The Budgets of Turbulent Kinetic Energy and Temperature Variance in the Atmospheric Surface Layer

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Abstract

Measurements of the shear production, buoyant production, turbulent transport (flux divergence) and dissipation terms in the budget of turbulent kinetic energy, and production and turbulent transport terms in the temperature variance budget are presented. Direct observations of the surface stress and heat flux over a horizontally uniform site enable presentation of the data in terms of surface layer similarity theory.

The dissipation term, obtained from differentiated hot-wire anemometer signals, agrees with estimates made from the inertial subrange levels of longitudinal velocity spectra with a value of 0.5 for the spectral constant. Under stable conditions dissipation essentially balances shear production, while turbulent transport and buoyant production are of secondary importance. Under unstable conditions, dissipation slightly exceeds the total production, and energy is also lost at a substantial rate due to upward export by the turbulence.

The large imbalance among the measured terms in the energy budget under unstable conditions is discussed. The cause of the imbalance cannot at this point be determined with certainty, but an interesting possibility is that pressure transport is significant under very unstable conditions.

The production rate of temperature variance exceeds its rate of vertical transport by an order of magnitude. Estimates of the universal temperature spectral constant were made with the assumption that temperature variance dissipation and production rates are equal; the average value, 0.8, falls within the range reported by other workers.

Abstract

Measurements of the shear production, buoyant production, turbulent transport (flux divergence) and dissipation terms in the budget of turbulent kinetic energy, and production and turbulent transport terms in the temperature variance budget are presented. Direct observations of the surface stress and heat flux over a horizontally uniform site enable presentation of the data in terms of surface layer similarity theory.

The dissipation term, obtained from differentiated hot-wire anemometer signals, agrees with estimates made from the inertial subrange levels of longitudinal velocity spectra with a value of 0.5 for the spectral constant. Under stable conditions dissipation essentially balances shear production, while turbulent transport and buoyant production are of secondary importance. Under unstable conditions, dissipation slightly exceeds the total production, and energy is also lost at a substantial rate due to upward export by the turbulence.

The large imbalance among the measured terms in the energy budget under unstable conditions is discussed. The cause of the imbalance cannot at this point be determined with certainty, but an interesting possibility is that pressure transport is significant under very unstable conditions.

The production rate of temperature variance exceeds its rate of vertical transport by an order of magnitude. Estimates of the universal temperature spectral constant were made with the assumption that temperature variance dissipation and production rates are equal; the average value, 0.8, falls within the range reported by other workers.

190 JOURNAL OF THE ATMOSPHERIC SCIENCES Vo~.u.~E28The Budgets of Turbulent Kinetic Energy and Temperature Variance in the Atmospheric Surface LayerJ. C. WYNOAARD AND O. R. COT~ Air Force Cambridge Researct~ Laboratories, Bedford, Mass.(Manuscript received 12 August 1970, in revised form 30 November 1970)ABSTRACT Measurements of the shear production, buoyant production, turbulent transport (flux divergence) anddissipation terms in the budget of turbulent kinetic energy, and production and turbulent transport termsin the temperature variance budget are presented. Direct observations of the surface stress and heat fluxover a horizontally uniform site enable presentation of the data in terms of surface layer similarity theory. The dissipation term, obtained from differentiated hot-wire anemometer signals, agrees with estimatesmade from the inertial subrange levels of longitudinal velocity spectra with a value of 0.5 for the spectralconstant. Under stable conditions dissipation essentially balances shear production, while turbulent transportand buoyant production are of secondary importance. Under unstable conditions, dissipation slightly exceedsthe total production, and energy is also lost at a substantial rate due to upward export by the turbulence. The large imbalance among the measured terms in the energy budget under unstable conditions is discussed. The cause of the imbalance cannot at this point be determined with certainty, but an interestingpossibility is that pressure transport is significant under very unstable conditions. The production rate of temperature variance exceeds its rate of vertical transport by an order of magni~rude. Estimates of the universal temperature spectral constant were made with the assumption that temperature variance dissipation and production rates are equal; the average value, 0.8, falls within the rangereported by other workers.1. Introduction In the summer of 1968 the Boundary Layer Branch ofthe Meteorology Laboratory of AFCRL made extensivemeasurements of wind and temperature in the first32 m of the atmospheric surface layer. The experimentsite was an extremely flat area in southwest Kansas,having 2400 m of uniform wheat stubble to the south,the prevailing wind direction during the experiments.The instrumentation included two drag plates forsurface stress measurements, three 3-component sonicanemometers and three platinum resistance thermometers for wind and temperature fluctuations at threelevels, three constant-temperature linearized, hot-wireanemometers for measurement of the streamwise windfluctuations to frequencies of 2000 Itz, and cup anemometers and resistance thermometers for measurementof mean wind and temperature profiles. This experiment has allowed a detailed study of thebudgets of turbulent kinetic energy and temperaturevariance in the surface layer. Previous efforts, summarized by Lumley and Panofsky (1964), have had anumber of limitations, so that until now only the roughoutlines of the budgets have been drawn. The Kansasexperiments were unique in having direct measurementsof production, turbulent transport, and dissipation ratesof turbulent energy, and the production and transportrates of temperature variance, allowing a quantitativedetermination of their imbalance over wide range ofstability conditions. In addition, the direct flux measurements have allowed the data to be presented in thecontext of similarity theory (Obukhov, 1946; Moninand Obukhov, 1954), which permits a concise representation of the effects of stability.2. Instrumentation, data collection and data re duction The sonic anemometers (Kaijo Denki PAT 311), hotwires (DISA 55D05-55D 15) and fast-response thermometers (Cambridge Systems Model 127) were mountedat 5.66, 11.3 and 22.6 m on the tower. Cup anemometersand thermometers for mean profile measurements wereplaced at 2, 4, 8, 16, 22.6 and 32 m, with cups also at 5.66and 11.3 m to allow comparison with sonic anemometerspeeds. The drag plates were located about 50 m eastsoutheast of the tower. A detailed description of the siteis given by Haugen et al. (1971). Data were digitized and stored on magnetic tape bythe computer-controlled data acquisition system described by Kaimal et el. (1966). The profile instrumentation was sampled once per second, and the other instruments 20 times per second. The hot-wire outputs weredifferentiated and low-pass filtered with filters of thetype described by Wyngaard and Lumley (1967). Thisprewhitening process amplified the high-frequencyturbulence components before recording on the FMchannels of an Ampex FR-1300 tape recorder andM.~{cn 1971 J. C. W Y N G A A R D A N D O. R. C O T I~, 191allowed subsequent data analysis out to 2000 Hz, whichwas sufficient to cover essentially the entire dissipationrange. By comparing the mean speeds of the cnp and sonicanemometers, it has been established by Izumi andBtu'ad (1970) that the cup speeds are about 10% high.This is due to the dsmamic nonlinearity of the cupanemometer (it reacts differently' to wind speed increasesand decreases) and to its sensitivity to the fluctuatingvertical wind component. Cup anemometer speeds weretherefore reduced by 10% for use on wind profiles. Since obtaining data from hot 'wires is less straightforward than with the other instruments, a brief discussion will be given here. The wires were 1.2 mm longand were mounted vertically. This length, approximately equal to the Kolmogorov microscale, wassufficiently short so that the attenuation in the meansquare velocity derivatives due to the finite wire length(Wyngaard, 1969) was no more than a few percent. Absolute calibration of the hot wires in the field wasnot attempted. Rather, wind tunnel testing was done to,~snre that the static response curve was indeed linearover the operating range. That is, the output /~ wasrelated to the longitudinal air speed U byThe set (5) reflects the fact that the mean hot-wireoutput is proportional to the mean horizontal windspeed S, and not the mean streamwise wind componentU. Eq. (2) becomes F S/v~ v~\-I e=b[u--K~OSq--~t~-~--~)]. (6)The second term within the bracket represents thesensitivity to temperature fluctuations; K~ was typically2.5X10-~(-C)-~ in this experiment. The final termcorrects for the difference between fluctuations ofhorizontal speed and streamwise velocity. The final correction to be made involves Taylor'shypothesis. Heskestad (1965) proposed that in highReynolds number turbulent shear flow, the higher orderterms in the frozen field expressionDu Ou Ou Ou Ou-- =0 =--+(S+u)--+v--+w--- (7)Dt Ot Ox Oy Ozbe retained. By assuming isotropy at small scales andindependence of structure at widely varying scales, heshowed that E=ad-bU.The fluctuating response was then(1)/ Ou\~ / Ou\ sF u~ 2v= 2w2-]~! = U~/--//i+--+T~+--/.\Ot / \Ox/ l_ U~ b ~ U2_1e=bu.Taylor's hypothesis implies that - bU--.Ot OxAssuming isotropy, the dissipation rate e is1 (Ou~~- 15~ (ae'~~ (4)(2) Similar expressions hold for the derivatives of v and w.The dissipation calculation started with the differentiated form of Eq. (6), which relates Oe/Ot andOu/Ot. The effects of the temperature and lateralvelocity fluctuation terms were found to be negligible.(3) Heskestad's modified Taylor's hypothesis [Eq. (8)] was then used to convert to mean-square streamwise derivatives, with the correction factor typically being 10-20%. Finally, the assumption of isotropy [Eq. (4)] gave dissipation rates from the streamwise deriva tive variances.To obtain the variance of the derivative of the fluctuating output signal, the recorded derivative signal wasreproduced and digitized at ~--3000 samples sec-L The above expressions for hot-wEre response andTaylor's hypothesis are accurat% strictly speaking, onlyfor vanishing turbulence levels. Slight corrections areneeded for application in the atmospheric surface layer.For hot-wire response, the expressions deri'ved by Rose(1962) were used. Eq. (1) was replaced by(5)3. The turbulent energy budget In high Reynolds number atmospheric turbulence,the budget of turbulent kinetic energy per unit mass isexpressed by (Lumley and Panofsky, 1964)10u~u~ U~ Ou~u~ OU~ g__ 1 Ouiu~u~ t }-u~u~ .... ~u~q2 O! 2 Oxs Oxs T 2 Ox~ Ou~Ou~ 1 Opu~ +p----+- =0. (9) Oxi Oxi p Ox~Here Ui and ul are the mean and fluctuating parts,respectively, of velocity in the xl direction, and repeatedindices are summed; T and O are the mean and fluctuating temperature; ~ is the mean density; p the fluctuating192 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME28-85_3 -2 -3,tF~c. 1. Dimensionless rate of shear production of energy.pressure; and g the acceleration of gravity. The subscript notation is convenient for Eq. (9), but in whatfollows we replace u~ and xg by (u, v, w) and (x, y, z),respectively. The energy budget simplifies considerably underhorizontally homogeneous conditions, where meanquantities depend only on the vertical coordinate z andthe mean velocity has only the streamwise (x) component U. We expect the smallest scales of motion tohave an isotropic structure, so the expression for dissipation can also be simplified. The steady-state budget isthen__OU g__ l Owq~ /Ou\~ l Opwuw-----wO+---+lsd--! +- =0, (~0) oz T 2 Oz \Ox/ p ozwhere q2 = u2q_vO.q_w2. The first four terms in Eq. (10) were measureddirectly. The first term is the rate of production ofenergy by the interaction of Reynolds stress with themean strain rate; we will call this shear production. Thesecond term represents the rate of working against thebuoyancy forces; we call this buoyant production, andnote that it is a source or a sink depending on the sign ofthe vertical heat flux. The third term is the divergenceof the turbulent ffux of kinetic energy, which we will callturbulent transport. The fourth term is the rate ofconversion of turbulent energy into internal energy bythe action of viscous forces, or simply the dissipationrate. The final term in Eq. (10) is another flux divergence; it represents the transfer of energy from one pointin space to another due to the fluctuating pressure, andwe call this pressure transport. The assumption of horizontal homogeneity wasprimarily responsible for the simplification of the energybudget from Eq. (9) to Eq. (10). 'Unfortunately, theKansas experiments included no direct measurements totest this assumption; it may seem reasonable because ofthe uniform surface conditions and le~ng fetch, but without measurements we cannot be certain. At this point itseems preferable to lump possible inhomogeneity effectstogether with the pressure transport and call this theimbalance term I. We take therefore as ore: model of theenergy budgetg__ 10wq~ /Ou\2 It has long been assumed, and recently demonstratedby Haugen et al. (1971), that to a good approximationthe surface layer is a "constant flrtx" layer. That is,the turbulent fluxes ~-& and wO are effectively independent of height from the surface. For these conditionssimilarity theory predicts that certain aspects of theturbulent structure are uniquely determined by theparameters wO, ~-~, g/T and z. Aozordingly, we nondimensionalize the terms in Eq. (11) with kz/u.a, whereu,, called the friction velocity, is the square root of thekinematic surface shearing stress, and k is yon Karmgn'sconstant. From similarity theory the terms in the budgetshould be universal functions of :;/L, where L, theObukhov length, is defined by--~,3TL= --_ (~2)kgwOThe friction velocity u. was derived from the average ofthe two drag plate readings, as discussed by Haugenet al. Since the observations revealed no significant systematic variation of heat flux with height, the average ofthe three measured values was used in calculating L.Von K~.rmgn's constant k was taken as 0.35, rather thanthe usual vales of 0.40, as a result of the profile analysisreported by Businger et al. (1971). We now proceed to a discussion of the behavior of theterms in Eq. (11). The results are presented in dimensionless form, and it should be pointed out that thechoice of kz/u.~ as a normalizing factor is not criticalfor the interpretation of the budget. Since all termsare normalized with the same factor, their relative magnitudes are unchanged. In all cases we refer the termsto the left side of Eq. (9), so that if a term representsan energy gain, it is negative.a. Shear production- Measurements of the dimensionless shear productionterm are shown in Fig. 1, each point representing a 1-hrrun. Reynolds stresses were computed for 15-rain blocksof data, and four of these values were aw.~raged to givethe stress for the run.M^~cn 1971 J. C. \VYNGAARD AND O. R. COTI~ 193 The average properties of this curve should be thoseof the nondimensional wind shear ~,~, where kz OU u, 8zThe obsmwations of -m from this experiment are presented in Businger et al. (1971). The scatter in Fig. 1 issomewhat greater than found in qb,,, due to the run-torun scatter of --~7-~/u,s about its average value of 1.0. From the profile analysis, it has been found that inunstable conditions qbm is represented well by theBusingcr-Dyer formtda qb,,-- (1 -- 15z/L)-I. (14)Under stable conditions, the form qS~ = l +4.7z/ L (15)is a good fit. These interpolation formulas are shown inFig. 1.b. Buoyant production Buoyant production simply non~alizes to kg gwO z (16) u,a T Land is therefore a gain under unstable conditions and atoss during stable conditions.c. Turbulent transport Hourly estimates of turbulent transport were made byapproximating the vertical derivative of wq~. Its valuewas twailable at the three sonic levels, and we usedkz a(wq~/2) kOwqe k = ~~~-~ , 07)'u,~ ~ u,aO(2 lnz) u, ~ 2 ln4which is eqtfivalent to fitting a least squares linearfimcfion of lnz to the data. The result w~ taken as thederivative at the midpoint of Inz, i.e., at 11.3 m. Thisprocedt~e differs insignificantly from the alternative offitting the data to z, not lnz, and assuming ~e approximation applies at 14 m, the midpoint in z. The results (Fig. 2) show some evidence of obeyings~ilarity, although the scatter is large. The turbulenttransport as calcxdated from high-pass filtered (5-minm~g average removed )fluctuation data is also sho~in Fig. 2. In general, the ~tered results are slightlysmaller but the scatter is not significantly ~fferent.Some of the scatter is probably due to the crude derivative approximation; had wq~ values been available atmore levels, higher order least square ~es could havebeen fitted to ~eld more accurate derivatives. Part ofthe scatter is probably also caused by the inherently I.O o-05 - 1.0 - 0.5 0.5 Fro. 2. Dimensionless rate of turbulent transport of energycalculated with and without a running-average filter on thevelocity components.larger uncertainty in third moments. To achieve thesame accuracy, increasingly longer records are neededfor higher moments (Lumley and Panofsky, 1964). Turbulent transport can also be written askz Owq2 k O(wq~/uJ)2u,s Oz 20ln(z/L)(18)The wq~/uJ data show no trend with z/L under stableconditions, but the unstable side is shown in Fig. 3.The dashed curve is a reasonable fit to the data, and ittogether with Eq. (18) implies that turbulent transportfollows the dashed line in Fig. 2. For --1.0x< z/L~< 0turbulent transport therefore behaves approximately as t~z Owq~ z2u,a Oz m'(19)with an uncertainty of the order of 4-0.25. The trend of the turbulent transport estimates on thestable side of Fig. 2 is less clear. While all but 2 of the 13unstable runs had wq~ monotonically increasing with z, 12 ~0 8~,.N. 4 2 0 -01 / o / t / 0 / 0 / o~:e~,5- o o.~' o/ oo -o o..' o o ~o o ~ .... 5--o o ~ ~ -0.~ -~ 0 -~0 ~ Fro. 3. Dimensionless turbulent energy flux under unstableconditions. Differentiation of the dashed curve gives the dashedline in Fig. 2.194 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLtairE280.51.0 0.5-0,4 i-o o o ~, I I ~-[.0 -0.5 0hL0 FILTEREO F~o. 4. Dimensionless rate of turbulent transport of vertical(upper figure) and horizontal (lower figure) components of energy,calculated with and without a running-average filter on thevelocity components.in almost all of the stable cases it was non-monotonicand had a maximum at 11.3 m. The transport rates inthe layers between 5.66 and 11.3 m and between 11.3and 22.6 m therefore tended toward losses and gains,respectively. This plot is not shown, but in similaritycoordinates looks much like the stable side of Fig. 2;there is scatter about zero for small z/L and a tendencytoward gains at larger z/L, but nearly all the estimatesare smaller in magnitude than z/L. It appears, therefore,for 0 ~< z/L x< 0.5, that turbulent transport is quite smallcompared to the dominant terms in the energy budget. The turbulent energy flux wq--'~ was positive in everyunstable case. In conjunction with the result (19), thisimplies that energy is exported upward by turbulence at.essentially the same rate it is produced by buoyancy. Aflux divergence term represents transfer from one regionin space to another, so that this exported energy has toappear as a local import, or gain, farther up. Recentmeasurements of Lenschow (1970) under unstable conditions and at heights of a few hundred meters do show alocal gain, as expected. A separation of turbulent transport into its contributions from horizontal and vertical energy components is:shown in Fig. 4. The scatter is again rather large, but itappears that under unstable conditions roughly twothirds of the transport is of the horizontal components.d. Dissipation The normalized dissipation results are shown in Fig. 5,in which each point represents a 1-hr run. The curvefitted through the unstable data has the form --= (~+0.51 z/LI ~)~. (20)The curve has been assumed to go through 1.0 at theorigin, in keeping with traditional thinking, but the datado not extend to sufficiently small z/L to 'test this point..Fig. 5 shows that dissipation exceeds the total produc-tion under moderately unstable conditions, with abalance being reached with increasing instability. Under stable conditions, the dissipation data arefairly well represented by kze -- = [1 + 2.5 (z/L)~/'~~/~. (21) The dissipation values and the: measurement andanalysis techniques which produced, them were checkedin three ways. First, as a check on the hot-wire performance, the values of u~ from hot wires and sonicswere compared. Over 52 hr of data the values agreedwithin 4%. There was some run-to-run scarer, probably due to hot-wire drift, but the overall agreement isgratif~ng. For the second check, the direct dissipation estimateswere used to test the universality of the sonic anemometer velocity spectra in the inertial submnge. In largeReynolds number turbulence, it has been suggested(Lumley, 1967) that inertial subrange spectra depend ina universal way on local spectral :flux. Spectral flux isthe rate at which energy is transferred through thespectrum, from large scales to smaller ones, ultimatelyto be dissipated by viscosity at the smallest scales. Inthe absence of energy sources or drains, the spectral fluxis simply the dissipation rate, and we have the Kolmogorov law for the inertial subrange. For the onedimensional u spectrum this isPrevious measurements of the universal constant a~ ;/ ~ ..'; !/--...:...-.~., ~- _~,~, ,,ooo~,,o,,x.~ ~-~i .... ~ ~O~ I ~ ~ ~ ~ -2 -I O ~/tFm.~5. Comparison of dimensionless rates ofdissipation and production of energy.MaRCX~1971 J.C. WYNGAARD AND O. R. COT]~ 195from a wide variety of turbulent flows suggest that itsvalue is --~0.5. An extensive spectral analysis program is now underway, using the fast Fourier transform technique, andwill be reported in detail later. For the present purposes,sonic anemometer u spectra were calctdated for each ofthe 1-hr runs for which there also was a hot-wire dissipation value. The spectra were averaged over an hourand also over frequency bands, with the bandwidthlilnited by the requirement that the spectral distortiondue to spectral curvature be negligible. The resultingspectra were very smooth, with typical scatter aboutthe --5/3 power taw behavior of no more than a few percent. Cospectra of Reynolds stress and heat flux in thespectral range considered (typically 0.5--5 Hz) werevery small and falling faster than the velocity spectrmn.In this range the change in spectral flux due to dissipation is negligible; the peak of the dissipation spectrum is at much higher frequencies, typically a fewhundred hertz, if we assume that the effects of pressureand the third-moment terms are also small in this wavennmber range, the data meet the requirements for usein Eq. (22). Taylor's hypothesis in the form ~= 2~rf/U was usedin conjunction with the measured ~ values and Eq. (22)to derive estimates of m, with the results shown in Fig.6. The mean value of ax is 0.52, with a standard deviation of 0.04. These ax estimates are probably systematically high due to the effect of the fluctuating convection velocity on Taylor's hypothesis. Lumley's (1965) model shows that incrdal subrange spectra, and therefore a~ values, are overestimated by an amount depending on turbu lence level The correction is less than that already made to the dissipation rates, but shows that the average of 0.52 for a~ is probably nearer 0.49. Spectral analysis has also been carried out on selected hot-wire runs, giving an a~ value of 0.53=t:0.02 over a wide range of z/L values. The reduced standard devia tion is a result of the more accurate calibration of the hot wires obtained by matching sonic and hot-wire inertial subrange spectral levels. It appears, therefore, that much of the scatter in the a~ estimates of Fig. 6 is due to hot-wire drift. Considering the fluctuating con vection velocity effect, the two a~ values differ by about 10%, and if the larger value (which, corrected, is -~0.54) is correct, the dissipation data of Fig. 5 are about 15% high on the average, with the overestimate probably largest under stable conditions. The third check involves the isotropy assmnption inherent in the dissipation calculation. Isotropy also implies that the inertial subrange levels of the w and u spectra should be in a i' ratio; about 20 hr of sonic data from the upper two levels, with a stability range --1.6<z/L~0.7, were used to test this prediction. The average ratio was within a few percent of { for this data, with a standard deviation of 0.06. For 10 hr of data from 5.66 m, the ratio averaged about 1.2, but it is possible,7.6.4.3.~.I -~ --I 0 ,-Fro. 6. Observations of the one-dimensional spectral constant.that at higher frequencies the ~ behavior exists. Theseresults then give some justification for assuming isotropyin calculating dissipation.e. The imbalance Our results indicate that under unstable conditions,buoyant production and turbulent transport areapproximately in balance, while under stable conditionsturbulent transport is small. The budget thereforereduces to uw--+~+I.~.O, --1.O~<z/L(O Oz__OU g__uw--+~----Ow+I~O, O(z/L(0.5Oz T(23)where I is the imbalance. Our data show that undercertain conditions the imbalance is substantial, andseveral factors arise as possible contributors. First, it is recognized that any budget term found bydifference contains the accumulated errors of the measured terms, which do show scatter about their fittedcurves. Buoyant production, a second moment and apoint measurement, shows the least scatter and is themost accurate. Turbulent transport, a spatial derivativeof a third moment, shows the most scatter and itsfitted curve is the least certain. While the use of fittedcurves reduces the effects of random errors on theimbalance, residual systematic errors could be present.Broadly speaking, an overall accuracy figure of 15% isprobably applicable, so that in the worst case, whereerrors are additive, the uncertainty in the imbalancecould be appreciable. This is probably the origin of a substantial part of thestable imbalance, which Eq. (23) shows to be a gain andabout 30% of the dissipation rate. The stable dissipationrates, as mentioned earlier, could be 15-/o high; thetransport data of Figs. 2 and 4 suggest that it could be asmall gain; and the shear production rate of Eq. (15)could be slightly low. These errors could account formost of the imbalance under stable conditions.196 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME28-2-I IMBALANCE BUOYANTPRODUCTION SHEARPRODUCTION-TURBULENT TRANSPORT DISSIPATION 2 -I.0 -0.5 OFro. 7. Dimensionless energy budget under unstable conditions. Under unstable conditions, however, the imbalance isconsiderably larger. Eqs. (14) and (20) for shear production and dissipation imply that the imbalance isk~--I = (1 -- lSz/L)-,~- (lq-O.S~z/LI ~)~. (24)Eq. (24) is plotted along with the measured terms in Fig.7. At z/L=- 1.0, the imbalance is about 75% of thedissipation rate and larger than any of the other terms;an explanation other than measurement error wouldseem to be needed to account for it. A second possible contributor to the imbalance ishorizontal inhomogeneity. If the mean flow depends onx as well as z, the energy budget has additional terms.The largest of these reflect additional production ofenergy and the advection of energy by the mean flow. Ifthese are the cause of the imbalance, we haveu~--+w~---+- Ox Oz 20x 20z(25)The mean continuity equation in the formOU OW--+.- =0, (26)Ox Ozand the assumption that 0 U/Ox is independent of z, give_ __OU UOq~ zOUOq~(u~--w~)----3 =I. (27) Ox 20x 20x Oz Results of an analysis of the turbulent velocityvariances, to be reported in detail in a forthcomingpaper, are useful at this point. It is found that q2 variessufficiently slowly with z in the surface layer underunstable conditions that the final term in (27) can bedropped in comparison with the first. Expressing thevariances in terms of ~ then gives, at z/L- --1.0,q~ OU U Oq~ u.~-- ---]--- -- = I-~ --1.3--.30x 20x kz(28)If, for example, two-thirds of the imbalance is due to thefirst term in Eq. (28), we can write the following expression for the required streamwise length scale of meanwind speed changes:U ~Vkzl~ = . ~---. (29)0~UxU 2'6u*aIf the second term in Eq. (28) causes essentially all theimbalance, the required length scale for streamwisechanges in energy is found to be the same. At z/L=--1.0, ~26u.~, and with typical conditions ofU=500 cm sec-~, u.=30 cm sec-~, z = 10 m, the scale is---600 m. This means that a mean wind speed decreasein the streamwise direction of 10% per 60 m wouldaccount for two-thirds of the observed energy budgetimbalance at z/L=- 1.0. The same rate of decrease ofq~, on the other hand, would explain all of the imbalance. The required gradients are severe, judging from thework of Dyer (1968) over apparently homogeneousterrain in Australia. He measured mean wind speeddifferences over horizontal separations of 200 m andfound l~ values about 20 times greater than our calculati.ons require. The required inhomogeneity can. also be comparedwith that expected due to normal boundary layergrowth. The integrated form of the x-component equation of motion, neglecting pressure gradients andCoriolis forces and assuming two-dimensi.onality, isOx(30)If h is chosen as the height where stress vanishes, weh ave --~h= --u.2, (31) Oxwhere a tilde denotes a representative value for thelayer. The scaling length for U is I I~ I = O~h/u,~' 8 X 10a m (32)for typical conditions and h assumed cortservatively tobe only 300 m. This is more than two orders of magnitude greater than the required length scale, and it seemsclear that any normal boundary layer growth had anegligible effect. As mentioned earlier, the fetch fox' 2400 m wasuniform wheat stubble; sections beyond this were inM.~act4 1971 J. C. WYNGAARD AND O. R. COTI~ 197var34ng crops, each with presumably different surface -Locharacteristics. This patchwork undoubtedly caused a -.scomplicated inhornogeneity in both u,2 and U, but zjudging from the boundary layer response calculations ~-.~of Pcterson (1969) the long uniform fetch would have oreduced the inhomogeneity far below the level required -.4to account for the imbalance. -.zThe foregoing discussion has concerned what mightbe called large-scale inhomogeneity, since we interpretedthe required OU/ax in terms of a 10% change in U over60 m. It is also equivalent to a 1% change over 6 m, and .athis type of small-scale inhomogeneity could have ~> .4existed upwind of the tower. However, a turbulence ~element is swept through 6 m in a thne short compared .ewith typical turbrdent time scales, so that we would notexpect the turbulence structure to respond in such a short '~t~e, although its intensity would. If, for exmnple, thereis a small-scale inhomogeneity, the neglected terms inu~u)OUi/Ox., would contribute to the energy budgetlocally, so the turbulence level leaving the region wouldbe slightly different from that entering. This introducesa local advection term U~:O(~/2)/Ox~ to cancel theadditional production. The other terms in the budget,and their knbalance, are substantially unaffected by the~nhomogeneity.A third possible contributor to the energy budgetimbalance is nonstationm'ity. If we attribute the imbalance to time changes in the turbulent energy, we findthat the required time scale is, at z/L= --1.0, Oq ~ ~r=~ / ~ 3.Sz/u../(33)At z=10 m and with u,=30 cm sec-~, r~2 rain; thismeans a rate of decrease of mean turbulence energy of10% per 12 sec is required to account for the imbalance.Observation indicates that this rate is perhaps twoorders of magnitude too high, so that we can eliminatenonstationarity from consideration. Although little is known about the fourth possiblecontributor, pressure transport, there arc bits ofevidence that suggest it could be important. Batchelor(1951) used a simplified turbulence model to predictthai: the magnitudes of pressure and velocity fluctuations are related by(p2)~=O.2pq~. (34)Applying this to surface layer turbulence at z/L = --1.0and taking the pressure-vertical velocity correlationcoefficient as 0.5, we have]pw ] ~ 5pu,a. (35)A vertical scaling length can be taken from the turbulent transport measurements. At z/L=--l.O, Wq2 is I- I I~ Iu0 I~/ I I -I.0 -0.5Fro. 8. Dimensionless components of the energybudget imbalance under unstable conditions.typically 6u.a and the length is0__l=wq~ --wq~ ~ 2z. /tOz I(36)If this is also the length scale for vertical variations inp~, an estimate of pressure transport is10pw 1 Ipw[ u,~ --~- ~0.9--.(37)This is the same order of magnitude as the imbalancegiven by evaluating Eq. (24) at ~/L= --1.0, i.e.,I~ -- 1.3u,~/(kz). (38) Kaimal and Businger (1970) recently analyzed thestructure of a convective plume in some detail andfound that the observed vertical accelerations could notbe explained by the existing temperature gradient. Theyconcluded that substantial vertical pressure gradientsmust exist to produce these accelerations. Further evidence that fluctuating pressure fields cansignificantly influence turbulence dynamics underunstable conditions is given by the budget of ~-~, alsoreported fully in a forthcoming publication. It is wellknown that pressure forces redistribute energy amongcomponents, and destroy ~-~, in neutral boundary layers,and the measurements of the production and transportterms in the ~ budget show that the rate at which stressis destroyed increases with instability. At z/L=- 1.0,the results show that1 O~ 10p u,a-w---t--u-- ~ --2.5--,p Ox p Oz ks(39)198 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME28Fro. 9. Dimensionless rate of production of temperature variance.which is the same order of magnitude as the energybudget imbalance. A formal expression for pressure transport can bewritten, involving an integral of velocity-temperatureand triple velocity correlations over the entire flowfield. At this point, however, our knowledge of thesecorrelations in the surface layer is so limited that it doesnot seem possible to make even a good approximation tothe solution. Therefore, while it is clear that pressurefluctuations can be important in turbulence, it is still anopen question whether they account for the imbalanceof our energy budget {~nder unstable conditions.f. The imbalance components It is interesting to consider the imbalance in thebudgets of the energy components. These budgets are l Ou2 __OU l Owu2 e - ---Fuw--+ +-+I~ =0 20t Oz 2 Oz 310v~ 1 Owv~ e- ---[-- +-+I~ =020t 20Z 31 Ow~ g__ 10w~ e wO+-- +-+I~=020t T 20z 3(40)The measurements indicate that turbulent transport isapproximately equally divided among the three component energies. Knowing this and using the results forproduction and dissipation, we can solve Eqs. (40) forfor the component imbalances. The results are shown indimensionless form in Fig. 8. It is generally accepted that the imbalances underneutral stability are caused by the pressure terms; thisimplies at z/L = 0 that 1 p Ox 1 I~ =- vop- .. (41) p Oy 10p p OzAt neutral the imbalance components sum to zero, sothat pressure forces simply transfer energy from u2 tothe other components at equal rates without affectingthe total energy. This transfer exists, of course, becauseat neutral the only energy source is shear production,but it produces only u2; pressure forces provide thenecessary energy source for the other two components. The behavior of the imbalance components changesdrastically under unstable conditions, as shown in Fig. 8.The u component increases and is a gain at z/L-- --1.0,while the w component decreases and is essentially zeroat that point. The v imbalance remains a gain butsteadily increases. At z/L=--I.O, therefore, theimbalance in the energy budget is essentially confined tothe horizontal components. It follows that if pressureforces cause the large imbalance under unstable conditions, their input is largely received by u and v throughaccelerations caused by horizontal pressure gradients.It is interesting to note that in his study of the effectsof heating on a laboratory boundary layer, Nicholl(11970) found evidence of significant pressure disturbances which he attributed to intense local convectiveactivity. He suggested that the pressure gradients-2.0-I.5 ~-I.O % ,~*-0.5 A o 0.5 -I.0 -0.5 0 0,5 ~,~Fro. 10. Comparison of dimensionless rates of turbulenttransport and production of temperature variance.M~acrt 1971 J. C. ~VYNGAARD AND O. Ro COTI~ 199accelerated cooler air into the regions between therising convective columns. Perhaps a similar situationexists in the unstable surface layer. The proper interpretation of Fig. 8 is not entirelyclear at this point, however. Fuller understanding mayhave to wait until a set of data including measurementsof horizontal homogeneity are available, and perhaps~mtil direct measurements of the pressure-velocitycovarlances have been made.4. The temperature variance budget The general form of the budget of temperature variance is (Lmnley and Panofsky, 1964) l O0-7 U,:O'Oi __O0 l OuiO2 O0 O0 ----t- .... +usO--+ .... +D----=0. (42) 20t 20xi Oxi 2 aa:~ OxiEq. (42) is actually the budget of 0=/2; the factor - isincluded for consistency with the energy budget. Understationary, horizontally homogeneous conditions, Eq.(42) simplifies to wO--+- +D-- --=0. (43) Oz 20z Oxi Ox~The first term in Eq. (43) is the rate of production bythe interaction of vertical heat flux and vertical meantemperature gradient; we call this production. Thesecond term represents the divergence of the turbulentflux of temperature variance, or simply turbulenttransport. The final term is the rate of destruction offluctuations by smoothing due to the molecular diffusivity D, or the temperature dissipation rate. Forconsistency with previous work, we label the dissipationterm N; the symbol X is often used for the dissipationterm in the budget of ~, and x=2N. We can nondimensionalize the budget with u,, z anda temperature scale T,, defined by T, = --Ow/u,, (44)1.3I.I.9.7 o.5 o o % o o % -- o o o o o ~o--~_ o ~, .~?g-o o -;,o.. o ~ ,oo oo oo'--. =~ / o- o 80 o ~ tt o o- ~~'I / - 0 - It# t~~ A ~~~ I -.OI - O. I - L0 --10 % Fro. 11. Dimensionless t~perature variance fl~ under unstable conditions. Differentiation of the dash~ curve gives thedashed curve in Fig. 12.o. o-0.1 0.1 0.2 -hO ,-0.5 ,0 0.5 Fro. )2. Dimensionless rate of turbu]ent transport of temperature variance, c~lculated wkh and without ~ running-averagefilter on the vertical velocity and temperature fluctuations.and we note that T, is independent of height in thesurface layer. It is convenient to include yon Kirmln'sconstant k in the normalization, so that productionbecomes ~z __O0) ~z 00 --.wO ....... ~,, (45) u,T,~ Oz T, Ozor simply the nondimensional temperature gradient.The analysis of the profiles from this experiment showsthat under unstable conditions, the formula ~=0.7 4(1--9z/L)-2 (46)fits the data well. Under stable conditions, the datafollow ~=0.74+4.7z/L. (47)These interpolation formulas are shown with the datain Fig. 9. The turbulent transport was calculated with thesame three-point approximation used in the energybudget, with the results shown in Fig. 10. Again, thetransport term can also be written as kz OwO~ k O(wO=/u,T,~) , (48)2u, T,= Oz 2 O ln(z/L)which suggests another way of estimating this term. Thestable values of ~/(u,T,~) show no evidence of a trendwith z/L, but similarity is followed fairly well underunstable conditions, as shown in Fig. 11. The dashedcurve in Fig. 1! corresponds, through Eq. (48), to thedashed curve in Fig. 12, and suggests that the transportterm is about an order of magnitude smaller than production. The trend of the dashed curve should perhapsnot be taken too seriously because of the scatter in thedata on which it is based, but there is a hint thattransport changes sign and becomes a gain under veryunstable conditions. This is consistent with observations (Deardorff, 1966) that under very unstable con200 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLU~U~;28ditions a positive heat flux can exist in the presence of avanishing or counter-gradient of potential temperature(O0/Oz/> 0). In this case the temperature variance production term vanishes or becomes a loss, and anothermechanism is needed to maintain the temperaturefluctuations. Deardorff suggested that the turbulenttransport term takes over this role, giving support tothe trend in Fig. 12 under very unstable conditions. An analysis of the effects of horizontal inhomogeneityshows that under typical conditions a mean temperaturevariation with x of 10C per 100 m, or a temperaturevariance change of 10% per 10 m, would be required tobalance the production rate of temperature variance.These rates of change are far greater than expected overthe site. Since nonstationarity also cannot contributesignificantly, the form (43) of the budget can be takenas a reasonable approximation, with the imbalancebetween production and turbulent transport a measureof the temperature dissipation rate N. The temperature dissipation rates derived in thismanner were used in conjunction with inertial subrangetemperature spectra to obtain estimates for the spectralconstant fi~. In this interval the temperature spectrumhas the form(49)There were 27 one-hour runs for which there were directe measurements and temperature spectra. With the approximation N = --~-~0 O/Oz, the -- 5/3 ranges (generallyin the band 0.5--5 Hz) of the spectra were used to derivefi~ estimates. The averages of the 27 estimates was 0.79with a standard deviation of 0.10. This fi~ value is in general agreement with earlierresults. Grant et al. (1968) found 0.62 in the ocean;Gibson and Schwarz (1963) report 0.70 in laboratoryturbulence; and Panofsky (1969) cites a Russian surveywhich gives values in the range of 0.41-0.88. The largescatter in these estimates is perhaps partly due to thegreater difficulty, in directly measuring fi~ compared tom. In addition, many of the meteorological observationsof fi~ are indirect, involving dissipation rates inferredfrom measurements of fluxes, profiles, or other quantities. Such a scheme then requires knowledge of similarityrelations or universal constants; the problem is that,generally speaking, the latter are far from wellestablished.5. Conclusions The turbulent energy budget in the surface layerappears to follow Obukhov similarity quite well. Therelative importance of the various terms in the budgetdepends on stability. Under stable conditions, in therange O<z/L-<0.5, shear production and viscousdissipation are the dominant terms and are essentiallyin balance. The buoyant term is a small loss, and turbulen.t transport is also very small. Under unstable conditions, --1.O<z/L<O, all four measured terms aresignificant. Shear production gradually becomes lessimportant as instability increases, and buoyant production assumes a dominant role as the energy source.Viscous dissipation approximately balances the sum ofthese two production terms. In addition, e. nergy is lostat a rate increasing with instability, due to verticalexport by the turbulence. 'Under very unstable conditions the four measuredterms are far out of balance, and the accuracy is felt tobe sufficient to justify taking the trend of tlhe imbalanceas real. 'The unmeasured terms, those reflecting possiblehorizontal inhomogeneity and pressure transport, wouldnaturally be suspected of causing the imbalance. However, calculations show that very strong horizontalgradients are required to explain the imbalance, and itwould be surprising if such inhomogeneity could existover the long uniform fetch of the site. Very little isknown about pressure transport, but what little information we have does not rule out its being important underunstable conditions. The production rate of temperature variance, --qb~,also follows similarity very well in the surface layer. Itexceeds the vertical turbulent transport term by anorder of magnitude in the range -- 1.0 < z/L < 0.5, whichimplies that to a good approximation the productionand molecular dissipation ra.tes of temperature varianceare equal. The estimates of the universal spectral constants, made with the direct dissipation measurements,are in general agreement with other observations. The a~value was near 0.5, while the temperature constant~ averaged 0.8. .Acknowledgments. This paper was made possiblethrough the efforts of all the members of the BoundaryLayer Branch of AFCRL, who carried out the fieldexperiments and helped in the data processing. We areparticularly grateful for the efforts of Mrs. J. O'Donnellin computer programming and Miss S. Tourville intyping the manuscript. Drs. Niels E. Busch, Duane A.Haugen and J. Chandron Kaimal made several helpfulcomments on the manuscript for which we are mostgrateful. REFERENCESBatchelor, G. K., 1951: Pressure fluctuations in isotropic turbu lence. Proc. Cambridge Phil. Soc., 47, 359-374.Businger, J. A., J. C, Wyngaard, Y. Izumi and E. F. Bradley, 1971: Flux-profile relationships in the atmospherk: surface layer. J. Atmos. Sci., 28, 181-189.Deardorff, J. W., 1966: The counter-gradient heat flux in the lower atmosphere and in the laboratory. J. Atmos. Sci., 23, 503-506.Dyer, A: J., 1968: An evaluation of eddy flux variation in the atmospheric boundary layer. J. Appl..~Ieteor., 7, 845-850.1971 J. C. WYNGAARD AND O. R. COTg 201Gibson, C. I1~.~ and W. H. Schwarz, 1963: The universal equilib rium spectra of turbulent velocity and scalar fields. J. Fhdd Mech., 16, 365-384.Grant, I~I. L., B. A. Hughes, W. M. Vogel and A. Moilliet, 1968: The spectrum of tea~perature fluctuations in turbulent flow. J. Fluid Mech., 34, 423442.F[augen, D. A., J. C. Kaimal and E. F. Bradley, 1971: An ext~eri mental study of Reynolds stress and heat flux in the atmospherlc surface layer, Quart. J. ttoy. Meteor. Soc. (in press).Heskestad, G., 1965: A generalized Taylor's hypothesis with appli cation for high Reynolds number turbulent shear flows. J. Appl. Mech, 87, 735-739.Izumi, Y., and M. L. Barad~ 1970: Wind speeds as measured by cup and sonic anemometers and influenced by tower structure. J. ,4 ppL M~teor., 9, 851-856.Kaim,'d, J. C., and J. A. Businger, 1970: Case studies of a convec- tive plume and a dust devil, J, Appl. Meteor., 9, 612-620.--, D. A. Haugen and J, T. Newman, 1966: A computer controlled mobile micrometeorological observation system. J. gppl. ~}feteor., 5, 4l 1-420.Lenschow, D. H., 1970: Airplane measurements of planetaryboundary layer structure, J. Appl. Meteor., 9, 874-884.Lumley, J. L., 1965: Interpretation of time spectra measured in high-inta~sity shear flows. Phys. Fhdds, 8, 1056-1062. -., 1967: Similarity and the turbulent energy spectrum. ~Phys. Fluids, 10, 855-858. , and H. A. Panofsky, 1964: The Structure of Atmospheric Turbulence. New York, Interscience, 239 pp.Monin, A. S., and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the ground layer of the atmosphere. Tr. Geofiz. Inst. Akad. Nauk SSSR, 151, 163-187.Nicholl, C. I. H., 1970: Some dynamical effects of heat on a tur- bulent boundary layer. J. Fluid Mech., 40, 361-384.Obukhov, A. M., 1946: Turbulence in an atmosphere with inhomogeneous temperature. Tr. Inst. Teoret. Geofiz. Akad. Nauk SSSR, 1, 95-115.Panofsky, H. A., 1969: The spectrum of temperature. Radio Sd., 4, 1143-1146.Peterson, E. W., 1969: Modification of mean flow and turbulent energy by a change in surface roughness under conditions ofneutral stability. Quart. J. Roy. Meteor. Soc., 95, 561-575.Rose, W. G., 1962: Some corrections to the linearized response of a constant temperature hot-wire anemometer operated in a low-speed flow. J. Appl. Mech., 29, 554-558.Wyngaard, J. C., 1969: Spatial resolution of the vorticity meter and other hot-wire arrays. J. Sci. Instr., 2, 983-987. , and J. L. Lumley, 1967: A sharp cut-off spectral dif ferentiator. J. A ppl. Meteor., 6, 952-955.

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