Vorticity in the Wake of Palau from Lagrangian Surface Drifters

Kristin L. Zeiden aApplied Physics Laboratory, University of Washington, Seattle, Washington

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Daniel L. Rudnick bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Jennifer A. MacKinnon bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Verena Hormann bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Luca Centurioni bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

Wake eddies are important to physical oceanographers because they tend to dominate current variability in the lee of islands. However, their generation and evolution has been difficult to study due to their intermittency. In this study, 2 years of observations from Surface Velocity Program (SVP) drifters are used to calculate relative vorticity (ζ) and diffusivity (κ) in the wake generated by westward flow past the archipelago of Palau. Over 2 years, 19 clusters of five SVP drifters ∼5 km in scale were released from the north end of the archipelago. Out of these, 15 were entrained in the wake. We compare estimates of ζ from both velocity spatial gradients (least squares fitting) and velocity time series (wavelet analysis). Drifters in the wake were entrained in either energetic submesoscale eddies with initial ζ up to 6f, or island-scale recirculation and large-scale lateral shear with ζ ∼ 0.1f. Here f is the local Coriolis frequency. Mean wake vorticity is initially 1.5f but decreases inversely with time (t), while mean cluster scale (L) increases as Lt. Kinetic energy measured by the drifters is comparatively constant. This suggests ζ is predominantly a function of scale, confirmed by binning enstrophy (ζ2) by inverse scale. We find κL4/3 and upper and lower bounds for L(t) are given by t3/2 and t1/2, respectively. These trends are predicted by a model of dispersion due to lateral shear. We argue the observed time dependence of cluster scale and vorticity suggest island-scale shear controls eddy growth in the wake of Palau.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Oceanic Flow–Topography Interactions Special Collection.

Corresponding author: Kristin L. Zeiden, kzeiden@uw.edu

Abstract

Wake eddies are important to physical oceanographers because they tend to dominate current variability in the lee of islands. However, their generation and evolution has been difficult to study due to their intermittency. In this study, 2 years of observations from Surface Velocity Program (SVP) drifters are used to calculate relative vorticity (ζ) and diffusivity (κ) in the wake generated by westward flow past the archipelago of Palau. Over 2 years, 19 clusters of five SVP drifters ∼5 km in scale were released from the north end of the archipelago. Out of these, 15 were entrained in the wake. We compare estimates of ζ from both velocity spatial gradients (least squares fitting) and velocity time series (wavelet analysis). Drifters in the wake were entrained in either energetic submesoscale eddies with initial ζ up to 6f, or island-scale recirculation and large-scale lateral shear with ζ ∼ 0.1f. Here f is the local Coriolis frequency. Mean wake vorticity is initially 1.5f but decreases inversely with time (t), while mean cluster scale (L) increases as Lt. Kinetic energy measured by the drifters is comparatively constant. This suggests ζ is predominantly a function of scale, confirmed by binning enstrophy (ζ2) by inverse scale. We find κL4/3 and upper and lower bounds for L(t) are given by t3/2 and t1/2, respectively. These trends are predicted by a model of dispersion due to lateral shear. We argue the observed time dependence of cluster scale and vorticity suggest island-scale shear controls eddy growth in the wake of Palau.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Oceanic Flow–Topography Interactions Special Collection.

Corresponding author: Kristin L. Zeiden, kzeiden@uw.edu

1. Introduction

When flow past an island generates a vorticity wake, eddies typically dominate current variability downstream (e.g., Caldeira et al. 2014). Substantial knowledge gaps about their generation and evolution persist due to the difficulty in obtaining high-resolution and sustained observations of what is typically an intermittent process. Two open questions are 1) what are the scale and strength of wake eddies and 2) how do they evolve downstream? The observations in this study are part of a larger Office of Naval Research (ONR) field campaign, Flow Encountering Abrupt Topography (FLEAT), designed to characterize the generation and properties of the Palau wake with a wide range of instruments and over a broad range of spatial and time scales (Johnston et al. 2019). Here, 2 years of current data from ∼100 surface drifters deployed into the Palau wake are used to characterize the relative vorticity (ζ = ∂xυ − ∂yu), kinetic energy (KE), and dispersion associated with wake eddies.

Wake eddies are generated when incident currents flowing around an island meet an adverse pressure gradient on the downstream side. The current separates and vorticity generated along the coast is injected into the interior ocean. Return flow forms in the lee to satisfy continuity and wake eddies develop, continuously fed by the separated shear layer. When the incident flow is relatively steady, wake eddies grow until they reach island scale and are then advected downstream (e.g., Chang et al. 2013; Heywood et al. 1990). The vorticity of a wake eddy is determined primarily by its scale and the strength of the incident current. It is useful to discuss the vorticity of geophysical eddies in terms of their Rossby number, i.e., the ratio of their relative to planetary vorticity (Ro = ζ/f). Additional variability is often present due to the complex nature of geophysical flows, but these basic principles undergird the generation of wake eddies.

Eddies in the ocean are reasonably well described as having a solid-body core which rotates at a fixed rate, surrounded by an irrotational outer region in which the velocity decays with distance (Chelton et al. 2011; Olson 1991). Mesoscale eddies typically have Ro < 1 and are in quasigeostrophic balance, with a pressure gradient force either away from or toward the center of the eddy and the Coriolis force in the opposite direction. By mesoscale we mean at the scale of the local Rossby radius of deformation or larger. Due to this balance, mesoscale eddies can be stable on the order of months to years (Chelton et al. 2011). More energetic submesoscale eddies with Ro > 1 are likely to be in approximate cyclogeostrophic balance, where centrifugal acceleration and the Coriolis force balance the pressure gradient force. In either case, cyclonic/anticyclonic eddies rotate about low/high pressure regions. Although their kinetic energy is dominated by their azimuthal velocities, secondary circulations can develop in the form of upwelling/downwelling in the interior of cyclonic/anticyclonic eddies (Sangra et al. 2007). Numerical studies suggest that the vorticity of isolated wake eddies tends to decrease with time (e.g., Dong et al. 2007; Jagannathan et al. 2021). The more stable mesoscale eddies are susceptible to mergers between successive vortices and viscous decay, while submesoscale eddies are additionally subject to rotational instabilities due to their high Ro (Dong et al. 2007).

There are observational challenges in determining the processes responsible for the decay of eddies in the real ocean. Eddy trajectories are difficult to predict a priori, and tracking them requires either broad spatial coverage or targeted adaptive sampling once an eddy has been identified. For example, the sea surface height (SSH) signature, large spatial scales, and longevity of mesoscale eddies enables tracking via satellite altimetry (e.g., Sun et al. 2019). Because mesoscale eddies are in approximate geostrophic balance, altimetry also provides information about their kinematics. This approach is not as effective for studying submesoscale eddies, which have much smaller spatial scales and a weaker SSH signature. Some studies have observed submesoscale eddies using high-resolution satellite images of chlorophyll and sea surface temperature (SST) (e.g., Andrade et al. 2013; Cheng et al. 2020), but additional in situ measurements are required to study their kinematics. Drifters have been useful in understanding the statistical distributions of submesoscale eddies in major ocean basins (e.g., Dong et al. 2011; Li et al. 2011; Henaff et al. 2014; Lumpkin and Flament 2001). However, chance encounters between drifters and eddies usually only last a few turnover time scales, and so information about the lifetime of individual eddies is limited. Furthermore, the error in vorticity estimated with a single drifter completing a few loops is high (Dong et al. 2011). Studies which combine observations from satellites with drifters intentionally seeded in eddies have provided some of the most comprehensive observations (e.g., Alpers et al. 2013). In these studies, typically only one or two eddies are sampled and their generation source is unknown. One successful strategy to address this gap has been to repeatedly deploy drifters downstream of islands and headlands known to generate vorticity wakes (e.g., Chiswell and Stevens 2010). These studies have predominantly been conducted in coastal regions on short time scales, often with a focus on tidal flows. Studies of deep ocean island wakes are rarer. Flament et al. (2001) repeatedly seeded individual drifters in the wake of Hawai’i over a period of 2 years and successfully tracked the evolution of multiple mesoscale wake eddies. In conjunction with satellite altimetry, they were able to demonstrate vortex merging of successive wake eddies likely contributed to a decrease in their vorticity over time. Tracking the evolution of deep ocean submesoscale wake eddies not resolved by altimetry requires more intensive in situ sampling.

In this study, clusters of five surface drifters each were repeatedly released in a location typically populated by wake eddies generated by flow past the archipelago of Palau in the tropical North Pacific. Palau is located at about 8°N, 134.5°E, roughly 1000 km due east of the Philippines (Fig. 1). Although the archipelago is comprised of hundreds of small islands, the subsurface reef connecting them extends almost 200 km in the north–south direction and rises to 20 m at the rim (filled contour in Fig. 1a). For the purposes of this study, we consider Palau a contiguous topographic obstacle (i.e., an island). The use of clusters rather than individual drifters enabled least squares estimates of vorticity, as well as lateral diffusivity downstream. Clusters were deployed over a 2-yr period at intervals of weeks to months, in a known submesoscale eddy generation site at the north end of the island (Wijesekera et al. 2020). Previous FLEAT studies have shown that vorticity is generated across a broad range of scales, but eddies are typically O(10) km and have Ro > 1 close to the island (Zeiden et al. 2021; Rudnick et al. 2019; Merrifield et al. 2019). We use the drifters to calculate time series of vorticity in two ways: 1) by computing spatial gradients in the velocity field (least squares method) and 2) by identifying peak frequencies in time series of velocity (via wavelet analysis). The first calculation is done at each time step using all drifters in a cluster, while the second is done on time series from individual drifters. Thus, they are truly independent methods of estimating vorticity. We will show that these methods give similar estimates of the vorticity for drifters entrained in wake eddies and use case studies to understand their differences. We also calculate diffusivity and use this information in conjunction with the kinematics to argue large-scale shear controls the growth of wake eddies downstream.

Fig. 1.
Fig. 1.

Regional bathymetry overlaid with (a) mean OSCAR surface currents over the observational period (October 2016–December 2018) and (b) drifter trajectories over the first 20 days of each cluster deployment. OSCAR currents are 1/3° resolution. Quivers in (a) are colored by zonal velocity magnitude, with red/blue indicating eastward/westward. To the north of Palau the NEC flows westward, while to the south the NECC flows eastward. Inset in (b) shows the bathymetric detail of Palau, with 1000- and 100-m isobaths in black. The 100-m isobath around Velasco is highlighted in blue. Drifters were deployed in a diamond grid pattern at the northern tip of Palau [red dots in (b) inset] and were advected westward by the NEC on average. The black line in (a) is the trajectory of a single drifter in the C7 cluster, which was entrained in a strong cyclonic wake eddy. Drifter trajectories in (b) are colored by time, referenced to their initial release (i.e., days since release). (c) Mean local current conditions during each release are depicted with quivers by averaging OSCAR surface currents in a 150-km box centered on the drifter release location. Note that clusters are numbered up to 21, because an additional two clusters (C1 and C3) were not released from the north end and are not mentioned further in this study. Thin vertical lines in (c) denote the release date of each drifter cluster, colored to indicate whether the drifters were advected westward (cyan), westward and into an eddy (magenta), or eastward (gray). Two clusters split into two or more groups upon release (green).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

2. Data and methods

a. Oceanographic context

Ocean Surface Current Analysis (OSCAR) near-surface currents averaged over the drifter observational period (October 2016–December 2018) give the mean regional circulation around the island (Fig. 1a). Palau lies in a transition region between two major surface intensified geostrophic currents: the westward North Equatorial Current (NEC) and the eastward North Equatorial Countercurrent (NECC). Weaker undercurrents flow below the pycnocline in opposing directions, but we do not sample these currents in this study. West of Palau, the NEC bifurcates near the Philippine coast and feeds the southward Mindanao Current (MC) as well as the northward Kuroshio (Schönau and Rudnick 2017). The MC reconnects with the NECC at ∼7°N via the Mindanao Eddy (ME; cf. Heron et al. 2006). East of Palau the NECC weakly recirculates back to the NEC, thus Palau sits in a region of predominantly cyclonic geostrophic circulation.

Drifter clusters were released at the northernmost end of the island, where the mean flow is westward (8.5°N, 134.25°E, red dots in Fig. 1b, inset). At this latitude, the NEC is surface intensified (above the mean pycnocline depth of about 200 m) and reaches up to ∼0.5 m s−1. However, strong seasonality in the regional currents can have a significant impact on the local flow (Heron et al. 2006; Hsin and Qiu 2012). Both the NEC and NECC are stronger in the first half of the year, when they are more northward, and Palau sits directly between them in a region of relatively weak flow. As a result, there is westward flow past its north end and eastward flow past its south end. In the second half of the year, both currents are weaker and more southward. The NEC engulfs Palau, leading to westward flow around both ends. Hsin and Qiu (2012) attribute this seasonality to westward propagating Rossby waves, while Heron et al. (2006) emphasize annual zonal elongation of the ME. Drifter trajectories show the impact of the latter, with southward deflection occurring near the coast or as far east as 130°E depending on these currents (Fig. 1b). On shorter time scales, mesoscale eddies advected westward by the NEC can disrupt the local flow (Qiu and Lukas 1996; Chelton et al. 1998; Schönau and Rudnick 2015). Time series of local zonal and meridional velocity obtained by averaging OSCAR surface currents in a 150-km box around the drifter deployment location give a sense of this variability (Fig. 1c). The predominantly westward currents reach up to 0.3 m s−1, but occasionally reverse (e.g., weak eastward flow in spring 2017). We note that interannual variability spurred by El Niño/La Niña events often exceeds the annual and intra-annual variability (Kashino et al. 2008; Zhao et al. 2013; Schönau et al. 2019). However, no events occurred during the course of the drifter observations, as the 2015/16 El Niño transitioned to neutral conditions in spring 2016 (Schönau et al. 2019).

b. Dataset description

In this study 19 clusters of surface drifters were released 6 km from the tip of Velasco, a submerged reef ∼70 km long at the northernmost end of Palau (Fig. 1b, inset). Note the clusters are numbered up to 21, because an additional two clusters which were not released from the north end are not used in this study. The incident westward flow has been shown to separate from the island at this location (MacKinnon et al. 2019; Wijesekera et al. 2020; Zeiden et al. 2021). Clusters were deployed at intervals ranging from weeks to months between October 2016 and December 2018 (thin vertical lines in Fig. 1c). Each cluster consisted of five drifters, initially in a diamond configuration with a separation of about 5 km. Drifters used in this study were Surface Velocity Program (SVP) drifters, which consist of a surface float connected to a holey sock drogue centered at 15-m depth (Centurioni 2018). SVP drifters have often been deployed in cluster formation to facilitate the calculation of spatial gradients (e.g., Hormann et al. 2016). The drifter positions are reported via satellite using Iridium Short Burst Data (SBD) telemetry with temporal resolutions ranging from 5 min to 1 h. The time series presented in this study have been smoothed to a resolution of 1 h using a moving Hanning window filter, to match the lower end of this range, and because we are interested in eddy time scales and greater. We estimate a lower bound for the eddy time scale to be ∼6 h, based on a maximum incident current of ∼0.5 m s−1 and minimum length scale of ∼10 km. Drifter lifetimes are up to 2 years, but here we use only the first 60 days of each cluster deployment, as cluster scales reach the scale of the regional circulation as early as 20 days into the deployment (note orange and red colors in the drifter trajectories in Fig. 1b). Velocities are calculated by finite differencing the drifter positions in time. We supplement the drifter position data with OSCAR surface currents. OSCAR data are provided by the Jet Propulsion Laboratory (JPL) Physical Oceanography Distributed Active Archive Center (POODAC) and developed by Earth and Space Research (ESR). Velocities combine geostrophic, Ekman and Stommel currents but do not include local acceleration and nonlinearities (Bonjean and Lagerloef 2002). Thus, we use the OSCAR currents to understand the large-scale, quasigeostrophic flow around Palau. To characterize mean properties of the Palau wake, we present averages of relative vorticity, kinetic energy, scale, and diffusivity downstream taken over the clusters which were advected westward into the wake (in all 15 of 19 clusters are presented and discussed). We also examine case studies to illustrate the wake structures which contribute to these averages.

c. Vorticity from least squares fit to velocity

We employ two methods of estimating vorticity (ζ) in this study. The first method involves calculating spatial gradients in velocity at each time step using all drifters in a cluster. We first assume that velocity gradients at the scale of the cluster are linear (i.e., a plane). This assumption is valid when the separation between drifters is small relative to the dominant length scales of the flow. A model for the velocity of each drifter is given by
ui=u0+u0x(xix¯)+u0y(yiy¯)+uiυi=υ0+υ0x(xix¯)+υ0y(yiy¯)+υi,
where u0 and υ0 are the model zonal and meridional velocities with constant spatial gradients and ui and υi are turbulent velocities (Okubo and Ebbesmeyer 1976; Molinari and Kirwan 1975). Drifter positions (zonal xi and meridional yi) are with respect to their mean (x¯), such that ui = u0 at the center of the cluster. Turbulent velocities include any features which are unresolved by the drifter array, such as waves and finer-scale eddies, as well as instrument error. The vorticity at each time step is given by ζ = ∂υ0/∂x − ∂u0/∂y. If we assume the drifters are entrained in an eddy, we may further constrain the model to reflect solid-body rotation (Rudnick et al. 2015). In this case,
ui=u0+ζ2(yiy¯)+ui,υi=υ0+ζ2(xix¯)+υi.

The solid-body model is also valid for lateral shear flows, although the estimate of ζ will be off by about a factor of 2. An advantage of this model is that it gives an estimate of the eddy center and consequently the radial position of each drifter in the eddy. In the limit of shear flow the distance to the eddy center approaches infinity, a trait we exploit in the discussion to identify an instance where a wake eddy is strained by the ambient geostrophic shear (section 4c).

In either case, the task is to solve the linear algebra problem U = MX + E by minimizing the mean square of the model misfit E at each time step. Here U is the vector of observed velocities, M is the vector of model parameters, and X is the position matrix, the forms of which are dependent on the chosen model. The solution is given by M = (XTX)−1XTU. In the plane-fit model (1), separate equations are solved for the zonal and meridional components of velocity, with U given by [ui], X by [1 xi yi] and M by [u0xu0yu0]T. If N is the number of drifters, the dimensions are N × 1, N × 3, and 3 × 1, respectively. In the solid-body model (2), one equation is solved with U given by [ui; υi], X given by [1 0 − yi; 0 1 xi] and M by [u0 υ0 ζ/2]T. The dimensions are 2N × 1, 2N × 3 and 3 × 1, respectively. In both cases then the model misfit E is the vector of turbulent velocities [ui].

An example time series of vorticity measured by C7 is shown in Fig. 2a. The trajectories of the drifters in this cluster suggest they were entrained in a wake eddy advected to the northwest (a single drifter trajectory from this cluster is plotted in black in Fig. 1a). The plane-fit solution is plotted in green, and the solid-body fit is plotted in black. Also plotted in Fig. 2a is the Okubo–Weiss (OW) parameter (gray), defined as OW2 = σ 2ζ2 (Okubo 1970; Weiss 1991). The OW is a measure of the relative strength of strain and vorticity. Here the strain is given by σ 2 = (∂xu + ∂yυ)2 + (∂yu + ∂xυ)2. The sign of the OW value determines whether streamlines in the flow spiral (negative) or form nodes/saddles (positive). Thus, regions of negative OW are often used to detect the presence of eddies in satellite altimetry and numerical models. In this case, the two estimates of ζ are in good agreement and the OW value is negative, validating our assumption that the drifters are entrained in a wake eddy in approximately solid-body rotation. There is greater high-frequency variance in the plane-fit estimate of ζ, which suggests that the solid-body constraint acts as a filter for other high-frequency currents such as tides and inertial oscillations which do not fit the model. This effect becomes increasingly apparent as the strength of ζ decreases with time. For C7, after 15 days the semidiurnal band contains 48% of the plane-fit vorticity variance but only 18% of the solid-body fit vorticity variance. Further details of this cluster are discussed in section 3c.

Fig. 2.
Fig. 2.

(a) Example vorticity time series obtained by least squares fitting a simple model of vorticity to the velocities of all drifters in C7, as well as example (b) anticyclonic and (c) cyclonic rotary scalograms obtained by performing a wavelet transformation of velocity from a single drifter in the cluster. White dotted lines in (b) and (c) indicate the local Coriolis frequency, which changes as the drifter moves north or south. Black lines indicate the region possibly contaminated by edge effects. Vertical axis labels D1 and D2 indicate the diurnal and semidiurnal tidal periods, respectively. Blue and black lines in (a) are the vorticity obtained from the simple plane fit and the solid-body fit, respectively (described in section 2c). Vorticity has been normalized by the local Coriolis frequency (Ro = ζ/f), and the dashed gray lines indicate Ro = 1. Also plotted is the Okubo–Weiss value (gray), which gives the relative strength of strain and vorticity. Negative values indicate the presence of an eddy. The wavelet shows anticyclonic rotation is dominated by strong peaks in the semidiurnal band, and no energetic peaks at frequencies lower than the inertial frequency. Cyclonic rotation is dominated by a strong superinertial peak which decays with time, moving to lower frequencies. This is in good agreement with the vorticity time series, which shows strong (Ro > 1) values prior to day 10 and a negative OW value, indicating the presence of an eddy.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Finally, the error in our estimate of vorticity can be computed directly if the velocity error σu is known. Assuming velocity errors across drifters are uncorrelated, the model parameter covariance σM2 is given by σu(XTX)−1, where X is the position matrix defined above (Okubo and Ebbesmeyer 1976). For the plane fit, Spydell et al. (2019, see their appendix) demonstrated that the vorticity covariance can be written as a function of the geometry of the cluster, σζ2=(σU2/N)(1+1/α2)(1/b2). Here α is the aspect ratio of the cluster, and b is the larger of two position covariance matrix eigenvalues (i.e., a measure of the cluster scale). Essink et al. (2022) used synthetic drifter clusters in a numerical model of submesoscale turbulence to confirm this inverse relationship with aspect ratio and scale. Similarly, the vorticity covariance for the solid-body fit can be written as σζ2=(σU2/N)(1/σL2). Here σL2 is the sum of the zonal and meridional drifter position variances, σx2+σy2, a different measure of the cluster scale. The aspect ratio of the cluster does not factor into the vorticity error for the solid-body fit because in this case the velocity gradient does not depend on direction. The velocity error σu has a component due to uncertainty in the GPS drifter positions and a component due to small-scale motions which are unresolved by the array. For SVP drifters, the former is ≈0.01 m s−1 (Essink et al. 2022). We do not know the magnitude of unresolved currents, but numerous observational studies have shown the near-surface kinetic energy wavenumber spectra falls off approximately as k−2 for wavelengths O(10) km and smaller (e.g., Shcherbinaet al. 2013; Callies and Ferrari 2013; Qiu et al. 2017). Mesoscale current variability near Palau is O(0.1) m s−1, which suggests that currents below the array scale are likely O(0.01) m s −1 or less. Using 0.01 m s−1 as the velocity error and given σL is initially O(1) km, the error in our estimate of vorticity is O(0.1f ) at early times and decreases with time as the cluster scale increases. On average, σζ reduces to 0.01f after 10 days. Thus, for strong eddies such as that sampled by C7, the error is a few percent of the vorticity magnitude.

d. Angular frequency from rotary wavelet transformations of velocity

The second method of estimating ζ exploits the relationship between vorticity and angular frequency (ω) for eddies in solid-body rotation, ζ = 2ω. An individual drifter orbiting an eddy will have peaks in kinetic energy variance at the angular frequency of the eddy. We may expect ω to change with time as the radial position of the drifter changes and/or the kinematic properties of the eddy evolve. By performing a wavelet transformation of the complex velocity time series, u + , we can estimate ω as well as its sense of rotation (anticyclonic or cyclonic). This method has been used in previous drifter studies to algorithmically identify eddies from individual drifter velocity time series (Lilly and Pérez-Brunius 2021). The wavelet transformation in this study follows Todd et al. (2012). In short, the method consists of convolving a function localized in both time and frequency space with the velocity time series. Here we use a Morlet wavelet which is a sinusoid modulated by a Gaussian envelope, ψ(t)=ei2πkte(t)2/2. The parameter k controls the number of oscillations contained in each wavelet and scaling the argument t changes its width (scale). Thus, wavelets allow an analysis of velocity as a function of time and frequency. Performing the wavelet transform of velocity over a range of wavelet scales gives the KE scalogram. The scalograms presented in this manuscript have been normalized such that integrating them returns the total variance of the original velocity time series. Because the vorticity generated by westward flow past Palau is cyclonic, we expect peaks in positive frequency. This makes distinguishing wake vorticity from internal waves feasible, as internal waves are more energetic at negative frequencies due to their elliptical polarization. Velocity hodographs of internal waves are clockwise (anticyclonic) in the Northern Hemisphere, with ellipticity that increases with frequency (Kundu and Cohen 2007).

An example rotary scalogram from a single drifter in C7 is given in Fig. 2 (trajectory plotted in Fig. 1a, black line). The presence of a strong cyclonic eddy is evident in the asymmetry between anticyclonic and cyclonic KE. A strong peak in cyclonic KE early in the time series coincides with the strong initial vorticity given by the least squares fit (Fig. 2c). This energy has a period of ∼30 h, or a frequency of ∼3f. If we take this frequency to be ω, this suggests the eddy has Ro ∼ 6. Thus, the wavelet estimate of vorticity provided by this single drifter velocity time series is a factor of 2 higher than the estimate given by the least squares fit to the velocity measured by all drifters at each time step. These differences are explored in sections 3a and 3c. Their temporal trends are similar, however; around day 9 the peak drops to subinertial frequencies and continues to decrease for the rest of the record. Anticyclonic KE is dominated by strong peaks in the tidal bands, likely a reflection of ambient internal waves.

e. Scale and diffusivity

The change in drifter cluster scale with time is a direct reflection of horizontal stirring in the ocean (LaCasce 2008). Here we define the cluster scale to be the mean distance of each drifter from the cluster center of mass at each time step,
L=1Ni=1N[(xix¯)2+(yiy¯)2]1/2,
where N is the total number of drifters and x (y) is the drifter position in the zonal (meridional) direction. If the drifters were to inscribe a circle, evenly spaced, L would be equivalent to the circle radius. The relative diffusivity, κ is the rate of change of the squared scale with time,
κ(t)=12ddtL2(t),
i.e., the change in the cluster area with time (Vallis 2006). Diffusivity is typically presented in terms of scale, as many theories give a prediction for κ(k), where k is inverse scale (1/L). To compare with these predictions, we also compute pseudo diffusivity wavenumber spectra by binning κ by k.

As noted in section 2c, the model of solid-body rotation used in the least squares fit provides an estimate of the location of the eddy center in addition to the vorticity. With this we may in turn estimate a mean radial distance of the drifters from that center, R. In the case of drifters evenly distributed around an ideal eddy at a fixed radius, the mean drifter position is collocated with the eddy center and R is equivalent to L. In the limit of unidirectional shear, the “eddy center” goes to infinity and thus R → ∞ ≫ L.

3. Results

Here we present estimates of relative vorticity, kinetic energy, and diffusivity from drifter clusters entrained in the Palau wake to the west, although our primary focus is vorticity. We consider 15 out of the 19 clusters released in this study (indicated by the colored vertical lines in Fig. 1c). Four clusters are not included in our analysis (gray lines in Fig. 1c). Two were advected to the east during anomalous regional currents. The other two were advected to the west but split upon release and therefore are not good candidates for the least squares method of estimating vorticity, nor estimates of cluster scale and diffusivity. We first present averages of wake vorticity and angular frequency as a function of time taken over all selected clusters. Time is always referenced to the initial release date (i.e., days since release). We then identify two dominant wake regimes, an eddy wake and an island-scale shear wake (cyan and magenta lines, respectively, in Fig. 1c). To understand their contributions to the average vorticity, we examine case studies of clusters entrained in each type.

In this section we discuss both vorticity and angular frequency (ω = uθ/r, where uθ is the azimuthal eddy velocity). In the case of solid-body rotation, ζ = 2ω. To differentiate between the two variables, we refer to vorticity in terms of the Rossby number, i.e., ζ/f → Ro, and angular frequency as a fraction of f, i.e., ω = Cf, where C is a constant.

a. Average wake vorticity

On average, vorticity in the westward wake of Palau is cyclonic, with initial Ro > 1 (Fig. 3a). This is expected, as the shear generated by westward flow to the north of an island is necessarily positive. Vorticity then decays with time as ζt−1. This average includes vorticity due to coherent wake eddies as well as any ambient lateral shear. Different wake types will be discussed in section 3b. The magnitude of vorticity variability is comparable to the mean and similarly decays with time (gray shading in Fig. 3). Because vorticity is a function of both scale and velocity, we must consider how the cluster size and mean kinetic energy change with time. Cluster scale (L) increases linearly and KE has no clear time dependence, although both the mean KE and variance increase significantly after about 40 days (Figs. 3b,c). Here we do not plot the mean drifter radial distance (R) as given by the least squares fit, because this estimate becomes nonphysical in the limit of unidirectional lateral shear. These trends suggest the decay of vorticity is primarily related to changes in scale. A pseudo enstrophy (ζ2) wavenumber spectrum obtained by binning vorticity by inverse cluster scale (k) confirms this relationship. A best-fit line to the data in enstrophy–wavenumber space gives a slope which corresponds to ζ2k2 (Fig. 4).

Fig. 3.
Fig. 3.

Averages of (a) vorticity (ζ), (b) kinetic energy (KE), and (c) cluster scale (L) taken over all drifter clusters which were advected westward. Gray shading in all is standard error on the mean. Red dashed lines in each give for comparison vorticity inversely related to time in (a), kinetic energy constant in time in (b), and scale increasing linearly with time in (c).

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Fig. 4.
Fig. 4.

Enstrophy (ζ2) from all westward clusters binned by wavenumber (inverse scale), giving probability density. A least squares fit trend to the data gives a power law of ζ2k1.9 (magenta line). Black line is ζ2k2 for comparison. The white dashed line indicates Ro = 1.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

We note that KE ranges from ∼0.02 to 0.1 m2 s−2, with an absolute mean value of 0.034 m2 s−2 (Fig. 3c, red dashed line). This corresponds to a mean azimuthal velocity of 0.18 m s−1, which is similar to the strength of the surface intensified NEC in this region (Zeiden et al. 2019). This value of KE does not include the mean velocity of the cluster, i.e., the rate at which the cluster is advected downstream. Thus, it is an indication of the kinetic energy associated with either the lateral shear or eddy azimuthal velocity in the wake. The slight increase in mean KE and significant increase in KE variance after ∼40 days can be attributed to clusters which ultimately become entrained in the Mindanao current system and are swiftly advected southward in a clockwise fashion. This is also reflected in the deviation of the mean scale growth at later times from its linear trend.

A rotary KE scalogram averaged over all westward-advected clusters shows how strong asymmetry between cyclonic and anticyclonic energy is induced by the presence of the island (Fig. 5). This scalogram is obtained by first computing the scalograms of each individual drifter, which are then averaged over each cluster. These cluster scalograms are further averaged together to obtain a total average. Similar to the example wavelet shown in Fig. 2, anticyclonic energy is dominated by strong peaks in the tidal bands likely due to internal waves. Anticyclonic tidal energy decreases with time, likely due to increased absolute distance from the island. Internal waves generated around topography spread radially and dissipate energy as they propagate from their source (Rudnick et al. 2003). The greater relative strength of the diurnal peak is consistent with the magnitudes of the barotropic diurnal and semidiurnal tide in this region (Zeiden et al. 2021; MacKinnon et al. 2019). There is a sharp decrease in energy at frequencies lower than the local inertial frequency, consistent with the limit for internal waves.

Fig. 5.
Fig. 5.

As in Fig. 2, but averaged over every drifter cluster which was initially advected westward. The average (1/2) vorticity obtained via least squares fit to the model of solid-body rotation is plotted in black. The scalogram shows on average there is a strong initial peak in rotational period of about 2.5 days which decays inversely with time. The vorticity follows the lower bound of this peak in time. This is a reflection of the greater energy of superinertial eddies, which dominate the mean scalogram.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Cyclonic rotation has comparatively less energy in the tidal bands and the dominant cyclonic feature is a strong, broad peak between the diurnal and inertial frequency at early times (first ten days). The angular frequency of this peak is ∼3f, which corresponds to a vorticity of Ro = 6. However, the peak is broad and skewed toward lower frequencies, up to periods just slightly longer than the local inertial period. The peak decays in both angular frequency and magnitude with time, reaching subinertial frequencies after ∼10 days. This trend is roughly inverse with time, consistent with the vorticity time series obtained via the least squares method (note how the peak in cyclonic energy follows the edge-effect cutoff given by t−1, white lines in Fig. 5).

At all times the angular frequency of the cyclonic peak in the scalogram corresponds to a vorticity which is greater than that given by the least squares method (black line in Fig. 5). This can be explained by differences between the two methods of estimating vorticity. Consider the case of drifters in an eddy with a core in approximately solid body rotation. Beyond the edge of the core the vorticity goes to zero over some finite distance. Drifters inside the core measure the same rotation rate, while drifters outside the core measure a progressively lower rotation rate with increasing distance from the center. The least squares estimate of vorticity will always be low relative to the true core vorticity because it depends on the rotation rate measured by all drifters. In contrast, each scalogram depends only on the rotation rate measured by an individual drifter. Those from drifters within the core will have peaks at the true angular frequency of the eddy, while those from drifters outside of the core will have peaks at a lower angular frequency. An average over these scalograms will have a peak which is broad, but not necessarily lower than the core angular frequency. The peak frequency will depend on the kinetic energy measured by each drifter, which is greatest at the edge of the core. Understanding these differences, the two methods are in agreement as to the order of magnitude of the vorticity as well as how it decreases with time.

b. Wake regimes

Close examination of the 15 clusters entrained in the westward wake reveals two dominant regimes: those entrained in cyclonic wake eddies and those which experience only cyclonic lateral shear as they are advected downstream. We identify the latter from clusters where 1) the drifters do not rotate about their center of mass as they are advected west and 2) the eddy center given by the least squares fit to the solid-body model is unrealistically far at all times and highly variable. Hereafter we refer to this shear as “free shear layers,” consistent with existing wake literature (e.g., Chang et al. 2019). The distribution was roughly even, with 7 clusters entrained in wake eddies for multiple turnover time scales and 6 clusters entrained in shear layers. Two additional clusters split soon after release, but otherwise experienced only strong cyclonic shear and no eddies. There was a degree of seasonality in the occurrence of each wake type. Shear layers were only sampled between January and June, while five of the seven wake eddies were generated in the second half of the calendar year. Three of the observed eddies were characterized by Ro > 1 during the first week of deployment (C5, C7, C13; Ro ∼4) and three by Ro < 1 at all times (C9, C14, C21). One eddy had Ro > 1 only in the first day (C12). Four shear layers had initial Ro > 1 (C8, C11, C16, C19; Ro ∼2), which dropped below 1 after one day. The other two shear layers had Ro < 1 at all times (C17, C18).

OSCAR surface currents averaged over the first 20 days of each wake type show strong differences in the concurrent regional flow (Fig. 6). Eddies were observed on average when the NEC was relatively weak, but its lateral extent reached south of the island (Fig. 6a). Thus, the incident current was northwestward, diverged around 8°N, and flowed westward around both ends of the island. This is intuitive, as flow blocking necessarily creates a pressure gradient across the island and leads to current separation at either end. Twin regions of cyclonic and anticyclonic vorticity subsequently develop in the lee (Fig. 6c). In contrast, the free shear layer wake type is revealed to be part of the regional-scale cyclonic circulation around the island. On average, both the NEC and NECC were more northward and intensified, such that Palau was in an intermediate region of relatively weak currents without a strong uniform direction. There was comparatively little flow blocking as a result, and therefore no means to generate a pressure gradient across the island and subsequent wake eddies. The currents associated with each wake type are reminiscent of the seasonality in the regional geostrophic circulation described in section 2a, where the NECC and NEC are both more northward and surface intensified in the first half of the year. This is consistent with our observations, where shear layers and eddies were predominantly observed in the first and second half of the year, respectively.

Fig. 6.
Fig. 6.

OSCAR surface currents averaged over the first 20 deployment days of each drifter cluster entrained into (a),(c) wake eddies and (b),(d) free shear layers. Vorticity is contoured in (a) and (b), and kinetic energy in (c) and (d). Black quivers overlaid on all are corresponding surface current vectors. Drifter trajectories are colored by time as in Fig. 1.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

c. Wake eddy case studies

To understand their contribution to wake vorticity, we examine two case studies of drifter clusters entrained in cyclonic wake eddies. These eddies were chosen because their strong vorticity enables comparison between the least squares estimate of vorticity and angular frequency obtained via rotary kinetic energy wavelet analysis (sections 2c and 2d). This comparison is not typically feasible in drifter studies, because it requires the entrainment of at least three drifters in a single eddy for at least a few eddy turnover periods. Also, the predominance of cyclonic rotation caused by westward flow makes it easy to distinguish between internal waves and eddies with vorticity greater than f. As previously noted, internal waves rotate clockwise (anticyclonic) in the Northern Hemisphere. Thus, this dataset is uniquely suited to compare the two methods.

The first case is cluster 5 (C5) which was released in November 2016 during strong northwestward flow around the island. The incident flow and subsequent wake can be seen in OSCAR surface currents averaged over the first 20 days of the deployment (Figs. 7a,b). A region of weak flow (relative to the incident current) extends ∼300 km downstream of the island, outlined by bands of high KE to the north and south. Drifter trajectories move in a cyclonic pattern coincident with a region of cyclonic vorticity extending west of the island. In the first day, vorticity increases rapidly from zero to Ro ∼4. Over the next few days vorticity decreases slowly before dropping abruptly below Ro < 1 on day 5. Over the same period KE increases from ∼0.1 to ∼0.4 m s−1 before also dropping abruptly on day 5 (Fig. 7d). Initially, the mean drifter radial position given by the solid body fit (R) is unrealistically large, exceeding 100 km (Fig. 7e). This likely reflects the initial collinear distribution of the drifters due to the shear, when they are not yet distributed about the center (see section 2e regarding the definition of R). The R reduces to the cluster scale (L) at the same time vorticity peaks on day 1. Afterward, both L and R increase approximately linearly with time (cf. growth to Lt plotted with black dashed lines in Fig. 7).

Fig. 7.
Fig. 7.

OSCAR surface currents averaged over the first 20 days of the deployment of C5, plotted over corresponding (a) kinetic energy and (b) vorticity, as well as time series of (c) vorticity, (d) KE, and (e) scale obtained via the least squares estimate method. Drifter trajectories in (a) and (b) are colored by time as in Fig. 1. Vorticity in (c) is estimated using two different fits to the velocity data: 1) linear velocity gradients (blue lines) and 2) solid-body rotation (magenta lines). The gray line in (c) is the Okubo–Weiss value obtained from the plane fit. Kinetic energy in (d) is the mean squared velocity obtained from the fit, which gives the eddy azimuthal velocity in the case of solid-body rotation. In (e) the mean drifter radial distance with respect to the estimated eddy center (R, magenta) is compared to the cluster scale (L, black line). The dashed black line in (e) gives Lt for comparison.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Estimates of the eddy angular frequency (ω) given by cyclonic KE scalograms from each drifter in C5 agree well with the least squares estimate of vorticity (magenta lines in Fig. 8). Here we have divided the vorticity time series by 2 to make a direct comparison with ω (see section 2d). All five drifters give a similar value for ω, apparent in a strong peak around ∼1.3f during the first 5 days. This value for ω corresponds to Ro ∼ 2.6, in direct agreement with the least squares estimate of vorticity during the first week. The angular frequency of the peak also drops below f around day 5 and continues to decay with time. However, the cluster-mean angular frequency from day 10 to day 30 is a factor of 2 higher than given by the least squares estimate (Fig. 8f). This suggests that at later times the model of solid-body rotation misses a portion of the vorticity, whether because the drifters have moved beyond the solid-body core or there is some external shear superimposed on the eddy.

Fig. 8.
Fig. 8.

Cyclonic rotary wavelets for (a)–(e) all drifters in C5 as well as (f) their average. Plotting scheme is the same as in Figs. 2a and 2b. Rotational period (i.e., half the vorticity) estimate for C7 from the plane fit (blue lines) and solid-body fit (magenta line) are overlaid in each. Corresponding velocity time series are plotted above each scalogram, with meridional velocity plotted in red and zonal velocity in blue.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

The second case is cluster 7 (C7), which was released in March 2017 under similar regional flow conditions, although the incident current was stronger (Figs. 9a,b). As a result, the wake extends to the northwest almost 600 km and the drifters were advected about twice as far downstream in the first 20 days. The drifters also move in a cyclonic pattern coincident with a region of cyclonic vorticity extending downstream. The vorticity, KE, and scale time series from C7 are overall similar to those from C5. Vorticity exhibits rapid growth in the first day and then peaks around Ro ∼ 6 on day 1.5. Vorticity decreases over the next week, followed by an abrupt drop to Ro < 1 around day 8. However, there are a few notable differences. During the first week when Ro > 1, both vorticity and KE from C7 are highly variable. Both peak around day 1.5 and then drop briefly between day 2–3, before increasing again (from Ro ∼ 6 to 2, and KE ∼ 0.35 to 0.1 m2 s−2). Interestingly, when vorticity and KE both drop abruptly on day 9, R increases substantially to unrealistic values (hundreds of kilometers). This increase is short lived, and R relaxes back to the cluster scale by day 10.

Fig. 9.
Fig. 9.

As in Fig. 6, but for C7.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Cyclonic KE scalograms from all five drifters in C7 reveal a significant amount of variability in their estimates of peak angular frequency (Figs. 10a–e). All five scalograms are characterized by a strong initial peak in the subtidal, superinertial frequency band, followed by an abrupt drop below f around day 10. Overall, this decay in frequency mirrors the trend given by the least squares estimate of vorticity, especially as seen by drifter 5 (Fig. 10e). However, the fit tends to comparatively underestimate the peak frequency, which differs by a factor of about 3 between the different drifters. In order of decreasing frequency, drifters 5, 2, and 4 give angular frequencies higher than the least squares estimate. Drifter 1 gives the same estimate, and drifter 3 gives a lower estimate. The resulting mean peak angular frequency is broad, extending between the diurnal and inertial frequencies. The least squares estimate of the vorticity follows its lower bound in time.

Fig. 10.
Fig. 10.

As in Fig. 8, but for C7. The estimate of peak angular frequency varies substantially between drifters in the cluster. Strong cyclonic rotation occurs at initial periods which range between the diurnal and inertial period. On average, the initial rotational period is superinertial and the trend compares well with estimates from the plane fit to all drifters.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

In section 4 we expand upon these observations of Palau wake eddies to make inferences about their dynamic properties. Here we note that the initial rapid increase in vorticity from zero to peak values is likely a reflection of the time it takes for the drifters to become distributed around the eddy center (Figs. 7a and 9a). When the drifters are first deployed, the least squares estimate of vorticity is likely an underestimate while velocity differences between each drifter are small. The measured velocity differences will increase and reach a maximum when the drifters are distributed around the center of the eddy. For example, the vorticity measured by C7 peaks around 1.5 days (Fig. 9a). At the same time, the least squares estimate of the eddy radius decreases to the scale of the drifter cluster, suggesting the drifters have become distributed around the eddy center (Fig. 9a). This time scale is consistent with the peak eddy vorticity, which corresponds to a rotation period of about one day. Velocity time series from the individual drifters in C7 confirm they each complete the first revolution after about 1–2 days (Fig. 10).

d. Shear layer case study

Briefly, we examine a case study of a cluster entrained in a shear layer strongly characteristic of the mean shown in Fig. 6, cluster 17 (C17, Fig. 11). The NECC flows past the southern tip of Palau, and the island sits in the center of strong regional cyclonic circulation (Fig. 11a). Thus, the island scale flow is not blocked and there is no clear vorticity wake pattern (Fig. 11b). The drifters are simply advected downstream and dispersed by the lateral shear of the regional flow. The least squares estimate of vorticity varies just up to Ro ∼ 1 around day 2 and then decays (Fig. 11c). KE measured by C17 is weak compared to C5 and C7. Although the cluster scale (L) increases with time, the estimate of the mean radial drifter positions from the solid body fit (R) is highly variable and remains unrealistically large over the course of the observation (Fig. 11d). As explained in section 2e, this is a consequence of fitting a model of solid body rotation to a unidirectional shear layer. We do not show the wavelet scalograms for each drifter in C17, because there is no cyclonic peak associated with an eddy to compare to the least squares time series of vorticity.

Fig. 11.
Fig. 11.

As in Fig. 7, but for C17.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

e. Dispersion in the wake

Drifters are particularly well suited to calculate dispersion in the ocean, as this calculation requires no assumptions about the underlying physics. The limitation is the number of drifters, and we must keep in mind the intentional bias produced by releasing the drifters in a known eddy formation region. Nonetheless, the dependence of cluster scale on time can tell us something about the dynamics of the wake. As shown previously, the increase in drifter scale is linear with time on average. Plotting scale in log–log space reveals that initially (day 1) the mean scale growth is exponential (Fig. 12a, black line). Averages taken over only the subset of clusters entrained in either lateral shear (Fig. 12b) or wake eddies (Fig. 12c) indicate that drifters disperse more rapidly in the first 10 days when entrained in shear layers than wake eddies. Eddies necessarily place a limitation on drifter dispersion. Around day 10, most of the clusters entrained in shear layers have been advected far west by the strong mean flow and begin to feel the influence of the regional circulation (note the southward deflection of drifter trajectories around day 10 in Fig. 6b). As a result, dispersion of these clusters is weaker at later times. Wake eddies are generated during comparatively weak ambient mean flow (Fig. 6a). As a result, clusters deployed in wake eddies are advected slowly downstream and do not feel the effects of the regional circulation until much later.

Fig. 12.
Fig. 12.

Cluster scale (L) dependence on time for (a) all clusters advected westward, (b) clusters entrained in lateral shear layers, and (c) clusters entrained in wake eddies. In all subplots, the mean scale growth over each subset is plotted in black. In (a), individual cluster scales are also plotted in gray, overlaid by theoretical curves discussed in the text for exponential and linear growth (red dotted and dashed, respectively), as well as dispersion controlled by shear (magenta dotted) and stochastic processes (green dotted). In (b) and (c), the mean scale growth is decomposed into along-stream (coral) and cross-stream (light blue) components. Transparent shaded areas are standard error on the mean. Gray shading in each indicates the period over which power laws have been fitted to the along-stream (red dotted) and cross-stream (blue dotted) growth.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

The scales of each individual cluster are also plotted in Fig. 12a (gray lines). At times greater than 1 day, lower and upper bounds are given by Lt1/2 (green line) and Lt3/2 (magenta line). The lower bound is consistent with diffusion due to a stochastic process, i.e., the well-known Fick’s law of diffusion (Okubo 1971). The upper bound is consistent with both diffusion due to a turbulent cascade and diffusion due to lateral shear (LaCasce 2008). Our observations of island scale cyclonic shear extending west of Palau implicate the latter, but we can eliminate the possibility of a turbulent cascade by decomposing scale growth into along-stream and cross-stream components. Turbulent diffusion is isotropic, while shear dispersion leads to extrusion in the along-stream direction only. For clusters entrained in free shear layers, growth is more rapid in the along-stream direction and a best-fit power law to the data matches the prediction for shear dispersion (light blue and blue dashed lines, Fig. 12b). The fit was done after the period of exponential growth and before the effects of the regional circulation are felt. Growth in the cross-stream direction is slower, with Lt0.7. Okubo (1968) described a model of dispersion where a background linear shear is superimposed with a stochastic process in the direction orthogonal to shear. Scale in the along and cross-shear direction increase as t3/2 and t1/2, respectively. The total area thus increases as t2, which corresponds to a linear growth in scale, consistent with the average growth in Fig. 12a. Thus, the combined stochastic-shear model of dispersion agrees well with our observations. The same decomposition for clusters entrained in wake eddies shows cluster scale increases in the cross- and along-stream direction at the same rate (Fig. 12c). This is necessitated by the azimuthal symmetry of eddies. We discuss the growth of wake eddies further in section 4c, here only noting that after the first week both along- and cross-stream scale also increase as ∼t3/2.

Different theories of dispersion also give predictions for the relationship between diffusivity and scale (Okubo 1971). We estimate the diffusivity wavenumber spectrum by binning observations in diffusivity–wavenumber space (Fig. 13). Here diffusivity is the rate of change of squared cluster scale with time [κ = (1/2)∂L2/∂t], and wavenumber is the inverse cluster scale (k = 1/L). There is a clear slope to the spectrum, estimated to be κk4/3 from a least squares fit line to the data. This relationship is consistent with dispersion controlled by lateral shear (LaCasce 2008).

Fig. 13.
Fig. 13.

Diffusivity (rate-of-change of L2) from all westward clusters binned by wavenumber (k = L−1). A least squares fit to the data gives a power law of κk1.3 (magenta line). The theoretical power law for shear dispersion κk4/3 is plotted in black.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

4. Discussion

In this manuscript we have presented observations of the Palau wake obtained from drifter clusters released from the northern end of Palau, a known wake eddy generation site (MacKinnon et al. 2019; Wijesekera et al. 2020; Zeiden et al. 2021). Fifteen of the nineteen clusters released were advected westward after their initial deployment, reflecting time-mean westward flow past the island (Fig. 1). By fitting a model of solid-body rotation to drifter velocities at each time step, we obtain time series of vorticity due to both eddies and free shear layers in the wake. We also compute rotary kinetic energy scalograms via wavelet transformations to velocity time series from individual drifters entrained in wake eddies. Estimates of vorticity from the two methods agree well. On average, wake vorticity is initially Ro > 1 and decreases inversely proportional to time (Fig. 3). Kinetic energy is comparatively constant. Cluster scale increases linearly in time, which suggests that vorticity is inversely proportional to scale. This is confirmed with an enstrophy wavenumber spectrum showing ζk2 (Fig. 4). A similar diffusivity wavenumber spectrum shows a k−4/3 slope (Fig. 13). Two case studies of drifter clusters entrained in wake eddies (C5 and C7) both exhibit initial periods of strong vorticity (Ro > 1) followed by an abrupt drop below Ro < 1 after approximately one week.

Here we discuss implications of the observations described above. In section 4a, we estimate the scale of superinertial eddies generated by westward flow past Palau. To do so we compare the observed time series of KE and vorticity with radial profiles of vorticity and KE predicted by two simple analytic models of an eddy. Then in section 4b, we infer the dynamic balance of the eddies using observations to scale terms in the momentum and continuity equations. Finally, in section 4c, we discuss the observed dispersion of drifter clusters and what it suggests about the evolution of eddies downstream of Palau.

a. Eddy scale

We can estimate the scale of wake eddies with strong vorticity (i.e., Ro > 1) by comparing time series of vorticity, KE, drifter radial position (R), and cluster scale (L) with two simple kinematic models of an eddy; a Rankine vortex and a line vortex (Kundu and Cohen 2007). The Rankine vortex is in solid-body rotation up to a fixed radius, beyond which the vorticity goes to zero and the azimuthal velocity decreases like r1, i.e., an irrotational outer region (blue lines in Figs. 14a,b). The resulting radial profiles of azimuthal velocity, vorticity, and (2×) angular frequency have discontinuities in their first derivatives at the eddy radius. Although vorticity goes to zero outside the core, the angular frequency is nonzero (dashed blue line in Fig. 14b). The second model, a line vortex, is the solution for an initially irrotational point vortex allowed to decay in time due to friction (orange lines in Figs. 14a,b). This model aims to account for the fact that discontinuities are likely smoothed by viscosity in the case of a real eddy. Well inside the core and outside the eddy edge the solution approaches the Rankine vortex model, but with a continuous transition at the eddy edge. Comparison between the two models suggests the simple Rankine vortex likely overpredicts the velocity and vorticity in the vicinity of the eddy radius. The increase in velocity with radial distance from the center of a realistic eddy is slightly less than linear, and vorticity decreases with increasing radial distance even within the core.

Fig. 14.
Fig. 14.

(a) Azimuthal velocity for two eddy models, a Rankine vortex (solid-body plus irrotational core) (blue line), and a line vortex smoothed by friction (orange line). (b) Corresponding vorticity (solid lines) and angular frequency (dashed lines) for each model. (c),(d) Model kinetic energy scalograms from two drifters moving outwards from a Rankine vortex due to an arbitrary positive radial velocity. One was deployed outside the vortex core in (c) and the other inside in (d). Black dashed line in each is the core rotation rate and the red dotted line is the rotation rate at the drifter position. The magenta and blue lines in each are angular frequency obtained from a least squares plane fit (blue) and solid-body fit (magenta) to a cluster of five synthetic drifters.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

Comparing our observations with the two models suggests drifters are moving outwards from within their eddy cores. We observed an abrupt drop in vorticity as measured by C5 from Ro > 1 to Ro < 1 around day 5 (Fig. 7c). KE also drops abruptly, while mean radial drifter distance (R) and cluster scale (L) steadily increase. Prior to this point, KE increases while vorticity decreases. Both models indicate that such abrupt drops in vorticity and KE would occur as drifters exit the eddy core. When the drop occurs, R and L measured by C5 are both ∼20 km. Recall that both estimates of scale are akin to radii when entrained in an eddy. Therefore, it is likely the eddy diameter is ∼40 km. We can estimate a radial velocity from the time it takes the drifters to reach the eddy edge, ∼5 days. This gives Ur ∼ 0.05 m s−1, an order of magnitude weaker than the azimuthal velocity. The drop in KE on day 5 of ∼0.2 m s−1 is sufficient to explain the change in vorticity of Ro ∼ 2, at a radius of ∼20 km. Another superinertial wake eddy, C13, has similar time series characteristics to C5. We estimate the scale of that eddy to also be ∼40 km. In that case, the drop in vorticity and KE occurs around day 7, so the corresponding radial velocity is Ur ∼ 0.03 m s−1. The significance of positive radial velocities is discussed in section 4b.

There are similar drops in vorticity and KE observed by C7 on about day 9, but key differences suggest this event does not signal drifters exiting the eddy core (Fig. 9). At the same time, R increases abruptly from 25 to 1000 km (Fig. 9e). Further, KE slowly decreases before dropping abruptly on day 9. These characteristics are inconsistent with both simple models. However, there is another sharp drop in both KE and vorticity which may signal drifters reaching the eddy edge around day 2, almost immediately after being entrained. KE scalograms from the different drifters in C7 give highly variable estimates of ω at early times, which suggests the drifters are orbiting near the edge of the eddy (Fig. 8). In order of highest to lowest angular frequency, drifters 5, 2, 4, 1, and 3 are ∼8, 9, 12, 16, and 20 km away from the estimated eddy center, respectively. Thus, differences in ω are likely due to this variation in their radial distance, with inner drifters having higher ω than outer drifters. This suggests a core radius of ∼13 km. We confirm this interpretation using model scalograms computed from synthetic drifters exiting a Line vortex with an arbitrary azimuthal velocity (Figs. 14c,d). Drifters inside the core overestimate the vorticity obtained from the least squares fit, while drifters outside give an underestimate. In this case it is not feasible to estimate a radial velocity from the observations, since the drifters were likely deployed close to the eddy edge.

The 10–40-km scale estimate for strong eddies with Ro > 1 is consistent with observations from previous FLEAT studies (Rudnick et al. 2019; Merrifield et al. 2019; Zeiden et al. 2019). We are unable to estimate the scale of eddies with Ro < 1 in the same manner. In those cases, the estimate of R is unrealistically large, suggesting that the scale of the cluster is much smaller than the eddies they are entrained in. There are no sudden kinematic shifts which signal drifters move beyond their eddy cores. It is likely these eddies are a few times larger than the superinertial eddies described above, but they are limited by the scale of the island [O(100) km].

b. Dynamic balance

The high Ro observed with C5, C7, and C13 suggest the balance of these eddies is cyclogeostrophic (rather than geostrophic) to first order. This balance is given by
fuθ=1ρ0pruθ2r,
where uθ is the azimuthal velocity and r is the distance from the eddy center (e.g., Shakespeare 2016). The key factor distinguishing between cyclogeostrophic and geostrophic balance is the relative strength of the centrifugal term and Coriolis term, i.e., here between U2/R and fU, if U is a characteristic azimuthal velocity and R is the eddy radius. Using observations from C5, U ∼ 0.5 m s−1, R ∼ 2 × 104 m, f = 2 × 10−5 s−1, and g = 10 m s−2. This gives U2/R = 1.25 × 10−5 and fU = 1 × 10−5, suggesting the balance is indeed cyclogeostrophic. A pressure gradient term of similar magnitude implies the sea surface displacement associated with these eddies is likely to be on the order of a few to tens of centimeters, consistent with observations of mesoscale eddies with an SSH signature (e.g., Yang et al. 2013).

We have shown that drifters move outwards from their eddy centers over time with a radial velocity about 1/10 the strength of the azimuthal eddy velocity. Such weak secondary circulation can arise due to slight imbalances in Eq. (5). For a cyclonic eddy, terms on the LHS and RHS of Eq. (5) are directed radially outward and inward, respectively. Induction of weak positive radial velocities, such as those observed here, would require an imbalance between the terms in Eq. (5) in favor of the Coriolis term. Assuming an initial steady state balance, either an increase in eddy radius or decrease in the absolute sea surface displacement (or both) would lead to the requisite imbalance. An increase in eddy size is probable, either due to friction or straining by larger scale currents, as a relaxation of the sea surface. We discuss the evolution of wake eddies further in section 4c. Continuity in turn ensures positive radial velocities induce upwelling in the interior, and as a result cyclonic eddies are typically cold core (e.g., Olson 1991; Mittelstaedt 1987; Martin and Richards 2001). A snapshot of sea surface temperature (SST) over the first week of C7 from the Moderate Resolution Imaging Spectroradiometer (MODIS) on board NASA’s Aqua satellite shows a strong temperature front extending northward along the wake edge, coincident with the drifter trajectories (Fig. 15). To the northwest of the island, the strong cyclonic wake region corresponds to an SST low. This is consistent with observations of cold core mesoscale eddies with upwelling in their interiors (e.g., Olson 1991; Mittelstaedt 1987; Martin and Richards 2001). It seems likely the C7 submesoscale eddy is similarly cold core.

Fig. 15.
Fig. 15.

(a) MODIS SST averaged over the first 8 days of C7 and (b) vorticity from OSCAR surface currents averaged over the same period. Black quivers in both are corresponding OSCAR velocity vectors, and green lines are the trajectories of the drifters. Warm surface water is advected toward the island and accelerated around its northern end, providing an outline of the wake region. To the northwest of the island, a region of cyclonic circulation is coincident with cooler surface temperature.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

c. Eddy evolution

Vorticity downstream of Palau is the result of drifters entrained in eddies as well as free shear layers extending westward from the island (Fig. 6). The Palau wake is highly variable, but two characteristics are remarkably robust. KE is approximately constant at all scales and times, and vorticity is inversely related to scale. The latter is confirmed by computing pseudo enstrophy wavenumber spectra over each wake type separately (not shown). The relationship breaks down when drifters are directly within the strong eddy cores as detailed previously, but persists outside their cores at later times. Once ejected from the core of strong eddies (after ∼1 week), drifters continue to orbit cyclonically as their scale separation increases (e.g., the trajectories shown in Figs. 7 and 9). However, the observed KE remains constant, which is inconsistent with drifters simply exiting an isolated eddy. In that case, both KE and vorticity should decrease quadratically with scale (Fig. 14).

These observations suggest the eddies themselves are evolving with time as the drifters continue to orbit them. The scale of clusters not entrained in wake eddies is controlled by the large-scale free shear layer downstream of the island (Fig. 12). It is possible that shear controls the growth of wake eddies downstream as well. After the first week, the clusters entrained in wake eddies increase in scale as ∼t3/2, similar to the along-stream growth of clusters entrained in the free shear layers (Fig. 12c). As discussed previously, this growth rate is consistent with dispersion due to lateral shear (LaCasce 2008). This would explain the steadiness of KE with increased scale. A classic paper by Brown and Roshko (1974) demonstrated that eddies embedded in a shear layer entrain ambient fluid, growing in scale as they do. In that case, continual entrainment of constant velocity fluid from the free-stream stabilizes the kinetic energy of the vortices. Entrainment is regularly observed in oceanic eddies, visible in concentrated surfactants and chlorophyll (e.g., Munk et al. 2000). A parameter which gauges the ability of eddies to exchange fluid with the exterior ocean is the ratio of their translation speed to their azimuthal velocities (Chelton et al. 2011). If this ratio is less than one, exchange is likely to occur. The probability density function of the two velocities for all observed eddies indicates that on average this parameter falls below unity, with a mean value of 0.6 (Fig. 16). This suggests it is likely the Palau wake eddies entrain external fluid as they are advected downstream, growing in scale while their kinetic energy remains relatively constant.

Fig. 16.
Fig. 16.

(a) Probability density function of number of observations binned by eddy azimuthal velocity (Ue) and translation speed (Uc). The ratio of these values gives a nonlinearity parameter which indicates whether eddies are likely to exchange fluid with the exterior region (yes if Ue/Uc < 1). (b) As in (a), but with vorticity as the binned variable.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0252.1

There is significant variance in the incident geostrophic flow, and occasionally submesoscale wake eddies may also be significantly strained by mesoscale meanders and eddies advected into the region by the NEC. This can be seen when the vorticity observed by C7 drops abruptly below Ro = 1 on day 9 (Fig. 5a). At the same time, the mean drifter radial position (R) increases sharply to a nonphysical 1000 km and the vorticity becomes briefly anticyclonic. The location of the (nonphysical) eddy center during this period is to the northeast of the cluster. OSCAR surface currents averaged over the first 8 days of the deployment indicate the mesoscale shear in this region at this time is anticyclonic (Fig. 15b). It seems likely the drifters are being strained by this shear. Around day 10, the drifters resume their cyclonic rotation at a lower frequency and larger radius. It is unclear whether they are still entrained in the original eddy or have been entrained in a different cyclonic wake eddy.

5. Summary

In this study we have presented observations from clusters of drifters released over a 2-yr period at the northern end of the island Palau, where the incident large-scale flow separates from the island. These clusters were often entrained in cyclonic wake eddies, and at other times advected downstream in cyclonic large-scale shear layers. We calculate vorticity from drifter velocities in two ways: 1) by computing spatial gradients in the velocity field and 2) by identifying peak frequencies in time series of velocity. Eddies characterized by strong Ro > 1 and O(10) km in scale are likely in cyclogeostrophic balance, with weak positive radial velocity at the surface. As a result, drifters are ejected from the eddy cores on the order of 1 week. Weaker, subinertial eddies were larger, up to O(100) km. Vorticity in the shear layers was mostly Ro < 1. Averaged over all clusters advected into the Palau wake, vorticity is inversely dependent on cluster scale, which in turn increased linearly with time (Lt). In contrast, kinetic energy is not dependent on scale or time. Upper and lower bounds for the cluster scale growth were given by Lt 3/2 and t 1/2, and diffusivity increases with scale κL4/3. These observations suggest that shear dynamics dominate dispersion in the wake of Palau, including the growth of wake eddies downstream. OSCAR surface currents reveal that while wake eddies are the result of flow blocking when currents are directly incident on the island, the observed shear layers were part of a regional-scale cyclonic circulation around Palau and thus their vorticity was not generated by flow past the island. It is likely wake eddies are preferentially generated in the second half of the year, due to annual changes in the regional flow around Palau.

Acknowledgments.

This work was funded by the Office of Naval Research under Grants N00014-15-1-2286_215E1A, N00014-17-1-2517_21AEBA, N00014-15-1-2488, N00014-18-1-2406, N00014-15-1-2592, and N00014-15-1-2264. We thank Pat and Lori Colin from the Coral Reef Research Foundation in Palau, as well as the captains and crews of the R/V Revelle during the FLEAT cruises, for their assistance in the repeated smooth and successful deployment of the drifters. We also thank the engineers in the Lagrangian Drifter Lab for their technical expertise in development and preparation of the drifters, and two anonymous reviewers for their helpful suggestions.

Data availability statement.

Drifter data can be made available by the authors upon request. For access please contact LC at lcenturioni@ucsd.edu or VH at vhormann@ucsd.edu.

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