10.8: Lagrangian Measurements of Currents

Oceanography and fluid mechanics distinguish between two techniques for measuring currents: Lagrangian and Eulerian. Lagrangian techniques follow a water particle. Eulerian techniques measure the velocity of water at a fixed position.

Basic Technique

Lagrangian techniques track the position of a drifter designed to follow a water parcel either on the surface or deeper within the water column. The mean velocity over some period is calculated from the distance between positions at the beginning and end of the period divided by the period. Errors are due to:

1. The failure of the drifter to follow a parcel of water. We assume the drifter stays in a parcel of water, but wind blowing on the surface float of a surface drifter can cause the drifter to move relative to the water.
2. Errors in determining the position of the drifter.
3. Sampling errors. Drifters go only where drifters want to go, and drifters want to go to convergent zones. Hence drifters tend to avoid areas of divergent flow.

Satellite Tracked Surface Drifters

Surface drifters consist of a drogue plus a float. Its position is determined by the Argos system on meteorological satellites (Swenson and Shaw, 1990) or calculated from GPS data recorded continuously by the buoy and relayed to shore.

Argos-tracked buoys carry a radio transmitter with a very stable frequency \(F_{0}\). A receiver on the satellite receives the signal and determines the Doppler shift \(F\) as a function of time \(t\) (figure \(\PageIndex{1}\)). The Doppler frequency is \[F = \frac{dR}{dt} \frac{F_{0}}{c} + F_{0} \nonumber \]

where \(R\) is the distance to the buoy and \(c\) is the velocity of light. The closer the buoy to the satellite, the more rapidly the frequency changes. When \(F = F_{0}\) the range is a minimum. This is the time of closest approach, and the satellite’s velocity vector is perpendicular to the line from the satellite to the buoy. The time of closest approach and the time rate of change of Doppler frequency at that time gives the buoy’s position relative to the orbit with a 180\(^{\circ}\) ambiguity (B and BB in the figure). Because the orbit is accurately known, and because the buoy can be observed many times, its position can be determined without ambiguity.

Figure \(\PageIndex{1}\): System Argos uses radio signals transmitted from surface buoys to determine the position of the buoy. A satellite receives the signal from the buoy B. The time rate of change of the signal, the Doppler shift \(F\), is a function of buoy position and distance from the satellite’s track. Note that a buoy at BB would produce the same Doppler shift as the buoy at B. The recorded Doppler signal is transmitted to ground stations E, which relays the information to processing centers A via control stations K. After Dietrich et al. (1980: 149).

The accuracy of the calculated position depends on the stability of the frequency transmitted by the buoy. The Argos system tracks buoys with an accuracy of \(\pm (1-2) \ \text{km}\), collecting 1–8 positions per day depending on latitude. Because \(1 \ \text{cm/s} \approx ≈ 1 \ \text{km/day}\), and because typical values of currents in the ocean range from one to two hundred centimeters per second, this is an very useful accuracy.

Holey-Sock Drifters

The most widely used, satellite-tracked drifter is the holey-sock drifter. It consists of a cylindrical drogue of cloth 1 meter in diameter by 15 meters long with 14 large holes cut in the sides. The weight of the drogue is supported by a float set 3 meters below the surface. The submerged float is tethered to a partially submerged surface float carrying the Argos transmitter.

The buoy was designed for the Surface Velocity Program and extensively tested. Niiler et al. (1995) carefully measured the rate at which wind blowing on the surface float pulls the drogue through the water, and they found that the buoy moves \(12 \pm 9^{\circ}\) to the right of the wind at a speed \[U_{s} = (4.32 \pm 0.67) \times 10^{-2} \frac{U_{10}}{DAR} + (11.04 \pm 1.63) \frac{D}{DAR} \nonumber \]

where \(DAR\) is the drag area ratio defined as the drogue’s drag area divided by the sum of the tether’s drag area and the surface float’s drag area, and \(D\) is the difference in velocity of the water between the top of the cylindrical drogue and the bottom. Drifters typically have a \(DAR\) of 40, and the drift \(U_{s} < 1 \ \text{cm/s}\) for \(U_{10} < 10 \ \text{m/s}\).

Argo Floats

The most widely used subsurface floats are the Argo floats. The floats (figure \(\PageIndex{2}\)) are designed to cycle between the surface and some predetermined depth. Most floats drift for 10 days at a depth of 1 km, sink to 2 km, then rise to the surface. While rising, they profile temperature and salinity as a function of pressure (depth). The floats remains on the surface for a few hours, relays data to shore via the Argos system, then sink again to 1 km. Each float carries enough power to repeat this cycle for several years. The float thus measures currents at 1 km depth and density distribution in the upper ocean. Three thousand Argo floats are being deployed in all parts of the ocean for the Global Ocean Data Assimilation Experiment (GODAE).

Figure \(\PageIndex{2}\): The Autonomous Lagrangian Circulation Explorer (ALACE) floats is the prototype for the Argos floats. It measures currents at a depth of 1 km. Left: Schematic of the drifter. To ascend, the hydraulic pump moves oil from an internal reservoir to an external bladder, reducing the drifter’s density. To descend, the latching valve is opened to allow oil to flow back into the internal reservoir. The antenna is mounted to the end cap. Right: Expanded schematic of the hydraulic system. The motor rotates the wobble plate actuating the piston which pumps hydraulic oil. After Davis et al. (1992).

Lagrangian Measurements Using Tracers

The most common method for measuring the flow in the deep ocean is to track parcels of water containing molecules not normally found in the ocean. Thanks to atomic bomb tests in the 1950s and the recent exponential increase of chlorofluorocarbons in the atmosphere, such tracers have been introduced into the ocean in large quantities. See Section 13.4 for a list of tracers used in oceanography. The distribution of trace molecules is used to infer the movement of the water. The technique is especially useful for calculating velocity of deep water masses averaged over decades and for measuring turbulent mixing discussed in Section 8.4.

The distribution of trace molecules is calculated from the concentration of the molecules in water samples collected on hydrographic sections and surveys. Because the collection of data is expensive and slow, there are few repeated sections. Figure \(\PageIndex{3}\) shows two maps of the distribution of tritium in the north Atlantic collected in 1972–1973 by the Geosecs Program and in 1981, a decade later. The sections show that tritium, introduced into the atmosphere during the atomic bomb tests in the atmosphere in the 1950s to 1972, penetrated to depths below 4 km only north of 40\(^{\circ}\)N by 1971 and to 35\(^{\circ}\)N by 1981. This shows that deep currents are very slow, about 1.6 mm/s in this example.

Because the deep currents are so small, we can question what process are responsible for the observed distribution of tracers. Both turbulent diffusion and advection by currents can fit the observations. Hence, does figure \(\PageIndex{3}\) give mean currents in the deep Atlantic, or the turbulent diffusion of tritium?

Figure \(\PageIndex{3}\): Distribution of tritium along a section through the western basins in the north Atlantic, measured in 1972 (Top) and remeasured in 1981 (Bottom). Units are tritium units, where one tritium unit is \(10^{18}\) (tritium atoms)/(hydrogen atoms) corrected to the activity levels that would have been observed on 1 January 1981. Compare this figure to the density in the ocean shown in figure \(13.4.5\). After Toggweiler (1994).

Another useful tracer is the temperature and salinity of the water. I will consider these observations in Section 13.4 where I describe the core method for studying deep circulation. Here, I note that AVHRR observations of surface temperature of the ocean are an additional source of information about currents. Sequential infrared images of surface temperature are used to calculate the displacement of features in the images (figure \(\PageIndex{4}\)). The technique is especially useful for surveying the variability of currents near shore. Land provides reference points from which displacement can be calculated accurately, and large temperature contrasts can be found in many regions in some seasons.

There are two important limitations.

1. Many regions have extensive cloud cover, and the ocean cannot be seen.
2. Flow is primarily parallel to temperature fronts, and strong currents can exist along fronts even though the front may not move. It is therefore essential to track the motion of small eddies embedded in the flow near the front and not the position of the front.

Figure \(\PageIndex{4}\): Ocean temperature and current patterns are combined in this AVHRR analysis. Surface currents were computed by tracking the displacement of small thermal or sediment features between a pair of images. A directional edge-enhancement filter was applied here to define better the different water masses. Warm water is shaded darker. From Ocean Imaging, Solana Beach, California, with permission.

The Rubber Duckie Spill

On January 10, 1992 a 12.2-m container with 29,000 bathtub toys, including rubber ducks (called rubber duckies by children), washed overboard from a container ship at 44.7\(^{\circ}\)N, 178.1\(^{\circ}\)E (figure \(\PageIndex{5}\)). Ten months later the toys began washing ashore near Sitka, Alaska. A similar accident on May 27, 1990 released 80,000 Nike-brand shoes at 48\(^{\circ}\)N, 161\(^{\circ}\)W when waves washed containers from the Hansa Carrier.

The spills and eventual recovery of the toys and shoes proved to be good tests of a numerical model for calculating the trajectories of oil spills developed by Ebbesmeyer and Ingraham (1992, 1994). They calculated the possible trajectories of the spilled toys using the Ocean Surface Current Simulations oscurs numerical model driven by winds calculated from the Fleet Numerical Oceanography Center’s daily sea-level pressure data. After modifying their calculations by increasing the windage coefficient by 50% for the toys and by decreasing their angle of deflection function by 5\(^{\circ}\), their calculations accurately predicted the arrival of the toys near Sitka, Alaska on November 16, 1992, ten months after the spill.

Figure \(\PageIndex{5}\): Trajectories that spilled rubber duckies would have followed had they been spilled on January 10 of different years. Five trajectories were selected from a set of 48 simulations of the spill each year between 1946 and 1993. The trajectories begin on January 10 and end two years later (solid symbols). Grey symbols indicate positions on November 16 of the year of the spill. The grey circle gives the location where rubber ducks first came ashore near Sitka in 1992. The code at lower left gives the dates of the trajectories. After Ebbesmeyer and Ingraham (1994).