Lagrangian Time Scales and Eddy Diffusivity at 1000 m Compared to the Surface in the South Pacific and Indian Oceans

Transcription

1 2718 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 Lagrangian Time Scales and Eddy Diffusivity at 1000 m Compared to the Surface in the South Pacific and Indian Oceans STEPHEN M. CHISWELL National Institute of Water and Atmospheric Research, Wellington, New Zealand (Manuscript received 12 February 2013, in final form 5 September 2013) ABSTRACT Argo floats cannot be regarded as true Lagrangian drifters because they periodically rise to the surface. Hence, previous estimates of eddy diffusivity at depth using single-particle statistics have been limited to one submerged cycle. However, unless the Lagrangian time scale is significantly shorter than the Argo cycle time, this single-particle calculation can have a large bias. Here, eddy diffusivity computed from single-particle statistics using Argo data is compared to that computed by assuming that Eulerian scales at depth are the same as at the surface, and that the relationship between Lagrangian and Eulerian time scales derived by Middleton is valid. If the methods provide the same answer, one can have confidence in both methods. Eddy diffusivity calculated from the single-particle statistics shows the same spatial structure as that computed from inferred time scale, but is smaller by a factor of about 2. It is suggested that this is because the deep Lagrangian time scale over much of the region is comparable to, or longer than, the 10-day Argo submergence cycle. 1. Introduction Many ocean processes are Lagrangian by nature, and several studies have compared modeled and/or observed Lagrangian and Eulerian statistics of the surface ocean (e.g., Garraffo et al. 2001; McClean et al. 2002; Sallee et al. 2008). For subtropical regions, Lagrangian time scales are typically about 3 4 days, which is up to an order of magnitude shorter than the Eulerian time scales (Chiswell and Rickard 2008; Lumpkin et al. 2002). Surface eddy diffusivities are scale dependent, ranging from O(100 m 2 s 21 ) in the coastal zone (e.g., Chiswell and Stevens 2010) to O(10 4 m 2 s 21 ) in the tropical ocean (Zhurbas and Oh 2004). There have been fewer studies that have considered the behavior of the ocean at depth. LaCasce and Bower (2000) computed eddy diffusivities from pairs of subsurface floats for a limited area of the North Atlantic Ocean. Later, Lumpkin et al. (2002) computed eddy time and length scales in the North Atlantic Ocean from surface and subsurface drifters and concluded that, while near-surface Lagrangian time scales are considerably shorter than the Eulerian time scales (their frozen Corresponding author address: Stephen M. Chiswell, National Institute of Water and Atmospheric Research, P.O. Box 14901, Wellington, New Zealand. s.chiswell@niwa.cri.nz field), in much of the deep ocean Lagrangian and Eulerian time scales are nearly equal (their fixed field). Middleton (1985) provides a theoretical explanation for this observation: at the surface, Lagrangian drifters quickly move into regions where the local (Eulerian) velocity is decorrelated with the velocity at the initial location, so that the Lagrangian velocity decorrelation time scale is short, however, at depth Lagrangian drifters become decorrelated from their initial positions more slowly so that the Lagrangian time scales are longer. Davis (2005) used a limited number of autonomous floats to estimate mean velocities, seasonal variability, and eddy diffusivity at a nominal depth of 900 m for the Pacific and Indian Oceans. Since 2000, however, the Argo global array of deep temperature-/salinity-profiling floats (Gould and Turton 2006) has provided increasing coverage of the ocean at 1000-m depth allowing for potentially improved estimates of these quantities. From a Lagrangian perspective, the eddy diffusivity is often estimated as the product of the velocity variance multiplied by the Lagrangian time scale (e.g., Middleton 1985) K 5 u 02 T L, (1) where the notation is defined in the next section. For convenience, we refer to this as the Lagrangian timescale method of estimating diffusivity. DOI: /JPO-D Ó 2013 American Meteorological Society

2 DECEMBER 2013 C H I S W E L L 2719 Argo floats cannot be regarded as true Lagrangian drifters because they rise to the surface every 10 days to broadcast their data, and so they cannot be used to compute the Lagrangian time scale. However, Davis (1991, 2005) provides two methods of estimating the eddy diffusivity from single-particle statistics by considering an ensemble of single particles passing through a location of interest x 0. A generalized form of the Taylor (1921) diffusivity in statistically inhomogeneous flows (where notation, including angle brackets, is defined in the following section) is given by K jk 52hy 0 j (t 0 j x 0, t 0 )d0 k (t 0 2 t j x 0, t 0 )i. (2) Alternatively, the eddy diffusion tensor can be written K jk t hd0 j (t 2 t 0 j x 0, t 0 )d0 k (t 2 t 0 j x 0, t 0 )i. (3) Davis (2005) suggests that, in general, Eq. (2) provides the more robust estimate. However, Eq. (2) requires the product of displacement anomaly d 0 atthetimeofobservation t 0 2 t multiplied by the velocity anomaly y 0 of the particles at the time they pass through x 0. It is generally considered that calculations of diffusivity made from Argo data should be limited to one cycle (e.g., Katsumata and Yoshinari 2010). If so, there is no estimate of the float s velocity as it passes through x 0, and as a result, estimates of diffusivity from Argo data can only be made using a form of Eq. (3) limited to one cycle (see next section). Both Eqs. (2) and (3) provide estimates of the eddy diffusivity that converge to the underlying ocean diffusivity only when the observations are made for observation times, several times longer than the Lagrangian time scale (Davis 1991). But if the results of Lumpkin et al. (2002) hold, it is likely that the Lagrangian time scale at 1000 m is comparable to, or longer, than the Argo submergence cycle. In this case, estimates made from Eq. (3) will be in considerable error. Thus, it becomes important to estimate both deep Lagrangian time scale and diffusivity from other methods. Even though one cannot make a direct estimate of the Lagrangian time scale from Argo data, an indirect estimate of the Lagrangian time scale at depth can be made if one has an estimate of the Eulerian time scale at depth, and one assumes that the theory relating Lagrangian to Eulerian time scales provided by Middleton (1985) is valid. Here we argue that, if the ocean is vertically coherent, the Eulerian time scale at depth will be the same as (or, at least close to) that at the surface. Where the ocean variability is dominated by barotropic or vertically coherent baroclinic processes, this assumption is likely to be valid, and so this this assumption is used here both to infer the deep Lagrangian time scale and to test whether the eddy diffusivity derived from one submerged cycle of Argo data is robust. We argue that, if the eddy diffusivity calculated from single-particle statistics agrees with that using the inferred Lagrangian time scale, then one can have some confidence in both estimates of eddy diffusivity. Surface Eulerian time scales can be derived from satellite altimeter data. Surface Lagrangian time scales can be derived from Global Drifter Program (GDP) drifters. Thus, in this article, we first show that the two methods (i.e., single-particle and Lagrangian time scale) provide comparable estimates of the eddy diffusivity at the surface. This gives confidence both in the methods and in the estimates of the surface Eulerian time scales. We then compare estimates of the diffusivity at depth and show that the two methods only give the same result when the single-particle estimate is corrected to account for the inferred Lagrangian time scales, which are comparable to the Argo sampling period. 2. Theory a. Eddy diffusivity from Lagrangian time scale For simplicity, theory is presented for the zonal case, and subscripts such as x, y, 0, or 1000 are used when necessary to differentiate between zonal, meridional, surface, and 1000-m values, respectively. At any location x, the Eulerian velocity u(x, t) can be expressed as having a mean and anomaly u(x, t) 5 u(x) 1 ~u(x, t). (4) The velocity scale u 0 is defined to be the standard deviation p of the Eulerian velocity at that location u 0 5 ffiffiffiffiffiffiffiffiffiffiffiffiffi var( ~u). The Eulerian and Lagrangian time scales are defined as the integral of the respective velocity autocorrelation functions (e.g., Rupolo 2007). For times greater than the Lagrangian time scale T L the diffusivity is given by (e.g., Middleton 1985) K 5 u 02 T L. The Lagrangian time scale and length scale L L are related by the velocity scale L L 5 u 0 T L. (5) By analogy with Eq. (5), it is convenient to define a characteristic velocity (sometimes called advection or evolution speed; e.g., Lumpkin et al. 2002) u*, so that L E 5 u*t E. (6)

3 2720 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 b. Eddy diffusivity from single-particle statistics For an ensemble of particles that pass through a location x 0 in a turbulent ocean, the eddy diffusion tensor is given by (Davis 1991) K jk 52hy 0 j (t 0 j x 0, t 0 )d0 k (t 0 2 t j x 0, t 0 )i, where y 0 and d 0 are the velocity and displacement departures from the respective Lagrangian means, and the angle brackets indicate the ensemble average over all particles. The notation a(t j x 0, t 0 ) denotes the value of a at time t for a particle passing through location x 0 at time t 0. Alternatively (e.g., LaCasce and Bower 2000; Zhurbas and Oh 2003), following (Taylor 1921), suggest that for times longer than the Lagrangian time scale T L, the diffusivity K jk canbeestimatedfromtherateofchange of the single-particle patch displacement variance K jk t hd0 j (t 2 t 0 j x 0, t 0 )d0 k (t 2 t 0 j x 0, t 0 )i, where the term in the angle brackets is the displacement variance s jk at time t 5 t 2 t 0 of the ensemble of particles passing through x 0. If calculations from Argo floats are limited to one cycle, then Eq. (2) cannot be used because there is no estimate of the velocity of the float as it passes through x 0 [if one calculates this velocity as the displacement divided by the cycle time then Eq. (2) is numerically twice Eq. (3)]. Thus for Argo floats, we use the discrete form of Eq. (3) K jk 5 1 hd 0 j (t 0 1Gjx 0, t 0 )d0 k (t 0 1Gjx 0, t 0 )i, (7) 2 G where G is the duration of one cycle (;10 days). Davis (1991) shows that for a homogenous ocean having an underlying eddy diffusivity K inf 5 u 02 T L, singleparticle estimates of diffusivity approach K inf only when the observation time t(5t 2 t 0 ) is significantly longer than the Lagrangian time scale. One can estimate the behavior of the diagonal components of the diffusivity K jj (which represent the magnitude of the diffusivity, whereas the cross terms represent the correlation between zonal and meridional components) from Monte Carlo random-walk simulations of a numerical ocean having prescribed Lagrangian time scales and variance (discussed in the next section, and shown in Fig. 1). Davis (1991) notes that the behavior for Eq. (2) can be crudely modeled as K jj /K inf 5 [1 2 exp(t/t L )]. A heuristic fit to the behavior FIG. 1. The K jj /K inf derived from Monte Carlo simulations of Lagrangian trajectories for four numerical oceans. The oceans are specified to have T L and u 0, so that the underlying ocean diffusivity for each ocean is K inf 5 u 02 T L.TheK jj from Eqs. (2) and (7) are shown normalized by K inf for each ocean (solid lines). The fits pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (dashed lines) are K jj /K inf 5 [1 2 exp(t/t L )] and K jj /K inf 5 f1 2 exp[t/(5t L )]g, for Eqs. (2) and (7), respectively. The four oceans have u 0 5 1, and T L 5 2, 6, 10, and14 days, respectively. p for Eq. (7) made here is K jj /K inf 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f1 2 exp[t/(5t L )]g. When t equals the Lagrangian time scale, Eq. (2) returns an estimated ;63% of the true value, and Eq. (7) returns an estimated ;40% of the true value. For GDP drifters, the eddy diffusivity can be calculated for times up to about 60 days (see methods section), which is times the Lagrangian time scale, so that both single-particle estimates should be robust. To be consistent with our Argo methodology, we use Eq. (3) for GDP drifters. The diffusivity tensor K jk can be rotated to provide along- and cross-axes diffusivities (e.g., Davis 1991; Zhurbas and Oh 2003), which can be represented by an ellipse having major E 1 and minor E 2 axes, and orientation u given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1 5 (K 11 1 K 22 )/2 1 (K 11 2K 22 ) 2 /4 1 K12 2, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 5 (K 11 1 K 22 )/2 2 (K 11 2K 22 ) 2 /4 1 K12 2, and u 5 0:5 tan 21 (2K 12, K 11 2 K 22 ). (8) MONTE CARLO SIMULATIONS Monte Carlo simulations were made for several numerical oceans. For each ocean, the Lagrangian time scale T L and velocity scale u 0 were prescribed, then a 5000-member ensemble of pseudorandom velocity times series y 0 j 5 u0 R TL,0,1 was created, where R TL,0,1 is a random time series having a mean of 0, a standard

4 DECEMBER 2013 C H I S W E L L 2721 deviation of 1, and an autodecorrelation time scale of T L. These random time series were generated by summing lagged Gaussian-distributed random time series, where the coefficient on each lagged time series is given by an exponential autodecorrelation function r exp(2t/t L ). Thus, each numerical ocean comprises an ensemble of 5000 randomly generated velocity series y 0 that represent an ocean having an underlying diffusivity given by K inf 5 u 02 T L. Displacement anomalies d 0 were computed for each pseudorandom velocity series, and two estimates of ensemble-average diffusivity were computed as a function of observation time t as specified by Eqs. (2) and (7). Figure 1 shows the results for four oceans, each having u 0 5 1, but with T L ranging from 2 to 14 days. c. Middleton relationship Middleton (1985) assumed that the relationship of Corrsin (1959) between Lagrangian and Eulerian correlations holds, and by analyzing data over a range of scales, suggested that the Lagrangian and Eulerian time scales were approximately related by the function h 5 q(q 2 1 a 2 ) 21/2, (9) where q 5 (p/8) 1/2, and h and a are the ratios and h 5 T L /T E (10) a 5 u 0 /u*. (11) Equation (9) is plotted in Fig. 2. The theory of Middleton (1985) has been found to hold reasonably well for numerical models of the surface ocean (Lumpkin et al. 2002) and data (Chiswell and Rickard 2008). 3. Data a. Global Drifter Program GDP drifters are drogued at a nominal 15-m depth and transmit their position by satellite. GDP data used here are quality controlled, and an interpolation method known as kriging is used to create 6-hourly time series of positions and velocities (Hansen and Poulain 1996). The 4 day 21 sampling rate is adequate to estimate surface Lagrangian time scales, which are typically about 3 5 days (e.g., Chaigneau and Pizarro 2005). GDP data used here had to be within our domain, have speeds less than 2 m s 21, and have both zonal and meridional velocity errors less than 0.01 m s 21. Only FIG. 2. Theoretical Middleton (1985) relationship between h 5 T L /T E and a 5 u 0 /u*. Horizontal and vertical lines illustrate why the Lagrangian time scales are shorter near the surface and longer at depth for an ocean that is vertically coherent (u* constant with depth), but slower at depth (u 0 decreases with depth). data after the launch of the Ocean Topography Experiment (TOPEX)/Poseidon altimeter (14 October 1992) are used. b. Argo floats Most Argo floats are set to have a nominal parking depth of 1000-m depth, and profile to the surface about every 10 days, measuring water properties (Gould and Turton 2006). During their time at the surface, Argo floats transmit their latitude and longitude via satellite. Quality-controlled Argo data are available from Argo s two Global Data Assembly Centers (GDACs) in Brest, France, and Monterey, California. Here, data were chosen to be all floats in the Southern Hemisphere Indian and Pacific Oceans that had parking depths between 750 and 1250 m. The nominal 1000-m velocities were computed from difference between last surface position before dive and first position after resurface. These velocities were assigned to the midpoint between the dive and resurface positions. There will be an error in the velocity measurement because of the few hours it takes the float to sink to, and rise from, the parking depth. This error is likely to be highest in regions of high vertical shear. Park et al. (2005) aim to reduce these errors by removing a mean drift and inertial motions derived explicitly from least squares fits to the Argos positions for each surface cycle separately. However, here, it was judged that making corrections to the Argo velocity would be likely to introduce errors as large as the original errors themselves. In addition, the main source of error in single-particle statistics is a result of having small ensembles.

5 2722 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 Occasionally Argo floats ground themselves temporarily. Velocities from floats that were judged to be grounded were removed from the analysis. c. Altimeter Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO) Maps of Sea Level Anomaly and associated geostrophic surface velocity derived from merged TOPEX/Poseidon and the European Space Agency s European Remote Sensing Satellite (ERS) data are provided by AVISO/Altimetry, Space Oceanography Division, France. AVISO data are made available as anomaly fields at 7-day intervals and with 1 /38 spatial resolution. AVISO data start in October Methods a. Single-particle statistics The single-particle displacement tensor s jk was computed from the GDP drifters and Argo floats at each x 0 on a 18 by 18 latitude by longitude grid. In practice few trajectories pass exactly through any grid point, so all trajectories that passed within a specified distance (100 km) of the grid point were collected to provide the ensemble of tracks. These trajectories were adjusted to pass through the grid point (i.e., ensuring that the initial displacement is zero). For GDP drifters, ensemble variances increase approximately linearly with time typically up to about 60 days. Thus, diffusivities were computed from Eq. (3) where the time derivative of the variance was derived from linear regressions over 60 days. For Argo data, the single-particle estimate was computed from one submergence cycle of each float according to Eq. (7). For both GDP and Argo, the mean velocities (u and y), velocity scales (u 0 and y 0 ), and eddy kinetic energy EKE 5 0:5(u 02 1 y 02 ) at each grid point were computed from the ensembles. This is equivalent to simple bin averaging for these properties (e.g., Fratantoni 2001). b. Time-scale statistics 1) SURFACE Surface Eulerian time scales were computed at each x 0 from AVISO-derived surface velocities using conventional methods, that is, by removing the time mean and integrating the autocorrelations following the procedures documented in Chiswell et al. (2007). Surface Lagrangian time scales were computed for GDP drifters using methods presented previously (Chiswell and Rickard 2008). Lagrangian velocities were calculated for all drifter tracks that lasted longer than 50 days (longer tracks days were split into several 50-day tracks). Autocovariances were computed for each 50-day record and assigned to x 0 closest to the mean location for the track, thus providing an ensemble of auto covariances at x 0. These ensemble-average autocovariances were then scaled by the surface velocity scale u 0 0 to produce autocorrelations, which were then integrated to provide the Lagrangian time scale T L0. The surface characteristic velocity u 0 * can then be inferred from the ratio of time scales h 0 5 T L0 /T E0 using Eq. (9). Surface eddy diffusivities were computed from the Lagrangian time scale multiplied by the velocity variance [Eq. (1)]. 2) 1000 M The characteristic velocity and Eulerian time scales, u 1000 * and T E1000, were assumed to be the same as at the surface (u 1000 * 5 u 0 * and T E T E0 ). The velocity scale u was determined from the Argo binned mean, thus allowing us to calculate the ratio a u /u 0 * at each x 0. This value, combined with the Middleton (1985) relationship [Eq. (9)], was then used to infer the value of h 1000, and hence the Lagrangian time scale was inferred as T L h 1000 T E0. The deep diffusivities were computed from the Lagrangian time scale multiplied by the velocity variance as K u T L Results a. Mean velocity and eddy kinetic energy It has not been the main focus of this work to produce the best mean velocity fields, because the mean fields are derived primarily to compute the velocity scales u 0 and are not presented here. LaCasce (2008) describes in detail some of the potential problems associated with computing mean velocities from geographical bins, including the issues of uneven data coverage, which are particularly appropriate for Argo data. Nevertheless, our mean fields, (not shown to conserve space) are not too different from Davis (2005) and Katsumata and Yoshinari (2010), and reflect well-established understanding of mean circulation in these oceans. Surface EKE derived from drifters or altimetry has already been published by other authors (e.g., Stammer 1997), and our results (Fig. 3) are similar to these previous studies. High EKE is found associated with the equatorial current system and Antarctic Circumpolar Current (ACC), and with the Agulhas and East Australian Current (EAC) western boundary currents, respectively. There is also a small lobe of elevated EKE associated with the Leeuwin Current off the west coast of Australia.

6 DECEMBER 2013 C H I S W E L L 2723 FIG. 3. EKE derived from drifter velocities in latitude longitude bins (see text). Fields are plotted on a log10 scale. (a) Surface EKE derived from GDP drifters (m 2 s 22 ), (b) EKE at 1000 m derived from Argo floats (m 2 s 22 ), and (c) the ratio of EKE at 1000 m to EKE at the surface (dimensionless). The red-dashed line schematically indicates the core of EKE in the ACC (see text). A band of relatively high EKE (; m 2 s 22 ) extending eastward ;258S from Australia is probably an expression of the subtropical counter current (STCC; Qiu and Chen 2004). A similar band appears in the Indian Ocean. For comparison with later plots, we have included a line indicating the core of eddy kinetic energy. Eddy kinetic energy at 1000 m reflects some, but not all, of the surface EKE features, although deep EKE values are less than at the surface by an order of magnitude or two. A notable difference between the surface and 1000-m-deep EKE is the relative absence at depth of eddy variability east of Australia. This is consistent with Qiu and Chen (2004) who suggest that STCC is strongly baroclinic and does not extend more than about 400 m deep. Elsewhere, elevated deep EKE is seen along the equator and in bands associated with the ACC, Agulhas, and East Australian Currents. The lowest EKE at 1000 m is in the Pacific Ocean, but it is proportionately even weaker than in the surface ocean. The Argo-derived velocity variances (u 02 and y 02 ) at the Bryden and Heath (1985) mooring are generally about 20% lower than estimates made from the current meters (Table 1). The ratio of deep to surface EKE (EKE 1000 /EKE 0 ; Fig. 3c) emphasizes the relative decrease in the central gyres of both oceans, where the ratio is typically Highest relative ratios occur in the ACC, where EKE 1000 is typically of the surface value. b. Surface Eulerian and Lagrangian time scales Surface Eulerian time scales are typically days and days for the zonal and meridional components,

7 2724 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 TABLE 1. Mean and eddy statistics from Argo data and from Bryden and Heath (1985) for the northern edge of the ACC. Mean and eddy statistics derived from the bin averages of Argo data compared with those from 527-day measurements from a current meter at 1000 m on mooring P from Bryden and Heath (1985). Quantity Argo Bryden and Heath (1985) values from 1000 m u (m s 21 ) T Eu (days) 24* 29.8 y (m s 21 ) T Ey (days) 26* 16.3 u 02 (m 2 s 22 ) y 02 (m 2 s 22 ) EKE (m 2 s 22 ) * Eulerian time scales are estimated from surface AVISO satellite data. respectively (Figs. 4 and 5). The spatial structure is slightly different between components, although both components show higher values south of about 458S and in the equatorial western Pacific. Surface Lagrangian time scales (Figs. 4 and 5) show strong bands of higher values centered on the equator for the zonal component and off-equator for the meridional component. As for the Eulerian time scale, the meridional component is about half the zonal value. Zonal values of the ratio of Eulerian to Lagrangian time scale h 0 5 T L0 /T E0 are typically 0.1 south of 208S, rising to about twice this value north of 208S. There is a band of elevated h 0 along the equator. According to Eq. (9), h cannot exceed 1, but there are regions in the Pacific subtropics where the calculated values exceed this value, suggesting either Lagrangian time scales maybeoverestimatedoreulerian time scales are underestimated. Meridional values of h 0 are generally higher than the corresponding zonal values and there is a broad region north of 208S wherethemeridional values approach or exceed 1. There is little evidence of h 0 showing any relationship to the mean circulation. FIG. 4. Surface zonal time scales. (a) T E derived from satellite altimeter (AVISO) observations of surface current, (b) T L derived from GDP drifter observations of surface current, and (c) the ratio between the surface Lagrangian and Eulerian time scales.

8 DECEMBER 2013 C H I S W E L L 2725 FIG. 5. As in Fig. 4, but for the meridional direction. Surface eddy diffusivities calculated from single-particle statistics and from the time scale show spatial patterns that are for the most part similar (Fig. 6), although values calculated from the single-particle method tend to be a little higher than those from the time-scale method. The domain-wide ratio of diffusivities calculated by the two methods has a median value of 1.25 (not shown). Zonal diffusivity is high along the equator, along the ACC, and within the East Australian Current. A band of relatively high diffusivity, O(from to m 2 s 21 ), is associated with the STCC east of Australia. The ocean is anisotropic, with meridional diffusivities generally much less than their zonal counterparts, except west of Australia in the Leeuwin Current region. The velocity scales (u 0 0 and y0 0 ) at the surface are not shown here because they essentially mimic the eddy kinetic energy distributions (Fig. 3). Zonal and meridional characteristic velocities (u 0 * and y 0 *) are shown in Fig. 7. Characteristic velocity peaks in equatorial and subequatorial regions for both components. The domain-wide mean meridional component (0.18 m s 21 ) is about twice that of the zonal component (0.09 m s 21 ). c m Lagrangian statistics The ratio of Lagrangian to Eulerian time scale at 1000 m [derived assuming the Middleton (1985) relationship holds and that the Eulerian time scales and characteristic velocities at depth are the same as at the surface] is larger than at the surface (cf. Fig. 8 with Figs. 4 and 5). Consequently, the inferred deep Lagrangian time scales are longer and show quite different spatial distribution than their surface equivalents (Fig. 9). Zonal time scales in the Indian Ocean are about 10 days in the southwest with a band of elevated time scale of days extending westward from Australia. Zonal time scales in the Tasman Sea are less than 25 days, but in the western equatorial Pacific, they reach as high as 50 days. Meridional deep time scales are less than their zonal equivalents, and are typically days in the Indian Ocean with highest values in the western equatorial Pacific.

9 2726 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 43 FIG. 6. Surface zonal and meridional eddy diffusivity calculated from GDP drifters using two different methods. (a) Zonal diffusivity calculated from single-particle statistics, (b) meridional diffusivity calculated from single-particle statistic, (c) zonal diffusivity calculated as the product of velocity variance times the Lagrangian time scale, and (d) meridional diffusivity calculated as the product of velocity variance times the Lagrangian time scale. Deep eddy diffusivities, which we denote as Kxx and Kyy when derived from the single-particle method, and as Kx and Ky when derived from the inferred time scale are shown in Fig. 10. Both zonal components (Kxx and Kx ) show higher values in the western equatorial Indian Ocean than in the eastern equatorial Indian Ocean, and lobes of higher diffusivity in the ACC around the flanks of the Campbell Plateau. In both cases, the structure of the spatial scale tends to mimic that of deep EKE (cf. with Fig. 3). The meridional components (Kyy and Ky ) show similar values to their respective zonal

10 DECEMBER 2013 CHISWELL 2727 FIG. 7. Characteristic velocity derived from observations of surface Eulerian and Lagrangian time scales, velocity variance, and the Middleton (1985) relationship (see text). (a) Zonal characteristic velocity u* (m s21) and (b) meridional characteristic velocity y* (m s21) are shown. components in the ACC, but are lower along the equator. Overall, however, Kxx and Kyy show much lower values than Kx and Ky. The spatial scale inherent in the computation of the Lagrangian time scale from GDP tracks (used to estimate u*) is longer that the spatial scale of the single-particle statistic. As a result, direct estimates of the ratios of diffusivity calculated from both methods (i.e., Kxx /Kx and Kyy /Ky ) are somewhat noisy. Figure 11 shows these ratios plotted in space and FIG. 8. Ratio of Lagrangian to Eulerian time scales h 5 TL /TE at 1000 m. (a) Zonal and (b) meridional ratios are shown.

11 2728 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 FIG. 9. Inferred Lagrangian time scale at depth derived by assuming that the Eulerian time scales at depth are the same as at the surface, and using the ratios of Lagrangian to Eulerian time scales shown in Fig. 8. (a) Zonal and (b) meridional time scales are shown. as histograms after the spatially smoothing the singleparticle estimate over 58 bins in an attempt to compensate for this effect. Domain-wide median values are 0.51 and 0.65 for the zonal and meridional cases, respectively, although these values are sensitive to the degree of spatial smoothing employed. The ratio K xx /K x is lower where inferred T L is higher (Fig. 9), which is what one would expect from Fig. 1. One can correct K jk assuming the relationship shown in Fig. 1 using the inferred Lagrangian time scale, and Fig. 11 shows the corrected values K corr jk 5 K jk /r(g/t L ), (12) where the correction r is derived from the fit shown in Fig. 1. Median values of Kxx corr/k x and Kyy corr/k y are 1.18 and 1.29, respectively. Single-particle diffusivity for the surface and corrected diffusivity for 1000 m are shown in ellipse form in Fig. 12 (e.g., Zhurbas and Oh 2003, 2004). Surface diffusivity in general is more anisotropic than deep diffusivity (surface ellipticity tends to be higher than deep ellipticity), with the difference being pronounced in the ACC. Deep eddy diffusivity is typically an order of magnitude less than at the surface, apart from within the ACC, where is about one-third that of the surface. 6. Summary and discussion The main findings of this work are that, at the surface, Lagrangian time scales are shorter than Eulerian time scales. Eddy diffusivity computed from single-particle statistics is about 25% higher than that computed from the Lagrangian time-scale method, giving us reasonable confidence in both estimates. At 1000 m, however, Lagrangian time scales are close to Eulerian time scales, and are comparable to, or longer than, the Argo submergence cycle. As a result, single-particle estimates of diffusivity derived from one cycle of Argo data are biased low. When corrected for using the bias predicted from Monte Carlo simulations, the single-particle estimates of deep diffusivity show about the same 25% increase over those computed from the Lagrangian timescale method as at the surface. Our surface Eulerian time scales are generally comparable to estimates made by Stammer (1997), although he provides a single isotropic value. Elevated time scales in the Indian Ocean and south of Australia, and low values west of South America and east of New Zealand (Fig. 4) all feature in both our and Stammer s (1997) surface time scales (his Fig. 18a). On the other hand, some of the features in the Pacific Ocean are different, for example, we show a band of high time scale extending east from Australia at about 308S, whereas Stammer (1997) has low time scale in this region. Zonal Eulerian

12 DECEMBER 2013 CHISWELL 2729 FIG. 10. Deep eddy diffusivity calculated two different ways. (a),(b) Zonal and meridional components calculated from single-particle statistics using one submerged cycle of Argo float data; and (c),(d) zonal and meridional components calculated as the product of velocity variance times inferred Lagrangian time scales shown in Fig. 9. time scales in the Tasman Sea are consistent with our previous estimates (Chiswell and Rickard 2008). Our estimates of surface eddy diffusivity derived from single-particle statistics are about 25% higher than those computed from the Lagrangian time-scale method. It should be pointed out that these two methods do not necessarily provide the same result unless the field is statistically homogeneous, and some of this 25% difference may result from real ocean inhomogeneity because differences are larger where the surface diffusivity is anisotropic, for example, along the equator and in the ACC (Fig. 12). Nevertheless, both estimates are in essential agreement with previous findings. Zhurbas and Oh (2004) shows lateral diffusivity for the South Pacific Ocean derived from single-particle statistics and GDP data prior to His values are similar to ours derived

13 2730 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 43 FIG. 11. Ratio of diffusivity at 1000 m computed by using single-particle method to that computed from inferred time scale. (a) Ratios K xx /K x and K yy /K y, (b) histograms of the ratios, (c) ratios corrected for effect shown in Fig. 1, Kxx corr/k x and Kyy corr/k y, and (d) histograms of the ratios are shown. from the time scale, both his and our values showing, for example a broad band of diffusivity O( m 2 s 21 ) extending out to the east from Australia at about 258S and similar values in the ACC. Our equatorial values, however, are higher than his by a factor of almost an order of magnitude. Sallee et al. (2008) plot their eddy diffusivity in along- and cross-stream components, which makes direct comparison a little difficult, but it appears that their values are similar to ours in particular in the band of high diffusivity seen east of Africa and south of Australia (Fig. 6). However, they comment on a region of high diffusivity east of New Zealand, which we do not see. Comparison of our inferred deep Lagrangian time scale (Fig. 9) with previous works is difficult because of lack of published data for this region. Lumpkin et al. (2002) show deep simulated Lagrangian time scales for the Atlantic Ocean that are shorter than ours by a factor of up to 2 although they compute their values by removing low-frequency variability and suggest their values are perhaps artificially low. Other estimates of deep Lagrangian time scale of about 10 days in highlatitude North Atlantic by Lavender et al. (2005) are consistent with those presented here for similar latitudes and about twice those of Lumpkin et al. (2002) for the same latitudes. Lagrangian time scales from 700 m in the Gulf Stream determined by Veneziani et al. (2004) range from 6 to 20 days, and, similarly, broadly agree with the magnitude of the inferred values presented here. The inferred time scale relies on the assumption that the Eulerian time scale is constant with depth. To the extent that the ocean is defined by vertically coherent mesoscale eddies, the Eulerian time and length scales (hence also, u*) will be the same at depth as at the surface. Where the ocean is dominated by one vertical mode, the Eulerian time and length scales will be constant with depth, but where there is significant energy in a number of vertical modes, this assumption may not be realistic. In particular, it is possible that deep Eulerian

14 DECEMBER 2013 CHISWELL 2731 FIG. 12. (a) Surface and (b) deep eddy diffusivities calculated from single-particle statistics using GDP and Argo float data, respectively. Deep eddy diffusivities have been corrected using the function shown in Fig. 1 and inferred Lagrangian time scale shown in Fig. 9. time scales are longer than surface Eulerian time scales because wind-driven variability is unlikely to get below the Ekman layer. Comparison of our satellite-derived values with the deep mooring estimates of Bryden and Heath (1985) is inconclusive on whether Eulerian time scales vary with depth (Table 1). Zonal deep Eulerian time scales from the mooring are shorter than surface time scales derived from AVISO (24 versus 30 days), whereas deep meridional time scales are longer than surface (26 versus 16 days). However, Lumpkin et al. (2002) show u* derived from a model is lower by about 20% 30% at 1806 m than at the surface (lower u* would result from higher Eulerian time scale if the Eulerian length scale remains the same). Hence it may be that deep Eulerian time scales are longer than estimated here if so, our estimates of deep Lagrangian time scale will be low. Regardless of whether our inferred Lagrangian time scales at depth are accurate, the deep Lagrangian time scales are almost certain to be longer than at the surface as illustrated by the Lumpkin et al. (2002) view that the deep ocean is fixed field, and thus comparable to the Argo submergence cycle. As a result, the single-particle estimate made from Argo data will almost certainly be biased low, unless corrected for using the results presented in Fig. 1. However, these corrections are large (Kjj /K inf is about 0.4), and one needs to examine how robust they are. Our estimate of Kjj /K inf for Eq. (2), agrees with Davis (1991), even though he used a completely different method to simulate the ocean than used here. Davis (1991) does not present Kjj /Kinf for Eq. (3) so we do not have independent validation of our result. Nor have we been able to derive an analytical function for Kjj /Kinf, and thus can only assume that the second set of curves shown in Fig. 1 adequately model the true relationship. Katsumata and Yoshinari (2010) computed eddy diffusivity at 1000 m from one submerged cycle of Argo floats, using Eq. (2), but estimated the velocity at the origin as the displacement over the submerged cycle divided by the submergence time (K. Katsumata 2013, personal communication), that is, they calculated the velocity, y0j (t0 j x0, t0 ), as d0j (t0 1 G j x0, t0 )/G. This is numerically equivalent to using Eq. (3) multiplied by a factor of 2. Examination of Fig. 1 shows that this procedure would give the correct value of diffusivity if the Lagrangian time scale were about 15 days. This value is reasonably typical of much of the South Pacific Ocean (Fig. 9). Thus, by chance, their published values largely are comparable to our corrected values. Finally regardless of whether the single-particle diffusivity is properly corrected the diffusivity at depth is

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