Broadband Submesoscale Vorticity Generated by Flow around an Island

a. Oceanographic setting

Palau is a meridionally elongated archipelago in the western North Pacific. The archipelago is composed of over 300 small islands which are connected by a submerged reef extending ~200 km from its northern to southern end. In this manuscript, when we refer to the “island” of Palau we mean this submerged reef. Palau lies in the path of two opposing geostrophic currents. At the northern end, the North Equatorial Current (NEC) flows between 8° and 17°N with maximum speeds of 0.5 m s⁻¹ (Qiu and Lukas 1996; Schönau and Rudnick 2015). On average, the NEC is westward and surface intensified, but extends down to the pycnocline centered at about 150-m depth close to Palau (Schönau and Rudnick 2015). Below the pycnocline there are weak eastward undercurrents, the North Equatorial Undercurrents (NEUCs), with average current speeds less than 0.1 m s⁻¹ (Qiu et al. 2013). Transport in these undercurrents is correlated with transport of the NEC (Schönau and Rudnick 2015), although this relationship weakens downstream of Palau (Zeiden et al. 2019). At the southern end, the North Equatorial Countercurrent (NECC) flows eastward at variable latitude between the equator and 7°N with maximum currents that can exceed 1 m s⁻¹ (Hsin and Qiu 2012).

Data presented in this manuscript were collected at the northern end of Palau, where on average the NEC and southernmost NEUC are incident on the archipelago. Here the steep-sided island boasts bathymetric gradients up to 45°, culminating in a 40-km-wide subsurface reef called Velasco. The reef rises to just under 20 m around the rim and blocks the incident flow almost completely to the surface. A concurrent FLEAT study quantified the mean mesoscale currents around Velasco and their variability using repeat glider surveys (Zeiden et al. 2019). Mean incident currents were westward and surface intensified (~0.2 m s⁻¹ within the upper 200 m) with eastward return flow in the lee of Velasco. Significant variability on subinertial time scales was likely due to the mesoscale eddy field advected westward by the NEC (Schönau and Rudnick 2015; Hsin and Qiu 2012). Strong northward currents occurred on occasion due to recirculation of the NECC, likely spurred by westward propagating Rossby waves (Heron et al. 2006).

b. Dataset description

Fourteen moorings were deployed around Velasco over an 11 month period from April 2016 to May 2017 (Fig. 1). The array was designed to sample high-frequency currents with adequate spatial resolution to resolve their submesoscale structure around the island. A subselection of these moorings was used in this study. Four far-field (greater than 10 km from the reef) deep-water moorings were used to characterize the ambient flow field (F1, M1, M2, and M5 deployed in 3390, 2600, 2610, and 3690 m, respectively). Three near-field (within 10 km) moorings were used to calculate vorticity, F2, F3, and F4. These moorings were deployed in 1515, 1666, and 880 m, respectively. Each had two ADCPs mounted around 60-m depth; an upward-looking 300-kHz RDI Work Horse Monitor and a downward-looking 75-kHz Work Horse Long Ranger. F2 had an additional upward-looking Long Ranger deployed at 770 m and a downward looking Work Horse at 1470 m. F3 had a downward looking Long Ranger at 760 m, and a downward-looking Work Horse at 1570 m. F4 had one additional upward looking Long Ranger at 880 m. These moorings form an equilateral triangle ~7-km scale and centered 5 km from the northernmost promontory of Velasco (Fig. 1c). Moorings were each equipped with a series of thermistors, CTDs and ADCPs. In this study we examine the ADCP data. F moorings each have ADCP coverage of over 80% of the water column at their respective depths and M moorings have coverage from the surface to 700 m. For all deep-water moorings, upward looking ADCPs deployed near the surface achieved a vertical resolution of 4 m and a minimum temporal resolution of 16 min, enough to resolve tidal motions. Velocities were smoothed and interpolated to a 10 m × 1 h grid.

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Mooring locations plotted over (a) regional and (b),(c) local bathymetry. Insets (b) and (c) are marked in (a) and (b), respectively, with dashed black lines. The filled tan contour in (a) is land above sea level. Black contours in (a) and (b) are 100- and 1000-m isobaths, and 100-m intervals in (c). Mooring locations are in red. Blue and orange vectors in (b) are mean surface currents during strong eastward (August, orange boxes in Fig. 3) and westward (December, blue boxes in Fig. 3) flow. Topographic steering is apparent during westward flow. Moorings were deployed for 11 months between May 2016 and April 2017 around the northern end of Palau, where the NEC encounters the island on average. Moorings F1, F2, F3, F4, M1, M2, and M5 are used to characterize the flow field, while moorings F2, F3, and F4 are used for a detailed vorticity analysis.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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To examine the depth dependence of vorticity around Palau we perform a layer-average in the surface (0–100 m) and subpycnocline (hereafter subsurface, 300–400 m). The surface layer was chosen to capture the variability of the NEC, which extends through the base of the pycnocline but decays substantially below 100 m (Schönau and Rudnick 2015). The subsurface layer similarly corresponds to the mean depth of the peak NEUCs velocities at this latitude, well below the pycnocline. An example of zonal and meridional velocity in the upper 1400 m during a period of westward surface flow at F3 gives an idea of the temporal and vertical scales included in these averages (Fig. 2). Vertically sheared geostrophic currents have length scales of O(100) m in the surface (NEC) and hundreds of meters in the subsurface (NEUCs). Energetic tides can lead to flow reversals despite the strong geostrophic westward flow. There is likely additional vertical variability on finer scales which has been filtered by our gridding process. The surface layer encompasses the mixed layer and the upper half of the pycnocline, although ADCP data in the surface 10 m were often discarded due to sidelobe contamination. The mean mixed layer depth (MLD) during the observation period was 40 m, with a standard deviation of 12 m (obtained with CTD data from concurrent glider surveys). Correlation values between velocity at any two depths across the MLD are no less than 0.5, and mostly greater than 0.8, despite a potential dynamic shift across the base of the mixed layer. Correlation values are highest in the semidiurnal band (R ≥ 0.7), but also strong in the diurnal band (R ≥ 0.6). Correlation values are lowest in the inertial band, most likely due to inertial oscillations confined to the mixed layer (R = 0.5). Nonetheless, the main results of our analysis are not sensitive to varying the base of the surface layer from 50 to 150 m. The subsurface layer is well below the pycnocline and above the bottom boundary. Correlation values within this layer are above 0.8 in all bands due to the large vertical length scales of the NEUC and tide, and absence of energetic mixed layer currents. Correlation values between velocity at any depth level in the surface and subsurface layer are less than 0.3, which supports treating them independently.

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Example section of (a) zonal and (b) meridional velocity from 0 to 1400 m at F3 during a strong westward subinertial surface current event from 15 Nov 2016 to 1 Jan 2017. Ticks are 1-day intervals. The black dot–dash line in both panels indicates the base of the surface layer (0–100 m). Black dash lines indicate the subsurface layer (300–400 m). Strong tidal velocities retard the subinertial current and often lead to flow reversals, especially at depth where the subinertial current is weaker. Correlation values between velocity at different depths within each layer are strong (R > 0.5 and 0.8 in the surface and subsurface layers, respectively).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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In this manuscript we often refer to subinertial low passes of each time series (bold lines overlaying each time series in all figures). These low passes are obtained by convolving a 10-day Hanning window with the time series. Although not plotted, we also refer to superinertial time series. These are obtained by subtracting a 2-day Hanning window filtered time series from the full time series.

c. Vorticity calculation

The spatial gradients necessary to compute the vertical component of relative vorticity at the center of a triangular array can be approximated by fitting a plane to zonal and meridional velocity at each time step and calculating their slopes (i.e., ζ = υ_x − u_y). Equilateral triangles provide the lowest error estimate of spatial gradients via this method (Davies-Jones 1993). This is intuitive because acute/obtuse triangles sample spatial gradients over different length scales in each direction and therefore give an over/underestimate of the magnitude of vorticity at the average scale of the array. The triangle formed by F2, F3, and F4 is nearly equilateral, with side lengths of 5.87, 5.82, and 5.87 km (in the clockwise direction starting at F2, Fig. 1). Together they inscribe a circle 6.8 km in diameter.

The implicit assumption when using a plane-fit to calculate vorticity is that the observed velocity variability is due to spatial gradients which are near the scale of the array. The error associated with this assumption is dependent on the relative energy at spatial scales smaller than the triangle. Without knowing the actual velocity field, it is difficult to estimate this error (Davies-Jones 1993). While KE spectra in the open ocean are typically red, it is unclear how proximity to Palau may impact this distribution. For example, MacKinnon et al. (2019) observed the generation of tidal eddies ~1 km in scale during ship surveys around the northern tip of Velasco. If these eddies were to propagate away from their generation site and cross a single mooring in the array, it would be aliased as vorticity variance at the scale sampled by the triangle. To better understand the spatial structure of currents included in our vorticity estimate, we perform an empirical orthogonal function (EOF) analysis of the velocity time series in each layer.

d. EOF analysis

Previous studies have used EOFs to identify the statistical modes with maximum covariance across an array (Rudnick and Davis 1988; Rudnick and Weller 1993; Fischer et al. 2002). In this study we calculate EOFs of velocity across the triangular array. The EOFs are orthogonal eigenvectors of the data covariance matrix. Each describe a certain percentage of the total variance, given by their eigenvalues. The projection of each EOF back onto the data gives its time-dependent amplitude. While EOFs should not automatically be interpreted as real physical modes, there is no more efficient way to describe the velocity variance. Typically, velocity time series are reconstructed using only the most energetic EOFs. In this study we are specifically concerned with calculating vorticity, so we also consider the Okubo–Weiss value (OW) of each EOF. OW is a measure of the relative magnitude of strain and vorticity and is given by σ² − ζ², where σ² is the magnitude of horizontal strain given by (u_x − υ_y)² + (υ_x + u_y)² and ζ² is the enstrophy (Okubo 1970; Weiss 1991). This value springs from approximating the horizontal flow field around a singular point as a power series, where the first-order terms are a function of local linear velocity gradients. The sign of OW determines whether the flow field follows spiral trajectories (negative OW) or nodal/saddle trajectories (positive OW). Regions of negative OW are frequently used to identify coherent vortices in modeling and satellite data (e.g., D’Addezio et al. 2020).

To minimize the impact of finescale features on the vorticity calculation, we select for currents with spatial scales close to that of the array. First, we identify the EOFs in each layer which describe either currents in phase across the array (i.e., in the same direction), or with vortical structure near the array scale (i.e., currents out of phase, but with a clear sense of rotation characterized by negative OW). We then “reconstruct” the velocity time series by multiplying these EOFs by their respective amplitudes and adding those time series. We find that the subsequent reconstructed velocity time series retain 94% and 89% of the total velocity variance in the surface and subsurface layers, respectively. We then proceed to analyze the vorticity obtained via plane-fit to these velocity time series with confidence that our estimates are not significantly aliased by finescale motions. Further details of the EOFs are examined in the following section, as well as their individual contributions to the vorticity.

a. Broadband currents around Palau

Surface currents around Velasco are broadband in frequency, as seen in time series of surface velocity (0–100 m) from a selection of the moorings counterclockwise around the reef: M2, M5, F3, F1, and M1 (Fig. 3). In the mean, M2 is upstream of Velasco and M1 is downstream. Four strong subinertial current events dominate the variance with velocities of ~0.5 m s⁻¹. Here and throughout the text, we will often refer to, e.g., “eastward flow” when discussing subinertial currents. Strong eastward flow occurs in late July and strong westward flow in September and December. Note that the strong northward component which accompanies westward flow is the result of topographic blocking. Using repeat glider surveys east and west of Palau, Zeiden et al. (2019) found the time-mean incident westward flow diverges at ~8.2°N. The current above that latitude gains a strong northward component as it circumvents the island. This effect can be seen clearly in average surface velocity vectors during the westward flow event in December (Fig. 1b). A fourth event in early March begins as strong northward flow on the eastern side of the island (at M2 and M5) and then transitions to westward flow. Subinertial velocities are well correlated across the array (e.g., R ≥ 0.6 between F3 and the other moorings), suggesting they have spatial scales of at least 60 km. This is consistent with glider observations which indicate that subinertial currents often have spatial scales exceeding 100 km. Weak downstream currents are further evidence of topographic blocking. For example, the strong eastward current in July appears in velocity data at M1, F1, F3, and M5 but does not appear downstream of Velasco at M2 (orange shaded boxes in Fig. 3). Similarly, currents are negligible downstream at M1 during westward flow (blue shaded boxes in Fig. 3). Thus, topographic blocking reduces the strength of large-scale, subinertial currents by almost an order of magnitude.

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Surface (0–100-m depth-averaged) zonal (blue lines) and meridional (red lines) velocity time series at M2, M5, F3, F1, and M1 (counterclockwise around the reef). Thick lines are 10-day low-passed. Four major subinertial current events dominate the variance and show topographic blocking of large-scale flows: strong eastward flow in August (absent downstream at M2, marked with red boxes) and strong northwestward flow in September, December, and March/April (absent downstream at M1, marked with blue boxes). Strong northward flow to the east of Velasco precedes the fourth westward event, appearing first at M2 and M5 before transitioning to northwestward flow. Significant high-frequency variability frequently leads to flow reversals.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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Subinertial currents are weaker in the subsurface layer (300–400 m), with strong events reaching ~0.25 m s⁻¹ (Fig. 4). One sustained westward flow event spanning August and September 2016 stands out in the time series of M2 and M5 (orange shaded boxes in Fig. 4), but otherwise high-frequency currents dominate the variance. Note that although the northward current is actually stronger than the zonal component here, concurrent glider observations show the incident large-scale subinertial flow is westward during this time (not shown). None of the five subinertial current events observed in the surface layer have a signature at depth, consistent with a weakened relationship between the NEC and NEUC close to the island (Zeiden et al. 2019). Correlation in space is divided between moorings to the east and west of Velasco (Fig. 4). For example, subinertial zonal currents at M5 are strongly correlated with those at M2 (both east of Velasco, R = 0.7) but are uncorrelated with those at F1 (west of Velasco). Interestingly, subinertial subsurface velocity at the tip of the reef is correlated with currents both to the east and west of Velasco (R > 0.6 between F3 and all other moorings).

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As in Fig. 3, but for the subsurface (300–400-m depth averaged). One main subinertial northwestward current event is apparent east of Velasco at M2 and M5 in August and September (marked with red boxes), but otherwise high-frequency currents are often significantly stronger than the subinertial flow at depth.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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In both layers, high-frequency currents are energetic enough to induce flow reversals except during the strongest subinertial current events. Their contribution to the total velocity variance is quantified with surface and subsurface kinetic energy spectra at the moorings in the triangular array (Figs. 5a,b). Subsurface KE variance is higher at F4 because it is shallower and therefore has stronger depth-mean stratification. Spectra are red with broad peaks around the local inertial (f) frequency, and sharp peaks at the diurnal (D1) and semidiurnal (D2) frequencies. These bands account for 10%, 9%, and 12% of the total variance in the surface layer, and 12%, 14%, and 32% in the subsurface layer. The greater percentage of KE variance contained in these bands at depth (>50%) reflects the weak subinertial currents there.

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(a) Surface and (b) subsurface kinetic energy spectra at F2 (green), F3 (red), and F4 (blue) are red, with strong peaks in the inertial, diurnal, and semidiurnal bands. The dotted gray line in (a) and (b) gives the Garrett–Munk internal wave spectrum for reference. (c) Time series of array-averaged surface velocity and (d) a corresponding scalogram of kinetic energy reveal that energy in these bands is nonstationary in the energetic surface layer. Conversely, (e) subsurface array-averaged velocity and (f) corresponding kinetic energy scalogram have steady tidal energy, modulated only by spring–neap variability.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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At the surface, tidal currents are highly irregular. A scalogram of array-mean surface KE shows sharp increases in tidal variance that occurs on time scales of about a week (Fig. 5d). The most energetic tidal event occurs in late July, coincident with strong subinertial eastward flow. In the subsurface, tidal KE variance is comparatively stationary (Fig. 5f). A strong spring–neap cycle is apparent in the semidiurnal band, and there is weak intermittency in the diurnal band. This spring–neap cycle reflects the barotropic tide (i.e., depth average, not shown). It is likely internal waves contribute more to the irregularity in surface tidal KE. In another FLEAT study, Voet et al. (2020) observe the formation of lee waves from a combination of subinertial and tidal flows over the ridge north of Velasco. However, their observations suggest it is unlikely these waves contribute appreciably to upper-ocean velocity. In an earlier study of Palau, Wolanski et al. (2004) observed isotherm shoaling at tidal periods. These internal tides were synchronous the local spring–neap cycle, suggesting they were locally generated. In our observations, strong peaks in tidal surface KE are not synchronous with the local spring tide (seen in Fig. 5f). Changes in the local stratification could cause detuning between the barotropic tide and locally generated internal tides, but remotely generated internal tides may better explain the observed variance. To the west of Palau, the Luzon Strait is a known generation site of strong internal tide beams which may encounter Palau during favorable current conditions (Zhao et al. 2013). Propagation speeds of mode-1 internal waves in the tropical western North Pacific are ~3 m s⁻¹ (Rainville and Pinkel 2006), which corresponds to a lag between the Luzon Strait and Palau on the order of 1 week. The arrival of such remote internal tides would not likely have a regular pattern that corresponds with the local spring–neap cycle.

There are similar irregular peaks in the inertial band at the surface, most notably twice in November and once in early March. This intermittency is consistent with wind forced inertial oscillations, which are sporadic by nature. In another FLEAT study, Voet et al. (2020) characterize the inertial currents around Palau using moorings from both the northern and southern ends of the island. Although only one inertial event is easily explained using local wind data, they argue that the large decorrelation length scales of the other inertial events suggest they are wind generated. At depth, near-inertial internal waves likely contribute to the variance as well. In early November there is a weak peak around ~2 days, as well as a smattering of energy in the superinertial, subtidal band in September and October. Andres et al. (2020) observed a similar signal in acoustic travel time records from pressure-sensor-equipped inverted echo sounders (PIES) around the island. This signal propagated through their array in the southeast direction, and they concluded near-inertial internal waves likely propagate into the region from higher latitudes.

b. Submesoscale vorticity time series

Here we present time series of submesoscale (~7 km) vorticity obtained via plane fit to the EOF filtered velocity time series from the triangular array (see section 2 for methodology). Because the reconstructed time series retain over 94% (89%) of the total velocity variance in the surface (subsurface) layer, we present the time series first and then the EOFs to help interpret the signals in them. Here and throughout the manuscript we often refer to “shear,” by which we mean lateral shear. When discussing vertically sheared currents, we use “vertical shear.”

Vorticity in the surface layer has a clear relationship to the subinertial flow. Magnitudes are O(1), reaching up to Ro ~ 6 during westward flow events (Figs. 6d and 7d ). This is consistent with synoptic observations in the same region obtained by Wijesekera et al. (2020) during the period of strong westward flow in early December. Over the course of a week they conducted repeat ship surveys in box patterns O(10) km around the northern end of Velasco. These surveys captured the flow accelerating and separating near F4, with strong recirculation immediately to the west. The recirculation was ~5–10 km in scale and had vorticity O(1) (see their Figs. 10 and 11). This suggests that vorticity here also reflects flow separation and recirculation at the scale of the array. We will show this interpretation is supported by the EOFs (section 2c). Subinertial vorticity is highly correlated with subinertial velocity, especially meridional velocity at F3 (R = 0.9). During the periods of energetic westward flow, vorticity increases from zero to Ro ~ 5. An ~0.1 m s⁻¹ increase in velocity corresponds to an increase in vorticity of Ro ~ 1. It is noteworthy that the flow event with greatest total kinetic energy does not have a strong vorticity signature (eastward flow in late July). The reason for this is evident in the relative lack of array-scale shear, which distinguishes it from the other events.

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Time series of array-mean surface (a) zonal velocity, (b) meridional velocity, (c) kinetic energy, and (d) vorticity. Velocity time series were reconstructed using EOFs (section 3b). Bold lines are 10-day low passes of the respective time series, except for (a) and (b), which are 10-day low passes at each of the three moorings in the array. Subinertial vorticity varies linearly with subinertial velocity during westward flow. A key result of this study is that high-frequency vorticity variance also increases during strong westward flow. Eastward flow does not have a strong vorticity signature.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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As in Fig. 6, but in the subsurface (300–400 m). Note the changes in vertical axes from Fig. 6. Unlike surface velocities, at depth the current is often dominated by high-frequency variance. One subinertial even stands out with strong shear across the array evident in meridional velocity in September and October. Vorticity is steady and significantly weaker than at the surface, except during this subinertial event when it increases both in the subinertial band and at higher frequencies.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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Surface vorticity is highly broadband, and a novel observation of this study is that high-frequency vorticity variance increases substantially during strong westward flow. This can be seen by comparing the range of high-frequency vorticity prior to and during the westward event in December (Fig. 6d). Initially, values range from $\left|\text{Ro}\right|\mathrm{\lesssim }1$. At maximum flow, this increases to $\left|\text{Ro}\right|\mathrm{\lesssim }4$. We define a low-frequency envelope for this variance by 10-day low-passing the magnitude of superinertial vorticity (not shown). This envelope is strongly correlated with subinertial velocity, especially meridional velocity at F3 and F4 (R = 0.8). The relationship between high-frequency vorticity and corresponding low-frequency envelopes of high-frequency velocity is weaker, with R ≤ 0.5 at all moorings other than F3, where R = 0.6.

Similar relationships between the subinertial current and vorticity exist in the subsurface layer as well (Fig. 7). Subinertial vorticity is strongest during the westward flow event in September and October, and high-frequency vorticity variance is nonstationary. Recall that close to the topography, currents become rectilinear and the meridional component dominates during westward flow (Fig. 4). For most of the time series, subinertial vorticity is negligible and high-frequency vorticity is $\left|\text{Ro}\right|\mathrm{\lesssim }1$. During the westward flow event this increases to $\left|\text{Ro}\right|\mathrm{\lesssim }4$. The correlation between high-frequency vorticity and subinertial velocity is weaker than at the surface. The strongest correlation is with meridional velocity at F4 (R = 0.7), but otherwise less than 0.5. It is clear that the strength of the current is not the distinguishing factor leading to elevated vorticity. Instead, strong shear in the meridional component across the array separates this period from the rest. As with the surface layer, the period with greatest kinetic energy (here January) does not have a vorticity signature due to a lack of array-scale shear.

c. Dominant statistical modes

The EOFs of velocity used to filter out unresolved spatial variability give the horizontal structure of currents around Velasco and their contributions to the vorticity (Fig. 8). At the surface, the most energetic EOF accounts for 72% of the total velocity variance and reflects large-scale flow around the island (Fig. 8a). Current vectors are in phase with similar magnitude. Velocities are strongest at F4 where the flow is forced to accelerate around the promontory, and weakest in the lee at F2 where it must decelerate. Because its vorticity is the result of shear rather than a coherent vortex, the OW value of the first EOF is weakly positive (Fig. 8b). The second most energetic EOF contains 17% of the total velocity variance and is similar in character to the first EOF. Together these EOFs describe large-scale currents forced to circumvent the island from either direction, producing shear at the array scale. This shear contains 36% and 13% of the total vorticity variance, respectively. The third most energetic EOF contains just 5% of the total velocity variance, but nearly half (45%) of the total vorticity variance. Current vectors rotate counterclockwise around the center of the triangle with similar magnitudes, and the OW value is strongly negative. This suggests the third EOF describes vortex motion at the scale of the array, which likely reflects a combination of attached recirculating flow and fully separated eddies.

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Empirical orthogonal functions of velocity in the (a) surface layer (0–100 m) and (e) subsurface layer (300–400 m). Colored contours in (a) and (e) are 100-m interval isobaths from 0 to 1000 m, and the black contour is the depth of the EOFs. (b),(f) Corresponding normalized magnitudes of vorticity, strain, divergence, and OW value. (c),(d),(g),(h) The percentage of the total variance described by each EOF. In both layers the two most energetic EOFs (red and blue lines and vectors, respectively) describe currents in phase with similar magnitude. The third EOF (magenta lines and vectors) describes vortex motion at the scale of the triangle. The three least energetic EOFs are characterized by strong relative divergence and strain. Subsequent vorticity time series are calculated using only the first three EOFs.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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The EOFs of subsurface velocity have a similar variance distribution and similar structures (Figs. 8e–h). The two most energetic EOFs account for 64% and 18% of the total velocity variance and describe large-scale flow around the island. The third EOF accounts for only 7% of the velocity variance and describes eddying motions at the scale of the triangle. This EOF is responsible for a bulk of the vorticity variance. While the first two EOFs account for only 5% and 4% of the total vorticity variance, the third EOF accounts for 75% of the total vorticity variance. An important difference between the surface and subsurface EOFs is that vectors of the first subsurface EOF follow the topography. Additionally, the second EOF describes flow from the east but has a westward component at F4 and therefore significant convergence. These two dissimilarities suggest that the large-scale subsurface flow often does not separate, reflected in the negligible vorticity of the first two EOFs.

Both in the surface and subsurface layers, the three least energetic EOFs are dominated by strain and divergence. We interpret these EOFs as currents which are not properly resolved by the triangle and they were not included in the reconstruction of velocity as a means of scale filtering.

d. Nonstationary vorticity variance

The EOFs enable us to decompose vorticity into contributions from 1) currents in phase across the array and 2) eddy motion at the scale of the array (Fig. 9). Our goal in this section is to gain physical insight into the processes leading to vorticity generation around Palau, i.e., to understand which motions are responsible for the observed nonstationary vorticity variance. For both layers, we define a “shear” component of vorticity as due to the first two EOFs (in phase) and an “eddy” component of vorticity as due to the third EOF (eddy motion). However, it is important to keep in mind that these are statistical modes, not true physical modes, and the real currents are a linear superposition of them (plus a mean).

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Time series of (a),(b) surface and (c),(d) subsurface vorticity calculated using velocity from EOFs characterized by lateral shear in (a) and (c) and EOFs characterized by eddy motion in (b) and (d). Bold lines are 10-day low passes of each. (e),(f) Corresponding vorticity power spectral densities are plotted, along with kinetic energy spectral densities in gray for comparison. The shear component at the surface is dominated by subinertial variability and steady high-frequency variability. The eddy component contains greater relative high-frequency variance, which is nonstationary. In the subsurface layer, the shear component is weak, steady, and dominated by tidal variance. The eddy component is responsible for a bulk of the vorticity as well as nonstationary vorticity variance.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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In the surface layer, shear and eddy components contribute 49% and 45% to the total vorticity variance, respectively. This equitable distribution is reflected in their individual time series (Figs. 9a,b). Both components range up to $\left|\text{Ro}\right|\mathrm{\lesssim }4$. However, their frequency content differs substantially. Shear vorticity contains greater relative subinertial variance, while eddy vorticity has greater relative superinertial variance. This is evident from the time series but is quantified in their corresponding power spectral densities (Fig. 9e). The spectrum of shear vorticity closely resembles kinetic energy, with strong inertial and tidal peaks (gray line in Fig. 9e). Eddy vorticity has similarly strong tidal peaks, but they are less prominent due to the broadband elevation in high-frequency variance. It is clear from the surface time series that nonstationary high-frequency variance is primarily due to the eddy component of vorticity (Fig. 9a). The range of values increases abruptly from $\left|\text{Ro}\right|\mathrm{\lesssim }1$ up to $\mathrm{\lesssim }4$ at the end of November, concurrent with the onset of strong westward flow. There is a corresponding low-frequency increase in the shear component of vorticity. High-frequency variance in the shear component is comparatively steady.

In the subsurface layer, the shear component contributes only 9% to the total vorticity variance while the eddy component contributes 75%. As a result, the eddy vorticity time series closely resembles the full vorticity time series (Fig. 7d). The eddy component ranges up to $\left|\text{Ro}\right|\mathrm{\lesssim }1$ for most of the record but increases up to $\left|\text{Ro}\right|\mathrm{\lesssim }4$ in September and October during westward flow. Shear component vorticity is characterized by a weak spring–neap cycle and strong peaks in the tidal bands, with close adherence to the spectrum of KE. As in the surface layer, eddy vorticity has greater relative variance at superinertial frequencies and as a result its tidal peaks are less prominent.

As with KE, we examine the spectral distribution of total vorticity as a function of time using scalograms (Figs. 10 and 11). We do not show scalograms of each component because our goal is to quantify the temporal variability of the true vorticity around Palau. However, we note that the scalogram of shear vorticity nearly mirrors KE. Differences between the scalograms of total vorticity and KE reflect nonstationary variance in the eddy component (as noted above). At the surface, the scalogram reveals that elevated variance during westward flow is the result of strong peaks in both tidal bands, especially the diurnal band in December and March (Fig. 10c). Energy increases at supertidal frequencies as well. Peaks in near-inertial variance appear to loosely coincide with the waxing and waning of the subinertial current. At depth, there are clear peaks in the tidal and inertial bands, but the increase in high-frequency variance is more broadband than at the surface. This is predominantly due to elevated variance in the superinertial, subtidal band, especially a strong peak at a period of ~2 days in early November. As mentioned in section 3a, this may be related to superinertial waves observed by Andres et al. (2020).

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Time series of surface (a) subinertial velocity at all three moorings and (b) vorticity, along with (c) the scalogram of vorticity gives time variability of its frequency content. Bold lines are 10-day low passes of each time series. Scalograms are normalized by the total variance contained at frequencies higher than (10 days)⁻¹. High-frequency vorticity variance is highly episodic and strongly correlated with the subinertial flow. Increases in surface tidal variance occur during strong subinertial westward flow in December and March. Kinetic energy (Fig. 5d) in the tidal bands is comparatively stationary, and the strongest peaks do not coincide with those in vorticity.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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As in Fig. 10, but for the subsurface layer. High-frequency vorticity variance increases dramatically in September and October during a subinertial current event characterized by strong shear. This is dissimilar to kinetic energy at depth, which has a strong spring neap cycle (Fig. 5f).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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These vorticity scalograms differ substantially from corresponding KE scalograms (described in section 3a, see Fig. 5). Surface tidal KE is irregular, and its low-frequency modulation only loosely coincides with westward flow. For example, KE in both bands increases in mid-October, over a month before the onset of westward flow in late November. In contrast, tidal vorticity variance only increases during strong westward flow. Also, peaks in tidal band KE mostly do not coincide with peaks in tidal band vorticity. At depth these differences are starker. Superinertial KE is dominated by a steady spring–neap cycle in the semidiurnal band, but vorticity is entirely event driven. These differences reflect the asymmetric distribution of velocity and vorticity variance between the shear and eddy components. The statistical modes which are responsible for most of the KE variance contain vorticity in the form of lateral shear. Therefore, the relationship between shear vorticity and KE is approximately linear across all frequencies. However, most of the vorticity is due to eddy motions, which increase nonlinearly in the tidal bands during strong westward flow events.

a. Nonlinear broadband eddy formation

The observations detailed above are consistent with wake eddies generated by a combination of oscillatory and steady flows past a headland. The source of vorticity in the recirculating wake is shear across the upstream boundary layer, which is a function of total incident flow strength. We see this dependence reflected in the EOFs with shear in the cross-shore direction (Fig. 8). These EOFs contain most of the KE variance and thus shear vorticity is strongly correlated with the total velocity (R = 0.7 in both layers, Figs. 9a,c). This attached recirculating flow forms eddies if the flow reverses and generates an opposing shear layer inshore of the wake which disrupts the upstream supply of vorticity (Signell and Geyer 1991; Pawlak et al. 2003). In our observations, tidal currents are often strong enough to either arrest or reverse the low-frequency flow (Figs. 3 and 4). Thus oscillatory currents force the formation of eddies, while low-frequency currents modulate their vorticity. An example snapshot of surface velocity vectors at both flood and ebb tide during strong westward flow encapsulates this process (Fig. 12). The velocity is decomposed into the shear and eddy components as discussed in section 3d. At flood tide, shear is generated by the superposition of strong subinertial and tidal currents. Later at ebb tide, the zonal component of velocity close to the reef at F2 has reversed, introducing oppositely signed shear and generating an eddy near the scale of the array. We note that elevated tidal vorticity variance occurs in this study when the free-stream flow does not fully reverse, but is substantially weakened by the opposing tide (Fig. 6). Weakened or slack currents may be sufficient to allow an inshore shear layer to grow due to friction between the attached recirculation and the headland (Davies et al. 1990).

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An example of surface velocity during (a) flood and (b) ebb tide during strong westward flow. Dark purple vectors in each panel are the total velocity. Purple vectors are the velocity due to the EOFs with lateral shear, and magenta vectors are velocity due to the EOF with vortex motion. A time-mean northwestward current contributes to the total velocity but is not plotted here for clarity. A 6-day window of (c) array-averaged velocity and (d) vorticity due to the shear and vortex EOFs provide context for the snapshots. Black dotted lines indicate the timing of the flood and ebb snapshots. During flood tide in (a), vorticity is due to the sheared component, but on the opposite phase of the tide in (b) an eddy is released as the direction of the zonal component of the tide reverses close to the topography, cutting off the shear produced during flood tide. Note that in (a) the magenta vectors do not appear because the amplitude of the EOF with vortex motion is zero at that instance, as is indicated by the first back line in (d).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0161.1

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Two recent synoptic observational studies off the coast of Taiwan provide further support for this interpretation. Using a combination of drifter and satellite data Cheng et al. (2020) captured the evolution of three successive submesoscale eddies generated by strong northward flow past the southern cape of Taiwan. These eddies appeared to be generated at a diurnal frequency with similar scales and vorticity to those observed in this study. Using a realistic model of the region, the authors show that eddies are indeed likely shed at tidal frequencies with vorticity modulated by the tide and subinertial current strength (see their Fig. 6). At smaller scales, synoptic surveys Chang et al. (2019) captured the generation of eddies by the flow of the Kuroshio past Green Island, a 7-km island northeast of the cape. These synoptic observations suggest eddies shed close to a diurnal period, and they observed strong tidal peaks in measurements of velocity from two downstream moorings. However, they estimate that the natural shedding frequency of the island (i.e., the Strouhal period) was also close to a tidal period, and so there was a high degree of uncertainty in their estimate of the wake period.

In our case, the observations are inconsistent with eddy shedding at a natural frequency. Although difficult to predict due to its dependence on the Reynolds number, we can estimate an upper limit for this frequency. Maximum values of St are 0.45 for Re > 10⁵ (Bearman 1969). Mean length scales in the wake on these longer time scales reflect the width of Velasco, so L ~70 km (Zeiden et al. 2019). Peak velocities observed during westward flow are 0.5 m s⁻¹ (Fig. 3). This gives a maximum shedding period of 86 h, many times either the diurnal or semidiurnal tidal period. We also consider the possibility that eddy–eddy interactions could modulate the near-field vorticity, as in Callendar et al. (2011). However, this modulation would likely reflect a spring–neap cycle (see their Fig. 6). In our observations tidal vorticity is not modulated by a spring–neap cycle, even at depth where the spring–neap forcing is strong.

b. Internal waves and vorticity variance

In the simple case of regular tides (i.e., barotropic) plus subinertial currents, the latter is responsible for modulating the eddy strength. At depth this explains most of the vorticity and velocity variance. Subsurface semidiurnal tidal currents dominate tidal KE variance with a clear spring–neap cycle (Fig. 5f). Tidal band vorticity is weak and steady, except during a period of subinertial westward flow which generates array scale shear (Fig. 7). This simple relationship is reflected in the scalogram of subsurface vorticity (Fig. 10e). At the surface, however, there is significant intermittency in tidal band KE (Fig. 5d). This is likely due to internal waves, which are not steady in time. Here we consider how internal waves may contribute to nonstationary tidal vorticity variance, whether directly via aliasing or indirectly via modulation.

First, we consider remotely generated internal tides which may propagate into the region. The horizontal length scales of freely propagating mode-1 internal tides in the tropical North Pacific are O(100) km (Zhao et al. 2013). At these scales remote internal tides would likely appear at the array as currents in phase with almost uniform magnitude. Our EOF analysis has shown that tidal band vorticity is mostly due to rotational eddy motion across the scale of this triangle. Therefore, remote internal tides with these characteristics do not likely contribute directly to eddy vorticity. However, they may modulate the strength of boundary layer shear in a manner analogous to the subinertial current. There are a few instances which match this description for internal tides. For example, in late December/early January KE variance increases sharply in the diurnal band (Fig. 5d). Tidal velocities increase simultaneously at far-field moorings F1 and M5, suggesting large length scales (Fig. 3). This is one of the few times when tidal band vorticity variance increases in both the shear and eddy component (not shown). The diurnal KE event in mid-March has similar characteristics, and notably both events occurred during strong westward flow. Two additional tidal KE events fit the description of an internal tide (abrupt increases in KE variance, large length scales), but neither have strong vorticity signatures. The first occurred in July during strong eastward flow, and the second in November when the subinertial current was weak. This may suggest the combination of tidal and westward flow is essential to generating eddies, or this may reflect the location of the array just west of the separation point (Fig. 1c).

Locally generated, high-mode internal waves with small horizontal scales have the potential to alias directly into the observed eddy vorticity. These waves might be perceived to contain vorticity, but only if they have a wavelength close to the scale of the array. Another FLEAT study observed the generation of internal waves over the ridge extending north of Velasco, at the same latitude as F3 (Voet et al. 2020). These waves were dependent on the combined strength of the subinertial current and the semidiurnal tide, which is the dominant component of the local barotropic tide (Fig. 5f). However, the observations of Voet et al. (2020) suggest it is unlikely the waves propagate through the strong vertical shear at the base of the surface layer. Further, high-mode internal waves do not typically contain much energy (Pinkel 1984). Finally, as previously noted, the observed variance in tidal band KE is weakly modulated on subinertial time scales but does not align closely with the local flow. Thus, it seems unlikely that local internal waves contribute significantly to the observed eddy vorticity.

c. Implications for the Palau wake

Here we speculate on implications for the Palau wake. Submesoscale eddies observed in this and other FLEAT studies are O(10) km. This is consistent with advective length scales estimated from typical velocities over a tidal period during strong westward flow (Fig. 5c). Given U ~ 0.25 m s⁻¹, eddy length scales are L ~ 10 and 20 km for the semidiurnal and diurnal tidal period, respectively. If vorticity is similarly approximated as ζ ~ U/L, these eddies have Ro ~O(1). This is consistent with vorticity observed in this and the synoptic study by Wijesekera et al. (2020). In the case of steady flow, wake eddies are proportional to the island scale (Heywood et al. 1990; Dong et al. 2007). Were eddies to develop on subinertial time scales to this scale [O(100) km] they would have Ro ~O(10⁻¹). Thus, when tidal currents control the formation rate they likely reduce the scale and increase the vorticity of wake eddies by an order of magnitude. FLEAT observations suggest both an island scale wake and submesoscale eddies are generated around Palau. Mean currents downstream of Velasco show island scale recirculation with Ro ~O(10⁻¹) at the surface (Zeiden et al. 2019). This wake pattern is often visible on time scales as short as a few days, but there is significant variance. Wake vorticity occasionally reaches Ro ~O(1) with 10-km scales and smaller, possibly the result of smoothing over many smaller eddies.

Submesoscale eddies likely relate dynamically to the island scale wake in one of two ways. First, they may act as agents of momentum transfer. This is the process which motivates the use of an “eddy” viscosity in lieu of molecular viscosity when comparing island wakes to idealized wakes (e.g., Teague et al. 2005). For example, Delandmeter et al. (2017) observed the development of finescale eddies in the wakes of shallow islands in the Great Barrier Reef. They argued these eddies control momentum transfer between the outer flow and the wake by enhancing the viscosity of the shear layer. Second, submesoscale eddies may behave like two-dimensional turbulence and merge, undergoing an inverse cascade to larger scales. A numerical study by Molemaker et al. (2015) depicts submesoscale anticyclones in the California Undercurrent coalescing until stabilizing near the local Rossby radius of deformation. In another study by Callendar et al. (2011), eddies generated by tidal flows coalesce on spring–neap time scales to larger scales. In these scenarios, submesoscale eddies are a direct source of island-scale wake vorticity. To what degree either (or both) of these processes influences the development of an island scale wake behind Palau is an open question.