a more consistent pattern in the winter months when of course the winds assist by tending to blow towards the west; this flow is generally

called the Northeast Monsoon Current. Thus far we have discussed the portions of the record, and they constitute the majority of it, in which the currents at the surface and at thermodine depth are in the same direction, i.e. in which the entire upper layer moves approximately as a block. The only strong and sustained example of a higher order baroclinic structure is found in March, 1973; individual profiles including that period of time are shown in Fig. 7. Those for January and February show no significant zonal currents. On 2 March we find a 'classic' equatorial undercurrent profile, with westward flow at the surface and eastward flow in the thermodine. The temperature profile is roughly linear, erasing the previously abrupt thermodine. This situation persists only a few weeks; by 29 March the nearsurface flow tends towards the east, and by 5 April the eastward jet is established throughout the upper 100 m or so. Because of the unfortunate loss of data in February, we can say only that the classic profile was present for between I and 2 months. A similar development did not occur in 1974, and there are differences in the winds between the 2 years which are at least consistent with this. Figure 8 gives a more detailed picture of the wind data, presenting daily average stress in component form. Whereas the early months of 1974 exhibited no consistent westward winds, and were preceded by the strongest episode of eastward winds, we see that in late January and February, 1973, there were continuous though gentle northeast trade winds and that the autumn eastward winds of 1972 were only strong for brief periods of time. Thus the winds at Gan in late 1972 and early 1973 would seem more favorable than those a year later for creating an accumulation of water at the African coast and an east-directed pressure force in the thermocline to drive an undercurrent against a westward surface flow.I Whether or not the Gan *The agreement is not significantly improved by allowing the layer thickness H to vary in time. tNote added in revision: preliminaryviews of raw data from early 1975 seem to show that a northeast trade regime and a weak 'classic' undercurrent existed for several weeks.

A long series of measurements of Indian Ocean equatorial currents near Addu Atoll

217

TEMP (*C) 20

30

"' 50

30

30

30

50

30

30

30

200

300

U (cm s-') O

2~)-

g3oO

0

~0

~00

I00

~

~.

,

18 JAN.

25 dAN.

Z F~B.

8 MAR.

I I~AR.

16 MAR,

22 MAR

29 MAR

- 5 APR.

Fig. 7. Profiles of temperature and zonal velocity in early 1973. Temperature profiles are offset by 10°C. Note that visual impressions of the comparative levels of small-scale variability between profiles may be misleading, as sinking speed of the profiler, and hence depth interval between samples, varies from profile to profile. winds were representative of those farther to the west remains an open question. Climatic atlases (NEOERLANDS M~EOROLOGISCH INs~Ttrtrr, 1952) suggest that in fact northeast trade winds may prevail longer into the spring over the western ocean than at Gan. Figure 9 is redrawn from that atlas and shows the mean monthly wind vectors for the two degree squares adjacent to the equator for March and April, and we see clearly the tendency of the eastward winds to appear first in the eastern ocean. Since the currents respond to the winds (Fig. 5) in a time much less than that

of these monthly charts, it seems possible that an undercurrent profile like those o f Fig. 7 might be found in the western ocean at the same time as an eastward jet in the central and eastern ocean. WYRTKI'S (1973) m a p of the jet currents also shows that the strong eastward surface flow is not found in the western ocean. The zonal pressure gradient is a key force in theories of the (steady) undercurrent (PHILANDER,1973; GILL, 1972), and in the O'BRIEN and HURLBURT (1974) timedependent model of the jet. Here again there is some evidence o f a significant difference between

~'y

"0.5

dr

cm'2

1.0 o0

o

. . . .

v

~

,

" 0 - 5 ~

,,s,,L?r:N~plz~t+L..~.~.~,

tgzzllgy 3

~

J

A

s

=....,...a....,... , o ~

N IOIJ.fF~M~,I,,A,1MIO

19Z3 rgz4

J ~ArS

'" ~

........

Fig. 8. Daily average wind stress components, computed as in equation (3) apart from change of averaging interval.

218

R.A. KNox

~/'~"

~

v

4. d'

v

4,

,I,

IOkt I 40* E

50*

60 ~

70"

80 °

9&

60 °

70 °

80 °

90 °

I00"

/

0° / ~ ,

~ a

I0

IO°S 4 & E Fig.

9.

50 °

ktI IO&

Climatic monthly mean equatorial winds (velocities) for March and April, adapted from NEDERLANDS METEOROLOGISCHINSTITUI2T(1952). Gan is at X.

the western region and the rest of the ocean. TAFT and KNAUSS (1967), using an equatorial hydrographic section conducted from 10 April to 24 March, 1963, near a monsoon transition time, found the sea surface sloping upward to the east eastward of 61°E while the 50 dbar surface sloped upward to the west, westward of 71°E; the remainders of these surfaces were level (relative to 400 dbar). No corresponding change in the vertical structure of currents in the region 61° to 71°E was found; the authors report an undercurrent beneath a westward surface flow all along the equator at this time. Clearly, highfrequency sampling of wind, current, and hydrographic properties simultaneously at a number of longitudes is required to separate and study these variations in time and space more effectively. Features of the temperature structure in Fig. 4 do not seem to follow such a pronounced twice yearly cycle as the currents. The temperature

of the mixed layer has a clear annual variation over the record. There is some indication that the depth of the mixed layer increases with each onset of the eastward winds, but the signal is noisy and the effect is only really clear and sustained in the fall of 1973, the strongest of the three wind episodes. During this time the mixed layer temperature also falls and this might be viewed as evidence of vertical mixing, but during the other two episodes the temperature rises. In the Pacific, where the equatorial surface water is colder than that immediately to the north or south, such a temperature increase might indicate winddriven Ekman convergence, but the result to be expected from such a convergence in the Indian Ocean is less clear, since the shallow temperature field (surface and 100 m levels; W ~ T g L 1971) is more complex, with pronounced variation in longitude as well as latitude. It does not seem possible to account satisfactorily for the observed

A long series of measurements of Indian Ocean equatorial currents near Addu Atoll

temperature variation o n the basis o f these data alone.* 3. SUMMARY AND CONCLUSIONS T h e m a i n points which emerge f r o m consideration o f this record are: (1) the d o m i n a n t p h e n o m e n a are the strong eastward winds of fall a n d spring a n d the resultant eastward jet i n the ocean, the time response o f which is very fast. T h e numerical m o d e l o f O'BRIEN a n d HtmLaURa" (1974) seems to a c c o u n t at least qualitatively for m o s t features o f the jet. (2) A n eastward u n d e r c u r r e n t b e n e a t h a westward surface flow was f o u n d in 1973, when there were consistent northeast trade winds, b u t n o t i n 1974 when such winds were only sporadic. The northeast m o n s o o n lasts longer over the western ocean, a n d the u n d e r c u r r e n t m a y do likewise. (3) Because the equatorial ocean responds so quickly to the winds, a n d because the winds vary strongly in space as well as time, refinement o f the picture presented here a n d more sensitive testing o f d y n a m i c models will require wind, current, a n d hydrographic data t a k e n at a n u m b e r of longitudes a n d at short time intervals.

PHILANDER S. G. H. (1973) Equatorial undercurrent: measurements and theories. Reviews in Geophysics and Space Physics, 11, 513-570. TAFT B. A. and J. A. KNAUSS (1967) The equatorial undercurrent of the Indian Ocean as observed by the Lusiad expedition. Bulletin. Scripps Institution of Oceanography, 9, 163 pp. WYRTKI K. (1971) Oceanographic atlas of the International Indian Ocean Expedition, National Science Foundation, 531 pp. WVRTKX K. (1973) An equatorial jet in the Indian Ocean. Science, 181, 262-264. APPENDIX

Estimation of absolute velocities The technique used to infer the absolute velocities for profiles on which radar tracking was not possible is a simple application of a more general method termed 'data adaptive linear estimators' by Davis (personal communication). Davis and colleagues plan a paper on the use of such techniques in designating oceanographic experiments, and some of the theory is contained in Appendix A4 to Chapter 4 of JENKINSand WATTS(1968). A brief exposition is given here. We suppose we have N data values q, and we wish to use these to estimate some other quantity Q. Choices of what constitute the q~ and Q may take many forms, and judicious selection is part of the art. In our case we see from the raw records that the deep currents tend to be small, so we choose to attempt to estimate these from available data. As the q~ we therefore select 1

Symposium on numerical models of ocean circulation, Durham, New Hampshire, October 17-20, 1972, National Academy of Sciences. JENKrNS G. M. and D. G. WATTS (1968) Spectral analysis and its applications, Holden-Day, 525 pp. KNOX R. A. (1974) Reconnaissance of the Indian Ocean equatorial undercurrent near Addu Atoll. Deep-Sea Research, 21, 123-129. NEDERLANDS METEOROLOGISCH INSTITUUT (1952)

lndische Oceaan Oceanografische en Meteorologische Gegevens, Publ. No. 135, Vol. 2, 24 charts. O'BRmN J. J. and H. E. HtmLntmT (1974) Equatorial jet in the Indian Ocean: theory. Science, 184, 1075-1077.

N

q~ = u,~ -- ~-r 1~ud,, ~v

REFERENCES DUING W. and D. JOHNSON (1972) High resolution current profiling in the Straits of Florida. DeepSea Research, 19, 259-274. GILL A. E. (1972) Models of equatorial currents. In:

219

i~l

(AI)

"

where the ud~are the depth-averaged deep currents --h 1

uh = ~ f u(z, h)dz.

(A2)

--(H+h)

Here h = 150 m, H + h is the greatest depth on the profile, generaUyabout 300 m, and h is the time of the profile. Note that in (AI) we have removed the sample mean; this is convenient in the present example but not a necessary feature of the general method. The problem now is to approximate the unobserved A value QI3 ofq at some other time tfl, using an estimator QI~ which will be a linear combination of the q~. That is, we wish to use N d13 = ,~1 ¢ti3,q, (A3) to approximate the unobserved item

*Note added in revision: J. C. Swallow (personal communication) has suggested that a reason for the annual signal in mixed layer temperature may be eastward advection of water from a source region near the African coast. There the annually reversing Somali Current alternately supplies warm water of northern origin and cooler water from the south.

1 N

Qfl = udfl -- ~ t~lud,,

(A4)

by choosing the N weights ¢tfl,to minimize the mean square error. We demand in general that such estimators have the correct mean value

220

R . A . KNOX

<~

-

or =

Q~>

c(~) I.o 0.8 i~

= 0

N X ¢tl3~, t:l

(A5)

0.6

where < > is an ensemble average. We now assume a stationary process so that < u ~ > = u0 independent of i, and then (A5) is satisfied identically, by (A1). i n more general applications, as with non-zero mean data, (AS) represents an additional constraint on the weights and must be incorporated into subsequent minimization of the mean square error, by use of a Lagrange multiplier. The mean square error El3 is N Ep = <(QI~ - QI3)'> = X

N Z up~al3J < q ~ q ~ >

0.4 02

-0.4

t~

-

i~lj~-I

-

0.6

--

/R

X // \

b

X II o

N --

2 X a ~ + . i=1

(A6)

Minimizing El3 by setting all partial derivatives with respect to the Ctl3~to zero yields N equations of the form

aE~

N

2~=~"S

0 = OCff~----~=



-- 2,(A7)

F i g . A1. Autocorrelation function for the deep current parameter defined by equation (A1). Dashed line connects computed estimates, solid curve is continuous function A sin o y [ ~ y ultimately used in approximating various correlations as described in text.

and El3 m m = [1 -- ap. 013]

(A13)

or one equation in matrix form = < q Z > [1 -- at3" 0[3]. K a13 = q913,

(A8)

where K is the N × N m a t r i x of autocorrelations , (1)13is the N element column vector of the , and a13 is the column vector of the ctl3~. The desired solution for the weights is a13 = K-1 ~PI~,

(A9)

and (A6) then gives the value of the minimized error as Eli mm= < Qp~ > - al~" q~13= < QI3~> -

N X ctfl~¢pl3¢(A10)

1=1

Different results (A9) and (A10) for the weights and error are obtained for each value of 1~considered; Kremains the same. If instead we wish to use normalized autocorrelations, let C be the N × N matrix of normalized values Cq

.

A

=

C -~

013,

(AI2)

with A = 0.9 and co = 0.095 days -~; these give a best leastsquares fit. At y = 0, of course, we must take c(0) = 1. Thus we can compute the necessary quantities in (A11) to (A13):

00, = V,( < Q p ~ > )



sin ¢o y

(All)

Then instead of (A9) al~

We now must furnish values of the correlations in (A8) to (A10) or of the normalized equivalents in (A11) to (A13) in order to do the computations; in practice we calculate estimates of the normalized sample time autocorrelation function c(T) at discrete lags k a y , k = 1. . . . M, using the data values, as in (4); this constitutes an ergodic assumption. Such a discrete computation will not in general give values of c(y) at all lags needed to evaluate the 0~ and Co o f ( A l 1) to (A12), nor will the discrete Fourier transform of C ( k AT) generally be non-negative, as it must be fora valid autocorrelation function. The easiest way to solve both these difficulties is to choose a continuous function e(y) which is a valid autocorrelation function and which approximates the discrete values c(kAT). Figure A1 shows the discrete estimates obtained from the data together with the continuous function

. , since =

Cq

A sin oa l fi - - tj [ N , i#j ~ ~. qk=~ oa [ t, -- ts [

1



"~" k=l / 1

, i = j

(AI4)

A long series of measurements of Indian Ocean equatorial currents near Addu Atoll

221

6O x

,5O

x 40

w

~O,

wx x tx

w

ltttjt11 ltt Fig. A2.

t

Resultant interpolated deep zonal velocities, with sample mean added back in. Error bar at each interpolated value extends to value -t- E m~, as defined in text. on matters of statistical calculations and to A. E. GILL for helping to clarify the discussion surrounding equations

A sin co I h -- t~ [

0~, (=, 1

q,~)tolt,-t~l

N

Z

~/Vk=

1

(s) to (7).

1

Results of the computations are shown in Fig. A2. Points without error bars are the data from days with radar tracks (mean added back in) and points with error bars are those interpolated by the above scheme. The procedure does not reduce the error greatly from its maximum possible value (the sample standard deviation), except at a few points, since the original data are sparse. There is the suggestion of a low frequency variation in the deep current which is neither annual nor semiannual, but much better measurements, e.g. by moored instruments, will be needed to say more about this. F o r the meridional velocities a similar procedure is possible in principle, but the autocorrelation estimates were low, and the interpolation was made on the assumption of zero correlation at non-zero lags; from (A12) and (AI3), this sets the ctt~~ = 0 and E~ ~t~ = < q ~ > . Table A1 gives the sample means and variances.

Acknowledgements--As in KNox (1974) I continue to be indebted to H. M. STOU~L for interesting discussions about the data and for encouragement in the work. W. DtnNo, D. MooR~, and other colleagues engaged in planning for I N D E X have given me the benefit of several stimulating conversations. I am grateful to R. E. DAVIS for assistance

Table A1.

tt

None of this work would have been possible without the remarkable level of competent assistance I have received from the British Meteorological Office and the Royal Air Force. i cannot possibly name all the men who have played a role in collecting the data; indeed, they have done the job so well that sometimes full 9-month tours of duty have elapsed between crises calling for my presence at Gan, and so I have never met some of them. I must, however, mention R. H. BENTLEYand J. G. LEWTAS,technical officers responsible for the Aanderaa meters and the crucial checks of same after each profile, and Sqn. Ldr. R. PARKIN, 1125 Marine Craft Unit, whose officers and men not only carried on the weekly profiling but helped perform a difficult special bathymetric survey in connection with a plan, unfortunately since made obsolete by the impending closure of R.A.F. Gan, to extend this work. Finally, Mr. D. W. RHEAD at Meteorological Office Headquarters in Brackneli, U.K., has since 1970 attended to dozens of problems of customs, shipping, air travel, urgent cables, forwarding of supplies, etc., and has as much as anyone ensured that the everyday operational needs of the program were met. The continued support of the Office of Naval Research through contract N00014-69-A-0200-6006 to the University of California, Scripps institution of Oceanography, is gratefully acknowledged.

Statistics for deep current components ua, Vd, from the 28 profiles with direct radar track determination of absolute velocities. Overbar indicates sample mean value. ;(i¢~/~) i0.9

Tj~m/~) Ud-'~¢~.~2/~ 2) -4.7

483.6

~(c~2/~z) lO1.5

u--~d¢c,.Z/s~2) -103.5