Dispersion of surface drifters and model-simulated trajectories

Ocean Modelling 39 (2011) 301–310

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Dispersion of surface drifters and model-simulated trajectories Kristofer Döös a,⇑, Volfango Rupolo b, Laurent Brodeau a a b

Department of Meteorology, Stockholm University, SE-10691 Stockholm, Sweden Volfango Rupolo, Climate Department, ENEA, Via Anguillarese 301, 00123 Rome, Italy

a r t i c l e

i n f o

Article history: Received 5 May 2010 Received in revised form 3 May 2011 Accepted 16 May 2011 Available online 2 July 2011 Keywords: Lagrangian trajectories Dispersion Model resolution Surface drifters

a b s t r a c t From a data set encompassing the years 1990–2008 pairs of surface drifters with maximum initial separations of 5, 10 and 25 km have been identified. Model trajectories have been calculated using the same initial positions and times as the selected pairs of surface drifters. The model trajectories are based on the TRACMASS trajectory code and driven by the ocean general circulation model NEMO. The trajectories are calculated off-line, i.e. with the stored velocity fields from the circulation model. The sensitivity of the trajectory simulations to the frequency of the stored velocity fields was tested for periods of 3 and 6 h as well as 5 days. The relative dispersion of the surface-drifter and model trajectories has been compared, where the latter was found to be too low compared to the relative dispersion of the drifters. Two low-order trajectory sub-grid parameterisations were tested and successfully tuned so that the total amplitude of the relative dispersion of the model trajectories is similar to that associated with the drifter trajectories. These parameterisations are, however, too simple for a correct simulation of Lagrangian properties such as the correlation time scales and the variance of the eddy kinetic energy. The importance of model-grid resolution is quantified by comparing the relative dispersion from an eddy-permitting and a coarse-resolution model, respectively. The dispersion rate is halved with the coarse grid. The consequences of the two-dimensionality of the trajectories is evaluated by comparing the results obtained with the 2D and the Lagrangian 3D trajectories. This shows that the relative dispersion is 15% stronger when the trajectories are freely advected with the 3D velocity field. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Model-simulated trajectories are increasingly used for studies of the ocean circulation. The primary advantage of the Lagrangian view, as compared to the Eulerian one, is the knowledge of ‘‘water particle history’’ from the origin to the destination of a given water mass. Useful applications are found both in the study of the general ocean circulation and in the biological field, where establishing the origin of a given water mass may be particularly important. By associating each trajectory with a small volume of water, Döös (1995) showed that one can follow water masses and not only water particles if a sufficient number of trajectories is generated. In addition Blanke et al. (1999) introduced the Lagrangian stream function, which is calculated by summing over the selected trajectories that belong to the chosen paths of interest. These model-trajectory tools have proved appropriate for analysing the ocean circulation (Blanke et al., 2001; Speich et al., 2002; Drijfhout et al., 2003; Döös et al., 2008), but have also been useful in the biological context of studying the interrelationship between basins, cf. Pizzigalli and Rupolo (2007) and Serra et al. (2010). Such applications, ⇑ Corresponding author. Tel.: +46 8 161734. E-mail address: [email protected] (K. Döös). 1463-5003/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2011.05.005

mainly based on the knowledge of a single water-mass path, belong to what is known as the single-particle problem. Particle-pair dispersion statistics have become a common method to analyse and describe tracer mixing and transport-barrier dynamics (d’Ovidio et al., 2009). For practical applications, knowledge of mixing properties is of paramount importance. Many of these applications are based on the numerical output of highlyresolved regional models, which are becoming increasingly available. The knowledge of mixing properties via the reliability of the representation of relative dispersion statistics may also be of interest for global OGCMs (ocean general circulation models), which must be of lower resolution, e.g. for investigations of the spatial distribution of planktonic species on global scale (Cermeño and Falkowski, 2009). There have been, however, few systematic attempts to validate model trajectories on the basis of observations, this due to the intrinsic difficulties given the scarceness and extremely high non-stationarity of Lagrangian data. Moreover, in climatological studies these trajectories are three-dimensional, and often used on time-scales ranging from decadal to centennial. Studies by Garraffo et al. (2001), Lumpkin et al. (2002) and McClean et al. (2002) compared Lagrangian statistics from surface drifters in the North Atlantic and trajectories from an OGCM. This was done with

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focus on the Lagrangian time- and length-scales, which do not require pairs of trajectories. Pizzigalli et al. (2007) calculated statistics of dispersion over the Mediterranean from model trajectories. By comparing these results with those from surface drifters they found deficiencies in the representation of particle dispersal, mainly in the short-time regime (O (hours)) due to the lack of energy in the high space- and time-domains. Lacorata et al. (2001) examined surface-drifter data in the Adriatic Sea using a simple chaotic model and showed that relative dispersion is mainly driven by advection at sub-basin scales until basin saturation sets in due to the limited area of the Adriatic. Generally speaking the representation, even statistical, of the Lagrangian dispersion is a very difficult task and a lot of work remains to done, both in the validation of models and also for the understanding of the dependence of Lagrangian dynamics represented by the model on the model setting. This latter subject has been addressed (Iudicone et al., 2002; Griffa et al., 2004) and also considered in a recent work by Poje et al. (2010) undertaken within a two-particle statistical context using a hierarchy of ocean models, from ideal ones to an eddyresolving (1/12°) model of the Gulf stream. Mainly focusing their attention on the asymptotic behaviour of relative dispersion, these researchers found that relative dispersion is strongly influenced by the energetic mesoscale structure, which may be independent of finer resolution. Also many of the relative-dispersion investigations based on field data have dealt with this important issue, both in the analysis of atmospheric weather balloons and oceanographic drifters as well as floats (Lacorata et al., 2004; Rupolo, 2007b). The purpose of the present study is to provide a first and basic estimate of the realism of the representation of surface-drifter trajectories in a global OGCM. The present work is hence a validation of the OGCM’s Lagrangian properties such as the relative and absolute dispersion. A more traditional way to estimate the capacity of the OGCM to simulate realistically the ocean circulation would be from an Eulerian perspective. This has been done for the NEMO/ ORCA025 configuration in studies such as Barnier et al. (2006, 2007), Renner et al. (2009) and Penduff et al. (2009). Since we are interested in comparing surface drifter data with data simulated with an ‘‘eddy-permitting’’ OGCM, which does not resolve scales finer than 25 km, we did not attempt to examine the small-time-scale asymptotic behaviour but instead focused on mesoscale space/time ranges (from 25 to 30 km, times up to 32 days). Relative-dispersion data were obtained by making an inventory of all the trajectories from the Global Drifter Program and isolating the original and the chance ’pairs’. Focus will be on the relative-dispersion properties and how this depends on the temporal and spatial resolution of the velocity fields. Two low-order trajectory sub-grid parameterisations have been tested to determine how the discrepancies between the dispersion of the computed and observed trajectories can be reduced. One of these approaches has been to add random horizontal turbulent velocities u0 , v0 to the horizontal velocity components U, V from the OGCM. Another attempt has been to add a random displacement to the trajectory position, this in order to incorporate a sub-grid parameterisation of the non-resolved scales. Both subgrid parameterisations were tuned so that the total amplitude of the relative dispersion of the model trajectories is similar to that of the drifter trajectories. The two sub-grid parameterisations are, however, too simple to yield a correct simulation of Lagrangian properties such as the time-scales and the variance of the eddy kinetic energy. The work is organised as follows. In Section 2 we describe both the pair-selection procedure from the Global Drifter Program data set and the numerical tools used to produce synthetic data. Hereafter we compare observational and modelled results. In Section 4, we compare relative dispersion from OGCMs of different resolution and model trajectories using 2D and 3D velocity fields. In

Section 5, we summarise and discuss the results. The definitions of the relative and absolute dispersion are given in Appendix A, and the sub-grid parameterisations are explained in Appendix B.

2. Real and synthetic-pair data 2.1. Surface drifters The surface-drifter data used in the present study have been assembled by the Atlantic Oceanographic and Meteorology Laboratory in Miami (AOML) and downloaded from a data base maintained by the Canadian Department of Fisheries and Oceans. The drifters are tracked by the Argos satellites with a positional accuracy of about 150–1000 m. The AOML Drifter Data Assembly Center undertakes a first quality control of the data (Lumpkin and Pazos, 2007) and then interpolates the raw fixes to uniform 6-h intervals using an optimal procedure known as kriging (Hansen and Poulain, 1996). The data obtained between 1990 and 2008 comprise 9385 surface drifters drogued at 15 m depth. Pairs of surface drifters that had at some point been ‘‘close’’ to each other in order to qualify as pairs were hereafter selected. This ‘‘close’’ distance should hence not exceed a certain value d0 in order for the two drifters to be defined as a pair. d0 is therefore the maximum initial distance allowed between the two drifters in order for them to be regarded as a pair, but they can of course be much closer than d0. The initial time t0 of a pair is set to the last time at which the pair is not separated more than the maximum initial distance d0. This means that the pairs are selected in such a way that there is only one distance that is smaller than d0 for each pair so that the initial distances are as similar as possible. This will hence include not only deployed pairs or clusters but also pairs of drifters that happen to be next to each other at a certain time but which have been deployed at different times and locations (‘chance pairs’). A possible disadvantage of this method of choosing pairs of drifters that happen to be next to each other, in contrast to truly deployed pairs, is that the dispersion may be biased due to convergence zones of the surface currents. In order to minimise this effect, the pairs were chosen by tracing all combinations of two trajectories backward until the maximum initial distance (d0) was found (Ollitrault et al., 2005). If, in contrast, the search is forward in time, there is, on average, a small convergence during the first hours or days of the pairs. We selected pairs that had data for at least 32 days. The number of pairs increases for larger maximum initial distances d0. With a maximum initial distance of 5 km we found 916, with 10 km 1646 with 25 km 3696, and with 50 km 5854 pairs. The present study will focus on a comparison of the relative dispersion obtained by seeding particles in a (numerical determined) velocity field from a model with a horizontal resolution of 1/4°. In order to maximise the number of pairs that nevertheless have a distance comparable to the grid size, we have chosen to use the 3696 pairs that qualify in terms of the 25 km initial-distance criterion. From Fig. 1(a) showing the selected pairs, it is evident that these are not uniformly distributed in the ocean. There are some oversampled areas: the northern hemisphere in general, the Equatorial regions, and the highly energetic areas characterised by a prevalence of coherent structures (Rupolo, 2007b), viz. the Gulf Stream and Kuroshio regions and the Falkland Shelf. This bias is an important factor that has to be taken into account when interpreting data. The different relative dispersions, (see Eq. (2) in the next section for a definition), are presented in Fig. 2. Note that the time evolution of the dispersion wobbles somewhat for an initial distance of 5 or 10 km while, increasing the number of pairs, the curves become smoother for initial distances of 25 and 50 km. Our choice of a

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Fig. 2. The relative dispersions as a function of time for the maximum initial distance (d0) of 5, 10, 25 and 50 km.

Fig. 1. The surface-drifter and TRACMASS-trajectory pairs started in the same positions and times with a maximum initial distance of 25 km and followed for 32 days. Each pair has a red and a blue trajectory. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

25-km initial distance has the advantage of making use of a large number of pairs, but is at the same time too large for separating the dispersion into two regimes, one for distances much smaller than the external Rossby radius of deformation (10 to 100 km in the World Ocean), the other for much larger scales (Ollitrault et al., 2005).

2.2. The TRACMASS trajectories The simulated trajectories in the present study are computed with the TRACMASS trajectory model. The trajectories are calculated off-line, i.e. after the OGCM has been integrated and the velocity fields have been stored. This makes it possible to calculate many more trajectories than would be possible on-line (i.e. simultaneously with the OGCM run). TRACMASS has been applied to many different general circulation models, both for the ocean and the atmosphere. The original feature of the method is that it

solves the trajectory path through each grid cell with an analytical solution of a differential equation which depends on the velocities on the grid-box walls. The scheme was originally developed by Döös (1995) and Blanke and Raynaud (1997) for stationary velocity fields and hereafter further developed by Vries and Döös (2001) for time-dependent fields by solving a linear interpolation of the velocity field both in time and in space over each grid box, this in contrast to the Runge–Kutta methods (Butcher, 2008) where the trajectories are iterated forward in time with short time steps. The TRACMASS code has been further developed over the years and used in many studies (Döös et al., 2004; Jönsson et al., 2004; Engqvist et al., 2006; Döös et al., 2008). The ocean velocity field has been obtained using the NEMO Ocean/Sea-Ice general circulation model (Madec, 2008). The model configuration employed, ORCA025, has a tripolar grid with a 1/4° horizontal grid resolution of 27.75 km at the equator. The grid is finer with increasing latitudes, yielding a 13.8 km resolution at 60° S and 60°N. The spatial resolution is hence only eddy permitting, not truly eddy resolving. The water column is divided into 75 levels, with a grid spacing ranging from 1 m near the surface to 200 m at the bottom. The configuration is described by Barnier et al. (2006) who demonstrated its capacity for representing strong currents and eddy variability, even compared with more highly resolved models. The Laplacian lateral isopycnal tracer diffusion coefficient is 300 m2/s at the equator and decreases poleward in proportion to the grid size. The atmospheric forcing is derived by Brodeau et al. (2010) and based on the ERA40 reanalysis. The surface salinity has been restored with a damping coefficient of 26.9257 mm/day. The velocity fields were time-mean archived every 3 and 6 h as well as 5 days in order to test the sensitivity of the off-line trajectory calculations on the velocity-forcing frequency. The model is fully prognostic, which means that it does not use assimilated data (neither satellite altimetry nor in situ temperature or salinity), and therefore can differ substantially from observations in some areas. On the other hand, there are no sources or sinks in the model domain and the dynamics are hence consistent with the momentum equations. Pairs of TRACMASS trajectories have been initiated in the same positions and at the same times as the surface drifters and followed for 32 days and are presented in Fig. 1(b), where each pair has a red and a blue trajectory. The trajectories are integrated off-line using the model velocity fields without adding any parameterisation of sub-grid scale processes except those of the OGCM itself. A smaller degree of separation of the numerical pairs is evidenced by a comparison with the surface-drifter results in

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Fig. 1(a). This is even more clearly borne out by Fig. 3(a) and (b), where the separation distance of each surface-drifter pair is presented as a function of time. These figures clearly show that the model trajectories have too weak a separation rate, probably due to the lack of spatial scales below the model grid size (0.25°). This motivates us to test, via stochastic terms, two simple parameterisations of sub-grid-scale processes. Both add an effect of diffusion

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to the trajectories themselves and are explained in detail in Appendix B and are presented in Figs. 1(c) and 3(c). 3. Dispersion of surface drifters and model trajectories The relative dispersion of the pairs, defined by Eq. (2) in Appendix A, is shown in Fig. 4(a) for both surface drifters and TRACMASS trajectories with and without added sub-grid parameterisations. The relative dispersion of the modelled trajectories is clearly too weak in comparison with the dispersion of real drifter pairs. Although the relative dispersion of the modelled trajectories is 34% stronger when updating the velocity fields every 6 h instead of every 5 days, it is still only around half of the surface drifters (Table 1). Trajectories were also computed with velocity fields updated every 3 h. This did not, however, lead to any changes of the Lagrangian statistics compared to the 6-h case, which might be explained by (i) the atmospheric forcing of the OGCM was only updated every 6 h, (ii) a 1/4° resolution is not high enough to generate these time scales. Single particle statistics have also been computed in order to evaluate how realistic the individual model trajectories are. The average absolute dispersion, defined by Eq. (3) of all the individual trajectories belonging to the pairs is shown in Fig. 4(b). The absolute dispersion of the TRACMASS trajectories is close to that of the surface drifters when the velocity fields are updated every 6 h and is only 6% too weak in comparison with the surface drifters. This also shows that the 5-day mean velocities are clearly to smooth in order to advect the particles far enough, which results in an absolute dispersion that is 32% too weak. The average power spectrum of all the drifter and TRACMASS trajectory velocities has also been computed and is shown in Fig. 4(c). It demonstrates a better agreement between TRACMASS and drifter trajectory velocities when the modelled trajectories are driven by the 6-hourly velocities rather than with the 5 day velocities even if it is still too weak. It is interesting to notice that NEMO manages to simulate some of the diurnal variability with a small peak when the velocities are updated every 6 h, but would require a tidal potential in order to simulate the amplitude correctly. In Table 1 we show for the different data sets the values of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P  2 and the correlation time standard deviation ru ¼ i ðui  uÞ R1 scales T L ¼ 0 RðtÞdt for the zonal and meridional components and their semi-sum. R(t) is the velocity-correlation function, values are computed from single trajectories, and Lagrangian time scales are estimated using the first zero-crossing technique, viz. considering the integral of the velocity autocorrelation function until the first time when it reaches the zero value. Real-drifter data sets with an initial separation of 5 and 25 km display similar values, manifesting an anisotropy with zonal values slightly larger than the meridional ones.

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The relative dispersion rates and the single particle statistics have also been computed for the TRACMASS trajectories with the two simple sub-grid parameterisations derived in Appendix B. The results are shown in Fig. 4 and Table 1 together with the other data. The coefficients of the two sub-grid parameterisations have been tuned in order to match the magnitude of the relative dispersion of the surface drifters after 32 days. The ‘‘diffusion’’ parameterisation, which adds a stochastic term to the trajectory in accordance with Eq. (5), attains realistic relative dispersion rates for AH = 2500 m2/s. By calibrating the amplitude of the extra horizontal turbulent velocities u0 , v0 (cf. Appendix B), the turbulence

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Table 1 Mean values with standard deviations of the Lagrangian correlation time (TL), the square root of the eddy kinetic energy (r) and the relative (DR) and absolute (DA) dispersion after 32 days. See the text for definitions. Statistics are computed from the single values of each trajectory of the surface drifter pairs with a maximum initial distance (d0) of 5 km and 25 km as well as those for the TRACMASS trajectories with velocity updates every 5 days and 6 h without sub-grid parameterisation. The two last columns pertain to TRACMASS trajectories with 6-hourly velocities with ‘‘turbulence’’ and ‘‘diffusion’’ sub-grid parameterisations. Surface drifters

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2.2 ± 1.3 1.9 ± 1.1 2.1 ± 1.0

2.2 ± 1.1 1.9 ± 1.0 2.0 ± 0.9

4.0 ± 1.5 3.8 ± 1.5 3.9 ± 1.1

2.7 ± 1.6 2.4 ± 1.4 2.5 ± 1.2

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Lagrangian correlation time. In other words, realistic particle-separation rates are obtained using a large diffusivity value, but at the cost of totally changing correlation properties and energy partitioning in the frequency domain. This is recognised from Fig. 4(c), where the total velocity spectrum is displayed for each trajectory data set. A more realistic representation of the unresolved scales would require a higher order sub-grid parameterisation. Griffa (1996) showed that a random walk does not describe the turbulent dispersion behaviour of ocean tracers and that a better quantitative agreement can be reached using an Ornstein–Uhlenbeck process. This work has been refined by Pasquero et al. (2001) who observed that the Ornstein–Uhlenbeck model assumes Gaussian velocity distributions, while the ocean displays exponential-like tails associated with the mesoscale dynamics (Bracco et al., 2000). Those tails are common to 2D turbulent flows (Bracco et al., 2000b) and to Lagrangian trajectories in an oceanic eddy-resolving model (Bracco et al., 2003). Based on these similarities Pasquero et al. (2001) built a family of two-process stochastic models that provided a better parameterisation of turbulent dispersion in rotating barotropic flows. Berloff and McWilliams (2002), Berloff et al. (2002) also explored in detail the issue of (horizontal) stochastic parameterisations for oceanic flows, suggesting an alternative model to the one of Pasquero et al. (2001). It is therefore to be expected that the zeroth-order Markov process used in the present study will not provide a good representation of the surface drifters. The relative dispersion rates can hence only be tuned to match the total value at a particular moment. The shape of the model-velocity power spectrum without parameterisations is therefore more realistic in its shape even if too weak. 3.2. Anisotropy

Fig. 4. Time evolution of the mean dispersion of the drifter- and model-trajectory pairs. (a) The relative dispersion, (b) the absolute dispersion (c) total velocity power spectrum. Surface drifters in black and TRACMASS trajectories in colour. In red and blue with no sub-grid parameterisation but with velocity fields updated every 5 days (red) and every 6 h (blue). In green with added extra turbulent velocity (u0 , v0 ) and in purple with added Laplacian diffusion. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

parameterisation reaches also realistic values. The absolute dispersion is not much affected when the diffusion parameterisation is added, but gives far too high values for the ‘‘turbulence’’ sub-grid parameterisation. The modelled trajectories with added diffusion/ turbulence also manifest values of the residual velocities which are similar to real data, but with decidedly smaller values of the

In Fig. 5, we have decomposed the relative dispersion (Eq. 2) in the zonal and meridional directions in order to judge the ability of the modelled trajectories to be anisotropic with respect to the zonal/meridional directions. The zonal and meridional components as well as the total relative dispersion are shown in Fig. 5(a), with the surface-drifter results in black and the TRACMASS trajectories advected with 6-hourly updated velocities in red. The dispersion is initially isotropic, but after a few days becomes clearly anisotropic with a stronger zonal than meridional component for both the surface drifters and the modelled trajectories. The spatial distribution of the anisotropy of the surface drifter pairs has also been investigated by repeating the same calculations as the ones shown in Fig. 5(a) but for different regions. There is a

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during the first 8 days. The 5-day simulation has even a reversed isotropy with a stronger meridional component while the trajectories with velocities updated every 6 h have a much more realistic evolution compared to the surface drifters. This might be due to that the trajectory pairs without sub-grid parameterisation and with time smoothed velocities tend to remain at a distance from each other near the radius of deformation and that the trajectories with sub-grid parameterisation have an added diffusion, which is isotropic in itself.

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4. Grid-resolution dependence and 3D Lagrangian numerical trajectories 4.1. Grid resolution In order to investigate the Lagrangian-dispersion dependence on model resolution, we have also calculated trajectories using a coarser model. So far we have used the velocity field from the NEMO model on the ORCA025 grid, with a resolution of approximately 1/4°. We will now compare these results from the same model, but with a grid 4 times as coarse. ORCA1 has a horizontal resolution of around 1° and a lateral mixing coefficient of 5000 m2/s instead of the 300 m2/s used in the ORCA025 configuration. Apart from the horizontal resolution and the diffusion coefficients, the model is set up and forced with the same fields as the more highly resolved ORCA025 version. The trajectories are released, as in previous experiments, at the same points and times as the surface drifters and shown in Fig. 6. The Lagrangian integrations are made without sub-grid parameterisation. The relative dispersion after 180 days is 509 km with the eddypermitting model and 228 km with the coarse-resolution model. Using the coarse grid the dispersion DR is hence less than half of that resulting from the fine grid. 4.2. Three-dimensional Lagrangian trajectories

clear anisotropic dispersion in favour of the zonal direction at all latitudes except in the Arctic Ocean (north of the polar circle), where the dispersion is isotropic. The zonal component of the relative dispersion after 32 days ðD2x ð32 daysÞÞ is in global average 2.5 stronger than the meridional one ðD2y ð32daysÞÞ. The strongest anisotropy is found in the Southern Ocean with a ratio between the zonal and meridional dispersion of 8.3 and in the Equatorial region with a ratio of 5.8, which can be explained by the dominating zonal currents in these two regions. The same ratio is at mid-latitudes is typically between 1 and 2. The Equatorial region is, hence, clearly contributing to the global average anisotropy but despite its overrepresentation and zonal currents not the only reason. A possible explantation of the zonal amplification of the dispersion could instead be attributable to the b effect, a process which takes place where the bottom is flat (LaCasce and Speer, 1999). The time evolution of the proportionality between the meridional and zonal components of the relative dispersion is shown in Fig. 5(b) for both the surface drifters and all the different TRACMASS simulations with and without sub-grid parameterisations. The TRACMASS trajectories capture the anisotropy well, even if it is weaker than that associated with the observations. The dispersion is originally isotropic, which occurs when the distance between the pairs is not greater than the Rossby radius (LaCasce and Ohlmann, 2003; LaCasce, 2008). However, the anisotropy grows as the distance between the particles increases, but faster for the surface drifters. The modelled trajectory pairs with velocity fields updated every 5 days and 6 h show significant differences

The surface drifters remain fixed in the vertical since they are drogued at a depth between 12 and 18 m. In the present study all the TRACMASS trajectories have hence so far been calculated with a depth integrated velocity between 12 and 18 m in order to simulate the surface drifters. Here, as in the previous sections, we will initiate them at the same locations and times as the surface drifters, but let them flow passively with the 3D-velocity fields.

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Fig. 5. The zonal and meridional components of the relative dispersion.

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Fig. 6. The time evolution of the relative dispersion in km of the TRACMASS trajectories with no sub-grid parameterisation using two different horizontal grids: 1/4° eddy-permitting resolution (red curve) and 1° coarse resolution (blue curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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They will hence qualify as true ‘‘Lagrangian’’ trajectories here and the impact is shown in Fig. 7. The separation rates are very similar during the first 20– 30 days. After this, the dispersion rate of the 2D trajectories locked in the surface layer is weaker than that of the Lagrangian 3D trajectories permitting vertical displacement. The dispersion rate of the 3D trajectories is thus clearly stronger than that of the 2D trajectories. After 180 days the relative dispersion is 15% stronger for the 3D trajectories than for the 2D trajectories. It is, however, only 6% stronger after 32 days, which is the time period used in the present study to compare the model trajectories with the surface drifter results.

5. Discussion and general conclusions In this study we have compared the relative dispersion of surface drifters and model trajectories computed with the trajectory model TRACMASS and the general circulation model NEMO. Surface-drifter pairs were obtained by screening the Global Drifter Program data set and joining original and ’chance’ pairs. The dispersion rate of the surface drifters is compared to that of simulated numerical pairs released in the same positions as the observed ones. The relative dispersion (DR) of the model trajectories is about half that associated with the surface drifters. This can be explained by the insufficient spatial resolution of the model grid and by the absence of tidal forcing. The absolute dispersion, on the other hand, is only slightly weaker for the modelled trajectories compared to the surface drifters, which is an indication that the model-current velocities have the right strength. The relative dispersion of the modelled trajectories increased when the velocities were updated every 6 h instead of every 5 days. Trajectories were also computed with velocity fields updated every 3 h. This did not, however, lead to any changes of the Lagrangian statistics compared to the 6-h case, which might be explained by the atmospheric forcing of the OGCM only being updated every 6 h and that an eddy-permitting model is still to coarse to generate these time scales. It would be interesting to in the future incorporate a tidal potential in the OGCM in order to generate tidal currents, which will require that the velocities be stored at least every three hours, this in order to capture the semi-diurnal M2 component. The model-resolution dependence was tested by repeating the trajectory calculations with a four times as coarse a model grid,

550

which resulted in twice as weak a relative dispersion (DR). In the future the integrations of a truly ‘‘eddy-resolving’’ model (1/12° horizontal resolution) will be used instead of the ‘‘eddy-permitting’’ model (1/4°) employed in the present study. This will make it possible to explore to what extent the relative dispersion would increase and how much less sub-grid parameterisation would be needed. These OGCM (ORCA12) integrations are, however, not yet available. It is important to distinguish between the sub-grid parameterisation of the horizontal mixing of the Lagrangian trajectories with that of the OGCM itself. The velocity fields are simulated by the OGCM with a Laplacian lateral isopycnal diffusion and a corresponding coefficient of 300 m2/s at the equator that decreases poleward in proportion to the grid size. The mixing is hence included in a trajectory as it progresses and changes its tracer properties by contact with its surroundings (Koch-Larrouy et al., 2008). It is therefore possible to argue that adding a component to this velocity field would be redundant, since the mixing has already been included in the OGCM. The modeled trajectory solutions always include implicit large-scale diffusion due to along-trajectory changes of temperature and salinity and by the models parameterisation of turbulent mixing in the momentum equations. These trajectories in themselves do not, however, explicitly represent sub-grid-scale turbulent motion since they are passively advected by the model-simulated currents with no sub-grid scales apart from the linear interpolations of the velocities between the grid points. We have therefore tested the effects of two different subgrid parameterisations of low order on the trajectories, this in order to evaluate the effects of the absent small scales. One technique consists of adding random horizontal turbulent velocities u0 , v0 to the horizontal velocities U, V from the OGCM. The other method is based on instead adding a random displacement (isotropic on a circular disk) to the trajectory position. By adjusting the corresponding eddy-diffusion coefficients it has been possible to tune the model trajectories to have a similar relative dispersion as the surface drifters. It is, however, important to note that we can not evaluate or validate the OGCM itself when we add a sub-grid parameterisation to the model trajectories, since we are adding something which is not included in the OGCM. Both sub-grid parameterisations were tuned so that the total amplitude of the relative dispersion of the model trajectories was similar to that of the drifter trajectories. The two sub-grid parameterisations are, however, too simple to yield a correct simulation of Lagrangian properties such as the time-scales and the variance of the eddy kinetic energy. In order to compare the model trajectories with the surface drifters we locked the model trajectories at a depth of 15 m, which is the depth of the drifter drogues. The trajectories are hence not

450 400

j

350 300

(x1,y1 )´

250

(x1 ,y1 ) Latitude

Relative dispersion (km)

500

200 150

R

(x1 ,y1 )

(x1,y1 )´

r xd

yd

100 50 0

j-1 0

20

40

60

80 100 Days

120

140

160

180

Fig. 7. The relative dispersion in km as a function of time of the 2D (red) and 3D (blue) TRACMASS trajectory pairs initiated at the same times and positions as indicated in previous figures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(x0 ,y0 )

i-1

Longitude

i

Fig. 8. The added displacement due to diffusion. Left panel shows in blue the original trajectory and in light blue the changed one due to the added displacement. Right panel zooms in on the added random displacement. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 9. Example of one pair of surface drifters in dark and light blue. TRACMASS trajectories started in the same positions without sub-grid parameterisation in orange and red, and with sub-grid parameterisation in purple and green. The yellow disk shows the starting position. The mesh is the ORCA025 C-grid on which TRACMASS has been applied. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

strictly Lagrangian since that would require a complete passive advection by the model 3D-velocity fields. After 180 days the relative dispersion is 15% stronger when the trajectories are freely advected with the 3D velocity field compared to only a few percent during the first month. This can be explained by the fact that when a 2D trajectory enters a region with horizontal convergence it cannot ‘‘escape’’ with a vertical displacement; instead it remains stuck in this region. This effect also applies to the surface drifters and it would hence be of interest to compare surface-drifter properties with those of 3D floats in the deep ocean. A further development of the present work would be to undertake statistical investigations of the drifter pairs for different geographical regions or dynamical regimes. Trajectories in an energetic regime characterised by coherent structures might e.g. be more difficult to model than those in a region with a more quiescent turbulent background. This could be investigated by dividing the trajectories into different classes as undertaken by Rupolo (2007a,b). In the present work we have seen how the relative dispersion, for both the surface drifters and the model trajectories, is anisotropic with a dominating zonal component. This might be explained by the b effect, which tends to favour zonal dispersion where the bottom is flat. The anisotropy with respect to f/h should also be studied in the future on the basis of the present combination of surface drifters and model trajectories. Hereby it would be possible to evaluate the capacity of the OGCMs to simulate the anisotropy correctly, and whether the preferential spreading along contours of (barotropic) f/h (LaCasce and Bower, 2000) is correctly represented in the OGCMs. The absolute dispersion, which uses single-particle statistics, would here be a better measure of the anisotropy. Another possible extension of the present study would be to investigate the separation rate at distances smaller than the Rossby radius, viz. on the grid-size scales. This would, however, have the drawback of much fewer pairs of surface drifters. The choice made in the present study of a rather extended initial distance between the drifters enabled us to use a large number of pairs, but the initial separation is at the same time too large for dividing the dispersion into two regimes (Ollitrault et al., 2005), with one region for pair distances much smaller than the external Rossby radius of

deformation (25 and 100 km in the World Ocean) and another for much larger scales. Acknowledgements This work has been financially supported by the Bert Bolin Centre for Climate Research and by the Swedish Research Council. The NEMO integrations were done on the computer cluster Ekman at the Centre for high performance computing in Stockholm. The authors furthermore wish to extend their thanks to Peter Lundberg as well as two unknown reviewers for constructive comments. Appendix A. Definitions of relative and absolute dispersion The mean position of the m-th trajectory cluster is defined as

xm i ðtÞ 

N 1X xn;m ðtÞ; N n¼1 i

ð1Þ

where t is the time, N the total number of trajectories contained in the m-th cluster and i the spatial coordinate index (i.e. the zonal, meridional or vertical position of the n-th trajectory xn;m ðtÞ). In i the present study, we only consider the horizontal dispersion. The vertical dispersion is, however, an important measure of the vertical mixing in the ocean but beyond the scope of the present study. The relative dispersion, often denoted the r.m.s of the separation distance of particles, is defined as the mean-square displacement of the trajectories relative to the time-evolving mean position:

D2R ðtÞ 

M X N X 2  2 1 1 X xn;m ðtÞ  xm : i i ðtÞ M N  1 m¼1 n¼1 i¼1

ð2Þ

The absolute dispersion is defined in the same way, but relative to the initial position of the cluster:

D2A ðtÞ 

M X N X 2  2 1 1 X xn;m ðtÞ  xm ; i i ðt 0 Þ M N  1 m¼1 n¼1 i¼1

where t0 is the initial time of the trajectory.

ð3Þ

K. Döös et al. / Ocean Modelling 39 (2011) 301–310

In the present study we have considered pairs of trajectories, for which definition (2) holds with N = 2.; in this case D2R ðtÞ ¼ ðx1 ðtÞ  x2 ðtÞÞ2 =2.

B.1. Turbulence parameterisation The sub-grid turbulence parameterisation, which was introduced by Döös and Engqvist (2007), adds random horizontal turbulent velocities u0 , v0 to the horizontal velocities U, V from the OGCM. These are the instantaneous OGCM velocities U, V, which are updated every 3 h, 6 h or 5 days in the present study and from which the trajectories are calculated when no sub-grid parameterisation is added. The turbulent velocities u0 , v0 are added to each horizontal grid-cell wall for each trajectory calculation and changed at every trajectory time step (Dt). The trajectories are hence calculated with the TRACMASS code as it is, but with a velocity field, u = U + u0 , that is somewhat shaken, resulting in a stirring of the trajectory particles. The amplitude of the random turbulent velocity is proportional to the velocity of the circulation model velocity U so that u0 = RU. R is a random number uniformly distributed between a and a, with standard deviation equal to (3  a2)1/2. This amplitude was set to the constant a = 1 in Döös and Engqvist (2007), but has here been tuned to obtain a relative dispersion similar to that of the surface drifters. The amplitude was furthermore adapted so that the trajectory time step (Dt) in the TRACMASS code did not affect the results, which was obtained by setting a = b/(Dt)1/3. The best fit for an amplitude of the relative dispersion similar to that of the surface drifters was obtained for b = 160. Using this scheme in practice we add a random noise with a standard deviation on the order of (3  a2)1/2ru, where ru is the Lagrangian standard deviation of the unperturbed velocity field. B.2. Diffusion parameterisation This isotropic diffusion parameterisation adds a random displacement to the trajectory position in order to incorporate the non-resolved sub-grid scales. The scheme was first introduced in TRACMASS by Levine (2005). The horizontal advection–diffusion equation is

ð4Þ

where AH is the horizontal eddy viscosity coefficient. Eq. 4 is equivalent (see e.g. Rupolo (2007a)) to the zeroth-order Markov process:

dxi dw : ¼ U i þ ð2  AH Þ1=2  dt dt

ð5Þ

Here the stochastic impulse is represented by the increment  1=2 dg ¼ x2d þ y2d ¼ ð2  AH Þ1=2 dw, where w is a Wiener process with a zero mean and a second order moment hdw  dwi=2  dt The corresponding Gaussian distribution is

 2  x þ y2d 1 Pðxd ; yd ; DtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  d : 2AH Dt 2pAH Dt

ð6Þ

Fig. 8 illustrates the displacements added to the original position of the particle after each time step of length Dt. The added random walk for the particles in the two directions is given by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AH Dt logð1  q1 Þ cosð2pq2 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yd ¼ AH Dt logð1  q1 Þ sinð2pq2 Þ;

xd ¼

ð7Þ ð8Þ

where qn are random numbers between 0 and 1. The added displacement in the horizontal plane will hence be

309

ð9Þ

with a standard deviation

RH ¼

Appendix B. Parameterisations of sub-grid scale processes

@P @P @P þU þV ¼ r  ðAH rPÞ; @t @x @y

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rH ¼ x2d þ y2d ¼ DtAH logð1  q1 Þ; pffiffiffiffiffiffiffiffiffiffiffi DtAH :

ð10Þ

This implies that about 70% of the particles will be within this distance from their original positions and that the new velocity field will be characterized by an extra standard deviation on the order of (AH/dt)1/2, where dt is the Lagrangian integration time step. In Fig. 9 we show an examples of trajectories released at the same point and integrated with and without the sub-grid scale parameterisation described here. It is important to distinguish between this sub-grid parameterisation of the horizontal mixing of the Lagrangian trajectories with that of the OGCM itself. The velocity fields are simulated by the OGCM with a Laplacian lateral isopycnal diffusion and a corresponding coefficient of 300 m2/s at the equator, which decreases poleward in proportion to the grid size. The mixing is hence included in a trajectory as it progresses and changes its tracer properties by contact with its surroundings (Koch-Larrouy et al., 2008). One could therefore argue that adding a component to this velocity field would be redundant since the mixing has already been included in the OGCM. These trajectories in themselves do not, however, explicitly represent sub-grid scale turbulent motion since they are passively advected by the model-simulated currents with no sub-grid scales apart from the linear interpolations of the velocities between the grid points. On the other hand, Lagrangian trajectories are the equivalent of integrating Eq. 4 with no effects of velocity scales under the grid scale, which clearly must exist in the real ocean. Furthermore when Eq. 4 is discretised and integrated in an OGCM for the tracers it will also include the numerical diffusion, which is not the case for our trajectories since they are exact analytical solutions to the velocity fields in TRACMASS. It is however important to note that we can only evaluate or validate the OGCM itself when we do not add any sub-grid parameterisation to the model trajectories. Appendix C. Volfango Rupolo Volfango Rupolo died on 5 April 2010. This article was meant as the first in a series of papers on Lagrangian dispersion by the present authors. The next article would have been on how realistically the model trajectories are represented by the different trajectory classes developed by Rupolo (2007b) using both single-particle statistics and pairs. References Barnier, B., Madec, G., Penduff, T., Molines, J., Treguier, Anne-Marie, Le Sommer, J., Beckmann, A., Biastoch, A., Böning, C., Dengg, J., Derval, C., Durand, E., Gulev, S., Remy, E., Talandier, C., Theetten, S., Maltrud, M., Mcclean, J., De Cuevas, B., 2006. Impact of partial steps and momentum advection schemes in a global ocean circulation model at eddy permitting resolution. Ocean Dynam. 56, 543– 567. Barnier, B., Brodeau, L., Le Sommer, J., Molines, J.-M., Penduff, T., Theetten, S., Treguier, A.-M., Madec, G., Biastoch, A., Bning, C., Dengg, J., Gulev, S., Bourdall Badie, R., Chanut, J., Garric, G., Alderson, S., Coward, A., de Cuevas, B., New, A., Haines, K., Smith, G., Drijfhout, S., Hazeleger, W., Severijns, C., Myers, P., 2007. Eddy-permitting ocean circulation hindcasts of past decades. CLIVAR Exchanges 12 (3), 810. Berloff, P., McWilliams, J., 2002. Material transport in oceanic gyres. Part II: Hierarchy of stochastic models. J. Phys. Oceanogr. 32, 797–830. Berloff, P., McWilliams, J., Bracco, A., 2002. Material transport in oceanic gyres. Part I: Phenomenology. J. Phys. Oceanogr. 32, 764–796. Blanke, B., Raynaud, S., 1997. Kinematics of the Pacific Equatorial Undercurrent: a Eulerian and Lagrangian approach from GCM results. J. Phys. Oceanogr. 27, 1038–1053. Blanke, B., Arhan, M., Madec, G., Roche, S., 1999. Warm water paths in the equatorial Atlantic as diagnosed with a general circulation model. J. Phys. Oceanogr. 29, 2753–2768.

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