SOFAR Floats Reveal Midlatitude Intermediate North Atlantic General Circulation. Part II: An Eulerian Statistical View

Michel Ollitrault Laboratoire de Physique des Océans, IFREMER Centre de Brest, Plouzané, France

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Alain Colin de Verdière Laboratoire de Physique des Océans, Université de Bretagne Occidentale, Brest, France

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Abstract

Part I of this paper has given a descriptive view of the trajectories of 26 SOFAR floats drifting near 700-m depth in the central North Atlantic during the mid-1980s, as part of the TOPOGULF experiment. Here an Eulerian analysis of the 53.4 collected float years is performed by grouping data in 2° latitude × 4° longitude boxes. The mean circulation lacks a significant southward Sverdrupian flow and shows instead zonal bands of alternating westward and eastward currents (except in the Canary Basin). West of the ridge and north of 38°N, the general northeastward flow of the Gulf Stream system is recovered while, south of 38°N the westward recirculation observed from historical float data between 70° and 55°W is shown to extend as far as 40°W. A mean Azores Current is observed both west of the ridge near 33°N and east of 30°W near 34°N. In between, east of the ridge, a tongue of high eddy energy indicates stronger eddy activity and local instabilities of the Azores Current. Eddy kinetic energy and eddy potential energy (the latter inferred from temperature measurements) are equipartitioned on the scale of the eddy field and show a tenfold increase from 33°N, 33°W to 38°N, 50°W.

Lateral diffusivity increases westward (1.5 103 m2 s−1 in the Canary Basin, 3.5 103 m2 s−1 in Newfoundland Basin, 4.1 103 m2 s−1 near Corner Rise Seamounts) and scales approximately as eddy velocity times the first baroclinic Rossby radius of deformation.

Corresponding author address: Michel Ollitrault, Laboratoire de Physique des Oceans, CNRS-IFREMER, Universite, 6 avenue Le Gorgeu, BP 809, Brest Cedex, France. Email: Michel.Ollitrault@univ-brest.fr

Abstract

Part I of this paper has given a descriptive view of the trajectories of 26 SOFAR floats drifting near 700-m depth in the central North Atlantic during the mid-1980s, as part of the TOPOGULF experiment. Here an Eulerian analysis of the 53.4 collected float years is performed by grouping data in 2° latitude × 4° longitude boxes. The mean circulation lacks a significant southward Sverdrupian flow and shows instead zonal bands of alternating westward and eastward currents (except in the Canary Basin). West of the ridge and north of 38°N, the general northeastward flow of the Gulf Stream system is recovered while, south of 38°N the westward recirculation observed from historical float data between 70° and 55°W is shown to extend as far as 40°W. A mean Azores Current is observed both west of the ridge near 33°N and east of 30°W near 34°N. In between, east of the ridge, a tongue of high eddy energy indicates stronger eddy activity and local instabilities of the Azores Current. Eddy kinetic energy and eddy potential energy (the latter inferred from temperature measurements) are equipartitioned on the scale of the eddy field and show a tenfold increase from 33°N, 33°W to 38°N, 50°W.

Lateral diffusivity increases westward (1.5 103 m2 s−1 in the Canary Basin, 3.5 103 m2 s−1 in Newfoundland Basin, 4.1 103 m2 s−1 near Corner Rise Seamounts) and scales approximately as eddy velocity times the first baroclinic Rossby radius of deformation.

Corresponding author address: Michel Ollitrault, Laboratoire de Physique des Oceans, CNRS-IFREMER, Universite, 6 avenue Le Gorgeu, BP 809, Brest Cedex, France. Email: Michel.Ollitrault@univ-brest.fr

1. Introduction

Between July 1983 and June 1989, 53.4 float years were collected near 700 m in the central North Atlantic, with 26 SOFAR floats, as part of the TOPOGULF experiment. The experimental context and setup and a description of the mesoscale motions as revealed by the individual quasi-Lagrangian trajectories has been given in a companion paper (Ollitrault and Colin de Verdière 2002, which is Part I of this paper). Basically, float trajectories paint a very turbulent ocean circulation near the base of the subtropical thermocline, with evidently little communications between both sides of the Mid-Atlantic Ridge (MAR). Float dispersion from their initial clusters (situated on either side of the MAR axis: one near 36°N, 40°W, the other near 33°N, 33°W) favors zonal over meridional directions over a long time (after one or two years).

In this paper (Part II), float data are analyzed statistically in an Eulerian fashion.

Box statistics of Eulerian means and variances of float velocities and temperatures are presented in section 2. Following Taylor's 1921 seminal paper, horizontal diffusivities can also be obtained directly from the statistics of particle trajectories under the assumptions of homogeneous and stationary turbulence. Obviously these assumptions are rarely met and care must be exercised to apply Taylor's concepts in realistic geophysical situations. Horizontal diffusivity estimates are presented in section 3.

2. Mean velocity, eddy kinetic energy (EKE), and eddy potential energy (EPE)

In contrast to the Lagrangian averages (see Part I), which depend upon initial positions and are directly related to tracer distributions, the large geographical areas covered by the dispersing floats allow us to calculate Eulerian averages that exhibit meaningful patterns and can be compared with results obtained from hydrographic measurements. If floats are treated as roving current meters, averaging the observed velocities in fixed boxes gives a mean circulation and its variance in which the time history of the trajectories is forgotten. Such a procedure pioneered by Richardson (1983) has been widely used, the most extensive mapping at 700 db being that of Owens (1991) for the western North Atlantic. Of course meaningful Eulerian means depend entirely on the number of degrees of freedom, that is, the number of independent observations in each selected box. It is usually assumed to be given by N/2τ, with N the number of float days and τ the Lagrangian integral timescale (see the appendix for details). Because the zonal direction is favored by the circulation, box averages were constructed over 1° latitude × 2° longitude bins. The number of float observations (float days) in each bin is shown in Fig. 1. Before discussing statistics, we propose that such a map may have a straightforward and useful information content in terms of tracer concentration or probability distribution. Since all floats of a given cluster are present at initial time in only one box, we assign a unit probability (or concentration) to that launching box and zero elsewhere. To find out the probability of passage of one float in a given box as a function of time, one must simply count the number of float days in that box and divide by the total number of float days since launching time. Apart from this normalization, the map gives therefore direct information of the “transition probability” from the launching box to any other box.

We hypothesize that a tracer launched as the floats would be distributed in space after several (say 3 to 5) years much as these transition probabilities that reveal the more energetic structure of tracer transport. That would be true if we had a great number of realizations and a great number of floats (Batchelor 1949). Each realization with a great number of floats would give us the stirred tracer blob as a function of time. The ensemble average would give us the mean tracer concentration after mixing. We have only one realization and not many floats. We thus implicitly make a bold ergodic-like assumption based on the fact that physically small-scale mixing acts to blur off the stirring action of mesoscale turbulence (Ledwell et al. 1998).

Knowing that the floats remained almost exclusively in each of their respective subbasins, the western and eastern clusters can be interpreted independently. The distribution in the western basin shows a spreading away from the MAR in a general southwest–northeast direction which is also the large-scale orientation of the boundaries of this subbasin (North American coast and MAR south of the Azores Plateau). The diffusion is more pronounced in the zonal than the meridional direction, a feature that is shared but marginally by the eastern cluster. Note that for this eastern cluster the maximum concentration does not occur in the initial box but significantly to the east, a distribution that cannot be obtained from pure diffusion and calls for net eastward advection by a coherent field, in this case the Azores Current. (This is apparent also on dispersion diagrams of the third launching. See Fig. 11 in Part I.)

A crude measure of the integral timescales (for the velocity field) being 5 days (10 days), respectively, for the western (eastern) basin (see section 3), we can estimate the number of degrees of freedom for each of the 1° × 2° adjacent bins in Fig. 1. Only 22 bins have at least 6 months of data giving a minimum of 18 (9) degrees of freedom. To improve the statistical significance of the mean currents (but at the expense of a lesser spatial resolution), averages were assembled for 2° × 4° bins by grouping four of the original 1° × 2° bins. To further improve the large structures of the mean flow, an overlapping average was conducted by shifting all 2° × 4° bins halfway in both latitude and longitude. With this choice, 63 bins have at least one year of data and 5 bins have a 95% confidence interval less than ±1 cm s−1 for the mean (Fig. 2 and Table 1). Davis (1991) has shown, however, that if the float density distribution is not uniform, the Eulerian means so computed are biased in the presence of diffusivity. This Lagrangian sampling bias written as −Kij(∂/∂xj)(ln C) (where Kij is diffusivity and C the float concentration) shows immediately that the launch from tight clusters is far from optimal since it naturally produces strong gradients of float concentration. However, these gradients are smoothed in time due to the dispersion process. Also, Kij is not easy to estimate accurately (more in section 3). A factor of 2 in the diffusivity induces a factor of 2 in the bias, which may change the mean circulation significantly if bias is of the same order of magnitude as the mean. We estimate crudely the lateral diffusivity tensor as K11 = 4000 (2000) m2 s−1 and K22 = 2000 (2000) m2 s−1 west (east) of the MAR. Table 1 shows that the bias is generally smaller than the mean and its standard deviation but not by much. Fortunately, the large-scale structure of the mean field does not change significantly from the uncorrected one, although locally there can be noticeable changes. Consequently we present only the uncorrected map, keeping in mind that the spatial distribution of float density must be used when interpreting Eulerian maps and that areas near initial launch sites or the edges of the mapping must be scrutinized. Because the velocity variances have lower statistical requirements (see the appendix), bins with a minimum of 45 floats days are presented for the eddy kinetic energy per unit mass (EKE) map (Fig. 3). There is no need to show principal components of eddy velocity because the field was generally found to be isotropic at the 5% significance level (Table 1). Given that the cloud of floats (Fig. 1) extends more in the zonal direction than in the meridional, this may seem odd. Although the underlying turbulent velocity field is nearly isotropic, its diffusive properties are not and the zonal dispersion is favored for long timescales (see section 3).

The floats measuring temperature (with an accuracy of ±0.05°C), the mean and eddy temperature field were estimated in the same way as velocities. Temperature variance is shown in Fig. 4. From this we may estimate the eddy potential energy per unit mass EPE = N2h2/2 (with N the mean Brunt–Väisälä frequency and h′ the eddy vertical displacement of a given potential density surface) as follows. To obtain density information we need a T/S relation. Harvey and Arhan (1988) show that there is a good central water T/S relation for temperatures above 11°C becoming less tight deeper. Quite clearly the operating depth of our floats is just marginal for this purpose. Using dS/dT = 0.2 psu °C−1, thermal (haline) expansion coefficients respectively 1.8 10−4 °C−1 (7.5 10−4 psu−1) yields the local conversion:
i1520-0485-32-7-2034-eq1
The eddy potential energy becomes in terms of temperature:
i1520-0485-32-7-2034-eq2
and for a value of Tz = ∼1–2 × 10−2 °C m−1, we obtain the following conversion between EPE in cm2 s−2 and T2 in °C2:
i1520-0485-32-7-2034-eq3
Eulerian statistics in the (600–800)-db interval are summarized in Table 1 (for 2° × 4° bins with a minimum of one float year).

a. Results

West of the MAR, and starting from the north, a northeastward mean flow is found between 39° and 42°N and between 40° and 45°W. It paints the large-scale picture of the active branches and fronts characteristics of the Gulf Stream extension region. Eddy kinetic energy is high along the route of the Gulf Stream, indicative of the mesoscale turbulence associated with its eastward mean drift. East of 40°W and north of 39°N, the eastward mean flow encompassing 4° of latitude seems to split north (see also float 13 trajectory in Fig. 2 of Part I) and south, near 40°N when reaching the Azores plateau, although without statistical confidence. Less expected is the broad and intense (∼5 cm s−1) westward flow that extends from about 35° to 38°N and 40° to 55°W. West of 50°W this return flow is well known and part of the inertial recirculation cell south of the Gulf Stream as can be found, for instance, in the summary circulation diagrams of Hogg (1992), Schmitz and McCartney (1993), and the mean float velocities of Owens (1991) at 700 m. The extended coverage of the present data set shows that this westward return flow extends much farther to the east. The eddy kinetic energy is still high (100–200 cm2 s−2). Positive 〈uυ′〉 appear south of 38°N in this whole region indicative of eddy interactions with the mean flow. This westward flow decreases in the vicinity of the MAR and appears to be fed by flows originating from the north and south along the western flanks of the ridge. Finally a rather restricted Azores current is found between 45° and 40°W along 33°–34°N with no apparent continuity farther east (between 38° and 34°W).

East of the MAR and west of 32°W, between 32° and 35°N, lies a region of rather weak mean flow whose directions are generally not well defined. However, although the Azores current is not seen west of 33°W, a tongue of eddy kinetic energy that extends up to 30°W along 33°–34°N, indicates that this area is probably a favored site for the instability of the Azores front and that not enough floats were present to reconstruct the mean position of a highly unstable jet. Actually, the individual trajectories show without doubt that a strong Azores Current exists in this region (see Part I) but it may be possible that even with more floats seeded there the mean eastward flow would emerge only slowly through our Eulerian box averaging. The area of the Seamounts (29°W) is exceptional because anisotropy occurs: just east of the Seamounts meridional eddy velocities are significantly larger than zonal ones (floats appear to be trying to find their way through the gaps between the Seamounts). A further distinct hydrodynamical regime appears east of the region of Seamounts between 28° and 22°W. The eddy kinetic energy decreases and the Azores current reappears along the same 33°–34°N latitude band. It turns southwest then and fans out southward between 20° and 26°W, with even a very slow but well defined westward flow at 30°–31°N just east of the Seamounts, which may be due to entrainment by underlying Mediterranean Water flowing west there (Spall et al. 1993). North of 35°N, the mean flows are weak and ill defined. Nevertheless, a southwestward flow may be noticed following the topographic contours on the eastern flank of the MAR south of the Azores in an area of weak eddy activity. It forms the westward branch of a cyclonic gyre centered at 36°N, 30°W.

Finally, and in the continuity of the very slow westward flow east of the Great Meteor Seamount, the mean velocity field shows a quasi-continuous westward flow near 30°N between 30° and 45°W, thus crossing the MAR into the western basin. But that needs confirmation with more data.

Mean currents as well as float trajectories do not show penetration south of 30°N (except east of 25°W), which appears to be a natural barrier at that depth. The eddy kinetic energy decreases sharply when this region is approached.

The general features of the temperature variance and hence eddy potential energy (Fig. 4) generally confirms that given by the eddy kinetic energy: high values in the Gulf Stream extension region, a tongue that penetrates into the eastern basin over the MAR, and a drop south of 32°N. Multiplying temperature variance by 100 (as discussed previously) gives EPE values in cm2 s−2 that are close to the EKE values. Dantzler's (1977) distribution of eddy potential energy constructed from the depth distribution of the 15°C isotherm obtained from an historical XBT database (at shallower levels than our float data) already included this isolated maximum due east of the MAR at the latitude of the Azores current (his Fig. 3). Dantzler's large-scale spatial variations compare well with our float estimated EPE values. Our values which are smaller than Danzler's by a factor of 4 around the MAR reflect the near-surface intensification of the eddies since the mean position of the 15°C isotherm that he uses is closer to 300 m in that area. There is also an underestimation of the temperature variances by the SOFAR floats because they tend to rise (sink) as temperature decreases (increases) (Ollitrault 1994). Since the float data provide estimates of EKE and EPE, their ratio gives information on the lateral scale of the geostrophic turbulence (Gill et al. 1974). This ratio which does not depart much from O(1) west and east of the MAR confirms the “visual” impression of similar spatial scales based on the trajectories. This comparison underscores the value of float temperature measurements in addition to the traditional kinematic information provided by the floats when a T/S relation exists.

b. comparison with other observations

The only long term measurement in the periphery of the area under investigation was made by Müller and Siedler (1992). Almost 9 yr of data were gathered at 33°N, 22°W, west of Madeira at Mooring site Kiel 276. The mean velocity vector at 670-m depth [0.3 ± 1.1(2σ) cm s−1, −0.9 ± 0.8(2σ) cm s−1] in the zonal and meridional directions, respectively, compares well with the float values at that location 0.9 ± 1.6(2σ) cm s−1, −0.8 ± 2.0(2σ) cm s−1 obtained from 516 days of data in the bin surrounding the mooring site (see Table 1). Similarly, the current meter and float measurements give eddy kinetic energy values of respectively 17 ± 9(2σ) and 22 ± 9(2σ) cm2 s−2. Along with spatial averaging, the floats appear to be advantageous from a statistical point of view. The above Eulerian time series have long integral timescales respectively 75 and 21 days in the zonal and meridional directions that are well in excess of the Lagrangian ones, typically of order 10 days in this basin. Henceforth fluid parcels decorrelate more quickly in time, allowing to obtain more rapid statistical convergence. In effect, Lagrangian arrays of circulating floats paint up the Eulerian mean with greater efficiency than current meter arrays because of the spatial averaging characteristics of the floats quickly circulating around eddies that move far more slowly across fixed arrays.

Most of the past hydrographic cruises carried out south of the Azores had as objectives a better description of the Azores front and associated current systems. As a thermocline current between the surface and 800 m, the Azores current runs along 34°N with a transport of about 10 Sverdrups (1 Sv = 106 m3 s−1) to widen and turns southward west of Madeira (Gould 1985; Stramma 1984; Klein and Siedler 1989; Gana and Provost 1993; Alves and Colin de Verdière 1999). This transport estimate was confirmed by hydrographic measurements taken by R/V Le Suroît one month after deployment of the western cluster (The TOPOGULF Group 1986). Geostrophic velocities referred to 2500 m allowed to satisfy mass conservation around a quasi-synoptic (1–17 August 1983) box of 59 stations (see Fig. 1 in Part I). They compared well with the float velocities at 700 m and showed that float motions captured the signature of the base of the Azores Current. Consistent with the convoluted float tracks, some short-scale, surface-intensified structures—more representative of eddies than semipermanent fronts—are found in the geostrophic velocity profiles.

Given that the flow structure revealed by the floats is a large-scale average over several years, it is also tempting to compare the present results with circulation inferred from tracers and hydrography. The Fukumori et al. (1991) atlas provides distribution of temperature–salinity–nutrients from high quality hydrographic sections of the early 1980s (including those aforementioned). Nutrients show nearly constant values between 30° and 40°N at 700 db that decrease sharply to the south at 25°N. Apparently this tracer front is the only common feature with the mean float distribution of Fig. 1, the associated EKE–EPE southern front in Figs. 3 and 4 and Dantzler's (1977) EPE map. Although we do not know the dynamical origin of this front, the float data are at least broadly consistent with the tracer distribution by indicating reduced stirring across the zonal front. We have already mentioned the rare events of floats crossing the MAR and we wondered whether this could be seen as well in the characteristics of water masses on each side of the ridge. Indeed, large-scale potential vorticity contours (on σθ = 27.0 and 27.3, which correspond roughly to the depth range of the floats) oriented meridionally and homogeneous values found on either sides (McDowell et al. 1982), are consistent with the weak particle exchange found across the MAR. Furthermore Harvey and Arhan (1988) who analyzed the long hydrographic sections carried out in 1983 as part of the TOPOGULF experiment note that the T/S relation of central waters east of the MAR is shifted to more saline values than west of the MAR by about 0.1 psu.

Recently, a 0.1° eddy-resolving numerical model (Smith et al. 2000) has produced EKE distributions near 700 m, which agree quite well with our float map: maxima of 200–500 cm2 s−2 along the path of the Gulf Stream east of 60°W, a tongue of relatively high values of 20–50 cm2 s−2 penetrating east of the MAR around 33°N and between 40° and 30°W, and low values of less than 20 cm2 s−2 south of 30°N. Two 3-yr EKE distributions corresponding to years 1989–91 and 1995–97 were estimated showing no difference statistically.

The model mean currents near 700 m (averaged over the same 3-yr periods and 2° lat × 4° lon) can also be compared to our mean float velocities. Although there is some variability between the two 3-yr averages, the model shows an Azores current around 34°N between 25° and 35°W of order 2 cm s−1, fed apparently from west of the MAR, but with no clear continuity above the ridge for the period 1989–91. Around 37°–38°N, and between 40° and 45°W, there is a westward flow of a few cm s−1, in rough agreement with the float mean currents, the latter being stronger. West of 50°W, and around 35°–36°N, both the model and floats show a westward flow (still somewhat stronger for the floats) which turns southeastward west of 58°W. But between 45° and 50°W, in contrast to the floats, the model shows no continuity between these two westward flows. There is however a well-defined westbound flow south of the Azores Plateau near 36°–37°N and between 25° and 35°W, which seems to continue west above the MAR, not unlike the float currents in the region. Finally the model also shows a westward flow near 31°N in the vicinity of the Great Meteor et al. Seamounts. Away from the Gulf Stream, the mean circulation is mainly zonal with an alternating current structure but generally narrower than the one revealed by the floats (1°–2° lat wide instead of 2°–4°). The variability in the model mean current field suggests the possibility of longer timescale motions.

3. One-particle diffusion

In order to estimate horizontal diffusivities and characteristic Lagrangian timescales within each basin, the trajectories were separated in two groups (33.2 float years for the western basin, and 19.3 float years for the eastern basin). To preserve as far as possible homogeneity, the two northernmost trajectories (the green and blue ones in Fig. 2 in Part I) were excluded from the statistics.

Statistical distribution of “instantaneous” velocities (sampled every 12 h) is quasi-normal and quasi-isotropic : 〈u′21〉 ≈ 〈u′22〉 ≈ 115 ± 10(2σ) cm2 s−2 for the west [respectively, ≈25 ± 4(2σ) cm2 s−2 for the east], 〈u1u2〉 ≈ 6 ± 5(2σ) cm2 s−2 for the west [respectively, −1 ± 2.5(2σ) cm2 s−2 for the east]. Skewness coefficients (〈u′3i〉/(〈u′2i3/2) are between −0.1 and 0.3, kurtosis coefficients (〈u′4i〉/(〈u′2i2) between 3.3 and 5.3. As usual, brackets stand for the stochastic mean, and ui = 〈ui〉 + ui (i = 1, 2) are the east and north velocity components.

Lagrangian (i.e., following one particle) covariance functions 〈u1u1(τ)〉, 〈u2u2(τ)〉, and 〈u1u2(τ)〉 were estimated with standard error of a few cm2 s−2 and for 0 ≤ |τ| ≤ 240 days. The error was estimated given the Gaussian behavior of the velocity components (see the appendix for details).

Although statistically not much different from zero after 30 days, meridional covariances show a negative lobe near 50 days (Fig. 5). This behavior is no surprise since many meanders and eddies are observed on generally zonal mean currents.

Integral timescales Iij(t) and one-particle diffusivities Kij(t) (i, j = 1, 2) result from integrating the covariance functions:
i1520-0485-32-7-2034-eq4

Under assumptions of spatial homogeneity and temporal stationarity (Taylor 1921; Batchelor 1949) Iij(t) and Kij(t) should tend toward constants (Iij and Kij) for tIij.

It is also possible to directly estimate diffusivities with Kij(t) = (1/2)(d/dt)〈Xi(t)Xj(t)〉, where Xi, i = 1, 2 is the particle displacement components and 〈Xi(t)Xj(t)〉 the dispersion tensor.

To increase the stability of the Lagrangian dispersion statistics, trajectories were reinitialized every 30 days, a time interval sufficiently large compared to the Lagrangian time scale to consider each 30-day segment of the trajectories as independent samples. (This artificial randomization is not without effects on the small time behavior of the diffusivities as discussed later in this section.) A second requirement is to correct for mean flow effects. We simply compute at any given time the mean position (center of gravity) of the floats that are used in the average. Mean position is then subtracted from instantaneous float positions to construct dispersion as a function of time (Fig. 6).

Both diffusivity estimations give roughly similar results (Fig. 7), giving some credence to the stationarity and homogeneity assumptions used by Taylor to relate the two. Turbulent dispersion is quasi-isotropic (K11K22, K12/(K11K22)1/2 ≪ 1) over the first 20–30 days and zonal and meridional diffusivities grow rapidly to reach a “plateau” by the end of the period. Thus, at t ∼ 30 days western diffusivities are twice as large than eastern diffusivities (K11K22 ≈ 5 103 m2 s−1 west of the MAR, vs 2.2 103 m2 s−1 east of the MAR).

Western integral timescales (at t ∼ 30 days) are half as long than eastern ones (I11I22 ≈ 5–6 days west of the MAR, versus 10–12 days east of the MAR). This means simply that turbulent (Lagrangian) velocities remain correlated twice as long in the eastern basin : since rms velocities (υ2rms = 2 EKE) are twice as large in the west than in the east (15 vs 7 cm s−1) this suggests a diffusivity scaling as υrms ℓ with ℓ a mixing length that would be similar on both sides of the ridge and the order of an eddy scale (33 km). This idea is consistent with the EKE/EPE rather constant ratio mentioned earlier and also with the observation that mesoscale eddies (in the thermocline at least) scale with the Rossby radius of deformation (Stammer 1997) which is constant on either sides of the MAR at these latitudes (Emery et al. 1984). As we show next however what is observed at 30 days may not be a good way to parameterize eddy mixing on longer timescales.

In the eastern basin, between 30 and 180 days, zonal diffusivity is approximately constant (K11 ≈ 2.3 103 m2 s−1 with corresponding integral time scale I11 ≈ 12 days) but the meridional diffusivity decreases to 1.2 103 m2 s−1 (integral timescale I22 ≈ 7 days) at 90 days, then keeping rather constant. Beyond 180 days, although we have enough samples to adequately estimate the diffusivities, the spatial homogeneity assumption breaks down (particle displacements may reach over 500 km). Furthermore, due to the limited size of the basin, both diffusivities slowly decrease (they should eventually reach zero at t infinite).

In the western basin, things are less clear, beyond 30 days. The greater variations of diffusivities (compared to their eastern counterparts) may mean they are less well (or rather ill-) defined. Clearly the homogeneity assumption is not satisfied, since float trajectories cover a very large region with strong mean currents (see section 2). Nevertheless, as in the eastern basin, meridional diffusivity decreases while zonal diffusivity keeps rather constant (direct estimate shows even a slight increase with time). This reflects the observational fact that floats disperse zonally over long time periods (see also Part I, section 3). Thus there is a random walk regime for the zonal dispersion from roughly 30 to 120 days, both in the eastern and the western basins, less well-defined however in the latter. Besides variations of zonal and meridional dispersions (hence diffusivities), Fig. 6 also displays the dispersion ellipse area (i.e., πab with a2 and b2 the principal dispersions), its major axis direction and the correlation coefficient {i.e., 〈X1(t)X2(t)〉/[〈X′21(t)〉〈X′22(t)〉]1/2}. When the correlation coefficient is small (say between −0.1 and +0.1), zonal and meridional dispersions are almost uncorrelated. When the correlation coefficient is not small however, analysis should be done in a rotated frame with uncorrelated longitudinal and transverse dispersions (hence diffusivities). This may not be rewarding however if the major axis direction varies rapidly (because we must choose a fixed frame over a given time range). But, in any case, the ellipse area gives us the 2D dispersion, thus an equivalent isotropic diffusivity defined as KO = (1/2π)(d/dt)(πab). Now the random walk regime stands out more clearly (between 20 and 120 days in the west; between 30 and 200 days in the east), with KOwest ≈ 5.4 103 m2 s−1 and KOeast ≈ 1.9 103 m2 s−1.

Let us refine the analysis by considering smaller hence more homogeneous subregions (less than 1000 km wide). Boxes CB, MAR, CRS, and NFB are centered respectively at 32°N, 22°W; 34°N, 38°W; 35°N, 50°W; and 40.5°N, 38°W and named after their geographical locations (Canary Basin, Mid-Atlantic Ridge, Corner Rise Seamounts, and Newfoundland Basin). Table 2 and Fig. 9 (later) give the exact limits of these boxes.

Dispersion and diffusivities were estimated using the direct method only. For each box, zonal and meridional dispersions are displayed on the left side of Fig. 8, longitudinal and transverse dispersions in the chosen rotated frame on the right side. Two criteria were used for defining the new frame and estimating the diffusivities. First, as the random walk regime is seen (on the ellipse area curves) to occur between roughly 30 and 90 days, we rotate the axes so that the correlation coefficient be zero near 60 days. Although the correlation coefficient then lies between −0.1 and +0.1 over the 30–90-day interval, the principal direction may still vary over several tens of degrees, which is not totally satisfying. Principal ellipses are plotted in Fig. 9 at 30, 60, and 90 days. Second, the outer ellipses plotted in Fig. 9, which are 2σ 90 days ellipses [they contain 86% of the points, whereas the principal (1σ) ellipses contain only 39%] must be contained within the corresponding boxes for homogeneity.

Before discussing the results and to understand properly the meaning of the temporal variations of diffusivities so presented, one must remember that trajectories were reinitialized every 30 days in order to obtain a sufficiently large number of degrees of freedom (reinitializing every 90 days gives exactly the same results but with a lesser number of degrees of freedom). In reality our floats were launched together and examination of trajectories shows that pairs of floats may remain correlated for a long period of time (up to 6 months, for a few pairs; see Part I, their section 3) and thus the physical dispersion (of a cluster of particles, or of a dyed cloud) occurs on a timescale much longer than the above values obtained after reinitialization. It is only after a year or so that we should recover the above diffusivities (obtained herein after order of one month because of the artificial randomizing of parts of trajectories). By that time smaller-scale dispersion processes will have eventually separated particles, which would then experience larger-scale straining motions. This is very well demonstrated by the Ledwell et al. (1998) dye experiment in the North Atlantic.

In the CB and NFB boxes, dispersion tends to grow in a northwest–southeast direction. An F test at the 5% significance level, however, reveals that CB dispersion is anisotropic but NFB is quasi-isotropic after 90 days. Equivalent isotropic diffusivities near 60 days (see Table 2) are 1.5 103 m2 s−1 (CB box) and 3.5 103 m2 s−1 (NFB box). Ledwell et al. (1998) and Sundermeyer and Price (1998) found (K11, K22) = (1.5, 0.7) 103 m2 s−1 at 300-m depth near 25°N, 30°W, thus southeast of the CB box and at shallower depths. However their EKE value of 16 cm2 s−2 is similar to the CB 19 cm2 s−2, which lends some credence to the comparison. Dispersion in the CRS box is clearly zonal (K11 ≈ 6 103 m2 s−1) while meridional diffusivity decreases after 30 days eventually reaching zero after 90 days, which is what was already observed globally in the western basin (but the decrease was less pronounced because of the NFB share). Note that the strong zonal enhancement in the CRS box is in very good agreement with the western float cluster dispersions (presented in Part I, section 3) 12–18 months after launch. Such a predominantly zonal diffusivity is also evident in the values given by Price (1983, unpublished manuscript, hereafter PR83) near 31°N, 69°W (K11 ≈ 8 103 m2 s−1, K22 ≈ 5 103 m2 s−1) and Riser and Rossby (1983) near 25°N, 65°W (K11 ≈ 4.5 103 m2 s−1, K22 ≈ 1.8 103 m2 s−1).

Why do NFB diffusivities not show the same phenomenon? LaCasce and Speer (1999) and LaCasce (2000) argue and present data showing that floats tend to follow f/H contours at least statistically. Although their analysis is slightly different from our classical two component analysis (they project the float motion on a locally f/H aligned frame), it gives us a clue for the NFB box. We see that depth contours (f/H contours are similar) encircle the region, thus suggesting a possible topographic constraint. It is not clear what happens for the CB box although f/H contours [see map in LaCasce (2000)] also encircle the box to the north. But since the flow is baroclinic it is not obvious if a barotropic argument still applies (the bottom is more than 4000 m deep). Note that Spall et al. (1993) have observed wavelike oscillations with northwest–southeast orientation, there, with floats launched at 1100 db. More simply, a mean northwest–southeast shear (not considered in our calculation) may explain part of the CB box anisotropy. West of the MAR and south of the Gulf Stream, f/H contours tend to be latitude lines, thus leaving only the β effect at work. Finally, in the MAR box centered at the latitude of the Azores Current (34°N), dispersion is quasi-isotropic up to 60 days then diffusivity parallel to the ridge axis stands at 3.5 103 m2 s−1 apparently up to 200 days, while transverse diffusivity decreases toward zero. These are only tendencies since after 90 days the area covered by the dispersing floats may well be outside our MAR box (which is actually an ellipse oriented along the MAR with 660 km and 330 km for greater and smaller axes). Thus the MAR seemingly affects diffusivity for larger scales and longer times.

4. A diffusivity discussion

Table 2 summarizes our results. We concentrate on the more statistically stable diffusivity values, that is the equivalent isotropic diffusivities, but quote also zonal and meridional or longitudinal and transverse diffusivities. Equivalent isotropic diffusivities are estimated as (1/2π)(ΔAt), ΔA being the dispersion ellipse area variation over the time interval Δt of the well-defined random walk regime, generally found between 30 and 90 days (but beginning sooner in the western basin). Other diffusivities are estimated similarly around 60 days. No confidence intervals for the diffusivities are given but 95% confidence intervals for dispersions given in Figs. 6 and 8 may help. Comparisons are made with 700-m zonal and meridional diffusivity estimates given by Riser and Rossby (1983) in the POLYMODE east area and PR83 for the LDE experiment.

While Rossby et al. (1983) suggested that float diffusivity scales with velocity variance, and integral timescales of order 10 days, Böning (1988) proposed instead a scaling with eddy zonal velocity and zonal length scales of 70 km (30 km) at the base of (below) the thermocline. In the above table, the length scale K11/u1 is not constant but drops by a factor of 2 between the POLYMODE and TOPOGULF region. The first baroclinic Rossby radius of deformation as computed by Emery et al. (1984) also shows a similar decrease from 25° to 35°N (due in large part to the northward decrease of the mean stability). Following the accumulating evidence from satellite altimetric observations that the mesoscale eddy length scales vary in the thermocline as the first baroclinic Rossby radius (see Stammer 1997 and references therein) the present comparison suggests that the “velocity × length” diffusivity scaling proposed by Böning (1988) may be modified as:
KOνrmsR
where ℓR is the first baroclinic Rossby radius and the rms speed νrms = (2 EKE)1/2. Table 2 (last column) shows encouraging support for this scaling. The fast increasing number of float experiments may help to test this particular scaling proposed for lower thermocline motions. It will also be of interest to find out if the result of the small asymptotic value for the meridional diffusivity found west of MAR and the polarization of the diffusivity tensor, seemingly related to patterns of f/H, in spite of the near isotropy of the underlying eddy velocity field, can be confirmed by further data or other tracers. What are particularly needed are interior values away from strong mean currents whose effects are always difficult to subtract in diffusivity calculations.

Diffusivity estimates are of particular value for understanding mean property distributions. Armi (1979) and Armi and Haidvogel (1982) have argued that the zonal salt tongue of Mediterranean origin, which extends westward across the MAR, may be the result of favored zonal over meridional diffusivity with little need for a westward zonal mean flow as postulated by Needler and Heath (1975). The Spall et al. (1993) float experiment at the Mediterranean Water level to the southeast (30°N, 25°W) of the TOPOGULF eastern cluster revealed a dominance of zonal, low-frequency eddy motions in the core of the salt tongue, and a general weakness of the mean flow. They also argued that these low-frequency motions play a major role in the spread of the Mediterranean Water (MW). If our 700-m observations carried out farther west are representative of the situation at 1000 m, they would support the idea that the MW tongue extent may be the result of anisotropic diffusion. The westward extension of the salt tongue into the western basin near 30°N shows the MAR (1000 m deeper at 30°N than at 36°N) does not appear to be a barrier at that latitude and 1000-m depth, in contrast to what our floats show at 700 m, north of 32°N. What is the share however between diffusion and mean flow in this salt tongue extension still remains an open question.

5. Concluding remarks

Because most of what is known or suspected about the oceanic circulation is cast in an Eulerian framework, the float velocities have been separated in mean and eddy parts in 2° lat × 4° lon boxes. The statistics so obtained are not truly Eulerian since there is an added mean flow component in this representation when the float distribution is inhomogeneous. This bias (computed following Davis' 1991 prescriptions) is not so large as to preclude all comparisons with other determinations of the mean Eulerian circulation. The data show no southward Sverdrup flow at depth 700 m except in the Canary Basin, but instead alternating bands of predominantly zonal mean flows. West of the MAR, northeastward (southwestward) flow is found north (south) of 38°N. A weak mean Azores current (33°N) seen west of the MAR, disappears on crossing the MAR to reappear in the seamounts region around 30°W and finally turns to the southeast, east of 25°W. Individual trajectories (see Part I) do show the Azores Current between 40° and 30°W but with a great temporal variability (unstable meanders and recirculations) precluding its determination by the larger-scale averaging used. In this area, a comparison with one of the longest Eulerian time series available obtained by the IFM Kiel (Müller and Siedler 1992) demonstrates the fast statistical convergence of the float averages and the unique advantages of the floats for determining the absolute circulation. As a word of caution, note that the lack of repeat releases widely separated in time might bias our mean flow picture (Rossby et al. 1983), and the alternating zonal flows revealed also by the Smith et al. (2000) numerical model might be the signature of a long period variability as demonstrated for example in the wind driven forced model of Sturges et al. (1998), although at shallower depths. Eddy kinetic energy and eddy potential energy (obtained from temperature measurements) are similar, supporting the notion that the eddies scale with the internal Rossby radius (Stammer 1997). The eddy kinetic energy map agrees and complements that of Owens (1991) farther to the west. Higher values of eddy energy penetrate as a tongue across the MAR in the region where the (binned mean) Azores current disappears suggesting the area to be a site of favored instability of that current.

Lateral diffusivity, defined as the equivalent isotropic diffusivity; that is, (1/2π)(dA/dt) with A the dispersion ellipse area) increases westward from ∼1500 m2 s−1 in the Canary Basin to 4100 m2 s−1 at 35°N, 50°W. However the diffusivity tensor is anisotropic with vanishingly small meridional diffusivity in the subtropical western basin, and small transverse diffusivity above the MAR as time increases. Anisotropy of the diffusive properties (which occurs with near isotropy of the eddy velocity field) has also been noted at the same depth by Riser and Rossby (1983) in the southwest corner of the North Atlantic gyre, in the region south of the Gulf Stream recirculation pointing out the possible influence, in this case, of β on the characteristics of mesoscale turbulence shown by Rhines (1975) to promote zonal eddy motions. LaCasce and Speer (1999) show that in numerical experiments with anisotropic eddies only the zonal diffusivity persists with the beta effect acting to cancel the meridional dispersion, but in contrast to their eddy fields, ours are isotropic at the scale of our 2° × 4° boxes. However the dispersion at large time is dominated by the lowest Lagrangian frequencies, and these are likely to be associated with motions at scales larger than our Eulerian boxes. These large-scale motions (although weaker than those in the mesoscale band) are anisotropic as the mean flow maps (Fig. 2) and displacement maps (see Fig. 11 in Part I) show. Including the POLYMODE east and Local Dynamics Experiment sites, we suggest that the lateral (isotropic equivalent) diffusivity scales as the eddy rms speed times the first baroclinic Rossby radius of deformation but we hasten to add that this has been observed at 700 m only, presumably as long as the size of these thermocline intensified eddies scales with the internal Rossby radius. If baroclinic instability is indeed the favored process for eddy generation, the Rossby radius appears as the dominant natural length scale of the eddy variability and the scaling should not be too surprising. The data analyzed so far suggest that it can be used as a mixing length, an idea proposed earlier by Stone (1972) in the context of eddy heat flux parameterization in planetary atmospheres.

Acknowledgments

We thank all the persons that have helped to make this float project successful and we gratefully acknowledge the continued support of IFREMER throughout the years. We had fruitful discussions with Phil Richardson during the final writing of the paper. One anonymous reviewer provided many suggestions which greatly improved the presentation.

REFERENCES

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APPENDIX

Statistical Estimates

Covariance functions

We used biased estimates of Lagrangian velocity auto or cross covariance functions. For example Ĉ(k) = 1/n Σnki=1 [u(i) − uall][υ(i + k) − υall] is the estimate of the cross-covariance function 〈uυ′(k)〉 for 0 ≤ kn − 1 and a continuous portion of trajectory of n points. With Δ denoting the time sampling interval, 〈uυ′(k)〉 is a short-hand notation for 〈uυ′(kΔ)〉. u and υ denote orthogonal components of horizontal velocity vector and uall and υall are sample means over all the trajectories found inside the study area. We assume that uall and υall give a good approximation to 〈u〉 and 〈υ〉 and consequently ignore the effect of estimating these true mean values by the sample means.

The “overall” covariance functions for all the continuous portions of trajectories, found inside the given study area are then obtained from the weighted averages of the corresponding individual estimated covariance functions.

For example, with q portions of trajectory:
i1520-0485-32-7-2034-eqa1
where n1 is the number of points and Ĉuυ,1 the estimated covariance function of the first continuous portion of trajectory, and so on.

Variances of sample means

For a continuous portion of trajectory with n points,
i1520-0485-32-7-2034-eqa2

Gaps within TOPOGULF trajectories being always greater than 5 days, we assume any velocity component of a given continuous portion is uncorrelated with any velocity component of another continuous portion. The same is also assumed for two different trajectories belonging to two different floats.1

Thus,
i1520-0485-32-7-2034-eqa3

Asymptotically, then, var(uall) ≈ 1/N Σ+∞r=−∞uu′(r)〉 if each nΔ is much greater than the Lagrangian integral time scale I11 defined as I11 = 1/〈uu′(0)〉 0uu′(τ)〉 so that Σn−1r=−(n−1) [1 − (|r|/n)]〈uu′(r)〉 is practically equal to Σ+∞r=−∞uu′(r)〉 = (2/Δ)I11 · 〈uu′(0)〉. This means that the &ldquo=uivalent number of independent u velocity components” also called the number of degrees of freedom for the u velocity component, is given approximately by ndf = NΔ/2I11 (see, e.g., Priestley 1981).

TOPOGULF velocity components being quasi-Gaussian distributed, a 95% confidence interval for 〈u〉 is given by [uall − (1.96 S), uall + (1.96 S)], with S2 = 〈uu′(0)〉2I11/NΔ ≈ uu2I11/NΔ (uu is sample variance over all trajectories found inside the study area).

Variance of υall and confidence interval are obtained similarly.

Variances of covariance functions

Assuming a Gaussian process, which implies in particular that
i1520-0485-32-7-2034-eqa4
it can be shown after a few algebraic manipulations that
i1520-0485-32-7-2034-eqa5
Similarly, variance of auto-covariance function estimate Ĉuu(k) is given by
i1520-0485-32-7-2034-eqa6
If we now assume that the covariance functions estimated from the different continuous portions of trajectory are uncorrelated (although the formal justification along the same lines as presented above for the sample mean values is not known to us) we have
i1520-0485-32-7-2034-eqa7
As a particular case,
i1520-0485-32-7-2034-eqa8
Asymptotically, var(uυ) = 1/N Σ+∞m=−∞uu′(m)〉〈υυ′(m)〉 if we neglect the contribution of the cross-covariance products 〈uυ′(m)〉〈υu′(m)〉 and if nΔ is much greater than the integral time scale J12 defined as J12 = 1/〈uu′(0)〉〈υυ′(0)〉 0uu′(τ)〉〈υυ′(τ)〉 . Along the same lines, one obtains
i1520-0485-32-7-2034-eqa9
Asymptotically, var(uu) = 2/N Σ+∞m=−∞uu′(m)〉2 if nΔ is much greater than the integral time scale J11 defined as J11 = 1/〈uu′(0)〉2 0uu′(τ)〉2 dτ.

Comparing with the classical result for the sample variance ŝ2 (which estimates σ2) when we have N′ independent normal observations [var(ŝ2) = 2σ4/N′], we see that as far as the estimation of 〈uu′(0)〉 (≈uu) is concerned, the &ldquo=uivalent number of independent observations” is N′ = [2〈uu′(0)〉2]/var(uu) ≈ NΔ/2J11 (see Priestley 1981). Note that if kΔ is much greater than J11, var[Ĉuu,all(k)] ≈ (1/2) var(uu) (Bendat and Piersol 1971).

The sample variance uu being chi square distributed, since it results from squares of Gaussian variables, a 95% confidence interval for 〈uu′(0)〉 is {[(N′ − 1)uu]/[q0.975], [(N′ − 1)uu]/[q0.025],} where q0.025 and q0.975 are the 0.025th and 0.975th quantiles of the chi square distribution with N′ − 1 degrees of freedom. An approximate 95% interval for 〈uu′(0)〉 is also given by
i1520-0485-32-7-2034-eqa10
if the number of degrees of freedom is large enough to apply the central limit theorem.
Variance of Ĉυυ(k) and confidence interval for υυ are obtained similarly. To estimate a 95% confidence interval for 〈uυ′(0)〉, we can no longer rely on chi squares, but we can still use the central limit theorem, which gives
i1520-0485-32-7-2034-eqa11

TOPOGULF integral timescales

Integration of covariance functions gives I11 ≈ 10–12 days and I22 ≈ 6–10 days in the east. In the west, I11 ≈ 5–7 days but I22 ≈ 2–5 days is not well defined (see section 3). On the contrary, integration of estimated squared covariances results in integral timescales well-defined and almost identical: J11J22J12 ≈ 3.5 days in the west and ≈7 days in the east. Whence our gross assumption in Table 2 that I = J ≈ 5 days in the west and ≈10 days in the east.

EKE variance

EKE is estimated naturally as (1/2)(u2 + υ2). Let us show that , where the underbar refers to the new frame defined by principal axes.

Since and are Gaussian and uncorrelated by construction, they are independent. Of course, (1/2)(u2 + υ2) = (1/2)(). Independence of and implies that and are also independent (cf., Mood et al. 1974, p. 151), so var(). Here the orthogonal velocity components are relative to the same time. Let us consider a nonzero time lag, and assume that and are also independent (a most plausible result), then

Now,
i1520-0485-32-7-2034-eqa12
The last term in the right-hand side of the above expression is given by 2 (1/N2) ΣNi=1 ΣNj=1 cov() (cf., Mood et al. 1974, p. 179), which is zero because the individual covariances are all equal to zero (see above). Finally,
i1520-0485-32-7-2034-eqa13

Fig. 1.
Fig. 1.

Number of float days within 1° lat × 2° lon bins. Four adjacent bins were grouped together to obtain more reliable statistical data, which are given in Table 1. Bins with more than 90 float days are shaded

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 2.
Fig. 2.

700-db Eulerian mean velocity vectors for 2° lat × 4° lon bins with at least 3 months of float data. Associated error ellipses (0.63 probability) are given for bins with at least 1 yr of float data. A uniform integral timescale of 5 days was assumed

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 3.
Fig. 3.

700-db EKE distribution (unit: cm2 s−2)

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 4.
Fig. 4.

700-db temperature variance distribution (unit: °C2)

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 5.
Fig. 5.

700-db covariance functions Cuu(τ), C(τ), and Cυυ(τ) estimated from float velocity time series (a) west of the MAR, (b) east of the MAR

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 6.
Fig. 6.

(a) 700-db western basin zonal, meridional, and cross-dispersions as a function of time: X1X1(t), X2X2(t), and X1X2(t) {the latter is given as a correlation, i.e., normalized by [X1X1(t) · X2X2(t)]1/2}. Also shown is the area of the dispersion ellipse, i.e., where and are principal dispersions (the angle of the major axis is given in degrees relative to zonal direction). The random walk regime is identified within vertical dashed lines. The asymptotic behavior of the ellipse area is extrapolated to the right and for the K0 value given. (b) 700-db eastern basin dispersions

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 7.
Fig. 7.

(a) 700-db western basin zonal, meridional, and cross diffusivities estimated from integration of the covariance functions (referred to as COV in the figure) and from one-half of the dispersion derivative (referred to as DISP in the figure). Angular momentum of fluctuations is also given. It is the integral of C12(τ) − C21(τ). Scale in days on the right results with u1u1 = 115 cm2 s−2 and applies for T11 and T22 as well, because u2u2u1u1. (b) 700-db eastern basin diffusivities. Integral timescale results with u1u1 = 25 cm2 s−2

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 8.
Fig. 8.

Dispersion diagrams for the four regions selected (Canary Basin, MAR, NFB, and CRS). On the left side, dispersion is relative to zonal and meridional directions, while on the right side it is relative to a rotated coordinate frame. Angle in degrees gives the longitudinal direction relative to zonal one.

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 8.
Fig. 8.

(Continued)

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Fig. 9.
Fig. 9.

Dispersion (1σ) ellipses (they contain ∼39% of the points) after 30, 60, and 90 days. 2σ 90-day ellipses (they contain ∼86% of the points) are seen not to exceed the corresponding box sizes

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2034:SFRMIN>2.0.CO;2

Table 1.

Statistics for 700-dbar floats in 2° lat × 4° lon bins (1 float-year at least)

Table 1.
Table 1.

(Continued)

Table 1.
Table 2.

Horizontal diffusivity at 700 db.

Table 2.

1

This is obviously wrong for the TOPOGULF clusters during the first months after launching. In consequence, we have excluded the correlated velocities from the estimation by suppressing order of 50 days at the beginning of the trajectories that resembled closely to a trajectory already considered. This amounted to suppressing 663 float days for the western basin, and 400 float days for the eastern basin.

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  • Fig. 1.

    Number of float days within 1° lat × 2° lon bins. Four adjacent bins were grouped together to obtain more reliable statistical data, which are given in Table 1. Bins with more than 90 float days are shaded

  • Fig. 2.

    700-db Eulerian mean velocity vectors for 2° lat × 4° lon bins with at least 3 months of float data. Associated error ellipses (0.63 probability) are given for bins with at least 1 yr of float data. A uniform integral timescale of 5 days was assumed

  • Fig. 3.

    700-db EKE distribution (unit: cm2 s−2)

  • Fig. 4.

    700-db temperature variance distribution (unit: °C2)

  • Fig. 5.

    700-db covariance functions Cuu(τ), C(τ), and Cυυ(τ) estimated from float velocity time series (a) west of the MAR, (b) east of the MAR

  • Fig. 6.

    (a) 700-db western basin zonal, meridional, and cross-dispersions as a function of time: X1X1(t), X2X2(t), and X1X2(t) {the latter is given as a correlation, i.e., normalized by [X1X1(t) · X2X2(t)]1/2}. Also shown is the area of the dispersion ellipse, i.e., where and are principal dispersions (the angle of the major axis is given in degrees relative to zonal direction). The random walk regime is identified within vertical dashed lines. The asymptotic behavior of the ellipse area is extrapolated to the right and for the K0 value given. (b) 700-db eastern basin dispersions

  • Fig. 7.

    (a) 700-db western basin zonal, meridional, and cross diffusivities estimated from integration of the covariance functions (referred to as COV in the figure) and from one-half of the dispersion derivative (referred to as DISP in the figure). Angular momentum of fluctuations is also given. It is the integral of C12(τ) − C21(τ). Scale in days on the right results with u1u1 = 115 cm2 s−2 and applies for T11 and T22 as well, because u2u2u1u1. (b) 700-db eastern basin diffusivities. Integral timescale results with u1u1 = 25 cm2 s−2

  • Fig. 8.

    Dispersion diagrams for the four regions selected (Canary Basin, MAR, NFB, and CRS). On the left side, dispersion is relative to zonal and meridional directions, while on the right side it is relative to a rotated coordinate frame. Angle in degrees gives the longitudinal direction relative to zonal one.

  • Fig. 8.

    (Continued)

  • Fig. 9.

    Dispersion (1σ) ellipses (they contain ∼39% of the points) after 30, 60, and 90 days. 2σ 90-day ellipses (they contain ∼86% of the points) are seen not to exceed the corresponding box sizes

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