## Abstract

A truly Lagrangian float would follow all three components of oceanic velocity on all timescales. Progress toward this goal is reviewed by analyzing the performance of nearly Lagrangian floats deployed in a variety of oceanic flows. Two new float types, described in this paper, are autonomous with durations of months, can alternate between Lagrangian and profiling modes, relay data via satellite, and can carry a variety of sensors. A novel hull design is light, strong, and has a compressibility close to that of seawater.

The key to making floats accurately Lagrangian is an improved understanding of the factors that control float buoyancy and motion. Several insights are presented here. Anodized aluminum gains weight in seawater due to reactions between its surface and seawater. At low pressure the buoyancy of floats with O-ring seals varies as if attached bubbles of air were being compressed. The volume of “air” decays exponentially with a decay scale of a few days from 10 to 30 cc at deployment to an asymptotic value that depends on pressure. The drag of floats moving slowly through a stratified ocean is dominated by internal wave generation and is thus linear, not quadratic. Internal wave drag acting on an isopycnal-seeking float will cause the float to be Lagrangian for frequencies greater than about *N*/30, where *N* is the buoyancy frequency.

These floats have proven most useful in measuring the turbulence in ocean boundary layers and other regions of strong turbulence where the ability of the floats to be Lagrangian on short timescales matches the short timescale of the processes and where the size of the turbulent eddies exceeds the size of the float. On longer timescales, the floats successfully operate as isopycnal followers. Because truly Lagrangian floats are highly sensitive to minor perturbations, extension of the frequency band over which the floats are Lagrangian will require careful control of float buoyancy and thus a detailed understanding of the float's equation of state.

## 1. Introduction

Lagrangian techniques for measuring the ocean are becoming increasingly common (Davis 1991). Most of the recent emphasis has been on using large numbers of inexpensive surface (Niiler et al. 1995) and subsurface (Davis et al. 1992; Rossby et al. 1986) instruments to adequately sample the large-scale, low-frequency circulation. Such drifting instruments, especially when combined with a profiling capability and stable temperature and salinity sensors, are becoming a major component of proposed operational programs to monitor the World Ocean.

Typically, subsurface floats are much less compressible than seawater. They therefore become more buoyant with depth and, in the limit of fixed volume, come to rest on a surface of fixed density (or the surface or bottom). Since most of the change in density of seawater is caused by pressure, this fixed density surface is nearly a surface of fixed pressure and the term “isobaric” is often used for such floats. Some floats (Rossby et al. 1985) have used a separate “compressee” to increase their compressibility to nearly match that of seawater. These will come to rest on a surface of constant potential density and are thus often called “isopycnal.” Oceanic flow follows neither surfaces of constant pressure nor constant potential density, particularly in the presence of mixing. An instrument that followed the water exactly would be called a Lagrangian float. This paper describes recent progress in building instruments that approximate this goal. Despite their imperfections, we call them Lagrangian floats.

## 2. Float design and uses

D'Asaro et al. (1996) describe the design of the first generation of Lagrangian floats. These are meter-sized devices designed to have the same velocity as a well-defined average of the water surrounding them. On the meter scale the acceleration of water is due primarily to pressure gradients, not viscosity, so a float with the same density as seawater will be accelerated by the same pressure gradients as the seawater and thus follow a similar trajectory. The main design challenge for Lagrangian floats is thus to accurately match the density of the surrounding water. Floats with the wrong density will have a net buoyancy and will thus fall or rise through the water. In a stratified water column, this shifts the floats to the wrong density level. In an unstratified region, buoyant (heavy) floats tend to accumulate near the surface (bottom) of the layer, and thus oversample this region (D'Asaro et al. 1996; Harcourt et al. 2002). Floats with the wrong density also have a different inertia, which causes them to follow trajectories that diverge from those of the water (Harcourt et al. 2002). Section 5 describes our efforts to understand and control the buoyancy of the floats.

The relative velocity of the float and the water can be reduced by increasing the drag of the float. This is done using a large horizontal drogue (Figs. 1 and 2). The floats described by D'Asaro et al. (1996) used a perforated metal disk. Large drag, however, also means that it is difficult to intentionally profile the floats, either to move them to a desired level or to obtain profiles of stratification. It also makes the motion of the float insensitive to buoyancy, by intent, and thus makes it difficult to determine the float buoyancy. Our recent designs, therefore, have used folding cloth drogues. Estimation and control of the relative motion of the float and the water therefore depends on the drag law for the float at low velocity. Section 6 describes our improved understanding of drag.

The most basic measurement made by Lagrangian floats is the position of the floats. The vertical position is measured from pressure and the vertical velocity from its rate of change. The horizontal velocity is more difficult to measure; acoustic tracking has been used in some experiments (D'Asaro and Lien 2000; Steffen and D'Asaro 2002). Ideally, the float's velocity should equal the local water velocity. Both the float's buoyancy and its finite size cause the float's motion to differ from that of the water. Accordingly, most Lagrangian float deployments have been made in strongly turbulent flows where the magnitude of the turbulent velocity is much larger than the buoyancy-induced velocity and the overturning scales of the flow are larger than the size of the float. In such flows, there is a strong theoretical basis (inertial subrange theory) for understanding the form of the Lagrangian frequency spectra of velocity at high frequency. Lien et al. (1998) derive the spectral forms assuming that the float velocity is a spatial average of the water velocity surrounding the float. The resulting model spectral form depends on the turbulent kinetic energy *E* or its dissipation rate ɛ, the large eddy frequency *ω*_{0} that is the typical overturning time of the largest eddies of the turbulence, and the size of the float. Measurements made in a variety of energetic turbulent flows (Lien et al. 1998; D'Asaro and Lien 2000) show remarkably similar spectral shapes that accurately fit the Lien et al. (1998) form. Spectra are isotropic within the inertial subrange, also in agreement with theory. Lien et al. (1998) show that the values of ɛ estimated from Lagrangian spectra in a convective mixed layer are close to those expected given the measured buoyancy flux. These studies provide some confidence that for these flows the floats are approximately Lagrangian, that the effects of finite float size can be modeled, and that estimates of vertical velocity, ɛ, and *ω*_{0} computed from the float data are meaningful. Due to the differences in sampling, accurate direct comparisons with other observational techniques have been difficult, although further attempts are currently under way. Further progress in understanding the motion of the float relative to the water is likely to come from direct measurement. This has been a major motivation for the development of floats that can carry a larger sensor suite.

Measurement of scalars (temperature and salinity) on the float is relatively simple. For turbulent flows, however, there is little existing theoretical guidance on how to interpret these signals. Harcourt et al. (2002) and D'Asaro et al. (2002) compare float and model estimates of all the terms in the horizontally averaged boundary layer heat budget and conclude that floats can provide accurate measurements of these. Lagrangian measurements offer the prospect of using the Lagrangian heat (or salt) equation *Dθ*/*Dt* = *κ*∇^{2}*θ,* and thus directly measuring the local rate of heating due to mixing. This is a powerful diagnostic. For example, in turbulent boundary layers D'Asaro et al. (2002) show that the surface heat flux *Q* = *ρC*_{P} ∫_{surface layer }*Dθ*/*Dt **dz,* where *ρC*_{P} is the volumetric heat capacity of the water, and confirm this using numerically simulated floats. Figure 3 shows the average profile of *ρC*_{P }∫^{0}_{−z }*Dθ*/*Dt **dz* from 11 floats deployed for 17 days in a convecting mixed layer in the Labrador Sea in 1997. Cooling of the ocean is confined to the top 10 m; the computed surface heat flux is about 500 W m^{−2}, which compares well with the heat flux of 400–600 W m^{−2} computed from the upper ocean heat budget. In the stratified ocean, Lien et al. (2002) provide evidence that the spectral level of *Dθ*/*Dt* in the inertial subrange frequency band is proportional to *χ,* the rate of dissipation of temperature variance. If confirmed, this could be a very powerful tool for estimating mixing rates, particularly if it could be used for both salinity and temperature. Ultimately, sufficiently Lagrangian floats could be used to measure *Dθ*/*Dt* and *DS*/*Dt* on long timescales in the stratified ocean. Section 7 discusses the prospects for creating such truly Lagrangian floats.

## 3. Instruments

The Lagrangian floats described by D'Asaro et al. (1996) were designed to be deployed from ships for periods of 36 h or less. This greatly limited the amount of data that could be gathered. Accordingly, newer floats have been autonomous, with mission durations of months, and have used satellite communications to relay data and aid in recovery if necessary. This paper will discuss results from two types of floats.

The Deep Lagrangian Float (DLF; Fig. 1) was designed for the Labrador Sea deep convection experiment of 1997 and 1998 (Lab Sea Group 1998; Steffen and D'Asaro 2002; Harcourt et al. 2002). It measures temperature and pressure, can operate to 2-km depth, has about 30 cc of active volume control out of a total volume of about 15 000 cc, and relays its data, after the mission, via Argos. The volume control is accomplished by extruding a small piston out of the bottom of the float. The piston also acts to unfold the drogue after launch and to release the drogue and a weight at the end of the mission. The Labrador Sea floats were tracked acoustically by the RAFOS system (Rossby et al. 1986). DLFs have also been used in a study of the equatorial undercurrent in 1998 (Lien et al. 2002) and air deployed into Hurricane Dennis in 1999 (D'Asaro 2003).

The Mixed Layer Lagrangian Float second generation (MLFII) is shown in Fig. 2. It is a larger instrument, about 50 000 cc, with a more limited depth capability (250 db), but has the ability to surface repeatedly using 750 cc of active volume control. The buoyancy control is again accomplished using an extruding piston. At first, the piston was also used to fold and unfold the drogue, but a separate system for drogue control was added in 2001. On each surfacing, the float uses GPS to determine its position and uses the Orbcomm satellite system (and more recently the faster Iridium system) to transmit data and receive instructions. MLFII can carry a large instrument suite including a Doppler sonar, altimeter, conductivity–temperature–depth (CTD) sensors, accelerometers, photosynthetically active radiation (PAR) sensor, and fluorometer. Storage of this data requires several hundred megabytes, so the float must be usually recovered to retrieve the bulk of the data. The use of the Iridium satellite system, with faster data throughput, will enable up to several megabytes of data to be transmitted. MLFII was used in a study of the wintertime North Pacific mixed layer in 2000, in studies of upwelling off Oregon during 2000 and 2001, and in an air-deployed study of Hurricane Isidore in 2002. An MLFII with a fluorometer and downward irradiance sensor was used to study biophysical interactions off Oregon in 2001.

## 4. Operations

### a. Float ballasting

These floats have the ability to control their buoyancy. This allows them to continuously match their density to that of the surrounding water and thus autonomously operate for long periods of time. It also enables them to make vertical profiles in a manner similar to autonomous Lagrangian current explorer (ALACE) floats (Davis et al. 1992).

There are several useful ways to use this capability. All of them rely on the equation describing the buoyancy *b* of a float in seawater:

where the float's mass is *M* and its volume *V* depends on a reference volume *V*_{0}, the change in volume due to the active buoyancy control *B,* the temperature *T*_{f} referenced to *T*_{0}, the thermal expansion coefficient of the float *α,* the compressibility of the float *γ,* the volume of trapped air *V*_{air} at atmospheric pressure *P*_{atmos}, and the density of the water *ρ,* which depends on the water's temperature *T,* salinity *S,* and pressure *P.* Note that *P* is taken to be zero at the ocean surface. A residual Δ*b* accounts for additional unknown terms. Note that the exact values of *α* and *γ* depend on the volume by which they are normalized. Here *V*_{0} is used, although *V*_{0} + *B,* or some other variant, might be more exact. These differences are minor since the changes in volume are small compared to *V*_{0}. Equation (1) is very similar to those used by Swift and Riser (1994) and Goodman and Levine (1990) with the addition of the *V*_{air} and Δ*b* terms.

If there is no mixing, *S* and potential temperature *θ* are conserved along Lagrangian trajectories. Assuming constant *M* and *V*_{0}, and *T*_{f} = *T,* the float needs only to adjust *B* as a function of pressure to compensate for differences between *γ* and the compressibility of seawater, and the differences between *T* and *θ.* When mixing occurs, *θ* and *S* change along the Lagrangian trajectory. All the terms in (1) must then be accurately known and *B* appropriately adjusted if the float is to remain Lagrangian.

Some insight into (1) is obtained by setting *B, **V*_{air}, Δ*b,* and *T*_{0} to zero for simplicity and rewriting (1) as

where *ρ*_{w0} and *ρ*_{f0} are the densities of the water and float at *T* = 0 and *P* = 0, respectively; *γ*_{f} and *γ*_{w} are the compressibilities of float and water, respectively; and *α*_{f} and *α*_{w} are the respective thermal expansion coefficients of float and water. The value of *ρ*_{w0} is a function of *S.* The value of *ρ*_{f0} = *M*/*V*_{0}. Equation (2) shows that the float's buoyancy is independent of the water temperature if the expansion coefficient of the float matches that of seawater. Similarly, the float's buoyancy is independent of pressure if its compressibility matches that of seawater. The point of neutral buoyancy, *b* = 0, is independent of the mass or volume of the float and depends only on their ratio, that is, the float's density.

The goal of float ballasting is to adjust *B* using Eq. (1) so that *b* = 0 for water with a specified *S, **T,* and *P* or for whatever *S* and *T* are present at a specified *P.* The float's mass *M* is known as are approximate values of *γ* and *α* (see sections 5b and 5c). If *b, **ρ,* and *T* can also be determined at a depth where the *V*_{air} term is insignificant, (1) can be solved for *V*_{0}. This may have to be done often if Δ*b* changes significantly (see section 5d).

Several modes of operation, described below, have been developed. The algorithms are relatively complex with a large number of parameters controlling various aspects of the float behavior. Accordingly, the possibility of human errors or ignorance leading to unexpected float behavior is significant. The float control code is therefore used to drive a float simulation program, which includes a simulated ocean with realistic temperature and salinity profiles and simulated turbulence, internal waves, tides, eddies, and the response of the float to these factors and its own buoyancy changes. The mission parameters are developed within this simulation environment. This also allows problems, such as that described in section 6, to be simulated and changes in parameters developed and transmitted to the float during the mission.

### b. Isopycnal-Lagrangian operation

Figure 4 shows the operation of MLFII as a midwater isopycnal-Lagrangian float. During most of the mission (“Lagrangian Drift” in the figure), the float is isopycnal. In this mode, potential temperature *θ* and *S* are fixed and *B* is adjusted to bring the float to a target isopycnal on a timescale of about 1 day. Once on its rest isopycnal, the float will remain there despite vertical excursions of the isopycnal because the float's compressibility matches that of seawater. The float can be moved off the isopycnal by mixing or by changes in Δ*b.* If it is displaced by mixing, the float naturally returns to its isopycnal at a rate governed by the balance of buoyancy and drag. In section 7b(1) the timescale for return is shown to be about an hour. Changes in Δ*b* are compensated by the adjustment of *V*_{0}. The float is thus isopycnal on long timescales and Lagrangian on short timescales.

Periods of Lagrangian drift alternate with vertical profiles to the surface. These provide a depth–time section along the float track and thus some spatial context for the drift measurements. At the top of each profile, the float obtains a GPS fix and transmits a small subset of its data to the shore via the Orbcomm satellite system. It can also receive instructions via Orbcomm. These can be used to adjust the mission parameters. In practice, this allows much of the final mission adjustment to happen after the float is deployed. This has proven very valuable. However, the data transmission system is not perfect and degrades in high sea states. Typically, at least 1 day elapses between the receipt of data indicating a problem, the analysis of this data, the formulation of a solution, and the receipt of the instructions by the float. Shore-based operators can only guide the float behavior; on the short term it is autonomous.

### c. Mixed layer operation

#### 1) With CTD

Figure 5 shows the operation of MLFII as a mixed layer float. The goal is to make the float neutrally buoyant in the mixed layer. The mixed layer itself is turbulent with vertical velocities of several centimeters per second (D'Asaro 2001), so it is difficult to determine the buoyancy of the float by examining the pressure and/or density data alone. Instead, the float is allowed to settle to a deep density surface once per day at the bottom of a vertical profile. Each of these “settle” operations provides a new estimate of *V*_{0}. Settling is done at depth to ensure a quiet ocean and to eliminate rapid near-surface changes in compressibility (see section 5h for examples). The settling is done with the drogue folded, which greatly speeds the equilibration. If the measured *V*_{0} is stable during the last part of the settling, the float's density is assumed to equal that of the water, *b* is assumed to be zero, and (1) used to compute *V*_{0}. The settle at 282.8 fails this test; the one at 283.8 passes. After settling, the float then profiles upward into the mixed layer until it reaches 20 db, which is always in the mixed layer. It then measures the density at 20 db and uses (1) to compute the proper value of *B* to set *b* = 0. This process compensates for both changes in mixed layer density and changes in Δ*b.*

During the Lagrangian drift modes between profiles, the float operates isopycnally: *θ* and *S* are held constant and *B* is adjusted only in response to changes in *P.* Thus the float does not immediately respond to changes in mixed layer *θ* and *S.* This prevents instabilities [see section 7b(2)] that might result from using improper values of *α, **γ,* or *V*_{air}. Usually, the mixed layer density changes slowly and readjustment of the float's density once per day is sufficient.

The accuracy of this ballasting scheme can be assessed using the Doppler sonar on the float, which measures the velocity of water just above the top of the float relative to the float. The simplest measure of ballasting accuracy is the velocity in the along-float direction, which, because the float remains aligned to within a few degrees of vertical, is nearly the vertical velocity of the float relative to the water. The measured velocity, averaged over each drift mode, has a standard deviation of 1.2 mm s^{−1} and a mean, over all drifts, of 0.5 mm s^{−1}. This is small compared to the typical 10–20 mm s^{−1} vertical velocity of the float within the mixed layer. The vertical velocity corresponds to a float buoyancy of about 1 g using the quadratic drag law discussed in section 6.

#### 2) Without CTD

A simpler version of this scheme was used for DLF deployments in the Labrador Sea (Fig. 6). DLFs do not measure density, only pressure and temperature. The floats were initially weighed in the ballasting tank. The values of *b, **T, **S,* and *M* were computed from tank data, yielding a value of *V*_{0}. Equation (1) was then used with hydrographic data from the Labrador Sea to compute the value of *B* necessary for the float to settle at 1000 db. After deployment *B* moved to the computed value. For the next week *B* was adjusted to maintain the float at 1000 db using the measured pressure to provide feedback. Generally, the changes in *B* during this week were only a few grams, verifying that our initial ballasting was accurate. A week was long enough for the initial transients in Δ*b* to relax (see section 7d). The value of *B* was then decreased by a fixed amount calculated to bring the float into the mixed layer. One day later the drogue was opened and the float began isopycnal operation. However, the compressibility was adjusted to be slightly less than that of seawater, equivalent to about 1 g (1000 db)^{−1} stabilization, in order to ensure that it was not accidentally unstable. Furthermore, *B* was slowly increased during the mission to compensate for an expected slow decrease in Δ*b* (see section 7d). This approach was only possible because the stratification of the Labrador Sea is weak, relatively uniform, and because real-time CTD data telemetered from ALACE floats (Lab Sea Group 1998) was available shortly before the deployment.

A variant of this scheme was used for DLF deployments in Hurricane Dennis (D'Asaro 2003). In this case the stratification was large and highly variable and the density was known only within wide bounds. After deployment in front of the storm, the float was allowed to equilibrate at 70 m for 1 day. The value of *B* was then varied so that the float moved upward at a constant rate of 6 mm s^{−1} with the drogue closed. This required about 1 g of buoyancy. When the float reached 15 m, assumed to be within the mixed layer, the drogue was opened and the float was assumed to be properly ballasted. With the drogue open, 1 g of buoyancy would produce only about 1 mm s^{−1} of vertical motion assuming a quadratic drag law, far less than the 0.06 m s^{−1} rms vertical velocity in the hurricane. This scheme worked well enough for two of the three floats deployed to allow useful data to be obtained. The third float was kept below 15-m depth by the mixed layer currents long enough that it became too buoyant.

### d. Isohaline and isothermal misadventures

Some float ballasting strategies have been unsucessful.

Intense mixing in hurricanes causes the mixed layer density to increase by up to 1 kg m^{−3} during the storm passage, enough to change the buoyancy of a DLF by 9 g. Changes in the measured temperature and an assumed temperature–salinity correlation were used to partially compensate for this effect during the 1999 Hurricane Dennis deployments. Unfortunately, it also caused the floats to occasionally move into the much colder waters of the thermocline and remain there for several hours. Similar problems occurred in the initial MLFII deployments off the Oregon coast. The floats were programmed to be isohaline. It was hoped that an isohaline float would be more Lagrangian than an isopycnal float because the effects of solar heating would be minimized. Although the floats remained nearly isohaline (see section 6 for additional problems), occasional salinity inversions lead to rapid depth fluctuations at times.

In both of the above cases, examination of historical CTD profiles showed monotonic profiles of both temperature and salinity, so that the observed unstable behavior was unexpected. Apparently, temperature and/or salinity can be stable in the vertical; but unstable along float trajectories in regions with significant variability in the temperature–salinity relationship.

## 5. Float physical parameters

### a. Overview

Equation (1) defines the important physical parameters controlling float buoyancy. Below, each of these is considered in detail.

### b. Compressibility

Rossby et al. (1985) first attempted to make floats more Lagrangian by modifying their compressibility using an external “compressee.” D'Asaro et al. (1996) describe how this can be done using cylindrical hulls by adjusting the hull dimensions. The compressibility of a long hull with rigid endcaps depends on the ratio of the hull radius *r* to its thickness Δ*r.*

The MLFII hull is a simple cylinder, but with a larger diameter (*r* = 12.7 cm) than used in the D'Asaro et al. (1996) floats in order to provide more endcap space for sensors. The resulting wall is quite thick (1.27 cm) so that the hull accounts for about half the total float weight. For the complete hull, about 25% of the compressibility is contributed by the endcaps.

Design of the DLF hull was more challenging. The float needed to match the compressibility of seawater to pressures of 2000 db. Simple cylindrical hulls with a large enough compressibility will fail at pressures far less than this. One solution is to use ring-stiffened cylindrical hulls, which have rings placed periodically along the hull to prevent buckling. This effectively turns one long cylindrical hull into several shorter ones. DLF used a variant on this. An aluminum tube was machined into a series of rings connected by cylindrical arched bays (Fig. 7). The rings prevent buckling of the hull and the arches transfer the pressure forces acting on the hull to the rings while maintaining a nearly constant stress within the material. The resulting hull is strong, compressible, and very light since most of the unnecessary material has been removed. If the distance between the rings is small enough, the hull fails when the stress in the arches exceeds the strength of the material rather than by buckling. This is desirable because material failure, unlike buckling, is easy to model and predict. The hull was therefore constructed of 7075 T6 aluminum, which is about twice as strong as the more common 6061 T6 alloy. It is, however, difficult to obtain in tube stock. A stock of 7075 tubing was salvaged from vintage 1970s deep sea pressure cases donated by numerous colleagues. The final hull design was achieved through a combination of numerical modeling and destructive testing. A test hull segment failed by rupturing, not buckling, at approximately 2900 db. The DLF hull is only 30% of the instrument weight, compared to 46% for MLFII, yet can withstand pressures 10 times greater. Somewhat smaller improvements would be realized for a 6061 hull of similar design. The Seaglider AUV (Eriksen et al. 2001) uses such a hull.

The compressibility of each DLF was measured using a strain gauge scale in a freshwater pressure tank as described in D'Asaro et al. (1996). The accuracy of the scale is limited by the hysteresis and slow creep of the material used to waterproof the strain gauge. This effect can be minimized by cycling the tank pressure over a 150–500-db range and removing the overall trend. This increases the scale precision to about 30 mg rms. At low pressures, the float weight decreases rapidly with decreasing pressure, as discussed below. Above about 150 db, the weight varies approximately linearly with pressure. The compressibility is computed over this linear region. Corrections are made for a uniform trend in time and a contribution from air bubbles to yield an average *γ* = 3.73 × 10^{−6} db^{−1} for the 1998 floats. This is about 85% of the compressibility of seawater; final tuning of the compressibility of the float is done using the ballasting piston under software control. The variability in *γ* estimated from duplicate measurements on the same floats is about 1.5 × 10^{−8} db^{−1} in compressibility or 0.2 g over 1000 db of pressure change. The variation in compressibility between 12 floats prepared for the 1998 deployments was about 3 × 10^{−8} db^{−1} rms. The maximum difference between floats was 8 × 10^{−8} db^{−1}.

### c. Air

D'Asaro et al. (1996) plot the weight of floats in the ballasting tank as a function of pressure. At low pressures the weight decreases rapidly. They attribute this to air trapped in bubbles on the outside of the float, in O-ring grooves or in other cavities because the variation of weight with pressure approximated that expected of air as shown in (1). Typically, a few cubic centimeters of air at atmospheric pressure explained the measurements. DLF had an average estimated *V*_{air} of about 4 cc, with a variability between measurements that was comparable to the mean. MLFII had *V*_{air} of about 8 cc.

Much of the volume change attributed to “air” may be due to O-ring compression. The main seal on the float endcaps is a quad-ring (not O-ring) face seal that is precompressed by tightening the bolts that hold the endcap to the main hull. These seals are typically packed full with grease to reduce the amount of air. The bolts must be repeatedly tightened until no more grease is extruded to ensure a metal-to-metal seal. If this is not done, the low pressure compressibility of the float still fits the air model well, but with a larger value of *V*_{air}, 25 cc rather than 8 cc for MLFII. It may be difficult, therefore, to distinguish between air compression and seal compression. Field measurements of *V*_{air} are discussed in section 5g.

### d. “Creep”

Rossby et al. (1975) noted a steady sinking of aluminum hulled isopycnal SOFAR floats deployed in the ocean thermocline. Typical rates were 0.85 m day^{−1} for the approximately 430-L floats, corresponding to about 1.6 g of buoyancy loss per day or a fractional volume loss of about 4 × 10^{−6} per day. Voorhis and Benoit (1975) attributed this to slow creeping of the aluminum under pressure, which slowly decreased the float's volume. Sullivan (1975) made measurements of aluminum creep rates supporting this hypothesis and predicted a strong dependence of the creep rate on pressure. As a result, the slow descent of floats at depth has generally been attributed to metal creep, despite the lack of an observed correlation between float descent rates and pressure (Richardson and Schmitz 1993) and despite the fact that the Sullivan (1975) creep rates are far higher than those found in the literature (J. Osse 1996, personal communication).

Figure 8 shows the depth and piston volume for DLF number 2 deployed in the Labrador Sea in the summer of 1995. The float spent most of the mission at very close to 1300 db. The central Labrador Sea at this time was very homogeneous with only 0.004°C change in temperature and 0.001 psu change in salinity between 1300 and 1400 db (J. Lazier 1994, personal communication). During the 45-day mission the piston extruded about 9 cc. This volume change could not have been balanced by a change in the density of the water, as this would have required a 0.6 kg m^{−3} density change. The change in piston volume must have been balanced by an increase in float mass or decrease in its volume. The rate, about 0.2 g day^{−1} is equivalent to 13 × 10^{−6} day^{−1}, that is, grams of weight change per gram of float per day. This is about 3 times larger than that observed by Rossby et al. (1975) on a per volume basis and about 500 times larger on a per surface area basis.

The buoyancy of a DLF hull was measured once a week for 7 weeks in our freshwater ballasting tank. Between weighings it was pressured in freshwater at about 2000 db for the first 3 weeks. For the remaining 3 weeks it was pressurized at about 200 db. The temperature of the ballasting water varied by up to 0.6°C, which caused less than 1 g of weight change. With the temperature effect removed, the float gained 2 g in the first week. The average weight gain during the second and third weeks at 2000 db was 0.5 g (week)^{−1}; the average weight gain during the 3 weeks at 200 db was 0.43 g (week)^{−1}. The overall buoyancy gain was 4.9 g in 43 days, about half that observed for DLF 2 in the Labrador Sea. The rate was nearly constant in time and showed no dependence on pressure. It does not appear to be due to metal creep.

The float's hull was “hardcoat anodized” (Shreir et al. 1994, chapter 15.1) to inhibit corrosion, unlike the SOFAR floats, whose hulls were untreated 6061 aluminum. This process consists of electrochemically forming a highly porous layer of aluminum oxide Al_{2}O_{3} on the metal surface in an H_{2}SO_{4} solution. The surface is then “sealed” by immersion in hot water. This hydrates the aluminum oxide causing the pores to swell, thus closing them. The sealing process is left incomplete in order to avoid an unsightly white powder on the surface. The sealed and anodized surface is still highly porous and easily absorbs fluids. It was hypothesized that the observed float weight gain was due to continued reactions between the anodized aluminum and seawater.

A series of small hard-anodized 7075 and 6061 coupons (i.e., pieces of metal) were placed in 1-L dark bottles filled with deionized water, artificial seawater, or boiled seawater from Puget Sound. The coupons were dried and weighed periodically for times up to 13 months. All anodized coupons gained weight with time. Unanodized 6061 coupons or painted coupons did not gain weight. The rates of weight gain were generally less than that observed in Fig. 8 on a per area basis, but could be increased dramatically by increasing the pH of the water above about 8. The rate of weight gain decreased with time while the pH of the water typically decreased with time to about 7.6. The weight gain was larger in real seawater than in artificial seawater. These observations suggest that the coupons altered the pH of the water and thus decreased the rate of weight gain over time. The weight gain in seawater was often accompanied by hard white deposits on the coupons and white precipitate in the water. Although these tests did not identify the mechanism of weight gain they clearly showed that anodized aluminum gains weight in water and that this can be prevented by painting or otherwise sealing the surface. Our subsequent experience with floats (section 5h) shows that painted floats gain weight at rates much less than shown in Fig. 8.

### e. Thermal expansion

It has been difficult to make direct measurements of the thermal expansion coefficient of the floats. However, since the floats are mostly constructed of aluminum alloy with small amounts of other metals, the thermal expansion coefficient of the float can be estimated from the known expansion coefficients of its external components. The major uncertainties are due to the contribution from plastics, used as potting for transducers and antennas, which typically have thermal expansion coefficients 2–5 times larger than that of aluminum with significant uncertainties. Estimated thermal expansion coefficients are 7.27 ± 0.1 × 10^{−5} °C^{−1} for DLF and 7.41 ± 0.15 × 10^{−5} °C^{−1} for MFLII. The uncertainty is comparable to the variation between different MLFII models. These values are 3%–4% larger than that of aluminum.

### f. Water properties

Accurate estimation of a float's buoyancy requires that the density of the surrounding water be known accurately. For a 15-L (50 L) float, an error of 0.01 kg m^{−3} in density results in a 0.15-g (0.5 g) error in buoyancy. The CTD sensors measure density with an accuracy that does not exceed 0.001 kg m^{−3}. Thus absolute ballasting of a float to accuracies significantly better than grams requires high quality CTD measurements. MLFII uses a SeaBird SBE41 pumped CTD designed for use on profiling floats. These sensors are specified to 0.005 psu accuracy. In practice, they are often stable to better than this, based on comparision of the two CTDs. However, they are subject to occasional jumps in conductivity calibration of up to 0.5 kg m^{−3} equivalent, probably due to injestion and ejection of a millimeter-sized plankton. Typically, the salinity becomes noisy for a period of hours to days and then becomes stable at a new calibration.

Additional error is caused by the placement of CTD sensors on the float. Practically, these must be located on the endcaps. The float buoyancy, however, is controlled by the volume-averaged density over the entire float, which will be close to the density at the center of the float. For a 0.5-m separation of the density sensor from the float center and a stratification *N* = 0.01 s^{−1}, the offset of the density sensor results in an error of 0.005 kg m^{−3} in density and a 0.25-g error in buoyancy for a 50-L float. These errors are not constant, but vary as the local stratification is stretched and compressed by the passage of internal waves. These variations are comparable to the mean stratification. A simple but expensive solution to this problem is to use two CTD sensors, one on each endcap, and average the values to estimate the central value. Sherman and Pinkel (1991) present a statistical model of the meter-scale variations in stratification. Using this model the use of two CTDs reduces the mean bias in density to negligible values. However, it only reduces the rms fluctuation in density error by half, to about 0.003 kg m^{−3}.

### g. Buoyancy control

The float's buoyancy is controlled by extruding a piston in and out of the bottom of the float using a motor-driven leadscrew system. This changes the volume of the float. The extruded volume must be known to an accuracy of about 0.1 cc. For DLF the piston was quite small, with a 1.58-cm diameter and a maximum volume of 30 cc. This was later increased to 58 cc, or about 4 kg m^{−3}, by increasing the piston diameter. For this system an accuracy of 0.1 cc requires about 0.1% accuracy in positioning, which was achieved using a linear resistance potentiometer read to 12-bit accuracy. For MLFII, the piston is used for both buoyancy control and to bring the float to the surface. This requires much more volume change. A 5-cm-diameter piston is used with a total volume change of 750 cc. No more than 400 cc, or about 8 kg m^{−3}, is available for buoyancy control. The larger range implies a much higher positioning accuracy to retain 0.1-cc precision. Accordingly, an optical counter on the motor is used, along with a zeroing switch. Intelligent use of this buoyancy control is discussed in sections 4 and 7.

### h. Field verification

#### 1) DLF compressibility

Figure 6a shows the depth of seven DLFs deployed in the central Labrador Sea in January 1998 (Steffen and D'Asaro 2002). From days 1 to 6 the floats adjusted their volume to seek a pressure of 1000 db (see section 4b). They then increased their volume in order to rise into the convective layer. The convective layer, however, was shallower than expected and the floats instead remained in the stratified water beneath it for about 25 days before being entrained. During this period the floats' density matched that of the stratified water in which they floated. Equation (1) with *b* = 0 is therefore used to compute the residual Δ*b* as shown in Fig. 6b. The water density is computed from any of seven CTDs taken at the time of the floats' deployment. The results are insensitive to the CTD profile used. Once floats are entrained into the convective layer, the initial CTD profiles are not appropriate and the value of Δ*b* is invalid. The data are plotted as gray in Fig. 6b. The thermal expansion coefficient *α* = 7.27 × 10^{−5} °C^{−1} is set as described above; there is little temperature change in the water column so the results are insensitive to the value of *α.* The reference volume *V*_{0} is set by assuming zero buoyancy at day 5.7.

The difference in pressure between the end of the autoballast period (day 6) and the start of the scientific data (day 7) provides an accurate estimate of the compressibility *γ.* The value *γ* = 3.62 ± 0.02 × 10^{−6} db^{−1} is chosen to provide the smoothest curve of weight change across the transition. This value is about 3% less than that estimated from the tank measurements (see section 5a). The reason for the difference is not understood.

#### 2) DLF creep

The estimated value of Δ*b* in Fig. 6b increases monotonically. Either the floats' masses are increasing or their volumes are decreasing. During the first few days, Δ*b* decreased exponentially with an *e*-folding time of 0.92 days for a total change of about 2.5 g. This rapid initial change was the main reason for the long initial autoballasting period. Thereafter, Δ*b* increased steadily at a rate of about 1 g (70 days)^{−1}, 10^{−6} day^{−1}, about 7% of that seen in Fig. 8. Painting the float hulls has greatly decreased the amount of weight gain. There was also a transient in weight and pressure from days 7 to 10 as the float settled into its new depth. The causes of this are not known, but may be due to O-ring adjustment as discussed in section 5c.

#### 3) MLFII air

MLFII number 6 was deployed in the northeastern Pacific on day 270.8 of 2000. It carried a Doppler sonar that measured the water velocity relative to the float and CTDs at the top and bottom of the float. The mission concentrated on measuring turbulence in the upper ocean boundary layer. When the float's buoyancy is nonzero, it will move relative to the water. The sonar can therefore be used to select times when the float is not moving relative to the water and is therefore most likely to have the same density as the water. The three sonar beams project upward from the top of the float (Fig. 2). The sum of the velocities along the beams, corrected for beam angle, gives the velocity component along the main axis of the float. This “along-float” velocity will be used to select times of low relative motion between the float and water.

The sonar was used to select times when the 1000-s average of along-float velocity was less than 1 mm s^{−1} and the potential density difference between the top and bottom of the float was less than 0.1 kg m^{−3}. Times of high stratification were not used because the average density of the water surrounding the float was not well known. At all chosen points, the float buoyancy was computed using (1) assuming *α* = 7.41 × 10^{−5} °C^{−1} estimated from the float components, *γ* = 3.3 × 10^{−6} db^{−1} estimated as described above, and nominal values of *M* and *V*_{0}. The value of *V*_{air} was assumed zero. The resulting buoyancy is plotted against pressure in the top panels of Fig. 9.

On days 271–279, the computed float weight increased near the surface, consistent with an unmodeled upward buoyancy. The solid curves in each panel show a fit of this buoyancy to the *V*_{air} term in (1). The model usually fits the data well. The bottom curve shows the estimated air as a function of time. Immediately after deployment *V*_{air} is large, with the first good value giving *V*_{air} = 20 cc on day 272. The value decays with an *e*-folding time of about 4 days (solid curve in bottom panel) and becomes a few cubic centimeters, barely distinguishable from zero, after day 280.

In contrast, MLFII deployments 8 and 10 off Oregon analyzed in a similar manner, found asymptotic “air” concentrations of 10 and 7.5 cc, respectively. These floats were much shallower than float 6 with respective maximum depths of 70 and 45 db.

Some additional tank experiments were conducted to understand the interaction between pressure and “air.” MLFII 14 was placed in the freshwater tank and weighed continuously for 4 days. For 8 h each day it was cycled from 10 to 140 db; during the rest of the day the pressure was constant at 80, 130, 50, and 0 db. The value of *V*_{air} decreased from 18 to 9 cc in the first 3 days, but increased to 10 cc during the last.

These data indicate that the behavior of *V*_{air} can be characterized by an exponential decay with a timescale of many days to an asymptotic level whose level depends on pressure. If the pressure is sufficiently large, perhaps 150 db, *V*_{air} can decrease to 10% of its initial value and become nearly negligible. This behavior is perhaps due to the diffusion of gas out of O-ring cavities or other components, with perhaps an additional component due to slow deformation of O-ring rubber. An understanding of the details is likely to require considerably more data and analysis.

#### 4) MLFII creep

In the fall 2000 deployments MLFII did a “settling” maneuver once per day, descending to 160–190 db, folding the drogue, fixing the ballasting piston, and letting itself settle onto an isopycnal (see Fig. 5). The heavy dots in Fig. 10 show the value of Δ*b* computed assuming *b* = 0, *α* = 7.41 × 10^{−5} °C^{−1}, *γ* = 3.3 × 10^{−6} db^{−1}, and nominal values of *M* and *V*_{0}. The value of Δ*b* increased by 4 g over the 41 days of measurement, or 2 × 10^{−6} day^{−1}, about twice that in Fig. 6 on a per volume basis.

#### 5) MLFII compressibility

The light dots in Fig. 10 are the same points shown in Fig. 9, but compensated for the buoyancy of air using *V*_{air} = 20*e*^{−(Td−272)/4} cc. The 1-day median filter of this data is shown by the dashed line. This provides an estimate of the near-surface float buoyancy. Typical temperatures are 12°–18°C. The heavy dots show the buoyancy after “settling.” Pressures are 160–190 db and temperatures are near 6°C. The difference between the shallow and deep buoyancy is small indicating that the values of *γ* and *α* used to compute the weight are correct. That is, (1) can be used both at 170 db and at 20 db with a single, time-varying value of Δ*b* and constant values of *γ* and *α.*

The parameters in (1) were varied to minimize the difference between the deep (dots) and shallow (lines) data in Fig. 10. The data do not constrain both *α* and *γ,* only their ratio. Fortunately, *α* is constrained to better than 10% by the float materials (see section 5e), which constrains *γ* to 3.2–3.35 × 10^{−6} db^{−1}. Similar analysis on floats 8 and 10, neither of which went as deep as float 6, yield 3.35–3.45 × 10^{−6} db^{−1} and 3.75–3.95 × 10^{−6} db^{−1}, respectively. Ballasting tank measurements on float 14 described in section 5h(3) yield 3.74–3.77 × 10^{−6} db^{−1}. The consistency between the float 10 and 14 measurements, both of which had the same sensor and mechanical configuration, suggests that the differences with earlier measurements are due to changes in the float design. Highly reliable values of *γ* will require more measurements.

## 6. Drag

Floats tend to be Lagrangian both because they are accelerated by the same pressure gradients as the water and because drag between the float and the water tends to minimize the difference in their velocity. It is thus important to understand the drag law for floats. The usual assumption is that float's vertical velocity *W* relative to the water is related to its buoyancy *b* by a quadratic drag law

where *g* is the acceleration of gravity and *C*_{D}*A* is an effective cross-sectional area of the float that includes both the true area and the drag coefficient. For a homogeneous fluid this relationship should apply as long as the Reynolds number Re = *WA*^{0.5}/*ν,* where *ν* is the viscosity of water, is greater than about 100. For *A* = 1 m^{2}, *W* = 10^{−3} m s^{−1}, and *ν* = 10^{−6} m^{2} s^{−1}, Re = 1000. Even for very small float velocities the Reynolds number remains large:

For a stratified fluid additional drag is caused by internal waves (iw) as first discovered by Larson (1969). Torres et al. (2000) find the drag on a sphere of radius *r* moving vertically through a stratified fluid for Re > 100 to depend on the Froude number F = *W*/*Nr.* For large F (3) applies. For F < 1 internal wave drag (*C*_{iw}) becomes important. In this regime the Fig. 11 of Torres et al. (2000) shows that *C*_{D} = *C*_{iw}/F, with *C*_{iw} = 4.2. The drag law is linear:

For *N* = 0.01 s^{−1} and *r* = 1 m, F = 1 for *W* = 0.01 m s^{−1}. Internal wave drag should become the dominant cause of drag on a float for speeds less than about 0.01 m s^{−1}.

Figure 11 shows the relationship between buoyancy and vertical velocity for an MLFII deployed off Oregon in June/July 2000. This float changed *B* with the goal of remaining on the *S* = 33.3 isohaline. Instead, it executed a slow oscillation around this isohaline with a period of about 12 h. Simulation studies revealed the following about the float's behavior. The control algorithm was designed assuming (3). The actual drag was much larger, similar to (4), leading to a resonance in the algorithm at close to the M_{2} tidal frequency. This resonance was excited by the internal tidal strain acting on the difference between the float center of buoyancy and the location of the CTD sensor, approximately 0.7 m below the center of buoyancy. This understanding allowed revised parameters to be transmitted to the float during the mission, which eliminated the oscillation. Fortunately, the oscillation also produced several weeks of data with very small *b* and *W,* allowing the accurate estimation of the drag law at low speed. Although the computed *W* are absolute, not relative to the water, there is sufficient data to average out the oscillatory internal wave motions, which are the primary source of vertical water motion. Much larger values of *b* and *W* are obtained during the float profiles. Both high and low speed data are shown in Fig. 11a with a detailed inset, Fig. 11b, for low speeds.

For |*W*| > 0.1 m s^{−1} the drogue is always folded. The drag law is approximately quadratic as shown by the dashed lines. Different values of *C*_{D}*A* are required for upward and downward motion because the folded drogue faces upward and tends to flair outward when the float is moving downward. (Note that the drogue folds downward in Fig. 2; this is a more recent design.) For |*W*| < 0.01 m s^{−1} the drogue is always open. The drag law is linear down to the smallest resolved velocities, about 5 × 10^{−4} m s^{−1}. Using the *C*_{D}*A* = 0.8 m^{2}, appropriate for the open drogue and *C*_{D} = 1, *r* = 0.5 m, and *N* = 0.015 s^{−1}, a typical value for this deployment, *C*_{iw} = 4.0, unreasonably close to that found by Torres et al. (2000). The drag at low speed is linear and appears to be due to internal waves. The solid line in Fig. 11 is a hybrid drag law, consisting of the sum of the drag in (3) and (4). This appears to model the drag across both regimes adequately.

Float performance within nearly unstratified turbulent boundary layers will also depend on the drag law in this environment. It seems likely that the drag in a turbulent fluid is larger than that in an unstratified laminar fluid since the “eddy viscosity” due to small-scale turbulence will force the falling float to accelerate more of the surrounding fluid. Presently, there are no data to confirm this idea.

## 7. Performance

### a. Overview

Ideally, a Lagrangian float should follow the three-dimensional velocity of a viscous-scale water parcel. Real floats do this imperfectly. The most important imperfections are buoyancy of the float relative to the water, the motion of the float relative to the water resulting from this buoyancy, the finite size of the float, and the imperfect location of sensors mounted on the float. The ability of the float to make useful Lagrangian measurements depends both on these imperfections and their interaction with the fluid flow. Different problems dominate at different timescales.

### b. Internal wave and turbulent timescales

One important application of Lagrangian floats is the measurement of oceanic mixing rates. In this case, the floats need to be Lagrangian on the relatively short timescales associated with mixing processes. On longer timescales, the floats need only remain in the location where the processes of interest occur.

#### 1) Stratified ocean

For a density-stratified ocean float, ballasting is not a difficult problem. In the absence of other factors, the float will settle to the level where its density matches that of the seawater. Small errors in the float density cause the float to settle onto a slightly different density surface. A simple control algorithm can be used to slowly move the float back to a target isopycnal by moving the ballasting piston. This also compensates for changes in float density due to biofouling, corrosion, or other factors.

The finite size of the float will be an important limiting factor in a stratified ocean, because the overturning scales of the turbulence are limited by the stratification to approximately the Ozmidov length *L*_{Oz} = (ɛ*N*^{−3})^{1/2} = (5*K*_{ρ}/*N*)^{1/2}. Here we have used the Osborn (1980) expression for *K*_{ρ} = 0.2ɛ/*N*^{2}. For the typical upper-thermocline values, *N* = 5 × 10^{−3} s^{−1} and *K*_{ρ} = 10^{−5} m^{2} s^{−1}, *L*_{Oz} = 0.1 m, much smaller than the size of the float, 2*L* = 1 m. The same result can be expressed in terms of Lagrangian frequency (Lien et al. 1998). The frequency at which float size effects become important is *ω*_{L} = (ɛ/*L*^{2})^{1/3}. The “large-eddy” frequency is *ω*_{0} = 0.5*N* in a stratified fluid (D'Asaro and Lien 2000) and *ω*_{L} = 0.3*N* (Lien et al. 1998), which is less than *ω*_{0}. Under these conditions, the Lien et al. (1998) float response functions attenuate the float velocity spectra by a factor of about 3 at *ω*_{0}. The inertial subrange, extending from *ω*_{0} to *ω*_{N}, cannot be measured. The overturning scales of turbulence in the main oceanic thermocline are too small to be reliably measured by 1-m-size floats.

Floats become most useful only at much higher mixing rates (and/or smaller *N*). For *K*_{ρ} = 10^{−2} m^{2} s^{−1}, ɛ = 10^{−6} W kg^{−1}, *L*_{Oz} = 3 m, and *ω*_{L}/*ω*_{0} = 7; the Ozmidov scale is larger than the float and an inertial subrange a decade long can be measured with only a weak reliance on the float response function. Thus, for example, floats appear capable of measuring mixing rates in the equatorial undercurrent, a stratified, but strongly mixing region (Lien et al. 2002).

Mixing in a stratified fluid will displace fluid parcels (and floats) away from their initial level and mix them away from their initial density. A Lagrangian float will therefore find itself in water of a different density after a mixing event. If the float is programmed to be isopycnal, it will retain its initial potential density after a mixing event and therefore be lighter or heavier than the surrounding water by an amount *N*^{2}*ζM,* where *ζ* is the vertical displacement. It will return to its target isopycnal at a rate governed by the drag law (4), where *W* = *dζ*/*dt.* For a linear stratification and in the absence of other forcing, *ζ* will decay exponentially with a time constant *τ* = *N*^{−1}*C*_{iw}*Ar*/*V.* The high drag of these floats prevents the vertical oscillations around their rest isopycnal described by Goodman and Levine (1990); such oscillations are never seen. Thus the float is Lagrangian for frequencies much greater than

and isopycnal for frequencies much less than this. For a spherical float *ω*_{LG} = *N.* For Lagrangian floats *ω*_{LG}/*N* is much smaller because of the large drogue, about 0.03 for MLFII and perhaps as low as 0.005 for DLF. The large internal wave drag at low speeds causes isopycnal Lagrangian floats to return to their isopycnal at rates much slower than *N*^{−1}. Since turbulent mixing occurs primarily at frequencies greater than *N,* sufficiently small floats will be Lagrangian rather than isopycnal at turbulent frequencies.

#### 2) Unstratified boundary layers

Lagrangian floats were originally designed for use in turbulent oceanic boundary layers and generally work well in this environment. The turbulent scales in boundary layers are almost always larger than the float so that the float's motion can accurately reflect the behavior of the larger eddies in the boundary layer. Since these eddies carry most of the fluxes and energy, floats can often provide useful estimates of these turbulent quantities (Harcourt et al. 2002).

The primary difficulty in operating floats in the upper ocean boundary layer is achieving and maintaining neutral buoyancy. Heavy floats tend to episodically sink out of the boundary layer and inhabit the stratified region at the mixed layer base. Light floats tend to oversample the near-surface region. D'Asaro et al. (2002) show how this also causes the floats to oversample the largest downward-going plumes and thus bias turbulent statistics computed from the floats at all depths. Techniques for achieving neutral buoyancy (see section 4) determine the float's volume either by allowing it to settle to equilibrium or from tank measurements, and then compute the volume necessary for neutral buoyancy using (1).

Table 1 summarizes the present uncertainties in the coefficients in terms of their contribution to the float volume. The largest uncertainty is due to the unknown and variable amount of trapped “air.” If the float is sufficiently deep, this effect probably decreases with time to a few grams at most. The float mass *M* is only know to a few grams and the reference volume *V*_{0}, determined from *M* at ballasting, has a similar uncertainty. However, float buoyancy is only sensitive to the ratio of *V*_{0} and *M,* not their exact values. “Creep” effects are small if the float is reballasted every few days; otherwise they can be important. The remaining uncertainties in buoyancy due to *γ* and *α* are less than 1 g as long as the changes in pressure and temperature are not too large. The uncertainties could be reduced further by more tank experiments and/or short float missions designed to measure the coefficient values. It appears that under favorable conditions float properties can be known well enough to maintain the buoyancy of floats in the upper ocean boundary layer to 1 g or less.

### c. Long timescales

#### 1) The challenge

At present the Lagrangian floats are operated mostly as isopycnal-Lagrangian or mixed layer floats. Misadventures with more complex schemes are discussed in section 4c. A more challenging task for floats would be to remain Lagrangian on longer timescales in the stratified ocean. This would allow, for example, the diapycnal velocity to be measured from the rate that floats cross isopycnals. The floats would need to remain Lagrangian for times ranging from many hours, for the average deepening of a mixed layer in a storm or the net tidal mixing in an estuary, to months, for the diapycnal velocity in the Equatorial Undercurrent, to years, for the mean upwelling of the thermocline.

A float will be Lagrangian if its density matches that of the seawater surrounding it and if it is sufficiently small. The size issue has been addressed above. Because the existing floats measure density, it should be possible to program them to change their density to match that measured by their density sensors and thus, in principle, produce a nearly Lagrangian float. There are several significant difficulties in implementing this idea.

The observed increase in float weight (creep; Table 1) will cause the float to cross isopycnals. For MLFII, 1 cc is equivalent to 0.02 kg m^{−3} density change. The observed rate, 0.1 cc day^{−1}, is probably negligible on a 1-day timescale, but will become increasingly important at longer times.

#### 2) Instability

As the density of the water changes, the float must use its equation of state (1) to adjust its buoyancy. Errors in the equation of state will result in errors in the new buoyancy. Consider a float programmed to be Lagrangian at rest in a quiescent ocean. If it is displaced upward it will adjust its density to match that of the water at the new level and not return to its initial level. If the equation of state used to make this adjustment is in error, the float may be slightly buoyant at the new level, causing it to continue to move upward. This is an instability. If the buoyancy adjustment is decreased slightly, the float will be stable. Thus a truly Lagrangian float exists at the edge of instability and is highly sensitive to imperfections in its equation of state.

Consider, for example, a float that obeys (1) has perfect measurements of density, pressure, and temperature; and suffers from no creep, but uses a value of *γ* that results in the float being a fraction ɛ more compressible than seawater. Using the drag law (4) a float will exponentially accelerate away from its initial location with an *e*-folding time of

For ɛ = 1%, *N* = 0.01 and the parameters of MLFII *τ*_{unstable} are about 8 days. The initial *e*-folding time may be longer than this due to the low Reynolds number at the initial small velocities.

#### 3) Diffusion

Another manifestation of the sensitivity of a truly Lagrangian float will be its tendency to diapycnally diffuse in response to random perturbations. The most obvious perturbation is due to imperfect measurement of the surrounding water's density. The float's buoyancy is due to an average density of the surrounding water. Even with two CTDs internal wave strain limits the accuracy to which the average density of the water surrounding the float can be estimated to about *δρ* = 0.003 kg m^{−3} (see section 5f). If, for example, the measured density is *δρ* heavier than the average density, the float will incorrectly increase its density and sink. The sum of such random perturbations will cause the float to diffuse vertically away from its initial position.

Consider for example a float that obeys (1) and (4). For MLFII a density error of 0.003 kg m^{−3} will cause a vertical velocity of *δW* = 4 × 10^{−5} m s^{−1} using (4). This may, in fact, be too large because of the low Reynolds number at these small velocities. The Lagrangian correlation time *τ*_{str} of internal wave strain is not well known, but is probably larger than *N*^{−1} and smaller than *f*^{−1}. Taking 1000 s as a rough guess, the float will diffuse at a rate *δW*^{2}*τ*_{str} = 10^{−6} m^{2} s^{−1}. This number is smaller than the average diapycnal diffusivity of the thermocline, suggesting that internal wave strain will not limit the performance of a Lagrangian float. Other sources of random perturbation may be more important and cause float diffusion to be much larger.

## 8. Summary

Over the past decade neutrally buoyant floats designed to follow the three-dimensional water trajectories have been developed. The present models, described in this paper, are autonomous, with durations of months, can alternate between Lagrangian and profiling modes, relay data via two-way satellite communications, and can carry a large suite of sensors. A novel hull design is light, strong, and has a compressibility close to that of seawater.

Floats yield the most information about ocean circulation and mixing if they follow water motions in three dimensions. The key to doing this is to accurately match the density of the floats to that of the water, understand and compensate for factors that change the float's buoyancy, and understand the effect of float imperfections on float motion. Recent work has revealed the following.

Anodized aluminum gains weight in seawater due to reactions between its surface and the seawater. The rate of weight gain is much faster for the first few days and then becomes linear. The weight gain is of order 0.5 g day

^{−1}m^{−2}. The rate is sensitive to surface treatments and water pH. Properly painting the surface eliminates this effect.At low pressure the buoyancy of floats with O-ring seals varies as if bubbles of air were being compressed. The volume of “air” needed to describe the compressibility decreases exponentially after deployment with a decay time of several days to a value that depends on pressure.

At high pressure the float buoyancy varies linearly and can be modeled with a compressibility that is constant with time and pressure. The compressibility can be measured to an accuracy of a few percent in laboratory tanks.

The drag of floats moving slowly through a stratified ocean is dominated by internal wave generation and is thus linear, not quadratic. Internal wave drag becomes important when the Froude number F =

*W*/*Nr*is less than one. Here*W*is the vertical velocity of the float relative to the water,*N*is the stratification, and*r*is the radius of the float or float drogue. For F ≪ 1 the drag is given by (4).A float with a constant potential density (i.e., an isopycnal float) displaced from its rest isopycnal and acted upon by internal wave drag will return to its rest isopycnal with an exponential decay time of

*τ*=*C*_{iw}*Ar*/*VN,*where*C*_{iw}= 4,*A*is the frontal area of the float, and*V*is its volume. For typical floats with meter-sized drogues this time is about 30/*N.*An isopycnal float in a stratified fluid will follow water motions accurately in three dimensions if they are larger than the float and have frequencies less than

*ω*_{LG}=*τ*^{−1}. Turbulence in the stratified ocean is dominated by frequencies greater than*N.*Water motions that are slower than*ω*_{LG}will be accurately followed by an isopycnal float only if they are adiabatic.

These floats have proven most useful in measuring the turbulence in ocean boundary layers and other regions of strong turbulence, where the ability of the floats to be Lagrangian on short timescales matches the short timescale of the processes and where the scales of the turbulence are larger than those of the float. On longer timescales, the floats successfully operate as isopycnal floats. The timescales over which the floats are Lagrangian can probably be extended by continuously matching the float's density to that of the water using the onboard density measurements. Such a scheme is highly sensitive to perturbations and may easily become unstable or produce spurious isopycnal diffusion of the float. Although our initial calculations indicate that these effects are not large, great care is warranted.

## Acknowledgments

This work was supported by NSF Grants OCE 9711650 and OCE 011741 and ONR Grants N00014-94-1-0024 and N00014-94-1-0025. The engineering and support staff at the APL-UW has been essential in conducting this work, as have the crews of numerous research and commercial vessels.

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## Footnotes

*Corresponding author address:* Dr. Eric A. D'Asaro, Applied Physics Lab, University of Washington, 1013 NE 40th Street, Seattle, WA 98105. Email: dasaro@apl.washington.edu