The Ocean Boundary Layer Below Hurricane Dennis

MARCH 2003 D'ASARO 561

The Ocean Boundary Layer below Hurricane Dennis

ERIC A. D'ASARO Applied Physics Laboratory and School of Oceanography, University of Washington, Seattle, Washington

(Manuscript received 4 February 2002, in ®nal form 3 September 2002)

ABSTRACT Three neutrally buoyant ¯oats were air deployed ahead of Hurricane Dennis on 28 August 1999. These ¯oats were designed to accurately follow three-dimensional water trajectories and measure pressure (i.e., their own depth) and temperature. The hurricane eye passed between two of the ¯oats; both measured the properties of the ocean boundary layer beneath sustained 30 m sϪ1 winds. The ¯oats repeatedly moved through a mixed layer 30±70 m deep at average vertical speeds of 0.03±0.06 m sϪ1. The speed was roughly proportional to the friction velocity. Mixed layer temperature cooled about 2.8Њ and 0.75ЊC at the ¯oats on the east and west sides of the northward-going storm, respectively. Much of the cooling occurred before the eye passage. The remaining terms in the horizontally averaged mixed layer heat budget, the vertical velocity±temperature covariance and the Lagrangian heating rate, were computed from the ¯oat data. Surface heat ¯uxes accounted for only a small part of the cooling. Most of the cooling was due to entrainment of colder water from below and, on the right-hand (east) side only, horizontal advection and mixing with colder water. The larger entrainment ¯ux on this side of the hurricane was presumably due to the much larger inertial currents and shear. Although these ¯oats can make detailed measurements of the heat transfer mechanisms in the ocean boundary layer under these severe conditions, accurate measurements of heat ¯ux will require clusters of many ¯oats to reduce the statistical error.

1. Introduction cold deep water into the mixed layer (Price et al. 1994; Jacob et al. 2000). The mixing is caused by strong shears Hurricanes draw their energy from the sensible and in the upper ocean forced by the hurricanes winds. latent heat supplied by warm ocean waters. Their in- Emanuel (1999) shows that simple models of hurricane tensity is therefore highly sensitive to the sea surface intensity that include this feedback produce greatly im- temperature (SST). Emanuel (1999) notes that proved predictions compared to those that do not. . . . there has been . . . little advance in predictions of Despite this, there are only a few observations of the [hurricane] intensity (as measured, for example, by max- upper ocean beneath a hurricane. Such measurements imum surface wind speed), in spite of the application of are dif®cult to make due to the large surface waves, sophisticated numerical models. The best intensity fore- winds, and currents, as well as the highly intermittent casts today are statistically based. Most of the research and unpredictable locations of hurricanes. Targeted literature on hurricane intensity focusses on the pre-storm studies (Sanford et al. 1987; Shay et al. 1992) have sea surface temperature and certain properties of the at- generally used air-deployed pro®lers. Other measure- mospheric environment. . . . This remains so, even ments have resulted from long-term observations that though it is well known that hurricanes alter the surface have been fortuitously overrun by hurricanes (e.g., temperature of the ocean over which they pass and that Dickey et al. 1998). Here, we describe the ®rst use of a mere 2.5 K decrease in ocean surface temperature near air-deployed neutrally buoyant ¯oats to observe mixing the core of the storm would suf®ce to shut down energy beneath a hurricane. production entirely. Simulations with coupled atmo- Another major uncertainty in hurricane prediction re- sphere±ocean models con®rm that interaction with the sults from the poor understanding of air±sea heat ¯uxes ocean is a strong negative feedback on storm intensity. at high winds. Extrapolation of existing parameteriza- tions to hurricane force winds results in heat ¯uxes that Cooling of the upper ocean by hurricanes, or the lack are insuf®cient to sustain a hurricane (Emanuel 1999). thereof, is important to hurricane dynamics and predic- The additional ¯uxes may result from the evaporation tion. The cooling results primarily from the mixing of of spray (Andreas 1998). Direct measurements of air± sea ¯uxes under hurricane conditions on the air side of the interface may be dif®cult, particularly because sen- Corresponding author address: Dr. Eric A. D'Asaro, Applied Phys- ics Laboratory, University of Washington, 1013 NE 40th St., Seattle, sible, latent, and spray components must be measured. WA 98105. Here, we report on heat ¯ux measurements made on the E-mail: [email protected] water side of the interface.

᭧ 2003 American Meteorological Society

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Furthermore, the hurricane environment offers a Turbulent velocity ¯uctuations in the oceanic bound- unique opportunity to study the physics of upper-ocean ary layer under a hurricane are many centimeters per turbulence. This is important because turbulence in the second, much smaller than the meter per second veloc- ocean boundary layer is forced by a combination of ities of the surface waves. This presents a formidable wind stress, surface waves, surface buoyancy ¯ux, and measurement challenge. However, because pressure shear. However, it is often dif®cult to separate the effects ¯uctuations are zero along particle paths for linear sur- of waves from those of wind because the wave prop- face waves and surface waves are nearly linear, the sur- erties are often highly correlated with the wind. Hur- face wave pressure ¯uctuations measured by Lagrangian ricane winds change in both magnitude and direction ¯oats are greatly attenuated from their Eulerian values so rapidly that the surface waves are far from equilib- (D'Asaro et al. 1996; D'Asaro 2001). This allows ac- rium (Wright et al. 2001). Furthermore, the hurricane curate measurements of vertical velocity to be made forcing is so strong that the effects of preexisting oce- from pressure measured on Lagrangian ¯oats. anic currents and swell on the boundary layer turbulence These ¯oats are not perfectly Lagrangian; they do not are diminished. Under these conditions, it may be much follow water parcels exactly. There are two major sourc- easier to untangle the interactions between wind, waves, es of error. First, the ¯oats are not perfectly neutrally and upper-ocean turbulence. This paper presents initial buoyant. For the ®rst day after the ¯oat is deployed it steps toward this goal. adjusts its buoyancy to make itself neutrally buoyant in Section 2 describes the Lagrangian ¯oats used in these the mixed layer (see DAS for a detailed discussion). measurements, and the methods by which air±sea ¯uxes Scienti®c data collection starts only after this operation were estimated from operational hurricane data. Section is complete. Even if the ¯oats are exactly neutral when 3 describes Hurricane Dennis and the basic properties they ®rst enter the mixed layer, the rapid cooling of the of turbulence observed beneath it. Section 4 describes mixed layer during the hurricane will cause them to the methods of extracting heat ¯ux pro®les from La- become lighter. A 2ЊC cooling will produce about 8 g grangian ¯oat data. Section 5 applies these methods to of buoyancy. The ¯oat compensates for this effect using the data to produce estimates of the surface and en- the measured temperature and the known expansion co- trainment heat ¯uxes. Section 6 places these observa- ef®cients of seawater and its aluminum hull. It cannot tions in the context of the issues raised above and sum- compensate for changes in the mixed layer density due marizes the results. to salinity caused, for example, by entrainment of water into the mixed layer. Typical ¯oat buoyancies are a few grams. The effect 2. Instrumentation and methods of this buoyancy on the ¯oat motion is greatly reduced by a circular cloth drogue with a frontal area of about a. Lagrangian ¯oats 1m2. With the drogue open, a buoyancy of 5 g results in an upward motion of 5 mm sϪ1 relative to the water Measurements of water trajectories and the temper- assuming a quadratic drag law (DAS). Without mea- ature along these trajectories were made using high- surements more detailed than are available on a DLF, drag, neutrally buoyant ¯oats [Deep Lagrangian Floats the buoyancy of a ¯oat cannot be directly determined. (DLFs)] constructed at the Applied Physics Laboratory, However, light ¯oats tend to concentrate near the ocean University of Washington. D'Asaro (2003, hereinafter surface; heavy ¯oats tend to concentrate near the bottom DAS) describes the construction and performance of of the mixed layer. Therefore, the histogram of ¯oat these ¯oats in detail. Three ¯oats were air-deployed depth can be used as a diagnostic for the ¯oat buoyancy ahead of the hurricane and gathered data for 4 days. (DAS). Pressure was measured every 20 s with an accuracy of The second major source of ¯oat error results from 0.1 dbar from which depth, vertical velocity, and ac- the approximately 1-m size of the ¯oat. This is much celeration were computed. Temperature was measured larger than the smallest scales of turbulence but smaller every 20 s to an accuracy of approximately 1 mK using than the largest scales, which are comparable to those the electronics and thermistor from a Seabird CTD. At of the mixed layer itself. The ¯oat's ®nite size reduces the end of the mission, the ¯oats dropped a weight, its response to velocity ¯uctuations smaller than itself surfaced, and relayed their data via the ARGOS satellite as quanti®ed by Lien et al. (1998). Fortunately, the larg- system. er scales contain most of the energy so that the ¯oat Data packets, once converted to pressure and tem- measures vertical velocity accurately as long as the perature data, required little additional processing. A mixed layer is substantially larger than the ¯oat size. small number of missing data points were linearly in- This is probably also true for vertical ¯uxes, although terpolated onto the 20-s sampling grid. The measured this is less well quanti®ed. pressure was corrected for the changes in atmospheric A similar error results from the placement of the pres- pressure associated with the hurricane; these corrections sure and temperature sensors on the top of the ¯oat, are small, about 0.4 dbar at the most, but signi®cant approximately 0.3 m above the center of buoyancy and when examining ¯oat trajectories very near the surface. 0.6 m above the drogue. These sensors therefore do not

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follow a Lagrangian trajectory even if the center of the Qhhϭ CU⌬T␳airCp, (1) ¯oat does. where U is the wind speed, ␳air is the density of air, Cp Ϫ3 is its heat capacity, and Ch ϭ 1 ϫ 10 is the transfer b. Hurricane Dennis deployments coef®cient. Note that Price et al. (1996) use Ch ϭ 1.3 Ϫ3 Ϫ3 Hurricane Dennis was the third hurricane of the 1999 ϫ 10 while Jacob et al. (2000) use Ch ϭ 1 ϫ 10 . season (Lawrence et al. 2001). It was a larger-than- There are no direct measurements of ⌬T, the air±sea average hurricane with a large and poorly formed eye temperature difference. Cione et al. (2000) compile data structure (Fig. 1). It formed over the eastern Bahamas from many hurricanes and ®nd the average ⌬T ϭ 2.5 on 24 August and reached hurricane strength on the Ϯ 1.5ЊC within a few degrees of the storm center. Price 26th. Maximum strength (90 kts, 962 hPa) occurred 28± et al. (1994) use 3ЊC. The average values from Cione 30 August as Dennis moved northward just east of Flor- et al. (2000) are used in (1). ida. The storm then turned northeast, stalling off North Latent heat ¯ux Ql was computed using the bulk ex- Carolina and making landfall on 4 September. pression

Three Lagrangian ¯oats were deployed between 2200 Qlhϭ CU⌬q␳airL, (2) and 2300 UTC 27 August 1999 ahead of Hurricane where L is the latent heat of evaporation and C ϭ 1.2 Dennis using a chartered King Air skydiving aircraft. h ϫ 10Ϫ3 is the transfer coef®cient. Note that Price et al. The storm was forecast to travel north along 79ЊW. The (1994) and Jacob et al. (2000) both use C ϭ 1.3 ϫ three ¯oats were therefore deployed in a line centered h 10Ϫ3. There are no direct measurements of ⌬q, the dif- on 79ЊW; the actual track was about 1Њ east of the fore- ference between surface humidity and saturation hu- cast. The eye of the storm passed between the deploy- midity. Cione et al. (2000) ®nd that the relative humidity ment locations of ¯oats 36 and 37. There is no additional is close to saturation (97%) near the storm center and information on the ¯oats' positions until they surfaced decreases to 85% at 4 from the center. An interpolated four days later. Њ version of this pro®le is used to compute ⌬q. This results in about 30% less latent heat ¯ux than if a constant 85% c. Air±sea ¯uxes relative humidity is assumed. 1) DATA Andreas (1998) argues that the evaporation of wind- blown spray will make large contributions to the sensible Air±sea ¯uxes were computed using bulk formulas and and latent heat ¯uxes at high wind speeds. The resulting operational surface wind speed maps produced by the Hur- sensible and latent heat ¯uxes are proportional to ⌬T and ricane Research Division of the National Oceanic and At- ⌬q, respectively, but have a much stronger dependence mospheric Administration using observations from ships, on wind speed than that used in (1) and (2). These spray moored buoys, and research aircraft ¯ight level and drop- ¯uxes can therefore not be parameterized using (1) and sonde winds (Powell et al. 1998). Wind speed contours (2). Estimates of the spray ¯uxes were computed based from each map were hand-digitized and bilinearly inter- on a ®t to the ®gures in Andreas (1998): polated in space. An example of the resulting wind speed map is shown in Fig. 1. Wind direction at a given location U 5 Q ϭ 40 ⌬T [W mϪ2 ], (3) in these maps was found to be very close to 112.6Њ coun- hs 32 terclockwise from the center of the storm. This constant ΂΃ turning angle is undoubtedly an artifact of the Powell et U 5 1 Ϫ R Ϫ2 al. (1998) mapping scheme, but was used nevertheless. Qls ϭ 790 [W m ], (4) 32 1 Ϫ 0.8 The vector wind was therefore computed using the inter- ΂΃ polated wind speed and the computed direction. Vector where R is the relative humidity. Equation (4) uses the surface winds at other times were computed by linearly relationship ⌬q ϳ (1 Ϫ R) to extrapolate the 80% hu- interpolating in time between values computed from the midity in Andreas (1998) to other relative humidities. two nearest maps after correcting for the motion of the The latent heat ¯ux is much larger than the sensible heat storm. ¯ux. Comparison with a more recent parameterization indicates that (3) and (4) may overestimate the spray ¯uxes (E. Andreas 2002, personal communication). 2) BULK FORMULAS Thus the correct heat ¯ux probably lies between that Wind stress was computed from the surface wind us- computed using the sum of (1), (2), (3), and (4) and ing the Large and Pond (1981) neutral drag coef®cient. that computed using only the sum of (1) and (2). Price et al. (1994) found that this drag law produced modeled mixed currents in reasonable agreement with 3) INERTIAL CURRENTS AND FLOAT POSITIONING the observations from three hurricanes. The drag law error is probably at least 20%. Although the ¯oats' positions are known only at the

Sensible heat ¯ux Qh was computed using the bulk ¯oat launch time and from ARGOS ®xes starting 4 days expression later, these plus modeled hurricane currents constrain

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FIG. 1. AVHRR image of Hurricane Dennis at 1338 UTC 29 Aug 1999 (image provided by Ocean remote sensing group, The Johns Hopkins University, Applied Physics Laboratory). The image is annotated with the hurricane center at 1330 UTC, the coastline of southeastern United States and the Bahamas (yellow), the track of Hurricane Dennis (red) with circles every 12 h, wind speed contours from the NOAA/AOML/HRD wind analysis at 1330 UTC, and the deployment location (colored circles) of each Lagrangian ¯oat. The ¯oats were deployed 2200±2300 UTC 28 Aug, began making measurements in the mixed layer at approximately 0000 UTC 29 Aug, and surfaced early on 1 Sep. The ARGOS locations of the ¯oats after surfacing are shown color coded for each ¯oat. The black ``o'' on each track marks the ¯oat position 4 days after surfacing. The dashed, color-coded ellipses show the estimated uncertainty (1 std dev) of the ¯oat position at 1330 UTC. the possible ¯oat trajectories. In the end, however, the a ®xed layer depth H. The wind stress near ¯oat 36, on errors in the ¯oat position remain large and cause large the east side of the storm, rotated clockwise with time uncertainties in the estimated air±sea ¯uxes. at a rate nearly resonant at the inertial frequency. With The wind stress was used to force a slab mixed layer H ϭ 100 m, the model predicts 1.2±1.5 m sϪ1 inertial model (Pollard and Millard 1970; D'Asaro 1985) with current amplitudes and 17±22-km inertial displacement

Unauthenticated | Downloaded 09/28/21 04:48 PM UTC MARCH 2003 D'ASARO 565 amplitudes. These are comparable to the mixed layer inertial currents observed in similar hurricanes (Price et al. 1994; Jacob et al. 2000). Float 36 underwent inertial oscillations of about 5-km amplitude after it surfaced; these decayed with a half-life of 2 days. Boldly extrapolating back to 29 August yields wind-forced inertial oscillations of about 20-km amplitude, consistent with the model. This model guidance thus has ¯oat 36 ex- ecuting a near circle under the in¯uence of the hurricane winds, with a 20-km diameter starting to the northwest and ending about 10 km east of its starting position. Floats 37 and 38 were on the east side of the hurricane where the winds rotated anticlockwise with time. There was no inertial resonance and the inertial amplitudes after surfacing were about half those seen at ¯oat 36 and showed about the same decay timescale. The model predicts horizontal displacements of about 10 km south to southwest for ¯oat 37, with the direction dependent on the exact location of the ¯oat, and about 10 km southwest for ¯oat 38. Based on these estimates and the variability of the ¯oat track after surfacing, the best-guess position of each ¯oat during the storm passage was taken as the deployment position plus 1.5 days of advection at the mean of the average velocity from deployment to surfacing and the average velocity during the ®rst 3 days after surfacing plus a small additional displacement to FIG. 2. Heat ¯ux and friction velocity in the ocean (ϫ104) at each represent the wind-forced motions: for ¯oat 36, 13 km ¯oat computed from NHC operational maps, bulk formulas and es- northward and 11 km westward; for ¯oats 37 and 38, timated ¯oat position. Two heat ¯ux curves are shown: a lower one 8 km southward and westward. The uncertainty is es- showing the sum of sensible and latent heat ¯uxes computed using timated from the variability of the ¯oat trajectories after the usual bulk formulas, and an upper one showing the effect of this plus sensible and latent heat ¯uxes due to spray. Error bars show one surfacing. It is modeled as a Gaussian with 0.2Њ standard standard deviation due to uncertainty in ¯oat position. There is ad- deviation in both latitude and longitude for all ¯oats ditional uncertainty due to errors in the atmospheric wind, temper- except for a 0.9Њ standard deviation in the latitude un- ature, humidity, and in the bulk formulas. certainty for ¯oats 37 and 38. The resulting spread of ¯oat positions are shown in Fig. 1. Estimates of the air± sea ¯uxes and their errors are shown in Fig. 2. hand side of a hurricane (¯oat 36) lead to much stronger shears, mixing, upward entrainment of colder water, and thus much more cooling of the mixed layer. 3. Data overview Figure 4 shows a potential temperature/depth trajec- Figure 3 shows pressure and potential temperature for tory for an approximately 1-h segment of data from ¯oat all three ¯oats. Floats repeatedly cycled across a layer 36. If the ¯oat is Lagrangian, changes in the measured 20±50 m deep thus de®ning the layer of near-surface temperature are due to heating or cooling of the water turbulence and rapid mixing. Typical vertical velocities parcel being tracked. This allows a simple diagnosis of were 0.02±0.06 m sϪ1; maximum vertical velocities the processes of heat transport (Fig. 4). The ¯oat usually were about 0.2 m sϪ1. Section 5a examines the statistics cooled when it encountered the bottom of the mixed of vertical velocity in more detail. layer (segments CD and H); this is the entrainment heat The mixing layer de®ned by the ¯oat trajectories was ¯ux. It also cooled near the surface (B and G); this is nearly isothermal and was thus very similar to the mixed the surface heat ¯ux. While transiting the mixed layer, layer de®ned by temperature. The dashed line in Fig. 3 the ¯oat temperature sometimes remained the same (seg- shows the estimated layer depth based on the ¯oat tra- ment BC); this is vertical heat transport. Sometimes, jectories and temperature. The temperature of the mixed however, strong cooling occurred within the mixed layer layer for all three ¯oats decreased with time, but the (segment EF); this cannot be the result of vertical pro- magnitude of the decrease varied from almost 3ЊCat cesses because there are no signi®cant heat sources with- ¯oat 36, to about 1ЊC at ¯oat 37, to about 0.8ЊC at ¯oat in the ocean interior and therefore must represent hor- 38. This is consistent with previous hurricane obser- izontal mixing. Section 5b quanti®es these various heat vations and models (Price et al. 1994; Jacob et al. 2000) transport processes. showing that the strong inertial currents on the right- The occasional deep excursions of ¯oats below the

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FIG. 3. Potential temperature and pressure data for all three ¯oats. Bottom panel for each ¯oat shows potential temperature; thick shaded line shows polynomial ®t to all points with depths shallower than 30 m. Top panel for each ¯oat shows pressure; thick shading indicates deviation of temperature from polynomial ®t; the shading for temperature deviations less than about 0.05ЊC is white and invisible. The dashed line shows the estimated mixing layer depth based on the depth of ¯oat excursions and the temperature deviations. The ®gure axes are the same for all three ¯oats except for an offset in temperature.

mixed layer (Fig. 3) are probably due to the ¯oat im- properly making itself denser than the mixed layer. This can occur because the ¯oats correct for density changes due to temperature, but cannot correct for salinity changes. DAS discusses this in more detail.

4. Analysis methods a. Goals and limitations The temperature, pressure, and vertical velocity data from the ¯oats were used to generate a variety of statistics on the properties of the turbulence in the mixed layer beneath Hurricane Dennis. However, the amount of data was quite limited; the hurricane passed over the ¯oats in about a day, the meteorological ¯uxes were not constant during this time and the ¯oats spent some of this time beneath the mixed layer. The number of degrees of freedom for these statistics is governed by the FIG. 4. Potential temperature/depth trajectory of ¯oat 36 during an approximately 1-h period. The water cools at the surface, at the bot- number of times that a ¯oat transits the mixed layer; tom of the trajectories (entrainment), and in occasional middepth this was only 30±50 for each ¯oat during the hurricane events (horizontal mixing). passage. Thus, at best, average pro®les of various sta-

Unauthenticated | Downloaded 09/28/21 04:48 PM UTC MARCH 2003 D'ASARO 567 tistics could be computed and even these had large error 0 D␪ ⌽ (z) ϵ dz. (12) bars. Consistency tests between different quantities were D ͵ Dt used to check for gross errors due to statistical problems z Ό΍ or methodological biases. The heat equation (9), integrated from z ϭ 0, can therefore be written b. Vertical velocity ⌰ tAz ϭ⌽(z) ϩ⌽ D(z), (13) The vertical acceleration can be written where w2ץ w 1ץ Dw d⌰ ml (ϭϩ ϩ١H ´(uw), (5) ⌰ϵ , (14 z t dtץ t 2ץ Dt where is the horizontal gradient operator. Averaging ١H where w ϭ 0 at the surface and ␪ is assumed to equal this in time and horizontally so the second and last terms ⌰ . Equation (13) can be written as a ¯ux divergence, vanish yields ml ץ (w2 ⌰ϭ [⌽ (z) ϩ⌽ (z)], (15 ץ Dw zץz) ϭ (z), (6) tAD) z Ό΍2ץ Ό΍Dt so that the total vertical ¯ux is the sum of the advective where the ͗͘denotes the averaging operation. Thus the ⌽A and diffusive ⌽D components. pro®le of ͗w 2͘(z) can be computed either directly, which The Lagrangian heat equation is will be denoted as␴2 , or from w D␪ (Dw ϭ ␬ٌ2␪, (16 0 ͗w2͘(z) ϭ 2(z) dz, (7) Dt ͵ Dt z Ό΍ where the difference between real and potential tem- 2 where w ϭ 0atz ϭ 0, which will be denoted as␴ wA . perature has been ignored. At the surface, w ϭ 0so⌽A 22 The comparison of␴␴ww andA provides a test of the ϭ 0. The surface heat ¯ux is entirely diffusive: accuracy of the measurements and of the horizontal av- ␳ץ eraging. Harcourt et al. (2002) shows an example of this Q0 ϵ lim ␬ ϭ lim ⌽D(z), (17) z z→0ץ in a numerical simulation. ␳C z→0 p []Ό΍ where ␳C converts temperature ¯ux (m sϪ1 ЊC) to heat c. Heat ¯uxÐEulerian perspective p ¯ux (W mϪ2). D'Asaro et al. (2002) show, using a sim- The potential temperature equation can be written ulated boundary layer, that the limit in (17) is nearly singular; ⌽ (z) increases very rapidly in a near-surface w␪ Dץ ␪ץ D␪ ϭϩ ϩ١H ´(u␪). (8) boundary layer of width ␦s. The surface heat ¯ux is :z evaluated just outside this boundary layerץ tץ Dt Averaging this in time and horizontally, but allowing Q0DDpϭ lim ⌽ (z)␳C (18) for the possibility of a temperature trend yields z→0; zϾϪ␦s w␪͘ D␪͗ץ ␪ץ (z) ϭϪ (z) ϩ (z). (9) de®nes the surface heat ¯ux evaluated from ⌽D. Cor- ,zDt Ό΍ rections for the ®nite size of ␦s/H may also be necessaryץ tץΌ΍ hence the retention of the limit notation in (18). Prac-

The terms are, in order, the Eulerian heating rate, the tically, this limit is evaluated by a linear ®t to ⌽D in a vertical divergence of the advective ¯ux, and the La- region away from the boundary layer, which is extrap- grangian heating rate. olated to z ϭ 0. Letting H denote the depth to which the heat budget In the interior of the mixed layer, the diffusive ¯ux is to be computed, the mixed layer temperature is is small; all the ¯ux is advective. The ¯ux at z ϭϪH, 0 1 the entrainment ¯ux, evaluated from ⌽A,is ⌰ϭml ͗␪͘ dz. (10) H ͵ QEA ϭ⌽Ap(ϪH)␳C (19) ϪH

If⌰ml is steadily changing it is more useful to rewrite (a diffusive estimate ⌽ED is described below). Similarly, the advective heat ¯ux in (9) using ␪Јϭ␪ Ϫ ⌰ ml since the surface heat ¯ux can be evaluated from ⌽A just its statistics are more stationary. De®ne the advective outside of the boundary layer: heat ¯ux Q0AApϭ lim ⌽ (z)␳C . (20) z→0; zϽϪ␦ ⌽A(z) ϵ͗w␪͘(z) ϭ͗w␪Ј͘(z), (11) s since ⌰ ml is a constant at any given time and ͗w͘ϭ0. D'Asaro et al. (2002) show that both Q0A and Q 0L equal De®ne the diffusive ¯ux Q0 in a simulated mixed layer.

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All three terms in (13) will be computed from the in layer X divided by the total time. Since the entire

¯oat data as well as Q 0A, Q 0D, QEA, and QED. The con- region is sampled uniformly sistency of these estimates will serve as a check on the t ␦ accuracy of the measurements. P(s) ϭϭss, (22) TH d. Heat ¯uxÐLagrangian perspective tee␦ P(e) ϭϭ. (23) TH 1) AVERAGING Assume that the region cools at an average rate ⌰ The above derivation of Q is problematic near the t 0D driven by ¯uxes at the surface Q and bottom Q ,so sea surface, particularly under hurricane conditions. The 0 E that surface is not ¯at, so the upper limit of integration in (12) is not well de®ned, nor is the diffusive model of QQ ⌰ ␳C ϭϪ0 E , (24) heat transport in (16) easy to apply to the frothy and tp HH wavy air±sea interface under high wind conditions. It is useful, therefore, to derive the results based on a where Q 0 represents true heat input to the region, while purely Lagrangian view of the boundary layer that is QE represents exchange of particles with the underlying free of these limitations. water. The total cooling must also equal the average rate of cooling of the Lagrangian trajectories: De®ne a Lagrangian averaging operator { }X, which takes a time average along the Lagrangian track for all times satisfying the condition speci®ed by X. In addition, D␪ ⌰ϭt . (25) if multiple Lagrangian trajectories are available, { }X av- Ά·Dt all erages these together. If the statistics of the problem are The Lagrangian heating can be divided by layer suf®ciently steady and/or there are enough trajectories then this Lagrangian average can be equated with a hor- D␪ D␪ D␪ izontal average ϭ P(s) ϩ P(i) Ά·Dtall Ά· Dtsi Ά· Dt 1 Z2 {} ϭ͗͘dz. (21) Z12ϽzϽZ D␪ Z21Ϫ Z ͵ Z1 ϩ P(e). (26) Dt D'Asaro et al. (2002) and Harcourt et al. (2002) show Ά·e that (21) is accurate for numerically simulated turbulent The surface term represents the heating of the layer due boundary layers. to the surface ¯ux, The vertical advective heat ¯ux ⌽A, and thus Q 0A, can be easily written using (21) as ⌽A(Z) ϭ {w␪Ј}zϭZ. D␪ QODϭ P(s)H ␳C p, (27) Ά·Dt s 2) LAGRANGIAN HEATING which by (22) implies Consider an ensemble of Lagrangian trajectories that D uniformly sample the region 0 Ͼ z ϾϪH so that (21) ␪ Q0Dspϭ ␦␳C , (28) is true for ¯oats in this region. Trajectories may also Ά·Dt s exit this region through the bottom boundary. Because which is equivalent to (18). the volume of the region is constant, the number of If, as in the hurricane data, all mixed region trajec- trajectories must be constant and each exiting trajectory tories return to the mixed layer, then QE can be computed must be replaced by an incoming trajectory. If, however, by a variant of this method. Equation (26) is exactly the identities of the outgoing and ingoing trajectories true, since the total heating on each trajectory in each are switched so that the exiting trajectory ``bounces'' layer ⌬␪X must sum to the overall total heating on that off the region bottom and assumes the properties of the trajectory. Heating below H, the third term in (26), is incoming trajectory, then all trajectories can be made assigned to Q : to stay within the region. These new trajectories may E rapidly change their properties, such as temperature, at ⌬␪eeee⌬␪ t ⌬␪ Q ϭ ␳CH (e) ϭ ␳CH ϭ ␳CH z H. This change can be considered to occur in a ED pppP ϭϪ ttTTee thin layer of thickness ␦e. Divide the region into three layers: s for surface, e ⌬␪si for entrainment, and i for interior. Surface heating oc- ϭ ␳CHpt⌰Ϫ , (29) ΂΃T curs in layer s, z ϾϪ␦s. Exiting particles switch identity in layer e, z ϾϪH ϩ ␦e. The remainder of the region which has no Eulerian equivalent. The last two expres- is layer i. Let P(X) be the probability of trajectories sions show two different ways to compute QED. Note being in layer XÐthat is, P(X) ϭ tX/T, the time spent that QED as expressed above is not necessarily the same

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FIG. 5. (a) Idealized potential temperature/depth trajectory for Lagrangian paths in a layer

cooled from above. (b) Pro®le of ⌽D for this trajectory. (c) Same trajectory but for deviation

of potential temperature from layer mean. (d) Pro®le of ⌽A.

as ␳Cp[⌽D(Ϫϱ) Ϫ⌽D(ϪH)]. Similar methods could of depth H that is cooling under the in¯uence of a heat be used to compute the surface ¯ux. These would yield ¯ux Q 0 applied at the upper boundary (Fig. 5). This results that differ from (18) if P(z) is not uniform. cools the entire layer at a rate ⌰t ϭ Q0/H. Lagrangian There are three important results of this analysis: trajectories traverse the layer with depth Z(t) and tem- First, the Lagrangian heating rate in the surface layer perature T(t). In the interior, there is no net heating or can be used to compute the surface heat ¯ux. Second, cooling so dT/dt ϭ 0. All of the Lagrangian temperature the results are independent of the details of the La- change therefore occurs in the surface layer of thickness grangian path through the surface layer and the mech- ␦ . The temperature changes by ⌬␪ during each surface anism of heating; all that matters is the average time s encounter. Trajectories require time td to traverse the that is spent in the surface layer and the average change layer going downÐthat is, from Z ϭϪ␦ to Z ϭϪHÐ in temperature that occurs during this time. This is ex- and time t to traverse it going up. Trajectories take time plicit in the variant (29). Third, these results require u t to traverse the surface layer, that is, from Z ϭϪ␦ to (22); the Lagrangian trajectory must spend equal s Z ϭ 0toZ ϭϪ␦. The trajectory must spend an equal amounts of time at each depthÐthat is, P(z) must be time at all depths so uniform for z ϾϪH.

tttuds 3) AN IDEALIZED PROBLEM ϩϭ. (30) H Ϫ ␦sssH Ϫ ␦␦ An idealized problem will help show the relationship between temperature changes along Lagrangian trajec- The resulting trajectories T(Z) are shown in Fig. 5a. tories and heat ¯uxes. Again, consider a surface layer Because the layer is heated only at the surface,

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Q ⌬⌰ ⌬⌰ t e. Computational issues ⌰ϭ0 ϭ ϭ s t Htϩ t ϩ tttϩ t ϩ t sud ssud The measured ¯oat temperature and pressure often change signi®cantly between samples (see Fig. 4). Es- D⌰ ␦ Q ϭϭs 0D , (31) timates of D␪/Dt and w at a given time will therefore Ά·Dts H H depend on how the data are interpolated and averaged. showing that Q ϭ Q . Here, ⌽ (z) rises from zero at The values of ⌽D and ␴wA are particularly sensitive to 0D 0 D these details. the surface to Q 0 at z ϭϪ␦s as shown in Fig. 5b. The advective heat ¯ux computed in a depth bin of A set of uniformly spaced depth bins of width ␦Z were de®ned. The values of potential temperature and width ␦Z at depth z is ⌽A(z) ϭ {w␪Ј}| zϪ␦z | Ͻ␦Z/2 . This average is the sum of contributions from the upgoing pressure were assumed to vary linearly between data and downgoing legs of the trajectory each weighted by points, thus de®ning a continuous function of time for the time spent in the bin ␦t (see Fig. 5c): each. All derived quantities were computed using this function exactly; ␪ and P are continuous; D␪/Dt and w wuu␪Ј␦t uϩ w dd␪Ј␦t d are discontinuous at data points and constant between ⌽ϭA , (32) ␦tduϩ ␦t them; vertical acceleration is a series of delta functions. The time average of these functions in computed in each where wu ϭ (H Ϫ ␦s)/tu and wd ϭϪ(H Ϫ ␦s)/td are the bin remembering that a trajectory can pass through each upward and downward velocities, respectively;␪Јu and bin many times. Note that with this scheme there is ␪Јd are ␪Ј for the upward- and downward-going legs, contribution to a given bin even if there are no data respectively; and ␦tu ϭ⌬Z/wu and ␦td ϭ⌬Z/wd are the points in the bin. times spent in the bin by the upward- and downward- There is only one free parameter ␦Z. Its value is lim- going legs, respectively. Using these de®nitions, ited by the approximate size of the ¯oat on the low end, ␪ЈϪ␪Ј since the ¯oat is not Lagrangian on scales smaller than ⌽ϭ du. (33) A 11 this, and by the size of the variation of the functions of ϩ interest on the high end. The calculations use 1 m and wwdd are not sensitive to the choice. The temperatures of the upward- and downward-going water parcels are the same (see Fig. 5a). Their perturbation temperatures ␪Ј differ only because the mixed f. Dissipation rates and diapycnal ¯uxes layer has changed temperature at a rate ⌰t in the time Lien et al. (1998), D'Asaro and Lien (2000), and Lien that it took for the trajectory to move between the two et al. (2002) use spectra of Lagrangian acceleration crossings of this bin (Fig. 5c). Substituting this time (H ⌽ (␻) to estimate the rate of turbulent kinetic energy Ϫ1 Ϫ1 ww ϩ z)(wwuuϩ ) into (33) yields dissipation ␧. Measurements of ⌽ww(␻) using Lagrang- H ϩ z ian ¯oats in a variety of both strati®ed and unstrati®ed ⌽ϭ⌰(H ϩ z) ϭ Q . (34) At 0 H environments show a clear pattern of spectral shapes. At Lagrangian frequencies ␻ above a large-eddy fre- 2 1/3 The advective heat ¯ux decreases linearly from Q 0 at quency ␻ 0, and below a frequency ␻L ϭ (␧/L ) set the surface to 0 at z ϭϪH (Fig. 5d) as expected. Note by the ¯oat size L, there is a turbulent inertial subrange from (33) that the advective heat ¯ux depends only on in which ⌽ww(␻) is white with a spectral level ␤␧, where the difference between the temperatures of the upward- ␤ is a Kolmogorov constant with a value of 1.7±2.2 and downward-going water parcels and the geometrical (Lien and D'Asaro 2002). We use 1.8. In the strati®ed mean of their velocities. thermocline ␻0 is about 0.5N. The diapycnal eddy dif- Another simple Lagrangian interpretation results fusivity can be computed following Osborn (1980), and from writing the depth average advective heat ¯ux as a host of others: 0 11 ␧ ⌽A(z) dz ϭ {␪Јw}all ϭ lim ␪Јwdt K ϭ 0.2 . (36) HT͵ T→ϱ ͵ O 2 ϪH N

1 Lien et al. (2002) suggest that the rate of temperature ϭ lim ␪Ј dz. (35) variance dissipation ␹ can be computed in a similar → T T ϱ Ͷ manner. In the inertial subrange, the Lagrangian fre-

The ¯ux is the area enclosed by the z Ϫ ␪Ј trajectory. quency spectrum of D␪/Dt, ⌽␪˙␪˙ (␻) is white with a spec- Clockwise loops in z Ϫ ␪Ј space imply a ¯ux of cold tral level ␤T␹, where ␤T is a Kolmogorov constant with water upward or warm water downward. This provides a value of about 1/␲. In the strati®ed thermocline the a useful qualitative diagnostic (D'Asaro et al. 2002) of diapycnal eddy diffusivity can be computed following the ¯ux. For the simple problem in Fig. 5, ␦ and ts are Osborn and Cox (1972) and Winters and D'Asaro small and the area equals H⌬⌰/2 ϭ ⌰ t/2. (1996):

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FIG. 6. (a)±(c) Time series of 20 000-s-averaged, half-overlapped rms vertical velocity ␴w (symbols) in the mixed layer (above H ϭ 40 m) with 95% ␹2 con®dence limits (vertical lines) and scaled friction velocity ͙1.8u with one std dev (®lled) (the std dev of u is the same as * * in Fig. 2). (d) Scatterplot of u with ␴w for all three ¯oats. Error bars as in (a)±(c). Each ¯oat * 22 uses a different symbol, as shown in (a)±(c). Dashed lines show ratio ofu to␴ w as labeled. The shorter, thicker dashed line shows the ratio and range of u found by D'Asaro* (2001). The data point labeled ``Tcline'' is a data point from below the mixed* layer (¯oat 37); no error bars for

␴w are shown. For data in the mixed layer, degrees of freedom for ␴w were computed as Tav/Tc

͗1/␶͘ ␶, where Tav ϭ 20 000 s is the averaging time, Tc ϭ 750 s is the estimated correlation time of vertical velocity, and ␶ ϭ u /H and 1/͗1/␶͘ϭ1100 s is its typical value. *

␹ a similar relationship, but with a large scatter. Time KOC ϭ 2 , (37) 2⌰ series plots for the individual ¯oats in Figs. 6a±c show z only a weak relationship between u and ␴ . Note that * w where ⌰ z is an average vertical gradient of potential the value ofA appears smaller at the start of the storm temperature. Several different ways of computing ⌰ z than near the end and that the decrease in wind stress have been suggested (Winters and D'Asaro 1996), as ¯oat 36 passed close to the eye is not re¯ected in which result in different interpretations of K . Prac- ␴ . Given the large errors in u caused by the uncer- OC w * tically, the differences are probably small (D'Asaro et tainty in ¯oat position, it is unwise to draw more detailed al. 2002). conclusions. Heat ¯ux will be computed using both (36) and (37). Figures 7a, 8a, and 9a show the average pro®le of 22 These results will be compared with other methods and ␴␴wwA(z) for each ¯oat, the pro®le of (z) computed using will act as another check on the accuracy of the data (7), and the probability distribution of ¯oat depth P(z). and analysis methods. All three ¯oats show a near-surface maximum in vertical kinetic energy. 5. Results The probability distribution is largest near the surface and decreases with depth for all three ¯oats. Kinemat- a. Energetics ically, this is consistent with an increasing deviation of ␴␴22from Ðthat is, an anomalous tendency for ¯oats 1) VERTICAL KINETIC ENERGY wA w starting at the surface to turn upward with depth, while

Figure 6 compares the rms vertical velocity ␴w from their water parcels continue downward. Dynamically, ¯oat data in the mixed layer with the estimated friction there are two possible explanations. First, the ¯oats may velocity u*. D'Asaro (2001) ®nds a strong correlation be buoyant, which is consistent with the fact that they between these with ␴22ϭ Au and a best-®t value of all ¯oat to the surface after the hurricane passes. Har- w * A ϭ 1.35 Ϯ 0.07. The hurricane data (Fig. 6d) show court et al. (2002) and D'Asaro et al. (2002) show ex-

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2 FIG. 7. (a) Pro®le (black) of vertical variance␴ w from ¯oat 36 on 29 Aug. Pro®le (red) of 2 ␴ wA computed from vertical acceleration using (6). Probability distribution of ¯oat pressure plotted as bar plot. (b) Pro®les of terms in (13) as labeled. Surface heat ¯ux computed from advective

Q0A (20) and diffusive Q0L (18) terms are plotted at negative pressures for ease of viewing. These are computed from linear ®ts over 5±25 dbar to the appropriate curve and extrapolating to the surface. Bulk ¯uxes from Fig. 2 are also shown. Error bars for all quantities are 0.03%, 0.5%, and 0.95% points computed from the distribution of 300 resamplings of the data in each bin, i.e., bootstrapping (Efron and Gong 1983).

FIG. 8. Same as Fig. 7, but for ¯oat 37.

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FIG. 9. Same as Fig. 7, but for ¯oat 38. amples of this behavior. Alternatively, the ¯oats may be at shallow depths because the mixed layer is more often con®ned to a mixing layer whose depth varies rapidly shallow than deep. D'Asaro (2001) shows a wind-driven with time because of the strong vertical or horizonal boundary layer with these properties. entrainment of heavier water. Floats spend more time By these measures, ¯oat 36 appears to be the most Lagrangian and ¯oat 38 the least Lagrangian. Although the relationship between these statistics of ¯oat error and the ability of the ¯oats to measure other Lagrangian statistics is not well understood, similar unpublished calculations made using the data described in D'Asaro (2001) indicate that ¯oats 36 and 37 are suf®ciently Lagrangian to make accurate heat ¯ux measurements.

2) KINETIC ENERGY DISSIPATION Figure 10 shows spectra of vertical acceleration for ¯oat 36 in three depth ranges. These were computed using a maximum-overlap wavelet method (Percival and Guttorp 1994) of the lowest order difference estimate of acceleration, which yields spectral estimates with periods of 40, 80, 160 . . . s every 20 s. The individual estimates were then averaged in depth bins to form spectra. The spectra were corrected for the response function of the differencing, although the correction is imperfect for the highest frequency wavelet. Note that the fre-

FIG. 10. Spectra of vertical acceleration from ¯oat 36 during the quency at which the ¯oat moves through the mixed layer period of strongest winds, day 29.35±29.85. Spectra are computed is about 0.01 sϪ1. At frequencies comparable to or less from wavelet coef®cients averaged into depth bins as indicated by than this the energy in a given wavelet is spread out the legend. Error bars show 95% con®dence limits of a ␹2 distribution. over the entire boundary layer and all depth bins have Thick shaded line shows spectral form computed by Lien et al. (1998) for kinetic energy dissipation rate ␧ϭ1.3 ϫ 10Ϫ5 m2 sϪ3, large-eddy the same spectral level. Ϫ1 As described in section 4f, the rate of dissipation of frequency ␻0 ϭ 0.05 s , a ¯oat half-height of 0.5 m, and a pressure bit noise of 8 cm. turbulent kinetic energy ␧ is given by the level of the

Unauthenticated | Downloaded 09/28/21 04:48 PM UTC 574 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 acceleration spectrum within the inertial subrange of linear in the center of the mixed layers and were there- frequencies. The thick gray line in Fig. 10 shows the fore ®t with a least squares line from 5 to 25 m. The spectral form of Lien et al. (1998) ®t to the 9±30-m slope of the line gives the net heating of the mixed layer; spectrum using ␧ϭ1.3 ϫ 10Ϫ5 m 2 sϪ3. Assuming isot- its ¯ux extrapolated to the surface gives the surface heat Ϫ1 2 ropy and a vertical kinetic energy of about (0.06 m s ) ¯ux Q0A (20); its ¯ux extrapolated to 30 m gives the 2 in the mixed layer, the dissipation time 1.5␴ w /␧ is about entrainment heat ¯ux QEA (19). 415 s, during which the rms velocity can move a water The diffusive heat ¯ux ⌽D(z) is expected to be con- parcel 25 m, about the depth of the mixed layer. stant except in a thin surface layer and in the entrainment Near the surface the acceleration spectra are larger zone (D'Asaro et al. 2002). The pro®les are also ®t with and bluer, presumably re¯ecting both a larger dissipation a least squares line from 5 to 25 m. The slope of the rate and a smaller eddy size. Our universal spectra are line is the net heating of the mixed layer due to this not appropriate in this case. At deeper levels, the spectral term; its value extrapolated to the surface gives the sur- levels are also high and somewhat blue, re¯ecting high face heat ¯ux Q0D (18). The total heating below H is dissipation levels but smaller eddy scales due to strat- used to compute QED (29). i®cation.

2) THE MIXED LAYER HEAT BUDGETS b. Heat ¯uxes (i) Accuracy 1) ANALYSES The top section of Table 1 lists the terms in the mixed Figures 7b, 8b, and 9b show the vertical integrals of layer heat budget for each ¯oat. The budgets close with the three terms in the upper ocean heat budget (13) for accuracies far less than the error bars; that is, the ``re- ¯oat 36 (east), 37 (west), and 38, respectively. All were sidual'' in line 4 of Table 1 is small. This may be a computed using a consistent interpolation and binning result of the self-consistent manner in which the quan- scheme, as described in section 4e. Each pass of the tities were computed. If so, the residual is not neces- ¯oat through each bin produced one number that con- sarily a measure of the accuracy of the methods. tributed to the averages of quantities in that bin. Con- The difference between the advective and diffusive ®dence limits were computed by randomly resampling estimates of the surface heat ¯ux may be a better es- these numbers to form 300 realizations of all computed timate of the errors. For ¯oats 36 and 37, this difference quantities. Con®dence limits were computed from the was 20%±40%, comparable to the statistical uncertainty distribution of these realizations for each quantity. The in the estimates. For ¯oat 38, the differences are much 70% con®dence limits were used to approximate the larger. This, combined with the large difference between 22 standard deviation of the various ¯uxes listed in Table ␴␴wwAand for ¯oat 38 and the surface peak in the ¯oat 1. No con®dence limits were computed for QED as it is depth distribution indicates that ¯oat 38 was buoyant dif®cult to generate independent realizations of (29) and that this buoyancy led to suf®cient non-Lagrangian subject to the constraint of (26). behavior to make the ¯ux measurements inaccurate.

The Eulerian heating term ⌰ t was evaluated from a Comparison between the surface heat ¯uxes estimated least squares linear ®t to all potential temperature mea- from the ¯oats and from bulk formulas are dif®cult due surements shallower than H ϭ 30 m for day 29. Potential to the large errors in both quantities. However, for both temperature is assumed uniform with depth so the in- ¯oat 36 and 37, the average of Q0A and Q 0L deviated tegral of the Eulerian heating term ⌰ tz is linear with by only about 5% from the corresponding bulk heat ¯ux depth. The negative of this is plotted as a green line, estimate. For both, the ``no spray'' estimate was less Ϫ2 multiplied by ␳Cp to cast it into units of W m . than either Q 0A or Q0L. This is consistent with the hy- The vertical advective heat ¯ux ⌽A ϭ {w␪Ј}(z) was pothesis that spray signi®cantly increases the air±sea computed using a high-pass ®lter of the potential tem- heat ¯ux during hurricanes. perature ␪. First,⌰ (t) was computed as a fourth-order polynomial ®t to all temperatures shallower than 30 m. (ii) Why does SST cool? Then ␪Љϭ␪ Ϫ ⌰ was computed. This was high-pass

®ltered using a sine ®lter of length Thp ϭ 5000 s to Clear differences between the heat budgets at the compute ␪Ј. Smaller values of Thp produced smaller val- three ¯oats emerge from these data. Float 37, on the ues of ⌽A while larger values resulted in larger errors left-hand (west) side of the storm, exhibited the classic with little change in the mean. entraining boundary layer pro®le. Here, ⌽A was linear

Normally we expect ⌽A(z) to vary linearly with depth with an entrainment heat ¯ux of about three times the within a mixed layer so that its depth derivative is uni- surface heat ¯ux, and ⌽D increased rapidly near the form with depth and it uniformly heats the mixed layer. surface, but was constant in the mixed layer interior. Deviations from linearity occur near the surface and at Thus in the layer interior, there was a clear balance the mixed layer base, where diffusive and/or radiative between heating and vertical advection.

¯uxes are nonzero. The pro®les of ⌽A(z) are nearly The ¯ux pattern for ¯oat 38 was similar to that of

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TABLE 1. Heat budgets of the upper ocean, day 29±30. All numbers are in units of watts per square meter although written as temperature ¯uxes. Error ranges (in parentheses) are Ϯ one std dev from mean. Float 36 37 38 0±30-m heating rate

(a) ⌰ tH Ϫ3802 (Ϫ3770 Ϫ3832) Ϫ1286 (Ϫ1256 Ϫ1316) Ϫ838 (Ϫ808 Ϫ868) 0 (b) ⌽A|Ϫ30 ; 5±25-m ®t Ϫ2196 (Ϫ1761 Ϫ2631) Ϫ1144 (Ϫ868 Ϫ1424) Ϫ628 (Ϫ468 Ϫ788) 0 (c) ⌽D|Ϫ30 ; 5±25-m ®t Ϫ1821 (Ϫ1208 Ϫ2444) Ϫ47 (Ϫ276 361) Ϫ205 (45 Ϫ455) Residual (a) Ϫ (b) ϩ (c) 215 (Ϫ536 966) Ϫ95 (323 Ϫ513) Ϫ5 (302 Ϫ292) Surface heat ¯ux

⌽A(0); 5±25-m ®t Ϫ605 (Ϫ406 Ϫ804) Ϫ436 (Ϫ317 Ϫ555) Ϫ602 (Ϫ523 Ϫ681)

⌽D(0); 5±25-m ®t Ϫ413 (Ϫ133 Ϫ693) Ϫ278 (Ϫ96 Ϫ460) Ϫ181 (Ϫ61 Ϫ301) Difference/mean 37% 44% 107% Bulk formulas Ϫ583 (Ϫ462 Ϫ704) Ϫ371 (Ϫ233 Ϫ509) Ϫ440 (Ϫ285 Ϫ595) Bulk formula, no spray Ϫ328 (Ϫ283 Ϫ373) Ϫ258 (Ϫ181 Ϫ335) Ϫ308 (Ϫ240 Ϫ376) Entrainment heat ¯ux

⌽A(Ϫ30m); 5±25-m ®t 1571 (1326 1816) 695 (541 849) 16 (97 Ϫ65)

QED 1502 852 Ϫ184 Difference 69 Ϫ157 Ϫ168 ⑀ using (36) 5070 (3550 6590) 442 (315 618) ␹ using (37) 6325 (3680 9660) 1717 (1073 2504)

¯oat 36. This is to be regarded with caution because age (Fig. 11) taken just after Dennis's passage suggests the ¯oat was poorly ballasted. a possible source for this cooling. A strong east±west Float 36, on the right-hand (east) side of the storm, temperature gradient is apparent; the warmest waters exhibited the largest cooling, equivalent to nearly 4000 were in the Gulf Stream, near ¯oat 38. A band of water WmϪ2 heat ¯ux divergence. This was not due to the about 2.5ЊC cooler extended 200 km eastward from surface cooling, but to large entrainment (1600 W mϪ2) Dennis's track. This was probably due both to cooling and Lagrangian heating (1800 W mϪ2) terms. The en- on the east side of the storm and to preexisting gradients. trainment is easy to understand; strong inertial currents A broad front, corresponding roughly with Dennis's on this side of the storm led to strong entrainment. The track, marked the western boundary of this colder water. Lagrangian heating, however, implies a heat source dis- Float 36 was deployed west of this front, on the warm tributed throughout the mixed layer. A satellite SST im- side, and surfaced east of it, in the colder water. The SST image suggests that a cyclonic eddy on the front advected the warm water containing ¯oat 36 into the colder region. In the Lagrangian frame following ¯oat 36, this would appear as a horizontal heat ¯ux divergence as the warm water is stirred into its colder sur- roundings. Water moving into the region and the cooling from E to F in Fig. 4 show an example of this cooling; the net Lagrangian cooling from 0 to 30 m at ¯oat 36 is explained as the sum of such events. Thus, the strong cooling at ¯oat 36 was primarily due to mixing with underlying cold water (entrainment) and mixing with surrounding cold water (horizontal mixing) with a small contribution from heat loss to the atmosphere.

3) MICROSTRUCTURE HEAT FLUXES Heat ¯uxes were computed during the short periods when ¯oats 36 and 37 were in the thermocline using (36) and (37) as described in section 4f. Although these ¯uxes are not averaged over the same period as the other ¯uxes, they serve as a useful comparison. FIG. 11. Sketch of a satellite SST image on 1231 UTC 1 Sep, 3 Figure 12a shows the pro®le of perturbation temper- days after Dennis's passage. The thin dashed line indicates the track ature for ¯oat 36; the deepest excursion occurred near of the hurricane. Launch positions (circles) of ¯oats, positions after surfacing (heavy lines), and interpolated tracks (dashed) are shown. day 29.5; the ¯ux was computed for this period. The Ϫ1 Interpolated tracks are guided by mesoscale features evident in the vertical temperature gradient was 0.04±0.07ЊCm ; ig- image. SST before the hurricane was 29Њ±31ЊC. noring salinity, this corresponds to a vertical strati®-

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FIG. 12. Pro®les of perturbation temperature ␪Ј for ¯oats 36 and 37. Lines are colored according to spectral level of D␪/Dt in wavelets with 80- and 160-s periods averaged over 100 s. Colored circles are drawn if ratio of spectral energy in the two wavelets is between 0.7 and 1/0.7, i.e., if the spectrum is approximately white. Thin dashed lines show approximate temperature gradients used to compute diapycnal ¯uxes. cation N ϭ 0.011±0.015 sϪ1. D'Asaro and Lien (2000) ertial currents and shears on the left-hand side of the ®nd that the inertial subrange in a turbulent strati®ed storm. In this case, the heat ¯uxes from ␹ are signi®- ¯uid begins at a frequency of about N, which includes cantly larger than those from ␧. the highest three wavelet frequencies. Using the mean of the second and third (40- and 160-s periods), for times c. Other features when the ¯oat is deeper than 30 m, ␧ϭ1.6 Ϫ 2.2 ϫ Ϫ5 2 Ϫ3 Ϫ2 1) THE SURFACE LAYER 10 m s , KO ϭ 0.014 Ϫ 0.036 m s , and the heat ¯ux is 3550±6590 W mϪ2. Heat is transferred from the atmosphere to the water The coloring in Fig. 12a shows the spectral density within the surface layer. Here, ⌽A is not linear and ⌽D of D␪/Dt for the second and third wavelets. The level increases rapidly. Figures 7b, 8b, and 9b suggest that is low in the mixed layer and high in the underlying this layer was about 5 m thick, comparable to that found strati®cation, as would be expected for ␹, the rate of in the Labrador Sea by Steffen and D'Asaro (2002). The dissipation of temperature variance. If the spectral level near-surface small clockwise loops in Fig. 4 suggest the of the two wavelets differ by less than 30%, the spec- presence of small heat-carrying eddies within this layer. trum is judged to be suf®ciently white to estimate ␹ and Within the top 2 m, a cool water layer was evident a circle is drawn. During the ®rst thermocline excursion (Fig. 12). The shallowest temperature measurement was of ¯oat 36 (day 29.22), the spectra were not suf®ciently often 0.1ЊC colder than the nearby measurements, in- white; during the second (day 29.5) they were. This dicating the presence of a thin surface layer cooled di- yields a value of ␹ ϭ 1.3±1.9 ϫ 10Ϫ4 C 2 sϪ1, and, using rectly by the atmosphere. Presumably this water is en- trained into cold, downward-going plumes that carry (37), K ϭ 0.013±0.06 m2 sϪ1, and a heat ¯ux of 3680± OC the advective heat ¯ux (Gemmrich and Farmer 1999). 6440 W mϪ2. The heat ¯uxes derived from ␹ and ␧ agree to within their rather large computational errors. Similar estimates for ¯oat 37 (see Table 1) yield much 2) SKEWNESS, FLUXES, AND ENTRAINMENT smaller entrainment heat ¯uxes, consistent with both the The vertical velocity within the boundary layer has mixed layer measurements and the expected smaller in- a clear asymmetry, with stronger velocity downward

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FIG. 13. Integrated Lagrangian heating for up (w Ͼ 0.03 m sϪ1), down (w ϽϪ0.03 m sϪ1), slow ( | w | Ͻ 0.03 m sϪ1), and all data. than upward. This is measured by the skewness ͗w3͘/ the ¯oat was rising or was slow. Perhaps this was be- 3 ␴ w, which for day 29 was about 0.5 at middepth in the cause the downgoing water had more small-scale energy boundary layer and rose to about 1 near the surface. and thus could more easily mix the cold water that was Alternatively, the average upward speed was 0.045 m intruding into its path from the side. sϪ1, while the average downward speed was 0.052 m sϪ1. (Speeds less than 0.01 m sϪ1 were excluded from both of these averages.) The maximum downward speed 6. Summary and discussion Ϫ1 was 0.22 m s , substantially larger than the maximum Three neutrally buoyant ¯oats were air-deployed Ϫ1 upward speed of 0.017 m s . ahead of Hurricane Dennis in late August 1999. These Water exiting the surface layer undoubtedly carried ¯oats were designed to accurately follow three-dimen- properties of the surface layer into the interior. Many sional water trajectories. The ¯oats measured pressure, of these, such as bubbles and dissolved gas, could not that is, their own depth, and temperature. All three ¯oats be measured by the DLF. However, the surface layer functioned, but one, deployed in the warm waters of the was also a source of small-scale turbulent kinetic energy Gulf Stream, was excessively buoyant, which signi®- and this is clearly carried downward into the layer in- cantly biased the results. The hurricane passed between terior. The energy in wavelet 2 (40-s period) was typ- the other two ¯oats, so that both measured the properties ically 3±10 times larger when a ¯oat was travelling of the oceanic boundary layer beneath sustained winds downward than when it was going upward. of about 30 m sϪ1, but on opposite sides of the storm. Figure 13 shows the distribution of Lagrangian heat- Major results include the following. ing between upgoing, downgoing and slow vertical velocities for ¯oats 36 and 37. Individual w estimates were • Boundary layer and sea surface temperature cooled sorted by depth and depth-integrated with no binning. about 2.8Њ and 0.75ЊC at the ¯oats on the east and This is simple and yields high depth accuracy, but is west sides of the northward-going storm, respectively. less accurate, especially near the surface, than the meth- Signi®cant cooling (45% and 75%) occurred before ods described in section 4e. Float 37 yielded no sur- the passage of the eye suggesting that such cooling prises; all the heating occurred near the surface; below can play an important role in hurricane thermodynam- 25 m the data were noisy. Float 36, however, showed ics (Emanuel 1999). that the interior heating occurred primarily when the • The ¯oats repeatedly moved through a mixed layer ¯oat was going down; much less heating occurred when 30±70 m deep at average speeds of 0.03±0.06 m sϪ1.

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This speed was approximately proportional to the fric- ¯oat clusters to reduce the statistical noise, as done suc- cessfully in the Labrador Sea (Steffen and D'Asaro tion velocity u* computed from operational hurricane wind products using the Large and Pond (1981) neu- 2002). tral drag coef®cient. The proportionality constant was close to that found by D'Asaro (2001) in the winter- Acknowledgments. This work was supported by NSF time North Paci®c. Grant OCE 9816807 and ONR Grant N00014-00-1- • All three terms in the horizontally averaged heat equa- 0893. The assistance of Peter Black in providing guid- tion were computed from the ¯oat data yielding con- ance into the world of hurricanes is gratefully acknowl- sistant estimates of the surface heat ¯ux and entrain- edged. Michael Ohmart built the ¯oats and air-deployment heat ¯uxes from the advective and diffusive ment system; without his cheerful and thorough work terms. 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