gas transport across an air/water interface populated with

Gas transport across an air/water interface populated with capillary wavesJ. R. Saylora) and R. A. HandlerNaval Research Laboratory, 4555 Overlook Avenue SW, Washington, D.C. 20375

~Received 28 January 1997; accepted 28 April 1997!

An experimental study of gas transport across an air/water interface, populated by a field of standingcapillary waves is presented. The experiments were conducted in a small tank containing distilledwater, enriched with carbon dioxide. The capillary waves were of the Faraday type, generated byproviding a small vertical vibration to the water tank. The frequency of excitation was varied from200 to 400 Hz, giving wavelengths from 3.62 to 2.26 mm~linear estimate!. The gas transport rateacross the interface increased by almost two orders of magnitude as the wave slope was increasedfrom zero to slightly above 0.2 m/m. A unique aspect of these experiments is that capillary waveswere isolated from the obfuscating effects of turbulence, aerosol generation, and other phenomenatypically present in wind/wave tunnel experiments. Consequently the large enhancement in gastransfer was due to the effects of capillary waves alone, demonstrating their importance in gasexchange processes. The maximum mass transfer coefficients obtained in these experiments are notachieved in typical wind/wave tunnel experiments below wind speeds of 10 m/s. ©1997American Institute of Physics.@S1070-6631~97!02408-2#

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I. INTRODUCTION

The oceans which cover the surface of the earth haveenormous capacity to absorb heat and dissolved gasescordingly, the rate at which heat and mass are transfeacross the air/sea interface greatly affects the weather asas the long term status of the environment. The developmof climatological models requires an understanding ofvarious parameters and phenomena which mediate the tfer of heat and mass across this interface. Waves haveshown to enhance transport across the air/water interfHowever, in spite of significant experimental and analytiefforts, an understanding of how and to what degree waaffect transport remains elusive. The objective of the expments described herein is to develop an understanding orole that capillary waves play in the transport of dissolvgases across the air/water interface. CO2 was chosen forthese experiments because of the ease with which its contration can be measured, and also because of its importas a greenhouse gas. It should be noted that the resultalso applicable to other dissolved gases and, to a ceextent, to the transfer of heat.

A large part of the experimental work dedicated to uderstanding air/sea transport has been performed in wwave tunnel facilities where transport processes in openenvironments are simulated by wind generated waveslong water tunnel. Figure 1 presents an example of thesults of such a study, where data due to Ocampo-Toet al.1 is presented. This figure is a plot of the mass transcoefficient for carbon dioxide,K, versus the wind speedu.The data show thatK is small and only weakly dependent ou when u is small, and thatK is much larger and moresensitive tou at higher wind speeds. A clear break in thdata, separating these two regimes, occurs at a critical wspeed, which for this study occurs atu>3 m/s. The authors

a!Corresponding author: Telephone:~202! 767-7081; Fax:~202! 767-3303;Electronic mail: [email protected]

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noted that this critical wind speed corresponds to the lowvalue of u at which waves are observed. This observatis found throughout the literature on this topic. Moreovit is almost always true that the waves which are obserat this critical wind speed are capillary waves. For exampKanwisher2 observed a sudden increase in the rateCO2 outgassing in a wind/wave tunnel, at a wind velocityabout 3 m/s, and found that this sudden increase occurreapproximately the same wind speed for which capillawaves first formed. Broeckeret al.3 found similar behaviorfor CO2 outgassing from water in a slightly deeper winwave tunnel, where the sudden increase inK occurred atslightly above 2 m/s, which also coincided with the formtion of capillary waves. Downing and Truesdale4 performedexperiments on the transport of O2 into water, in a wind/wave tunnel, and found a sudden increase in the transrate at a critical wind velocity of 3 m/s. They report that thsudden increase inK coincided with the appearance of wavlets and ripples. Discontinuities inK versusu data were alsoobserved by Hoover and Berkshire5 for CO2 and by Liss6 forboth CO2 and O2, however, no discussion of the wave fiewas provided.

Jahne et al.7 did experiments in a circular wind/wavtunnel for both CO2 and N2. In these experiments, a suddeincrease in the gas transport rate was observed whenwater surface went from ‘‘smooth’’ to ‘‘rough.’’ Althoughinformation concerning the size of the waves which rougened the surface was not provided, experiments conductethe same facility, using ethanol/water solutions, showebreak in the data and a roughening of the surface at a mlower wind speed. Since ethanol/water mixtures have a losurface tension than pure water, there is reason to belthat the sudden roughening of the surface and concomincrease in mass transfer coefficient at the lower wind spewere due to surface tension dominated waves, i.e., capilwaves.

The aforementioned experiments suggest that capilwaves play an important role in gas transport, however, s

252913/$10.00 © 1997 American Institute of Physicso AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html.

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eral questions have not yet been addressed. First, whiletrue that a sudden increase in transport occurs at the swind speed for which capillary waves are first observedcausal relationship has not been demonstrated, and thesibility that the two observations occur coincidentally hasbeen ruled out. Second, even if it is true that capillary wacontribute to the sudden increase in mass transfer at thecal wind speed, many other factors also contribute to traport, and the relative contribution due to capillary wavesunknown. It is not clear, for example, what fraction of thtransport is due to air and water turbulence, bubble entrment, etc. Finally, if capillary waves are in fact responsibfor a significant amount of mass transport, the physimechanism of their contribution needs to be identified. Fexample, it has been suggested that these waves may atrip the air flow from a laminar to a turbulent state,4 and asimilar conjecture for the water side could also be made.also possible that it is the inherent hydrodynamics ofcapillary waves themselves which cause the enhanced trport by, for example, rectified diffusion, as has been analcally investigated by Szeri8 for Crapper9 waves. Also, theincrease in surface area caused by capillary waves mustcontribute to the transport. In any event, determiningactual mechanism by which capillary waves contribute totransport is an outstanding issue.

None of the aforementioned questions can be adequaaddressed by wind/wave tunnel experiments alone; so mfactors which affect transport exist in parallel that the rowhich capillary waves play cannot be ascertained. Sowork has been done to isolate capillary waves and ascetheir contribution to gas transport. For example, Witting10

performed an analysis of capillary waves for the case of htransfer and found an enhancement of transport by as mas a factor of 9.0 for Crapper waves of limiting amplitudMacIntyre11 has analyzed Crapper waves and demonstrathat they enhance gas transport by as much as a factor oover that of a flat free surface. MacIntyre also presenexperiments for CO2 transport through a water surfac

FIG. 1. Typical plot of mass transfer coefficientK versus wind speedu in awind/wave tunnel. Data due to Ocampo-Torreset al.1 for CO2 transport.

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populated by capillary waves, in which the water was simtaneously stirred in the bulk via a magnetic stirrer. The cillary waves were created by exciting a flask with an acousspeaker at a frequency of 100 Hz, giving a wavelength;3 mm and a wave amplitude of;0.2 mm. The data reveathat capillary waves with stirring enhance CO2 transport by,at most, 30% over stirring alone. The meaning of thesesults is somewhat obscured by the lack of information ccerning the stirring. If the turbulence generated belowsurface was very strong, even a significant increase in traport due to the capillary waves would be seen as a relativsmall increase with respect to the turbulent transport. Sprisingly, the data of MacIntyre11 indicate that, for constanwave parameters, an increase in the stirring rate results iincrease in the relative contribution due to the capillawaves. An explanation for this result is not presented. In aevent, experiments which go beyond the preliminary workMacIntyre11 are needed. Specifically, experiments whethere is no externally imposed turbulence or bulk liquid floare needed so that the contribution of capillary waves canisolated.

The proposed experiment is an idealized case that dnot account for the nonlinear interactions which occur btween waves and turbulence, among other things. Moreoas noted by Ja¨hne et al.,12 since capillary waves obtain energy from larger scale gravity waves, it is also importantstudy gravity waves. Nevertheless, a large number of wiwave tunnel studies, along with existing analytical workseem to suggest that capillary waves play an importantin air/sea gas exchange. Consequently, experiments wisolate capillary waves and accurately ascertain their conbution to gas transport in a controlled environment areimportant first step toward understanding their role in tactual air/sea environment.

In the work described herein, we isolate capillary wavby generating them independently of any air or water mtion. A small tank of water is subjected to very small verticvibrations to create capillary waves via the Faraday instaity. There is no wind or bulk stirring of the liquid. Hence anenhancement of gas transport, above that of pure diffusand background natural convection, is due solely to theherent hydrodynamics of capillary waves.

II. EXPERIMENTAL METHOD

The capillary waves which were investigated in thwork were Faraday waves,13 generated by vibrating a smatank in the vertical direction. When excited in this fashiothe water surface experiences a parametric instabilitywaves are generated at one half the driving frequency.theory describing these waves has been investigated byeral authors14–20 and a review can be found in Miles anHenderson.21 Gollub and co-workers22,23 and Meron andco-workers24,25 have used Faraday waves as a platformstudying chaotic dynamics in spatially extended systeThe motion of particles and dye interfaces on the surfaceFaraday waves has also been investigated. Ramshaet al.26 used fractional Brownian motion to describe thtransport of particles on the surface, and RamshankarGollub27 used fractal dimensions to characterize the isoc

J. R. Saylor and R. A. Handlero AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html.

6.
FIG. 2. Schematic illustration of experimental setup. For details of the optics used between the laser exit and the PSD sensor, see Fig.
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centration contours of dye interfaces on the surface of Fday waves. Other than the preliminary work of MacIntyre11

however, Faraday waves have not been used as a platfor studying the effects of capillary waves on gas exchaprocesses.

The experimental apparatus used in the experimentillustrated in Fig. 2. Waves were generated in a small pleglass tank whose interior dimensions were 5 by 5 by 1.75deep. The tank was mounted on an electrodynamical sh~Vibration Test Systems, VTS65! which was powered by aTechron 5515 power supply amplifier. A sine wave wasto the amplifier from a function generator~HP 3325A! whichwas used to control the amplitude and frequency of the texcitation. A pH meter having an accuracy of60.005 pHunits~Orion, model 290A! was used to monitor the pH of thtank water. A temperature compensated pH electrodelocated below the water surface, and the pH values wused to measure the CO2 concentration, as described in thfollowing section. Data from the pH meter was continuousent to a computer which controlled data acquisition. Taccelerometers~Endevco, model 2221D! were mounted onthe water tank. One accelerometer was oriented in a direcparallel to the shaker motion, and the other in a directperpendicular to the shaker motion. These acceleromemonitored the motion of the tank in the on-axis~direction ofshaker motion! and off-axis direction~direction perpendicu-lar to shaker motion!. The signals from these two accelerometers were sent to Unholtz Dickie charge amplifiers~modelD22! whose output was monitored on an oscilloscope, sto an A/D converter, and stored on the computer. A phograph of a typical Faraday wave field obtained from tapparatus is presented in Fig. 3.

The primary results of the experiments described hwere plots of mass transfer coefficientK versus wave slopeS, for several excitation frequenciesf . The means by which

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these two quantities were measured will now be describe

A. Wave slope measurement

Several optical methods exist for the measurementwave slope. We chose to use the relatively simple metdescribed in Sturm and Sorrell,28 wherein a laser beam ivertically directed from beneath the water surface and a msurement of the beam displacement, caused by refractiothe angled air/water interface, is used to obtain the wslope. In wind/wave tunnel experiments, waves having aplitudes greater than 1 cm are common, and an error inmeasurement is introduced by the variation in the distabetween the water surface and the optical detector usemeasure the refracted beam displacement. More sophcated laser based wave slope measurement techniquesbeen developed to eliminate this error.29–34 Techniqueswhich permit the instantaneous measurement of wave sover a two-dimensional region have also been developed.35,36

Although a method for measuring the wave height wasutilized for the experiments presented herein, observatiothe wave surface, while holding a scale near the tank edshowed that the wave height was always less than 1 mConsequently, the distance between the water surface andetector varied very little, obviating the need for a moadvanced measurement technique. The maximum error induced by the variation in the distance between the wasurface and the detector due to the maximum possible pto-trough wave height of 1 mm was 2%.

A schematic illustration of the wave slope measuremmethod is presented in Fig. 4. A two-dimensional positisensitive detector~On-Trak, OT300 amplifier with 2L20SPdetector! was located a distanceL above the water surfaceThe two voltage outputs from the position sensitive detec~PSD!, Vx andVy , were linearly related to thex andy loca-

2531J. R. Saylor and R. A. Handlero AIP copyright, see http://ojps.aip.org/phf/phfcpyrts.html.

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FIG. 3. Photograph of the water tank, excited at a frequency off 588 Hz. The tank water is dyed for visualization purposes in this photograph, and aof white paper has been placed beneath the normally transparent tank floor.
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tion of the laser spot striking the detector surface viarelations

x5Vx /Cx ~1!

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y5Vy /Cy , ~2!

whereCx and Cy are the calibration constants relating toutput voltage to the laser spot displacement in the twoordinate directions. The PSD was mounted on a precisthree-dimensional stage~L. S. Starrett Co.! having a posi-tioning accuracy of 2.5mm, which was adjusted so thatVx

andVy were zero when the water surface was flat. Thus aobtainingCx and Cy the total displacement,d, of the laserbeam was obtained from the relation

d5A~Vx /Cx!21~Vy /Cy!2. ~3!

Using Snell’s law and the geometry shown in Fig. 4, twave slope at the location where the laser pierces the wsurface is given by the relation

S5d/L

12 ~nw /na!A11~d/L !2, ~4!

wherenw andna are the indices of refraction for water anair, respectively.

Calibration of the PSD to obtainCx and Cy was per-formed by translating the precision stage and recordingsition and voltage. A least-squares fit to the data was tperformed to getCx andCy . The PSD was mounted so ththex andy axes of the PSD were parallel to thex andy axesof motion for the translation stage. The voltage-displacem


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characteristics of the PSD were very linear and highly repducible. A sample calibration is presented in Fig. 5. Tcalibration was repeated periodically, and the maximvariation in the calibration constant was found to be less t1% over a two month period. The average calibration cstants wereCx5958.45 V/m andCy5956.44 V/m.Vx andVy were continuously sampled throughout the experimenthat the slopeS was continuously recorded.

Errors due to background room lighting and stray ligsources were virtually eliminated by mounting a laser libandpass interference filter in front of the PSD detector. Tcenter wavelength of the filter is tuned to the helium nelaser line,lc5632.8 nm and has a bandwidth of 12 nmeffectively filtering out all light except that correspondingthe laser beam.

An optical method was devised to provide an accurmeasurement ofL which, according to Eq.~4!, is necessaryto obtain the wave slope. A prism of known diopter val~centimeters of beam deflection at 1 m from the prism! wasplaced on the dry tank floor. The distance that the beamdeflected by the prism is measured by the calibrated PSDlocating the prism so that the laser beam passed througha location where its height had been measured using acrometer, the distance from the tank floor to the detector wcalculated using the diopter value of the prism~17.60 diopt-ers!. This distance was then used to compute the distafrom the water surface to the detector, using the known waheight. For all of the experiments presented herein, the wheight was 1.860 cm and the water surface to detectortance was 4.602 cm.

The laser beam used for the laser slope gauge~Uniphase,


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HeNe, 0.8 mW! had a beam diameter of 0.48 mm (1/e2

value!, and so it was necessary to focus the beam to achieadequate spatial resolution of the wave slope. The opticapparatus used to focus the beam down to a small measument volume is illustrated in Fig. 6. The beam is first ex-panded by a factor of 10 using the two lens assembly, L1 anL2. Lenses L1 and L2 have focal lengths of 10 and 100 mmrespectively, and are separated by a distance equal to the sof their focal lengths. This results in a collimated beam having a 1/e2 diameter of 4.8 mm. This expanded beam ispassed through an iris to remove any irregularities at thperipheral regions of the beam and is focused by lens L( f 5500 mm! down to a small measurement volume. A rightangle prism was located between lens L3 and the measument volume to redirect the beam into the vertical directionThe location of lens L3 was adjusted so that the focal poinwas located at the water surface. Exactly the same quantof water was used for all experiments, so the location of thfocal point did not have to be changed from run to run. Eacoptical element was adjusted to ensure that the laser beawas centered and normal to the lens. Additionally, the orientation of the shaker was adjusted so that the tank floor anthe water surface were normal to the laser beam, insurinthat the beam direction coincided with the gravity vector.

The diameter of the beam at the focal point was measured by mounting a razor blade onto a precision translatiostage~positional accuracy of 2.5mm!. The razor blade waslocated just above the water surface, and the stage was thtranslated in the horizontal direction. The position at which

FIG. 4. Geometry of the refraction of a laser beam as it pierces the watsurface.


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the blade just perturbed the image of the beam on a wscreen located well above the tank, and the position at whthe beam image was completely extinguished by the blawere recorded, and the difference was taken as the spoameter. A value of 250mm was obtained using this methodwhich yields a beam diameter considerably larger than1/e2 diameter typically cited. The 1/e2 diameter computedusing diffraction limited optics for a Gaussian beam37 was 84mm. The actual 1/e2 diameter lies somewhere between thetwo values. As demonstrated by Cox,36 satisfactory resolu-tion of the wave slope can be obtained when the radiusr 0 ofthe light spot is less thanl/6.8. The minimum water wavelength for the experiments presented here isl52.26 mm,requiring r 0,330 mm. Thus even our overestimate of thbeam diameter, which givesr 05125 mm, is sufficient toresolve wave slope for the parameter space explored hThe depth of focus at the focal point~length over which thebeam diameter varies by less than 5%! was computed to be5.6 mm. The peak-to-trough wave height was less than 1for all experiments indicating that the water surface wasways situated at the focal point of the laser beam.

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FIG. 5. Sample calibration of the PSD detector. The output voltagesVx andVy are plotted as a function of the displacement of the PSD in thex andydirections, respectively. This particular calibration yielded calibration cstants ofCx5960.281 V/m andCy5958.487 V/m.

FIG. 6. Optical apparatus for laser beam focusing.


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B. Measurement of mass transfer coefficient

The change in the CO2 concentration due to transpoacross the air/water interface is described by the equatio

]Cw

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H D , ~5!

whereCw is the concentration of CO2 in the water,Ca is theconcentration of CO2 in the air,h is the height of the waterand H is the Henry’s law constant which accounts for tdegree of solubility of CO2 in the water. Integrating Eq.~5!over time gives

Cw5S Cw,02Ca

H De2K/h t1Ca

H, ~6!

wheret is time andCw,0 is the concentration att50. Assum-ing the hydrodynamics do not change during the course oexperiment, Eq.~6! requires that theCw versust data yield astraight line of slope2K/h, when plotted on semilogarithmic coordinates. Values forK were obtained by performinga least-squares fit to the time traces of log@CO2#. For theseexperiments the log@CO2# time traces were relatively straigh~see Sec. III!. Nevertheless,K was computed by first breaking the time traces up into smaller segments and then cputing K from each subset of the time trace so that smfluctuations in the hydrodynamics of the system did not pduce erroneous values forK.

The CO2 concentration was obtained from the pH dausing the relationship which exists between@CO2# and pHfor pure water. Following Stumm and Morgan,38 the disso-lution of CO2 in water results in the equilibration of thspecies @H1#, @OH2#, @HCO3

2#, @CO322#, @H2CO3#, and

@HCO32#, governed by the chemical equilibrium equation

@CO2#1@H2O#@H2CO3#@H1#1@HCO32#2@H1#

1@CO322#, ~7!

where @-# denotes concentration in moles/m3. Equation~7!can be written as a set of five algebraic equations relatingconcentrations of all species present.

2@CO322#1@HCO3

2#1@OH2#5@H1#, ~8!

k15@CO2#

@H2CO3#, ~9!

kH2CO35

@HCO32#@H1#

@H2CO3#, ~10!

k25@CO3

22#@H1#

@HCO32#

, ~11!

kw5@H1#@OH2#. ~12!

Equation~8! imposes charge neutrality on the system, aEqs. ~9!–~12! impose chemical equilibrium for the specieconcentration via the constantsk1, k2, kH2CO3

, andkw . Com-bining these equations into a single relation for@CO2# yields

@CO2#5k1

kH2CO3

@H1#~@H1#22kw!

~2k21@H1# !. ~13!


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The values for the equilibrium constants arek15650,kH2CO3

52.89731024, k254.69310211, andkw51310214,all obtained at 25 °C.38

Converting pH to@H1# using the definition for pH,

pH52 log@H1#, ~14!

and then substituting@H1# into Eq. ~13! provides the@CO2# time traces necessary to computeK from Eq. ~6!.

Sample plots of wave slope time traces are presenteFigs. 7~a!–7~d!. As noted earlier, Faraday waves oscillatea frequencyf w , which is one-half that of the excitation frequency f . Hence in these figures, the dominant frequenwill be observed at one-half the indicated excitation frquency,f . In Fig. 7~a!, the first two seconds of a run at aexcitation frequency of 200 Hz is presented. A time tracethe same duration at an excitation frequency of 400 Hzpresented in Fig. 7~b!. A close up of the first 0.1 s of the timtraces are presented as well. The behavior observed in Fis mirrored in the physical appearance of the water surfduring the course of the experiment. An ordered collectionnodes and antinodes is observed over the entire surface.up and down motion of the waves is too fast for the eyeobserve, but is seen as the rapidly changing wave slwhich is apparent in Figs. 7~c! and 7~d!. The entire wavepattern is observed to slowly translate about the tank,this translation is manifested as the low-frequency modution of the wave slope, as seen in the time traces of Figs.~a!and 7~b!. This modulation occurs more rapidly as the amptude of shaking is increased, and as the frequency of exction is increased. This modulation also permitted an accusampling of the wave slope using our technique wherelaser, and therefore the measurement location, is fixedspace. Because the wave field drifted randomly overmeasurement location, all phases of the wave were msured. This modulation is due to a random drift current anprobably responsible for some degree of mixing, as willdiscussed in Sec. III.

C. Experimental procedure

Prior to starting the experiments, the pH meter was cbrated using calibration buffers~Sigma Chemical!. Care wastaken to thoroughly wash the pH probe with triply distillewater after calibration, since contamination of the tank wawith the buffer would have invalidated the relationship btween pH and@CO2#. In a separate flask CO2 was bubbledthrough triply distilled water for about 30 min. When the pin the flask reached;4.0, 300 ml of the water was transferred into the wave tank. The glassware used in this tranprocess, as well as all objects used near the wave tank,cleaned with methanol and then rinsed with triply distillewater and blown dry prior to each use in order to prevcontamination of the wave tank. Powderless latex glowere worn at all times when handling the glassware or woing near the tank.

After filling the tank, the location of the PSD detectorthe plane above the water surface was adjusted to zeroVx andVy . The most recent PSD calibration was loaded inthe program which controlled the data acquisition syste


FIG. 7. Time traces of wave slope for two runs. Two seconds of data are presented in~a! and~b! and a close up of the first 0.1 s of data are presented in~c!and ~d!.

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and the shaker was powered up and set at the desired atude and frequency. The system was run for 5 min to reacsteady state, and then data taking was initiated.

During the course of the experiments, the output frothe pH meter, the PSD outputs, and the on-axis and off-accelerometer outputs were continuously sampled. Atcompletion of the experiment, the pH time traces were cverted into@CO2# time traces using the relations describedthe previous section. The mass transfer coefficientsK wereextracted from this data, and the PSD outputs were conveto wave slopeS. An experiment typically lasted 20 to 3min.

As mentioned above, extreme care was taken to mainthe cleanliness of the system. The primary reason forwas to eliminate the introduction of anything that might cotaminate the water used in the system and thereby invalithe relationship between pH and@CO2#. We had also hopedto eliminate the presence of surfactants. However, in spitour precautions, it is most likely the case that a surfactmonolayer always existed on the water surface. It shouldnoted that the same experimental procedure was strictly


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lowed from experiment to experiment so that any surfactawhich were introduced might at least be introduced in apeatable fashion. No measurements of elasticity or surftension were made.

Experiments were run at excitation frequencies off 5200, 240, 280, and 400 Hz, resulting in wave frequencf w5 100, 120, 140, and 200 Hz, respectively. A summarythe values off , f w , and the resulting wavelengthl usedduring these experiments is presented in Table I. Notel is obtained from the linear dispersion relationship,39

c25H S gl

2p D F ~r2r8!

r G1F s

~r1r8!G S 2p

l D J tanhS 2p

l Dh,

~15!

where c is the phase speed of the wave,s is the surfacetension,g the gravitational acceleration,l the wavelength,r the water density,r8 the air density, andh the waterheight. The value ofl is obtained by substitutingc5l f w inthe left-hand side and then solving forl. The value used forsurface tension was 7.1831022 N/m.


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The ratio of the on-axis acceleration to the off-axis aceleration, as quoted by the shaker manufacturer is 5%.measured this ratio~rms on-axis acceleration to rms off-axacceleration! to be 6%, 5.7%, 25%, and 21% atf 5 200, 240,280, and 400 Hz, respectively. While large off-axis tank acelerations can result in ‘‘sloshing’’ modes, no such modwere observed in our experiments. Off-axis accelerationsalso induce secondary flows and bulk mixing beneathwave surface. The existence and effects of such bulk mixare discussed in Sec. IV.

For the largest wavelength, the ratio of the tank lengththe wavelength was 35, eliminating the effect of the wallsthe essential wave dynamics. The slopes varied from zeran average wave slope just above 0.2 m/m.

III. RESULTS

As described in the previous section, the mass trancoefficient K is directly obtained from the slope of th@CO2# time traces plotted on semilogarithmic coordinatFigure 8 is a plot of log@CO2# versus time for 35 sampleruns, obtained at an excitation frequency off 5400 Hz. Thetime traces were obtained at several different excitationplitudes. The data from each run are reasonably well-fit bstraight line. However, small variations in the local slopeeach time trace are present. To reduce the effect of this svariation on the data, each time trace was broken into sev

FIG. 8. Plot of log@CO2# versus time for sample runs at an excitation frquency off 5400 Hz. The@CO2# concentration is normalized to its value at50.

TABLE I. Values of the wave frequencyf w and l corresponding to eachvalue of the excitation frequencyf used in the experiments. Values forl areobtained from the linear dispersion relationship.

f ~Hz! f w ~Hz! l ~mm!

200 100 3.62240 120 3.19280 140 2.87400 200 2.26


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subsegments which were then used to obtain separate vof K. The resulting data were then averaged in wave slbins having a width of 0.01 m/m. A plot of the resulting dais presented in Fig. 9, whereK is plotted against the averagwave slopeS, for each of the four excitation frequencies.parabolic curve fit of the form

K5B11B2S2

~16!

is provided for each data set. The values ofB1 and B2 foreach excitation frequencyf are presented in Table II. Thlength of the error bars in Fig. 9 is equal to two standadeviations of the data set for that frequency.

Jahne12 states that the mass transfer coefficient is mappropriately correlated to the mean-square wave slope.unclear, however, whether the mean-square slope has therage wave slope subtracted or not. Consequently, we hconsidered two different definitions of this quantity, definin Eqs.~17! and ~18! asS8 andS9, respectively,

S85~S2S!2, ~17!

S95S2. ~18!

Figure 10 is a plot ofK againstS8, and Fig. 11 is a plot ofK againstS9. Note that they are qualitatively identical anthat the data are well fit by a straight line in both cases.

FIG. 9. Plot of mass transfer coefficientK versus average wave slopeS, forall four excitation frequencies. A parabolic curve fit is included for each dset. The error bars were obtained from the entire data set at each excifrequency.

TABLE II. Values of the coefficientsB1 andB2 used in the parabolic curvefit @Eq. ~16!# in Fig. 9.

f ~Hz! B1 B2

200 2.33 3 1024 0.0349240 3.77 3 1025 0.077280 4.37 3 1024 0.116400 3.36 3 1025 0.178


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In some of the higher amplitude runs~none of whichwere included in Figs. 9–11! the waves were steep enougthat the refracted laser beam was deflected off the deteThe data collection system registered these events by recing a warning message in a log file. Since the wave slmeasurements obtained while the laser beam is off the detor are wrong, the average slope obtained during these ruinvalid and was not used. The measurements ofK for theseruns, however, were valid. In an attempt to utilize this dathe wave slope for these ‘‘off-detector’’ runs was extraplated from the accelerometer data. Figure 12 is a plot ofS asa function of the on-axis accelerometer output for all of truns where the wave slope did not leave the PSD detecThese data are reasonably well fit by a straight line, andthe accelerometer output from runs where the wave slwas too large to be recorded by the detector was determ

FIG. 10. Plot of K versus mean-square wave slope defined

S85(S2S)2 for all four excitation frequencies. A linear curve fit is includefor each data set.

FIG. 11. Plot ofK versus mean-square wave slope defined asS95S2 for allfour excitation frequencies. A linear curve fit is included for each data


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by using this linear fit to extrapolate the actual wave sloThis method is crude; nevertheless, it permits the use of dthat would otherwise be discarded. The results are plotteFig. 13. Note that in this figure the data of Fig. 9 are beisupplemented with the extrapolated, higher wave slope dA parabolic curve fit of the form given in Eq.~16! is in-cluded in Fig. 13. The constantsB1 andB2 are presented inTable III. Note that the scatter in Fig. 13 is significantlarger than in Fig. 9, which is not surprising since extraplating the wave slope from the accelerometer output is mless accurate than a direct wave slope measurement.reason for presenting this data is simply to illustrate the vlarge mass transfer coefficients which were obtained frthese moderate slope waves, even though the exact valuewave slope at the high end of Fig. 13 are questionable.

s

t.

FIG. 12. Plot of S versus on-axis mean-square accelerometer outpuf 5240 Hz.

FIG. 13. Plot ofK versusS, where the large values ofS are extrapolatedfrom the accelerometer output.


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should mention that no data presented here was obtaineamplitudes for which the capillary waves broke up into drolets, or entrained bubbles.

IV. DISCUSSION

In this work capillary waves were generated via the Faday instability so that a controllable, standing wave ficould be investigated. The wave slope was varied, and itfound thatK was linearly related to the mean-square waslope.

Perhaps the most interesting result of this work concethe magnitude of the values ofK which were observed. Referring to Fig. 9, the magnitude ofK is on the order of 0.01cm/s for thef 5400 Hz data set. In fact, referring to Fig. 1K exceeds 0.01 cm/s for both thef 5280 Hz andf 5400 Hzdata sets. These values are put in perspective by compawith wind/wave tunnel experiments. The dataOcampo-Torres,1 plotted in Fig. 1, show thatK does notexceed 0.01 cm/s until the wind velocity exceeds 10 mHence in our experiments, capillary waves transfer CO2 at arate which is comparable to that which would occur inwind/wave tunnel facility at a wind speed of 10 m/s. Obvously, the capillary waves which we have generated infacility may differ from those which are observed in a winwave tunnel, or on the open sea~i.e., parasitic capillarywaves!. Also, the water surface in our tank was completecovered with capillary waves, while this is not the casethe ocean. Nevertheless, our results do demonstrate thatillary waves have the potential to transport dissolved gasea very high rate. Furthermore, since the wavelengths ofwaves which we have investigated are comparable to thof typical parasitic capillary waves, it is very possible thregions of the ocean surface populated by parasitic capilwaves represent localized regions of very high flux and ta remote sensing method for measuring wave slope calso be used to locate regions of high heat or mass flux.

In addition to the magnitude ofK, its variation is also ofinterest. Figure 9 shows thatK increases with increasinwave slope and with decreasing wavelength~increasing fre-quency!. The wavelength dependence will be addressed fiReferring again to Fig. 9, at constant wave slope,K increasesrapidly with a decrease inl. For example, at a constant slopof S50.15 m/m,K increases from 0.001 to 0.004 cm/sl decreases from 3.6 to 2.3 mm. Hence decreasing the wlength by a factor of 1.6 results in a fourfold increasetransport. This is an intriguing result since wind/wave tunexperiments typically show good correlation between mtransfer coefficient and wave slope in spite of the fact tmany different wavelengths are present. The results

TABLE III. Values of the coefficientsB1 andB2 used in the parabolic curvefit @Eq. ~16!# in Fig. 13.

f ~Hz! B1 B2

200 8.4 3 1024 0.0183240 2.6 3 1024 0.0672280 8.7 3 1024 0.0918400 6.9 3 1024 0.1376


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sented here show good correlation with wave slope also,only at constant wavelength. Exactly why transport variesdramatically with wavelength at a constant wave slope isclear at present. It is possible that the drift velocities whoccur on the wave surface~and are discussed in more detabelow!, become more vigorous as the wavelength decrea~frequency increases!. Actual velocity fields, obtained beneath the surface of these waves, are necessary to deterwhether or not this is the case. An experimental programobtain these measurements is currently being planned.

Another notable result of this work is the degreewhich the capillary waves enhance transport over that oflat surface. To illustrate this point the data of Fig. 9 is rplotted, on semilogarithmic coordinates, in Fig. 14. Tpoint having zero wave slope in this plot, was obtainedsimply taking data with the shaker turned off. The valueK for this point is slightly greater than 1.031024 cm/s,while the largest value ofK for all of the data falls just below0.01 cm/s, giving a range inK slightly less than two ordersof magnitude. Actually, if the ‘‘off-detector’’ runs~see Fig.13! were included, the range inK is greater than two orderof magnitude. Note that the zero excitation point doescorrespond to pure diffusive mass transfer, since thereinevitably buoyancy driven flows due to natural convectiin the tank.

The observed enhancement of two orders of magnitexceeds the experimental data of MacIntyre,11 as well as theanalytical predictions of Szeri8 and Witting10 for the en-hancement of transport due to capillary waves. As mentioabove, the exact mechanism for this large increase inK isnot clear, however, some statments about what is occurcan, nonetheless, be made.

It should first be noted that while there is an increasethe transport of CO2 due to increased surface area createdthe waves, this can only explain a small portion of the oserved increase inK. Because the wave profiles were nmeasured, the exact increase in surface area due to the wwas not computed. However, an upper bound on the incre

FIG. 14. Plot of log(K) versusS. The solid square symbol at zero wavslope corresponds to data taken with the electrodynamical shaker turne


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in the surface area of the waves can be computed usingmaximum peak-to-trough wave height of 1 mm and assuing the wave surface takes the form of two-dimensional sfunctions. Such a surface has an area larger than that wany capillary wave could create. The areaAe , of a surface,l on a side, populated by these ‘‘step function waves’’given by the relation

Ae5l214al, ~19!

where a is the peak-to-trough amplitude of the wave.patch of undisturbed surface,l on a side, will have an areof

A5l2. ~20!

The increase in the surface area over that of the flat surfadefined by the variableF,

F5Ae /A. ~21!

Substituting Eqs.~19! and ~20! into Eq. ~21! gives

F5114a/l2. ~22!

For a fixed value ofa ~1 mm in this case!, F will be amaximum for the smallest value ofl. Referring to Table I,the smallest wavelength for these experiments isl52.26mm. Substituting this into Eq.~22! gives F52.8 meaningthat the surface area is increased by, at most, a factor ofWhile this is a large increase~certainly larger than the increase in surface area that the waves in these experimactually produced!, it does not explain the two orders omagnitude increase inK which was observed.

A possible explanation concerns the drift velocitiwhich occur on the surface of these capillary waves.noted by Ramshankaret al.,26 while a perfect standing wavhas an extremely small drift velocity, even a very small stial modulation in the standing wave pattern can resultsignificant surface drift velocities. These authors show tthe surface velocities are random and have speeds onorder of several mm/s under conditions not dissimilar frothose of the current work. Flow visualization experimenconducted by the present authors qualitatively confirmedresults of Ramshankaret al.26 By placing a drop of dye onthe water surface, extremely rapid mixing in the horizondirection was observed. It is natural to expect that thesedom surface velocities will combine to form points of stanation and divergence at various regions on the surfaThese regions would act to exchange the surface fluid wthe bulk. While we were unable to prove or disprove thpoint, we did note that when dye was injected beneathwater surface, bulk mixing, while small, did occur at a rawhich was larger than that which was observed in thesence of waves. Because waves generate velocities wbecome negligibly small just one or two wavelengths bneath the water surface, it is reasonable to suggest thatbulk mixing is a result of the random surface velocities cobining to create hydrodynamic currents which exchangeface fluid with the bulk. The velocities at which the dye wconvected beneath the surface were much smaller thansurface velocities. Nevertheless, it is likely that this slomixing contributed to CO2 transport.


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If the spatial modulation of the capillary wave fieldindeed creating some sort of bulk fluid motion, then partthe increase inK which was observed may be due to the win which we make our comparisons. In the work of Szer8

for example, two comparisons are made. One comparisobetween pure capillary waves and pure diffusion, yieldingmaximum increase in transport of about 3. Then Szeri8 pre-sents a modified version of surface renewal theory, whesurface experiencing surface renewal in conjunction wcapillary waves is compared to a surface experiencingface renewal alone. In this case the capillary wavesshown to enhance transport by a maximum of 1.7. Inexperiments, we are comparing a surface with wavessome degree of induced bulk mixing to a surface withowaves and negligible bulk mixing~natural convection only!.This comparison is not inappropriate, since whatever crents are formed by the waves are part of the inherent phics of these capillary waves. However, it should be noted tour comparison is, strictly speaking, not the same as thaSzeri8 and Witting.10 We are unable to make a similar comparison since our ‘‘bulk mixing’’ is caused by the wavethemselves and can therefore never be eliminated. If weincluded some artificial bulk motion, such as the stirriwhich MacIntyre11 includes, the increase inK for a wavysurface with stirring over that for just a stirred system, woulikely have been smaller than what is reported here.

Finally, although detailed measurements of the waheights were not made, a comparison with the workCrapper9 yields a rough estimate of the magnitude of twave heights for this work. By differentiating Crapper’s anlytical solution to obtain slope, and then integrating, the aerage wave slope for Crapper wavesSc is obtained as

Sc52H 12A2

pAbarctanS 1

bD J , ~23!

where

A52l

paH S 11p2a2

4l2 D 1/2

21J , ~24!

b5A11A426A2

4A2, ~25!

and a is the peak-to-trough wave amplitude. The valuesS for the current experiments varied from zero to just abo0.2 m/m. In Fig. 15 four plots of the wave profiles obtainfrom the solution due to Crapper9 are presented. Each profilcorresponds to a value of the parametera/l, chosen so thatthe average wave slope for that profile correspondedSc5 0.05, 0.10, 0.15, and 0.2 m/m, thereby spanningrange ofS explored in this work. The largest wave heigshown in these plots has an amplitude ofa/l50.099. Thisvalue corresponds to a height ofa50.36 mm~for the largestwavelength!, not very different from our crudely measurevalue of 1 mm. Sincea/l must exceed 0.73 before bubbleor droplets are formed, the above result indicates that a gdeal of mass transfer can be obtained even at relativelywave slopes.


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V. CONCLUSION

Experiments on gas transport across an air/water inface populated by capillary waves were presented. Farawaves were used as a platform for investigating how calary waves affect transport in a controllable laboratory sting, where such complicating factors as turbulence and asol formation were eliminated. The average wave slovaried from 0 to about 0.2 m/m. It was demonstrated tmass transfer coefficients as large asK50.01 cm/s were ob-tained for average wave slopes of order 0.2 m/m. Such lamass transfer coefficients are typically obtained in a wiwater tunnel only when the wind speed exceeds 10 mThese results indicate that capillary waves have the poteto transport a large amount of dissolved gas~and presumablyheat!, raising the question of exactly how much transportthe open sea is due to capillary waves. While an exactplanation for the mechanism by which transport is effecby these waves is left for future work, it is suggested thatrandom surface drift velocities observed in this system mact to generate some degree of bulk mixing. Measuremof the velocity field below a capillary wave surface are neessary to determine the exact mechanism of the enhatransport which was observed herein.

ACKNOWLEDGMENTS

Helpful discussions with Andrew J. Szeri, GeoffreySmith, and Richard I. Leighton are gratefully acknowledgThe authors acknowledge financial support from the Offiof Naval Research and the National Research Council. Twork was performed while J. R. Saylor held a National Rsearch Council–Naval Research Laboratory Postdoctoralsearch Associateship.

1F. J. Ocampo-Torres, M. A. Donelan, N. Merzi, and F. Jia, ‘‘Laboratomeasurements of mass transfer of carbon dioxide and water vapousmooth and rough flow conditions,’’ Tellus46B, 16 ~1994!.

FIG. 15. Profiles of Crapper waves for four selected values ofSc. ~a!

Sc50.05 m/m, ~b! Sc50.10 m/m, ~c! Sc50.15 m/m, and~d! Sc50.20.These four values correspond toa/l5 0.025, 0.050, 0.074, and 0.099, respectively.


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for

2J. Kanwisher, ‘‘On the exchange of gases between the atmosphere ansea,’’ Deep-Sea Res.10, 195 ~1963!.

3H.-C. Broecker, J. Petermann, and W. Siems, ‘‘The influence of windCO2-exchange in a wind-wave tunnel, including the effects of monolers,’’ J. Marine Res.36, 595 ~1978!.

4A. L. Downing and G. A. Truesdale, ‘‘Some factors affecting the ratesolution of oxygen in water,’’ J. Appl. Chem.5, 570 ~1955!.

5T. E. Hoover and D. C. Berkshire, ‘‘Effects of hydration on carbon dioide exchange across an air–water interface,’’ J. Geophys. Res.74, 456~1969!.

6P. S. Liss, ‘‘Process of gas exchange across an air–water interfaDeep-Sea Res.20, 221 ~1973!.

7B. Jahne, K. O. Munnich, and U. Siegenthaler, ‘‘Measurements of gexchange and momentum transfer in a circular wind-water tunnel,’’ Te31, 321 ~1979!.

8A. J. Szeri, ‘‘Capillary waves and air-sea gas transfer,’’ J. Fluid Me332, 341 ~1997!.

9G. D. Crapper, ‘‘An exact solution for progressive capillary wavesarbitrary amplitude,’’ J. Fluid Mech.2, 532 ~1957!.

10J. Witting, ‘‘Effects of plane progressive irrotational waves on thermboundary layers,’’ J. Fluid Mech.50, 321 ~1971!.

11F. MacIntyre, ‘‘Enhancement of gas transfer by interfacial ripples,’’ PhFluids 14, 1596~1971!.

12B. Jahne, K. O. Munnich, R. Bosinger, A. Dutzi, W. Huber, and P. Libner‘‘On the parameters influencing air–water gas exchange,’’ J. GeopRes.92, 1937~1987!.

13M. Faraday, ‘‘On the forms and states assumed by fluids in contact wvibrating elastic surfaces,’’ Philos. Trans. R. Soc. London121, 319~1831!.

14J. W. Miles, ‘‘On Faraday waves,’’ J. Fluid Mech.248, 671 ~1993!.15D. M. Henderson and J. W. Miles, ‘‘Single-mode Faraday waves in sm

cylinders,’’ J. Fluid Mech.213, 95 ~1990!.16D. M. Henderson and J. W. Miles, ‘‘Faraday waves in 2:1 internal re

nance,’’ J. Fluid Mech.222, 449 ~1991!.17S. T. Milner, ‘‘Square patterns and secondary instabilities in driven c

illary waves,’’ J. Fluid Mech.225, 81 ~1991!.18J.-M. Vanden-Broeck, ‘‘Nonlinear gravity-capillary standing waves in w

ter of arbitrary uniform depth,’’ J. Fluid Mech.139, 97 ~1984!.19T. B. Benjamin and F. Ursell, ‘‘The stability of the plane free surface o

liquid in vertical periodic motion,’’ Proc. R. Soc. London, Ser. A225, 505~1954!.

20B. Christiansen, P. Alstro”m, and M. T. Levinsen, ‘‘Dissipation and ordering in capillary waves at high aspect ratios,’’ J. Fluid Mech.291, 323~1995!.

21J. W. Miles and D. M. Henderson, ‘‘Parametrically forced surfawaves,’’ Annu. Rev. Fluid Mech.22, 143 ~1990!.

22S. Ciliberto and J. P. Gollub, ‘‘Chaotic mode competition in paramecally forced surface waves,’’ J. Fluid Mech.158, 381 ~1985!.

23F. Simonelli and J. P. Gollub, ‘‘Stability boundaries and phase-space msurement for spatially extended dynamical systems,’’ Rev. Sci. Instr59, 280 ~1988!.

24E. Meron and I. Procaccia, ‘‘Theory of chaos in surface waves: Theduction from hydrodynamics to few-dimensional dynamics,’’ Phys. RLett. 56, 1323~1986!.

25E. Meron, ‘‘Parametric excitation of multimode dissipative systemsPhys. Rev. A35, 4892~1987!.

26R. Ramshankar, D. Berlin, and J. P. Gollub, ‘‘Transport by capillawaves. Part I. Particle trajectories,’’ Phys. Fluids A2, 1955~1990!.

27R. Ramshankar and J. P. Gollub, ‘‘Transport by capillary waves. ParScalar dispersion and structure of the concentration field,’’ Phys. Fluid3, 1344~1991!.

28G. V. Sturm and F. Y. Sorrell, ‘‘Optical wave measurement techniqueexperimental comparison with conventional wave height probes,’’ ApOpt. 12, 1928~1973!.

29B. A. Hughes, H. L. Grant, and R. W. Chappell, ‘‘A fast response surfawave slope meter and measured wind-wave moments,’’ Deep-Sea Re24,1211 ~1977!.

30A. M. Reece, ‘‘Modulation of short waves by long waves,’’ BoundarLayer Meteorol.13, 203 ~1978!.

31T. Hara, E. J. Bock, and D. Lyzenga, ‘‘In situ measurements of capillarygravity wave spectra using a scanning laser slope gauge and microradars,’’ J. Geophys. Res.99, 12593~1994!.

32P. A. Lange, B. Ja¨hne, J. Tschiersch, and I. Ilmberger, ‘‘Comparison b


a

fo

d

f

ids

’ J.

tween an amplitude-measuring wire and a slope-measuring laser wwave gauge,’’ Rev. Sci. Instrum.53, 651 ~1982!.

33G. Tober, R. C. Anderson, and O. H. Shemdin, ‘‘Laser instrumentdetecting water ripple slopes,’’ Appl. Opt.12, 788 ~1973!.

34S. R. Duke, L. M. Wolff, and T. J. Hanratty, ‘‘Slopes of small-scale winwaves and their relation to mass transfer rates,’’ Exp. Fluids19, 280~1995!.

35X. Zhang and C. S. Cox, ‘‘Measuring the two-dimensional structure o


ter

r

a

wavy water surface optically: A surface gradient detector,’’ Exp. Flu17, 225 ~1994!.

36C. S. Cox, ‘‘Measurements of slopes of high-frequency wind waves,’Marine Res.16, 199 ~1958!.

37M. Born and E. Wolf,Principles of Optics~Pergamon, Oxford, 1970!.38W. Stumm and J. J. Morgan,Aquatic Chemistry~Wiley, New York, 1970!.39G. Dietrich,General Oceanography~Wiley, New York, 1963!.


gas transport across an air/water interface populated with

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