Four numerical simulations are used to characterize the impact of submesoscale circulations on surface Lagrangian statistics in the northern Gulf of Mexico over 2 months, February and August, representative of winter and summer. The role of resolution and riverine forcing is explored focusing on surface waters in regions where the water column is deeper than 50 m. Whenever submesoscale circulations are present, the probability density functions (PDFs) of dynamical quantities such as vorticity and horizontal velocity divergence for Eulerian and Lagrangian fields differ, with particles preferentially mapping areas of elevated negative divergence and positive vorticity. The stronger the submesoscale circulations are, the more skewed the Lagrangian distributions become, with greater differences between Eulerian and Lagrangian PDFs. In winter, Lagrangian distributions are modestly impacted by the presence of the riverine outflow, while increasing the model resolution from submesoscale permitting to submesoscale resolving has a more profound impact. In summer, the presence of riverine-induced buoyancy gradients is the key to the development of submesoscale circulations and different Eulerian and Lagrangian PDFs. Finite-size Lyapunov exponents (FSLEs) are used to characterize lateral mixing rates. Whenever submesoscale circulations are resolved and riverine outflow is included, FSLEs slopes are broadly consistent with local stirring. Simulated slopes are close to −0.5 and support a velocity field where the ageostrophic and frontogenetic components contribute stirring at scales between about 5 and 7 times the model resolution and 100 km. The robustness of Lagrangian statistics is further discussed in terms of their spatial and temporal variability and of the number of particles available.
It has become apparent in recent years that stirring at the ocean surface is critically impacted by processes occurring at the so-called submesoscales. The term submesoscale applies broadly to the spatial and temporal scales at which circulations have a Richardson and a Rossby number of order O(1) (Thomas et al. 2008; McWilliams 2016), usually composed between 100 m and a few kilometers and hours to few days. Submesoscale circulations are the expression of the weakening of the geostrophic constraint, in turn intrinsically linked to the nonlocality of advective processes in the ocean, and are responsible for local stirring. In other words, submesoscale processes govern the relative separation of tracers for distances up to several kilometers (McWilliams 2008; McWilliams et al. 2009; Koszalka et al. 2009; Molemaker et al. 2010; Zhong et al. 2012; Zhong and Bracco 2013; Poje et al. 2014; McWilliams 2016).
The key role played by submesoscale processes in determining the horizontal mixing properties of near-surface tracers became evident between April and July 2010, when the northern Gulf of Mexico (GoM) was severely impacted by the largest oil spill in history, the Deepwater Horizon disaster, that released about 7 × 108 kg of oil in the open waters (Joye et al. 2011). During the spill, synthetic aperture radar (SAR) images of the surface oil and aerial photos revealed the presence of numerous submesoscale frontal structures that contributed to the oil patchiness (Walker et al. 2011). The pervasiveness of submesoscale structures in the northern GoM was further confirmed by the Grand Lagrangian Deployment (GLAD) conducted in August 2012 (Poje et al. 2014) and by the more recent Lagrangian Submesoscale Experiment (LASER; http://carthe.org/laser/) carried out in February 2016. In both experiments, a uniquely large number of surface drifters were released over the De Soto Canyon, with the goal of quantifying the role of local, submesoscale processes in the near-surface relative dispersion. These experiments provide spatial and temporal context for this investigation.
Models, in conjunction with observations, have demonstrated the key impacts of submesoscale processes on the distribution of material within and across the mixed layer (Koszalka et al. 2009; D’Asaro et al. 2011; Lévy et al. 2012; Zhong and Bracco 2013; Shcherbina et al. 2013; Poje et al. 2014; Haza et al. 2016, Beron-Vera and LaCasce 2016; Zhong et al. 2017). At the surface, for instance, submesoscale fronts that emerge alongside energetic mesoscale currents and strong density gradients concentrate material at their boundary (Zhong et al. 2012; Gula et al. 2014; Poje et al. 2014), while submesoscale motions are responsible for the leakage of tracers from mesoscale eddies (Haza et al. 2016). Here, we adopt a modeling perspective to investigate how the dynamics of Lagrangian particles depend on the strength of the submesoscale circulation. The strength of the submesoscale circulation is allowed to vary by using different grid resolutions and freshwater forcings.
This paper is the third in a series investigating Eulerian and Lagrangian implications of resolving, to different degrees, submesoscale circulations in the Gulf of Mexico. It complements the Eulerian analysis in Barkan et al. (2017a, hereinafter Part I) and the investigation of the Mississippi River jet region in Barkan et al. (2017b, hereinafter Part II). It focuses on 2 months, February and August, coinciding with LASER and GLAD, respectively. From a dynamical perspective, February is characterized by deep mixed layer depth and the strongest (in a climatological sense) heat fluxes into the ocean, and therefore the largest conversion of available potential energy (APE) into kinetic (EKE), while August is within the secondary summer peak in terms of submesoscale activity and APE conversion because of the intense lateral density gradients created by the freshwater inflow from the Mississippi River system. A detailed characterization of the seasonal cycle of submesoscale activity in the GoM can be found in Luo et al. (2016), while the energy conversion is discussed in Part I.
All quantities analyzed can be easily computed from large drifter deployment such as GLAD or LASER. We note, however, that for the most part in the following we use variables computed from the Eulerian model fields interpolated at the particle positions. Only in the last section do we present results using standard interpolation techniques based on Lagrangian quantities alone, with the objective of assessing sampling requirements for accurate Lagrangian estimations.
2. Model setup and simulation details
We consider a set of Regional Ocean Modeling System (ROMS) integrations representative of February (winter) and August (summer) conditions, at horizontal resolutions of 500 [high-resolution (HR) case] and 1500 m [low-resolution (LR) case] and in the presence/absence of riverine inputs (With-River and No-River solutions, respectively). Both sets of integrations support the development of submesoscale circulations but to a different degree. All runs are forced by a daily varying climatology of QuikSCAT winds (Risien and Chelton 2008) and CORE (Large and Yeager 2009) monthly heat fluxes. Because of the climatological nature of the wind field, near-inertial oscillations are not captured in those simulations. The freshwater atmospheric forcing fields are from HOAPS (Andersson et al. 2010) and have monthly frequency. In LR With-River, the Mississippi River system outflow is included based on monthly mean volume flux data (Dai and Trenberth 2002), while in the HR case the daily volume flux for the year 2010 from the USGS is imposed. ROMS uses a third-order, upwind, advection scheme that is equivalent to a fourth-order, centered, advection scheme augmented by hyperdiffusion based on the grid velocity values (Shchepetkin and McWilliams 2005). The numerical hyperviscosity is tuned to have a grid (hyper) Reynolds number of 6 with coefficients that vary with grid resolution. Further details on the ROMS setup and implementation can be found in Part I.
Lagrangian tracers are released at the surface in all simulations every other day from the second to the sixteenth of each month considered for a total of 16 deployments; 28 870 particles are initially uniformly distributed and shown as white dots in Fig. 1. They occupy an area where the water column is deeper than 50 m and have an initial separation of 0.02°. We purposefully avoided the shallow shelf where, even at 500-m horizontal resolution, we cannot resolve the mixed layer baroclinic deformation radius and the modeled mixed layer extends to the bottom during a large portion of the year (Luo et al. 2016; Part I). Additionally, 4 circles of 40 km in diameter are each seeded with 5019 tracers at an initial particle–particle distance of 0.005°. Finally smaller dots of 10-km diameter and a number of particles varying from 12 to 299 organized in triplets are randomly deployed within the circles, over scales closed to those sampled by the GLAD/LASER drifters. In all cases the tracer are confined to the ocean surface and are advected offline by the ROMS hourly averaged horizontal velocity field using the larval transport Lagrangian model (LTRANS), version 2b (North et al. 2011), for 2 weeks. Most statistics are, however, computed only on the first 10 days after release because of the significant number of tracers leaving the domain by then (>40%) and the focus on submesoscales. We verified that at the resolutions considered, the use of hourly average fields for the particle advection introduces an error small enough that it does not affect the Lagrangian statistics. More frequent temporal sampling would be, however, required at a resolution higher than ~500 m.
We present an investigation of the model representation of two quantities that impact lateral mixing, such as vorticity and surface horizontal convergence, and their Lagrangian mapping as function of resolution and presence/absence of freshwater fluxes for winter and summer separately. Both quantities can be calculated from drifter deployments whenever the number of drifters per unit area at the time of release is large enough, as in the case of GLAD and LASER.
We verified that the transport statistics vary with season but are independent of resolution. The representation of absolute dispersion is indeed nonlocally controlled, as postulated by Taylor (1921), and determined by the mesoscale circulations, which are well captured at both resolutions (not shown; see, e.g., Zhong and Bracco 2013). This is not the case for relative dispersion, as highlighted later in this work. We then focus on the temporal and spatial variability of the Lagrangian statistics, quantifying how they depend on the particle number for the most realistic simulation. Cluster C1 is in the northwest corner of the De Soto Canyon; C2 is influenced by the Mississippi River jet and in February is fully contained in the river plume; C3 is immediately offshore of the 50-m isoline to the east of the domain, within the reach of the Atchafalaya outflow; and C4 is offshore, over waters deeper than 2000 m and in an area often occupied by the Loop Current (LC). Figure 2 shows the distribution of the circles superposed on the mean salinity field or its gradient and the location of the particles in one of the deployments considered 3 days after release. Low-salinity anomalies associated with the riverine outflow extend farther offshore in summer, as to be expected, given the seasonal cycle of the riverine input (Cardona et al. 2016) but with much smaller gradients at the jet interface than in February. Particles released in February generally cover larger distances than in August over the same amount of time, in response to stronger winds. Additionally, tracers seeded in the Mississippi jet in February tend to remain in it and are carried effectively offshore, as seen in Fig. 2b; the Eulerian and Lagrangian dynamics of the river jet region in winter is evaluated in detail in one of the companion papers (Part II).
a. Lagrangian PDFs in the northern GoM: Seasonal dependence and riverine influence
Figure 3 shows the vorticity field normalized by the Coriolis parameter (ζ/f) on 9 February superposed with the tracers released 3 days earlier in the simulations with and without river inflow, in the HR and LR runs. While the position of the Loop Current and of the mesoscale eddies differ between the integrations, all four runs are characterized by a large number of submesoscale eddies and fronts that generate a nonuniform particle distributions (Zhong et al. 2012).
In winter, submesoscale eddies and fronts are numerous, as in other regions of the World Ocean (Mensa et al. 2013; Callies et al. 2015; Luo et al. 2016), because of the deep mixed layer depth. In this season, surface buoyancy gradients and vorticity filaments can intensify and undergo frontogenesis (McWilliams et al. 2009). Secondary circulations then develop in the vertical in the form of upwelling on the warmer side of the front and downwelling on the colder side in response to the increased strain rate (Capet et al. 2008), and submesoscale mixed layer instabilities (MLIs) contribute to the generation of small eddies by extracting energy from the mixed layer (Molemaker et al. 2005; Boccaletti et al. 2007; Fox-Kemper et al. 2008; Thomas et al. 2008). Submesoscale structures form everywhere, including around and within the LC and its detached loop eddies, except in the shallowest portions of the continental shelf. In LR, submesoscale instabilities are only partially resolved, fronts are wider (as to be expected), and ζ/f values are reduced compared to the HR counterpart. In all simulations, independently of the representation of the river outflow and of resolution, particles quickly converge into submesoscale fronts and cyclonic eddies. In the absence of freshwater input, the mixed layer is on average deeper, and results in an amplification of the conversion rates of available potential energy into eddy kinetic energy through submesoscale processes (Luo et al. 2016; Part I).
Differences in the representation of the submesoscale circulations and their impact on the Lagrangian statistics in the four simulations are quantified through probability density functions (PDFs) of ζ/f, shown in Fig. 4, and horizontal velocity divergence defined as , shown in Fig. 5.
From the PDF analysis, the following emerges:
For both fields, PDFs are indicative of energetic surface submesoscale circulations that manifest in positively skewed vorticity distributions and negatively skewed velocity divergence PDFs (Shcherbina et al. 2013). The Pearson’s first skewness coefficient or mode skewness, defined as (mean − mode)/STD, where STD is the standard deviation of the distribution, is best suited to compare asymmetry in PDFs for which the mean and the mode are significantly different. Mode skewness is as large as 0.77 (0.80) and −0.43 (−0.47) for the Lagrangian vorticity and horizontal divergence distributions 9 days after release in HR in February in the With-River (No-River) cases, respectively. For the Eulerian histograms, the mode skewness is only 0.41 (0.43) for vorticity and −0.12 (−0.17) for divergence.
The root-mean-square (RMS) of the ζ/f and −∂w/∂z PDFs also depends strongly on resolution, increases substantially from LR to HR, and is slightly higher in the absence of riverine forcing. For example, the RMS of vorticity for the Eulerian fields vary in With-River (No-River) from 0.52 (0.56) in LR to 0.88 (0.94) in HR and for the Lagrangian ones at day 9 after release from 0.64 (1.07) in LR to 2.93 (3.70) in HR.
Between the resolutions of 1.5 and 0.5 km, the strength of the submesoscale circulations increases such that Eulerian and Lagrangian distributions in winter differ significantly and consistently across both With-River and No-River cases, as tracers sample predominantly regions of elevated convergence and positive vorticity. Differences between Eulerian and Lagrangian statistics are larger for vorticity than for horizontal velocity divergence (Fig. 6).
At a given resolution, differences in surface statistics between simulations with and without riverine inflow are small but consistent across all deployments. Stronger horizontal velocity convergence and higher positive vorticity values are sampled by the particles released in the absence of riverine fluxes.
The convergence of Lagrangian tracers in submesoscale fronts and eddies happens quickly during the winter season. Differences in Eulerian and Lagrangian sampling are visible within few hours. By the end of day 3, all particle PDFs have approached their asymptotic state.
Figure 6 quantifies the differences in the respective Eulerian and Lagrangian distributions in the With-River simulations through the Kullback–Leibler divergence DKL measure (Kullback and Leibler 1951). Often used in information theory, DKL(E||L) is also known as relative entropy of E with respect to L and is defined as . It measures the expectation of the logarithmic difference between, in our case, the Eulerian E and Lagrangian L probabilities with i being the PDFs classes or, in other words, the amount of information lost when L is used to approximate E. It should be noted that DKL(E||L) does not equal DKL(L||E).
The DKL(E||L) increases proportionally to the strength of the submesoscale circulations and therefore with resolution for all quantities considered. In February, the growth of DKL(E||L) over time is quicker in the HR runs, where a plateau is generally reached by the end of the third day of particle integration, and slower in LR, where a steady increase can be seen for as long as a week.
The vorticity fields ζ/f in the middle of August, superposed with tracers released 9 days prior, are shown for the four runs in Fig. 7. In August, the vorticity field is weaker, and submesoscale eddies and fronts are mostly absent but for the 500-m With-River integration, where APE conversion and submesoscale instabilities are fueled by the abundant riverine inflow (Luo et al. 2016; Part I). The summer weakening of the submesoscale circulations is due to the smaller amount of available potential energy stored in the shallower mixed layer compared to winter and is confirmed by observations in the Gulf Stream region (Callies et al. 2015). However, the limited depth of the summer mixed layer (about 20 m on average in the Gulf of Mexico for the area with water depth greater than 500 m; Part I) causes the mean mixed layer deformation radius to be a few kilometers offshore and a few hundreds of meters over the shelf and smaller in the With-River case than in the absence of riverine inflow. The mixed layer deformation radius is therefore barely resolved in HR and mostly unresolved in LR (Part I). For the With-River case, where APE conversion is fueled by the riverine forcing, even at 500-m horizontal resolution it is not possible to determine if the weakening of the submesoscale circulations in August compared to February is due to seasonality or to insufficient resolution.
The resulting PDFs have smaller RMS and skewness than in winter (Fig. 8, shown only for vorticity). Differences between Eulerian and Lagrangian statistics are not significant independently of resolution except for the HR With-River case, where particles preferentially sample regions of high positive vorticity and horizontal velocity convergence. For example, the RMS of the Eulerian and Lagrangian vorticity distributions in the HR run are 0.34 and 0.35 in the No-River case and increase to 0.51 and 0.92 in the With-River run, while the addition of the riverine forcing modifies the mode skewness of the Lagrangian distributions 9 days after deployment from 0.28 to 0.38 (and from −0.05 or a nearly symmetric distribution to −0.23 for divergence).
The DKL(E||L) is significantly different from zero in the HR With-River simulation but nearly one order of magnitude smaller than in February for vorticity and 4 to 5 times smaller for divergence (Fig. 6). HR With-River is the only simulation in which submesoscale circulations energized in summer by the riverine forcing are resolved well enough to impact stirring, as shown also by the particle distributions in Fig. 7.
b. Lateral mixing analysis
Finite-size Lyapunov exponents (FSLEs; Artale et al. 1997) are used to quantify lateral tracer mixing. FSLEs measure local stirring by computing the average time for particle pairs to separate from a given scale δ to a scale αδ, where α >1 is a parameter. In this work, FSLEs are calculated using α = 1.2 and the fastest crossing methods, following Poje et al. (2010). A sensitivity investigation on the dependence of the results on the choice of α, on the deployment strategy, and on the role of turbulent subgrid motions using a simple random walk model (Visser 1997) was performed. Varying α from 1.05 to 1.4, or releasing particles in s-shaped triplets (Poje et al. 2014) or in spiraling shapes, did not change the FSLE slopes whenever the overall density of particles was kept constant. Adding a random walk to the particle advection to account for subgrid motions, on the other hand, modified the curves as discussed further below.
FSLEs allow for separating nonlocal from local stirring and, in the case of the ocean, mesoscale- from submesoscale-dominated mixing (d’Ovidio et al. 2009; Haza et al. 2012; Poje et al. 2014). Beron-Vera and LaCasce (2016) have shown that FSLE slopes (or analog quantities based on distance) may be altered in the presence of strong inertial motions, such as those characterizing observed drifter motions in the GoM in summer. Quantities that use time as an independent variable, on the other hand, are not affected by inertial oscillations. Here, we use daily climatological winds (daily means build from the QuikSCAT scatterometer dataset), and inertial oscillations do not develop in our integrations. Their presence and impact, however, should be accounted when comparing FSLE curves computed in this work with those based on observed trajectories and reported in Poje et al. (2014).
In a turbulent field with an energy power spectrum E(k) proportional to k−5/3, typical of surface quasigeostrophic flow at scales larger than the first deformation radius (Blumen 1978), the diffusion coefficient depends on the particle separation following the Richardson law K(D) ∝ D4/3 (Richardson 1926) that translates into an FSLE slope λ ~ δ−2/3. Such an exponent was recovered during the GLAD experiment (Poje et al. 2014), during which approximately 300 surface drifters were released in the northern Gulf of Mexico.
For flows with an energy spectrum E(k) ~ k−2, as commonly found in submesoscale-resolving simulations where ageostrophic velocities and frontogenesis are accounted (e.g., Capet et al. 2008; Badin 2013; Zhong and Bracco 2013), and in some oceanic observations (Callies and Ferrari 2013), the FLSE slope will decay as λ ~ δ−0.5. An exponential particle separation, theoretically expected for the enstrophy cascade regime [E(k) ~ k−3] translates in a flat plateau in FLSE. A slope λ ~ δ−2 for separation scales much larger than the characteristic length scale of the velocity field indicates diffusive behavior. Finally, in simulations of oceanic flows it is common to encounter FSLE slopes close to −1 for large separation distances because of the shear dispersion associated with particles moving along separate current systems (Iudicone et al. 2002).
Our Gulf of Mexico simulations support a submesoscale controlled k−2 energy spectrum at the ocean surface in all HR cases in winter and in the presence of riverine forcing in summer (Fig. 9). In the No-River case in August, the kinetic energy power spectrum is clearly steeper than k−2, and close to k−3, at large and intermediate scales. For asymptotic consistency, the associated FLSE exponents are close to −0.5 at scales composed between 10 and 70 km in winter and in HR With-River in summer (Fig. 10). We note, however, that the estimation error is not negligible even for deployments with more than 28 000 particles. Best fitting on single releases instead of the mean of 8 or extending the fit to 100 km introduce uncertainty in the scaling exponent.
In February, best fits are comprised between −0.52 and −0.62 in HR With-River and −0.58 and −0.70 in HR No-Rivers, making it difficult to clearly distinguish between regimes. Slopes are close to −0.5 in both solutions for particles released over water depths greater than 500 m, and differences are enhanced over water depths between 50 and 200 m, with flatter curves in the With-River case. This change in slope over the shelf may be linked to the presence of the Mississippi and Atchafalaya plumes that contribute to efficiently transport particles released over shallow waters offshore (Part II), but we cannot exclude that differences in the mesoscale circulations also play a role. The length of the integrations does not allow for separating between those processes in a definitive way.
In August, the best fit to the With-River curve in Fig. 10 is −0.50. The exponent is smaller than found for GLAD drifters, where it followed closely −2/3 (Poje et al. 2014). Regardless of the uncertainty in determining the FLSE exponent, the locality of processes responsible for lateral mixing whenever submesoscale circulations are present and properly resolved is unequivocal.
At 1500-m horizontal resolution, all curves are flatter than in HR in February and are comparable to the 500-m case in the No-River case in summer. For separation distances below 5 km, HR FLSE curves flatten, consistent with previous modeling experiments that have shown that the impact of local processes on stirring is captured only at scales 5–10 times larger than the nominal model resolution (Zhong and Bracco 2013; Bracco et al. 2016).
The inclusion of subgrid turbulent diffusion in the form of a random walk with diffusivity varying from 0.01 to 1 m2 s−1 modifies the FSLE slopes for small separation distances in agreement with previous analysis by Haza et al. (2012). The greater the diffusivity, the closer the exponent is to −1 for small distances (Fig. 11, shown only for With-River).
c. Spatial and temporal variability of PDFs and FSLEs
We focus next on the HR With-River integration, and we examine the spatial and temporal variability of horizontal velocity divergence PDFs and of FSLEs. Our goal is to establish how robust these statistics are for varying location and time of deployment. We consider the four particle clusters shown in Fig. 1, each containing 5019 tracers, or about 5 times the total number of drifters released during LASER, and again eight releases per season, with starting dates between 2 and 16 February or August. As shown in Fig. 12, 3 days after tracers are released, the Lagrangian PDFs are statistically indistinguishable for clusters C1, C3, and C4 in February and for clusters C1, C2, and C3 in August, whenever the standard error is accounted. In winter, the particles inside the Mississippi plume tend to sample lower divergence values, while in summer the tracers in the offshore circle are the least affected by submesoscale dynamics. In February, the standard error (twice the standard deviation) of the eight deployments is generally greater than in summer, indicating a higher temporal dependence that reflects the variability in the atmospheric forcing fields. Deployments in the De Soto Canyon display the largest standard error, as tracers can be effectively stretched along the submesoscale fronts generated along the riverine-induced salinity gradients or may sample the shelf area to the west, where surface divergence values are generally lower. In summer, the PDFs for the offshore cluster (C4) reflect the weakening of the submesoscale structures away from the region of influence of the riverine outflow.
In February, FSLEs reflect the abundance of submesoscale circulations with slopes composed between −0.5 and −0.7 at scales from 5 to 80 km in all cases (Fig. 13). Large separation scales are not well sampled by circle C4, as many particles exit the domain within a few days, while C1 slopes tend to be slightly steeper than all others for intermediate separations. In August, curves flatten at scales less than 8 km compared to winter deployments but maintain slopes indicative of local dynamics from 8 to 80 km, except for the offshore cluster. Such flattening is an indication of weak submesoscale activity, also echoed in the smaller tails of the PDFs. We cannot exclude that such flattening partially results from the limited horizontal resolution, given the shallow mixed layer depth in this season (see Part I for a detailed discussion of resolution requirements). In the case of circle C4, the plateau in the FSLE curve reflects theoretical expectations in the case of an enstrophy cascade regime [E(k) ~ k−3]. The standard error quantifies how different FSLE slopes may be for particles deployed at the same location, at most 16 days apart.
Figure 14 presents relative dispersion (mean square separation) curves calculated using all particle pairs with initial separation distance of 1 km for the circles during the winter season. All four curves confirm the submesoscale local stirring. The largest error is found for the C4 circle and is linked to the fast decreasing number of couples within the model domain for increasing time. Particle separation for pairs in C2, some of which are affected by the Mississippi jet, displays a steeper slope than all other relative dispersion curves, consistent with the presence of a ballistic separating process in the core of the jet, as identified in Part II.
d. Dependency on number of drifters
Next, we explore the dependence of the PDFs and lateral mixing statistics on the number of drifters. In doing so we also evaluate the biases associated with constructing PDFs from Lagrangian data alone, as done from releases of “real” drifters when the Eulerian velocity field is not known and dynamical quantities have to be determined from the evolution of the particle positions. We consider horizontal velocity divergence PDFs (but similar results are obtained for vorticity) and FLSEs for the circles, for February and August, and we evaluate their statistical convergence, varying the number of particles initially located in the smaller dots of 10 km of diameter. Each dot contains randomly distributed particles’ triplets, defined as any combination of three particles such that the distance between any two is less than 2 km. The number of particles varies from 12 to 299, with the highest triplets’ density being about 33 500 at deployment time and of the order of 100 000 2 days after release. In the following, we still consider up to 100 “dots” distributed randomly inside the larger circle and eight deployments separated by 2 days in the FSLE calculation. While appreciating that statistics are much larger than what can be obtained in a field campaign, we want to highlight the stringent statistical convergence requirements and uncertainty in the slope determination. We use, however, only 10 dots and one deployment time for the PDF analysis, as we verified that the error in the PDFs does not decrease in any significant way by increasing the number of dots above 10, and even 1 dot of 299 particles is sufficient to reproduce the “true” Eulerian distribution and capture the main characteristics of the histograms. We remind the reader that about 300 drifters were deployed in triplets during GLAD and in each phase of the LASER experiment.
Divergence distributions (Fig. 15) are calculated from the particle locations using the triplets and the least squares method introduced by Molinari and Kirwan (1975) and Okubo et al. (1976) and applied by Niiler et al. (1989), Paduan and Niiler (1990), and Righi and Strub (2001), among others. Triplets whose aspect ratio is less than 0.07 are removed from the PDF calculation; if not removed, the tails of the PDFs are strongly overestimated as shown in the figure. The aspect ratio (AR) is calculated from the variances along minor and major axes as STDminor/STDmajor using the eigenvalue decomposition, and the 0.07 threshold has been chosen empirically as the best one in reproducing the Eulerian “truth” in an exploration with AR from 0.05 to 0.2. Overall, the Lagrangian interpolation (LI) reproduces well the “true” PDFs obtained using the divergence at the particle location calculated from the Eulerian modeled fields (EI) at least up to 3 days (the daily average for day 2 past deployment is shown in the figure). Deterioration is found past day 4, such that the error in the PDF RMS becomes as large as 27% in single dots’ estimates. One reason for such deterioration is that triangles become too large and the estimation of vorticity and divergence less informative. In both seasons, the standard error is smaller to that found for deployments at different times (Fig. 12), indicating that temporal more than spatial variability controls the uncertainty. Overall, the error in the PDFs is small enough to conclude that even one deployment of 299 particles is sufficient to recover the general shape of the distributions once triplets with very low AR are neglected.
We compare FSLEs computing the sum of variance (SOV) defined, independently of the statistics considered, as
where yo(i) is the FSLE value in the class i for the maximum density case, and y(i) refers to the coarser case. In the following, statistical convergence is arbitrarily defined as SOV less than or equal to 0.5. Figure 16 presents, in the top panels, the SOV evolution in February and August. For 100 simultaneous deployments, a minimum of 60 particles per dot is required in winter and only 15 are required in summer to achieve statistical convergence in the FSLE slope for most locations. Additionally, in winter convergence is decisively faster for clusters more directly impacted by the Mississippi–Atchafalaya outflows (C2 and C3) and slower in the De Soto Canyon and offshore. All curves represent the mean of eight deployments of 100 dots for a given number of particles; each deployment lasts 15 days, and the error bars are twice the STD around the mean, shown for circle C2 only for clarity. The standard error associated with the temporal spread around one location is larger than the differences because of the initial particles’ position, again indicating that the temporal variability of the system, here simulated using climatological winds so likely underestimated, has to be accounted when planning any experiment or monitoring activity, especially during winter. Incidentally, the SOV calculated for the horizontal velocity divergence PDFs instead of FSLEs for the same particles and deployments indicates convergence independently of the number of tracers considered (SOV for PDFs is always <0.4).
Finally, we evaluate convergence for groups of 27, 45, 99, 197, and 299 particles and a number of dots varying from 1 to 40 during one deployment only (bottom panels in Fig. 16; shown for C1 only). The SOV is calculated with respect to the FSLE curve obtained using 299 particles and 100 dots. For a given deployment, convergence can be attained with approximately 20 dots, independently of location for particle numbers greater than 45, further confirming that the spread in winter is dominated by the differences between releases at different times.
4. Discussion and conclusions
We performed an investigation of basic Lagrangian statistics in high-resolution regional simulations of the northern Gulf of Mexico. Using integrations at two resolutions, 500 and 1500 m, with and without a representation of the riverine input, we have shown that Eulerian and Lagrangian probability density functions of dynamical quantities modified by submesoscale circulations, such as vorticity and horizontal velocity divergence, differ whenever the submesoscale circulations are sufficiently developed. Lagrangian tracers preferentially concentrate in regions of elevated positive vorticity and horizontal convergence, so that the mean distribution of the same fields over the general area encompassing the particles differs. In winter, differences develop quickly and are slightly greater in the absence of freshwater fluxes from the river system, in response to a more energetic submesoscale field (Part I). In summer, and at the horizontal model resolution used in this work, the presence of riverine forcing is a necessary ingredient for the submesoscale circulations to develop and achieve a strength sufficient to modify the Lagrangian statistics.
FSLE curves further provide a quantification of the (modeled) impact of submesoscale circulation on lateral mixing. The exponents in our runs capture the locality of the near-surface dynamics in winter and summer for the HR With-River case and in winter only in HR No-River. In this respect, they are in broad agreement with the observational results from the GLAD campaign (Poje et al. 2014); they do not provide, however, an exact match to the observed exponent and are generally flatter in our simulations. Furthermore, we found that in the modeled runs FSLE slopes can vary between approximately −0.5 and −0.7, depending on location and the time of deployment. As a result, it is not possible to robustly distinguish between −0.5 and −2/3 exponents or to unequivocally reconstruct the slope of surface eddy kinetic energy spectra from the FSLE curves alone.
Furthermore, we explored the geographical and temporal variability associated with Lagrangian PDFs and FSLE curves. In winter, the variability is high across deployments only 2 days apart, and the Mississippi plume region emerges as different from any other. More details about the dynamics at play in this area can be found in Part II. In summer, offshore statistics are strongly influenced by the patchy spreading of freshwater anomalies from the riverine input; submesoscale dynamics dominate the Lagrangian statistics wherever the freshwater anomalies are present, and quasigeostrophic balances control the spreading of tracers in their absence, in agreement with the energy spectra observed in the Gulf Stream region (Callies et al. 2015).
Finally, we evaluated the requirements to achieve convergence in the Lagrangian distributions and FSLE, finding, unsurprisingly, that convergence in winter is attained more slowly than in summer. About 300 particles are sufficient to capture the vorticity or horizontal velocity divergence PDF characteristics using triplets with a large enough aspect ratio and the least squares method to derive dynamical quantities from the particle positions over time. Such a number of particles are comparable to the drifters deployed during GLAD and in each phase of the LASER experiment. The convergence in FSLE slopes, on the other hand, requires a much larger number of particles (or dots, in our case). In February, the variability across deployments at different times is especially large and generally greater than across deployments at different locations. The interior of the Mississippi jet in winter and the far offshore waters in summer, where the riverine fluxes do not penetrate, are exceptions to this, as submesoscale circulations are generally weaker and do not impact the statistics as strongly as in other regions.
In reference to future work, we note that the riverine outflow estimates used here were for 2010. Given that the generation of submesoscale circulations in the northern GoM is affected by the amount of the freshwater input, and the Mississippi River outflow is highly variable on interannual AU8 scales (e.g., Cardona et al. 2016), further investigations are needed to assess how the variability of the riverine system is reflected in the Eulerian and Lagrangian flow statistics.
This research was made possible by a grant from the Gulf of Mexico Research Initiative through the CARTHE Consortium. Data are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) online (at https://data.gulfresearchinitiative.org; https://doi.org/10.7266/N7CN720V, https://doi.org/10.7266/N7PK0DK2, https://doi.org/10.7266/N7F18X4S, https://doi.org/10.7266/N75H7DQ5, https://doi.org/10.7266/N7JS9NVS, and https://doi.org/10.7266/N79885FW). RB, AFS, JCM, and JMM are further supported by ONR N000141410626. The Extreme Science and Engineering Discovery Environment (XSEDE) provided support for computing.
This article has companion articles which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-17-0035.1 and http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-17-0040.1