1 Introduction

The Agulhas Current is the Western Boundary Current of the southwest Indian Ocean. In March 1995, as part of the Agulhas Current Experiment (ACE), a section of fifteen full depth CTD/LADCP stations and shipboard ADCP data was collected across the current at 32S (figure 1). The LADCP measurements revealed an Undercurrent not observed before flowing equatorward beneath the core of the poleward flowing Agulhas Current. These observations prompted a closer investigation into comparisons between LADCP, shipboard ADCP and geostrophic estimates of the velocity field.

2 Estimating the measurement errors in geostrophic velocity

The dynamic method consists of vertically integrating the geostrophic approximation as expressed in pressure co-ordinates and is usually evaluated by first differences:

1

where = 1/ is the specific volume, /x is taken along constant pressure surfaces, = - (35,0,p) is the specific volume anomaly relative to a ìstandard oceanî specific volume and is the difference in dynamic height anomaly between, for instance, stations 1 and 2 separated by distance L. Thus, the error in the geostrophic velocity can be estimated assuming that the three sources of error are uncorrelated (Johns et al, 1989):

2

1 2 3

So the measurement errors in geostrophic velocity arise from uncertainties in the station spacing (2), the dynamic height measurements (1) and the reference velocity (3). The error in dynamic height anomaly, based on the measurement error in temperature and salinity, is at most 0.004dyn m. While a previous study in the Gulf Stream indicates that by far the most important source of error is the station spacing term, which contributes errors as large as 12cms^-1 when ship drift is assumed random (Johns et al, 1989), in assessing these errors for the Agulhas Current, it was found that most station pairs exhibited an on-station drift of similar magnitude and direction so that the drift ultimately makes little contribution to station spacing error.

Overall, the maximum baroclinic geostrophic velocity error across the Agulhas Current (terms 1 and 2 of equation 2) is less than 0.6cms^-1, except at station pair 11-10 where the error is as much as 3cms^-1. Thus the error associated with the approximation or measurement of a reference velocity will generally be the largest component of the measurement error in absolute geostrophic velocity.

Another important source of error is sampling biases which can result if the Agulhas Current shifted laterally during the seven day occupation of the ACE section. The resultant fractional error in the measured isopycnal slopes is equal to the ratio of the lateral velocity of the current path to the shipís velocity along the transect. Estimating the maximum meander lateral velocity to be 6kmday^-1 on the ACE section and taking the extreme case of stations 9 and 8, which have a station spacing of just 10km and were sampled 13 hours apart, then a maximum error of 30% could potentially arise in the geostrophic velocity.

3 LADCP measurement errors

The LADCP velocities have an accumulative error that derives from the stringing together of each shear profile over the depth of the cast (Firing and Gordon, 1990). Consider a bad velocity measurement occurring in a bin at the centre of a profile, with bins above and below it. This bad point produces a peak in the shear profile - in other words the shear is negatively correlated to either side of it. Now consider a bad velocity occurring in the last bin of a profile. In this instance the shear becomes offset and does not recover because there is no good data following the bad to produce a sharp gradient in the opposite sense. This error then, translates to low vertical wavenumbers, producing an offset which will affect the remaining shear data as it is added on. Assuming no instrument bias this error can be estimated by a random walk:

On average the SD of the velocity from each ping of the LADCP was 2.3x10^-3 cms^-1 (dependent on the set-up of the instrument). During ACE the ping rate = 1 s^-1 and the lowering rate of the instrument was approximately 1 ms^-1. The profiling range of the instrument varies with depth mainly as a result of the variability of scatterers in the water column. Taking a conservative average of 96 m (5 x 16 m bins plus a 16 m blank-after-transmit), then the number of pings per profile averages 96 and the standard deviation per profile is thus 0.24 cm s^-1. This error then accumulates to low vertical wavenumber motions, as the shear profiles are strung together over the depth of the cast, so that, for a cast of 2500 m depth (i.e. n = 26 profiles). This estimate of the errors is overly optimistic however, since it ignores bias, increase in SD with range, swimmers and the sampling of high frequencies. In practice LADCP errors at low vertical wavenumbers have been found to be up to an order of magnitude greater than this calculated SD (Firing and Gordon, 1990).

4 Comparing LADCP and shipboard ADCP

The time-averaged on station ADCP profiles are compared with LADCP velocities over the top 300m of the water column (figure 2). The standard deviations of the depth-averaged differences between LADCP and ADCP absolute velocities are summarised in the table below.

station	11	10	9	8	7	6	5	14	4	3	15	2
cross-track	-1.4	5.7	1.3	-2.6	3.5	2.6	0.4	3.4	4.0	3.6	2.3	-2.7
st.deviation	1.5	0.9	1.5	1.0	2.2	2.0	3.1	1.9	1.7	1.9	2.2	1.8
along-track	5.4	2.1	2.2	4.2	-0.4	1.2	-0.9	3.4	-2.4	3.1	-4.7	-0.2
st.deviation	1.8	1.3	2.9	0.9	2.5	2.0	2.2	1.2	1.4	2.1	2.6	1.6

Table 1 The depth-averaged difference between LADCP and ADCP on station, cross-track and along-track velocity profiles in cms^-1, plus the standard deviation of the difference.

The two independent measurements of cross-track velocity exhibit similar shear structure on all stations and with the exception of station 10, the cross-track depth-integrated difference between the two techniques is 4cms^-1 or less. There is no indication that the LADCP consistently under-estimates or over-estimates near-surface ocean currents. These results indicate that LADCP and shipboard ADCP are essentially showing the same pattern of ocean currents over the top 300m of the water column.

5 Comparing geostrophic and LADCP velocities

In order to make direct comparisons between the two methods, the LADCP measurements were linearly interpolated onto station pairs. The two sets of velocity profiles are in good agreement except close to the surface where the shears are sometimes dissimilar (figure 3). The geostrophic velocities have been referenced to the LADCP measurements by estimating the depth-integrated (barotropic) difference between the profiles from each method and adjusting them so that this difference is zero. The top 200m of the profiles were not included in the calculation since here the direct and geostrophic shears are not well correlated. The matched profiles are shown in figure 3 and the standard deviation of the fit and the resultant geostrophic bottom velocities are given in the table below.

Table 2 The standard deviation about the depth-mean difference (below 200m) between profiles of LADCP and geostrophic velocity on each station pair and geostrophic bottom velocities, given in cms^-1 for each station pair. Positive velocity indicates flow in the direction of the Agulhas Current.

The absolute geostrophic velocity as referenced to LADCP reveals north-eastward flow at depth, consistent with Undercurrent observations from LADCP and from moored current meter data - see figures 4 and 5.

The volume transport of the Agulhas Current from absolute geostrophic velocities is 73Sv (including bottom triangle adjustments) as compared to 75Sv from LADCP velocities alone.

6 Comparing geostrophic and shipboard ADCP velocities

On station time-averaged ADCP profiles were interpolated onto station pairs in the same manner as the LADCP profiles. Geostrophic and ADCP velocity profiles at four station pairs are shown in figure 6. The shears of the two sets of velocities are not well matched over the limited depth range of the ADCP. As a result when the geostrophic velocities are referenced to shipboard ADCP using the same method as described for the LADCP,the standard deviation of the fit is as high as 20 cms^-1. Further, the deep velocity field is unrealistic, with bottom velocities as large as 14cms^-1 180km from the coast and at adjacent station pairs 7-6 and 6-5 the bottom velocity switches from 7cms^-1 northward to 16cms^-1 southward. In conclusion we have been unable to use shipboard ADCP to obtain a geostrophic reference velocity which produces a consistent and reliable estimate of the absolute geostrophic velocities.

7 Ageostrophic flow

Figure 7 shows contours of , the difference between the direct LADCP measurements and matched geostrophic velocities. Below 300m v is generally within the measurement errors estimated above and therefore is not significant. The highest magnitudes of v occur in the upper 300m in the core of the Agulhas Current.

There are a number of phenomena that may explain geostrophic departures that cannot be accounted for through error considerations. Since the LADCP measures instantaneous currents, tidal velocities must contribute to the differences between observed and geostrophic velocities. However, from a tidal analysis of the moored current meter data, barotropic tidal velocities across the Agulhas Current were found to peak at just 1.5cms^-1 and therefore will have a negligible effect when compared to the magnitude of the measurement errors estimated above.

The near-surface geostrophic departures could be the result of non linear flows. The LADCP observations allow direct estimates of uu/x, the advective acceleration. In the surface core of the current (between stations 11 and 10) exists the largest gradient in the cross stream flow, about 10cms^-1 over 5km, which results in uu/x values of approximately 2% of fv.

Another way to estimate the non linear departure from geostrophy is to estimate the path curvature of the Agulhas Current. For instance, taking the gradient velocity, v_grad to include the centripetal term due to the curvature of the Agulhas Current then,

. 3

From examination of a chart, the radius of curvature of the continental shelf from Durban (30S) to Port Edward (31S) is estimated to be 400km. Assuming that the Agulhas Current follows the topography and taking v_g as 150cms^-1 in the surface core of the current then v_grad will be 8cms^-1 greater than the geostrophic velocity here. The general pattern of geostrophic departures due to v_grad would be a swath of positive values across the Agulhas Current with a maximum in the current core and values reducing away from the coast and with depth, consistent with the velocity field of the WBC. This pattern of super-geostrophic flow is not consistent with the pattern of geostrophic departures seen in figure 7 which are patchy, sometimes negative and do not appear to increase as the geostrophic velocity increases. This implies that there must be additional factors causing geostrophic departures near the surface.

8 Potential Vorticity

Potential vorticity (PV) can be calculated effectively from LADCP and CTD data since the measurements coincide in time and space. PV can be expressed in a natural co-ordinate system that reflects the geometry of the Agulhas Current. Using Ertel vorticity (Pedlosky, 1986) and neglecting unimportant terms it can be written:

4

where x is the cross stream co-ordinate (positive offshore) and z is depth (positive upwards). The first term on the right hand side is the planetary vorticity multiplied by the stratification, the second term is the shear vorticity (including stratification) along an isopycnal. Since the density surfaces in a WBC are not horizontal, the shear vorticity normal to an isopycnal has contributions from both the vertical and horizontal shear of the current. The vertical density gradient can be expressed in terms of the buoyancy frequency, N and the horizontal density gradient can be expressed in terms of the vertical shear using the thermal wind balance:

5 6

So that the potential vorticity equation becomes:

7

These three components of Q are shown in figure 8 and the total PV is shown in figure 9. At depths between 400m and 1400m and within about 80km of the coast the lateral gradient of total PV is high and the its magnitude becomes large. The transition between relatively uniform PV in the eastern end of the section and the inshore region of large lateral PV gradients is located just to the west of the Agulhas Current core and moves offshore with depth, as does the current core. This transition occurs at a PV value of about 1.310^-10m^-1s^-1 and between the 14C and 4C isotherms, above and below which the PV is more or less uniform along an isotherm across the whole section. Thus at intermediate depths there is a tendency for the PV distribution to behave like a barrier between the high velocity cyclonic core of the Agulhas Current and water farther offshore, impeding lateral mixing of intermediate water masses. Such a barrier may help explain why RSW and AAIW are so distinct along the same isopycnal in the Agulhas Current. The PV barrier in the Gulf Stream is stronger than its counterpart in the Agulhas Current and extends almost from the sea surface to about 7C (Leaman et al, 1989). Below 7C the PV is practically uniform across the width of the Gulf Stream, supporting other work which has suggested that there is a relatively free exchange of fluid across the current in the deep layers (Bower et al, 1985). In the Agulhas Current such a free exchange may be expected in waters of potential temperature less than 4C.

Although the high PV gradients appear to be coincident with the region of greatest velocity shear on the cyclonic side of the Agulhas Current, in fact they are primarily a result of increased stratification and not of increased relative cyclonic vorticity (see figure 8). At the foot of the continental slope the horizontal density gradient increases, doubling at mid-depths, resulting in a decrease in layer depth in the same region that the PV ëbarrierí occurs.

Various investigators have attempted to use PV as a conservative tracer to follow types of ëmodeí waters (e.g., McDowell et al, 1982; Talley & McCartney, 1982; Talley & Raymer, 1982). The PV distribution across the Agulhas Current is shown in figure 10 using potential density as the vertical co-ordinate and with contours of salinity and oxygen overlayed. The area of high negative PV is coincident with the relatively high salinities that indicate RSW next to the continental slope and is just above the oxygen minimum. This may suggest that the mid-depth negative PV maximum is acting as a tracer of RSW, much like the salty, mid-depth Mediterranean water outflow in the North Atlantic can be seen as a relative PV maximum.