Editorial Type:
Article
Article Type:
Research Article

Drifter Deployment Strategies to Determine Lagrangian Surface Convergence in Submesoscale Flows

H. M. Aravind aDepartment of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts

Search for other papers by H. M. Aravind in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0003-3544-7251
,
Helga S. Huntley bDepartment of Mathematics, Rowan University, Glassboro, New Jersey

Search for other papers by Helga S. Huntley in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0002-9225-4483
,
A. D. Kirwan Jr. cSchool of Marine Science and Policy, University of Delaware, Newark, Delaware

Search for other papers by A. D. Kirwan Jr. in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0002-2103-2146
, and
Michael R. Allshouse aDepartment of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts

Search for other papers by Michael R. Allshouse in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0001-7656-6746
Online Publication:
26 Jan 2024
Print Publication:
01 Jan 2024
DOI:
https://doi.org/10.1175/JTECH-D-22-0129.1
Page(s):
95–112
Free access

Abstract

Surface convergence in the ocean is associated with accumulation of buoyant pollutants as well as with vertical transport that is important to biological activity. Such surface convergence regions are marked by a high dilation rate, i.e., the finite time Lagrangian average divergence. Dilation-rate observations are most easily derived from the change of the area encompassed by a drifter swarm over time. The technological advances that have enabled the deployment of large numbers of drifters in a single experiment have raised new questions about optimal deployment strategies for extracting dilation-rate information with acceptable accuracy and as much spatial coverage as possible. Using a submesoscale-resolving operational model of the Mediterranean Sea, we analyze synthetic trajectories of drifter polygons to evaluate the impact of the number of drifters and their initial separation on the accuracy of the resulting dilation-rate estimates. The results confirm that estimates improve as the circumradius of the polygon decreases and as more drifters are added, but with only a marginal improvement for drifter polygons containing more than four drifters. Moreover, GPS positions obtained from drifters in the ocean are subject to uncertainty on the order of 2–50 m, and when this uncertainty is taken into account, an optimal circumradius can be identified that balances uncertainty from position measurements with that from the area approximations.

Significance Statement

Locating regions of convergence over a finite time interval on the ocean surface can help in pollution mitigation, locating biological hotspots, and even search-and-rescue operations. Finite time convergence can be quantified using the dilation rate, but it is hard to measure in the ocean. Hence, we present a method to estimate the dilation rate using trajectories of drifters, which are instruments widely used by oceanographers during field experiments to understand the local flow features. We show that even though the drifter-based dilation rates are prone to error as a result of a finite number of drifters and limited GPS accuracy, the estimates locate around 90% of the strongest convergent features in our model.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher's Note: This article was revised on 8 February 2024 to correct a distortion of the bin widths in Fig. A2 that was present when originally published. In addition, ORCID identifiers were added for coauthors Kirwan, Huntley and Allshouse.

Corresponding author: Michael R. Allshouse, m.allshouse@northeastern.edu

Abstract

Surface convergence in the ocean is associated with accumulation of buoyant pollutants as well as with vertical transport that is important to biological activity. Such surface convergence regions are marked by a high dilation rate, i.e., the finite time Lagrangian average divergence. Dilation-rate observations are most easily derived from the change of the area encompassed by a drifter swarm over time. The technological advances that have enabled the deployment of large numbers of drifters in a single experiment have raised new questions about optimal deployment strategies for extracting dilation-rate information with acceptable accuracy and as much spatial coverage as possible. Using a submesoscale-resolving operational model of the Mediterranean Sea, we analyze synthetic trajectories of drifter polygons to evaluate the impact of the number of drifters and their initial separation on the accuracy of the resulting dilation-rate estimates. The results confirm that estimates improve as the circumradius of the polygon decreases and as more drifters are added, but with only a marginal improvement for drifter polygons containing more than four drifters. Moreover, GPS positions obtained from drifters in the ocean are subject to uncertainty on the order of 2–50 m, and when this uncertainty is taken into account, an optimal circumradius can be identified that balances uncertainty from position measurements with that from the area approximations.

Significance Statement

Locating regions of convergence over a finite time interval on the ocean surface can help in pollution mitigation, locating biological hotspots, and even search-and-rescue operations. Finite time convergence can be quantified using the dilation rate, but it is hard to measure in the ocean. Hence, we present a method to estimate the dilation rate using trajectories of drifters, which are instruments widely used by oceanographers during field experiments to understand the local flow features. We show that even though the drifter-based dilation rates are prone to error as a result of a finite number of drifters and limited GPS accuracy, the estimates locate around 90% of the strongest convergent features in our model.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher's Note: This article was revised on 8 February 2024 to correct a distortion of the bin widths in Fig. A2 that was present when originally published. In addition, ORCID identifiers were added for coauthors Kirwan, Huntley and Allshouse.

Corresponding author: Michael R. Allshouse, m.allshouse@northeastern.edu
Keywords: Mediterranean Sea; Ageostrophic circulations; Convergence/divergence; Lagrangian circulation/transport; Buoy observations; Numerical analysis/modeling

1. Introduction

Recent field campaigns have established the key role of horizontal divergence in submesoscale dynamics (Mahadevan et al. 2020a; Esposito et al. 2021). Moreover, convergent regions assist with locating pollution hotspots (Cózar et al. 2021), algal blooms (Pitcher et al. 2010), phytoplankton communities (Hernández-Carrasco et al. 2018), and the aggregation regions of any type of flotsam (D’Asaro et al. 2018). Thus, identification of their locations and quantification of their intensities are of considerable importance both for scientific investigations and operational applications.

Unfortunately, achieving these two objectives has proved challenging. In the context of accumulating buoyant materials, as well as of sustained vertical displacement (Aravind et al. 2023), the processes act over finite time intervals. Hence, the Eulerian divergence, which provides instantaneous snapshots of the flow field, is not ideal. A Lagrangian measure is more appropriate, as it follows a parcel over time. One such measure is the dilation rate, which has been shown to capture clustering regions (Huntley et al. 2015) as well as regions where large vertical displacements are observed over finite time intervals (Aravind et al. 2023).

The dilation rate is defined as the Lagrangian average of horizontal divergence, i.e., the horizontal divergence of velocity averaged over time along a fluid trajectory (Huntley et al. 2015). It can be computed either from Lagrangian data or by integrating along synthetic trajectories from high-resolution Eulerian velocity fields. Coastal high-frequency radar systems (Kim 2010) as well as ship-mounted acoustic Doppler current profilers (Shcherbina et al. 2013) and X-band radars (Berta et al. 2020) provide high-resolution velocity fields, but these measurements are generally highly localized, severely limiting the extent of any synthetic trajectories. For Lagrangian data, drifters have proven the most useful, since dye patches are extremely hard to track accurately (Filippi et al. 2021).

A variety of methods have been developed to estimate divergence and other kinematic properties using drifter swarms (Molinari and Kirwan 1975; Okubo et al. 1976; Kawai 1985; Gonçalves et al. 2019). To compute dilation rate, methods based on the change of the area of a patch encompassed by a drifter swarm (Molinari and Kirwan 1975) are the most appropriate, since the Lagrangian averaging is built in. For these methods, a minimum of three drifters is required per swarm to make a single estimate. Because such a grouping maximizes the number of data points and minimizes resource expenditure, drifter triads have been popular for estimating kinematic properties (Berta et al. 2016; Dräger-Dietel et al. 2018; Huntley et al. 2019; Berta et al. 2020; Huntley et al. 2022). Other studies, though based on least squares algorithms, suggest a larger number of drifters for greater accuracy in instantaneous divergence estimates (Ohlmann et al. 2017; Tarry et al. 2021; Essink et al. 2022). Few drifter experiments have access to more than 20–30 drifters (Lumpkin et al. 2017) for the entire deployment. So, a trade-off has to be made between increased accuracy and increased sampling coverage within such resource constraints.

Drifter-based estimates of dilation rate are susceptible to error, primarily because the area of an evolving polygonal approximation of the fluid patch is not the same as its exact area. Increasing the number of drifters used to discretize the fluid patch boundary is an obvious solution, but limited resources mean that fewer patches can be tracked, resulting in fewer dilation-rate measurements. Quantifying the effect of boundary discretization is especially important for dilation-rate estimations in comparison with divergence and other kinematic properties because of the time intervals considered. Dilation rates are usually calculated over a few hours or days (e.g., Hernández-Carrasco et al. 2018; Aravind et al. 2023), which are at least an order of magnitude longer than the time intervals over which divergence estimation is usually performed (e.g., Tarry et al. 2021; Essink et al. 2022). Over these longer time intervals, processes like filamentation (Mahadevan 2006) can result in significant deformation of fluid patches, which may not be captured accurately using a finite number of drifters. Additionally, the drifter locations obtained from the attached global positioning systems (GPS) units are accurate only up to 2–50 m (appendix; Centurioni 2018; Novelli et al. 2017), resulting in further error in the dilation-rate estimates due to position uncertainty. The impact of both sources of error on dilation-rate estimation is directly related to the size of the fluid patch whose boundary is approximated using a drifter polygon, i.e., a polygon with drifters deployed at its vertices.

Quantifying the interrelations of these error sources is formidable, and, to our knowledge, has yet to be addressed. Clearly these issues are critical to the overall goals to identify and quantify regions of Lagrangian divergence and convergence, hereinafter referred to as L divergence and L convergence, respectively. To address this deficiency, we investigate the effects of fluid boundary discretization and position uncertainty on drifter-based estimates of dilation rate for fluid patches of various sizes. A detailed error analysis on a real-time data-assimilative model of the western Mediterranean is performed, which allows us to obtain optimal parameters for drifter deployments to determine L convergence in submesoscale flows. In other words, the error analysis reveals an optimal deployment radius for a drifter polygon to obtain a dilation-rate estimate. The analysis treats drifters as massless particles that follow the fluid precisely. This assumption is justified since modern drifters are designed to minimize the effect of windage and waves (Niiler et al. 1995; Novelli et al. 2017; Poulain and Gerin 2019; Poulain et al. 2022).

The next section summarizes the methods used in our analysis, including precise definitions of the algorithms employed and a description of the high-resolution general circulation model used here. The results of the analysis are presented in section 3. This includes analysis of effects of boundary discretization and position uncertainty, and optimal deployment parameters. A discussion on how to generalize our method to other systems follows. The report concludes with a summary of our conclusions and some commentary on implications and further research.

2. Methods

a. Methods to calculate dilation rate

The dilation rate can be calculated in a number of ways depending on what data are available. Below we define the “pointwise dilation rate” Δ for an infinitesimal patch, the “area-averaged dilation rate” ΔR for a circular patch of radius R, the “polygonal dilation rate” ΔR,N for an N-gon approximating a circular patch of radius R, and the “drifter-based dilation rate” ΔR,N,σ that could be derived from imprecise drifter positions due to GPS uncertainty σ.

When smooth velocity fields are available, the dilation rate at a point can be computed using finite strain theory, as originally defined by Huntley et al. (2015). The pointwise dilation rate Δ is a measure of area expansion of an infinitesimal fluid patch initialized at a point, along its trajectory computed over the analysis time interval. Given a surface velocity field u = (u1, u2), the fluid trajectories are obtained by solving
dxdt=u(x,t),
where x = (x1, x2).
The two-dimensional flow map is defined as Ft0tf(x0) :=x(tf;x0,t0), which maps particles initialized at x0 = (x0,1, x0,2) at time t0 to their final positions at tf. The flow-map gradient tensor, ∇Ft0tf(x0)=∂xi/∂x0,j, which is a linearization of Ft0tf(x0), is then used to define the pointwise dilation rate as
Δ=log⁡(μ1μ2)tf−t0,
where μ1 and μ2 are the singular values of ∇Ft0tf(x0). Physically, μ1R and μ2R are the lengths of the semimajor and semiminor axes of the ellipse at time tf, onto which an infinitesimal circle (radius R Μ 1) at t0 is mapped after advection over a finite time interval tf − t0 (as presented in Fig. 1a). For the infinitesimal two-dimensional fluid patch, dilation rate is the Lagrangian average of divergence, since
Δ=log⁡(μ1μ2)tf−t0=log⁡(Af/A0)tf−t0=∫t0tf1A(τ)dA(τ)dτdτ∫t0tfdτ=∫t0tf∇⋅u[x(τ),τ]dτ∫t0tfdτ,
where A0 = πR2 and Af = πμ1μ2R2 are the areas of the fluid patch, A(t), at the initial time t0 and final time tf, respectively. Note that the dilation rate is zero for an incompressible flow. For positively buoyant particles at the ocean surface, however, it can be nonzero.
Fig. 1.
Fig. 1.

Schematic of the various areas considered for different dilation-rate calculations at the initial time t0 and final time tf: (a) evolution of an infinitesimal fluid patch into an ellipse, considered to measure the pointwise dilation rate, (b) evolution of a finite-sized fluid patch (blue) and an infinitesimal component (gray) used to compute the area-averaged dilation rate, (c) evolution of the area of a drifter polygon (beige) used to discretize the finite-sized fluid patch (blue), with the three dots representing positions of the drifter triad, considered to compute the polygonal dilation rate, (d) distorted area of the drifter polygon at tf (green) as a result of GPS position uncertainty, with the uncertainty zone for each drifter marked around it, used to compute the drifter-based dilation rate, and (e) observed area at tf (green) of the drifter polygon deployed at t0 (beige) to track the evolution of a finite-sized fluid patch (blue).

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

Although a finite-difference approach can be used to obtain the flow-map gradients (Farazmand and Haller 2012; Allshouse and Peacock 2015a), the relatively smooth oceanic velocity fields obtained from models allow us to employ the advected gradient approach (Huntley et al. 2015; Allshouse and Peacock 2015b) to more accurately compute the flow-map gradient. In conjunction with solving (1), we solve
ddt∂xi∂x0,j=∂xk∂x0,j∂ui∂xk
along the fluid trajectory, using the initial conditions
x(t0)=x0,∂x1∂x0,1=∂x2∂x0,2=1and∂x1∂x0,2=∂x2∂x0,1=0,
to obtain the four components of the flow-map gradient tensor ∂xi/∂x0,j.
Circular fluid patches approximated using drifter polygons during field experiments (R≳1), however, can no longer be assumed to deform into ellipses (Fig. 1b). Hence, we assume that the large fluid patch with initial radius R≳1 (and initial and final areas A0R and AfR, respectively) is composed of n infinitesimal patches, each with initial radius R ⪡ 1, as represented in Fig. 1b. Each infinitesimal patch has an initial area A0 and final area Aif, where i = {1, 2, 3, …, n}. The dilation rate for each component Δi is then computed using the advected gradient approach, thereby allowing us to define the area-averaged dilation rate for a finite-sized fluid patch of radius R≳1 by considering n infinitesimal components as
ΔR=1tf−t0log⁡(AfRA0R)≈1tf−t0log⁡(∑iAifnA0)=1tf−t0log⁡{1n∑iexp⁡[Δi(tf−t0)]},
where Δi=log⁡(Aif/A0)/(tf−t0) is the pointwise dilation rate for the infinitesimal fluid patch i, computed using the advected gradient approach. Here, Δi was computed at n uniformly distributed points inside each circular patch of radius R, such that n ≥ 13 and the maximum separation between adjacent points was 100 m. This resolution exceeds that of the underlying model grid, since structures in Lagrangian fields like dilation rate are more highly resolved than the velocity field they derive from, due to their integrative nature (Fang et al. 2020).
Using the advected gradient approach to measure the dilation rate is not feasible during field experiments, however, due to the unavailability of accurate and high-resolution velocity fields. When using drifters to measure dilation rate, we approximate the boundary of a circular fluid patch with radius R by an N-drifter polygon (Fig. 1c). The polygonal dilation rate for an N-gon can now be defined as
ΔR,N=log⁡(AfR,N/A0R,N)tf−t0,
where A0R,N and AfR,N are the areas of the N-sided polygons at the initial and final time instants, respectively. These areas are obtained in our study by computing synthetic drifter trajectories for the vertices of randomly oriented regular N-gons centered on the grid points and located on a circle of radius R, by solving (1) using a fourth-order Runge–Kutta scheme. Converged trajectories are obtained using a 60 s time step for integration and 3-hourly velocity fields from the operational model of the western Mediterranean.

The observed area of a drifter polygon (e.g., the area of the triangle in Fig. 1c) is not the same as the area of the circular fluid patch being approximated (area of the circle in Fig. 1c). A polygonal approximation of the boundary of the fluid patch also means that the observed final area of the polygon is likely different from the actual final area of the fluid patch, which can undergo substantial deformation over the lengths of time intervals considered for dilation-rate estimations. Hence, the polygonal dilation rate ΔR,N includes a discretization-based error that depends on how well the boundary of the fluid patch is resolved.

Drifter-based dilation-rate estimates are also susceptible to position uncertainty associated with the finite accuracy of the drifters’ GPS units. Here, we take the initial positions of drifters as given, but account for position uncertainty in their final positions (Fig. 1d). We consider 100 realizations of final drifter positions drawn from a normal distribution with a standard deviation σ, and compute the drifter-based dilation rate for each realization j as
ΔjR,N,σ=log⁡(Af,jR,N,σ/A0R,N)tf−t0,
where Af,jR,N,σ is the observed final area of the N-drifter polygon in the jth realization, which differs from AfR,N due to position uncertainty. The default choice of σ = 6 m for this study was made based on dock-side experiments to estimate drifter position uncertainty, as detailed in the appendix.

To summarize, four versions of dilation rate for a time interval tf − t0 = 24 h are computed on a regular 477 × 761 grid with a spacing of 500 m and compared. The pointwise dilation rate (Δ) is obtained from a synthetic trajectory computed at each grid point using the advected gradient approach. The area-averaged dilation rate ΔR at each grid point is computed by averaging pointwise dilation rates at 13 or more uniformly distributed points within a circle of radius R centered at the corresponding grid point, using (6). The polygonal dilation rate ΔR,N at each grid point is computed by tracking the evolution of a randomly oriented synthetic N-drifter regular polygon initialized on the circumference of the patch. Last, we compute drifter-based dilation rates, accounting for uncertainty in the final positions, (ΔR,N,σ) for 100 different realizations of the position error added to the final drifter positions, modeled as Gaussian white noise with standard deviation σ. All dilation-rate calculations are performed over a time interval of one day, which corresponds approximately to one inertial period for the region, since the model resolves submesoscale flow features.

b. Dilation-rate error definitions

To quantify the effects of discretization and GPS accuracy when computing drifter-based dilation rates, we consider three types of scaled dilation-rate error ϵ. First, to understand the effect of discretization (Fig. 1c), that is, approximating fluid patches as drifter polygons, we define
ϵD=|ΔR,N−ΔR|max⁡Ω|Δ|,
where the maximum is taken over the entire model domain Ω. The error due to position uncertainty that is in addition to the discretization error (Fig. 1d) is then isolated using
ϵP=|ΔR,N,σ−ΔR,N|max⁡Ω|Δ|.
Last, the total error due to both discretization and position uncertainty (Fig. 1e) is quantified as
ϵT=|ΔR,N,σ−ΔR|max⁡Ω|Δ|.
Spatial averages of the scaled errors defined in (9)–(11) computed over the full domain or specific subdomains are also of interest and are denoted using a bar (ϵ¯D, ϵ¯P, and ϵ¯T).

c. Ocean model

The model considered is a data-assimilative, real-time operational model of the western Mediterranean (Mahadevan et al. 2020b). This model resolves a rich set of dynamics at a high resolution and will allow this study to consider a wide range of submesoscale features simultaneously. The simulations were performed using the MIT Multidisciplinary Simulation, Estimation and Assimilation Systems (MIT-MSEAS) primitive equation ocean modeling system (Haley and Lermusiaux 2010; Haley et al. 2015). The computational domain spanned approximately 430 km × 267 km in the horizontal at a resolution of ∼500 m, with 70 optimized terrain-following vertical levels based on the SRTM15 + bathymetry (Tozer et al. 2019). Surface forcing for the simulations was based on the atmospheric fluxes from the ¼° NCEP Global Forecast System (Environmental Modeling Center 2003) product and tidal forcing on the high-resolution TPXO8-atlas (Egbert and Erofeeva 2002). Initial conditions for the simulation were obtained from the ∼(1/50)° Western Mediterranean Operational Prediction (WMOP) forecasting system (Juza et al. 2016; Mourre et al. 2018), which assimilates satellite sea surface temperatures and sea level anomalies, Argo temperature and salinity profiles, and surface currents from HF radar observations in the Ibiza Channel. The MIT-MSEAS simulation was corrected using observational data from Argo floats and, whenever possible, moorings.

In this domain, dense, salty water from the Mediterranean meets a jet of less dense water from the Atlantic entering through the Strait of Gibraltar, resulting in the formation of strong density fronts and basin-scale gyres (Tintoré et al. 1991; Viúdez et al. 1996). Signatures of the mesoscale gyres can be observed in the surface velocity field presented in Fig. 2a. Strong density fronts are expected near the periphery of these gyres, resulting in regions of convergence on the ocean surface, which can be seen in the horizontal divergence of the surface velocity field. Alternating regions of strong convergence and divergence are also observed north of Alboran Island (35°N, 3°W). The pointwise dilation-rate field in Fig. 2b presents features of similar length scales, but demonstrates that surface L convergence and L divergence in a Lagrangian sense is markedly different from the Eulerian perspective. Specifically, many regions that are instantaneously divergent in Fig. 2a are L convergent in Fig. 2b, including peripheries of the gyres and the region north of Alboran Island. These fluid patches are divergent at the initial instant, but experience greater convergence than divergence over the 24-h analysis window. The regions with the strongest 1% negative values of dilation rate have −1.144 ≤ Δ/f ≤ −0.5, which correspond to area contractions by a factor of 40–1000. The maximum value of the pointwise dilation rate (max⁡Ω|Δ|=8.5day−1) observed in this model is chosen as the dilation-rate scale, which is used when computing scaled errors.

Fig. 2.
Fig. 2.

(a) Modeled horizontal divergence of velocity ∇H ⋅ u, nondimensionalized using the Coriolis parameter f (=0.86 × 10−4 s−1) (color shading), plus the surface velocity field (vectors) at 1400:00 LT 25 Mar 2019. (b) Normalized pointwise dilation rates Δ/f computed over a 24-h interval, plotted at the initial positions at 1400:00 LT 25 Mar 2019.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

The MIT-MSEAS model presents a mean horizontal velocity magnitude of 0.44 m s−1 at 1400:00 UTC 25 March 2019, when our analysis begins. Thus, 24 h trajectories can be expected to be about 38 km long. Because this is two orders of magnitude larger than the model grid resolution, the derived Lagrangian structures have a much higher resolution (Fang et al. 2020) and are representative of the modeled processes.

3. Results

To design an optimal drifter deployment strategy for obtaining accurate dilation-rate estimates, we investigate the errors due to discretization and position uncertainty separately. But first, we discuss how the size of the initial fluid patch changes the patterns in the area-averaged dilation rate.

a. Dilation rate for finite-size fluid patches

When computed for finite-sized fluid patches, the dilation rate differs from the corresponding pointwise quantity. Figure 3 presents a comparison between fields of the pointwise dilation rate Δ and the area-averaged dilation rate ΔR for a variety of values of R. We focus on the region north of Alboran Island, where there are filamentary submesoscale structures (Mahadevan 2006) that are hundreds of meters wide. The Δ100 field in Fig. 3b resembles the Δ field in Fig. 3a, as expected in a model with 500-m resolution. Increasing R to 1 km (Fig. 3c) starts to smooth out some of the filamentary structures, as observed, for example, near (36°N, 3°W). Larger R results in further smoothing of the dilation-rate fields, as observed in Figs. 3d and 3e, with the L-divergent region growing in size but weakening in strength.

Fig. 3.
Fig. 3.

Excerpts in the region north of Alboran Island of (a) pointwise dilation rate Δ, along with area-averaged dilation rate ΔR for R = (b) 100 m and (c) 1, (d) 2, and (e) 5 km. Also shown are (f) dilation-rate transects at 2.93°W, shown by the thick vertical lines in (a)–(e), plotted as a function of meridional distance.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

Figure 3f presents dilation-rate transects. The location of the transect is highlighted by a line at 2.93°W in Figs. 3a–e and was chosen in an area of high small-scale variability. The Δ100 transect captures the variation in Δ accurately except for the maximum and minimum dilation-rate magnitudes. An R of 1 km results in significant differences near the maximum and minimum features, and shows a wider L-divergent region, but still identifies an L-convergent extremum. The Δ2000 and Δ5000 transects, however, indicate that measuring dilation rates for a larger initial fluid patch filters out the L-convergent extremum. These differences in dilation rates reflect the averaging process. Analysis of a higher-resolution model should introduce differences between the pointwise dilation rates and finite-patch results for smaller R. In this study, however, we focus on submesoscale features, which are resolved by the chosen operational model of the western Mediterranean.

b. Effect of discretization

The area-averaged dilation rates discussed in the previous section were computed using high-resolution velocity fields, which will not be available during field experiments, and will provide the baseline to measure errors resulting from drifter-based estimates. Using a finite number of drifters, we estimate dilation rates by tracking the evolution of the enclosed polygon’s area and vary the number of drifters used and the radius of the initial release. Figures 4a–c present the dilation-rate fields for R = 100 m computed using drifter triads, tetrads, and octads, respectively. All three fields reveal qualitatively similar features, but discrepancies appear near the strongest and sharpest features. The Δ100,3 field presented in Fig. 4a is noisy, especially in regions of strong L convergence and L divergence. When N is increased to 4, the Δ100,4 field is smoother (Fig. 4b). Drifter octads provide the best results, and the Δ100,8 field in Fig. 4c is qualitatively similar to the Δ and Δ100 fields in Figs. 3a and 3b, albeit with some noise.

Fig. 4.
Fig. 4.

Excerpts of normalized polygonal dilation rates for R = 100 m (Δ100,N) in the region north of Alboran Island, for (a) N = 3, (b) N = 4, and (c) N = 8. Also shown are (d) dilation-rate transects, shown by the thick vertical lines in (a)–(c), plotted as a function of meridional distance. The area-averaged dilation-rate transect for R = 100 m (Δ100) presented in green in Fig. 3f is also plotted for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

The dilation-rate transect in Fig. 4d, however, presents a striking result: polygonal dilation-rate estimates for all three N fail to capture the L-convergent extremum. This can be attributed to filamentation, that has been discussed in detail in the literature on submesoscale flows (Mahadevan 2006; Omand et al. 2015; Verma et al. 2019; Freilich and Mahadevan 2021). Filamentation results in fluid patches with very large aspect ratios, and discretization of the patch boundary with an N-gon can result in artificially large polygonal area, even when the actual fluid patch undergoes an area contraction. Aside from the misclassification of the L-convergent extremum as an L-divergent one, drifter tetrads and octads yield reasonable estimates of dilation-rate magnitudes.

To quantify how well these dilation-rate estimates perform, we compute the scaled error ϵD defined in (9) over various regions of interest in the flow domain. Figure 5a presents the ϵ¯D for the entire domain as a function of R for N = {3, 4, …, 8}. For all values of N, error increases as a function of R because the discretized boundaries perform progressively worse at capturing the behavior of the fluid patch as its initial size increases. On the other hand, a significant improvement is observed in the scaled error as N changes from 3 to 4, especially for larger R, and only a marginal improvement beyond, indicating that N = 4 may be a good balance between accuracy and resource intensity. The scaled errors averaged over the full domain, nevertheless, always remain below 0.04 for all values of N, even for large R.

Fig. 5.
Fig. 5.

(a) Error in dilation-rate estimates due to boundary discretization of the fluid patch averaged over the entire domain (ϵ¯D) for N = 3–8, plotted as a function of R. (b) Two-dimensional histogram of the error ϵD vs normalized baseline ΔR/f for (R, N) = (100, 4). A bin width of 0.05 is used for both variables. The mean error for wider bins of 0.10 in the x direction is also plotted, with the data point colored based on the bin count, along with the sided standard difference as error bars. The black dashed line separates L-converging regions from L-diverging ones. Within the L-converging or L-diverging regions, the dotted lines demarcate the top 1% L-converging or L-diverging regions. Also shown is ϵD averaged over (c) the top 1% L-converging and (d) the top 1% L-diverging regions for N = 3–8, as a function of R.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

To better understand the distribution of errors in the domain, a two-dimensional histogram of the scaled error ϵD is plotted in Fig. 5b as a function of the nondimensionalized Δ100 for tetrads released at a R = 100 m radius. The distribution of Δ100 indicates that the flow domain is composed more of L-convergent regions than L-divergent ones, consistent with Fig. 2b. The white region near (0, 0) indicates that the scaled error in dilation rate is negligible for a majority of the domain that is only weakly L convergent or L divergent. We also see the largest errors occur for the strongest negative Δ100. The bin-averaged ϵD, shown as a curve in Fig. 5b, increases as the magnitude of Δ100 increases. This means that the strongest L-converging and L-diverging features are more susceptible to error in dilation-rate estimation than are the regions with dilation rates close to zero.

Hence, we also consider the strongest L-converging and L-diverging regions in the domain separately, chosen as the portion of the domain with the top 1% values of the pointwise dilation rate, in Figs. 5c and 5d, respectively. As observed for the full domain, the average scaled errors for the strongest L-converging regions increase as a function of R, and N = 4 remains a good compromise between accuracy and resource intensity. However, the error magnitudes are now always greater than 0.04 even for the smallest R = 1 m, attaining a maximum value of 0.27 for N = 3 and R = 1 km. The errors for each N also seem to saturate for large R, potentially due to the filtering out of strongly L-convergent features due to the smoothing across the larger area as discussed in section 3a. The average scaled dilation-rate error ϵ¯D for the top 1% L-divergent regions (Fig. 5d), on the other hand, presents errors closer to the order of what was observed for the full domain (Fig. 5a). The error for the strongest L-divergent regions also monotonically increases with R, and always stays below 0.06. That the L-convergent errors are larger may be in part due to converging patches experiencing filamentation.

c. Effect of position uncertainty

Aside from discretization, another difficulty in using drifters to accurately track the evolution of a fluid patch is the limited accuracy of their GPS units. A comparison with the Δ100,3 field in Fig. 4a reveals that the submesoscale features are qualitatively captured well by the drifter-based estimate, Δ100,3,6, presented in Fig. 6a, although the uncertainty in drifter positions makes the results noisier. The error introduced by position uncertainty is primarily concentrated in the L-convergent regions, as can be seen in Fig. 6b, showing the dilation-rate transects of all 100 realizations of Δ100,3,6. This issue for L-convergent regions arises because GPS accuracy matters more for smaller final polygons. In the particular case of the Δ100,3 transect in Fig. 6b, the most negative value of dilation rate corresponds to a deployment radius of 14.6 m at tf, which is only about twice σ. For the L-divergent regions, however, σ will always be less than 10% of the size of the final polygon, which is always greater than 100 m by definition. Thus, there is negligible error for the L-divergent measurements.

Fig. 6.
Fig. 6.

(a) Excerpt of one realization of the normalized drifter-based dilation rate Δ1R,N,σ, for (R, N, σ) = (100, 3, 6) in the region north of Alboran Island. (b) Dilation-rate transects, shown using a thick vertical line in (a), plotted as a function of meridional distance for all 100 realizations (black). The polygonal-dilation-rate transect Δ100,3 presented in Fig. 4d is also plotted (blue) for comparison. Also shown is the ensemble averaged error in dilation-rate estimates due to position uncertainty ϵP averaged over (c) the entire domain and (d) the top 1% L-converging regions for N = 3–8, plotted as a function of R.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

Because uncertainty errors are primarily dependent on the size of the final polygon, the average scaled error in the dilation-rate estimates for the full domain due to position uncertainty alone [ϵ¯P, defined in (10)], for σ = 6 m decreases as a function of R for all values of N (Fig. 6c). The maximum error magnitudes are approximately 0.3, which is an order-of-magnitude larger than what was observed for ϵ¯D, and there is no clear benefit in using more drifters. Since the errors due to position uncertainty are especially important near the L-convergent regions, ϵ¯P curves for the top 1% L-converging regions in Fig. 6d show even larger magnitudes of error for all values of R, with the maximum error now close to 0.85. The cutoff dilation rate to belong to the top 1% L-converging regions in the domain corresponds to an area contraction by a factor of 41 (i.e., polygon size contraction by a factor of 6.4), which means that any fluid patch with R < 38 m will have a final size less than the GPS accuracy of σ = 6 m. In other words, if R < 38 m, the final areas of drifter polygons released in the strongest L-converging regions are dominated by the uncertainty in GPS readings, and errors for deployments at this scale are close to 0.3. However, the error due to position uncertainty drops to 0.1 for a larger R of 250 m, and deployment at this scale corresponds to an average size of 20 m for the final polygon, which is 3 times the value of σ considered. An interesting observation for the L-convergent regions is that N = 3 performs best, albeit only marginally, than larger values of N. This is possibly related to the formation of self-intersecting polygons due to position uncertainty for N > 3.

d. Optimal deployment radius

Drifter-based dilation rates are affected by both boundary discretization and position uncertainty, and Fig. 7a presents the average scaled error ϵ¯T [defined in (11)] computed for the full domain for drifter triads and tetrads. The average total error ϵ¯T is approximately the sum of the ϵ¯D and ϵ¯P curves, presented in Figs. 3 and 6, respectively. Because the position uncertainty and boundary discretization errors are inversely related with respect to initial release radius, a local minimum is observed in ϵ¯T, which allows us to determine an optimal deployment radius RΩ* for each N and σ. Over the full domain, for N = 3 or 4 and σ = 6, RΩ* is 250 m, which corresponds to an ϵ¯T of 0.01 (Fig. 7a). This means that the error between a dilation rate estimated using a drifter polygon deployed at R (ΔR,N,σ) and the corresponding area-averaged dilation rate for an initially circular fluid patch of radius R (ΔR) is minimized at R* = 250 m. It is important to note, however, that the error worsens from the optimum by less than 0.007 for 100 ≤ R ≤ 1000 m.

Fig. 7.
Fig. 7.

Total error ϵT in drifter-based dilation-rate estimates (solid lines with circles), along with ϵD (long dashes) and ϵP (short dashes), plotted as a function of R for N = 3 (blue) and N = 4 (red), averaged over (a) the full domain and (b) the top 1% L-converging regions. Note that the ϵ¯T curve may lie on the ϵ¯D and ϵ¯P curves, for small and large R, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

Similar trends are observed for the strongest L-converging regions in the domain as well with the ϵ¯T curve in Fig. 7b. As a result, local minima are observed at RC*=250 m for both N = 3 and N = 4, which suggests that drifter swarms deployed at R = 250 m can estimate L convergence most accurately in submesoscale flows. The error magnitudes do not worsen significantly for other R that are close to RC*, as was observed for the full domain. For the strongest L-converging regions, however, the error corresponding to the optimal deployment radius is close to 0.22, which is large relative to what was observed for the full domain.

e. Variation of optimal deployment radius

Thus far, the results have only considered drifters that have GPS units with an accuracy of σ = 6 m. Figure 8 presents the total error ϵT as a function of R and σ, with the optimal deployment radius for the full domain RΩ* and the strongest L-converging regions RC* marked using black circles. For both the full domain and the strongest L-converging regions, the optimal deployment radii increase as a function of σ because the larger the size of the drifter polygon, the less its area will be affected by GPS inaccuracy. Since the strongest L-convergent regions are the primary contributors to error, RΩ* can be expected to be close to RC*, but not necessarily equal. Here, we observe that RΩ* matches RC* exactly for all but one value of σ, namely, σ = 25 m, although the average error associated with the optimal deployment radii for the L-convergent regions is 0.2, as opposed to 0.01 for the full domain. While these radii are optimal, Fig. 8 reveals that for this range of σ, the error magnitudes barely worsen from the optimum for R∼RΩ*, both for the full domain and the L-convergent regions.

Fig. 8.
Fig. 8.

Total error ϵT in drifter-based dilation-rate estimates, averaged over (a) the full domain and (b) the top 1% L-converging regions, as a function of R and σ. The results are presented for R = 1, 2, 5, 10, 20, 50, 100, 250, 500, 750, and 1000 m and σ = 0.75, 1.5, 3, 6, 9, 12, 25 and 50 m. The optimal deployment radii for each sigma, RΩ* and RC* for the full domain and top 1% L-converging regions, respectively, are also marked with a black circle around the corresponding data point.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

f. Locating regions of strongest Lagrangian convergence

The L-convergent regions always exhibit substantial error magnitudes (close to 0.2), which hinders our ability to obtain accurate estimates of strong L convergence. However, it may be worthwhile identifying the regions of strongest L convergence (Aravind et al. 2023). Here, we consider the top 5% L-convergent regions in addition to the top 1% to evaluate how well drifter-based dilation rates perform when the threshold is relaxed. For this analysis, the pointwise dilation rate Δ is used as baseline to have the same reference for all drifter-based estimates ΔR,N,σ.

Figure 9 presents a comparison between the top 1% (Figs. 9a–c) and top 5% (Figs. 9d–f) L-converging regions identified by the pointwise dilation rate Δ and the drifter-based dilation rate ΔR,N,σ for N = 4, σ = 6 m and R = 50, 250 and 1000 m. The regions marked in green are the true positives (TP), which are regions identified as the strongest L converging by both Δ and ΔR,N,σ. Regions identified by Δ but not by ΔR,N,σ are false negatives (FN), plotted in red, while those identified by ΔR,N,σ alone are false positives (FP), plotted in blue. Because the same number of grid points are identified as true positives by Δ and by ΔR,N,σ, the number of false negatives and false positives are equal. Hence, the percent overlap can be calculated as TP/(TP + FP) × 100 or TP/(TP + FN) × 100. Before delving into the results, it is important to note that in practice a point-by-point comparison could be too strict. For example, even when the drifter-based estimate classifies most of the points inside an L-convergent region correctly, the percent overlap metric will penalize ΔR,N,σ for missing the remaining points.

Fig. 9.
Fig. 9.

Strongest L-converging regions in the domain, as identified by the drifter-tetrad-based estimates alone (blue), pointwise dilation rate alone (red), and both drifter-based and pointwise dilation rates (green) for the top (a)–(c) 1% and (d)–(f) 5% L-converging regions for (top) R = 50 m, (middle) R = 250 m, and (bottom) R = 1000 m. The percent of green regions, i.e., the percent overlap of L-converging regions identified by the drifter-based estimates and pointwise measurements, is also illustrated in a Venn diagram.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

Figure 9a shows only a 23% overlap between the top 1% strongest L-converging regions identified by Δ and ΔR,N,σ for a 50 m deployment, primarily because position uncertainty plays a significant role at such a small value of R. Increasing the R to the optimal value of 250 m (Fig. 9b), the percent overlap increases to 60%, indicating that a majority of the top 1% L-converging regions are located by the drifter-based dilation rates. This improvement is striking, since the dilation-rate errors in Fig. 8 only improves from 0.29 to 0.22, as R changes from 50 to 250 m. Increasing R more to 1000 m (Fig. 9c) does not qualitatively affect the results, although the percent overlap falls to 51%.

If the threshold is relaxed and the top 5% L-convergent regions are considered, larger fractions of overlap are observed in Figs. 9d–f than in Figs. 9a–c. The improvement is primarily because some false positives for the top 1% analysis are reclassified as true positives for the top 5%, as can be observed from a comparison between Fig. 9a and Fig. 9d. As a result, the percent overlap for R = 50 m improves from 23% for top 1%–58% for the top 5%, which is close to the value for the top 1% L-converging regions for the optimal deployment radius of 250 m. The percent overlap for R = 250 and 1000 m improve considerably as well, to 84% and 73%, respectively. This indicates that even though the dilation-rate magnitudes obtained from a drifter-based calculation may be prone to error, they can still be successfully used to locate regions of strongest L convergence.

To understand if the optimal deployment radius is affected from the perspective of locating the L-convergent regions as opposed to dilation-rate error magnitudes, we plot the ensemble mean of the percent overlap as a function of R for drifter triads and tetrads with position uncertainty in Fig. 10. The standard deviation for all σ are less than 1%, indicating that the percent overlap for each R is robust with respect to position uncertainty. For the top 1% L-converging regions in Fig. 10a, triads reveal a maximum overlap of 48% at R = 250 m, whereas tetrads achieve maximum overlap of 61% at R = 500 m. Reducing R to 250 m for the tetrads (60% overlap), however, does not result in a considerable drop in percent overlap. The curves for the top 5% L-converging regions show similar trends, but with larger percent overlaps of 73% for triads and 84% for tetrads, both at an optimal deployment radius of R = 250 m. The dashed lines present the percent overlap between the strongest L-converging regions in the fields of polygonal dilation rate and pointwise dilation rate, plotted here to highlight the significance of GPS accuracy in locating L convergence using drifter-based dilation-rate estimations.

Fig. 10.
Fig. 10.

Percent of (a) the top 1% and (b) the top 5% L-converging regions in the pointwise dilation-rate fields located by the drifter-based dilation rates for N = 3 (blue, solid) and 4 (red, solid), as a function of R. The percent overlaps for polygonal dilation rates are also plotted (dashed) to highlight the role played by position uncertainty.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

4. Discussion

The presented drifter deployment strategies are designed to retrieve accurate estimates of Lagrangian convergence (L convergence) in the western Mediterranean Sea and are particularly apt at locating regions where L convergence is strongest. The recommendation of R* = 250 m for drifter polygon deployment is optimal for the particular geographical region studied here and the assumed GPS uncertainty, for the ∼500 m resolution model considered. It may need to be adapted for other regions with different dispersion properties or for different GPS accuracy.

As presented in Fig. 7, GPS position uncertainty dominates the error for drifter-based estimation of dilation rate at small deployment scales. This is due to the large relative errors in the final area of the polygon when the size of the polygon is comparable to the position uncertainty. At large scales, the boundary discretization error dominates, although it grows more slowly with increasing scale than the position error does with decreasing scale. The optimal R* is thus a compromise scale that is small enough to capture the boundary deformation and large enough to overcome position uncertainty. The upper bound of acceptable deployment scales will depend on the coherence of the flow features being sampled. Figure 5 provides some general guidance, when averaging over the many flow features represented in the model. For the lower bound, the goal is to ensure that the final polygons have sizes sufficiently larger than the position uncertainty σ.

Thus, the optimal R* can be approximated by the deployment size R^ that results in a final polygon with size Rf = κσ, where κ > 1. The scaling factor κ is independent of flow regime and the expected relative error in final area as a result of position uncertainty scales as κ−2. Although this means that a larger κ reduces error due to GPS uncertainty, the choice of κ is bounded from above by the error due to boundary discretization. The scaling factor κ corresponding to the optimum can be estimated from our model analysis (see below). To relate R^ to Rf, an estimate of the dilation rate that is representative of the flow features of interest Δ^, which in our study corresponds to the strongest L-converging regions, is also required. Unlike κ, this is flow dependent and can be obtained either from model forecasts for the region of interest or estimated from historical or observed values of instantaneous divergence or dilation rate. Since polygonal area A scales as A ∼ R2, (8) gives
Δ^=log⁡[(κσ)2/R^2]tf−t0⇒R^=κσexp⁡[−Δ^(tf−t0)/2].
Since the objective of the paper is to locate the regions of strongest L convergence, an appropriate choice for Δ^ is the cutoff dilation rate for the top 1% L-converging regions. For the MIT-MSEAS simulation of the western Mediterranean analyzed in this paper, this corresponds to Δ^=−0.5f, where f = 0.86 × 10−4 s−1 is the Coriolis parameter. This represents an area contraction of drifter polygons by a factor of 41 over a day. As a result, the size of a drifter polygon deployed at the optimum R* = 250 m will contract to 39 m. This means that, for a GPS position uncertainty of σ = 6 m, the optimum R* corresponds to κ = Rf/σ ≈ 6.5, with an expected error of 2.4% in the area of the final polygon. Note that this error in final area is only for a dilation rate of −0.5f. Stronger L- convergence will result in final polygons with smaller Rf, resulting in larger inaccuracies in final areas, and hence larger error in dilation-rate estimates.

To summarize, for the MIT-MSEAS model of the western Mediterranean, R^=R* can be recovered from (12) for tf − t0 = 1 day and σ = 6 m if we anticipate that Δ^=−0.5f and set κ = 6.5. In other geographical regions, these parameters can be varied depending on the oceanic flow features of interest. Sometimes, the overestimation of Δ^ can result in R^ larger than the length scale of the flow feature of interest. In such cases, drifter deployment at R^ minimizes error in the dilation rate averaged over the area spanned by the drifter polygon, but this can potentially smooth out the Lagrangian flow feature of interest.

One distinction that needs to be made is that the R* for deployment discussed here differs from the length scale of oceanographic interest: whereas the former concerns the size of a single drifter polygon, the latter determines the distance between polygons and the number of polygons to be deployed. For example, Fig. 11 presents a strategy to use our result to sample the region near the L-convergent filament presented in Figs. 3, 4, and 6.

Fig. 11.
Fig. 11.

Deployment strategy to sample the L-convergent filament: (a) Forty-eight drifters (black dots) deployed as a 4 × 3 grid of drifter tetrads, each deployed at R = 250 m, to sample the region near the L-convergent filament observed in the normalized pointwise dilation-rate field (shading). (b) Drifter tetrads (black dots), along with the normalized dilation rates estimated using their trajectories (colors), assuming a 6-m position uncertainty.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

If we anticipate the existence of the filament revealed by the pointwise dilation-rate field in Fig. 11a (using sea surface temperature maps, for example) and approximately 50 drifters are available during the field experiment, drifter tetrads at R = 250 m can be deployed in a 4 × 3 grid to map out the L convergence and L divergence in the region. Here, the size of the drifter polygons is determined by our error minimization analysis, and the resolution of the drifter grid is determined by the expected length scale of the flow feature of interest.

Figure 11b presents the dilation rates estimated using each of the 12 drifter tetrads, for finite-sized fluid patches. Although the dilation-rate estimates are obtained by approximating finite-sized fluid patches as polygons of drifters that are susceptible to a GPS position uncertainty of 6 m, the 4 × 3 grid-like sampling correctly suggests the existence of a strong L-convergent filament next to an L-divergent feature. As discussed in section 3b, the L-convergent filament presented here is one of the most difficult features in the domain to capture using drifter-based estimations of dilation rate. In the event that a dilation rate averaged over a larger length scale is desired (like in Figs. 4c–e), our recommendation is to average multiple dilation rates estimated at R*, using (6).

5. Conclusions

This study has emphasized the dilation rate, which is a measure of the area expansion of a fluid patch over a finite time interval, as an appropriate diagnostic for identifying and quantifying strong regions of submesoscale Lagrangian convergence (L convergence) in the ocean. This diagnostic is readily calculated from polygons of surface or near-surface drifter observations. However, there are a number of errors associated with this paradigm, which were studied in detail. We quantified the error due to boundary discretization and GPS position uncertainty, and we evaluated the ability of the drifter-based dilation-rate estimates to identify regions of surface L convergence.

Dilation-rate error due to boundary discretization increased with the size of the initial drifter polygon and decreased with the number of drifters per estimate. The decrease in error with increased number of drifters revealed that drifter tetrads are a good compromise for dilation-rate estimation between accuracy and resource demand. The error due to position uncertainty, on the other hand, had negligible dependence on the number of drifters, and decreased with the size of the initial drifter polygon. As a result, the average total error for drifter-based dilation rates was minimized at an optimal initial deployment size. Even though this optimum was 250 m for drifter tetrads equipped with GPS units that are accurate to 6 m, the errors were not significantly worse for slightly larger or smaller tetrads. Nor did it significantly vary with changes in σ between 1 and 12 m.

Further, we found that the drifter-based estimates of dilation rate were most prone to error in strongly L-converging regions, corresponding to errors that were on average an order of magnitude larger than those in L-diverging regions, and amount to approximately 20% even for an optimum deployment at 250 m. This, of course, is an issue in estimating the intensity of L convergence. Importantly, however, this error does not materially impact the use of drifter-based estimates of dilation rate to locate regions of strongest L convergence, as drifter tetrads deployed at the optimum R* successfully located 84% of the top 5% strongest L-converging regions in the domain.

Our analysis covered a broad range of mesoscale and submesoscale features observed in the western Mediterranean Sea. While these results are directly applicable to other regions with similar flow features, the optimal release radius may be different for systems with significantly different dynamics. The length of the analysis time interval can also impact the results. The optimum in such scenarios, however, can be approximated by identifying the smallest initial deployment radius for which the final size of the drifter polygon is larger than the GPS uncertainty. The final size of the drifter polygon for such a calculation can be obtained from a baseline divergence or dilation-rate measure at the strongest converging regions in the domain, drawn from simulations or using test drifters.

During field experiments, of course, deployment of drifters at the vertices of a regular polygon, as in this modeling study, may not be feasible. Deviation of the initial polygon from the ideal regular shape constitutes an additional potential error source. Huntley et al. (2022) proposed shape criteria for triads to remove unreliable dilation-rate estimates. Given our finding that tetrads are preferable to triads, those results should be extended to tetrads. The shape criteria, in combination with the results presented here, can also be useful in estimating dilation rates from “chance polygons” composed of drifters that were previously deployed.

When investigating chance polygons, however, drifter position uncertainty needs to be accounted for even at the initial time instant, since a more accurate initial positioning of drifters relative to the vessel (in the case of drifter deployments from a ship, for example) may not be available. While the errors due to position uncertainty can be reduced by more accurate GPS units, drifter trajectory filtering (e.g., Niiler 2001; Poulain et al. 2009; Yaremchuk and Coelho 2015) may be a viable alternative. Since the sampling frequency impacts the accuracy of filtering methods, cost–benefit analyses performed on the GPS accuracy versus battery life could also be insightful for oceanographic applications.

Acknowledgments.

The authors acknowledge Pierre F. J. Lermusiaux and Patrick J. Haley for running the MIT-MSEAS model simulations. Authors Aravind and Allshouse were supported by ONR Grant N00014-18-1-2790. Authors Huntley and Kirwan were supported by ONR Grant N00014-18-1-2461.

Data availability statement.

The datasets generated and/or analyzed can be accessed at the following DOI: https://doi.org/10.5281/zenodo.7262224. The observational dataset used for the GPS accuracy analysis is available from the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) under DOI 10.7266/2H3H2KHC.

APPENDIX

Drifter Position Accuracy

The accuracy of drifter positions in a dataset depends on the type of positioning system used by the drifter, the reliability of its satellite communications, and the postprocessing. Scientists use a variety of different positioning systems for drifters, but three of the most popular ones are based on Argos, Iridium, and Globalstar communications. Argos is a system of satellites that was specifically conceived for communicating environmental data in the 1970s (Ortega 2003). In contrast to standard GPS, Argos relies on the Doppler shift in consecutive transmissions to locate the transmitter and therefore only requires a single satellite (Benson 2012). Argos states that the most accurate resulting positions still contain errors up to 150 m, and practically, errors have been found to be even larger (Costa et al. 2010). Therefore, modern drifters relying on Argos for communications generally use a standard GPS to determine the positions, although Argos tags remain popular for tracking animals (Douglas et al. 2012). However, Argos also tends to be slow, allowing for at most hourly location fixes, with latency of up to 2 h (Lumpkin et al. 2017). For this reason, the main international long-term drifter program, the Global Drifter Program (GDP) is transitioning to Iridium, which can provide more frequent positions (Lumpkin et al. 2017). Both Iridium and Globalstar are commercial systems. Globalstar offers relatively simple (and hence more affordable) GPS units, while Iridium serves broader communication needs, including GPS, as well as more comprehensive global coverage. Within its coverage area, Globalstar is frequently used for the types of large, high-density drifter deployments needed for estimating velocity gradients (Lumpkin et al. 2017).

The accuracy of the reported positions can be degraded by local conditions—for example, if the transmitter is shielded from the satellites by nearby structures such as large ships or drilling platforms. Atmospheric conditions can also play a role. Occasional mistransmission of the data can lead to exceedingly large outliers, many tens or even hundreds of kilometers off. These, however, are easily identified and generally removed during postprocessing prior to any data analysis. Thus, we will not concern ourselves here with them. Smaller errors, within the range of reasonable motion, however, can be difficult to detect and their processing can become somewhat subjective. This is why drifter trajectories are frequently filtered to some degree as part of the postprocessing as well (e.g., Niiler 2001; Poulain et al. 2009; Yaremchuk and Coelho 2015) to reduce position errors.

For the dilation-rate calculations in this study, only initial and final positions are needed, with no intermediate sampling. For this reason, coarse (in time) sampling is adequate. However, that hampers filtering efforts. Therefore, we consider here the position accuracy without any filtering, but after the removal of blatant outliers.

For this purpose, we consider the dataset collected in preparation for the Grand Lagrangian Deployment (GLAD) experiment (Poje et al. 2014), when 10 SPOT Messenger II GPS units, using the Globalstar communications system, were tied next to each other to a stationary dock with a clear view of the sky (Huntley et al. 2020). During the first test, only 9 units reported data. Positions were recorded every 5 min for two roughly 12-h periods, for a total of 2516 positions.

The left panel of Fig. A1 shows the zonal and meridional positions relative to the median position over all drifters in each of the tests. The first test showed some significant outliers with errors of over 35 m in both the zonal and the meridional directions and total errors greater than 35 m; the second test had only a single point in this regime. The root-mean-square (rms) relative zonal positions for the two tests are 6.2 and 4.1 m, respectively; over both tests together the rms relative zonal position is 5.2 m. Similarly, the rms relative meridional positions for the individual tests are 8.1 and 3.4 m, and for both tests combined 6.1 m. These findings are in line with manufacturer’s specification of an accuracy of 6.4 m, especially if the single largest outlier from the first test is removed, resulting in rms zonal and meridional positions of 5.5 and 7.4 m. Although these statistics suggest that the error is not entirely isotropic, we will assume it is isotropic in the open ocean.

Fig. A1.
Fig. A1.

(left) Recorded positions relative to the median position over all units in each of the GPS accuracy tests. Dark-blue circles show the first test, and red triangles show the second test. The gray circle has a radius of 35 m. (right) Differences in the empirical distributions from the observation and the two types of models, two independent Gaussian distributions (blue) and normally distributed magnitude with uniformly distributed angle (red) for a range of different standard deviations σ.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

The total position errors can be modeled in different ways. We explore two options here: 1) the zonal and meridional errors are assumed to be drawn from identical but independent Gaussian distributions and 2) the total error magnitude is assumed to be drawn from a Gaussian distribution while its direction is drawn from a uniform distribution over [0, π). The mean for the Gaussian distribution is set to 0, assuming unbiased position measurements. A range of different values for the standard deviation σ are tested, and the resulting distributions of 10 000 samples each are compared with the distribution of the observed errors. For this purpose, modeled and observed samples of total errors are binned into 12 bins of size 2 m, with the first bin at [−1, 1] m and the last bin at [19, 21] m. The fraction of samples falling into each bin is recorded. The rms difference between the model and the observations is then plotted in the right panel of Fig. A1.

The figure suggests that the optimal model uses two independent normal distributions and a standard deviation of 3.5 m. The other type of model produces the smallest differences for a choice of σ of 5.5 m in the instance shown. However, other random samples of 10 000 draws can result in minima for σ = 5 or 6 m. A comparison of the distributions for the zonal, meridional, and total errors for the observations with those from the two models is shown in Fig. A2. Note that the observations show a slight negative bias in the zonal error that is not modeled, as discussed above. The figure also shows that the smallest meridional errors are somewhat underrepresented in the model, especially positive ones. For the total error, both models fail to capture the large peak in the [3, 5]-m bin, The double Gaussian model (top) also underpredicts very small error (<1 m) and large errors (>9 m), whereas the Gaussian-uniform model (bottom) overpredicts almost all bins, except the [3, 5]-m bin, with very small underpredictions in the [5, 7]- and [17, 19]-m bins.

Fig. A2.
Fig. A2.

Histograms, scaled as a probability density function (pdf), for the (left) zonal, (center) meridional, and (right) total errors for (top) the model with Gaussian zonal and meridional errors, using σ = 3.5 m and (bottom) the model with Gaussian signed error magnitude and uniformly distributed angle, using σ = 5.5 m. Observed results are plotted in blue, modeled results are plotted orange, and overlaps of the histograms appear brown. The bright-red curves in the top-left and top-center panels show the corresponding Gaussian distributions, illustrating that the empirical histograms capture the pdf well.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0129.1

In the main body of this report, we choose a Gaussian-uniform position error model, even though it appears to perform somewhat less well here. This is because we hypothesize that the errors in the open ocean will be more isotropic than those exhibited by the GPS units on the dock. Moreover, it provides a more conservative test of the error impacts, as larger errors are represented more frequently in this model. Similarly, we have chosen a standard deviation of 6 m, which is more conservative than the other options and closer to the accuracy specified by the manufacturer for the SPOT Messenger II units.

REFERENCES

  • Allshouse, M. R., and T. Peacock, 2015a: Lagrangian based methods for coherent structure detection. Chaos, 25, 097617, https://doi.org/10.1063/1.4922968.

    • Search Google Scholar
    • Export Citation

  • Allshouse, M. R., and T. Peacock, 2015b: Refining finite-time Lyapunov exponent ridges and the challenges of classifying them. Chaos, 25, 087410, https://doi.org/10.1063/1.4928210.

    • Search Google Scholar
    • Export Citation

  • Aravind, H. M., V. Verma, S. Sarkar, M. A. Freilich, A. Mahadevan, P. J. Haley, P. F. J. Lermusiaux, and M. R. Allshouse, 2023: Lagrangian surface signatures reveal upper-ocean vertical displacement conduits near oceanic density fronts. Ocean Modell., 181, 102136, https://doi.org/10.1016/j.ocemod.2022.102136.

    • Search Google Scholar
    • Export Citation

  • Benson, E., 2012: One infrastructure, many global visions: The commercialization and diversification of Argos, a satellite-based environmental surveillance system. Soc. Stud. Sci., 42, 843–868, https://doi.org/10.1177/0306312712457851.

    • Search Google Scholar
    • Export Citation

  • Berta, M., A. Griffa, T. M. Özgökmen, and A. C. Poje, 2016: Submesoscale evolution of surface drifter triads in the Gulf of Mexico. Geophys. Res. Lett., 43, 11 751–11 759, https://doi.org/10.1002/2016GL070357.

    • Search Google Scholar
    • Export Citation

  • Berta, M., and Coauthors, 2020: Submesoscale kinematic properties in summer and winter surface flows in the northern Gulf of Mexico. J. Geophys. Res. Oceans, 125, e2020JC016085, https://doi.org/10.1029/2020JC016085.

    • Search Google Scholar
    • Export Citation

  • Centurioni, L. R., 2018: Drifter technology and impacts for sea surface temperature, sea-level pressure, and ocean circulation studies. Observing the Oceans in Real Time, Springer, 37–57, https://doi.org/10.1007/978-3-319-66493-4_3.

  • Costa, D. P., and Coauthors, 2010: Accuracy of Argos locations of pinnipeds at-sea estimated using Fastloc GPS. PLOS ONE, 5, e8677, https://doi.org/10.1371/journal.pone.0008677.

    • Search Google Scholar
    • Export Citation

  • Cózar, A., S. Aliani, O. C. Basurko, M. Arias, A. Isobe, K. Topouzelis, A. Rubio, and C. Morales-Caselles, 2021: Marine litter windrows: A strategic target to understand and manage the ocean plastic pollution. Front. Mar. Sci., 8, 571796, https://doi.org/10.3389/fmars.2021.571796.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., and Coauthors, 2018: Ocean convergence and the dispersion of flotsam. Proc. Natl. Acad. Sci. USA, 115, 1162–1167, https://doi.org/10.1073/pnas.1718453115.

    • Search Google Scholar
    • Export Citation

  • Douglas, D. C., R. Weinzierl, S. C. Davidson, R. Kays, M. Wikelski, and G. Bohrer, 2012: Moderating Argos location errors in animal tracking data. Methods Ecol. Evol., 3, 999–1007, https://doi.org/10.1111/j.2041-210X.2012.00245.x.

    • Search Google Scholar
    • Export Citation

  • Dräger-Dietel, J., K. Jochumsen, A. Griesel, and G. Badin, 2018: Relative dispersion of surface drifters in the Benguela upwelling region. J. Phys. Oceanogr., 48, 2325–2341, https://doi.org/10.1175/JPO-D-18-0027.1.

    • Search Google Scholar
    • Export Citation

  • Egbert, G. D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183–204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Environmental Modeling Center, 2003: The GFS atmospheric model. NCEP Office Note 442, 14 pp., http://www.lib.ncep.noaa.gov/ncepofficenotes/files/on442.pdf.

  • Esposito, G., M. Berta, L. Centurioni, T. M. S. Johnston, J. Lodise, T. Özgökmen, P.-M. Poulain, and A. Griffa, 2021: Submesoscale vorticity and divergence in the Alboran Sea: Scale and depth dependence. Front. Mar. Sci., 8, 678304, https://doi.org/10.3389/fmars.2021.678304.

    • Search Google Scholar
    • Export Citation

  • Essink, S., V. Hormann, L. R. Centurioni, and A. Mahadevan, 2022: On characterizing ocean kinematics from surface drifters. J. Atmos. Oceanic Technol., 39, 1183–1198, https://doi.org/10.1175/JTECH-D-21-0068.1.

    • Search Google Scholar
    • Export Citation

  • Fang, L., S. Balasuriya, and N. T. Ouellette, 2020: Disentangling resolution, precision, and inherent stochasticity in nonlinear systems. Phys. Rev. Res., 2, 023343, https://doi.org/10.1103/PhysRevResearch.2.023343.

    • Search Google Scholar
    • Export Citation

  • Farazmand, M., and G. Haller, 2012: Computing Lagrangian coherent structures from their variational theory. Chaos, 22, 013128, https://doi.org/10.1063/1.3690153.

    • Search Google Scholar
    • Export Citation

  • Filippi, M., R. Hanlon, I. I. Rypina, B. A. Hodges, T. Peacock, and D. G. Schmale III, 2021: Tracking a surrogate hazardous agent (rhodamine dye) in a coastal ocean environment using in situ measurements and concentration estimates derived from drone images. Remote Sens., 13, 4415, https://doi.org/10.3390/rs13214415.

    • Search Google Scholar
    • Export Citation

  • Freilich, M., and A. Mahadevan, 2021: Coherent pathways for subduction from the surface mixed layer at ocean fronts. J. Geophys. Res. Oceans, 126, e2020JC017042, https://doi.org/10.1029/2020JC017042.

    • Search Google Scholar
    • Export Citation

  • Gonçalves, R. C., M. Iskandarani, T. Özgökmen, and W. C. Thacker, 2019: Reconstruction of submesoscale velocity field from surface drifters. J. Phys. Oceanogr., 49, 941–958, https://doi.org/10.1175/JPO-D-18-0025.1.

    • Search Google Scholar
    • Export Citation

  • Haley, P. J. Jr., and P. F. J. Lermusiaux, 2010: Multiscale two-way embedding schemes for free-surface primitive equations in the “Multidisciplinary Simulation, Estimation and Assimilation System.” Ocean Dyn., 60, 1497–1537, https://doi.org/10.1007/s10236-010-0349-4.

    • Search Google Scholar
    • Export Citation

  • Haley, P. J. Jr., A. Agarwal, and P. F. J. Lermusiaux, 2015: Optimizing velocities and transports for complex coastal regions and archipelagos. Ocean Modell., 89, 1–28, https://doi.org/10.1016/j.ocemod.2015.02.005.

    • Search Google Scholar
    • Export Citation

  • Hernández-Carrasco, I., A. Orfila, V. Rossi, and V. Garçon, 2018: Effect of small scale transport processes on phytoplankton distribution in coastal seas. Sci. Rep., 8, 8613, https://doi.org/10.1038/s41598-018-26857-9.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., G. Jacobs, and A. D. Kirwan Jr., 2015: Clusters, deformation, and dilation: Diagnostics for material accumulation regions. J. Geophys. Res. Oceans, 120, 6622–6636, https://doi.org/10.1002/2015JC011036.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., and A. D. Kirwan Jr., 2019: Anisotropy and inhomogeneity in drifter dispersion. J. Geophys. Res. Oceans, 124, 8667–8682, https://doi.org/10.1029/2019JC015179.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., E. Ryan, B. Haus, and T. Özgökmen, 2020: Accuracy test for SPOT GPS units used in the surface drifters deployed in the Grand Lagrangian Deployment (GLAD) experiment, collected 26–27 June and 3–4 July 2012 in Miami, Florida. GRIIDC, accessed 15 July 2022, https://doi.org/10.7266/2H3H2KHC.

  • Huntley, H. S., M. Berta, G. Esposito, A. Griffa, B. Mourre, and L. Centurioni, 2022: Conditions for reliable divergence estimates from drifter triplets. J. Atmos. Oceanic Technol., 39, 1499–1523, https://doi.org/10.1175/JTECH-D-21-0161.1.

    • Search Google Scholar
    • Export Citation

  • Juza, M., and Coauthors, 2016: SOCIB operational ocean forecasting system and multi-platform validation in the western Mediterranean Sea. J. Oper. Oceanogr., 9, s155–s166, https://doi.org/10.1080/1755876X.2015.1117764.

    • Search Google Scholar
    • Export Citation

  • Kawai, H., 1985: Scale dependence of divergence and vorticity of near-surface flows in the sea. J. Oceanogr. Soc. Japan, 41, 157–166, https://doi.org/10.1007/BF02111115.

    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., 2010: Observations of submesoscale eddies using high-frequency radar-derived kinematic and dynamic quantities. Cont. Shelf Res., 30, 1639–1655, https://doi.org/10.1016/j.csr.2010.06.011.

    • Search Google Scholar
    • Export Citation

  • Lumpkin, R., T. Özgökmen, and L. Centurioni, 2017: Advances in the application of surface drifters. Annu. Rev. Mar. Sci., 9, 59–81, https://doi.org/10.1146/annurev-marine-010816-060641.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., 2006: Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales? Ocean Modell., 14, 222–240, https://doi.org/10.1016/j.ocemod.2006.05.005.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., A. Pascual, D. L. Rudnick, S. Ruiz, J. Tintoré, and E. D’Asaro, 2020a: Coherent pathways for vertical transport from the surface ocean to interior. Bull. Amer. Meteor. Soc., 101, E1996–E2004, https://doi.org/10.1175/BAMS-D-19-0305.1.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., and Coauthors, 2020b: CALYPSO 2019 cruise report: Field campaign in the Mediterranean. Woods Hole Oceanographic Institution Tech. Rep., WHOI-2020-02, 123 pp., https://doi.org/10.1575/1912/25266.

  • Molinari, R., and A. D. Kirwan Jr., 1975: Calculations of differential kinematic properties from Lagrangian observations in the western Caribbean Sea. J. Phys. Oceanogr., 5, 483–491, https://doi.org/10.1175/1520-0485(1975)005<0483:CODKPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mourre, B., and Coauthors, 2018: Assessment of high-resolution regional ocean prediction systems using multi-platform observations: Illustrations in the western Mediterranean Sea. New Frontiers in Operational Oceanography, E. Chassignet et al., Eds., GODAE OceanView, 663–694.

  • Niiler, P., 2001: The World Ocean surface circulation. Ocean Circulation and Climate, G. Siedler, J. Church, and J. Gould, Eds., Academic Press, 193–204.

  • Niiler, P., A. S. Sybrandy, K. Bi, P. M. Poulain, and D. Bitterman, 1995: Measurements of the water-following capability of holey-sock and TRISTAR drifters. Deep-Sea Res. I, 42, 1951–1964, https://doi.org/10.1016/0967-0637(95)00076-3.

    • Search Google Scholar
    • Export Citation

  • Novelli, G., C. M. Guigand, C. Cousin, E. H. Ryan, N. J. M. Laxague, H. Dai, B. K. Haus, and T. M. Özgökmen, 2017: A biodegradable surface drifter for ocean sampling on a massive scale. J. Atmos. Oceanic Technol., 34, 2509–2532, https://doi.org/10.1175/JTECH-D-17-0055.1.

    • Search Google Scholar
    • Export Citation

  • Ohlmann, J. C., M. J. Molemaker, B. Baschek, B. Holt, G. Marmorino, and G. Smith, 2017: Drifter observations of submesoscale flow kinematics in the coastal ocean. Geophys. Res. Lett., 44, 330–337, https://doi.org/10.1002/2016GL071537.

    • Search Google Scholar
    • Export Citation

  • Okubo, A., C. C. Ebbesmeyer, and J. M. Helseth, 1976: Determination of Lagrangian deformations from analysis of current followers. J. Phys. Oceanogr., 6, 524–527, https://doi.org/10.1175/1520-0485(1976)006<0524:DOLDFA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Omand, M. M., E. A. D’Asaro, C. M. Lee, M. J. Perry, N. Briggs, I. Cetinić, and A. Mahadevan, 2015: Eddy-driven subduction exports particulate organic carbon from the spring bloom. Science, 348, 222–225, https://doi.org/10.1126/science.1260062.

    • Search Google Scholar
    • Export Citation

  • Ortega, C., 2003: Argos capabilities for global ocean monitoring. Building the European Capacity in Operational Oceanography, H. Dahlin et al., Eds., Elsevier, 317–324, https://doi.org/10.1016/S0422-9894(03)80051-X.

  • Pitcher, G. C., F. G. Figueiras, B. M. Hickey, and M. T. Moita, 2010: The physical oceanography of upwelling systems and the development of harmful algal blooms. Prog. Oceanogr., 85, 5–32, https://doi.org/10.1016/j.pocean.2010.02.002.

    • Search Google Scholar
    • Export Citation

  • Poje, A. C., and Coauthors, 2014: Submesoscale dispersion in the vicinity of the Deepwater Horizon spill. Proc. Natl. Acad. Sci. USA, 111, 12 693–12 698, https://doi.org/10.1073/pnas.1402452111.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., and R. Gerin, 2019: Assessment of the water-following capabilities of CODE drifters based on direct relative flow measurements. J. Atmos. Oceanic Technol., 36, 621–633, https://doi.org/10.1175/JTECH-D-18-0097.1.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., R. Gerin, E. Mauri, and R. Pennel, 2009: Wind effects on drogued and undrogued drifters in the eastern Mediterranean. J. Atmos. Oceanic Technol., 26, 1144–1156, https://doi.org/10.1175/2008JTECHO618.1.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., L. Centurioni, and T. Özgökmen, 2022: Comparing the currents measured by CARTHE, CODE and SVP drifters as a function of wind and wave conditions in the southwestern Mediterranean Sea. Sensors, 22, 353, https://doi.org/10.3390/s22010353.

    • Search Google Scholar
    • Export Citation

  • Shcherbina, A. Y., E. A. D’Asaro, C. M. Lee, J. M. Klymak, M. J. Molemaker, and J. C. McWilliams, 2013: Statistics of vertical vorticity, divergence, and strain in a developed sub-mesoscale turbulence field. Geophys. Res. Lett., 40, 4706–4711, https://doi.org/10.1002/grl.50919.

    • Search Google Scholar
    • Export Citation

  • Tarry, D. R., and Coauthors, 2021: Frontal convergence and vertical velocity measured by drifters in the Alboran Sea. J. Geophys. Res. Oceans, 126, e2020JC016614, https://doi.org/10.1029/2020JC016614.

    • Search Google Scholar
    • Export Citation

  • Tintoré, J., D. Gomis, S. Alonso, and G. Parrilla, 1991: Mesoscale dynamics and vertical motion in the Alborán Sea. J. Phys. Oceanogr., 21, 811–823, https://doi.org/10.1175/1520-0485(1991)021<0811:MDAVMI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tozer, B., D. T. Sandwell, W. H. F. Smith, C. Olson, J. R. Beale, and P. Wessel, 2019: Global bathymetry and topography at 15 arc sec: SRTM15+. Earth Space Sci., 6, 1847–1864, https://doi.org/10.1029/2019EA000658.

    • Search Google Scholar
    • Export Citation

  • Verma, V., H. T. Pham, and S. Sarkar, 2019: The submesoscale, the finescale and their interaction at a mixed layer front. Ocean Modell., 140, 101400, https://doi.org/10.1016/j.ocemod.2019.05.004.

    • Search Google Scholar
    • Export Citation

  • Viúdez, Á., J. Tintoré, and R. L. Haney, 1996: Circulation in the Alboran Sea as determined by quasi-synoptic hydrographic observations. Part I: Three-dimensional structure of the two anticyclonic gyres. J. Phys. Oceanogr., 26, 684–705, https://doi.org/10.1175/1520-0485(1996)026<0684:CITASA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yaremchuk, M., and E. F. Coelho, 2015: Filtering drifter trajectories sampled at submesoscale resolution. IEEE J. Oceanic Eng., 40, 497–505, https://doi.org/10.1109/JOE.2014.2353472.

    • Search Google Scholar
    • Export Citation
Save
Cover Journal of Atmospheric and Oceanic Technology
Journal of Atmospheric and Oceanic Technology

  • Allshouse, M. R., and T. Peacock, 2015a: Lagrangian based methods for coherent structure detection. Chaos, 25, 097617, https://doi.org/10.1063/1.4922968.

    • Search Google Scholar
    • Export Citation

  • Allshouse, M. R., and T. Peacock, 2015b: Refining finite-time Lyapunov exponent ridges and the challenges of classifying them. Chaos, 25, 087410, https://doi.org/10.1063/1.4928210.

    • Search Google Scholar
    • Export Citation

  • Aravind, H. M., V. Verma, S. Sarkar, M. A. Freilich, A. Mahadevan, P. J. Haley, P. F. J. Lermusiaux, and M. R. Allshouse, 2023: Lagrangian surface signatures reveal upper-ocean vertical displacement conduits near oceanic density fronts. Ocean Modell., 181, 102136, https://doi.org/10.1016/j.ocemod.2022.102136.

    • Search Google Scholar
    • Export Citation

  • Benson, E., 2012: One infrastructure, many global visions: The commercialization and diversification of Argos, a satellite-based environmental surveillance system. Soc. Stud. Sci., 42, 843–868, https://doi.org/10.1177/0306312712457851.

    • Search Google Scholar
    • Export Citation

  • Berta, M., A. Griffa, T. M. Özgökmen, and A. C. Poje, 2016: Submesoscale evolution of surface drifter triads in the Gulf of Mexico. Geophys. Res. Lett., 43, 11 751–11 759, https://doi.org/10.1002/2016GL070357.

    • Search Google Scholar
    • Export Citation

  • Berta, M., and Coauthors, 2020: Submesoscale kinematic properties in summer and winter surface flows in the northern Gulf of Mexico. J. Geophys. Res. Oceans, 125, e2020JC016085, https://doi.org/10.1029/2020JC016085.

    • Search Google Scholar
    • Export Citation

  • Centurioni, L. R., 2018: Drifter technology and impacts for sea surface temperature, sea-level pressure, and ocean circulation studies. Observing the Oceans in Real Time, Springer, 37–57, https://doi.org/10.1007/978-3-319-66493-4_3.

  • Costa, D. P., and Coauthors, 2010: Accuracy of Argos locations of pinnipeds at-sea estimated using Fastloc GPS. PLOS ONE, 5, e8677, https://doi.org/10.1371/journal.pone.0008677.

    • Search Google Scholar
    • Export Citation

  • Cózar, A., S. Aliani, O. C. Basurko, M. Arias, A. Isobe, K. Topouzelis, A. Rubio, and C. Morales-Caselles, 2021: Marine litter windrows: A strategic target to understand and manage the ocean plastic pollution. Front. Mar. Sci., 8, 571796, https://doi.org/10.3389/fmars.2021.571796.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., and Coauthors, 2018: Ocean convergence and the dispersion of flotsam. Proc. Natl. Acad. Sci. USA, 115, 1162–1167, https://doi.org/10.1073/pnas.1718453115.

    • Search Google Scholar
    • Export Citation

  • Douglas, D. C., R. Weinzierl, S. C. Davidson, R. Kays, M. Wikelski, and G. Bohrer, 2012: Moderating Argos location errors in animal tracking data. Methods Ecol. Evol., 3, 999–1007, https://doi.org/10.1111/j.2041-210X.2012.00245.x.

    • Search Google Scholar
    • Export Citation

  • Dräger-Dietel, J., K. Jochumsen, A. Griesel, and G. Badin, 2018: Relative dispersion of surface drifters in the Benguela upwelling region. J. Phys. Oceanogr., 48, 2325–2341, https://doi.org/10.1175/JPO-D-18-0027.1.

    • Search Google Scholar
    • Export Citation

  • Egbert, G. D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183–204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Environmental Modeling Center, 2003: The GFS atmospheric model. NCEP Office Note 442, 14 pp., http://www.lib.ncep.noaa.gov/ncepofficenotes/files/on442.pdf.

  • Esposito, G., M. Berta, L. Centurioni, T. M. S. Johnston, J. Lodise, T. Özgökmen, P.-M. Poulain, and A. Griffa, 2021: Submesoscale vorticity and divergence in the Alboran Sea: Scale and depth dependence. Front. Mar. Sci., 8, 678304, https://doi.org/10.3389/fmars.2021.678304.

    • Search Google Scholar
    • Export Citation

  • Essink, S., V. Hormann, L. R. Centurioni, and A. Mahadevan, 2022: On characterizing ocean kinematics from surface drifters. J. Atmos. Oceanic Technol., 39, 1183–1198, https://doi.org/10.1175/JTECH-D-21-0068.1.

    • Search Google Scholar
    • Export Citation

  • Fang, L., S. Balasuriya, and N. T. Ouellette, 2020: Disentangling resolution, precision, and inherent stochasticity in nonlinear systems. Phys. Rev. Res., 2, 023343, https://doi.org/10.1103/PhysRevResearch.2.023343.

    • Search Google Scholar
    • Export Citation

  • Farazmand, M., and G. Haller, 2012: Computing Lagrangian coherent structures from their variational theory. Chaos, 22, 013128, https://doi.org/10.1063/1.3690153.

    • Search Google Scholar
    • Export Citation

  • Filippi, M., R. Hanlon, I. I. Rypina, B. A. Hodges, T. Peacock, and D. G. Schmale III, 2021: Tracking a surrogate hazardous agent (rhodamine dye) in a coastal ocean environment using in situ measurements and concentration estimates derived from drone images. Remote Sens., 13, 4415, https://doi.org/10.3390/rs13214415.

    • Search Google Scholar
    • Export Citation

  • Freilich, M., and A. Mahadevan, 2021: Coherent pathways for subduction from the surface mixed layer at ocean fronts. J. Geophys. Res. Oceans, 126, e2020JC017042, https://doi.org/10.1029/2020JC017042.

    • Search Google Scholar
    • Export Citation

  • Gonçalves, R. C., M. Iskandarani, T. Özgökmen, and W. C. Thacker, 2019: Reconstruction of submesoscale velocity field from surface drifters. J. Phys. Oceanogr., 49, 941–958, https://doi.org/10.1175/JPO-D-18-0025.1.

    • Search Google Scholar
    • Export Citation

  • Haley, P. J. Jr., and P. F. J. Lermusiaux, 2010: Multiscale two-way embedding schemes for free-surface primitive equations in the “Multidisciplinary Simulation, Estimation and Assimilation System.” Ocean Dyn., 60, 1497–1537, https://doi.org/10.1007/s10236-010-0349-4.

    • Search Google Scholar
    • Export Citation

  • Haley, P. J. Jr., A. Agarwal, and P. F. J. Lermusiaux, 2015: Optimizing velocities and transports for complex coastal regions and archipelagos. Ocean Modell., 89, 1–28, https://doi.org/10.1016/j.ocemod.2015.02.005.

    • Search Google Scholar
    • Export Citation

  • Hernández-Carrasco, I., A. Orfila, V. Rossi, and V. Garçon, 2018: Effect of small scale transport processes on phytoplankton distribution in coastal seas. Sci. Rep., 8, 8613, https://doi.org/10.1038/s41598-018-26857-9.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., G. Jacobs, and A. D. Kirwan Jr., 2015: Clusters, deformation, and dilation: Diagnostics for material accumulation regions. J. Geophys. Res. Oceans, 120, 6622–6636, https://doi.org/10.1002/2015JC011036.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., and A. D. Kirwan Jr., 2019: Anisotropy and inhomogeneity in drifter dispersion. J. Geophys. Res. Oceans, 124, 8667–8682, https://doi.org/10.1029/2019JC015179.

    • Search Google Scholar
    • Export Citation

  • Huntley, H. S., B. L. Lipphardt Jr., E. Ryan, B. Haus, and T. Özgökmen, 2020: Accuracy test for SPOT GPS units used in the surface drifters deployed in the Grand Lagrangian Deployment (GLAD) experiment, collected 26–27 June and 3–4 July 2012 in Miami, Florida. GRIIDC, accessed 15 July 2022, https://doi.org/10.7266/2H3H2KHC.

  • Huntley, H. S., M. Berta, G. Esposito, A. Griffa, B. Mourre, and L. Centurioni, 2022: Conditions for reliable divergence estimates from drifter triplets. J. Atmos. Oceanic Technol., 39, 1499–1523, https://doi.org/10.1175/JTECH-D-21-0161.1.

    • Search Google Scholar
    • Export Citation

  • Juza, M., and Coauthors, 2016: SOCIB operational ocean forecasting system and multi-platform validation in the western Mediterranean Sea. J. Oper. Oceanogr., 9, s155–s166, https://doi.org/10.1080/1755876X.2015.1117764.

    • Search Google Scholar
    • Export Citation

  • Kawai, H., 1985: Scale dependence of divergence and vorticity of near-surface flows in the sea. J. Oceanogr. Soc. Japan, 41, 157–166, https://doi.org/10.1007/BF02111115.

    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., 2010: Observations of submesoscale eddies using high-frequency radar-derived kinematic and dynamic quantities. Cont. Shelf Res., 30, 1639–1655, https://doi.org/10.1016/j.csr.2010.06.011.

    • Search Google Scholar
    • Export Citation

  • Lumpkin, R., T. Özgökmen, and L. Centurioni, 2017: Advances in the application of surface drifters. Annu. Rev. Mar. Sci., 9, 59–81, https://doi.org/10.1146/annurev-marine-010816-060641.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., 2006: Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales? Ocean Modell., 14, 222–240, https://doi.org/10.1016/j.ocemod.2006.05.005.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., A. Pascual, D. L. Rudnick, S. Ruiz, J. Tintoré, and E. D’Asaro, 2020a: Coherent pathways for vertical transport from the surface ocean to interior. Bull. Amer. Meteor. Soc., 101, E1996–E2004, https://doi.org/10.1175/BAMS-D-19-0305.1.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., and Coauthors, 2020b: CALYPSO 2019 cruise report: Field campaign in the Mediterranean. Woods Hole Oceanographic Institution Tech. Rep., WHOI-2020-02, 123 pp., https://doi.org/10.1575/1912/25266.

  • Molinari, R., and A. D. Kirwan Jr., 1975: Calculations of differential kinematic properties from Lagrangian observations in the western Caribbean Sea. J. Phys. Oceanogr., 5, 483–491, https://doi.org/10.1175/1520-0485(1975)005<0483:CODKPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mourre, B., and Coauthors, 2018: Assessment of high-resolution regional ocean prediction systems using multi-platform observations: Illustrations in the western Mediterranean Sea. New Frontiers in Operational Oceanography, E. Chassignet et al., Eds., GODAE OceanView, 663–694.

  • Niiler, P., 2001: The World Ocean surface circulation. Ocean Circulation and Climate, G. Siedler, J. Church, and J. Gould, Eds., Academic Press, 193–204.

  • Niiler, P., A. S. Sybrandy, K. Bi, P. M. Poulain, and D. Bitterman, 1995: Measurements of the water-following capability of holey-sock and TRISTAR drifters. Deep-Sea Res. I, 42, 1951–1964, https://doi.org/10.1016/0967-0637(95)00076-3.

    • Search Google Scholar
    • Export Citation

  • Novelli, G., C. M. Guigand, C. Cousin, E. H. Ryan, N. J. M. Laxague, H. Dai, B. K. Haus, and T. M. Özgökmen, 2017: A biodegradable surface drifter for ocean sampling on a massive scale. J. Atmos. Oceanic Technol., 34, 2509–2532, https://doi.org/10.1175/JTECH-D-17-0055.1.

    • Search Google Scholar
    • Export Citation

  • Ohlmann, J. C., M. J. Molemaker, B. Baschek, B. Holt, G. Marmorino, and G. Smith, 2017: Drifter observations of submesoscale flow kinematics in the coastal ocean. Geophys. Res. Lett., 44, 330–337, https://doi.org/10.1002/2016GL071537.

    • Search Google Scholar
    • Export Citation

  • Okubo, A., C. C. Ebbesmeyer, and J. M. Helseth, 1976: Determination of Lagrangian deformations from analysis of current followers. J. Phys. Oceanogr., 6, 524–527, https://doi.org/10.1175/1520-0485(1976)006<0524:DOLDFA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Omand, M. M., E. A. D’Asaro, C. M. Lee, M. J. Perry, N. Briggs, I. Cetinić, and A. Mahadevan, 2015: Eddy-driven subduction exports particulate organic carbon from the spring bloom. Science, 348, 222–225, https://doi.org/10.1126/science.1260062.

    • Search Google Scholar
    • Export Citation

  • Ortega, C., 2003: Argos capabilities for global ocean monitoring. Building the European Capacity in Operational Oceanography, H. Dahlin et al., Eds., Elsevier, 317–324, https://doi.org/10.1016/S0422-9894(03)80051-X.

  • Pitcher, G. C., F. G. Figueiras, B. M. Hickey, and M. T. Moita, 2010: The physical oceanography of upwelling systems and the development of harmful algal blooms. Prog. Oceanogr., 85, 5–32, https://doi.org/10.1016/j.pocean.2010.02.002.

    • Search Google Scholar
    • Export Citation

  • Poje, A. C., and Coauthors, 2014: Submesoscale dispersion in the vicinity of the Deepwater Horizon spill. Proc. Natl. Acad. Sci. USA, 111, 12 693–12 698, https://doi.org/10.1073/pnas.1402452111.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., and R. Gerin, 2019: Assessment of the water-following capabilities of CODE drifters based on direct relative flow measurements. J. Atmos. Oceanic Technol., 36, 621–633, https://doi.org/10.1175/JTECH-D-18-0097.1.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., R. Gerin, E. Mauri, and R. Pennel, 2009: Wind effects on drogued and undrogued drifters in the eastern Mediterranean. J. Atmos. Oceanic Technol., 26, 1144–1156, https://doi.org/10.1175/2008JTECHO618.1.

    • Search Google Scholar
    • Export Citation

  • Poulain, P.-M., L. Centurioni, and T. Özgökmen, 2022: Comparing the currents measured by CARTHE, CODE and SVP drifters as a function of wind and wave conditions in the southwestern Mediterranean Sea. Sensors, 22, 353, https://doi.org/10.3390/s22010353.

    • Search Google Scholar
    • Export Citation

  • Shcherbina, A. Y., E. A. D’Asaro, C. M. Lee, J. M. Klymak, M. J. Molemaker, and J. C. McWilliams, 2013: Statistics of vertical vorticity, divergence, and strain in a developed sub-mesoscale turbulence field. Geophys. Res. Lett., 40, 4706–4711, https://doi.org/10.1002/grl.50919.

    • Search Google Scholar
    • Export Citation

  • Tarry, D. R., and Coauthors, 2021: Frontal convergence and vertical velocity measured by drifters in the Alboran Sea. J. Geophys. Res. Oceans, 126, e2020JC016614, https://doi.org/10.1029/2020JC016614.

    • Search Google Scholar
    • Export Citation

  • Tintoré, J., D. Gomis, S. Alonso, and G. Parrilla, 1991: Mesoscale dynamics and vertical motion in the Alborán Sea. J. Phys. Oceanogr., 21, 811–823, https://doi.org/10.1175/1520-0485(1991)021<0811:MDAVMI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tozer, B., D. T. Sandwell, W. H. F. Smith, C. Olson, J. R. Beale, and P. Wessel, 2019: Global bathymetry and topography at 15 arc sec: SRTM15+. Earth Space Sci., 6, 1847–1864, https://doi.org/10.1029/2019EA000658.

    • Search Google Scholar
    • Export Citation

  • Verma, V., H. T. Pham, and S. Sarkar, 2019: The submesoscale, the finescale and their interaction at a mixed layer front. Ocean Modell., 140, 101400, https://doi.org/10.1016/j.ocemod.2019.05.004.

    • Search Google Scholar
    • Export Citation

  • Viúdez, Á., J. Tintoré, and R. L. Haney, 1996: Circulation in the Alboran Sea as determined by quasi-synoptic hydrographic observations. Part I: Three-dimensional structure of the two anticyclonic gyres. J. Phys. Oceanogr., 26, 684–705, https://doi.org/10.1175/1520-0485(1996)026<0684:CITASA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yaremchuk, M., and E. F. Coelho, 2015: Filtering drifter trajectories sampled at submesoscale resolution. IEEE J. Oceanic Eng., 40, 497–505, https://doi.org/10.1109/JOE.2014.2353472.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Schematic of the various areas considered for different dilation-rate calculations at the initial time t0 and final time tf: (a) evolution of an infinitesimal fluid patch into an ellipse, considered to measure the pointwise dilation rate, (b) evolution of a finite-sized fluid patch (blue) and an infinitesimal component (gray) used to compute the area-averaged dilation rate, (c) evolution of the area of a drifter polygon (beige) used to discretize the finite-sized fluid patch (blue), with the three dots representing positions of the drifter triad, considered to compute the polygonal dilation rate, (d) distorted area of the drifter polygon at tf (green) as a result of GPS position uncertainty, with the uncertainty zone for each drifter marked around it, used to compute the drifter-based dilation rate, and (e) observed area at tf (green) of the drifter polygon deployed at t0 (beige) to track the evolution of a finite-sized fluid patch (blue).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) Modeled horizontal divergence of velocity ∇H ⋅ u, nondimensionalized using the Coriolis parameter f (=0.86 × 10−4 s−1) (color shading), plus the surface velocity field (vectors) at 1400:00 LT 25 Mar 2019. (b) Normalized pointwise dilation rates Δ/f computed over a 24-h interval, plotted at the initial positions at 1400:00 LT 25 Mar 2019.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Excerpts in the region north of Alboran Island of (a) pointwise dilation rate Δ, along with area-averaged dilation rate ΔR for R = (b) 100 m and (c) 1, (d) 2, and (e) 5 km. Also shown are (f) dilation-rate transects at 2.93°W, shown by the thick vertical lines in (a)–(e), plotted as a function of meridional distance.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Excerpts of normalized polygonal dilation rates for R = 100 m (Δ100,N) in the region north of Alboran Island, for (a) N = 3, (b) N = 4, and (c) N = 8. Also shown are (d) dilation-rate transects, shown by the thick vertical lines in (a)–(c), plotted as a function of meridional distance. The area-averaged dilation-rate transect for R = 100 m (Δ100) presented in green in Fig. 3f is also plotted for comparison.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    (a) Error in dilation-rate estimates due to boundary discretization of the fluid patch averaged over the entire domain (ϵ¯D) for N = 3–8, plotted as a function of R. (b) Two-dimensional histogram of the error ϵD vs normalized baseline ΔR/f for (R, N) = (100, 4). A bin width of 0.05 is used for both variables. The mean error for wider bins of 0.10 in the x direction is also plotted, with the data point colored based on the bin count, along with the sided standard difference as error bars. The black dashed line separates L-converging regions from L-diverging ones. Within the L-converging or L-diverging regions, the dotted lines demarcate the top 1% L-converging or L-diverging regions. Also shown is ϵD averaged over (c) the top 1% L-converging and (d) the top 1% L-diverging regions for N = 3–8, as a function of R.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) Excerpt of one realization of the normalized drifter-based dilation rate Δ1R,N,σ, for (R, N, σ) = (100, 3, 6) in the region north of Alboran Island. (b) Dilation-rate transects, shown using a thick vertical line in (a), plotted as a function of meridional distance for all 100 realizations (black). The polygonal-dilation-rate transect Δ100,3 presented in Fig. 4d is also plotted (blue) for comparison. Also shown is the ensemble averaged error in dilation-rate estimates due to position uncertainty ϵP averaged over (c) the entire domain and (d) the top 1% L-converging regions for N = 3–8, plotted as a function of R.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Total error ϵT in drifter-based dilation-rate estimates (solid lines with circles), along with ϵD (long dashes) and ϵP (short dashes), plotted as a function of R for N = 3 (blue) and N = 4 (red), averaged over (a) the full domain and (b) the top 1% L-converging regions. Note that the ϵ¯T curve may lie on the ϵ¯D and ϵ¯P curves, for small and large R, respectively.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Total error ϵT in drifter-based dilation-rate estimates, averaged over (a) the full domain and (b) the top 1% L-converging regions, as a function of R and σ. The results are presented for R = 1, 2, 5, 10, 20, 50, 100, 250, 500, 750, and 1000 m and σ = 0.75, 1.5, 3, 6, 9, 12, 25 and 50 m. The optimal deployment radii for each sigma, RΩ* and RC* for the full domain and top 1% L-converging regions, respectively, are also marked with a black circle around the corresponding data point.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Strongest L-converging regions in the domain, as identified by the drifter-tetrad-based estimates alone (blue), pointwise dilation rate alone (red), and both drifter-based and pointwise dilation rates (green) for the top (a)–(c) 1% and (d)–(f) 5% L-converging regions for (top) R = 50 m, (middle) R = 250 m, and (bottom) R = 1000 m. The percent of green regions, i.e., the percent overlap of L-converging regions identified by the drifter-based estimates and pointwise measurements, is also illustrated in a Venn diagram.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Percent of (a) the top 1% and (b) the top 5% L-converging regions in the pointwise dilation-rate fields located by the drifter-based dilation rates for N = 3 (blue, solid) and 4 (red, solid), as a function of R. The percent overlaps for polygonal dilation rates are also plotted (dashed) to highlight the role played by position uncertainty.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Deployment strategy to sample the L-convergent filament: (a) Forty-eight drifters (black dots) deployed as a 4 × 3 grid of drifter tetrads, each deployed at R = 250 m, to sample the region near the L-convergent filament observed in the normalized pointwise dilation-rate field (shading). (b) Drifter tetrads (black dots), along with the normalized dilation rates estimated using their trajectories (colors), assuming a 6-m position uncertainty.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    (left) Recorded positions relative to the median position over all units in each of the GPS accuracy tests. Dark-blue circles show the first test, and red triangles show the second test. The gray circle has a radius of 35 m. (right) Differences in the empirical distributions from the observation and the two types of models, two independent Gaussian distributions (blue) and normally distributed magnitude with uniformly distributed angle (red) for a range of different standard deviations σ.

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    Histograms, scaled as a probability density function (pdf), for the (left) zonal, (center) meridional, and (right) total errors for (top) the model with Gaussian zonal and meridional errors, using σ = 3.5 m and (bottom) the model with Gaussian signed error magnitude and uniformly distributed angle, using σ = 5.5 m. Observed results are plotted in blue, modeled results are plotted orange, and overlaps of the histograms appear brown. The bright-red curves in the top-left and top-center panels show the corresponding Gaussian distributions, illustrating that the empirical histograms capture the pdf well.

Fig. 1.

Schematic of the various areas considered for different dilation-rate calculations at the initial time t0 and final time tf: (a) evolution of an infinitesimal fluid patch into an ellipse, considered to measure the pointwise dilation rate, (b) evolution of a finite-sized fluid patch (blue) and an infinitesimal component (gray) used to compute the area-averaged dilation rate, (c) evolution of the area of a drifter polygon (beige) used to discretize the finite-sized fluid patch (blue), with the three dots representing positions of the drifter triad, considered to compute the polygonal dilation rate, (d) distorted area of the drifter polygon at tf (green) as a result of GPS position uncertainty, with the uncertainty zone for each drifter marked around it, used to compute the drifter-based dilation rate, and (e) observed area at tf (green) of the drifter polygon deployed at t0 (beige) to track the evolution of a finite-sized fluid patch (blue).
Fig. 2.

(a) Modeled horizontal divergence of velocity ∇H ⋅ u, nondimensionalized using the Coriolis parameter f (=0.86 × 10−4 s−1) (color shading), plus the surface velocity field (vectors) at 1400:00 LT 25 Mar 2019. (b) Normalized pointwise dilation rates Δ/f computed over a 24-h interval, plotted at the initial positions at 1400:00 LT 25 Mar 2019.
Fig. 3.

Excerpts in the region north of Alboran Island of (a) pointwise dilation rate Δ, along with area-averaged dilation rate ΔR for R = (b) 100 m and (c) 1, (d) 2, and (e) 5 km. Also shown are (f) dilation-rate transects at 2.93°W, shown by the thick vertical lines in (a)–(e), plotted as a function of meridional distance.
Fig. 4.

Excerpts of normalized polygonal dilation rates for R = 100 m (Δ100,N) in the region north of Alboran Island, for (a) N = 3, (b) N = 4, and (c) N = 8. Also shown are (d) dilation-rate transects, shown by the thick vertical lines in (a)–(c), plotted as a function of meridional distance. The area-averaged dilation-rate transect for R = 100 m (Δ100) presented in green in Fig. 3f is also plotted for comparison.
Fig. 5.

(a) Error in dilation-rate estimates due to boundary discretization of the fluid patch averaged over the entire domain (ϵ¯D) for N = 3–8, plotted as a function of R. (b) Two-dimensional histogram of the error ϵD vs normalized baseline ΔR/f for (R, N) = (100, 4). A bin width of 0.05 is used for both variables. The mean error for wider bins of 0.10 in the x direction is also plotted, with the data point colored based on the bin count, along with the sided standard difference as error bars. The black dashed line separates L-converging regions from L-diverging ones. Within the L-converging or L-diverging regions, the dotted lines demarcate the top 1% L-converging or L-diverging regions. Also shown is ϵD averaged over (c) the top 1% L-converging and (d) the top 1% L-diverging regions for N = 3–8, as a function of R.
Fig. 6.

(a) Excerpt of one realization of the normalized drifter-based dilation rate Δ1R,N,σ, for (R, N, σ) = (100, 3, 6) in the region north of Alboran Island. (b) Dilation-rate transects, shown using a thick vertical line in (a), plotted as a function of meridional distance for all 100 realizations (black). The polygonal-dilation-rate transect Δ100,3 presented in Fig. 4d is also plotted (blue) for comparison. Also shown is the ensemble averaged error in dilation-rate estimates due to position uncertainty ϵP averaged over (c) the entire domain and (d) the top 1% L-converging regions for N = 3–8, plotted as a function of R.
Fig. 7.

Total error ϵT in drifter-based dilation-rate estimates (solid lines with circles), along with ϵD (long dashes) and ϵP (short dashes), plotted as a function of R for N = 3 (blue) and N = 4 (red), averaged over (a) the full domain and (b) the top 1% L-converging regions. Note that the ϵ¯T curve may lie on the ϵ¯D and ϵ¯P curves, for small and large R, respectively.
Fig. 8.

Total error ϵT in drifter-based dilation-rate estimates, averaged over (a) the full domain and (b) the top 1% L-converging regions, as a function of R and σ. The results are presented for R = 1, 2, 5, 10, 20, 50, 100, 250, 500, 750, and 1000 m and σ = 0.75, 1.5, 3, 6, 9, 12, 25 and 50 m. The optimal deployment radii for each sigma, RΩ* and RC* for the full domain and top 1% L-converging regions, respectively, are also marked with a black circle around the corresponding data point.
Fig. 9.

Strongest L-converging regions in the domain, as identified by the drifter-tetrad-based estimates alone (blue), pointwise dilation rate alone (red), and both drifter-based and pointwise dilation rates (green) for the top (a)–(c) 1% and (d)–(f) 5% L-converging regions for (top) R = 50 m, (middle) R = 250 m, and (bottom) R = 1000 m. The percent of green regions, i.e., the percent overlap of L-converging regions identified by the drifter-based estimates and pointwise measurements, is also illustrated in a Venn diagram.
Fig. 10.

Percent of (a) the top 1% and (b) the top 5% L-converging regions in the pointwise dilation-rate fields located by the drifter-based dilation rates for N = 3 (blue, solid) and 4 (red, solid), as a function of R. The percent overlaps for polygonal dilation rates are also plotted (dashed) to highlight the role played by position uncertainty.
Fig. 11.

Deployment strategy to sample the L-convergent filament: (a) Forty-eight drifters (black dots) deployed as a 4 × 3 grid of drifter tetrads, each deployed at R = 250 m, to sample the region near the L-convergent filament observed in the normalized pointwise dilation-rate field (shading). (b) Drifter tetrads (black dots), along with the normalized dilation rates estimated using their trajectories (colors), assuming a 6-m position uncertainty.
Fig. A1.

(left) Recorded positions relative to the median position over all units in each of the GPS accuracy tests. Dark-blue circles show the first test, and red triangles show the second test. The gray circle has a radius of 35 m. (right) Differences in the empirical distributions from the observation and the two types of models, two independent Gaussian distributions (blue) and normally distributed magnitude with uniformly distributed angle (red) for a range of different standard deviations σ.
Fig. A2.

Histograms, scaled as a probability density function (pdf), for the (left) zonal, (center) meridional, and (right) total errors for (top) the model with Gaussian zonal and meridional errors, using σ = 3.5 m and (bottom) the model with Gaussian signed error magnitude and uniformly distributed angle, using σ = 5.5 m. Observed results are plotted in blue, modeled results are plotted orange, and overlaps of the histograms appear brown. The bright-red curves in the top-left and top-center panels show the corresponding Gaussian distributions, illustrating that the empirical histograms capture the pdf well.
All Time Past Year Past 30 Days
Abstract Views 2007 1602 0
Full Text Views 1237 1120 103
PDF Downloads 296 141 28