Similarity Relationships in the Marine Atmospheric Surface Layer for Terms in the TKE and Scalar Variance Budgets*

Transcription

1 1JULY 1998 EDSON AND FAIRALL 11 Similarity Relationships in the Marine Atmospheric Srface Layer for Terms in the TKE and Scalar Variance Bdgets J. B. EDSON Woods Hole Oceanographic Instittion, Woods Hole, Massachsetts C. W. FAIRALL NOAA/Environmental Technologies Laboratory, Bolder, Colorado (Manscript received September 1996, in final form 5 October 1997) ABSTRACT Measrements of the momentm, heat, moistre, energy, and scalar variance flxes are combined with dissipation estimates to investigate the behavior of marine srface layer trblence. These measrements span a wide range of atmospheric stability conditions and provide estimates of z/l between 8 and 1. Second- and third-order velocity differences are first sed to provide an estimate of the Kolmogorov constant eqal to The flxes and dissipation estimates are then sed to provide Monin Obkhov (MO) similarity relationships of the varios terms in the trblent kinetic energy (TKE) and scalar variance (SV) bdgets. These relationships are formlated to have the correct limiting forms in extremely stable and convective conditions. The analyses concldes with a determination of pdated dimensionless strctre fnction parameters for se with the inertial dissipation flx method. The prodction of TKE is fond to balance its dissipation in convective conditions and to exceed dissipation by p to 17% in near-netral conditions. This imbalance is investigated sing the athors measrements of the energy flx and reslts in parameterizations for the energy flx and energy transport term in the TKE bdget. The form of the dimensionless energy transport and dimensionless dissipation fnctions are very similar to previos parameterizations. From these measrements, it is conclded that the magnitde of energy transport (a loss of energy) is larger than the pressre transport (a gain of energy) in slightly nstable conditions. The dissipation of SV is fond to closely balance prodction in near-netral conditions. However, the SV bdget can only be balanced in convective conditions by inclsion of the transport term. The SV transport term is derived sing or estimates of the flx of SV and the derivative approach. The behavior of the derived fnction represents a slight loss of SV in near-netral conditions and a gain in very nstable conditions. This finding is consistent with previos investigations. The similarity between these fnctions and recent overland reslts frther sggests that experiments are generally above the region where wave-indced flctations inflence the flow. The athors conclde that MO similarity theory is valid in the marine srface layer as long as it is applied to trblence statistics taken above the wave bondary layer. 1. Introdction Or nderstanding of the behavior of trblence in the atmospheric srface layer was vastly improved by a nmber of overland field experiments condcted dring the late 1960s and 1970s. These inclde the landmark 1968 Kansas (Izmi 1971) experiment, the 197 Minnesota (Champagne et al. 1977) experiment, and the 1976 International Trblence Comparison Experiment (Dyer and Bradley 198). These experiments led to the WHOI Contribtion Nmber 916. Corresponding athor address: Dr. James B. Edson, WHOI MS#10, 98 Water Street, Woods Hole, MA jedson@whoi.ed validation of a powerfl set of statistical tools derived from Monin Obkhov similarity theory. The semiempirical relationships derived from their careflly condcted measrements are now sed extensively in the lower bondary conditions of nmerical forecast models where one mst derive trblent qantities from the mean variables available from the model. Similarly, these relationships are often sed to estimate the desired trblent qantities from mean measrements over the ocean where direct measrement of the flxes is very difficlt. However, the se of overland measrements to infer srface flxes over the open ocean raises qestions abot the niversality of these relationships. There have been a nmber of experiments to investigate the strctre of atmospheric trblence in the marine bondary layer (Smith et al. 1996). These inclde the 1969 Barbados Oceanographic and Meteorological 1998 American Meteorological Society

2 1 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 Experiment (Kettner and Holland 1969) and the 1974 North Pacific Experiment (Schmitt et al. 1979). Both of these experiments were condcted aboard the Floating Instrment Platform (R/P FLIP), which allows researchers to make flx measrements over the open ocean from a stable platform. Thogh these experiment were sccessfl on many fronts, they both sffered from the now well-known (bt still difficlt to overcome) problem of sea salt contamination of their temperatre probes (Schmitt et al. 1978). Large and Pond (1981, 198) sed the Bedford Institte of Oceanography spar boy pls ship data in a landmark stdy of flxes and transfer coefficients over a broad range of wind speeds. The 1986 Hmidity Exchange Over the Sea (HEXOS) (Katsaros et al. 1987) main experiment deployed instrmentation designed to avoid environmental contamination problems. This effort inclded a nmber of instrments designed to remove the effects of sea spray (Katsaros et al. 1994), as well as the se of sonic thermometers to compte the temperatre flctations (Larsen et al. 199). The HEXOS investigations were condcted from a platform located 10 km off the Dtch coast. The reslts from this experiment confirmed a nmber of earlier reslts, for example, that the prodction of trblence kinetic energy (TKE) is closely balanced by its dissipation (Edson et al. 1991) and that the drag coefficient is a fnction of wave age at a given wind speed (Smith et al. 199; Maat et al. 1991). The experiment also demonstrated that the transfer coefficients for temperatre and hmidity were only weakly dependent on wind speed (DeCosmo et al. 1996), althogh the significance of these reslts remains a hotly debated topic (e.g., Katsaros and de Leew 1994; Andreas 1994; Andreas et al. 1995). In recent years, a nmber of researchers have made a great deal of progress in compting direct covariance flx estimates from ocean-going platforms (e.g., Fjitani 1981, 1985; Tskamoto et al. 1990; Bradley et al. 1991; Fairall et al. 1997; Edson et al. 1998). In the case of moving platforms, this reqires systems that are capable of removing platform motion from or velocity measrements before compting the flxes (Fjitani 1985). This generally involves the integration of linear accelerometers and anglar rate sensors to compte the highfreqency motion of the platform that are combined with the low-freqency velocity measrements from global positioning systems and/or crrent meters (e.g., Edson et al. 1998). Similar measrements from boy monted instrmentation can be fond in Anctil et al. (1994). For example, direct covariance flx estimates were collected from ship-monted systems dring the Tropical Ocean Global Atmosphere (TOGA) Copled Ocean Atmosphere Response Experiment (COARE) in the eqatorial Pacific Ocean (Webster and Lkas 199). These flx estimates have been sed to develop a blk algorithm based on the parameterizations proposed by Li et al. (1979), Smith (1988), and Godfrey and Beljaars (1991). The hybrid model incldes new semiempirical formlas that improve the performance of the code in very convective conditions as reported by Fairall et al. (1996a) and Fairall et al. (1996b). Despite the emergence of covariance flx measrements from ships, interest remains high in the inertial dissipation flx estimation method (Large and Pond 198; Fairall and Larsen 1986). Papers freqently appear in the literatre sing this method (e.g., Anderson 199; Yelland and Taylor 1996; Fairall et al. 1996a; Frederickson et al. 1997), which is now being sed extensively on boy-based systems (Skpneiwicz and Davidson 1991; Yelland et al. 1994). It is clear that inertial dissipation stress data give mch lower scatter than covariance estimates (Fairall et al. 1996a) and there is sbstantial evidence that covariance stresses are mch more inflenced by flow distortion (Edson et al. 1991; Oost et al. 1994). Inertial dissipation measrements still represent the vast majority of data sed to determine the drag coefficient over the open ocean at wind speeds greater than 0ms 1. The disadvantage of this method is that it relies on Monin Obkhov similarity and reqires specification of the dimensionless strctre fnction parameters. These are empirical fnctions and the method is ncertain to the extent that these fnctions are ncertain. Forms of these fnctions determined over Kansas are still being sed (Wyngaard and Cote 1971; Wyngaard et al. 1971a,b), bt demands for ever-increasing accracy in flx estimates and blk transfer coefficients reqire s to continally refine these fnctions. In this paper we present an analysis of data recently taken in the marine srface layer aboard the R/P FLIP and the R/V Colmbs Iselin. These new measrements featre direct eddy correlation stress and heat flx estimates, and TKE and scalar variance (SV) dissipation compted from the inertial sbrange of the velocity, temperatre, and specific hmidity spectra. The highfreqency velocity and temperatre measrements were made sing sonic anemometers thermometers, while the specific hmidity measrements were made sing infrared hygrometers. The data collected dring these two experiments cover a mch wider stability range than any of the above mentioned datasets. In particlar, the extremely nstable data will allow s to examine the behavior of the trblence statistics in the free convective limit. The focs of this investigation is the evalation of the components of the TKE and SV bdgets and their relationship to the inertial sbrange variables. Or goal is to verify and extend the overland reslts for application over the oceans inclding a closer look at the relative balance of trblent and pressre transport in the TKE bdget. In the sections that follow we begin with a brief overview of Monin Obkhov similarity theory. This is followed by a description of the field programs, the instrments, and the data processing techniqes in section. Dissipation estimates and evalation of the Kolmogorov constant sing the skewness rela-

3 1JULY 1998 EDSON AND FAIRALL 1 tions of Kolmogorov (1941) are discssed in section 4. The TKE bdget is evalated in section 5; the SV bdgets in section 6; and the dimensionless strctre fnction parameters are given in section 7.. Similarity theory The strctre of the trblent flow in the srface layer is inflenced by both mechanical and thermal forcing. Obkhov (1946) and Monin and Obkhov (1954) were the first to describe a similarity hypothesis abot the statistical natre of the trblent flow based on the relative strength of these two forcing mechanisms. Monin Obkhov (hereafter MO) similarity theory states that the strctre of trblence is determined by the height above the srface z, the boyancy parameter g/, the friction velocity, and the srface boyancy flx w 0, where is the virtal potential temperatre and g is the acceleration of gravity (e.g., Wyngaard 197). These last two terms are defined from the srface stress and heat flxes as 1/ 0, (1) [ ] Q0 E0 w 0.61T 0, () 0 c L p where 0 is the srface stress vector (the srface vale of the momentm flx), Q 0 is the srface vale of the sensible heat flx, E 0 is the srface vale of the latent heat flx, is the density of air, c p is the specific heat at constant pressre, and L e is the latent heat of vaporization of water. One of the basic assmptions of MO similarity theory is that these flxes are constant with height in the srface layer. In reality, this constant flx assmption is never trly valid in the atmospheric bondary layer. However, a good approximation to this constant flx assmption is obtained if we define the top of the srface layer as that height where the momentm flx is 90% of its srface vale. Since the momentm flx generally decreases linearly with height, this definition reslts in a srface layer that occpies the lowest 10% of the atmospheric bondary layer. The nstable marine bondary layers stdied in this paper were typically m in height, which reslts in a srface layer height of approximately 60 m. This allows s to estimate the srface flxes from or covariance measrements (made at abot 1 m height) as 0 iw jw, () Q Q c w, (4) 0 p E E L wq, (5) 0 e where,, and w are longitdinal, lateral, and vertical e velocity flctations, respectively; are temperatre flctations; and q are specific hmidity flctations. a. The marine srface layer In addition to the constant flx layer constraint, the application of MO similarity theory to the marine srface layer reqires some cation becase the scaling parameters are only meant to accont for the inflence of mechanical and thermal forcing on the trblence. Many investigations sch as those by Geernaert et al. (1986), Rieder et al. (1994), Donelan et al. (199), and Hare et al. (1997) have demonstrated that additional scaling parameters are reqired to describe trblent variables within the wave bondary layer (WBL). The WBL is defined in this paper as the layer where the total momentm flx, even if assmed to be constant with height, has appreciable trblent and wave-indced components. That is, within the WBL the momentm eqation can be written as z U ũw w 0, (6) z where primes denote trblent flctations, tildes denote the wave-indced flctations, U is the mean wind [i.e., U(t) U (t) ũ(t) U (t)], and the last term on the right-hand side represents the viscos stress where is the kinematic viscosity. To examine the applicability of MO similarity theory in the marine srface layer, we are limiting or analyses to observations where the flow is not expected to be inflenced by wave-indced flctations. We believe that we are meeting this constraint based on the fact that the measrement heights for the R/P FLIP (1 m) and R/V Iselin (11.5 m) are generally larger than the reciprocal wavenmber of the dominant wind waves, 1 k w, observed dring or respective crises (i.e., in gen- eral, k w z 1). Therefore, althogh we have reason to believe that we are experiencing wave-indced effects at or highest winds (see section 8), we believe that most of or dataset is characteristic of a srface layer where the trblent flxes dominate the total flx and MO similarity theory is applicable. b. Monin Obkhov scaling Monin Obkhov similarity theory is covered in detail in a nmber of texts, inclding Lmley and Panofsky (196) and Wyngaard (197). For the prposes of this paper we briefly describe the basis MO similarity theory by first combining the for governing parameters to form an additional velocity scale defined as 1/ zg w, (7) f 0 whose se is restricted to positive vales of the heat flx (i.e., convective conditions). The two velocity

4 14 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 scales, and f, are then sed to define two temperatre and moistre scales, w 0 w 0 T, Tf, (8) f wq 0 wq 0 q, qf, (9) f and a length scale now known as the Monin Obkhov length, L, (10) g w 0 where is the von Kármán constant. The magnitde of the MO length is determined by the relative strength of the mechanical verss thermal forcing, while its sign is determined by the sign of the boyancy flx; that is, it is negative in convective (nstable) conditions and positive in stratified (stable) conditions. The varios scales are not independent (Wyngaard 197) as they can be combined to obtain and 1/ f z (11) L 1/ Tf qf z (1) T q L in convective conditions. Therefore, it is common practice to select, T, and q as the velocity, temperatre, and moistre scales for both stable and nstable flows. The similarity hypothesis then states that varios trblent statistics, when normalized by these scaling parameters, are a niversal fnction of z/l. This hypothesis has been validated by a nmber of stdies in the atmospheric bondary layer over land. Notable examples inclde the stdies by Dyer and Hicks (1970), Wyngaard and Coté (1971), Kaimal et al. (197), Champagne et al. (1977), and Dyer and Bradley (198). In light winds conditions with appreciable heat flx, MO similarity theory reqires that the srface stress (i.e., ) is no longer a relevant scaling parameter and that the small-scale trblence variables approach the convective limits. In this limit the strctre of the marine atmospheric srface layer in the region between L z 0.1 z i shold approach that of local free convection and depend only on z, g/, and w 0 (Tennekes 1970). Under these conditions it is more appropriate to se the convective scaling parameters denoted by the sbscript f. Additionally, above the srface layer (i.e., z 0.1 z i ), stdies of the mixed layer have shown that many trblent processes scale with the height of the bondary layer, z i. In this region, z i replaces z as the appropriate length scale and one ses the free-convective velocity scale proposed by Deardorff (1970): 1/ 0 zg w i w. (1) Overland stdies involving this type are scaling analysis have been reported by Wyngaard et al. (1978), Wyngaard and LeMone (1980), and Højstrp (198).. Field programs and data processing In this paper we present analyses of data taken in two recent experiments in the marine srface layer aboard the R/P FLIP and the R/V Colmbs Iselin. These two experiments sed similar instrments, data acqisition systems, and processing techniqes to estimate the trblent statistics. One experiment was an open ocean location jst otside the islands off Los Angeles, California; the other was in the vicinity of the Glf Stream edge off Cape Hatteras. Details abot the instrments and basic processing are provided in this section. The overall qality of the covariance flxes is illstrated by comparison with blk flx estimates. The R/P FLIP was deployed off the west coast of sothern California for a -week period dring the second half of September 199 as part of the San Clemente Ocean Probing Experiment described in Kropfli and Clifford (1994). The R/P FLIP was positioned off San Clemente Island, California (N, 118W). Data were obtained in the wind-speed range from 0.5 to 1 m s 1, with sea air temperatre differences from 0 to 4C. The Environmental Technologies Laboratory s ship flx measrement system was sed. This system is described in detail by Fairall et al. (1997), so only a brief sketch will be given here. A sonic anemometer thermometer is sed to make measrements of the stress and boyancy flx, and a fast-response infrared hygrometer is sed with the sonic anemometer to obtain the latent heat flx. A dal inertial navigation system is sed to correct for ship motions (Edson et al. 1998). Flxes were compted sing covariance, inertial dissipation, and blk techniqes. Sea srface temperatre is derived from blk water measrements at a depth of 5 cm with a floating thermistor and the corrections for the cool skin effect described in Fairall et al. (1996b). Mean air temperatre and hmidity are derived from a conventional aspirated T/RH sensor; the infrared hygrometer provided redndant information. The instrments were deployed at the end of FLIP s 0-m-long port boom. Editing for radio interference and fog contamination yielded 180 sable 50- min averages. The second dataset was taken aboard the R/V Colmbs Iselin in Jne 199 as part of the Office of Naval Research s (ONR) High Resoltion Program. The experiment was designed to investigate the case of the srface featres seen in remotely sensed images of the sea srface arond the northern edge of the Glf Stream. Sea srface temperatres in this region ranged between and 9C and often reslted in a change in the sign

5 1JULY 1998 EDSON AND FAIRALL 15 FIG. 1. A comparison of the friction velocities,, estimated sing the blk aerodynamic vs direct covariance methods. The friction velocity estimates for the R/P FLIP are compted from 50-min averages, while the R/V Iselin estimates are taken from 0-min averages. of the stability as the ship crossed the Glf Stream edge. The advection of warm air over cooler water dring a portion of the experiment reslted in a nmber of measrements made nder stable conditions. As a reslt, the combined datasets cover a stability range 8 z/l 1, which allows s to investigate the scaling laws in the limits of both local free convection and extremely stable stratification. Or estimates of the srface stress, and therefore, are compted sing the eddy correlation techniqe. Or estimates of the dissipation rate of TKE are determined from the sonic velocity spectra as described in the following section. Or estimates of the dissipation rate of temperatre and hmidity variance are compted from the sonic temperatre and infrared hygrometer hmidity spectra, respectively. The Monin Obkhov lengths were determined sing the boyancy flx from the sonic anemometer and a von Kármán constant of 0.4. The R/P FLIP and the R/V Iselin data were corrected for contamination de to platform motion sing strapdown accelerometers and rate sensors as described in Edson et al. (1998). The corrections on FLIP were mch smaller than on the Iselin, reslting in somewhat more ncertainty in the Iselin measrements. Additionally, the Iselin data exhibit more scatter near netral stratification becase these conditions are sally the reslt of being near the Glf Stream edge where conditions are inhomogeneos. In fact, we have fond that the removal of a few data points where the sign of the heat flx does not match the sign of the sea air temperatre greatly redces the scatter in or comparisons. Therefore, we have taken this approach in the following analysis to remove data collected dring very inhomogeneos conditions. FIG.. A comparison of the latent heat flxes estimated sing the blk aerodynamic vs direct covariance methods. The averaging times are as in Fig. 1. We compare or direct covariance estimates of the scaling parameters with blk estimates calclated sing the TOGA COARE algorithm (Fairall et al. 1996a) in Figs. 1. The present data are comparable to the COARE reslts, with FLIP data being significantly better for, where motion corrections and flow distortion play a smaller role than in the ship-based measrements. Ths, the greater scatter in the Iselin estimates is most obvios in the comparison given in Fig. 1. In this figre, the Iselin momentm flx estimates have been redced by 15% (a 7% redction in ) based on the R/P FLIP verss ship comparison presented in Edson et al. (1998), which acconts for the effect of flow distortion by the ship s sperstrctre. Once we have applied this correction we obtain very good agreement between the direct covariance and blk momentm and heat flx estimates over a wide range of conditions. This is particlarly tre of the latent heat flx estimates shown in Fig., which has been a difficlt variable to measre directly (or even indirectly) over the open ocean. 4. Inertial dissipation and third-order strctre fnction estimates a. Inertial dissipation estimates The dissipation rate of trblent kinetic energy is one of the most widely sed variables in investigations of both atmospheric and oceanic bondary layers. Direct measrements of the dissipation rate are sally accomplished by taking the spatial or time derivative of hotwire anemometers signals. The se of hot wires is necessary becase the anemometers mst be fast enogh to measre flctations with a spatial resoltion that

6 16 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 to find estimates of the dissipation rates from or longitdinal velocity spectra if we know the vale of the Kolmogorov constant. Estimate of this constant have been determined from direct measrement of sing hot-wire anemometry in several overland experiments. A review of these measrements by Högström (1996) sggested a vale of 0.5. FIG.. A comparison of the sensible heat flxes estimated sing the blk aerodynamic vs direct covariance methods. The averaging times are as in Fig. 1. approaches the Kolmogorov microscale, ( /) 1/4, which is typically 1 mm in atmospheric flows. Unfortnately, the se of hot-wire anemometry over the ocean is severely limited by the contamination and freqent destrction of the wires by sea spray. As a reslt, dissipation rates in the atmospheric marine srface layer are sally compted from onedimensional variance spectra measred with sonic anemometers. In the inertial sbrange of isotropic trblence, the one-dimensional velocity variance spectrm can be expressed as a fnction of wavenmber magnitde, (k) / k 5/, (14) where is the dissipation of TKE into heat, k is the wavenmber, and is the one-dimensional Kolmogorov constant. These spectra can be related to the freqency spectra S, commonly measred in the field sing Taylor s hypothesis (here, k f/u) as / fs ( f ) f / k (k), (15) T U where T is a factor that corrects for inaccracies in sing Taylor s hypothesis to estimate the magnitde of wavenmber spectra in the inertial sbrange. We se the form given by Wyngaard and Clifford (1977), 1 ( w) T 1, (16) 9(U) (U) where the standard deviations are compted from the same time series that prodced the spectral estimate (Hill 1996). This correction redces or spectral estimates by an average of %. Therefore, (15) can be sed b. Third-order strctre fnction dissipation estimates Another method that can be se to measre and determine the Kolmogorov constants relies on the skewness coefficient for the distribtion of velocity differences as described by Kolmogorov (1941). The skewness B (r) S (17) [B (r)] / is compted sing the ratio of the second- and thirdorder velocity differences, B (r) [U(x) U(x r)], (18) B (r) [U(x) U(x r)], (19) where U(x) and U(x r) are instantaneos longitdinal velocity measrements separated by a distance, r, in the direction of the longitdinal wind. In the inertial sbrange these parameters are related to the strctre fnction parameter C and the dissipation rates by B (r) / C 4.0 (0) r / and B (r) 4. (1) r 5 Based on these expressions, Kolmogorov (1941) hypothesized that in the inertial sbrange the skewness shold remain constant. Using this hypothesis and the above eqations, Kolmogorov obtained an expression that can be sed to compte the one-dimensional Kolmogorov constant as / 1. () 10S The skewness of the longitdinal velocity flctations has been reported by a nmber of researchers (e.g., see the review by van Atta and Chen 1970). In general, these researchers have relied on hot-wire anemometry to compte these statistics. A notable exception involves the work of Paqin and Pond (1971), who prodced estimates of the Kolmogorov constants sing sonic anemometer measrements by invoking Taylor s hypothesis; for example, r Ut. Paqin and Pond (1971) describe in some detail the constraints that restrict the choice of the appropriate

7 1JULY 1998 EDSON AND FAIRALL 17 spatial separations for determination of the Kolmogorov constants. The two constraints reqired of all sch measrements are that r be mch smaller than the scale of the energy-containing eddies and mch larger than the Kolmogorov microscale in order to be in the inertial sbrange. The first constraint is generally met if we confine or measrements to separation distances that are smaller than or measrement heights, that is, r/z 1. The second constraint is never an isse becase r k, even at or lowest recorded wind speed. Additionally, the se of a sonic anemometer reqires r to be mch larger than the size of the instrment. This constraint, which arises becase of the spatial averaging associated with sonic anemometers (Kaimal et al. 1968; Larsen et al. 199), reqires s to limit or analysis to separation distances mch larger than the distance between the sonic transdcers, L. While the effect of spatial averaging on B (r) is negligible in the inertial sbrange (Stewart 196; Hill 1996), its effect on B (r) can be sbstantial. Stewart (196) and Hill (1996) have derived expressions to accont for the effect of spatial averaging on second-order statistics. They have shown that for r/l k 1, [ / ] L r 1 L 9 L / B (r) Cr 1 54 r 0 r / Cr V, () sch that the strctre fnction reqires an additional correction, [U(t) U(t t)] C. (4) / (U t) T V (L/r) We note that (k) forms a transform pair with B (r) (Tatarskii 1971), which reslts in the relationship given by (0). However, this also means that (k) reqires correction for spatial averaging. This correction is obtained from the Forier transform of () given by 1 5 L 5/ 1 (k) sin Ck 54 sin 1/ Ck 5/ 0.5Ck [1 0.08(kL) ], (5) FIG. 4. Estimates of the Kolmogorov constant compted sing () as a fnction of the dimensionless length r/z. The lines in the figre denotes a vale of 0.5 (solid lines) 0.04 (broken lines). where the first term is responsible for the constant in (0). The correction term hardly affects or inertial sbrange estimates of over the wavenmber range sed in or analyses, namely, kl 1. Finally, interpretation of or measrements is simplified if we confine or investigations to look at the correlation between signals with zero lags, that is, to variances and covariances that are compted from correlations between variables measred at the same instant in time. The second constraint avoids the complications that arise from spatial displacements cased by the platform that cannot be handled by Taylor s hypothesis (e.g., Lmley and Terray 198). Nonetheless, it appears that the FLIP data are sable to examine the lagged qantities mch better than the shipboard measrements owing to its small horizontal and vertical displacements. Therefore, we confine or investigation of the skewness to the FLIP dataset. The average of all of the individal FLIP measrements of () that met the constraints r/l 10 and r/z 1 gives a Kolmogorov constant of Additionally, if we first compte the average skewness sing these same constraints and then se the average skewness in (), we obtain a vale of These vales are in remarkably good agreement with those determined by Paqin and Pond (1971), who derived a vale of sing individal estimates of () and a vale of 0.54 sing the average skewness. The dependence of or estimates of the Kolmogorov constant on the constraints given above can be examined by averaging estimates of () in bins of r/l. These binaveraged estimates are shown in Fig. 4 along with the eqivalent vales of r/z. It is interesting to note that these estimates span the range of estimates reported in the literatre and approach a vale of 0.50 at large vales of r/l. This sggests that the varios constraints considered above are responsible for mch of the experimental ncertainty. We feel that we can redce some of the ncertainty in or estimate by sing the bin-averaged reslts. This approach gives more even weight to or estimates over the entire range of r/l. The mean of binaveraged data between r/l 10 and r/z 1 gives a Kolmogorov constant of Therefore, we se

8 18 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME in the analysis that follows and recognize that the difference between this vale and 0.57 (or 0.49) introdces an ncertainty of approximately 10% in or estimates of the dissipation rate. 5. The TKE bdget The TKE bdget eqation for stationary and horizontally homogeneos conditions in the srface layer above the wave bondary layer is U V g 1 wp we w w w 0, z z z z (6) where p denotes pressre flctations and e is the trblent kinetic energy defined as e 0.5 [ w ]. The first two terms on the left-hand side of (6) represent the generation of mechanical trblence throgh shear, while the third term represents the prodction (sppression) of trblence throgh convection (stratification). The final two terms neither prodce nor consme TKE; instead they act to redistribte TKE within the atmospheric bondary layer throgh pressre and energy transport. According to MO similarity theory, the varios terms in the TKE bdget are expected to be niversal fnctions of z/l after normalization by the appropriate scaling parameters, kz/, z () m () tp () te (), (7) where z/l, is the dimensionless dissipation fnction, tp and te are the dimensionless transport terms, and the dimensionless shear is defined as [ ] 1/ z U V m(). (8) z z Eqation (8) is consistent with or definition of and the following parameterizations for the momentm flx components: z U w (9) m() z and z V w. (0) m() z These parameterizations are common applications of MO similarity theory in first-order closre models. They offer a good example of why we are limiting or analysis to heights above the WBL, since we wold not expect these expressions to be valid in a region where the waves, in addition to the stability, can effect the velocity field and thereby modify the wind profiles. In the sections that follow we determine the form of the dimensionless fnctions in (7) sing or combined datasets. The wide range of stabilities present in this dataset allows s to examine the limiting forms of these fnctions in both the convective and stable limits. The behavior of the varios terms in the TKE bdget in these limits are predicted sing the approach otlined by Wyngaard (197). a. Dimensionless dissipation fnction The fnction that describes the stability dependence of the dimensionless rate of dissipation is a key ingredient in the inertial dissipation method of estimating air sea flxes (Fairall and Larsen 1986). A nmber of observation stdies condcted over the sea (e.g., Large and Pond 1981; Edson et al. 1991) have fond that the dissipation of TKE is very nearly in balance with its prodction, which redces (7) to () m (). (1) It is easy to show that this form of the dimensionless dissipation fnction is consistent with its predicted form in local free convection. In this limit we expect the dissipation rate to be proportional to f /z. This prediction can be combined with (11) to find its form in the local free-convective limit as z z f () a, 1, () f where a is a constant of proportionality. Since shear prodction and therefore m become negligible in this limit, (1) and () are identical if the constant of proportionality is eqal to one, which is the expected reslt if prodction trly eqals dissipation. The vale of this constant can be determined from or measrements by setting m () te () tp () 0 in (7), which reslts in 1 g a w. () The average vale of this constant from all estimates of a fond for 1.5 is Althogh the scatter is fairly large, this implies that prodction balances dissipation in the local free-convective limit. We now compare this reslt with or measrements of () in Fig. 5. In this figre, the solid line is drawn sing (1), and the broken line is drawn sing a 1 in (). Since we are not able to compte an estimate of the dimensionless shear with the present dataset, we have opted for consistency and sed a fnction that has the correct form in the convective limit, m () (1 15) 1/, 0, (4) as given by Carl et al. (197) and more recently by Frenzen and Vogel (199). This fnction agrees very well with the commonly sed Bsinger Dyer formlation in forced convection.

9 1JULY 1998 EDSON AND FAIRALL 19 FIG. 5. Estimates of the dimensionless dissipation rate of TKE as a fnction of z/l. The left-hand panel displays the raw data, while the right-hand panel shows the bin-averaged reslts. The error bars denote the standard error (standard deviation divided by the nmber of points). The lines drawn in the figre are identified by the eqation nmbers as given in the text. As one wold expect, the measrements are in excellent agreement with (1) in very nstable conditions since a 1. However, it is clear that this simple parameterization is not accrate for slightly nstable and near-netral conditions where or data indicate that prodction exceeds dissipation. This finding is consistent with recent measrements reported by Thiermann and Grassl (199), Vogel and Frenzen (199), and Frenzen and Vogel (199), who also fond that prodction exceeds dissipation in the slightly nstable regime. Their dimensionless dissipation fnctions are shown in Fig. 6. The average vale of () for this dataset is eqal to 0.70 over the stability range This vale is in good agreement with the reslts reported by Garratt (197), who obtained an average vale of 0.78 for measrements within the range sing a Kolmogorov constant of This vale wold be redced to 0.71 sing a Kolmogorov constant of 0.5. This slightly nstable estimate is also in agreement with the netral vales of dimensionless dissipation, (0), given by Frenzen and Vogel (199) and Vogel and Frenzen (199) in two separate experiments. Their netral vales ranged from 0.84 to b. Dimensionless transport terms FIG. 6. Fnctional forms of the dimensionless dissipation fnctions that inclde two recently pblished parameterizations as well as or own fnctions. The symbols denote bin-averaged reslts shown in Fig. 5. The solid line drawn in the figre are identified by the eqation nmbers as given in the text. The balance between prodction and dissipation in free-convective conditions implies that we are sing the correct vale for the Kolmogorov constant. This assmption frther implies that the imbalance at near-netral conditions is de to the exclsion of the transport terms in or simple parameterization. The direct determination of the energy transport terms in (6) reqires flx estimates at mltiple levels that are not available in or datasets. However, we can se the wide range of stability in or datasets to or advantage by estimating these terms sing the derivative method described in Wyngaard and Coté (1971) and Wilczak et al. (1995). In this method the normalized flxes are plotted as a

10 0 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 e () () 1/ (1 ) /, 0, (8) as shown by the solid line in Fig. 7. It provides a good fit to the rapid rise exhibited by the data in near-netral conditions, as well as the tendency toward a smaller slope as increases while retaining the correct local free-convective limit. This fnction is easily differentiated to obtain the dimensionless energy transport fnction in nstable conditions as FIG. 7. Estimates of the dimensionless energy flx as a fnction of z/l. The symbols represent or bin-averaged reslts. The error bars denote the standard error. The lines drawn in the figre are identified by the eqation nmbers as given in the text. fnction of to determine the dimensionless energy flx fnction, we e(). (5) This flx shold scale with f in the convective limit, which leads to the prediction f we we c, 1, (6) f where c is another nmerical constant. The derivative of this fnction is then combined with the definition of the dimensionless energy transport to obtain a prediction for local free convection, z we we te() c, 1, z (7) where we see that the constant is only modified by removal of the von Kármán constant. The dimensionless energy flxes are plotted in Fig. 7. The broken line in this figre is drawn sing the Kansas reslts reported by Wyngaard and Coté (1971). Wyngaard and Coté (1971) obtained a vale of c 1.0 sing a von Kármán constant of 0.5 sch that c/.9. Note that we have added an offset of 0.5 to this line [i.e., (7) becomes.9 0.5] and that or definition of the kinetic energy is related to Wyngaard and Coté s as q e. Or near-netral to moderately nstable reslts are also in good agreement with the data reported by Garratt (197), Banke and Smith (197), and Vogel and Frenzen (199) over a similar stability range. To inclde the behavior of this flx over the entire range of nstable conditions we introdce the fnction given by 4/ 1/ te() [4() (1 ) e()], 0. (9) The reslts from this stdy and previos investigations show that the trblent transport of TKE is a nonnegligible term in the bdget eqation. Therefore, the observed near-balance between prodction and dissipation in the free-convective limit sggests that the pressre transport term mst be nearly eqal in magnitde bt opposite in sign to the energy transport. This is in agreement with the findings of McBean and Elliot (1975). Under slightly nstable conditions or reslts indicate that prodction significantly exceeds dissipation. This sggests that the magnitde of the TKE transport term (a loss of TKE) is greater than the magnitde of the pressre transport term (a gain of TKE). Althogh we believe that the observed deficit is a reslt of a local imbalance in the transport terms, we recognize that the actal mechanisms responsible for the imbalance are debatable. A recent investigation by Edson et al. (1997) sing measrements from the 1995 marine bondary layer (MBL) experiment aboard the R/P FLIP indicate that this dissipation deficit is a strong fnction of sea state, implying that MO similarity theory is not valid even well above the srface nder high-wind conditions (see section 8). However, althogh there is some evidence for this effect nder the highest wind conditions, the agreement between or data and recent overland reslts sggests that the measrements are generally above the WBL. Therefore, we believe that we are in a region of the marine srface layer where MO similarity is a valid hypothesis. As sch, we recommend the following dimensionless dissipation fnction for se in shipbased inertial dissipation systems: (1 ) (), 0. (40) (1 7) This fnction represents a slightly modified form of the fnction given by Thiermann and Grassl (199), which retains their fnction s simplicity while providing better agreement with or data, as shown in Fig. 6. c. Stable regime As we move away from netral conditions into the stable regime, the boyancy force begins to restrict the

11 1JULY 1998 EDSON AND FAIRALL 1 prodction TKE throgh shear by limiting the size of the energy-containing eddies (i.e., their velocity flctations). This is a reslt of the restoring forces that limits the displacement of the parcels from their eqilibrim position. In extremely stable conditions, the size of the eddies are completely limited by the stability (Wyngaard 197) and they become naware of their distance from the srface. The scaling becomes heightindependent nder these conditions and the MO length becomes the only length scale. As a reslt, we often refer to sch conditions as z-less stratification. This also means that is the only velocity scale becase we cannot form another from the three remaining governing parameters. Under these conditions the variance terms in the TKE bdget shold go asymptotically as (Wyngaard 197). Or dimensionless energy flx analysis and the Kansas reslts indicate that the transport terms are trly negligible in stable conditions sch that (1) is an appropriate form of the dimensionless dissipation fnction. Additionally, the limiting form of the dimensionless shear in very stable conditions sggest that it shold go as m () 1 e, 0. (41) This reslts in a form of the dimensionless dissipation fnction given by () 1 (e 1), 0, (4) where a vale of e 6 gives good agreement between or data, as shown in Fig. 6. The form of all of the proposed fnctions in the TKE bdget eqation is smmarized in Fig. 8, where the pressre transport is derived by sbtracting (9), (40), and from (4). These fnctions are in good agreement with the consenss fnctions presented by Wyngaard (199). 6. The scalar variance bdgets The scalar eqivalents of the TKE bdget eqation are the potential temperatre and specific hmidity variance bdgets. In homogeneos and steady-state conditions, these bdgets are as follows: FIG. 8. The dimensionless TKE bdget terms proposed in this paper. The boyant prodction terms is simply z/l, the shear prodction is given by (4) and (41); the trblent transport by (9); and the dissipation by (40) and (4). The pressre transport is the residal of these fnctions. 1 w w N 0, (4) z z Q 1 wq wq Nq 0, (44) z z where the first terms represents the prodction of SV, the flx divergence terms again act to redistribte the variance, and N and N q are one-half the dissipation rate of potential temperatre variance and specific hmidity variance, respectively. These terms are made dimensionless by dividing these terms by z/ x where x, q.rearrangement of the dimensionless expressions reslts in the SV dissipation fnctions Nz N () h() t() (45) T and nqz N () w() tq(). (46) q q The dissipation, prodction, and transport terms in each of these eqations are expected to go as 1/ in the local free-convective limit (Wyngaard 197). The SV dissipation rates are compted sing or estimates of the SV spectra in the inertial sbrange. In this sbrange, the Kolmogorov variance spectrm for temperatre and hmidity is / fs x( f ) f 1/ k x(k) x N x, (47) T U xx where x is the Obkhov Corrsin constant and 1 1( w) xx T 1, (48) 9(U) (U) as given by Wyngaard and Clifford (1977). The investigations provided by Hill (1989a,b) and Andreas (1987) have shown that the temperatre and hmidity fnctions are eqal and mst share the same vale of the Obkhov Corrsin constant within the constraints of MO similarity theory. In this investigation we have sed vale of x 0.80 reported by Wyngaard and Coté (1971), Paqin and Pond (1971), Champagne et al. (1977), and Högström (1996). Or plots of the dimensionless scalar dissipation fnctions are shown in Fig. 9. The dotted line drawn in the

12 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 FIG. 9. Estimates of one-half the dimensionless dissipation rate of temperatre variance, N, and specific hmidity variance, N q, as a fnction of z/l. The left-hand panel displays the raw data, while the right-hand panel shows the the bin-averaged reslts. The error bars denote the standard error. The dotted line represents the Bsinger Dyer formlation of the dimensionless scalar gradient fnction given by (49). The other lines drawn in the figre are identified by the eqation nmbers as given in the text. FIG. 10. Estimates of the dimensionless temperatre variance flx, w, and specific hmidity variance flx, wq, as a fnction of z/l. The symbols denote or bin-averaged reslts. The error bars denote the standard error. The lines drawn in the figre are identified by the eqation nmbers as given in the text. figre represents the Bsinger Dyer formlation for the dimensionless scalar profiles, y () (1 16) 1/, 0, (49) where y h, w. Althogh the exponent sed in this fnction does not agree with or prediction in the convective limit, a nmber of field experiments (Bsinger et al. 1971; Dyer 1974; Dyer and Bradley 198) have shown that it provides a better fit to the data in near-netral conditions than a fnction with a () 1/ dependency. Or data also exhibit good agreement with this fnction in near-netral conditions. However, the agreement between or averaged data and the Bsinger Dyer formlation clearly worsens as the conditions become more convective. This sggests that the transport terms are no longer negligible in convective conditions. We can infer something abot the dimensionless form of the transport terms by sing or SV flx estimates with the derivative approach explained above. In the local free convective limit we have wx wx x f f 1/ x() B(), 1, x fxf x (50) where the nmerical constant incldes the von Kármán constant as B b 1/. The plot of this fnction is shown in Fig. 10, where the broken line in this figre is or convective limit prediction. The constant that gives the best agreement between or data and the prediction is given by B 1. The averaged data in this figre follow the free-convective prediction to abot 0.5 and then abrptly retrns to the origin. This behavior is consistent with the reslts reported by Wyngaard and Coté (1971). We can parameterize this behavior with a fairly simple fnction shown by the solid line in this figre, which is given by x () 18(1 8) 4/, 0. (51) This fnction provides good agreement with the data in nstable conditions and has the proper form in the freeconvective limit. The fnction is easily differentiated to

13 1JULY 1998 EDSON AND FAIRALL tx() [(4) (1 8) x()], 0. 6 (5) The behavior of this transport term is consistent with the observations of Deardorff (1966) and Wyngaard and Coté (1971). It represents a loss of SV in near-netral conditions and a slight gain in very nstable conditions, as shown in Fig. 11. The dimensionless dissipation fnction can then be parameterized by inserting (49) and (5) in (45) and (46). The sm of these two fnctions agrees very well with or averaged data, as shown by the broken line in Fig. 9. Unfortnately, besides being rather cmbersome, this fnction does not strictly have the proper convective form. However, we can se the good agreement between the Bsinger Dyer formlation and or near-netral data to determine an estimate of the dimensionless SV dissipation fnction that has the correct convective limit form and is in good agreement with the measrements over the entire range of data. This fnction is represented by the solid line in Fig. 9 and is given by 1/6 1/ N x () (1 ) (1 16), 0. (5) The prodction crve in Fig. 11 represents the sm of (5) and (5). Finally, or reslts have shown no clear trend in the dimensionless SV flxes nder stable conditions. This sggests that the derivative of this fnction (and therefore the trblent transport) is negligible nder these conditions, sch that prodction eqals dissipation. Althogh we have a limited amont of data in stable conditions, we obtain good agreement sing the same constant that we have sed for the dimensionless shear sch that N () h() 1 6, 0. (54) x This is consistent with the findings that the dimensionless shear and profile fnctions are identical nder stable conditions (e.g., Panofsky and Dtton 1984). FIG. 11. The dimensionless scalar variance bdget terms proposed in this paper. The trblent transport is given by (5) and the dissipation by (5). The prodction term is the sm of these two fnctions. The broken line is the Bsinger Dyer formlation for the prodction term. obtain a form of the dimensionless transport fnction given by 7/ 7. The strctre fnctions The method sed above to compte the dissipation rates of temperatre and hmidity variance reqires s to combine TKE dissipation rates (compted from or velocity spectra) with or scalar spectra. This approach can become problematic when the instrments sed to measre the velocity and scalar qantities are not collocated. Additionally, becase we sed inertial sbrange techniqes to determine the dissipation rates, or reslts on the TKE and SV bdgets depend on the choice of Kolmogorov constants. An approach that can be sed to avoid these problems is to se the strctre fnction parameter to investigate some of the characteristics of the trblence. The dimensionless strctre fnction parameters are known to obey MO similarity theory (Wyngaard et al. 1971b; Wyngaard et al. 1978; Friehe et al. 1975; Fairall et al. 1980; Edson et al. 1991). In fact, the relationship between the strctre fnction parameters and the dissipation rates given by C 4.0 x 1/ x N x (55) reslts in a form of the dimensionless strctre fnction parameter given as / Cz x / 1/ f x() 4.0x N () (), (56) x x where x is now eqal to,, and q. Using the Kolmogorov and von Kármán constants sed in this investigation and (40) and (4) for the dimensionless dissipation fnction, the dimensionless velocity strctre fnction parameter becomes and ] [(1 ) f ().9, 0 (57) 7) / f ().9(1 5), 0. (58) The agreement between this fnction and or data is very similar to that shown in Fig. 6. Or dimensionless scalar strctre fnction parameter estimates are shown in Fig. 1. In this figre, the solid line is drawn sing (56) with Eqs. (5) and (54) for N x () and (40) and (4) for (). Alternatively, the limiting forms of the dimensionless dissipation fnctions predict that this fnction shold be proportional /