1. Introduction
Dispersive processes can quickly redistribute both natural and anthropogenic tracer materials in the ocean, thereby impacting a wide range of practical and scientific concerns, from spill containment options to biological productivity. Because these processes involve complex interactions and operate over scales ranging from molecular to the mesoscale, a comprehensive theoretical description has yet to be achieved. Some progress has been made connecting the mesoscale with submesoscales (McWilliams 2016; Kunze 2019). However, much remains unanswered, especially as it relates to even smaller scales near the ocean surface, which are the focus here. These have classically been approached from the perspective of turbulence theory. However, some of the underlying assumptions, such as homogeneity and isotropy, may not hold in an ocean environment, particularly near the surface. Consequently, much of what is known of the ocean’s dispersive properties derives from empirical analyses (Okubo 1971; Poje et al. 2014; Rypina et al. 2016; Poje et al. 2017; Zavala Sansón et al. 2017). Here we report on quantitative dispersion estimates obtained from a unique dataset capturing trajectories at unusually small scales for an open ocean experiment.
Ocean dispersion experiments include both dye and drifter methodologies. Dye experiments generate concentration measurements and provide insight into turbulent mixing coefficients (e.g., Okubo 1971; Sullivan 1971; Anikiev et al. 1985; Watson and Ledwell 2000; Takewaka et al. 2003; Ledwell et al. 2016). Drifter dispersion studies generally focus on separation statistics of hypothetical fluid particles approximated by drifters and floats (e.g., LaCasce 2008, and references therein).
Both methods pose technical challenges, which boil down to limitations in the time and space scales that can be observed and the accuracy of the measurements. Quantitative measurements of dye concentrations rely on discrete ship-based samples (e.g., Watson et al. 2013; Ledwell et al. 2016). Drifter observations can be obtained autonomously using GPS at the surface or sonar subsurface (Lumpkin et al. 2017; LaCasce 2008). This permits sampling over a larger area and longer times. However, GPS uncertainty, alongside cost and logistical considerations, limits the lower end of the time and space scales over which they can be used for dispersion studies (Haza et al. 2014).
Dispersion estimates have been obtained for many areas of the world’s oceans including global surveys (Roach et al. 2018), the North Pacific (Kirwan et al. 1978; Zhurbas and Oh 2004), the Atlantic (LaCasce and Bower 2000; Ollitrault et al. 2005; Lumpkin and Elipot 2010), the Arctic (Koszalka et al. 2009; Mensa et al. 2018), the Southern Ocean (LaCasce et al. 2014; Balwada et al. 2016b), the Mediterranean (Lacorata et al. 2001; Schroeder et al. 2012), and the Australian (Mantovanelli et al. 2012), Finnish (Torsvik and Kalda 2014), and French (Porter et al. 2016) coasts. Substantial work has also been done in the Gulf of Mexico (e.g., LaCasce and Ohlmann 2003; Poje et al. 2014; Beron-Vera and LaCasce 2016; Zavala Sansón et al. 2017). However, due to the experimental constraints, little information is available on open ocean dispersion over time and space scales on the order of minutes and meters. Yet these scales are critical for the dispersion of pollutants such as oil and relevant to small-scale dynamics near oceanic fronts (D’Asaro et al. 2018). Here we seek to begin to fill that gap, using an innovative dataset of optically tracked floating bamboo plates (Carlson et al. 2018), which provides higher space and time resolution for the trajectories than available from classic Lagrangian drifter and float data.
Specifically, we focus on the dispersive properties during a Langmuir event. Langmuir circulation (LC) is characterized by counterrotating vortices aligned with the wind direction (Leibovich 1983) and results from an interaction of the wind with the wave field (Fig. 1). The phenomenon is important for dispersion, as it traps floating matter in narrow convergence zones at the surface (Thorpe 2004) and enhances vertical mixing of neutrally buoyant material and water properties (Kukulka et al. 2009; McWilliams and Sullivan 2000). Its dispersive properties have been primarily studied in models (e.g., Yang et al. 2015; Kukulka and Harcourt 2017; Shrestha et al. 2018) and generally from an Eulerian perspective. A few attempts have been made to derive estimates of diffusivities from sonar observations, which are thought to detect bubble bands associated with Langmuir circulation, from which velocities are derived (Thorpe et al. 1994; Li 2000).
Simple diagram of counterrotating Langmuir circulation cells of 20-m diameter with arrows indicating direction of rotation with wind direction shown by red arrow. Red plates are shown clustered in the convergence zones (above downwelling regions), which are spaced 40 m apart.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
The present study centers on particle-pair statistics, reporting relative dispersion and relative velocity statistics, and presenting velocity structure functions. This approach places the small-scale Lagrangian observations within the context of turbulence theory and permits direct (though limited) comparison with comparable observations from contemporaneous surface GPS-tracked drifters at larger scales. This extends previous work on near-surface structure functions using drifters with resolution down to approximately 100 m (Poje et al. 2014; Balwada et al. 2016a; Poje et al. 2017; Mensa et al. 2018).
Velocity fields in Langmuir circulation are well known to be anisotropic (e.g., Thorpe 2004). We quantify the anisotropies in the dispersion and identify convergence scales using the structure functions. The findings can inform future modeling and experimental studies by providing observed statistics as a baseline.
Details of the experimental setup are given in the next section. As for most optical techniques, custom tracking algorithms were developed to process the data. These are based on the methodology described in Carlson et al. (2018), and section 3 provides a summary. The cumulative effect of uncertainties in the camera position and orientation on the derived statistics is discussed in the appendix. Results are presented in section 4, followed by a summary and conclusions in section 5.
2. Experimental setup
a. LASER
The Lagrangian Submesoscale Experiment (LASER; D’Asaro et al. 2018) was a large, coordinated effort to study the transport properties of the DeSoto Canyon region in the northern Gulf of Mexico (Fig. 2). The expedition lasted from 18 January to 13 February 2016. It relied on two main vessels, the R/V Walton Smith, equipped with a modern suite of instruments and the 110-ft offshore supply boat Masco VIII, which was retrofitted to collect meteorological data, while primarily serving as a platform for drifter releases and aerostat operations.
Northeastern Gulf of Mexico with bathymetry contours. The blue box indicates the area where aerostat operations were permitted. Trajectories of GPS-tracked surface drifters with (green) and without (magenta) drogue from 1 Feb through 4 Feb 2016 are shown. The black square marks the location of the Langmuir experiment on 6 Feb 2016 near 28.75°N, 88.28°W.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
b. Plate experiments
The present analysis is based on trajectories of bamboo plates, observed from an aerostat tethered to a station-keeping ship, the Ship-Tethered Aerostat Remote Sensing System (STARSS; Carlson et al. 2018), illustrated in Fig. 3. The plates were chosen because they are cheap, biodegradable, and positively buoyant with nominally no windage. The plates have a diameter of 28 cm, draft of 1.75 cm, and thickness of 2 mm. Prior to the field experiment, they were tested for their float properties and ease of detection in imagery. Within a few minutes of being in the ocean, these plates soak up sufficient water to become submerged to 1–2-cm depth, with negligible extent above the surface that could be subject to direct wind forcing. While waves can flip the plates upside down, orientation was not found to affect the float properties in wave tank experiments. The main downside of using plates to measure surface currents is the relatively large effort required first to track the plates visually and then to extract accurate trajectories from the imagery; see section 3 for details. In addition, the observational window is limited by the time the camera can be kept aloft with the plates within its field of view, and nighttime observations are not possible.
Experimental setup. (a) Canon camera, GoPro camera, GPS-INS, and 5-GHz wifi antenna mounted to the aerostat. (b) Aerial view of the aerostat (upper right), ship (lower left), and bamboo plates (middle right). (c) Plates being released in the ocean by small boat. (d) Stack of painted plates.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
As part of LASER, hundreds of plates were released in a series of subexperiments. FAA regulations restricted operations to the box indicated in Fig. 2. The bamboo plates were tossed into the ocean from a small boat. About half were painted using red, yellow, and magenta biodegradable paint (Fig. 3d) to contrast with the ocean surface and to help distinguish between plates. Unpainted plates were a light beige color (Fig. 3c). A high-resolution (8688 × 5792 pixel) color image was taken every 15 s. The lens was set to a 17-mm focal length. The aerostat, a helium-filled balloon, was tethered at an altitude of about 150 m, resulting in a field of view (FOV) of approximately 320 m × 210 m. Total flight times per experiment for the aerostat were about 3–4 h. This resulted in just under 2.5 h of usable trajectories during the Langmuir event on 6 February 2016.
The camera look angles, ship speed, and ship heading were adjusted to keep the bamboo plates in the camera FOV as they drifted and dispersed. The GPS-INS (Inertial Navigation System) recorded the camera position and angle at 5 Hz. This information is used for absolute georectification (see section 3). The Masco VIII was equipped with two GT31 GPS units, recording ship position at 1 Hz. Environmental data were collected from the weather stations on the Masco VIII and the R/V Walton Smith, as well as from mobile platforms.
The focus here is on the experiment from the afternoon of 6 February 2016 (all times reported as local time, CST). Plates were deployed along two tracks perpendicular to the wind direction over less than 8 min, forming a 150 m (crosswind) × 30 m (downwind) patch of approximately uniform density with 1 plate per 8 m2 (Fig. 5a). Their motion was tracked for about 2.5 h.
c. Surface drifters
During LASER, over 1000 CARTHE-type surface drifters (Novelli et al. 2017) were deployed in multiple groups. They reported their GPS positions nominally every 5 min. Initially, they were drogued at 0.5 m, but many lost their drogues during rough storm events. This resulted fortuitously in a second, distinct set of surface drifters consisting only of a float, which extended about 5 cm into the water column (Haza et al. 2018). In section 4d, we report on the statistics of both sets from a deployment on 21 January 2016, approximately two weeks prior to the plate experiment. The sampling window chosen here is a calm 3-day period, 1–4 February 2016, exhibiting similar wind conditions to those seen during the plate experiment and preceding the drifters accumulating along a density front. By this time, the drifters—300 drogued and 168 undrogued—were dispersed over a 300 km × 100 km region in the northern Gulf of Mexico (Fig. 2).
d. Environmental conditions
During the 2.5 h of the experiment on 6 February, wind speed was steady between 5 and 7 m s−1 from the north, and net surface heat flux decreased from slight heating to cooling (Fig. 4a). Potential density ρ profiles derived from conductivity–temperature–depth (CTD) measurements from the R/V Walton Smith (Fig. 4b) indicate a mixed layer depth of approximately 40–50 m and a second stronger pycnocline around 70–80 m. The temperature profile at 1637 CST reveals an additional gradient at 20–30-m depth. This depth corresponds to the dominant Langmuir cell diameter during this experiment, as will be described in section 4.
Environmental conditions on 6 Feb 2016. (a) Wind speed (blue) and surface heat flux (red) from the R/V Walton Smith. (b) CTD temperature and potential density profiles. (c) Wind direction from the R/V Walton Smith (blue); spectral averaged wave direction (green) and surface Stokes drift (red) based on wave buoy data. Start and end times of plate observations are indicated in (a) and (c) by black dotted lines. (d) Stokes drift speed (left) and direction (right) calculated from wave buoy data, averaged over the 2.5 h of the Langmuir experiment. Directions are reported as where wind, waves, and Stokes drift originate from.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Two wave buoys deployed near the plates provided the wave spectrum, height, and direction. The wave buoy data were processed using methods described in Thomson et al. (2018). Data from one of the buoys was eliminated, since it was found to have too much low frequency noise. For the remaining buoy, only data from intermediate frequencies (0.15–0.50 Hz) were kept due to directional incoherence at the lower and higher frequencies. Significant wave heights overall were 0.67–0.80 m and somewhat lower (0.56–0.71 m) within this frequency band, compared with predictions of 0.5–1.0 m for the significant wave heights at the Pierson–Moskowitz limit (Young 1999) for observed wind speeds. The spectrally weighted wave direction differs by about 30° from the wind direction measured simultaneously on the R/V Walton Smith 1 km to the south-southeast, as shown in Fig. 4c.
To calculate the average profile of Stokes drift from the observed spectrum after Kenyon (1969), the contributions from frequencies above the resolved 0.5 Hz must be estimated. For this purpose, we assume f−4 frequency scaling of the energy spectrum extends to fx = 4fp, where fp = g/(2.4πU10) is the peak frequency in a fully developed sea state, and extends above fx as f−5. An f−4 wind sea scaling enhances surface Stokes drift, a transition to f−5 scaling around 4fp is selected because recent observations of f−4 scaling behavior in more well-resolved spectra (Thomson et al. 2013) are limited to approximately this range. This rather conservative assumption for the scale of a transition at fx implies that the Stokes drift spectrum between 0.5 Hz and fx continues as f−1, and above fx as f−2. The resulting average Stokes drift profile is shown in Fig. 4d. With a surface Stokes drift of US = 4.67 ± 0.49 cm s−1 and shear velocity of
CTD surveys in the nearby area earlier on 6 February reveal weak lateral density gradients, implying weak frontal and submesoscale activity. Therefore, Langmuir circulation should be the dominant dynamic feature present in the small scales measured during the plate experiment.
3. Data processing
Before attempting any quantitative analysis of the data, the raw images have to be processed by 1) detecting plates, 2) converting pixel coordinates to physical coordinates, and 3) linking plates between images. The procedure is a modification of that described in Carlson et al. (2018) and is briefly summarized below.
a. Plate detection
The first step in image processing is to mask out swaths along the edges of the photos that are known to contain problematic elements (the ship and sun glint) and are not likely to contain plates. The small boat used for deployment is masked manually. Plates are then identified iteratively in each image using (i) their color, (ii) their size and shape, and (iii) their persistence to distinguish them from other features such as sun glint, white caps, and seaweed.
At the start of subsequent rounds of detection, 5-pixel-radii circles centered on each identified plate are masked and a shape filter is applied by convolving the masked total intensity field with a kernel constructed by applying a 42 × 42 pixel Gaussian low-pass filter with standard deviation of 3 pixels (as implemented in MATLAB’S imgaussfilt.m) to a 2D step function that is 1 in a disk of radius 4 pixels and otherwise 0. After the first round, the minimum separation distance between plate center candidates is set to 4 pixels. Moreover, the total convolved intensity of the pixel at the center and of those surrounding it up to 2 pixels away must meet two thresholds: it must exceed 80% of the maximum value of the filtered total intensity field and 20% of the integral of the kernel. Detection is stopped when no additional plates are identified.
b. Position rectification
The detected plates are first referenced as pixel coordinates in each image. A high-quality Canon lens was used with a full-frame DSLR, resulting in minimal image distortion, allowing the approximation with a pinhole camera model (Kannala et al. 2008). With this assumption, given position, and orientation of the camera, the pixel coordinates are converted to real-world coordinates on a flat ocean surface (Mostafa and Schwarz 2001). This is termed absolute rectification.
Unfortunately, the heading data for the experiment analyzed here was not accurately recorded, which may also have contaminated the other INS data. Therefore, a relative rectification step was added. This process translates and rotates and potentially dilates the entire plate position field so that the sum of distances between each plate in one frame and the nearest plate in the next frame is minimized. Our implementation used the MATLAB functions knnsearch and fminsearch for the nested optimization problem. The initial guess centers the field of plates on the origin and aligns its main axis of variability with the x axis. In the sum of distances, the largest 10% are excluded to avoid contamination from erroneous “plates” (i.e., sun glint).
Relative rectification was used to improve the time syncing between images and INS data (by comparing dilation factors to the measured altitude time series), to determine bias corrections for pitch and roll, to remove false positives (by removing plates that moved more than 3.4 m in 15 s in the relatively rectified images), and to precondition the linking (see next section). For the latter two purposes, dilation was not included. Relative rectification and outlier removal was performed twice to remove the impact of outliers on the rectification. Relative rectification removes absolute position and velocity information. However, it does not impact the relative motion needed for the analyses in this paper.
c. Plate linking
Linking is required to connect individual plates between images so as to obtain plate trajectories. Most linking algorithms involve a global minimization of distance between associated plates. We used a MATLAB linking algorithm called “Simple Tracker” by J.-Y. Tinevez (https://www.mathworks.com/matlabcentral/fileexchange/34040-simple-tracker). After linking, plate velocities are estimated from forward finite differencing. Any link implying a velocity greater than two standard deviations from the average velocity in its neighborhood is eliminated. This is necessary, because the number of plates detected may change from image to image: plates exit the camera field of view or are not detected, while sun glint produces false identifications. Our experimental and processing techniques minimize but cannot fully eliminate these issues. For this dataset, complete trajectories were obtained for approximately 20% of the 600 plates, using a maximum linking distance of 3.4 m and a maximum gap of 8 image frames. These are the approximately 120 plates used in the relative dispersion calculations in section 4b. Data for all 600 plates can be used for the average velocity calculations (section 4a) and structure function calculations (section 4d), which do not require full trajectories. For this we linked using a maximum linking distance of 1.7 m and a maximum gap of 1 image frame. A shorter maximum linking distance reduces the maximum allowable velocity but produces fewer erroneous links.
4. Results
a. Observations of Langmuir circulation
Figure 5 shows five snapshots of the rectified plate positions. Consistent with the presence of Langmuir circulation, the initially uniformly distributed buoyant plates cluster along convergence zones. After 10 min (Fig. 5b), convergence zones with approximately 15-m spacing are visible. Over the next 10 min, some of the neighboring windrows approach each other and merge (see animation in supplemental material in the online version of this paper), so that after 20 min (Fig. 5c), the visible convergence zones are spaced approximately 40 m apart. This secondary merging of plates suggests the presence of LC with different magnitudes and length scales, where the initial spacing (Fig. 5b) reveals smaller scale LC and the secondary spacing (Fig. 5c) is driven by stronger, larger-scale LC. After 30 min (Fig. 5d), this spacing remains evident. After 125 min (Fig. 5e), convergence zones have separated slightly to approximately 50-m spacing. This final slow growth in spacing coincides with a transition from net surface heating to cooling in the late afternoon (Fig. 4a). The persistent 40–50-m windrow spacing implies a dominant LC cell diameter of 20–25 m (see Fig. 1), reflecting a well-mixed layer of the same depth, evident in the temperature profile at 1637 CST in Fig. 4b. In addition to the evolution of crosswind spacings, plates spread in the downwind direction from 30-m extent (Fig. 5a) to 100 m (Fig. 5e). Windrows are approximately aligned with wind directions.
(a)–(e) Relatively rectified plate positions after 0, 10, 20, 30, and 125 min, respectively. Black arrows show wind direction inferred from the heading of the aerostat, which was designed to point into the wind. Grid spacing is 10 m in both directions. An animated version of this figure is included as online supplemental material.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Convergence zone spacing changed rapidly from 15 (Fig. 5b) to 40 m (Fig. 5c) over the course of just 10 min. These multiple convergence zone spacings are evidence that LC is composed of the superposition of many scales of motion, an idea first proposed by Langmuir (1938), of which the dominant scale results in a LC cell diameter of 20 m (corresponding to the 40-m spacing). Alternatively, the dominant LC cell diameter may have grown in time from 7 m (corresponding to the 15-m spacing) to 20 m. However, this is less likely since the 10-min period over which we observe the change in spacing is much faster than expected for LC evolution, and there is no significant change in wind speed, wind direction, or wave direction over the course of the 2.5-h experiment (Fig. 4).
Clearly, the plates experienced anisotropic dispersion. This is illustrated by expressing our subsequent analyses in downwind (x) and crosswind (y) directions.
While the plate observations are significantly biased toward convergence zones, the spatial data density allows estimates of the average crosswind structure of the surface velocity field. Figure 6 shows profiles of cumulative plate count, average crosswind velocity, and average downwind velocity as functions of crosswind position at both early and late times. The mean quantities represent averages over both time and downwind position of all available observations. The velocities (u, υ) are taken relative to the moving center of mass of the plate distribution.
Binned cumulative plate count (black), average crosswind velocity (red dashed), and average downwind velocity (blue dashed) as functions of crosswind position, for (a) the first 10 min and (b) 50–90 min. The corresponding solid curves are low-pass-filtered velocities, with cutoff at 10 m for (a) and cutoff at 20 m for (b).
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Early time results clearly show organized, roughly sinusoidal behavior of the crosswind velocity υ consistent with counterrotating Langmuir cells. Regions of high plate accumulation are nearly aligned with convergence zones where ∂υ/∂y < 0. The average relative crosswind velocity is 1–2 cm s−1, slightly lower than both the 2–3 cm s−1 crosswind velocities observed on a lake under similar wind conditions by Langmuir (1938) and typical estimates of 1% of the wind speed. The derived velocities are consistent, however, with the early time evolution from a nearly uniform distribution to 40-m windrows in approximately 20 min. Unlike the crosswind velocity, there is very little discernible pattern in the averaged downwind velocity during the formation period.
Later time results, where downwind averaging occurs over longer spatial scales and data density outside the convergence zones is low, show the persistence of 1–2 cm s−1 mean crosswind velocities with strong correlation between plate density and crosswind convergence. In addition, there is an organized mean downwind velocity of similar magnitude with peak positive values roughly aligned with the convergence zones. Sometimes the peak downwind speeds are slightly displaced to the right, which could be a consequence of the tilting of the windrows relative to the instantaneous wind direction.
b. Relative dispersion
Figure 7 shows that RD (black) taken over all possible initial plate pairs, increases with time, albeit with noticeable fluctuations. Given the pronounced anisotropy in the plate distributions, the downwind,
Relative dispersion as a function of time for initially isotropic pair distributions. Average over all directions and scales (black), in the crosswind direction (red), and in the downwind direction (blue). Gaps reflect frames that could not be successfully processed.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Values of total relative diffusivity in Table 1 are calculated for early times (t1 = 0 min, t1 = 38 min) and late times (t1 = 50 min, t1 = 88 min) for pairs with initial separation scale
c. Two-point velocity statistics
Like relative dispersion, the longitudinal velocity increment δυl is invariant to translation and rotation of the plate fields. The transverse velocity increment δυt is invariant to translation but varies with rotation, so we will not use this metric. Figure 8 shows the normalized probability density functions (PDFs) of the longitudinal velocity increments for three ranges of separation r. Normalization is with respect to the standard deviations, approximately 4, 5, and 6 cm s−1 for separation scales 0–20, 40–60, and 80–100 m, respectively. All the PDFs exhibit Gaussian cores with smaller scales exhibiting larger relative tails, consistent with measurements at larger scales using surface drifters (Poje et al. 2017). Some uncertainty in the PDFs results from the choice of maximal linking distances: a shorter maximum linking distance reduces the maximum velocity but produces fewer erroneous links, while a longer maximum linking distance allows larger velocities but produces more erroneous links. Thus, heavier tails result from larger allowed linking distances.
Normalized PDFs of longitudinal velocity increments for three separation distance bins. Uncertainty is shown (shading) for maximum linking distances of 1.7–3.4 m, which correspond to maximum velocities of 11–22 cm s−1 relative to the mean. A normalized Gaussian is indicated by the black curve.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
d. Structure functions
Longitudinal first-order structure function for crosswind (red) and downwind (blue) components, subsampled at early (solid) and late (dashed) times. Shaded regions are 95% confidence intervals determined from standard error calculations.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Longitudinal second-order structure function. (a) Crosswind (red) and downwind (blue) components, subsampled at early (solid) and late (dashed) times. (b) For early time (solid black) and late time (dashed black) plates, plus drogued (green) and undrogued (magenta) drifters. Shaded regions are 95% confidence intervals determined from standard error calculations.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
The first-order structure function
Figure 9 shows the first-order structure function
If two plates are initially separated by less than the LC cell diameter, they will more likely end up in the same convergence zone and therefore have mean convergence. If two plates are initially separated by more than the LC cell diameter, they will more likely end up in different convergence zones and exhibit mean divergence. This is true up to an initial separation of two LC diameters. See the diagram in Fig. 1.
The early time crosswind component (solid red curve) indicates mean convergence for separations below 7 m and divergence at larger scales, consistent with the first observed windrow spacing of 15 m (Fig. 5b). At later times (dashed red curve), the zero crossing occurs at r = 20 m matching the observed convergence zone spacing of approximately 40 m shown in Figs. 5c and 5d. The averaged magnitude of crosswind relative velocity is higher at early times when the plate distribution is more homogeneous.
In the downwind direction, there is initially (solid blue curve) no significant trend, while at later times (dashed blue curve) there are clear indications of mean divergence, rapidly increasing with the separation distance. This behavior matches the formation of organized mean downwind velocities observed at later times as shown in Fig. 6. The strong anisotropy between the mean crosswind and downwind divergence at later times is also consistent with the slower growth in crosswind relative dispersion compared with downwind relative dispersion in Fig. 7.
The second-order structure function was introduced by Kolmogorov (1941b) to describe the inertial range behavior of homogeneous isotropic turbulence. More generally, the second-order structure function is the Fourier cosine transform of the kinetic energy spectrum, whose scale-dependent behavior is related to the nature of the underlying flow. Figure 10a shows that crosswind
LC motions are also reflected in the difference between crosswind and downwind
To connect the structure function observations at the small scales (r < 100 m) with larger scale dynamics, we also calculated
At the largest scales (r > 5 km), both drogued and undrogued drifters have similar behavior, roughly consistent with standard
The 2/3 power from the drogued drifters is consistent with the inertial range of a forward energy cascade in 3D turbulence (Kolmogorov 1941a,b), an inverse cascade in 2D turbulence (Kraichnan 1967), and other scalings applicable to oceanic flows (McWilliams 2016). Similar power law behavior was observed using CODE-style drifters (Davis 1985) with 1-m drogue depth deployed in summer 2012 in the Northern Gulf of Mexico (Poje et al. 2017). Deviation from this power law for the near surface structure functions (from plates and undrogued drifters) may be attributed to the nature of the observations: the computed structure functions are strictly 2D, involving only horizontal differences of horizontal velocity components despite evidence that the vertical to horizontal aspect ratio of the flow approaches unity at these separation scales. In addition, the Lagrangian measurements are inherently biased to oversampling horizontal convergence zones, and as such may produce different power law behavior than that derived from unbiased Eulerian measurements (Pearson et al. 2019). Despite these limitations, the plate observations clearly indicate a horizontally anisotropic and inhomogeneous flow field with coherent features on 10–100-m length scales. The change in the power law behavior of
5. Summary and conclusions
During the LASER field campaign in the northern Gulf of Mexico during January–February 2016, several small-scale dispersion experiments were performed using floating bamboo plates observed from a high-resolution camera mounted on a helium-filled aerostat. Here the focus is on the experiment that revealed Langmuir streaks roughly aligned with the wind. Individual plates were detected, rectified, and linked to extract their velocities on separation scales of 1–100 m. The large number of plates (nearly 600) provided sufficient statistics to evaluate relative dispersion and velocity structure functions.
These statistics show clear differences between the downwind and crosswind direction and are consistent with Langmuir circulation. In the crosswind direction, material spreads out on average at scales larger than the dominant Langmuir cell diameters (7 and 20 m in this experiment) and converges at smaller scales. In the downwind direction perpendicular to the direction of coherent Langmuir cell motions, spreading rate increases with separation distance, which is consistent with turbulent shear dispersion. At larger separation scales, spreading rate in the downwind direction is significantly higher than in the crosswind direction due to the constraining effect of Langmuir circulation. At submeter to tens of meters scales, the crosswind component of the second-order structure function is larger than the downwind component; additionally the crosswind component is larger during the early times than the later times, implying Langmuir motions are active simultaneously at multiple scales, as first hypothesized by Langmuir (Langmuir 1938). The time scales of evolution are too short relative to the changes in the environmental conditions (see Fig. 4) to support the alternate hypothesis of Langmuir cells of a single preferred scale growing over time.
The second-order structure function continuously extends results from similar structure functions obtained from surface drifter data. Energy input from wind and waves likely alter the magnitude and slope of the second-order structure function at small scales and shallow depths. At scales less than 1 km, there are clear differences between the second-order structure function at 0.5-m depth (measured by the drogued drifters) and at very shallow 1–5-cm depths (measured by the plates and undrogued drifters) where the structure function has a shallower slope and more energy. This implies that the properties of mesoscale and submesoscale dispersion cannot simply be extrapolated to describe phenomena driven at the small scales, such as Langmuir circulation.
We observed differences in early time velocity statistics, when the plates are more uniformly distributed, and the later time statistics, when most of the plates have organized into windrows. While the early time downwind spreading rate
We have shown that inexpensive, biodegradable bamboo plates along with high-resolution photography can generate field measurements of small-scale ocean surface dynamics in the open ocean. Analysis of the dispersion characteristics on these scales is essential to unraveling the dynamics across scales and to identifying routes to dissipation in the ocean. This dataset provides unique measurements of a Langmuir event that can be directly compared with models for Langmuir circulation. Such comparisons are underway and are expected to lead to improved understanding of Langmuir circulation and validation or improvement of the models.
The authors wish to thank the entire CARTHE LASER team, especially Chief Scientist Eric D’Asaro, for their contribution to a successful field campaign. Weather data was processed by Brian Haus; CTD profiles were produced by Alexander Soloviev and John Kluge; wave buoy data was processed by Jim Thomson; and drifter data was quality controlled and sorted for drogue status by Angelique Haza. We wish to thank Tobias Kukulka and Dong Wang for helpful discussions about Langmuir turbulence, as well as Pablo Huq for helpful discussions about structure functions. This work was made possible by grant support from The Gulf of Mexico Research Initiative for CARTHE and from the Office of Naval Research for CALYPSO (N00014-18-1-2461). All datasets are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) at https://data.gulfresearchinitiative.org under the following DOIs: 10.7266/N7M61H9P (plate observations), 10.7266/N7W0940J (drifter data), 10.7266/N7S75DRP, 10.7266/n7-93j3-mn56 (shipboard measurements), and 10.7266/n7-8h5w-nn91 (wave buoy data).
APPENDIX
Error Analysis
Known error sources for the calculations of
There is a factor of 2 here, because there are terms at t and at t + Δt, each squared separately, but then averaged over all times.
We compute
Integrated second-order structure function with error (dark blue) as a function of sampling window τ (5-frame average) and the best fit for the individual components and their sum of the right-hand side of (A17), as computed over 38 sampling windows.
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
Unfortunately,
Different estimates of the error statistics
Citation: Journal of Physical Oceanography 49, 12; 10.1175/JPO-D-19-0107.1
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