a. Bay of Bengal drifter dataset

A total of 46 drifters (one of which failed upon deployment) were released in the northern Bay of Bengal, at the edge of a strong cyclonic eddy and across a freshwater-dominated density front (Hormann et al. 2016; Essink et al. 2019) (Fig. 1). Surface Velocity Program (SVP) drifters drogued at 15-m depth minimize Stokes drift by surface gravity waves and wind slippage (Niiler et al. 1995; Centurioni 2018). Since mixed layers were shallow (see Fig. 1b), this drogue depth might affect the extent to which mixed layer currents are sampled. All drifters were equipped with a thermistor centimeters below the surface. Thirty-six drifters carried an additional conductivity sensor at about 0.5-m depth to infer salinity (Hormann et al. 2015, 2016) and allowing to compute near-surface density which is largely controlled by changes in salinity in the Bay of Bengal (e.g., Jaeger and Mahadevan 2018; Mahadevan et al. 2016).

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(a) Map of the Bay of Bengal on 23 Aug 2015, one week before the drifter deployments. Sea surface salinity from the Soil Moisture Active Passive (SMAP) mission is shown in color (Entekhabi et al. 2010). Light (dark) contours are positive (negative) sea level anomalies from delayed-time, gridded 0.25° Copernicus Marine Environment Monitoring Service (CMEMS) data (Traon et al. 1998; Ducet et al. 2000). The drifter release location is marked by the red triangle. (b) Example north–south CTD section of salinity with density contours collected on 15 Sep 2015 along a section shown as a thick black line in (a). (c) Two months of drifter trajectories released at the red triangle overlaid onto CMEMS sea level anomalies with days after deployment are shown in color.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The drifters were programmed to report GPS positions at 5-min intervals during the first month of deployment and at 30-min intervals thereafter, sufficient to resolve submesoscale variability. To resolve flow patterns at length scales from 1 to 30 km, the drifters were released in tight clusters of four drifters with $\mathcal{O}(1)\hspace{0.17em}\text{km}$ initial separation. Ten clusters formed a larger swarm spanning an area of about 30 km² (Hormann et al. 2016). The following analysis is an expansion of the work in Hormann et al. (2016), where this dataset was used to provide first estimates of the kinematic properties using the Saucier method.

b. Numerical model

The nonhydrostatic Process Study Ocean Model (PSOM) (Mahadevan et al. 1996a,b) was used to simulate a submesoscale flow field, initialized with a strong density front measured in the northern Bay of Bengal in 2013 with a profiling conductivity–temperature–depth (CTD) probe. The model initial condition is characterized by a north–south buoyancy gradient set up by riverine freshwater in the northern bay, similar to conditions during the 2015 field campaign (Jaeger and Mahadevan 2018) during which the drifter data were collected.

The model is in channel configuration, periodic in east–west and with solid boundaries in the north–south direction, spanning a 320 × 190 km² area with a 1000-m flat bottom. The horizontal grid resolution is Δx = Δy = 1 km, sufficient to resolve submesoscale variability. The vertical coordinate is exponentially stretched over 35 levels with Δz ≈ 2 m at the surface and Δz ≈ 100 m at the bottom. The simulation is stepped forward with a time step of Δt ≈ 5 min. A constant-diffusivity along-isopycnal mixing is used with κ ≈ 1 m² s⁻¹ (Redi 1982) and a “K profile parameterization” (KPP) for vertical mixing (Large et al. 1994).

The simulation is forced with realistic wind stress and heat fluxes measured by a mooring at 18°N in the Bay of Bengal (Weller et al. 2016). The surface forcing counteracts a loss of eddy kinetic energy. During initialization, the north–south front becomes unstable to shear and baroclinic instability and breaks up into smaller eddies with diameters of $\mathcal{O}(1\text{–}10)\hspace{0.17em}\text{km}$. After about 30 days, the domain-averaged kinetic energy changes slowly, mean mixed layer depth stabilizes at ∼50 m, and a fully eddying flow field emerges.

Examples of the spun-up density and vorticity fields are shown in Fig. 2. Stirring by eddies generates elongated filaments of sharp density gradient with vorticity of up to 5f. The distribution of vorticity is positively skewed; that is, larger positive flow features being more frequent than negative eddies. Furthermore, positive-vorticity filaments and fronts are associated with strong downward velocity in their vicinity (not shown).

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Three-dimensional image of the fully spun-up numerical simulation. Front and top faces of the domain are shown of (a) density anomaly, σ₀ = ρ − 1000 kg m⁻³, showing a ∼50-m mixed layer and strong cross-domain density gradients and (b) relative vorticity ζ normalized by the Coriolis frequency f showing strong filaments and eddies that reach vorticity magnitudes of up to 5f. Overlaid black dots are examples of drifter clusters at random positions. (c) Zoom onto a filament with drifter clusters, each made up of three drifters.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The model is a process study model with limited spatial domain. We do not claim that it models the Bay of Bengal accurately because it is missing the large-scale circulation context, topography, and large-scale atmospheric forcing that other regional models would include. All of these can change the nature and length scale of fronts. In the model, the smallest scales (or the sharpest fronts) are set by the resolved physics that provide frontogenetic tendencies and the resolved and unresolved processes that mix to dissipate fronts. Our model only parameterizes subgrid-scale mixing, so the frontal scale is close to the grid scale (like it is the case in many models).

c. Estimation of kinematic properties

The two-dimensional velocity gradient tensor G_ij = ∂u_i/∂x_j of the horizontal velocity u_i can be written as the sum of symmetric and antisymmetric parts, G^sym = (G_ij + G_ji)/2, and G^anti = (G_ij − G_ji)/2. The antisymmetric part describes the rotations of a fluid parcel. Its two nonzero components are ±ζ/2, where ζ is the vertical component of the relative vorticity. The trace of the symmetric part is the horizontal divergence $\delta =G_{ii}^{\text{sym}}$ and its off-diagonal components $G_{i\ne j}^{\text{sym}}$ are the shear and normal strain rates. The eigenvalues of G are the rates at which two particles separate, subject to the divergence, strain, and shear of the flow. The sum of the eigenvalues is equal to the divergence δ. Vorticity, horizontal divergence, shear, and normal strain rates are defined as ζ ≡ υ_x − u_y, δ ≡ u_x + υ_y, S_s ≡ υ_x + u_y, and S_n ≡ u_x − υ_y, respectively. Lateral strain rate is defined as $S\equiv \sqrt{S_s^2+S_n^2}$.

We focus on the three methods that have previously been applied to drifter observations: (i) the Saucier method based on the area rate of change of a drifter cluster first documented by Saucier (1953, 1955), (ii) the Kawai method (e.g., Kawai 1985a,b) based on Stokes’s and Gauss’s theorems, and (iii) the LS method (e.g., Molinari and Kirwan 1975). All of these methods require at least three measurements (i.e., N ≥ 3) of velocity and position in space.

Molinari and Kirwan (1975) and Okubo and Ebbesmeyer (1976) developed a bilinear fit to the drifter velocities to determine the velocity gradients. At each instant in time, six parameters are fitted: two mean-flow components $(\overline{u},\hspace{0.17em}\overline{\upsilon })$ and four two-dimensional velocity gradient components. At least three points (i.e., N ≥ 3) for each velocity component are needed to solve this system.

The coefficient vectors of the bilinear fit, C_u and C_ν, contain the mean-flow components and spatial derivatives of the horizontal velocity. These can be combined into the vertical component of the vorticity ζ, the divergence δ, and the lateral strain rate S.

a. Comparison of methods

The kinematic properties are calculated for synthetic drifter clusters, each of which can be described by the number of drifters N, the cluster length scale L, and its aspect ratio α as defined below. For each set of parameters, a total of 10 000 clusters of synthetic drifters are seeded in the model domain (Fig. 2a) at random locations and with random orientations.

Note that the synthetic drifters are not advected by the flow, so that natural deformation of drifter clusters by convergences or shear flows is eliminated entirely and drifters do not sample the flow in a biased way like actual drifters would (e.g., Pearson et al. 2019). Instead, synthetic drifters only sample the model’s velocity field with a cubic spline interpolation at the drifters’ position. The parameter space of cluster shapes will be explored systematically by choosing N, L, and α.

We define the length scale of a drifter cluster as the root-mean-square (rms) distance between all pairs of drifters $L\equiv \sqrt{\langle {({\mathit{x}}_i-{\mathit{x}}_j)}^2\rangle }$, where the angle brackets indicate the mean over all pairs in a cluster; L is normalized by the flow length scale L_u, which is determined from the horizontal wavenumber spectra of velocity and velocity gradients (Fig. 3). We expect L_u to be a fair representation for both velocity components because the grid is regular in both horizontal directions. In the model simulation the flow length scale is L_u ≈ 5 km.

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Wavenumber spectra of the model velocity |u|, vorticity ζ, divergence δ, and lateral strain rate S as a function of horizontal wavenumber k_h. Two spatially filtered vorticity fields, 2 and 5 km, are shown with dotted lines. One-dimensional spectra were computed from two-dimensional wavenumber spectra by transformation to polar coordinates and integration over azimuths, and bin averaging into 10 bins per decade. Each spectrum is normalized by its maximum value for better comparison of spectral slopes and rolloffs. Vertical line at k_h = 1/L_u = 1/5 cpkm indicates the wavenumber of the original model velocity, above which the kinematic property spectra are smooth, and the red vertical lines indicate L_u = 8 km and L_u = 15 km of the spatially filtered fields.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The aspect ratio α = λ_min/λ_max is defined as the ratio of minor and major eigenvalues λ_min/λ_max of the position covariance matrix cov(X, X^T), where X is the matrix of positions (cf. Choi et al. 2017; Spydell et al. 2019). A circular cluster has an aspect ratio of one, an ellipsoidal cluster α < 1, and a colinear cluster α ≈ 0.

The estimated kinematic properties are compared to the true model kinematic properties by averaging over the drifter cluster area; for example, for the vorticity $\overline{{\zeta }_m}\equiv 1/A\int \zeta dA$. We regress the estimate against the “true” values in the numerical model to compute the coefficient of determination R², where R is Pearson’s correlation coefficient defined as R = cov(X, Y)/[std(X)std(Y)] for two populations X and Y. R² determines how much of the total variance is captured by the regression model. A second, more practical metric is the rms error of this regression; the vorticity error is ${ϵ}_{\text{rms}}=\sqrt{{\left(\zeta -{\zeta }_m\right)}^2}$, where ζ_m is the area-averaged model vorticity.

All three methods estimate the kinematic properties from the model velocity field (Fig. 4) with high correlations between the estimated and model values; that is, R² = 0.96. However, even under ideal conditions (i.e., small length scales L/L_u = 0.2, and large aspect ratios α = 1), the Saucier and Kawai methods produce rms vorticity errors of 0.53 and 0.61, respectively. The sign and magnitude of the vorticity can be predicted, but the Saucier and Kawai methods underestimate the vorticity when the magnitude of the true vorticity is large, perhaps because the largest values occur at the smallest scales. The reason why the Saucier and Kawai methods underpredict vorticity without systematic bias at true zero vorticity may be that they are area-averaged estimators, whereas the LS method performs better at recovering the vorticity.

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Comparison of drifter-estimated, normalized vorticity to true model vorticity using (a) the Saucier method based on the area rate of change, (b) the LS method, and (c) the Kawai method based on Stokes’s theorem. For each comparison, 10 000 synthetic drifter clusters were released with length scale L/L_u = 0.2, aspect ratio α = 1, and N = 6 drifters per cluster. The coefficient of determination R², p value, and rms error ϵ_rms are shown in the upper-left corner of each panel.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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On the contrary, the LS method produces smaller rms errors, ϵ = 0.29, at the same correlation R² = 0.96. Especially when the vorticity is small, the LS method appears to perform more robustly. The error dependencies on N, α, and L also hold for lateral strain rate and divergence with similar functional forms (not shown). This is due to the fact that the error here is largely due to nonconstant velocity gradients at and below the cluster length scale L and systematic errors are zero. For the remainder of this section, we use the LS method, which we identify as the most robust method.

b. Impact of unresolved flow length scales

To vary the dominant length scale of variability of the velocity field, we filter the model fields with a two-dimensional Gaussian filter resulting in smooth velocity fields below L_u = 8 and 15 km (Fig. 3). These are typical length scales at submesoscale fronts (e.g., Hormann et al. 2016). Filtering the model output, by smearing out the vorticity field, might not be equivalent to varying the dominant length scale in evolving simulations and likely produces smoother fields below the filter length scale. Nevertheless, this exercise is illuminating to better understand the uncertainty in drifter cluster-based velocity gradient estimates.

As above, we use the coefficient of determination R² and the rms error ϵ_rms to evaluate the performance of the estimate (Fig. 5). Figure 5 shows the vorticity fields for the three test cases: the original model run with Δx = Δy = 1 km, the 2-km filtered field, and the 5-km filtered field. The LS estimate of vorticity in all three cases is well correlated with the true vorticity field, with R² values ranging from 0.8 to 0.99 and p values below 10⁻⁴. As long as the cluster length scales are small relative to the scale of the flow features, the estimates improve and R² increases. Analogously, when the cluster length scales are small relative to the scale of a flow features, the rms errors decrease.

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Vorticity normalized by f of the (a) original simulation with L_u = 5 km (as in Fig. 2b) and (b),(c) the Gaussian filtered field with L_u = 8 km and L_u = 15 km, respectively. (d)–(f) The performance of the LS vorticity estimates using drifters in the 5-km, 8-km filtered, and 15-km filtered model fields. A linear regression between the true model vorticity and estimated vorticity and the rms error ϵ_rms are used to evaluate the performance.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The relationship between the rolloff length scale of low-pass filtered model fields and the cluster length scale becomes more explicit in wavenumber spectra. As shown in Fig. 3, filtering removes variability below the filter length scale such that velocity and vorticity fields are smooth. In other words, velocity gradients are constant across drifter clusters, consistent with the LS assumptions of a linear velocity field. The original model fields drop off steeper than k⁻² below about 5 km. The two filtered fields drop off at 8 or 15 km, respectively, explaining why larger clusters were able to perform better in estimating the velocity gradients.

c. Sensitivity to shape

The shape of the drifter clusters significantly influences the estimates of the kinematic properties. In the numerical model, we vary the cluster shapes systematically, compute the kinematic properties, and evaluate the uncertainties.

For each shape configuration with L, α, and N, 10 000 synthetic drifter clusters are randomly positioned and randomly oriented in the model to compute the kinematic properties. The resulting distributions of the estimates allow us to determine the uncertainties and sensitivity to the cluster shapes.

Figure 6 shows the sensitivity of the LS estimates to the shape of the drifter clusters in the model. Each data point represents an experiment with 10 000 model clusters of fixed L, α, and N. As L/L_u increases (Figs. 6a,d), the vorticity errors grow logarithmically. As soon as the drifter clusters become larger than L/L_u > 2 and ϵ_rms ∼ 1, the estimates become increasingly meaningless as the variability at smaller scales is aliased into the signal. R² approximately decreases like 1/L, suggesting that the kinematic properties can only be estimated accurately if the clusters have small length scales that are smaller or approximately equal to L_u.

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Sensitivity analysis of the vorticity estimates based on drifter cluster shape. Each parameter set, consisting of normalized length scale L/L_u where L_u is set to 5 km, aspect ratio α, and number of drifters N, was used to estimate the vorticity from 10 000 synthetic clusters. (a)–(c) The coefficients of determination R². (d)–(f) The rms errors ϵ_rms normalized by the rms vorticity rms(ζ) between the modeled and estimated vorticities. Black vertical lines indicate the parameter limits, L/L_u = 2 and α = 0.1, used in the analysis of the observational drifter data in section 4.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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This relationship becomes clearer when considering the filtered velocity fields in Fig. 5. The linear regression works well if the clusters are smaller than the dominant length scale L_u, resulting in large R². The Gaussian filtering removes variability below the filter scale, such that L_u increases. Since L remains unchanged in these comparisons, L/L_u decreases and there is progressively less aliasing or impact of unresolved scales in the filtered cases.

The aspect ratio α also affects the velocity gradient estimates, but to a lesser extent than L/L_u. As the drifter clusters elongate and α decreases, R² decreases and ϵ_rms increases, particularly in the extreme cases with α < 0.2. In addition to the elongation of the clusters, the direction of elongation relative to the direction of the velocity gradients is important. If the clusters are elongated perpendicular to the velocity gradients, the ability to compute the gradients accurately is largely diminished. While this is possible to address in a model, the cluster orientation is challenging to determine in drifter observations because the direction of the velocity gradients is usually not known.

Last, the number of drifters N per cluster theoretically increases the accuracy of the estimates but as N increases, R² and ϵ_rms show only a weak or no dependence on N (Figs. 6b,c). We find that N = 3 is slightly worse than N = 6 or 9 but we cannot find a functional dependence. The impact of N, however, might be more pronounced when there is systematic error on the velocity data, which is absent here, because it could reduce the residual using least squares. Okubo and Ebbesmeyer (1976) determined that at least six drifters are needed to yield reasonable results for the vorticity and divergence estimates. It is worth mentioning that although an increase in N reduces the uncertainty of the LS method, this does not imply an increase in accuracy.

d. Estimating error functions

Summing up the findings of the previous section, we show in Fig. 6 that the vorticity error increases with L/L_u, where L_u is the decorrelation scale of the velocity field. In Fig. 5, we show that when L_u is increased, for example by low-pass filtering in space or by a shift in dynamic regime in the real ocean, the vorticity estimation improves.

2) Theoretical expectations

The LS method computes an average velocity gradient $\overline{u_x}$. If the true u_x is constant over the drifter cluster, $u_x(x_0)=\overline{u_x}$. If u_x is nonconstant, the LS methods generates errors. Since instrument errors are zero in the numerical model, all errors described in this section and in Fig. 6 are due to nonconstant velocity gradients.

For simplicity we consider the LS method as a spatial low-pass filter that computes an average velocity gradient over the length scale of a drifter cluster:

${\overline{u}}_x=\frac{1}{L}{\displaystyle {\int }_{x_0-L/2}^{x_0+L/2}\frac{du}{dx}}dx=\frac{1}{L}\left[u\left(x_0+\frac{L}{2}\right)-u\left(x_0-\frac{L}{2}\right)\right].$(9)

If we assume that the velocity gradient u_x(x₀) is distributed with a covariance function C(n) ≡ 〈u_x(x₀)u_x(x + n)〉, so that the lagged correlation function is ${\rho }_{u_x{u}_x}=C(n)/\langle u_x^2\rangle$, we can compute statistics on the average velocity gradient. The other components of the two-dimensional velocity gradient are assumed to behave analogously.

Using the expression above, the variance of the LS velocity gradient can be rewritten using the expression for the variance as

$\langle {[\overline{u_x}(x_0)]}^2\rangle =\frac{1}{L}{\displaystyle {\int }_{x_0-L/2}^{x_0+L/2}{\displaystyle {\int }_{x_0-L/2}^{x_0+L/2}\frac{du(x_1)}{d{x}_1}\hspace{0.17em}\frac{du(x_2)}{d{x}_2}\hspace{0.17em}d{x}_{1}d{x}_2}},$(10)

$=\frac{4\langle u_x^2\rangle }{L^2}{\displaystyle {\int }_0^{L/\sqrt{2}}{\displaystyle {\int }_0^{L/\sqrt{2}-s}{\rho }_{u_x{u}_x}}(n)d\mathit{nds}}.$(11)

The correlation between estimated $\overline{u_x}$ and true u_x is

$\langle \overline{u_x}(x_0)u_x(x_0)\rangle =\frac{1}{L}{\displaystyle {\int }_{x_0-L/2}^{x_0+L/2}\frac{du}{dx}\hspace{0.17em}dx},$(12)

$=\frac{2\langle u_x^2\rangle }{L}{\displaystyle {\int }_0^{L/2}{\rho }_{u_x{u}_x}(n)dn}.$(13)

Assuming a covariance function C(n) = exp(−|n|/L_u), where L_u is the flow decorrelation scale, we can solve for the variance and correlation, respectively:

$\langle {[\overline{u_x}(x_0)]}^2\rangle =\frac{4\langle u_x^2\rangle L_u^2}{L^2}\left[\frac{L}{\sqrt{2}{L}_u}+\exp \left(\frac{L}{\sqrt{2}{L}_u}\right)-1\right],$(14)

$\langle \overline{u_x}(x_0)u_x(x_0)\rangle =\frac{2\langle u_x^2\rangle L_u}{L}\left[1-\exp \left(\frac{L}{\sqrt{2}{L}_u}\right)\right].$(15)

The rms error is a combination of the two functions 11 and 13 above:

${ϵ}_{\text{RMS}}=\langle {[\overline{u_x}(x_0)-u_x(x_0)]}^2\rangle =\langle {[\overline{u_x}(x_0)]}^2\rangle -2\langle \overline{u_x}(x_0)u_x(x_0)\rangle +\langle u_x^2\rangle .$(16)

In Fig. 7, the variance, correlations and rms errors are shown as a function of L. At large L/L_u, that is, when the cluster length scale is much larger than the flow decorrelation scale, the error saturates. The rms error asymptotes to the total velocity gradient variance. At small L/L_u, the mean variance and correlation decrease like 1/L and the rms error increases like L. A similar behavior has been observed in the empirical curves suggesting that L/L_u is the dominant parameter with much larger influence on the error and correlation than α and N (Fig. 7c). The mismatch in magnitude between theoretical and empirical might stem from the fact that the theoretical curves are only derived for one-dimensional gradients and an symmetric covariance function around zero.

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Theoretical predictions from sampling a known, space-lagged covariance function of velocity gradients for (a) variance $\langle {[\overline{u_x}(x_0)]}^2\rangle$, (b) correlation $\langle \overline{u_x}(x_0)u_x(x_0)\rangle$, and (c) rms error ϵ_RMS as estimated in Eqs. (14)–(16). In (c), the empirical error from fitting to the numerical results is also plotted. All quantities are normalized by the total variance $\langle u_x^2\rangle$. These curves should be compared to Fig. 6. The vertical black lines indicate the L/L_u threshold above which the rms error saturates at its maximum.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The theoretical considerations here show that the primary sources of error are nonconstant velocity gradients below the scale of a drifter cluster. When the background gradient is not constant across a cluster, that is, if $u(x+x^{\prime },\hspace{0.17em}y+y^{\prime })\hspace{0.17em}\ne \hspace{0.17em}\overline{u}(x,\hspace{0.17em}y)+u_x(x,y)x^{\prime }+u_y(x,y)y^{\prime }$, the accuracy is limited by the LS assumption being violated and not by the noise.

a. Error due to GPS uncertainty

Drifter GPS accuracy of the position improves with runtime but the position components stabilize with a rms error of σ_x ≈ 5 m after several minutes. The propagation of position uncertainties yields estimates of the velocity errors of σ_uυ ≈ 0.01 m s⁻¹ (cf. Hormann et al. 2016; Maximenko et al. 2013; Centurioni 2018). For a regular (i.e., α = 1) six-drifter cluster of L = 5 km, the vorticity error due to GPS uncertainty is about σ_ζ = 0.03f while an elongated cluster with α = 0.1 gives σ_ζ = 0.2f. For small aspect ratios α < 0.1, the GPS uncertainty becomes large; that is, as large as the estimated eddy vorticity which is consistent with our findings for the method error ϵ_RMS determined above.

The error variability due to GPS uncertainty, within the range of scales and aspect ratios considered here, is lower than that of the rms error computed for each method above, which varies between 0 and 1f. Only when method uncertainty is low, the GPS uncertainty will control the total error. The GPS error function (17) increases with α and decreases with N and L.

b. Cluster geometry

The kinematic properties will be computed on clusters with N = 6 drifters. There are over 8 million possible combinations for 45 drifters, every 5 min. Reducing this number in a physically meaningful way is a challenge to large drifter releases and has computational relevance. In this study, we filter all combinations in space with the length scale and aspect-ratio criteria described above; that is, L/L_u < 2 and α > 0.1.

On average, the drifter cluster sizes grow monotonically over the first month after deployment in the Bay of Bengal (Fig. 8a). The mean length scale L initially grows rapidly (approximately as e^0.2t ). Here, the most energetic motions are mesoscale eddies that are about 100 km in diameter, larger than the deformation radius L_D = 60 km. The initial 3–4 days after deployment suggest exponential growth, consistent with the nonlocal regime found using pair dispersion statistics (cf. Essink et al. 2019). Similar behavior and an analogy to pair dispersion was found by LaCasce and Ohlmann (2003), who derived an expression to relate pair separations to cluster length scales.

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Time evolution of (a) mean length scales L/L_u and (b) mean aspect ratios α of all (N = 6) (blue) and selected (yellow) (α > 0.1, L/L_u < 2) in the Bay of Bengal. (c) Number of available clusters that fulfill the criteria (α > 0.1, L/L_u < 2). The shaded regions shows the 3-day window favorable for further analysis with enough clusters available to obtain meaningful statistics.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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After deployment at t = 0 days, when a large fraction of the drifter clusters has an aspect ratio of α < 0.5, the clusters elongate dramatically (Fig. 8b) as drifters wrap around the perimeter of the dominant mesoscale eddy (Fig. 1c), the length scales increase and the aspect ratios collapse most rapidly. Within 2 days, the clusters stretch out until at least 75% have an aspect ratio of α < 0.1, equivalent to a major axis 10 times larger than the minor axis. The exponential e-folding time with which this occurs is about two days, roughly consistent with the e-folding time scale computed from drifter-pair statistics (Essink et al. 2019). LaCasce and Ohlmann (2003) observed a similar change in cluster aspect ratios for drifters in the Gulf of Mexico. Their aspect ratio, defined as the ratio of the base and height of a triangle, becomes on the order of 10 (approximately inverse of α) with a time scale of 2 days.

Based on the sensitivity analysis (section 3), the ensemble-mean L and α (Fig. 8) predict low coefficients of determination R² and large rms errors ϵ_RMS for the estimated velocity gradients. Three additional challenges for drifters in the ocean can be hypothesized: (i) small-scale variability, with nonconstant velocity gradients across the cluster-scale L will violate the LS assumptions, (ii) velocity correlations between pairs and clusters of drifters decrease substantially within only a few days (cf. Beron-Vera and LaCasce 2016; Essink et al. 2019), and (iii) position and velocity measurements will be subject to noise. Therefore, the estimation of the kinematic properties can only be successful for a few days after deployment.

c. Composite maps of kinematic properties

By spatially binning the drifter-derived kinematic properties, we map vorticity, lateral strain rate, and divergence in space. The longitudinal and latitudinal length scales of each bin are chosen to be 10 km, consistent with Hormann et al. (2016) and Essink et al. (2019) and equal to 2L_u. The median of all data in each latitude–longitude bin is shown in color in Fig. 9. This binning procedure reduces the degree of redundancy because individual drifters can be part of multiple clusters. Plotting maps of the kinematic properties presumes that the vorticity, divergence, and strain rate vary slowly over the chosen ∼3-day period. The near-surface density estimates from the drifters are also shown in Fig. 9a. Consistent with other observations during the 2015 ASIRI field experiment, the freshwater-dominated density front stands out in the spatial maps. Relatively fresh waters originating from rivers and rain in the northern bay are advected southward in plumes and stirred by mesoscale eddies.

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Composite spatial maps over 3 days after the drifter deployments (i.e., 3–6 Sep 2015) of (a) density anomaly ρ₀, (b) vertical vorticity ζ/f, (c) length scale L/L_u, (d) divergence δ/f, (e) aspect ratio α, and (f) lateral strain rate σ/f. Maps show the median value of latitude–longitude (5 km × 5 km) hexagonal bins containing at least three data points. Overlaid are contours of the density anomaly ρ₀ in gray that are also shown in (a) and contours of sea surface salinity from SMAP (4 Sep 2015).

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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The vorticity and lateral strain rate maps reflect the north–south pattern in the density gradients (Figs. 9b,d,f). A sharp change of the vorticity values and sign occurs where the density transitions from denser to fresher waters, with the largest values of vorticity and lateral strain rate occurring at the edge of the density front. The vorticity distribution across the density front is consistent with a frontal jet in geostrophic balance, with negative vorticity on the lighter side and positive vorticity on the denser side of the front. This agreement between the density and vorticity patterns is robust since the density is derived from drifter sensor measurements while the kinematic properties are independently estimated from the drifters’ GPS positions. The divergence shows a less clear pattern when plotted in space. However, some of the structures are consistent with the vorticity and lateral strain rate fields.

The large-scale patterns further show the dominant cyclonic mesoscale eddy that was present during the period of the experiment (see Fig. 1c), which is also reflected in the drifter trajectories (cf. Hormann et al. 2016; Essink et al. 2019). Within the eddy, the vorticity field is mostly positive, especially at later times during the drifter experiment, when the influence of the density gradients weakens (see Fig. 9b). The cyclonic background vorticity likely biases the cluster-scale vorticity estimates, particularly when the small-scale vorticity is weak. Furthermore, a positively skewed vorticity distribution is expected as observed in previous experiments (e.g., Rudnick 2001; Shcherbina et al. 2013).

During the first few days, we observe a banded pattern in the vorticity and divergence maps that is not reflected in the density field (see Figs. 9b,d). This banded pattern has a horizontal wavelength of approximately 30 km, about 0.1f amplitude, and alternating positive and negative values with vorticity leading divergence by about 6–12 h. Although challenging to pinpoint with this dataset considering the Lagrangian space–time aliasing of spatial maps, this is likely due to diurnal heating and cooling of the surface ocean and an associated diurnal pattern of the eddy viscosity (e.g., Thushara and Vinayachandran 2014). Changes in the eddy viscosity in the surface layer modulate the ageostrophic frontogenetic flow, and thus the horizontal divergence. Previous experiments have attributed this phenomenon to frontal adjustment to the turbulent thermal wind balance (Dauhajre and McWilliams 2018; McWilliams 2016) and recent observations and simulations in the Gulf of Mexico provide similar evidence (e.g., Johnson et al. 2020a,b).

d. Temporal variability

Figure 10 shows time series of the kinematic properties computed as the mean of all drifter clusters available at each point in time. Before computing the mean time series over all clusters, the time series of each cluster are filtered with a 4-h running mean to focus on the diurnal and submesoscale variability and to remove high-frequency noise. The time series of the kinematic properties show high variability as indicated by the confidence intervals plotted around the mean values (Fig. 10). The tails of the distributions at each instant in time, particularly for submesoscale currents, reach $\mathcal{O}(f)$ vorticity and lateral strain rate magnitudes. Since the drifter clusters traveled as a coherent group during the initial phase of the experiment, sampled similar flow features, and drifter velocities are correlated (Essink et al. 2019), computing such time series is a reasonable approach.

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A 3-day time series of the drifter experiment for the period 3–6 Sep 2015 showing (a) the mean length scale L/L_u, where L_u = 5 km, (b) the mean aspect ratio, (c) the theoretical and empirical error calculated from section 3d, (d) the mean vertical vorticity ζ/f, (e) the mean divergence δ/f, and (f) the mean lateral strain rate S/f. The shaded regions indicate the 95% confidence interval determined directly from the distribution of estimates for each cluster. Vertical dotted lines indicate 24-h periods, and horizontal dotted lines indicate L/L_u < 2 and α > 0.1.

Citation: Journal of Atmospheric and Oceanic Technology 39, 8; 10.1175/JTECH-D-21-0068.1

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