Flux-Profile Relationships in the Atmospheric Surface Layer

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  • 1 Air Force Cambridge Research Laboratories, University of Washington, Seattle
  • | 2 Air Force Cambridge Research Laboratories, Bedford, Mass
  • | 3 Commonwealth Scientific and Industrial Research Organization, Canberra, Australia
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Abstract

Wind and temperature profiles for a wide range of stability conditions have been analyzed in the context of Monin-Obukhov similarity theory. Direct measurements of heat and momentum fluxes enabled determination of the Obukhov length L, a key independent variable in the steady-state, horizontally homogeneous, atmospheric surface layer. The free constants in several interpolation formulas can be adjusted to give excellent fits to the wind and temperature gradient data. The behavior of the gradients under neutral conditions is unusual, however, and indicates that von Kármán's constant is ∼0.35, rather than 0.40 as usually assumed, and that the ratio of eddy diffusivities for heat and momentum at neutrality is ∼1.35, compared to the often-suggested value of 1.0. The gradient Richardson number, computed from the profiles, and the Obukhov stability parameter z/L, computed from the measured fluxes, are found to be related approximately linearly under unstable conditions. For stable conditions the Richard on number approaches a limit of ∼0.21 as stability increases. A comparison between profile-derived and measured fluxes shows good agreement over the entire stability range of the observations.

Abstract

Wind and temperature profiles for a wide range of stability conditions have been analyzed in the context of Monin-Obukhov similarity theory. Direct measurements of heat and momentum fluxes enabled determination of the Obukhov length L, a key independent variable in the steady-state, horizontally homogeneous, atmospheric surface layer. The free constants in several interpolation formulas can be adjusted to give excellent fits to the wind and temperature gradient data. The behavior of the gradients under neutral conditions is unusual, however, and indicates that von Kármán's constant is ∼0.35, rather than 0.40 as usually assumed, and that the ratio of eddy diffusivities for heat and momentum at neutrality is ∼1.35, compared to the often-suggested value of 1.0. The gradient Richardson number, computed from the profiles, and the Obukhov stability parameter z/L, computed from the measured fluxes, are found to be related approximately linearly under unstable conditions. For stable conditions the Richard on number approaches a limit of ∼0.21 as stability increases. A comparison between profile-derived and measured fluxes shows good agreement over the entire stability range of the observations.

1971 BUSINGER, WYNGAARD, IZUMI AND BRADLEY 181Flux-Profile Relationships in the Atmospheric Surface LayerJ. A. B~JsrNo~.R,~ J. C. WYNOAARD,~ -. IZUM~2 ANO E. F. BRADLEY~ (Manuscript received 27 July 1970)ABSTRACT Wind and temperature profiles for a wide range of stability conditions have been analyzed in the contextof Monin~Obukhov similarity theory. Direct measurements of heat and momentum fluxes enabled deterruination of the Obukhov length L, a key independent variable in the steady-state, horizontally homogeneous, atmospheric surface layer. The free constants in several interpolation formulas can be adjusted to giveexcellent fits to the wind and temperature gradient data. The behavior of the gradients under neutral conditions is unusual, however, and indicates that yon K~rm/m~s constant is ~,~0.35, rather than 0.40 as usuallyassumed, and that the ratio of eddy diffusivities for heat and momentum at neutrality is ~-~1.35, comparedto the often-suggested value of 1.0. The gradient Richardson number, computed from the profiles, and theObukhov stability parameter z/L, computed from the measured fluxes, are found to be related approximately linearly under unstable conditions. For stable conditions the Richardson number approaches alimit of ~0.21 as stability increases. A comparison between profile-derived and measured fluxes shows goodagreement over the entire stability range of the observations.1. Introduction The impormuce of turbulent exchange processes in theatmospheric boundary layer to the general circulationof the atmosphere has long been recognized. Over a dry,flat, horizontally homogeneous land surface, the predominant processes are the vertical transport of momentum and sensible heat. A central and recurringfeature of much of micrometeorologlcal research hasbeen to establish means of deriving these fluxes fromwind specd and temperature profiles. Indeed, the literature abounds with profile fonuulas intended to accomplish this, particularly for the surface layer, the lowest50 m or so of the boundary, layer where the Coriolisforce can be ignored and the fluxes can be assumedconstant with height. Recent profile formulas reflect thesimilarity concepts put forward originally by Obukhov(1946). The first experimental evidence for these concepts was given by Monin and Obukhov (1954). The testing of similarity predictions and profileformulas is ideally done with directly measured fluxesas well as profiles, a requirement which is rarely satisfiedin micrometeorological field experiments. In the Kerangand Hay expeditions, for example, the results of whichhave been presented by Swinbank (1964) and Swinbankand Dyer (1968), and which have been extensivelyanalyzed in the literature by Charnock (1967), Dyer(1965, 1967, 1968) and Swinbank (1968), stress was notmeasured directly but was inferred from the profiles. ~ Work performed at Air Force Cambridge Research Laboratories on a National Research Council Associateship on leavefrom the University of Washington, Seattle. ~- Air Force Cambridge Research Laboratories, Bedford, Mass. ~ Commonwealth Scientific and Industrial Research Organization, Canberra, Australia. The purpose of this paper is to present an analysis ofa set of data that includes both profile and flux measnrements over horizontally uniform, flat terrain. The results have been analyzed in dimensionless form, usingthe dimensionless quantities: gOO/OzRiO(Or)/Oz)~Richardson number, a stability parameter (1)kzu, OzA dimensionless wind shear (2)z 00O, OzA dimensionless temper ature gradient (3) K~, w~O~ OU/Oz Ratio of the eddy transfera =. coefficients (4) K,~ u'w~ O~/Ozz kgw~O'zL Ou,aA dimensionless height (5) List of Symbolsg acceleration due to gravityk yon K~rm~n's constantu, v, w longitudinal, lateral and vertical components of the windmean horizontal wind speedmagnitude of the mean horizontal wind vectorvertical coordinatez0 roughness length0 potential temperature182 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME280.L~'0friction velocity E(r0/p)l~scaling temperatureObukhov length E--u.a/(kgw'O'i~specific heat of air at constant pressureair densitysurface shearing stressAn overbar denotes a time average and a prime thedeviation from the average.rather for experimental convenience., than for objectiveconsideration of steady-state periods. At typical runlength was one hour, occasionally :shortened to 30 or45 min during the actual experiment or during subsequent analyses. Twenty-eight of 'the runs to be discussed here were one hour long, two were 45 min, andfour were 30 min. However, unless otherwise noted,the results to be presented will be based, on the basic15-rain averaged data.2. Instruments and experimental procedures The data were obtained during the summer of 1968at a site located in wheat-farming country of southwestern Kansas. A description of the site, the instruments, and the' experimental procedures followed hasalready appeared in the literature (Haugen et al., 1971).Only a brief outline will be given here. Instruments were mounted on a 32 m tower locatedin the center of a one mile square field of wheat stubble~ 18 cm tall. The neighboring section was also coveredby wheat stubble, giving 2400 m of uniform fetch. Therewas no change in the surface during the experiments.Profile analyses to determine relevant characteristics ofthe surface included data from two masts of small cupanemometers (Bradley, 1969) with units mounted oneach mast at heights of 23, 36, 50, 70 and 100 cm. Theseanalyses suggest a surface roughnes length of ~2.4 cmand a zero plane displacement of ~--10 cm. Wind speeds on the 32 m tower were measured withthree-cup anemometers manufactured by ControlEquipment Corporation, mounted at heights of 2, 4,5.66, 8, 11.3, 16, 22.6 and 32 m. In addition, two ofthese anemometers were mounted at 0.5 and 1 m,displaced ~30 m upwind from the tower. Temperaturedifferences were measured with a high-resolution systemdeveloped by Stevens (1967), with sensors at 0.5, 1, 2,4, 8, 16, 22.6 and 32 m. The instrument outputs weresampled once a second and stored on magnetic tape bya computer-controlled data acquisition system (Kalmalet al., 1966). Fluctuating wind components were measured withthree-component sonic anemometers (Kaijo Denki,Model PAT 311) at 5.66, 11.3 and 22.6 m. Temperaturefluctuations at the same levels were obtained withplatinum wire resistance thermometers (CambridgeSystems Inc., Model 127). Surface stress was measuredwith two independent drag plates of the type describedby Bradley (1968). These instruments were sampled 20times per second and filtered, the former with a 10-Hzlow-pass filter and the drag plates with a 5-Hz low-passfilter.3. Data analysisa. Data set In all, 34 runs have been analyzed. The basic averaging period used was 15 rain, an averaging period chosenb. Determination of mean gradients Gradients of mean wind speed were fo'und by differentiating second-order polynomials, in ln(z) fitted tospeeds measured near the level in question. In mostcases, five levels were used for each gradient computation: the level itself, two above, and two below. An8 m wind shear, for example, was found from the 4,5.66, 8, 11.3 and 16 m wind speed:~. At 22.6 m, threespeeds below and one above the level were used. Thesame method was used to compute temperature gradients, except at 5.66 and 11.3 m, where only four levelswere used, two above and two below. Computer experiments were performed to test thistechnique. The Businger-Dyer formulas, which willlater be shown to fit the observatiorts well, were used togenerate sample unstable wind and temperature profiles,so that the gradients were known exactly. The gradientscalculated by the curve-fit method agreed well with theexact values, the deviations being at most 20'/o. Understable conditions the observed profiles are closely loglinear, as will be shown later. To te~t the method usingthe second-order polynomial in ln.(z), the gradients wererecalculated from the raw data by using as the polynomial, aq-bln(z)=cz, which fits log-linear profilesexactly. Here the discrepancies were somewhat larger,but neither the zero intercepts nor the slopes of thedimensionless gradients differed significantly betweenthe two methods. It was therefore concluded that thetechnique was satisfactory, and th~ gradients calculatedat 4, 5.66, 8, 11.3, 16 and 22.6 m were taken as the basicset of profile data to be considered. Temperature bath calibrations and on-site atmospheric comparisons indicate that the raw temperaturedata are accurate to ~ 0.02C or better, independent ofthe ambient temperature. The accuracy of the windspeed data, however, is more difficult to assess becausethe response of a cup anemometer to a fluctuatinghorizontal wind field is nonlinear. Typical cup anemometers also respond nonlinearly to the vertical wind component. These characteristics lead to overspeeding, orindicated wind speeds greater than the actual horizontalwind speed. The importance of assessing overspeedinghas been stressed by a few authors (e.g., Frenzen, 1966;MacCready, 1966) but quantitatiw~ estimates are rare. For the data at hand, a good estimate of overspeeding(:an be made by comparing the cup anernometer speedswith the mean horizontal wind speeds from the sonic1971 BUSINGER, WYNGAARD, IZUMI AND BRADLEY 183anemometers at the corresponding levels. Sonic anemometers are ideal for this purpose, being absoluteinstruments, inherently linear (in contrast to most otherwind speed instruments), and very accurate. Windttmnel calibration and side-by-side atmospheric cornpar[son tests (Kaimal and Haugen, 1969; Izumi andBarad, 1970) indicate that their mean wind speedreadings are accurate to within 1%. Izumi and Baradmade the speed comparison, and found that the cupspeeds are ~ 10% too high for all stabilities. Therefore,the calculated wind shears were reduced by 10%. Theresidual overspeeding error restdting from applying aconstant correction factor is of the order of ~:3%. It has become traditional in micrometeorologicalprofile research to treat the mean horizontal wind speed,,~, rather than the magnitude of the mean horizontalwind vector, [Y, although it could be argued that ~ ismore appropriate. They are rdatcd, to a second-orderapproximation, bywhere v'" is the variance of the lateral wind speedfluctuations. In prindple, the ~ values can be convertedto l~ values with Eq. (6); the correction is a 'weak function of height and stability and ranges from 1 toin the extreme. We have not made this correction, buthave elected to work with wind speed in the manner ofother investigators. Our point is simply that the windshears used in the analysis are perhaps very slightlyoverestimated.c. D~lermination of the surface shearing str,ss and heat fiux A detailed discussion of both the stress and heat fluxmeasuremenLs has been given by Haugen et al. (1971).These quantities were computed for each 15-rainperiod, as were the profile data. In addition, 1-hraverages of the Reynolds stress and heat flux werecomputed for the three tower levels. The average of thefour 15-rain results for each run was compared with thehourly average to determine the contribution fromlow-frequency fluctuations to the fluxes. Individualruns yielded differences of about 5% in the extreme,with the overall difference for the entire data set beingless than 1%. We therefore conclude that the 15-rainaverages provide adequate estimates of the Reynoldsstress and heat flux for the purposes of this paper. The results from the flux study of Haugen et al. whichare used here will be briefly summarized. Both heat fluxand Reynolds stress were found to be constant withheight, within the limits of observational accuracy,although, overall, the sonic anemometer measurementsindicated a decrease in stress between 5.66 and 22.6 mof about 60-/o. This is consistent with the decrease withheight which would be expected from consideration oftypical mean horizontal pressure gradients. The trend was extrapolated linearly to the surface toobtain an estimate of overall surface stress, which wasfound to be lower, by a factor of 0.67, than the overalldrag plate value. The discrepancy was ascribed to thesubjective technique required to match the drag platesample tray to its surroundings. No such uncertainty ofexposure exists in the case of the sonic anemometer, sothe overall sonic stresses were taken as a reference, andthe factor 0.67 used as a correction factor for the dragplate measurements. Although the drag plates did not measure accuratelythe absolute value of the stress, its temporal variationwas considerably less than that of the eddy-correlationtechnique. Haugen et al. took advantage of this fact toreduce the scatter in data diagrams involving u.. Thecomparison of their value of 1.3 for a(w)/u, at neutralitywith that of other workers is a useful justification of thestress correction procedure, particularly since it doesnot involve von K~rm~n's constant. The significance ofthis will become apparent in the next section. We find, also, that the drag plate measurements leadto very dosely ordered data diagrams, after correctionfor the overestimate, and are therefore the values usedin this paper. The heat flux value for each run is theaverage of the measurements at the three sonic levels.4. Resultsa. Dimensionless wind shear, In Fig. 1, the dimensionless wind shear is plottedagainst ~. The most remarkable aspect of this plot isthat 4~(0), the value at neutral stability, is about 1.15(see inset in Fig. 1) rather than 1.0 as required. If thedata are correct, we must conclude that yon Kfu-man'sconstant k is not 0.40 as assumed, but is ~0.35. It isperhaps worth noting that the latter value agrees wellwith a recent prediction of 0.34 by Tennekes (1968). The reduced value of k comes as a surprise, and itmight be well to review the steps leading to its determination. The basic data set incorporates correctionfactors for both the wind shear and surface stress. Whilewe feel that these factors are nearly completely effective, they are necessarily based on a limited data sampleand are subject to some uncertainty. However, had thewind shear values not been corrected for overspeeding,the derived k would have been even smaller at 0.32. Ifthe sonic anemometer stress values had been used toderive u., k would have been 0.34. It therefore appearsunlikely that these factors could account for the departure from the usually assumed value of 0.40. As havemost workers, we have assumed horizontal homogeneity, on the basis of visual observations of surfaceconditions and fetch; the lack of quantitative evidenceof homogeneity introduces some uncertainty in theinterpretations, but again it does not seem likely thatthis could account for the reduced k. Accordingly, wehave used k=0.35 in the calculations in this paper, and184 JOURNAL OF THE ATMOSPHERIC SCIENCES VOnUME28 k'0.35 k"0,40 1,4- -I,6 1.2- -I,4 -0.8 0.6- 5 - 0.6 0'4-_0.4 - 0.2--0.2 ~k=O,40 4' - '1 I I I- 0,10 -0.0,5 0 0.05 0.10..... ~m= {I-15~)3 2- 2.5 - 2.0 -t.5 -I.0 -0.5 0 0.5 1.0 1,5 2.0Fro. 1. Comparison of dimensionless svind shearobservations with interpolation formulas.look for further carefully performed experiments toverify this new value. Several expressions for qbm have been suggested, andthe present data can be used to test them. A comprehensive survey will not be attempted here, but twoformulas will be treated briefly; these are the so-calledKEYPS formula (see Panofsky, 1963; Lumley andPanofsky, 1964), defined by ~m4-3'1~*/~)m3: 1, (7)and a modification of the KEYPS formula suggested byBusinger (1966) and Dyer (unpublished), i.e., q~m = (1--3'20-t (8)These expressions are intended for unstable conditions,and have been chosen from the many described in theliterature because of their relative simplicity. Eachequation contains an adjustable constant. The freeconstants were determined by fitting Eqs. (7) and (8)to all the unstable observations and minimizing themean square error, with the result that -n=9 and72= 15. Fig. 1 shows that both of these expressions fitthe observations well.One aspect of similarity theory that has been discussed extensively in the literature is the behavior of qb~near neutral stability, ~'=0. A quadratic curve fitted'through the points shown in the inset in Fig. 1 hasthe form ~,~ = 1 +3.0~+ 10.2:,~, (9)which has a slope of 3.0 at ~ =0. Previous estimates ofthe slope vary widely, from 0.5 to near 5, reflecting thelack of agreement.in the literature on the shape of qS,~.In both the KEYPS and Businger-Dyer formulas theslope at the origin is 3'/4. Our values for the free constants then give slopes of 2,2,5 and. 3.75, respectively,which deviate equally on either side of the measuredvalue. Under stable conditions, Fig. 1 indicates that ~varies essentially linearly with ~ over the entire stabilityrange of the observations. The curve drawn has the form qb,~ = 1d-4.7L (10)and the scatter of the data appears to allow slopes inthe range 4.5--5.0. Eq. (10) is a good overall fit, but itsslope near neutrality is about 50% greater than indi~h c~h -1.2-. 70,40.2- 5- 'I I I I-0.10 -0.0 0 0.05 0.10 4- I/2..... qbh= 0.741~-9~ ) 2.... 4. :o.~(,*~1C I~'* ~~'' -2.5 -2.0 -I.5 -I.0 -0.5 0 0.5 1.0 1.5 2.0Fig. 2. Comparison of dimensionless temperature gradientobservations with interpolation formulas.MARCH 1971 BUSINGER, WYNGAARD, IZUMI AND BRADLEY 185cared by the observations; apparently the slope of ~b~changes rapidly as the neutral point is crossed.b. Dimensionless tempt~ralure gradient, A plot of -~, vs f is given in Fig. 2. Again we find thatthe results show remarkably little scatter, thus permitring a careful comparison with existing interpolationfox,as. Fkst we note that -n(0) does not appear to be equalto 1.0, but is ~0.74. This means that the ratio of theeddy diffusMties is not unity even for neutral conditlons, a point discussed in more detail later. The normalized hnction ~(f)/~(0) h~ been compared with the interpolation fm~as of Elliott (1966)and of Businger-Dyer (Businger, 1966), respectively,as follows: -,,(f)/Oa -) = (1 --~r)-~, (12)both of whi& are applicable to unstable conditions. These functions are very sitar to each other andagree rather well with the obse~ations if 7a=4 and,~=9, which were obtained with the previously mentioned m~imum square error technique. The behaviorof ~t~ near the neutral po~t has also been investigatedby fitting a quadratic curve through the near neutrMdata, as shown in the inset in Fig. 2. In this case thecurve is given by ~ =0.74+3.0r+9.2f~. (13)The slope at the origin is 3.0, in good agreement withthe value of 3.3 obtained from the Businger-Dyerfomula [Eq. (12)]. Elliott's fom~a kmplies an infinite slope at f =0 and is not meant to be valid there.A comparison M~ the corresponding expression for~ in Eq. (9) shows that the shapes of -n and ~,,, arevery shnilar in near-neutral con~fions. In the stable range, ~, is well represented by a linearfunction of ~. The curve in Fig. 2 is -~, =0.74+4.7f, (14) with an uncertainty of perhaps ~0.5 h~ O~.e slope due to scatter of the point. As with ~, we note that the slope changes rapidly tom neutr~ to stable conditions. The very u~table cases have been plotted separatdy in Fig. 3 in logdog coordinates to investigate ~e be haG. or when approa&ing free convection. It is evident from this plot ~at Ca approaches ~ b&aGor ra~er ~mn the ~-t predicted origin~ly by Prandd (1932) and later by Obukhov (1946) and also by Priestley (1954). This is in agreement with Dyer's (1965) analysis of the Kerang amt Hay data, and with Eqs. (11) and (12). As po~ted out by EHiott (1966), the obse~ed power law behaGor 1.0 O.B 0.6 0.4~f ~ I I i i I i I I i .... q~h ~ I~"~ -,,2 - ~ -h~l~l I I I 018I I. I I 0.2 014 016 . l0 2.0 4.0 -~Fro. 3. The dimensionless temperature gradienLunder very unstable conditions.implies, from the definitions of q~a and f, that 05 (w 0 )q:} u.,z-'. (16) Oz \0/On the other hand, the result ,-~ ,:,: r-~ (17)is based on the assumption that u. no longer enters ~ abasic variable, so that O0 , ,ig5-~ ,If the obse~ed behavior continues to the tree freeconvection state, where u. vanishes, we have a d~emma,since Eq. (16) indicates that the temperature gradientthen vanishes as well. In the atmosphere, Ge temperature gradient usually vanishes at 100 m or so underunstable conditions, and there is a tendency for thisheight to decrease with increasing instability, as can bededuced from the Cedar Hill observations (Kaim~,1966). Nevertheless, O0/Oz cannot vanish near thesurface. Therefore, if Eq. (16) describes the correctheight dependence for conditions approa&ing and including free convection, a new scale has to be introducedto replace u,. This implies, contrary to the assumptionleading to Eq. (18), that the independent parametersO'w', z and g/O are not sufficient to determine the temperature gradient. Perhaps the additionM scale neededis a velocity scale characteristic of the large convectiveeddies. Free convection observations in the atmosphereare very scarce, however, and the verification of thisspec~ation will have to wait for better documentationof free convection.c. Ratio of eddy di~usivities In a constant stress layer, the definition of a, K~ ~>O' arC/as K~ u-r-~w~(19)186 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME2$be gained by rewriting Eq. (19) as I I I I-2.5 -2.0 -I.5-I.0 -0.5-2.5 ~ 0 = 1,3,5 --2.0 I I I I I 0.5 1.0 1.5 2.0 2.5 CFro. 4. The dependence of the ratio of eddydiffusivities on stability.(1~9~ ?"(,-,5~)'~ I3.0 3'.5can be rewritten by replacing u'w' with --u.~, withthe result ~ =--. (20)Values of a found from Eq. (20) are shown in Fig. 4. Thescatter is significantly greater than in the q~= and ~hplots, because u. is the chief source of the scatter, andthe effect on a is greater since it contains u.2. Nevertheless, it is clear that a increases with increasing instability. It also appears, as we remarked earlier, thatin neutral conditions a is ~ 1.35, and not 1.0 as has beensuggested in the past. This value of 1.35 is in doseagreement with laboratory measurement (cf. Hinze,1959). Much of the previous work on a has been carried outwithout the benefit of direct stress measurements. Someinsight into the difficulty imposed by this approach canRim.o.r,~t;.(,[l,~ i;)''~ '/" {; {0.74 + 4.?/~.}.),5- -,.5 --' I.o.,oFro. 5. The dependence of Richardson number on stability. w'O' OU/Oz - a= . (2D --u.~ OO/OzIn the absence of direct measurements, u, can beinferred from wind profiles. At neutrality, the profile is,.;imply _ u, /z\ ln[,7o). (22)The roughness length z0 is found by exWapolating theprofile to zero speed; knowing k, n. can then be foundfrom Eq. (22). The important point here is that the use,of the nsual value of 0.40 for k would lead to u. valuesabout 15% high, in view of our finding that k=0.35.'Use of these high u. values in Eq. (21) then would givelow values for a; a true value of a=: 1.35 would decreaseto about 1.0. It is probable that this accounts for thedifference between our a value and sonm of the lower- values in the literature. For the unstable range, the Businger-iDyer equations(8) and (12) in Eq. (20) give 1.35(1--9~')~ a = (23) (t-lSD,-Eq. (23) has been plotted in Fig. 4. The fit is as good asthe scattered data would allow. The KE-?S descriptionassumes a = 1.0 and therefore does; not give a basis forcomparison, The data do not appear to indicate a specific trend inthe stable range. From Eqs. (10) and (14), however, wecan write 1.0+4.7~ a = (24) 0.74d-4.7g'which indicates that a decreases slowly with increasing~ under stable conditions, asymptotically approaching1.0. The a data (Fig. 4) show too much scatter for acritical test of Eq. (24), but it does appear that a isroughly constant, with a value between 1.0 and 1.35. Inthis stable regime, the scatter is reduced considerablyif a is calculated from the local measured values of heatflux and stress. In this case it is found that a decreasesvery slowly with increasing stability, much as predictedby Eq. (24), with an average value for the stable observations of ~ 1.2.d. The function Ri(~) Knowledge of the relationship between Ri and ~ isobviously of value in studying data which include nodirect flux measurements, and hence no ~ values. Wecan derive an expression which :fits our data well bystarting with the relation r =aqb,,Ri. (25)MARct~1971 BUSINGER, WYNGAARD, IZUMI AND BRADLEY 187Under unstable conditions, the Businger-Dyer formulas 5swere found to fit the observations well, and led to theexpressions for qbm and a, given in Eqs. (8) and (23),respectively; these together with Eq. (25) give 0.74r(1-- lSC)~ Ri = -- (26) (1-9r)~Eq. (26) fits well, as shown in Fig. 5. It has been suggested independently by Pandolfo(1966), Businger (1966), and Dyer (unpublished) thatRi =~' is a good practical approximation. Very close toneutral, we can write 0Ri Ri~ (--~-) ~' =q~,,(0)~' =0.74~-, (27) ~' 0so that here the approximation Ri=~' is only fair, asshown in the inset in Fig. 5. At g = --0.1, however, thedeviation between Ri=~ and Eq. (26) is only 15%, andit decreases asymptotically to about 4% as ~'-~--oo.The simple expression Ri=~ is therefore a good overallapproxhnation for unstable conditions.Under stable conditions, Eqs. (10), (24) and (25)imply that ~-(0.74-k4. TD Ri --, (28) (1+4.7D-~which, as shown in Fig. 5, fits the data very well. Windmrmel observations of this relation, by Arya and Plate(1969), although extending over a smaller stabilityrange, show the same behavior. Eq. (28) shows that Riapproaches a limit of 0.21 as ~--~. In an analogousway, Webb (1970) obtained a limit of about 0.20. If,as conditions become more stable, the heat flux andReynolds stress approach zero at the same rate, then Lapproaches zero because it contains these terms todifferent powers. Then ~-->o~ as the turbtdence dimirdshes and the flow becomes laminar, and we caninterpret the limiting value of Ri as the criticalRichardson number. It should be emphasized that our ~ovalue of 0.21 was obtained from a limited amount of data and more observations are needed for substantiation.e. Flux computat.ions from profiles Since both qb~ and Ca are described accurately byinterpolation formtdas, stress and heat flux estimatesobtained by fitting integrals of these formulas to theraw profiles would be expected to agree well with thedirectly measured values. However, this comparison isuseful in giving an indication of the accuracy with whichan individual run may yield its fluxes using profileformulas, as shown by the scatter about the one-toone lines. l~y fitting Eqs. (8) and (12) in integrated form to themeasured wind and temperature profiles for the unstable range, u,, 0., z0, and consequently the fluxes aswell can be determined. A detailed discussion of thistechnique has been given by Paulson (1967). Fluxescalculated in this manner from the 15-min profiles havebeen plotted vs the observed values in Figs. 6 and 7. Itis clear that the data scatter around a one-to-onerelation as expected. For the stable cases, fluxes were calculated by usingthe integrated froThS of Eqs. (10) and (14), i.e.,_ u,/ z \U =7[ln~q-4.7~), (29)~:00+0.(0.74 ln~+4.7~'),(30)with z0 = 2.44 cm, which was obtained from the analysesof the unstable cases. For each 15-min stable period,except those during transition periods, u. and L werefound by minimizing the mean square error betweenthe observations and Eq. (29). The calculated valnes of I I I I I I I ~ ~o ~'~5 . * ~.'; * :~ oo o-~, ~ ~ ~ ,* [ ~e 5 0 5 I0 15 20 ~5 30 35 40 45 OBSERVED H (row cm-z)Fro. 7. Comparison of pro~e-derived and observed heat[fluxes, unstable cases.188 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME285O4O0 0I0Fro. 8. I 0 20 ' 3O 40 50 OBS~'R~I~'D u ~ ( crn sec-/)Comparison of profile-derived and observed friction velocities, stable cases.u, are plotted against the observed values in Fig. 8, andit is obvious that the agreement is good. The heat flux was found similarly. Values of 0, and00 were found by minimizing the mean square errorbetween Eq. (30) and each 15-rain temperature profile,using L values found during the u. calculation. Fromthe definition of 0., the heat flux was then obtained byusing the profile-derived u.. The heat fluxes calculatedin this manner are compared with the observed heatfluxes in Fig. 9. The agreement is not as good as foundfor the unstable cases (Fig. 7). It should be mentionedthat in both Figs. 8 and 9 the data points with thesmallest values of u. and heat flux were those withRichardson numbers dose to the critical value. An independent set of observations will be requiredto test the validity of these results, since the free constants in the profile formulas were determined by ~-,Fro. 9. Comparison of profile-derived and observed heat fluxes, stable cases.optimizing, the fit to the data. TkLe relatively smallscatter does indicate, however, that the profile techniquecan be a simple and useful means of obtaining thefluxes.5. Conclusions The following conclusions appear warranted by theanalyses described above:1) The predictions of the similarity theory of Moninm~d Obukhov are well satisfied by the data. 2) The wind shear data indicate l:hat yon K~rm/[n'sconstant is ~0.35 rather than the widely used valueof 0.40. 3) The ratio of the eddy diffusivities for heat andm.omentum is greater than unity. The value at neutral is1.35; it decreases slightly in stable conditions, and showsa marked increase with instability. 4) The relationship between Ri mad ~' shows remarkably small scatter, and is nearly linear in unstable conditions; on the stable side, Ri approaches a limit ofabout 0.21 as ~-~m. 5) Profile-derived and measured fluxes agree wellover the entire stability range of the observations, butan independent set of observations is needed to verifythis conclusion. Acknowledgments. The authors v(~sh to acknowledgethe joint effort of all members of the Boundary LayerBranch of AFCRL, who planned a,nd carried out the1968 Kansas field trip which supplied the; data for thispaper. Particular thanks are due to Dr. D. A. Haugenfor his critical review of the manuscript, to Mr. R. Sizerfor preparing the drawings, and to Miss S. Tourville fortyping the manuscript. REFERENCESArya, S. P. S., and E. J. Plate, 1969: Modeling of the stably strati fied atmospheric boundary layer. J. Atmos. S-i., 26, 656-665.Bradley, E. F., 1968: A shearing stress meter for micrometeorological studies. Quart. ~r. Roy. Meteor. Soc., 94, 380-387.,1969: A small sensitive anemometer system for agriculturalmeteorology. Agric. Meteor., 6~ 185-1!)3.Businger, J. A., 1966: Transfer of heat ~nd momentum in the atmospheric boundary layer. Proc. Arctic Heat Budget and Atmospheric Circulation. 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C., 1964: The exponential wind profile. Quart. Roy. Meteor. So,., 90, 119-135. , 1968: A comparison between predictions of dimensional analysis for the constant-flux layer and observations in unstable conditions. Quart. J. Roy. Meteor. Soc., 94, 460467. - , and A. J. Dyer, 1968: Micrometeorological expeditions, 1962-1964. Tech. Paper No. 17, Div. of Meteor. Phys., CSIRO, Australia.Tennekes, H., 1968: Outline of a second-order theory of turbulent pipe flow. AIAA Journal, 6, 1735-1740.Webb, E. K., 1970: Profile relationships: The log-linear range and extension to strong stability. Quart. J. Roy. Meteor. 96, 67-90.

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