The Water-Following Performance of Various Lagrangian Surface Drifters Measured in a Dye Release Experiment
1. Introduction
Lagrangian measurement of ocean surface currents using drifters has a long history (LaCasce 2008; Lumpkin et al. 2017). In recent years a large number of drifter designs have been developed for various purposes. In particular, efforts have been made to develop physically small designs facilitating the deployment of large numbers of units, either in simultaneous releases or for sustained programs over longer periods (Novelli et al. 2017; Pawlowicz et al. 2019; Page et al. 2019; Meyerjürgens et al. 2019; van Sebille et al. 2021) or simply because they can be easily and cheaply constructed (Abascal et al. 2009; Choi et al. 2020; Tamtare et al. 2021). However, there is always a degree of “slip” (also called leeway) between a floating object and the water surrounding it, ultimately due to the wind. Drifter motions therefore do not solely reflect ocean currents, but the slip of many of these designs is unknown or is known only poorly. Not only does this affect the proper analysis of ocean research datasets, but this bias can also lead to difficulties in tracking oil spills and other floating debris using these drifters (De Dominicis et al. 2016; Laxague et al. 2018).
Potential mechanisms contributing to slip have long been qualitatively known (e.g., Geyer 1989) to include wind drag and wave effects. A direct drag from surface winds can occur since parts of the drifter often extend into the air above the water’s surface. Then, just below the surface there may be an Eulerian shear in the water column boundary layer, induced by wind drag, even within an apparent mixed layer in which scalar properties are homogenized. Near-surface water parcels are also subject to a Lagrangian shear, due to the additional effects of Stokes drift (van den Bremer and Breivik 2018) arising from surface waves. Other, more complicated and possibly resonant interactions between a buoyant drifter body and waves of particular frequencies can also lead to anomalous downwind motions in some circumstances (e.g., Novelli et al. 2017).
Many attempts have then been made to quantitatively measure slip for different drifter designs. Slip of the widely used Surface Velocity Program (SVP) and Coastal Dynamics Experiment (CODE) drifters has been well studied using both direct measurements of ambient currents from sensors attached to the drifter and from statistical analyses (e.g., Niiler et al. 1995; Poulain et al. 2009; Poulain and Gerin 2019). The new Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE) design has been comprehensively studied in wave tanks and in the field (Novelli et al. 2017). These studies report slips of about O(0.1–0.5)% of the wind speed. Although other designs are less well characterized, attempts to improve operational prediction models of the ocean (e.g., for search and rescue or oil spill management), along with interest in upper-ocean processes, have motivated a number of intercomparisons to obtain slip factors. For example, Röhrs et al. (2012) compare tracks of two drifter designs with ocean observations of wind, currents, and waves, and Blanken et al. (2021) perform a similar comparison for four drifter designs, but only using significant wave height and period information for waves. Sutherland et al. (2020) compare tracks from six drifter designs with pseudodrifter tracks in two different numerical ocean prediction models, with and without a Stokes drift component. Several unique drifter designs have had their motions compared with surface currents directly measured using high-frequency (HF) radar techniques so as to improve drift predictions (Abascal et al. 2009; van der Mheen et al. 2020) or for studies of near-surface currents (Morey et al. 2018). Typically, somewhat larger slips of 1%–3% of wind speed are found for “surface” drifters, and the addition of Stokes drift parameterizations improves comparisons. However, as there are many adjustable parameters of uncertain magnitude in these comparisons, the degree to which different physical mechanisms contribute to this slip is still under debate.
In 2020, the Tracer Release Experiment (TReX) was designed and carried out in the lower estuary of the St. Lawrence River in eastern Canada to investigate dispersion processes over a variety of length scales. In contrast to earlier field studies of drifter leeway, which generally involved a small number of drifter designs, here satellite-tracked drifters of eight different designs (including both commercially available and more crudely constructed “home-built” models that have nevertheless been widely used) were simultaneously co-deployed. Some of these drifter designs are drogued (i.e., they are designed with a subsurface drag element) whereas others are undrogued, in an attempt to better measure the surface currents, or the drift of objects on the surface. Another novelty was the additional deployment of rhodamine dye to act as a true water-mass tracer, for comparison with the drifter motions. The dye patch was monitored for many hours using aerial drones as well as satellite remote sensing and hydrographic profiling from several small vessels. Ancillary observations were made of full surface wave spectra, winds, and water column currents.
Taking advantage of this experimental design, here we evaluate and compare the performance of these different drifters under the particular set of wind/wave/stratification conditions that occurred and discuss the degree to which drifters both respond to the wind and act as water-mass tracers. We then develop a quasi-analytical model for drifter slip, independent of any drifter observations, which is successful at matching our observations. This model is specifically focused on details of the air–sea boundary layer rather than being adapted from, for example, a numerical general circulation model. Use of this model allows us to quantitatively separate out the different factors that are important in determining slip. As our model does not involve empirical fitting to drifter observations, we then generalize predictions of performance to a range of typical open-ocean conditions.
2. Methods
From 1145 to 1203 11 September 2020 (all times are UTC), 56 L of 20% rhodamine-WT mixed with water from the ship’s freshwater supply to a total volume of 680 L were released into the lower St. Lawrence estuary about 13 km offshore. Dye was released along a north–south line ∼1 km long from the stern of the Research Vessel (R/V) Coriolis II, using a gravity-fed system from a tank 1.7 m above water level (Fig. 1a).
(a) Dye deployment at 1156 UTC looking north from the stern of the R/V Coriolis II. UBC-ESD and SCT drifters can be seen to the left of the buoyed hose. (b) Dye patch looking south from an altitude of 112 m at ∼1353 UTC. In the foreground is the 8.2-m-long F.J. Saucier, and in the middle of the patch is the 6-m-long Mordax. The photograph in (a) was taken by C. Chavanne; that in (b) was taken by E. Dumas-Lefebvre.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
The dyed mixture had an estimated density of 1016 kg m−3, somewhat lower than surface seawater density of 1021.3 kg m−3, and initially the dye did float near the surface but was soon mixed throughout a surface mixed layer. At the same time, 19 drifters, 2 or 3 each of eight different designs (Table 1), were released as “drifter lines” along the dye line.
Drifter types in TReX. Freeboards (i.e., height above waterline) were calculated from given geometry and vendor specifications or were estimated visually. Drawings or photographs of the different drifters can be found in the references.
The dye was then surveyed in several ways over the rest of the day. First, the Coriolis II was used as a launch platform for DJI Mavic 2 Pro unoccupied aerial vehicles (UAV). Complete aerial photographic UAV surveys, each requiring 5–8 min, were carried out at 1201, 1337, and 1541 (Fig. 1b), and downward-looking images were georectified and combined into orthomosaics (Fig. 2a). Second, the dye patch was also observed in its entirety at 1539 with 10-m horizontal resolution by an overpass of the satellite-based Sentinel-2 multispectral optical imager. The dye was particularly apparent in band 3 (green) for this instrument.
(a) Dye patches at 1201, 1337 (both from UAV mosaics with the red band only shown in grayscale), and 1539 (from Sentinel-2 image Band 3) UTC. Note that all three survey vessels (the R/V Coriolis II as well as the smaller F.J. Saucier and Mordax) are visible in the satellite image at their corresponding GPS-tracked locations. Also shown are the drifter locations at (b) 1201, (c) 1337, and (d) 1541, plotted with the corresponding dye patch. Locations of drifters of the same design are joined with a line to guide the eye. The “Drift” lines show cumulative integrals of water column velocities at different depths taken from ADCP measurements in the patch, with integrations starting from the location of the center of the dye line at 1201. Squares show the location of the estimated displacement for each indicated depth at the time of each subplot. Wind arrows in (c) and (d) show mean winds at PMZA-RIKI between 1200 UTC and the image times [the location of this buoy is shown in the inset map in (c), relative to the outlined study area].
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
Additional surveying of the dye patch for the whole period was also carried out by two smaller vessels, one of which had a flow-through system to monitor rhodamine in the surface water as well as a Teledyne-RDI Sentinel V50 acoustic Doppler current profiler (ADCP) mounted in a towed body, and both of which had profiling conductivity–temperature–depth (CTD) probes with Turner Designs Cyclops-7F fluorometers designed for rhodamine detection.
A freely floating wave-measuring buoy (SOFAR Spotter) was deployed beside the patch from 1206 to 1508 to measure surface wave spectra at 30-min intervals, and a Sentinel V100 ADCP deployed on a separate drifting platform for the same period to measure water column currents to greater depths but with less resolution than the towed body system. Surface winds at a height of 3.5 m and other weather conditions were observed 25 km away to the northeast at the PMZA-RIKI Viking buoy (48° 40′N, 68° 35′W; Fig. 2c inset).
Surveying continued through the day, until around 1730 when the rhodamine dye could no longer be observed. Although rhodamine does degrade in sunlight (Tai and Rathbun 1988), these rates are small enough that little sunlight-related loss can be expected over a few hours, and it is more likely that the dye was lost because it subducted under an ∼5-m layer of lighter water to the north. Here we concentrate on observations over the period of 1201–1541.
3. Results
a. Direct observations
After the dye was released, it drifted southwest until about 1400 and then northwest, at speeds of about 30 cm s−1, slowly and unevenly spreading out in both the east–west and north–south directions (Fig. 2a). However, the patch still retained its general linear shape and north–south orientation throughout the experiment, suggesting that, while dispersion by small-scale turbulent processes occurred, there was no significant large-scale horizontal vorticity or shear. Optical imaging matched direct measurements of dye in the upper 1 m using a shipborne flow-through system (not shown here) and comparisons between red intensity in UAV images and vertical dye concentration profiles (also not shown) suggests our images are showing average dye concentration in the upper 3 m.
The drifters deployed with the dye also followed the same general path (Figs. 2b–d), moving southwest and then northwest, but appeared somewhat east of the dye patch soon after their initial deployment, with offsets (slips) that grew with time and were clearly different for different drifter designs. After nearly 4 h, these offsets were between 350 m and 2.2 km. During this time, winds measured at PMZA-RIKI averaged about 3.8 m s−1 roughly to the east-northeast. The drifter offsets were therefore about 30° to the right of directly downwind, although with some uncertainty because the wind measurement was about 25 km away.
1) Dye motion
First we evaluate the degree to which the dye patch itself acted as a useful water tracer. The water column containing the dye had a depth of about 35 m for the first 2.5 h of the experiment, increasing toward its end as the patch was advected northward into a deeper channel. Hydrographic profiling shows that this water column was approximately linearly stratified down to the bottom or about 40 m (whichever was shallower), but with a noticeable mixed layer of uniform density in the upper 4 m until about 1600, after which the mixed layer deepened to about 7 m (Fig. 3a). The dye was “lost” to sight soon after.
Profiles of (a) density and (b) rhodamine concentrations within the dye patch during this study, labeled with their times. Solid and dashed curves are used to indicate profiles from the two different small boats.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
Although the dye quickly filled this upper mixed layer, profiles before 1430 often showed increased concentration at the base of the layer (Fig. 3b). This could have occurred because the dyed fluid was slightly greater in density than the mixed layer so that it eventually concentrated at the base of the layer (despite our calculations), or it may reflect the presence of downwelling convergences (e.g., from Langmuir circulation) that could potentially trap the small boats used to obtain profiles. Notwithstanding the specific mechanisms at work, the dyed layer occupies the apparent mixed layer within the upper 4 m and can be seen as deep as 7 m in some profiles before 1600.
Next, we examine the water column velocity field using downward-looking ADCP measurements made from a towed body that repeatedly crossed the dye patch (Fig. 4; these measurements are virtually identical to those obtained from the drifting ADCP but extend in time past 1508). Zonal velocities are westward with little shear in the upper 10 m, although currents are less westward below that. Northward velocities however are sheared in the upper 5 m. In the upper part of the water column there is also a slowly changing mean meridional flow over the time of the study, at first southward and then northward, which matches the dye and drifter motions.
Water column velocities in the (a) eastward and (b) northward directions, as measured by an ADCP repeatedly towed across the dye patch. Overplotted magenta lines show actual velocity profiles at the times marked by vertical black bars.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
More quantitatively, if we time-integrate the measured velocities from the ADCP dataset to estimate the cumulative motions of the water column at different depths, we see that the dye locations at 1337 and 1541 very closely match the calculated cumulative drift at depths of less than 4 m to these same times (Figs. 2c,d). At greater depths, the cumulative motions end up north and east of the dye patch. This is especially true at 12–17 m, the nominal drogued depth of our SVP iridium (iSVP) drifters, which in consequence are not expected to follow the patch (note, however, that they do not follow the cumulative drift at 12–17 m either but lie between this cumulative drift location and the dye patch).
Any bias in the ADCP-derived velocities, in either magnitude or direction, will result in an offset in our time-integrated cumulative motions. Perfect agreement is therefore unlikely; however, the close match strongly suggests that the dye does in fact track the surface water, and that over the upper ∼4 m within the mixed layer the water column remains relatively unsheared, at least in comparison with the amount of turbulent horizontal spreading that occurs. Below this depth the shear is large enough [i.e., changes of O(10 cm s−1) over a few meters] to cause measurable changes in drift, although mostly in the north–south direction.
The basic conclusion is that, to a first approximation, the upper 4 m of the water column acts as a single slab, with dye spreading within that slab. Dye is also being mixed down another meter or so into the stratified region below, but within this stratified region a measurable northward shear is occurring. However, in principle, all drifter designs other than the deeply drogued iSVPs are within the slab and so we can usefully intercompare drifter motions with dye motions for all designs except the iSVP.
As a check on our conclusions, note that the size of the patch has grown to be about 300 m wide and 1500 m long at 1541, and with a depth of 5 m this volume would result in a dilution of the volume of rhodamine initially added to a mean concentration of about 5 ppb. This is similar to measured concentrations in profiles at that time (Fig. 3b) suggesting that all rhodamine has been accounted for.
A horizontal spread of ±150 m over the period of the experiment is equivalent to speeds of about ±1 cm s−1, which then provides a lower bound estimate of the scale of uncertainty in drifter-derived speeds as representatives of the mean flow.
2) Wind and surface waves
Measured surface winds at a height of 3.5 m during this period were not strong, between 2.8 and 4.7 m s−1 (mean 3.8 m s−1) approximately from the west-southwest (Figs. 2c,d). Air and sea surface temperatures differed by less than 0.5°C so that the stability was essentially neutral.
Observed 30-min-average omnidirectional surface wave spectra S(f) as a function of frequency f for the experiment show two prominent peaks (Fig. 5a). Using standard methods (Herbers et al. 2012) for determining the frequency-dependent direction of a single spectral component θ(f) from the various cross-spectra, the lower frequency peak at about 0.2 Hz is found to be associated with waves with a significant wave height of about 0.12 m propagating from the east-northeast (i.e., the more open Gulf of St. Lawrence). A second peak with frequency 0.47 Hz is associated with waves propagating from the west-southwest in a downwind direction, with a significant wave height of 0.20 m (Fig. 5b). The total significant wave height Hs ≈ 0.23 m.
(a) Omnidirectional 30-min-average wave spectra S(f) measured during the dye study. The dashed line shows the theoretical Phillips spectrum modeling spectral energy at frequencies above the peak. (b) The direction of dominant wave energy θ(f) for each frequency, with line colors matching those of the spectra in (a). In addition, symbols show the peak wave frequency and direction for a fully developed wind sea using fPM = 0.13g/U10 (Holthuijsen 2007), with wave direction taken as the direction of winds, every 30 mins from 1200 to 1530 UTC.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
3) Drifter slip
To summarize the drifter observations within the mixed layer (i.e., other than for the iSVPs), we plot them as eastward offsets from the center of the dye patch (Fig. 6). We focus on eastward rather than total offsets because the mean longitude of the patch is more easily determined than mean latitude, ADCP velocities suggest a slight degree of northward shear that we wish to avoid in later analysis, and, in any case, slips are nearly eastward (Fig. 2). We find that the iSphere drifters end up more than 2 km downwind after nearly 4 h, whereas the CODE and CARTHE drifters, as well as the home-built University of British Columbia expendable surface drifters (UBC-ESDs), are 350–500 m downwind. The disk-like XeosTechnologies, Inc., Oil Spill Kit Emergency Response (OSKER) drifters and home-built Institut des Science de la Mer de Rimouski (ISMER) designs with a very similar size and shape both end up about 1500 m downwind, whereas the Surface Circulation Tracker (SCT) drifters that have a disk-like surface float but also a small drogue are about 750 m downwind. These represent slip speeds of 2.6–17 cm s−1 relative to the dye, significantly faster than apparent motions from horizontal turbulence [found in section 3a(1) to be about ±1 cm s−1].
Mean eastward drifter motions relative to the dye patch. Error bars associated with drifter displacements, which have all been calculated at 1337 and 1541, represent the range of observed speeds and have been slightly shifted vertically to avoid overlaps between different designs.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
Taking cumulative integrals of the wind at a standard height of 10 m, U10 [calculation of which is described in section 3b(1)], a first crude estimate of these displacements is that they represent design-dependent drifts of 0.6%–4% of wind speed.
In addition to moving downwind, the drifters start to deviate from their initial north–south alignment with the dye line, with zonal separation distances between units of the same design that grow to a few hundred meters. However, these separation distances are similar to the zonal spread of the dye over the same time, and probably reflect dispersion by the same small-scale horizontal turbulent processes.
b. Theoretical calculations
Although our ancillary observations provide information on the coupled air–sea boundary layer, the drifters themselves are located several meters below the lowest direct wind measurement and several meters above the highest direct water-column velocity measurements. To understand the drifter motions, we must therefore interpolate between these measurements using a suitable, and ideally a simple, boundary layer theory to estimate air and water velocities at the vertical position of the drifters themselves.
The fully three-dimensional current field of the ocean surface boundary layer is first separated into a horizontally averaged mean vertical profile, and an additional part due to unresolved processes (both turbulence and wave orbital velocities). The turbulent processes act to disperse dye. Over a horizontal scale of 1 km, a parameterization by Okubo (1971) suggests that these processes can be described, within a factor of 3, by an effective oceanic horizontal eddy diffusivity of about 0.6 m2 s−1, which matches our observations. In the vertical, turbulent processes disperse dye over time scales significantly shorter than the length of our experiment.
1) Eulerian mean boundary layer
A number of researchers have investigated the flow structure at the air–sea interface (e.g., Madsen 1977; Lewis and Belcher 2004; Rascle et al. 2006; Samelson 2022, and others), although mostly in the context of developing an appropriate theory for the entire planetary boundary layer, including Coriolis effects. Although informative, a significant drawback of these theories is that their “spinup” occurs over inertial time scales, which are far longer than the time scale of our observations. They also provide information about the spiraling of velocities over the whole boundary layer, and investigate the angle between winds and surface velocities, which is generally rightward (Northern Hemisphere) but at angle that is somewhat sensitive to the details of the boundary layer and its time history.
Another more subtle problem with these earlier studies is that the surface velocity uE(0) is often specified as an input parameter (e.g., as a fixed percentage of wind speed at a standard height; Lewis and Belcher 2004; Samelson 2022) and used to set other boundary layer parameters, rather than obtained as a model output. Since we are concerned here with determining the surface velocity as an output to compare with drifter observations, that approach leads to circular arguments.
Instead, we shall develop a simpler model that is entirely independent of drifter observations. A critical simplification we make is to ignore the small angle between measured winds and drifter slip vectors, treating them both as “downwind” [i.e., taking cos(30°) ≈ 1]. Shorter-term modeling of a shear-driven atmospheric boundary layer near the surface, embedded within the planetary layer, over vertical scales small enough that no significant spiraling occurs, is well developed using Monin-Obukhov similarity theory (Csanady 2001). Here we shall adapt and extend that theory for additional use within the oceanic mixed layer to estimate mean velocities at the depths of our drifters.
A basic assumption in this approach is that the turbulent shear stress τ = ρν∂uE/∂z, related to the velocity shear with a vertical eddy viscosity ν and density ρ, is assumed to be approximately constant in magnitude and direction over the depth/height range in question (here −4 < z < 10 m). Although when seas are not fully developed the stress in the water is less than that in air (as some of the energy is being used to increase wave magnitudes), for our conditions of fetch and wave age the fraction of stress used for growing the surface waves is only a few percent (Donelan 1979; Mitsuyasu 1985) and this loss is therefore neglected. Air (density ρa ≈ 1.2 kg m−3) is present for z > 0 and water (density ρw ≈ 1021 kg m−3) is present for z < 0. Considering variations only in the shear direction, useful parameters are then the airside and waterside friction velocities,
Gradients on the water side are small but not negligible (Figs. 7a,c), as they result in shears of the same order as the Stokes drift terms considered below. The surface speed is 3.8 cm s−1 greater than uE (−4 m), or 2.8 cm s−1 greater than the mean over the slab
The (a) coupled surface boundary layer velocity profile, (b) air side only, and (c) water side only.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
In contrast, the mean velocity structure shows large gradients on the air side close to the water surface (Figs. 7a,b). U10 is only about 10% larger than u(3.5 m). However, winds at 10 cm above the surface are still 60% as large as U10, and even at 1 cm above the surface are 40% as large (Fig. 7b). There is then a very strong gradient in winds in the lower centimeter of the atmosphere, since the surface speed relative to the slab is 2.8 cm s−1 (i.e., only about 0.6% of U10). Such extreme gradients are a consequence of the very small roughness length za0. Shear is much more broadly distributed through the mixed layer below the surface because zw0 ≫ za0.
The results just described are numerically robust. Changes to the roughness length parameterizations within a range of acceptable uncertainties (or use of other parameterizations, e.g., the “Charnock” roughness length parameterization; Csanady 2001) causes changes of only up to about 15% in u0; such changes are almost unnoticeable in Fig. 7. Even tripling the wind speed increases u0 by less than a factor of 2.
A log-layer structure is also found to be embedded in full models of the ocean/atmosphere coupled boundary layer (Madsen 1977; Lewis and Belcher 2004), as long as eddy viscosities increase linearly away from the ocean surface and distances from the surface are less than scale distances. The vertical scale depths of the Ekman boundary layers in the air and water are
However, this does not necessarily mean that the Coriolis force can be completely neglected in the water. Our water side friction velocity
2) Lagrangian correction (Stokes drift)
However, practical application of this formula involves a number of problematic factors. The first is that, although nondirectional wave spectra (and the dominant wave directions) are known here (Fig. 5), the directional spread of wave energy is not well characterized. The second is that the Stokes drift, which is primarily affected by the highest-frequency waves due to the f3 weighting of the wave spectrum in Eq. (13), is strongly affected by assumptions about spectral levels at unmeasured frequencies above 0.8 Hz. In addition to actual wave processes at these unmeasured frequencies, a further difficulty is that the finite size of any drifting objects will begin to interact with wave effects at high frequencies (i.e., small wavelengths). A drifter 30 cm across would tend to filter out Stokes drift effects from surface waves with wavelengths of less than 30 cm, or deep-water wave frequencies higher than 2.3 Hz. On the other hand, these higher frequencies could still contribute a wave radiation pressure as they reflect from the drifter body, instead of a Stokes drift component.
A directional spread of wave energy reduces the total Stokes drift because the off-axis drift components of calculated drift will cancel if the spread is symmetric. After assuming a somewhat complicated and frequency-dependent canonical form for this spread, Webb and Fox-Kemper (2015) calculate a functional form for HDHH and find that it is generally somewhere between 0.8 and 0.9, although in practice one might expect values closer to 1 for a very unidirectional swell. However, in the absence of better information about the full 2D wave spectrum, we shall treat it as a constant ≈0.85, accepting that this gives rise to an uncertainty of O(5%) in the resulting Stokes drift estimates.
To show the effect of these considerations, we calculate the magnitude of the Stokes drift for both swell and wind peaks (Fig. 8). For the wind peak, a monochromatic assumption [i.e., Eq. (14)] predicts a surface drift of 1.2 cm s−1. Integrating over the observed wind sea frequency band results in a somewhat larger surface drift prediction of about 1.8 cm s−1. However, integration after extrapolating beyond the observed frequency band with an f−5 dependence results in a surface Stokes drift of ∼4.5 cm s−1 (equivalent to 1% of U10). If instead we limit this integration to an upper frequency limit of 2.3 Hz (i.e., assuming that the finite drifter size removes any dependence on waves with frequencies above this limit) the surface Stokes drift is only about 3.5 cm s−1. The two integrations differ over the upper 2 cm of the water column. Clearly these higher-frequency components are critical to the behavior of undrogued drifters, and in practice the “surface Stokes drift” is an ill-defined concept at best.
Stokes drift profiles uS(z) for both wind and swell peaks, calculated from the measured wave spectra with (a) logarithmic and (b) linear y-axis scales. Also shown is the Stokes drift averaged over depth
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
Notwithstanding the sensitivity of the estimated surface Stokes drift to various assumptions, this drift rapidly decays with depth and is an order of magnitude smaller at a depth of only ∼0.55 m. Thus, the depth-averaged Stokes drift associated with wind waves is only about 1.4 cm s−1 averaged over the upper 0.5 m and 0.85 cm s−1 averaged over the upper 1 m. This decay is very much more rapid than the decay of Eulerian mean velocities discussed in section 3b(1). Over the upper 4 m the mean wind-wave drift is only 0.23 cm s−1. The swell peak, on the other hand, is associated with a west-southwest drift of only about 0.07 cm s−1at the surface, and hence is negligible in magnitude relative to the drift resulting from wind waves.
3) Calculated drifter slip
Empirically, the dye patch moves as a slab in the upper 4 m. Dye may move in this way without matching the motion of a drifter at the top of the slab because dyed water parcels cycle through the whole mixed layer, moving at a speed uML on average. Thus, although the Stokes drift will affect transport in the upper ∼0.4 m of the water column, individual dye parcels will be affected by Stokes drift for only a small fraction of the time. If the Stokes drift layer is 0.4 m thick within a 4-m-deep mixed layer, parcels vertically cycling through the mixed layer will only be affected by the Stokes drift for 0.4/4 ≈ 10% of the time, while our drifters, floating at the surface, will be affected by Stokes drift for the entire time. Thus an eastward offset of even “perfect” surface drifters moving at speed u(0) need not be matched by a corresponding offset in the dye seen near the surface.
Drag calculations from Eqs. (18) and (19): (a) Vertical profiles of downwind velocities in the water side boundary layer, as well as the predicted downwind drift ud of the different drifter designs. The extent of the vertical line for each drifter marker shows the depth range occupied by that design. Calculations are also made while neglecting air drag, and these values are shown at drogue midpoints zc. The upper scale shows speed in excess of uML as fraction of U10. (b) Calculated range of speeds for each design associated with increases and decreases of 1 cm in the freeboard for all designs. Mean observed speeds are shown, with an error bar representing ±2 standard deviations for the expected range of speeds (i.e., an ∼95% confidence interval), assuming a dispersion with νH = 0.4 m2 s−1.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
The calculated slip speeds ud − uML are consistent with measured values for most designs, especially considering the sensitivity to freeboard for the faster-moving drifters. The undrogued ISMER and OSKER drifters in particular have predicted slips that range over a factor of nearly 2 from small changes in freeboard. However, the undrogued iSphere has a measured slip that is far beyond any prediction for a reasonable range in static freeboard, although it does have the largest predicted slip. Neither measured nor predicted slips for the undrogued drifters are consistent with the motion of water parcels alone at any depth. Instead, the slip from direct wind drag alone is between 0.3% and 3% of wind speed, depending on freeboard.
The motion of the drogued drifters, on the other hand, is roughly consistent with the motion of water parcels near the surface, although the speed they measure is somewhat larger than the layer average speed uML of the mixed layer itself, and there is an additional component related to the Stokes drift. Their speeds, however, lie within the range of minimum to maximum water speeds.
Drag calculations can also be carried out for the underwater component alone (Fig. 9a), that is, ignoring drag in the air. The calculated drift speeds, plotted at the depth of the drogue centers zc, fall reasonably closely to u(zc) and only slightly within the concave curvature of u(z), suggesting that effects arising from the nonlinear nature of u(z) in this calculation are small. Thus, the major discrepancy between drifter and water column motions arises from direct wind drag on the portion of the drifter above the surface. However, even though the drogued drifters move at a speed of approximately u(zc), this speed is generally different from that of the mixed layer as a whole (uML).
c. Predictions
Although we have only a single set of ocean conditions for model comparison, the boundary layer structure, Stokes drift, and drag calculations just described can be carried out for any desired wind conditions and wave field and so we can simulate a range of ocean conditions. Although the results should be treated with some caution until verified by actual measurements in other conditions (in particular at higher wind conditions when wave-breaking is common), it is hoped that they at least provide some guidance about expected behaviors.
Carrying out our calculations under these conditions, we find that all of our predicted drifter speeds increase relatively steadily with wind speed (Fig. 10a). The shallow drogued drifters (CODE, CARTHE, UBC-ESD, and SCT) in particular appear to measure the surface speed (Eulerian plus Lagrangian) u(0) reasonably well, especially at higher wind speeds, while the deeply drogued iSVP approximately measures a speed mostly between uML and u0 [= uE(0)], that is, does not measure the Stokes drift. The undrogued drifters (OSKER, ISMER, and iSphere) on the other hand are always affected by wind drag and move somewhat faster than any water parcels. Because of the depth of the mixed layer, the mean mixed layer velocity is generally only slightly larger than the speed at its base (about 0.3% of wind speed). The Stokes drift component is larger than the wind-driven speed at the surface except in the lightest of winds but is not more than double the size of the wind-driven component at even the highest winds.
(a) Calculated drifter speeds ud as well as mixed layer mean speed uML, surface Eulerian speed uE(0), and surface total water speed u(0) = uE(0) + uS(0) as a function of wind speed U10. (b) The same results after subtracting uML, plotted as a fraction of wind speed.
Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-23-0073.1
Considering speeds in excess of uML as a fraction of U10 (Fig. 10b), performance varies slightly at low wind speeds (less than ∼5 m s−1) for the shallow drogued drifters, but is reasonably constant at higher wind speeds, thus confirming the usual hypothesis that a constant fraction of wind speed is a reasonable model for drifter slip. The surface Eulerian current u0 = uE(0) is larger than uML by 0.5%–1.5% of the wind speed (consistent with our experiment) but the surface Stokes drift is fairly constant at about 1.5% of the wind speed over most of the range of wind speeds, somewhat larger than in our experiment where the wind sea was far from fully developed. The shallow drogued drifters thus slip at about 2% of the wind speed relative to the mixed layer as a whole, whereas the undrogued drifters slip at 3%–4% of the wind speed. The deeply drogued iSVP slips at only about 0.4% of the wind speed relative to the mixed layer (and is within 0.1% of wind speed to the water surrounding the drogue, although this is not shown here).
4. Discussion and conclusions
What do surface drifters actually measure? We released a large number of drifters of different designs along a linear rhodamine dye patch, and observed their subsequent motions, in a controlled attempt to answer this question. The number of different designs used was greater than in any previous intercomparison and we also attempted to make more comprehensive measurements of the oceanographic conditions than has been possible in many previous studies. Although dye and drifters have been co-deployed in several other studies recently (Romero et al. 2019; Choi et al. 2020), the focus on those studies was relative dispersion rather than investigating drifter performance as in this one.
First, we confirmed that the rhodamine dye patch acted like a tracer for the mixed layer. Over the course of our experiment the patch retained its north–south orientation and generally linear shape, suggesting that effects of large-scale vorticity and shear were minimal in this experiment. However, the dye also spread out horizontally and somewhat unevenly at a rate that can be described with an eddy viscosity of about 0.4 m2 s−1, which is entirely consistent with apparent diffusivities over scales of ∼1 km in the ocean according to data presented by Okubo (1971). The east/west (i.e., across-line) dispersion of different drifter types was also consistent with the same magnitude of small-scale diffusion. In our experiment, this diffusion is equivalent to a “noise” of about ±1 cm s−1, or about ±0.2% of wind speed U10. The degree to which this dispersion is important in design intercomparisons has not previously been highlighted.
Second, we found that our drifters all moved downwind of the dye patch, at a wide range of rates between 0.6% and 4% of wind speed. One surprising result of the experiment was that even the “best”-performing drifters moved away from the dye at >0.6% of the wind speed, somewhat in excess of the 0.1% of wind speed previously determined as the expected slip for CODE-style drifters (Poulain and Gerin 2019), so that even after a few hours they were clearly no longer tracking the dye patch, even in fairly light winds.
Although a wide variety of slips were found across the range of drifter designs, our independent calculations of slip magnitude, based on a physical modeling of the boundary layer and its interactions with each drifter design, match these observations for six of the seven designs that were vertically contained in the patch. The agreement between observed and predicted slip magnitudes at least suggests that the boundary layer theory we use for predictions is sufficiently accurate for this purpose, and hence may provide useful guidance in other situations. This is despite the fact that some aspects of this theory have clear deficiencies. The lack of Coriolis forcing has already been mentioned, as has been the lack of consideration of wave-breaking and skin friction effects in high winds and the uncertainty associated with the effects on Stokes drift of unmeasured high-frequency waves. The wind sea is taken to be fully developed in response to local winds only, and the mixed layer depth must be far greater than the drogue depth of drifters. Other simplifications include the use of a steady-state drag calculation as a function of height well within the height range of the surface wave field, and the use of a quasi-planar flow approach appropriate to long slender bodies, even for mostly flat undrogued drifters.
Acknowledging these deficiencies, however, we still conclude that the apparent mismatch between slips previously estimated to be around 0.1% for the best designs and our larger observations does not arise because the performance of drifters is worse than expected. Instead, the problem lies in an inaccurate, or at least largely unstated, expectation that a surface drifter should track a well-mixed water column. In fact, most of the observed slip arises in roughly equal proportions from the effects of Eulerian and Lagrangian shear in the upper O(0.5 m) of the water column, whose effects are minimized for the dye because water parcels in a mixed layer spend only a fraction of their time near the surface. The slip relative to water near the drogue, arising from drag in the air, is calculated to be only about 0.2% of the wind for all of the drogued drifters, almost the same as the 0.1% value obtained from earlier measurements. Thus, the ability of drifters to track patches of water in the mixed layer (e.g., to monitor purposeful injections of chemicals) is greatly overstated by existing “slip” predictions.
For drogued drifters with Aw/Aa > 40 [a design criterion first proposed by Niiler et al. (1995)], that is, for the UBC-EST and CODE designs (and nearly the CARTHE design), Eq. (22) suggests the drifter speed is larger than ub(zc) by only about 0.2% wind speed U10. However, for undrogued drifters, Aw ≈ Aa so that not only is the predicted drift speed greater than ub(zc ≈ 0) by about 1.5% of wind speed, but this slip is subject to great uncertainty because of its sensitivity to small changes in freeboard. This may hide additional sensitivities to microscale wave-breaking and skin friction effects (Sutherland and Melville 2015).
One caution with these predictions is that we do not know whether the Eulerian shear in the water column, at high wind speeds for which wave breaking may be significant, follows a log-layer shape very near the surface. Although studies of turbulent energy dissipation in this region suggest that this may not be completely accurate (Thorpe 2005; Sutherland and Melville 2015), the true shape of the shear profile near the surface is still uncertain.
In contrast to results for the first six small-sized drifter designs, the iSphere drifter motions are not well modeled in our theory, with observed slip about 50% larger than predicted. Large slips for iSphere drifters are generally found in other studies as well (Röhrs et al. 2012; Novelli et al. 2017; Blanken et al. 2021; Sutherland et al. 2020). However, missing from our model is any attempt to account for dynamic interactions between a buoyant drifter with the wave field. Such interactions may explain some of the excessive slip known to be associated with rigid-neck versions of the CARTHE drifters under certain wave conditions. Novelli et al. (2017) suggested that a flexible connection between the drogue and the surface float (a feature shared with the CODE and iSVP drifters) can correct for this issue; however, at least in this case the rigid-necked UBC-ESD design performs equally well without such modification. It may be that the particular wave field characteristics in our experiment avoided triggering such a response, but note also that the dimensions of the surface float for the UBC-ESD are about 50% smaller than for the CARTHE design, and so it may simply be less susceptible to wind drag with changing freeboard and/or tilt.
A possibly related fact is that the surface floats for most designs tested are relatively flat, so that a small change in freeboard will not only have large effects on buoyancy but “bobbing” motions are likely to be highly damped. The spherical iSphere, on the other hand, will have the greatest freeboard change for a given buoyancy change, and (due to its spherical shape) less damping resistance to bobbing, so that it might have significant dynamic variations in freeboard. This may be the reason for not only its large slip, but also our lack of success in predicting that slip. A careful examination of Fig. 9 also shows that our predictions for SCT and UBC-ESD designs are on the low end of the measured range of speeds, so it is possible that such effects arise at a less significant level even for those designs. The dynamic behavior of SVPs has been studied (Niiler et al. 1995) and perhaps more attention should be paid to this factor for other designs. In any case, for undrogued drifters, at present the combined uncertainties in both surface current, freeboard sensitivity, and these dynamic factors make it difficult to recommend them for quantitative tracking purposes.
For all drifters, the importance of Stokes drift in modeling drifter behavior, and the uncertainty in how best to handle the unmeasured high-wavenumber components of the wave field, highlights the fact that the “surface Stokes drift,” although a standard wave model product (e.g., The WAVEWATCH III Development Group 2019) is not a very well-defined concept. Calculated values depend rather critically on currently unmeasured, and theoretically uncertain, aspects of the wave spectrum at high frequencies. Direct measurements of the near-surface shear may be possible using drifters with drogues situated at different depths [as attempted by, e.g., Csanady (1984), Kudryavtsev et al. (2008), and Laxague et al. (2018)], ideally with the same surface float to reduce any bias from the effects of direct wind drag, but separating the Eulerian and Lagrangian contributions will be difficult if relying on drifters alone. Although attempts have already been made to do this in conjunction with HF-radar measurements (Morey et al. 2018; van der Mheen et al. 2020) the results are confusing. More well-controlled deployments with better information about ocean conditions, such as the experiment described here, as well as direct observations of the drift of surface slicks (natural or artificial) are probably necessary.
Last, although this experiment has highlighted many of the issues that govern drifter slip and may provide guidance on the proper choice of drifter design for different purposes, the winds during the experiment (≈4 m s−1) were relatively low. Additional experiments at the higher wind speeds more often found in the ocean are necessary before our extrapolations into such conditions can be fully trusted.
Acknowledgments.
Thanks are given to C. Bluteau for organizing the field program, as well as to C. Boutot, T. Tamtare, E. Dumas-Lefebvre, G. Dupéré, B. St-Denis, T. Leclercq, B. Cayouette, S. Blondeau, A. Dussol and the crew of the R/V Coriolis II for field operations; R. Hourston, D. Schillinger, and N. Soontiens for providing some of the (many) different drifters; and J. Clary for producing the rhodamine dye patches shown in Fig. 2 from the UAV surveys. Funding was provided by the Marine Environmental Observation, Prediction and Response (MEOPAR) network of centers of excellence for the Gulf of St. Lawrence Tracer Experiment grant 1-02-02-082.2 (authors Chavanne and Dumont), Réseau Québec maritime (RQM) and its Odyssée St-Laurent program for ship time OSL-MAC2020 (Chavanne and Dumont), and the Natural Sciences and Engineering Research Council through RGPIN-2022-03106 (author Pawlowicz), RGPIN-2018-06585 (Chavanne), and RGPIN-2019-06563 (Dumont).
Data availability statement.
The drifter dataset is publicly available in the Scholars Portal Dataverse (https://doi.org/10.5683/SP2/EFTSXM), and other measurements from the field program are publicly available from the St. Lawrence Global Observatory (https://slgo.ca/en/home-slgo/). Satellite imagery was obtained online [https://scihub.copernicus.eu (now https://dataspace.copernicus.eu)].
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