Wavelet analysis of the surface temperature field at an air–water interface subject to moderate wind stress

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A new analysis methodology based on the wavelet transform is used to estimate the crosswind scale statistics of high surface temperature events from two-dimensional infrared imagery. The method is applied to laboratory data obtained from an

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  Wavelet analysis of the surface temperature field at an air–waterinterface subject to moderate wind stress Nicholas V. Scott * , Robert A. Handler, Geoffrey B. Smith Naval Research Laboratory, Code 7233, 4555 Overlook Avenue, Washington, DC 20375, United States Received 19 March 2007; received in revised form 19 November 2007; accepted 20 November 2007Available online 15 February 2008 Abstract A new analysis methodology based on the wavelet transform is used to estimate the cross-wind scale statistics of high surface tem-perature events from two-dimensional infrared imagery. The method is applied to laboratory data obtained from an experiment con-ducted in 2001 at the University of Miami, Rosenstiel School of Marine and Atmospheric Science Air–sea Interaction SaltwaterTank (ASIST). For the case of positive heat flux, in which heat is transported from water to air and for all wind forcing conditions,results from the analysis of infrared images show that the number of high temperature events, when scanned in the cross-wind direction,occupies a wide range of scales. For all wind cases, at a low temperature fluctuation threshold, the number of high temperature eventsincreases with increasing wavenumber (decreasing scale). As the temperature fluctuation threshold is increased, this distribution collapsesaround a characteristic scale. In addition, as the wind speed increases this scale decreases. The statistical scale results are shown to beconsistent with estimates of streak spacing based on a standard parameterization of the turbulent dynamics as well as with the resultsfrom numerical experiments. Our results suggest that further statistical analysis may possibly allow for the estimation of wind stress frominfrared imagery.   2008 Elsevier Inc. All rights reserved. Keywords:  Wavelet transform; Coherent temperature groups; Temperature fluctuation threshold; Infrared imagery; Characteristic scale 1. Introduction The transport of heat and gas across the air–water inter-face is a complex problem whose importance spans a vari-ety of fields in science and engineering.We are particularly concerned in this work with the fre-quently occurring situation in which heat is transportedfrom the water column to the atmosphere. In this instance,a thin thermal boundary-layer, often called the ‘cool-skin’,developsatthewatersurface.Thisresultsinthesurfacetem-perature of the ocean being cooler than the bulk tempera-ture of the water column. The opposite situation is that inwhich heat is transferred from the atmosphere to the water.Inthiscasea‘warm-skin’isformedattheinterface.Thepro-cesses occurring within this thin thermal layer play a signif-icantroleindeterminingglobalmassandheatbudgetsfromthe very small scale to synoptic and decadal scales. Morespecifically, these processes are directly involved in deter-mining the flux of heat and gases at the air–sea interface.A major goal of research in this area has been to linkair–sea fluxes to other more accessible parameters such aswind stress. One investigative route is via point measure-ments of the evolution of gas concentration in a saturatedfluid layer. Since the Schmidt number of most gases inwater is much larger than one, the mass transfer is primar-ily controlled from the water side of the interface. In thiscontext, the classical measurement of the mass transfercoefficient has evolved from the measurement of thee-folding time scale associated with gas transport out of the water column. These measurements have revealed avariety of mechanisms which contribute to changes in mass 0142-727X/$ - see front matter    2008 Elsevier Inc. All rights reserved.doi:10.1016/j.ijheatfluidflow.2007.11.002 * Corresponding author. Present address: University of Florida, Civiland Coastal Engineering Department, Weil Hall, Gainseville, FL 32608.Tel.: +352 392 9537x1434. E-mail address:  nscott@whoi.edu (N.V. Scott). www.elsevier.com/locate/ijhff   Available online at www.sciencedirect.com International Journal of Heat and Fluid Flow 29 (2008) 1103–1112  transport. For example, Dickey et al. (1984) measured thegas transfer using this methodology along with the sub-sur-face turbulence statistics and found results that were con-sistent with the classical models of surface renewal andenergy dissipation scaling. This technique pointed to thesurface layer turbulent physics as being of great importancein controlling the gas and heat flux.More innovative techniques of mass transfer measure-ment have progressed from these simple measurementstowards identifying the enhancing effect of wind forcingand waves or surface geometry on the interfacial transport.For instance, Wanninkohf and McGillis (1999) used fieldmeasurements of gas transfer and found it to scale withthe cube of the wind speed. They attributed this scalingto breaking surface gravity waves. Bock et al. (1999) wenton to investigate the relationship between surface geometrynoting the strong correlation between gas transfer andmean square slope of short wind waves produced in thelaboratory. Melville et al. (1998) performed laboratorymeasurements of the generation and evolution of Lang-muir circulation. Though their experiment was not explic-itly designed to study the interfacial heat or masstransport, the occurrence of specific scales for Langmuircells and surface streaks point to the importance of thesestructures to the dynamics of the momentum and thermalboundary-layer which control flux processes. All of thesestudies have revealed not only the richness of the processesat the air–sea interface and their relationship to mass trans-fer and indirectly heat transfer, but also the importance of understanding air–sea interfacial dynamics.With the advent of new computer and optical technol-ogy, even greater progress has been made in the investiga-tion of the surface layer temperature field and heat andmass transport. For example, McKenna and McGillis(2004) used digital particle image velocimetry to make mea-surements of surface divergence and produced evidence of a strong relationship between the ensemble averaged valueof the magnitude of the surface divergence and theenhanced gas exchange coefficient. Infrared imagery, withits high spatial and temporal resolution along with the factthat it is a true ‘surface’ measurement, has been of vitalimportance. Jessup et al. (1997) used infrared technologyto remotely quantify breaking wave dynamics by examin-ing the surface disruption and recovery produced by break-ing waves. Zappa et al. (2004) used the same type of infrared thermography to understand the role of micro-scale breaking waves in the enhancement of gas and heattransfer. Numerical simulations have played a role inextending our knowledge of the processes at the air–waterinterface and their connection to airside forcing. In partic-ular, Handler et al. (2001) performed direct numerical sim-ulations of low Reynolds number flow under a shearedinterface with no surface waves. They found the presenceof coherent temperature structures which had the shapeof fish-scales throughout the two-dimensional spatialdomain. These structures were consistent with the observa-tions of  Smith et al. (2007) who found the same type of fish-scale structures in laboratory experiments. This range of work has revealed the complex processes involving the fluxof heat present near the interfacial region and the need fora fuller understanding of them.There has been a great wealth of infrared imageryobtained to this point but very few mathematical methodsof data analysis have been exploited in trying to investigatethe interfacial heat transport. Komori et al. (1993) used theVITA technique in an attempt to establish a correlationbetween the mass transfer velocity and the frequency of surface renewal events. Some research has been carriedout by Schimpf et al. (2004) who used Laplacian pyramidsto recursively filter infrared thermographic images accord-ing to scale to understand the contribution of the varioussize scales to the surface layer temperature variance. Inaddition, Garbe et al. (2004) used passive thermographictechniques and digital image processing techniques to notonly make measurements of oceanic and laboratory tem-perature fields but also to verify the surface renewal modelof air–water heat exchange. These methods yielded newways to infer interfacial heat flux. However, these studieshave not provided an in depth analysis of the naturallyoccurring coherent temperature structures which appear(Handler et al., 2001).Numerical simulations along with experimental investi-gations have shown a correlation between the change inscales of coherent thermal structures at the interface andwind shear (Handler et al., 2001; Tsai et al., 2005, andSmith et al., 2007). Such studies point to a need to developnot only a robust way to detect and quantify the tempera-ture structures in the surface layer, but a way that, in prin-ciple, allows for the extrapolation of wind stress from theirmodulation in size. This is the main motivation for thispaper. Direct observations of surface temperature havebeen made using a high resolution infrared sensor in a lab-oratory. We combine the results of this new technologywith a new data analysis technique based on the wavelettransform (Scott et al., 2005) to examine the relationshipof coherent surface layer temperature events to wind stress. 2. Experiment The laboratory experiment conducted at the Universityof Miami, Rosenstiel School of Marine and AtmosphericScience was designed to investigate the dynamics of the sur-face layer temperature field under wind forcing. One of theaims of the experiment was the attainment of spatial tem-perature data in a laboratory for the improvement of parameterizations of gas and heat transfer. The infraredimages were obtained from an Indigo Systems MerlinMid-Wave IR sensor. The imager possesses a rectangulararray size of 320  256 pixels which translates to a spatialresolution of 0.0907 cm/pixel in our experiment. The ima-ger obtains snapshots of the water surface at a rate of 60 Hz and has a thermal sensitivity of about 0.02   C. Fur-ther details concerning the camera and the experiment ingeneral may be found in Smith et al., 2007. 1104  N.V. Scott et al./Int. J. Heat and Fluid Flow 29 (2008) 1103–1112  A table of conditions for the five different cases is shownin Table 1. Four data sets were from the unstable scenarioof a cool-skin layer above a bulk layer of warm fluid wherethe temperature differences were on the order of 1.5   C.One data set from the stable case of a warm-skin layerabove a cooler fluid bulk layer was also used where thetemperature differences were on the order of 0.25   C. Eachdata set corresponds to a different wind speed and heat fluxand therefore a different flux-based Richardson numberdefined by:  Ri ¼  b  gq s m u  w   4 q 0 c ;  ð 1 Þ where the quantities  b ,  g  ,  q s ,  u  w ,  q 0 ,  c  and  m  are the coeffi-cient of thermal expansion, gravitational acceleration, heatflux, friction velocity in water, water density, the specificheat of water, and water viscosity respectively. The quan-tity  u  w  is defined as  u  w  ¼  ffiffiffiffi sq 0 q   where  s  is the shear stressat the air–water interface. From this point onwards theexperimental runs for each Richardson number will be des-ignated by case 1, case 2, etc. as defined in Table 1.At low to moderate mean wind speeds the surface tem-perature field, as viewed by the infrared camera, possesses acoherent fish-scale temperature pattern described above.This is illustrated in Fig. 1a. In this figure the camerawas oriented such that the wind moved from the upper leftof the image towards the lower right. The resulting struc-tures align themselves with the wind and exhibit a quasi-periodicity in the cross-wind direction. They appear to pos-sess a width that is hypothesized to be inversely propor-tional to wind stress and directly proportional toviscosity via the parameterization given in Handler et al.(2001). Simple Fourier analysis of the cross-wind structureof the infrared temperature field failed to clearly define aspatial scale, so a local analysis based on the wavelet trans-form was conducted. The methodology used here is verysimilar to that used by Scott et al. (2005), where a wavelettransform was applied to spatial measurements of the oce-anic surface wave field to detect steep surface wave crestsand to quantify their scale. Here we apply a similar typeof analysis to infrared surface temperature measurementstaken from the five data sets. Table 1Table of experimental parameters for data citedCool skin  U   (m s  1 )  u  w  ð m s  1 Þ  k þ  ¼ 100 m u   ð m Þ  T  air  –  T  w  (  C)  Q  (W/m 2 )  Ri ¼  b  gq s m ð u  w Þ 4 q 0 c Case1 1.925 2.2  10  3 4.55  l0  4  14.24 393 7.0  10  3 Case 2 3.977 4.3  l0  3 2.33  10  4  14.78 850 1.0  10  3 Case 3 5.0031 5.7  10  3 1.75  10  4  14.63 1144 4.0  10  4 Case 4 7.0552 9.3  10  3 1.08  10  4  14.69 1764 1.0  10  4 Warm skinCase 5 2.951 3.1  10  3 3.23  10  4 9.731   140   6.0  10  4 Fig. 1. (a) Infrared image of water surface with mean wind speed of 1.9 m s  1 and (b) rotated infrared image of the same water surface. N.V. Scott et al./Int. J. Heat and Fluid Flow 29 (2008) 1103–1112  1105  3. Data analysis 3.1. Wavelet transform The srcinal motivation for the use of the wavelet trans-form was the desire to obtain local cross-wind scale infor-mation from a temperature signal taken from infraredimages. The wavelet transform of a spatial temperaturesignal  s ( u ) in this study is defined as Ws ð a ;  x Þ¼ Z   11  s ð u Þ 1 a W  u   xa   d u :  ð 2 Þ Here  W ( a , x ) is the Morlet mother wavelet defined in thisstudy as W ð a ;  x Þ¼ W  xa   ¼ Real e i2 p  K  0  xa e  12  xa ðÞ 2 h i ;  K  0  ¼ 1 :  ð 3 Þ We note the srcinal mathematical form of the Morletwavelet as formulated by Goupillaud et al. (1984) has theform W ð a ;  x Þ¼ W  xa   ¼  Be  12  xa ðÞ 2 e i2 p  K  0  xa  j   ;  ð 4 Þ where j ¼ e  12  2 p  K  0 ð Þ 2 and  B ¼ p  14  1 þ e ð 2 p  K  0 Þ 2    2e  34 ð 2 p  K  0 Þ 2 Þ  12 .Since 2 p K  0  > 5,  j  < 10  5 is negligible and therefore thisterm is neglected in Eq. (3). The wavenumber of the Morletwavelet is  K  0 , the variable ‘ a ’ is a scaling parameter, andthe spatial coordinate is designated by the variable  x . Thewavenumber associated with the fluctuations in the struc-ture of the temperature signal is defined as k   ¼ 1 k ;  ð 5 Þ where k ¼  a K  0 :  ð 6 Þ The wavenumber ‘ k  ’ therefore has a one to one relationshipwith the scale parameter ‘ a ’.We note that the Morlet wavelet, as defined here, is thereal part of a complex quantity and is used since we areconcerned only with quantifying the cross-wind scale of the temperature fluctuations from the mean. The normali-zation used here is different from the classical  1  ffiffi a p   energy pre-serving normalization. The wavelet transform  Ws ( a , x )preserves the amplitude of the fluctuation of the signal, s ( u ), in the sense that signals of different scales but the samedeviation from the mean possess the same wavelet trans-form (Scott, 2005).The Morlet wavelet is a solution to the linear wavetheory equations and is used because of its characteristicfeature of resembling a compact periodic group of fluctua-tions. When applied to a signal it effectively searches thesignal and finds regions where the data has wave-likecharacteristics. In addition, if the signal is interpreted astemperature, the local peak value of the inner product of the signal and the Morlet wavelet produces a measureof the average temperature fluctuation over the supportof the wavelet. Thus, a signal with a high temperature eventwith scale  a 0  and position  x 0  will have a wavelet transformcharacterized by a large peak value of   Ws  at ( a 0 , x 0 ), andthe peak value is proportional to the average temperaturefluctuation of the signal in the neighborhood of   x 0 . In addi-tion, two sinusoidal signals whose temperature amplitudefluctuations are equal will have equal peak values of  Ws ( a , x ) at each of the signals’ respective scales. The pres-ervation of the signal’s average temperature deviation fromthe mean in the ‘local’ sense as described here allows for thedetection of high temperature events associated with coher-ent temperature groups in the data. 3.2. Wavelet analysis of the temperature field  The continuous wavelet transform, as delineated in Eqs.(2) and (3), is implemented in a discretized version usingthe Matlab 5.3 wavelet toolbox software. The continuouswavelet transform allows for the continuous and redundantunfolding of information in both scale and space. This inturn enables the dynamical tracking of coherent structures.Wavelet analysis was performed on temperature signalswhich were extracted from a set of images. Before thiswas done, a simple autocorrelation of single temperaturevalues in a sequence of images over time was performedto assess the point in time in which the images becamestatistically independent. Every 25th image was found tobe statistically independent and on this basis 45 statisticallyindependent images were selected to undergo wavelet anal-ysis. These images were subsequently rotated such that themean wind direction was directed vertically downward asshown in Fig. 1b.If we define a coordinate system such that  x  is the hor-izontal coordinate and  y  is the vertical coordinate parallelto the wind direction, a temperature–space series  s ( x ) canbe described by:  s ð  x Þ¼  K  ð S  ð  x ;  y  0 Þ S  ð  y  0 ÞÞ ;  ð 7 Þ where  S   is the raw thermal data,  S   is the mean value of thedata taken at a fixed vertical coordinate  y 0 , and  K   is a con-version constant (0.02   C/count). Thus the data analyzed, s ( x ), is in units of degrees Celsius.The Morlet wavelet transform was performed on eachtemperature–space series over a set of discrete scales (witha constant  D a  in log space) that extends from 0.1362 cm  1 to a limiting wavenumber of 4.2 cm  1 . This wavenumberrange was obtained from performing Fourier analysisand inspecting the power spectrum to find the biggest scalesresolvable in the data set and the smallest scales before thenoise floor occurs. The extracted temperature–space serieswere subject to edge effects at the boundaries, especiallyat large scales, due to the ‘start up’ process of the convolu-tion. This edge effect takes the form of a cone of influence(Addison, 2002) which produces a linearly decreasingregion of contamination as the scale is decreased. Wecalculated the cone of influence for the largest scale usedin the analysis. This was done by taking the largest scale 1106  N.V. Scott et al./Int. J. Heat and Fluid Flow 29 (2008) 1103–1112  of the wavelet transform and a sinusoid of correspondingscale and estimating the point at which the amplitude of the sinusoid decreased by  e  2 . This is similar to the workof  Grinsted et al. (2004). We chose to use only the statisticsthat started from the higher wavenumber of 0.2 cm  1 which corresponds to truncating 5 cm of data from eachside of the temperature space series. This truncatedregion represents the maximum contaminated region of the wavelet transform since it is associated with the largestscale.The conversion from the wavelet transform threshold tothe real temperature fluctuation threshold is accomplishedby first taking a pure sinusoid  s ( u ) =  A sin( ku ) of knowntemperature amplitude  A  and wavenumber  k  , and obtain-ing the wavelet transform of the signal,  Ws ( a , x ). The peakvalue of the wavelet transform is then related to thetemperature fluctuation amplitude in order to determinethe conversion constant  v : v ¼  A max ½ Ws ð a ;  x Þ ð 8 Þ With real temperature data, this constant is used to convertfrom the local maximum value of the wavelet transform tothe equivalent temperature fluctuation threshold. This tem-perature fluctuation threshold represents the temperatureamplitude, or temperature deviation from the mean, thata temperature sinusoidal signal of a scale equivalent tothe real temperature–space series would possess. The proof of this, albeit applied to surface gravity wave fluctuations,can be found in Scott (2005).A typical example of a wavelet transform is shown inFig. 2 along with a plot of the temperature signal from case1 from which it was computed. The wavelet transform con-tains an array of high temperature events designated bylocal maxima. In order to obtain a distribution of theseevents, a temperature fluctuation threshold is applied. Thusall points with a wavelet transform value above the setthreshold are selected. These points appear as aggregatesin local regions of   Ws ( a , x ) enclosed by the dashed linesshown in Fig. 2. From these groups of points, the highestvalue is sought. The highest values or local maximathroughout the wavelet transform are obtained by usinga nine-point box filter. The filter is moved throughout thetransform and from these groups, only those points thatare larger than the surrounding eight points are selected.The local maxima selected in the manner described aboveappear as asterisks in Fig. 2. Since the Morlet wavelet inthis study is a compact group of temperature fluctuations,these asterisks are defined as individual high temperatureevents associated with cohesive temperature groups in thesubsequent analyses.Each high temperature event in a temperature signal,denoted by asterisks in Fig. 2, has associated with it a scalevalue and spatial position. Thus after wavelet transforma-tion, each temperature–space series denoted by an index‘  j  ’ can be converted to  H   jT  ð k  Þ  j   = 1,2,3, ... ., the numberof events of wavenumber  k   exceeding a set high tempera-ture fluctuation threshold  T  . The final statistic  H  T  ( k  ) isobtained by averaging the individual distributions over100 temperature–space series per image and over 45 statis- Fig. 2. Wavelet transform of temperature signal from infrared image of water surface with mean wind speed of 2.9 m s  1 . Asterisks delineate hightemperature events with events in the contaminated regions not shown. Temperature threshold of 0.05   C is designated by dotted line. Contours drawn atevery 0.074 interval. N.V. Scott et al./Int. J. Heat and Fluid Flow 29 (2008) 1103–1112  1107
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