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Relating eddy correlation sensible heat flux to horizontal sensor separation in the unstable atmospheric… Black, T. Andrew; Lee, Xuhui Sep 30, 1994

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JOURNAL  OF GEOPHYSICAL  RESEARCH,  VOL. 99, NO. D9, PAGES 18,545-18,553, SEPTEMBER  20, 1994  Relating eddy correlation sensibleheat flux to horizontal sensor separation in the unstable atmospheric surface layer Xuhui  Lee  and T. Andrew  Black  Department of Soil Science, University of British Columbia, Vancouver, Canada  Abstract. Measurements of a scalar flux from an extended surface are frequently made with the eddy correlation technique consistingof a vertical velocity sensor and a sensorfor the scalar of interest. In many cases the two sensorshave to be mounted with a significanthorizontal separation to avoid flow interference. Consequently, the technique will underestimate the scalar flux. This paper addressesthe issue of flux underestimationdue to this separation. A model is developed in the framework of Monin-Obukhov similarity for the spatial covariance of vertical velocity and air temperature in the unstable surface layer. It allows the underestimationof sensibleheat flux to be assessedusing information on separation orientation relative to wind direction, atmospheric stability, measurementheight, and separation distance. The coefficient in the model is evaluated with observationsmade over a potato field and a clover field. The principles establishedhere should also be applicable to fluxes of scalars other than sensible heat.  1.  Prior to the analysisit is useful to review previous studies  Introduction  relevant to this work, which is done in section 2. Section 3  Measurements of the vertical fluxes of atmosphericscalar constituentsare important in studiesof processescontrolling the exchange of these scalars between land surfacesand the atmosphere. Over an extensive and uniform surface on flat terrain these measurements are frequently made with the eddy correlation technique consisting of a fast response velocity sensor for measuring the vertical velocity component and a sensorfor the scalar of interest. In many casesthe two sensorshave to be mounted with a significanthorizontal separation to avoid flow interference because either the scalar sensor is rather bulky or it has to be confined in a weather-proof enclosure to allow long-term operation. As a result of the separation this technique underestimatesthe flux. Previously, a few workers have expressed concern about this issue (see below). Their work appears to indicate that the degree of flux underestimation depends on separation distance, measurementheight, atmosphericstability and separation orientation relative to wind direction. However, as pointed out in the next section of this paper, there is no theory that relates the underestimation to all these factors to allow quantitative assessment. The main purpose of this paper is to develop a model for assessingthe underestimation of eddy correlation sensible heat flux due to horizontal sensorseparationin the unstable atmospheric surface layer. It is also hoped that the present study will improve our understanding of the horizontal turbulent structure near the ground surface. We chose to study sensible heat for the reason that measurements of temperature fluctuations can be easily made at multiple points with thermocouples. However, the principles established here should also be applicable to scalars other than sensible heat, based on the similarity of scalar transport processesin the atmosphere[Hill, 1989]. Copyright 1994 by the American Geophysical Union. Paper number 94JD00942. 0148-0227/94/94 JD-00942505.00  then  establishes  a model  in the  framework  of  Monin-  Obukhov similarity theory for assessingthe flux underestimation due to spatial separation. Experimental methods are describedin section 4. In section 5 the constantsthat appear in the model are evaluated from the observations. Finally, in section6 the implications of the model are briefly discussed. 2.  Review  The eddy correlation technique measures the flux of sensible heat by forming a covariance between vertical velocity (w) and temperature (T), w' (•) T' (• + •), where w'(•) and T'(• + •) are fluctuationsof vertical velocity at a horizontal position • and temperature at • + • and the overbar denotes temporal (Reynolds) averaging: Since horizontally homogeneous surfaces are the concern of this study, w'(•)T'(• + •) is independent of • and can be denoted as H(?). Ideally w' and T' should be measuredat the same point (• = 0), and the covariance (H(0)) is the kinematic sensibleheat flux density. Otherwise H(?) will be different from H(0). Because H(D is a spatial covariance, in principle its conservation equation can be constructed from the NavierStokes and heat conservation equations, in a manner similar to that of deriving the conservation equations of the spatial covariancesof velocity componentsor temperature [Hinze, 1975; Tatarskii, 1971]. Finding solutions to these conservation equations is difficult because of the closure problem (there are more unknown variables than the governing equations) and it is only possible in highly simplified situations. For example, provided that turbulence is isotropic, analytical expressions have been derived for the spatial covariances of velocity components and those of temperature [Hinze, 1975]. The assumption of isotropy, however, leads to zero covariance between velocity and any of the scalars[Tatarskii, 1971] and is not appropriate at all for w-T covariance.  18,545  18,546  LEE AND BLACK:  EDDY CORRELATION  Instead of assumingthat the flow is completely isotropic, Moore [1986] used the property of local isotropy in the  inertial•subrange inthewavelength domain. Hedefined a spectraltransfer function, Ts, associatedwith sensorseparation, in a similar manner as done to assessthe effect of line averaging [Silverman, 1968; Kristensen and Fitzjarrald,  SENSIBLE  HEAT FLUX 2  wherethe proportionality factor,Cq, calledthe structure parameter, is independent of separation distance, r. The velocity structure function also follows the 2/3 power law expression, but the direction of separation is important 2  [Tatarskii, 1971]. It iswellestablished nowtha•Cqcanbe related to surface layer parameters in the framework of  1984].He arguedthat if separationdistancer (= Il, m) is Monin-Obukhov similarity. A good illustration of this is the smalland only affectsturbulencefluctuations in the inertial expressionfor the temperature structure parameter derived subrange, the velocity cross spectra obtained for isotropic turbulencecan be usedto defineTs, which hasthe following working description for lateral separation:  Ts(f) = exp(-9.9f1'5),  (1)  from dimensional analysis  C2r= r,2Z•n2/3 g(Zm/L),  where T, is a temperature scale defined as the ratio of the kinematic sensibleheat flux to the friction velocity (u,)  wheref - nr/u, n being natural frequency(Hz) and u wind speed (m/s). Hence  H(r) =H(O) •:Ts(f)Swr(n) dn (2)  (6)  T, = - H(O)/u,,  (7)  and L is the Monin-Obukhov length scale [Wyngaard et al., 1971]. With the assumptionof horizontal homogeneitya little manipulation of (4)-(7) yields an expression for the spatial temperature covariance at small r  where Swr is the cospectrum of sensible heat flux in the surface layer. For longitudinalseparation,Taylor' s hypothesisof frozen turbulence will result in unrealistic longitudinal coherencein ßThe function# wasevaluatedexperimentally isotropic turbulence [Kristensen and Jensen, 1979]. Conse- wherex quently, the concept of eddy turnover time has to be by Wyngaard et al. [1971] and was derived by Panofs• and incorporated into spectral characteristics [Kristensen, 1979]. Dutton [1984] from balance equations of turbulence kinetic fluctuations. SinceTJ/T' 2 is alsoa Assumingthat r/u is small comparedto eddy turnover time energyandtemperature and the quadrature spectrum is negligible, Moore [1986] function of zm/L only, (8) reveals that temperature covariarrived at the following transfer function for longitudinal ance can be calculated from the measurements of L. It is tempting to assume that spatial decays of T and w-T covaseparation:  [  T'(x)T'(x +r)=r'2(x)1-2--•  dances  Ts(f) = cos (2rrf).  are identical  #  , (8)  so that  (3)  (On the other hand, G. W. Thurtell [see Heikinheimo, 1986] believed that (3) should represent the transfer function for lateral separation.) It can be shown from (1) (or (3)), (2), and the expression for Swr in the neutral and unstablesurfacelayer [Kaimal et al., 1972; Kristensen and Fitzjarrald, 1984] that the ratio,  H(r)=H(O) 1- 2•  g  .  (9)  This, however, will result in egors. In an unstable surface  layer, the decay of w spatial covariance with r is expected to be faster than that of T covariance, as the energy-containing range occursat larger wavelengthsfor the T power spectrum  H(D/H(O), is a functionof r/Zm, where Zm = Z -- d, with z thanfor the w spectrum. Thisdifference in the two decay the heightabovethe groundsurfaceand d the displacement rates was observed by Koprov and Sokolov [1973], who also height, and is independentof wind speed.The r/Zm depen- showed that the decay of w-T covariance was in between the dence has somewhat been confirmed by a few experiments [Heikinheimo, 1986; Koprov and Sokolov, 1973]. However, there are three significant problems with the above transfer function approach: it implies similarity between velocity components and w-T spatial cospectra, which, for the reason that will become clear later, is not valid; it is not appropriate when separationr is large; and it does not relate w-T covariance to separation orientation or stability. Monin-Obukhov similarity, on the other hand, may provide a more promising tool than the above approachesfor predicting turbulence structure over horizontally homogeneous surfaces. For example, it has been used for studying  two. But it is possible, as discussedin the next section, to develop a model relating w-T covariance to L in a manner similar to that done for T covariance.  In the past, several studieshave addressedexperimentally the issue of w-T spatial covariance. Leuning et al. [1982] observed that increasing the lateral separation distance between the temperature and vertical velocity sensors of their eddy co,elation system from 0.05 to 0.45 m reduced the measured sensibleheat flux by about 6% at a height of 2 m in the daytime. Based on limited observations, Koprov and Sokolov [1973] found that the dependence of w-T covafiance on lateral separation in slightly unstable condithe behaviorof the structurefunction,Dqq, of a certain tions (z/L = -0.02) could be described with the following quantity, q [Tatarskii, 1971], which is defined as equation:  Dqq= [q(•)- q(• + •)]2.  (4)  At separation distances of inertial subrange scale, the temperature or humidity structure function has the form •,2  2/3  Dqq= Cqr ,  (5)  H(r)/H(O) = exp (-r/Zm).  (10)  This equation was used by Heikinheimo [1986] as one of the means for estimating the loss of eddy cogelation fluxes due to lateral sensor separation. The prediction of (10) appeared to agree with his observations, although the stability during  LEE AND BLACK' EDDY CORRELATION  his experiment was probably not neutral (see section 6). These experimental studies dealt with specific cases. The question regarding the influence of wind direction and stability still remains unanswered. In next section, a model will be developed in the context of Monin-Obukhov similarity to allow the assessmentof the spatial decay of w-T covariance from the measurementsof all controlling factors.  3.  3.1.  18,547  qbe = 1 - zm/L.  (17)  Substituting(14)-(17) into (13) and using (7) and (12) yields  H(?)= H(0)[1-/3(/5, Zm/L)(r/Zm)4/3], (18) where  1•1(/5, Zm/L)= a(/5)(1- 16Zm/L)-l/2(1 - Zm/L)1/3. (19) This completes the derivation of the constraint of w-T covariance at small r. The coefficient a(/5)will be determined experimentally (section 5.2).  Model  Constraint at Small Separation Distance  3.2.  Tatarskil [1971] derived a one-dimensionalspectral density function from the structurefunction (4). By equatingthis spectral density function to the inertial subrange spectra, Wyngaard and colleagues[Wyngaard et al., 1971; Wyngaard and Clifford, 1978; Wyngaard and LeMone, 1980] obtained expressionsfor structureparametersof temperature, humidity, and velocity in the atmosphericsurface layer. Alternatively, one can use similarity argumentsto arrive at the same resultswithout knowing the details of the spectraldynamics [Tatarskil, 1971; Hill, 1989]. By analogy with (4), let us define a cross-structure function for w-T  SENSIBLE HEAT FLUX  as  Dwr = [w(.•) - w(.• + •)][T(.•) - T(.• + •)].  (11)  Apparently,the directionof separationis importantfor Dwr. Because turbulence is considered to be horizontally homogeneous, (11) can be written as  H(•) = H(O) - Owr/2,  (12)  where H(D = H(-•) is assumed.Wyngaardand Cote [1972] proposed a model for the w-T cospectrum in the inertial subrangethat is in excellent agreement with observations. They postulated that the cospectrum depends on vertical potential temperature gradient (aO/az), dissipation rate of turbulent kinetic energy (e), and wavelength. It can be deducedfrom this model that for r in the inertial subrange, Dwr shoulddependon aO/az, •, and r. Dimensionalanalysis  Constraint at Large Separation Distance  The w-T covariance should approach zero as separation distance  increases  H(•)/H(O) -• 0  as r--• •.  (20)  The exact form of H at large r is not clear. Here we choose to use the following exponential function:  H(•) = H(0) exp[-•(/5, Zrn/L)(r/Zrn)4/3].(21) The advantage of this function lies in its simplicity and its ability to match constraintsfor both small and large separation distances (equations (18) and (20)). As discussed in section 5, (21) fits rather well with the observed behavior of w-T covariance. With a separation distance in the inertial subrange, (21) differs only slightly from (18) (section 6).  4.  Experimental Methods  4.1.  Site Descriptions  The experiments were conducted in a potato field and a clover field located on level delta land at the mouth of the  Fraser River in Delta, British Columbia, Canada, in the summer of 1992. The potato crop was planted on E-W running ridges 0.20 m high, 0.41 m wide, and spaced0.84 m apart. The crop was near maturity. The canopy was very denseand the height of the stand, measuredfrom the bottom of the ridges, was 0.76 m. An open-lattice triangular instrument tower 0.25 m wide was positioned near the SE corner requires that of the field, which was very extensive. The roughness length, estimated from wind speed measured at two heights Dwr= 2k4/3a(8)lao/az • 1/3r4/3. (13) near-neutral stability, was about 6 cm. The clover field was 500 m (N-S) x 200 m (E-W). To the Here k (= 0.4) is the von Karman constant and a(/5) is a wind direction dependent coefficient, with /5 defined such east of the clover field was a potato field, to the west a bare that it varies between 0ø when wind is parallel to the field, to the north a field of beans in the early stages of direction of separation (longitudinal separation) and 90ø development, and to the south a farmhouse and a paved when wind is perpendicularto the direction of separation road. The clover canopy was 10 cm tall and covered about (lateral separation).Thus D wr doesnot follow the 2/3 power 10% of the ground surface. The rest of the surface was law of (5) or the 1/1 power law as suggestedby (10) at small covered by dry plant residue from the harvest of the previous pea crop, resulting in a significant mulching effect. The instrument tower was positioned at the center of the field. It is known that [Panofsky and Dutton, 1984] The estimated roughnesslength was about 3 cm.  c)Z --kZm •bh  ,  (14)  • =kz--• cb •  ,  (15)  where the similarity functions have the forms (L < 0)  qO h= (1 - 16zm/L) -1/2,  (16)  4.2.  Instrumentation  The same apparatus was used for both field experiments. The key instrumentation included a one-dimensional (vertical) sonic anemometer/thermocouple (Campbell Scientific, Incorporated, model CA27, Logan, Utah; unit 1138) and two linear arrays of thermocouples oriented perpendicular to each other, called TX and TY, each consisting of five sensorsat various distancesfrom the sonic path (Figure 1).  18,548  LEE AND BLACK:  EDDY CORRELATION 30-min  favorable  1143  wind  directions ßwind  ';  X X /  •.  vector  X  X  ! !  lm  st  Figure 1. Plan view of the instrument layout for the experiments in the potato and clover fields in Delta, 1992: onedimensional sonic anemometer/thermocouple(solid circle), thermocouple (crosses), cup anemometer (bow tie), fast response hygrometer (open circle), and wind vane (open triangle). Also shown are the angle between horizontal wind vector and each of the two thermocouple arrays (TX and TY). Refer to Table 1 for instrument heights and Table 2 for array separation distances.  intervals.  HEAT FLUX These  were  the variances  of and covari-  ance between vertical velocity and air temperature from unit 1138 and covariances between each of the two signalsfrom unit 1138 and those of the array thermocouples. A second eddy correlation unit was mounted at a level above unit 1138 (Table 1). It consisted of a one-dimensional sonic anemometer/thermocouple(1143) of the same type as unit 1138 and a fast responsekrypton hygrometer (Campbell Scientific, Incorporated, model K20). A second 21X data logger sampled signalsfrom this unit at 5 Hz and calculated fluxes  tower  SENSIBLE  of sensible  and latent  heat  over  30-min  intervals.  These measurements served as, among other things, a check of the performance of unit 1138. Air temperature and wind speed were measured at two levels with thermocouples of the same type as those of the arrays and sensitive cup anemometers (Thornthwaite Associates, Centerton, New Jersey, model 901-LED), for the purpose of determiningthe gradient Richardson number. Wind direction was monitored with a wind vane (Met-One, Incorporated, Grants Pass, Oregon, model 024A). Supplementary measurements included net radiation flux, heat flux into the soil, and wet and dry bulb temperature. After the completion of the two experiments, a threedimensionalsonic anemometer/thermometer(Applied Technologies,Incorporated, Boulder, Colorado, model SWS-211/ 3V) was operated at a height of 1.42 over the clover field from August 18-20, 1992, with the two-level thermocouples and cup anemometers still in operation. The MoninObukhov length scale, L, was calculated from the fluxes of momentum  and sensible heat measured  with this unit after  All thermocouples were made by cross-weldingchromel and  the momentum  constantan  the three-dimensionalanemometer probe [Lee, 1992].  wires.  The  diameter  of the wires  used for the  arrays was 26/am and that for the sonic anemometerwas 13 /am. The distance between the center of the sonic anemometer path and the thermocoupleof unit 1138 (r0) was 3.5 cm and was in the direction of TY. Both arrays were mounted at the height of the center of the sonic path. They were operated at one height (3.35 m) over the potato field and two heights (1.54 and 3.01 m) over the clover field (Table 1). The array thermocouples were supported on two 0.6-cmdiameter stainless steel rods. Their measurementjunctions were  about  interference  15 cm above  the rods  on the measurements  se the effect  of flow  was minimal.  Signals from unit 1138 and the arrays were sampled at 5 Hz with a data logger (Campbell Scientific, Incorporated, model 21X with extended software II). The reference junctions for the array thermocoupleswere at the terminal strip of the data logger. Statistics were calculated on line over  4.3.  flux was corrected  for the shadow  effect of  Data Processing  In the potato field the range of favorable wind directions was 2500-300ø, with fetches of 400-700 m. In the clover field the range was 120ø-270ø when the arrays were at the 1.54-m  height and 150ø-210ø when they were at the 3.01-m height. The corresponding fetches were 100-270 and 250-270 m. Tower  interference  with  the  measurements  was minimal  when wind direction was in these ranges (Figure 1). Only runs with  favorable  wind  direction  were  selected  for the  following analysis: The gradient Richardsonnumber (R i) was calculatedfrom !7 02--01  Ri -• -_  T (u2- Ul)2  ( z2 - z l)  '  (22)  where # is the acceleration due to gravity, T is the air temperatureaveragedover heightsz l and z2 (Table 1), and Table 1.  Instrument Heights for the Experiments in the  Potato and Clover Fields in Delta, 1992 Ri  measurement levels (zl andz2),za = [(zl - d)(z2- d)]¸.  TX, TY  Field Potato Clover  Date July 20-26 July 28 to Aug. 1 Aug. 1-4  1138 3.35 1.54 3.01  1143 3.50 3.37  zl 1.57 1.37  Oi and u i (i = 1, 2) are the potential temperatureand wind speedat heightzi. Ri calculatedfrom (22) is for the geometric mean height above the displacement height (d) for the two  z2 2.91 2.72  Dir 3.35 3.62  Heights are given for one-dimensional sonic anemometer/ thermocouple units (1138 and 1143), thermocouplearrays (TX and TY), wind speedand temperature sensorsfor measuringRichardson number (Ri), and wind vane (Dir). For the potato field all heights were measuredfrom the bottom of the ridge. Heights are in meters.  (One reviewer brought the work of Arya [1991] to our atten-  tion.AryaimpliedthatRi at heightza be calculated asRi -•  (#/•(02- O1)/(u2 - u•)2za In [(z2- d)/(Zl- d)],which is consistentwith Sellers [1965, p. 153]. For our height configuration (Table l) the difference between this formula and (22) is insignificant.) Since Ri is proportional to height in the surface layer under unstable conditions [Businger, 1966; Pandolfo, 1966], the following equation is used to obtain Ri  for the height of the arrays, Zm'  LEE AND BLACK: EDDY CORRELATIONSENSIBLEHEAT FLUX 0.15  Ztn  Ri z = •  Ri.  (23)  potatofield  zg 0.10  Ri z is then related to the Monin-Obukhovlength scale as  -  [Businger, 1966; Pandolfo, 1966]  1:1 '"  -  0.05  Ztn  -•-= Riz.  (24) 0  Thus the parameter,z m/L, which appearsin the similarity  0  functions,canbe replacedby Ri z. Kinematic  18,549  0.05  0.10  0.15  sensible heat flux was calculated from the 0.3  gradientmeasurements(Panofskyand Dutton [1984], combinationof their equations6.8.4, 6.9.3, and 6.10.2), H(0) = -  k2(z- d)2 OuO0  0.2  [qbrn(aiz)] 3 OZOZ  _ • ßo•5o =•  0.1  or  k2  (u2- u•)(02- 0•)  H(O) =-[q•m(Ri)]3 [In(z2-d)-In(z•- d)]2' (25) where •m is the similarity function for momentumflux and is related to qO h as 1/2  •m = •h  ß  (26)  Becauseof the sparseness of the canopy, d was set to zero for the cloverfield. For the potatofield it was estimatedto be 0.76 m (section5.1). The calculationsusing(25) were compared with w-T covariancedirectly measuredwith eddy correlationunit 1138to check the consistencyamongmeasurements.  i  0  0.1  i  /  0.2  0.3  øc m/s, 1143 Figure 3.  Comparison of w-T covariance (øC m/s) mea-  sured with the one-dimensional sonic anemometer/thermo-  coupleunits 1138and 1143 over the clover (1138 at height z = 1.54 (open squares) and 3.01 m (solid squares) and potato field in Delta, 1992.  sonicanemometerof unit 1138and the thermocoupleof TY at separationr from the anemometer.Similarlyfor array TX  Let/3x and•y be the decaycoefficients for TX andTY,  In[H(r)/H(ro)] = - [•x(r/Zm) 4/3+ [•y(ro/Zm) 4/3  respectively. It follows from (21) that  •  In [H(r)/H(ro) ] = -[3y[(r/Zm) 4/3- (ro/Zm)4/3],(27)  [3x[(r/Zm) 4/3_ (rO/Zm) 4/3].  (28)  andBywereevaluated bylinearlyregressing In for array TY, whereH(ro) is the w-T covariancemeasured Therefore/3x (r/Zm)4/3_ (rO/Zm)4/3 for individual with unit 1138with a slightseparation distance,r0 (= 3.5 [H(r)/H(ro)]against from each of the two arrays. cm) in the direction of TY, and H(r) was measuredwith the runs usingmeasurements 5.  Results  5.1.  Comparison of Measurements  There was good agreement between Ri calculated from  (22)andza/L fromthe three-dimensional sonicunit (Figure 2). This confirmed(24) and indicatedthat our techniquefor determining Ri was reliable. Good agreement was also  achievedbetweenthe values of w-T covariance(H(r0)) measured with the two one-dimensional  -0.5  -1.0  units over both  fields.The scatterin the plot was mostlywithin 10% of the 1:1 line (Figure 3). The kinematic sensibleheat flux, H(0), calculated from  -0.5  0  the gradient measurements(equation (25)) was compared with w-T covariance,H(ro), measureddirectly with eddy correlationunit 1138(Figure 4). This comparisonprovidesa rigorouscheck of our measurementsystem.No correction wasmadeto H(r o) to accountfor the separation distancer0. Our calculation  Zg/L  based on the model of w-T  covariance  indicated that this would only cause a very small error, Figure 2. Comparison of the gradient Richardson number typically 0.5%. As shown in Figure 4, the scatter for the (Ri) andMonin-Obukhov parameter (za/L) overtheclover clover field was rather large but overall was about 1:1 line. field in Delta, 1992. The situation for the potato field was rather complex  18,550  LEE AND BLACK:  EDDY CORRELATION  SENSIBLE  HEAT FLUX  TY, Bx/By,determined fromthe observations (section4.3), plottedagainstsin/•x. The influenceof the wind angleon the spatial behavior of w-T covariance can be clearly seen: the decay in the lateral direction is about twice that in the longitudinal direction. This is because turbulent eddies tend to be elongated in the mean wind direction in the surface layer [Panofsky and Dutton, 1984]. Since for a specificrun, stability is the same for both arrays, it follows from (19) that  theratiois a functionof/•x or/•y only.To furtherquantify 0  0  0.1  0.2  the effect of the wind angle, we proposethat turbulent eddies have an elliptical shape such that  0.3  a(•) = a(cos2 • + b sin2 •)2/3 where  a and b are two constants.  (29)  Hence  0.2  • x (cos2 /•x + b sin2/•x)2/3  • y (bcos 2/•x+sin2 /•x)2/3 ß  0.1  0  nø 0  I  I  0.1  0.2  0.3  øCm/s,eddycorrelation Figure 4. Comparisonof the kinematicsensibleheat flux (øC m/s), calculatedfrom gradientmeasurements,and w-T covariance measuredwith the eddy correlation techniqueusing unit 1138over the potato and over the clover field (1138at z = 1.54 (open squares)and 3.01 m (solid squares))in Delta, 1992.The displacementheight d was 0 for the clover field and was determinedto be 0.76 m for the potatofield. because of the unknown parameter, d. Using the conventional value of the ratio, d/h = 0.7 [Jarvis et al., 1976], where h (= 0.76 m) is the height of the stand, we found that H(0) from the gradient measurementswas 47% higher than  H(r o) from the eddy correlation measurements.This was unlikely a result of the roughnesssublayer effect becauseit would have resultedin an underestimationof •m compared to that calculated from (16) and (26) [Garratt, 1978], and hence the neglect of the effect would have caused an underestimateof sensibleheat flux from (25). To produce good agreementbetween the two techniques, a value of 0.76 m for d, or d = h, was required (Figure 4). This value is rather high, but it is believed to be reasonable because the potato canopy was very dense and completely closed. The dominant part of the sensibleheat flux is expectedto come from the thin layer very near the top of the stand. The displacement height, if interpreted as the average height of the source of sensible heat [Lee and Black, 1993], should therefore be close to the stand height. This d value was used  With a value of b = 2.4, (30) fits well the observations in Figure 5. A little manipulation of (19) and (29) yields  -•/2 (•3/2q-•y3/2•2/3 • _ a(1+ b)2/3(1-16Riz) x  ß(1 - Riz)1/3. 3/2  Figure 6 shows (fl•/2 + fly )  2/3  (31)  plotted against Riz. As  indicated by this figure, the decay of H was greatly influenced by stability. There was considerable scatter in nearneutral conditions,which was a result of large relative errors arising from the measurementsof H of small magnitudes. The trend is clearly predicted by (31)' lower decay occurred in more unstableconditions.Basedon the data presentedin  Figure6, a(1 + b)2/3wasevaluated to be2.67or a -- 1.18. In summarythe decay coefficient(/•) can be expressedas follows'  •3(•i,Riz)= 1.18(cos 2 •5+ 2.4 sin2 •i)2/3 ß((1 - 16Riz)-l/2(1 - Riz)•/3.  2.0  '  I  '•  o oo  1.0-  ß ß  L% - o  0.5•_  I •  0  eo  •  ß  -o-.q•  -  , 0  (32)  '  1.5-  in the subsequentanalysisof the w-T covariance data from the potato field. Sensitivity calculations indicated that the uncertainty in the d value would cause only a very small error in the analysis:for the typical value of Ri = - 0.2, the changeof d/h from 0.7 to 1.0 would only increaseH(F)/H(O) from (21) by less than 3% for all the separationdistancesin the potato field experiment. 5.2. Relating Decay Coefficient15to Wind Direction and Stability  (30)  •.;•  o o •  o  o  o  I 0.5  o  _  , 1.0  sin •gure 5.  The ratio of decay coefficientof array TX to that  of TY (fix/fly) plottedagainstthe sineof the anglebetween a•ay TX and the horizontalwind vector (•x) over the potato  Let/5x and/Sybe the windanglesfor arraysTX andTY, (squares, z - d = 2.59 m) and clover (open circles, z respectively, andnotethat/5x q- /Sy= 90ø(Figure1). Figure d = 1.54 m; solid circles, z - d = 3.01 m) fields in Delta, 5 shows the ratio of the decay coefficient for TX to that for  1992.The solid cu•e representsequation(30) with b = 2.4.  LEE AND BLACK: EDDY CORRELATION I  I  SENSIBLE  HEAT FLUX  18,551  each of the fields. These days were selected for comparison because continuous measurements were made throughout the day and there were wide ranges of wind angle (8) and  I  stability (Riz). The influenceof stability is obvious. For  o  0  • 0  -0.5  •  •  -1.0  -1.5  Riz Figure 6.  The decay coefficient plotted as a function of  Richardson number(Riz) over the potatoand cloverfieldsin Delta, 1992.The solid curve representsequation(31) with a = 1.18. Other variables and symbolsare the sameas in Figure 5.  example, runs 1330 and 1730 PST in the clover field had similar wind angles, but run 1330 experienced a smaller decay of w-T covariance becausethe air was more unstable. The influence of wind direction is best shown by Figure 7a. In the morning of July 26, wind direction was approximately aligned with array TX. The decay for TX with separation was evidently smaller than that for TY. Wind direction started to shift steadily around noon. At 1500 PST the wind angle was about the same for both arrays, and the decay was almost identical. The model of form (21) describes the observations rather well in general. In practice, (21) and (32) can be used to assess flux underestimation due to horizontal sensor separation and to correct the measurements.Based on these equations,H(ro)  (r0 = 3.5 cm) was estimatedfrom the measuredH(•) at various separation distances and the measurementsof 8 and 5.3.  Comparison of Calculationsof (21) and (32)  With  Observations  Ri z. The resultswere then comparedwith the direct measurementsof H(r o) made with the eddy correlation unit  Figure 7 compares the observed decay of w-T covariance with that calculated using (21) and (32) for 2 days, one for  1138. The relevant  information  is summarized  in Table  !  9:30 -1.331  8:00 -0.032  66ø  8ø  11:30 -0.892  10:00 -0.062  41ø  6ø  13'30 -0.640  12:00 -0.221  22ø  4ø 14:00 -0.169  15:30 -0.416  ..• 26 ø  "••'''"  28ø  15:00  :"'•. - ø-0.183 37 ø  17:30 -0.205 0  1  2.  There exists a good correlation between the estimated and  21ø  2  r 4/3.ro4/3 (m)  I  0  1 4/3  I  I  2  3  _ ro4/3 (m)  Figure 7. (a) Comparison of the decay of w-T covariance observed (TX, open squares; TY, solid squares) with that calculated using equations (21) and (32) (TX, dashed line; TY, solid line) at z - d = 2.59 m in the potato field in Delta, July 26, 1992. Also indicated are ending time of the 30-min runs (PST), Richardson number at the height of the covariance measurements,and wind angle for array TX. (b) Same as in Figure 7a except for z - d = 1.54 m in the clover field, July 31, 1992.  18,552  LEE AND BLACK: EDDY CORRELATION  Table 2.  SENSIBLE  HEAT FLUX  Slope of Regressionof w-T Covariance(H(ro)) EstimatedFrom (21) and (32) for Various Separation  Distances (r) Against That Measured With Eddy Correlation Unit 1138 Thermocouple Arrays TX  TX  TX  TX  TX  TY  TY  TY  TY  TY  0.40 0.99  0.80 0.98  1.60 0.99  1.90 1.01  0.20 0.99  0.40 1.00  0.80 0.99  1.35 1.02  1.73 1.02  Potato z - d = 2.59 m  r, m slope  0.20 1.00  n = 30  R  0.993  0.986  0.981  0.969  0.961  0.986  0.985  0.981  0.967  0.952  Clover z - d = 1.54 m  r, m slope  0.20 0.98  0.40 0.96  0.80 0.95  1.60 1.02  1.90 1.08  0.20 1.00  0.40 0.95  0.80 0.90  1.35 0.97  1.73 1.06  n = 32  R  0.995  0.994  0.990  0.965  0.943  0.994  0.994  0.984  0.973  0.938  Clover z- d = 3.01 m  r, m slope  0.20 0.98  0.40 0.95  0.80 0.96  1.60 0.93  1.90 0.93  0.20 0.98  0.40 0.96  0.80 0.96  1.35 0.97  1.65 0.97  n =  R  0.993  0.994  0.978  0.964  0.954  0.994  0.991  0.981  0.966  0.959  17  Also given are correlation coefficient (R), distance above the displacement height (z - d), and total number of runs (n).  directly measuredH(ro) for all height and separationdistance  combinations.  For  26 of the  30 combinations  the  agreement was within +-6%. 6.  Discussion  As shown by the studies of Wyngaard and Cote [1972], Kaimal et al. [1972], and Schmitt et al. [1979] on the vertical velocity and scalar cospectral characteristics in the surface layer under neutral and unstable conditions, the inertial  subrangeis establishedbeyond Zmkl • 3, where kl is wave number, which is roughly equivalent to r/Zm < 0.3 in the space domain. In this range the 4/3 power law, or the small separation constraint (18) differs from (21) by less than 7 and 3% at zm/L = 0 and 1 and 0.5% at zm/L =- 0.5 for longitudinal and lateral separations,respectively. Concern has been raised by a reviewer regarding the assumptionH(D = H(- ?) (section 3.1). For lateral separation the two covariances should be the same, but for longitudinal separationthe mean wind shearmay causethese to differ. With the measurement apparatus of the present study (Figure 1), it is not feasible to assess the possible asymmetry of the w-T covariance in the atmospheric surface layer. One alternative is to use w and T time seriesobserved at a single point and apply Taylor's frozen turbulence  pendent validation of (21) and (32), but it confirms the 4/3 scaling law. Assuming that 3% is the tolerable flux loss due to sensor separationduring eddy correlation measurements,the maximum allowable separation distance (rmax)will depend on measurementheight, stability, and wind direction (Figure 9). There is an advantage in elevating the measurement level:  the increasein z will decreasermax/Zrn andincreaseIzm/LI, both resulting in a longer safe separation distance. Of course, when doing so, one should be aware of the fetch requirement. 7.  Conclusion  The model of w-T spatial covariance, establishedhere for unstable atmosphere in the framework of Monin-Obukhov similarity, describes the observations satisfactorily. The decay of w-T covariance with separationfollows 4/3 power law at small separation distancesand is exponential at large distances. It is slower in more unstable air and is faster in the  lateral direction than in the longitudinal direction. The model can be used to assess the flux underestimation  due to sensor  separationfrom information on separationorientation, measurementheight, separationdistance, and stability. Assurn-  hypothesis suchthatH(r) = p-1 $•' w(t)T(t + r/u) dt and H(-r) = p-1 $•' w(t)T(t - r/u)dt for longitudinal  0  I  I  separation, where P is the averaging interval and u is the longitudinal velocity component. Taylor's hypothesis is expected to be adequate for Zmkl > 3 (S. Pond, personal communication, 1993). H(r) and H(-r) were then calculated with r = 0.7 m from the time series obtained by Lee [1992] at z = 2.25 m over a bare field. For a total of 29 runs there  is no appreciable difference between them, as shown by the  • -0.2  regression resultH(-r) = 0.992H(r) (R 2 = 0.9989). Figure 8 compares predictions of (21) and (32) with daytime observations made by Heikinheimo [1986] at two heights over an alfalfa field and averaged over 2 days. The separationwas in the lateral direction, with r 0 = 5 cm at Zm = 8 m and r0 = 20 cm at Zm = 3.5 m. Heikinheimo [1986] did not report stability for his experiment, although from the magnitude of the standard deviation of air temperature (0.3ø-0.8øC), it is likely that the atmosphere was  moderatelyunstable.A valueof Ri z =-0.2 wasusedin our calculation to provide a good fit with Heikinheimo's [1986] observations. This comparison is not a rigorous and inde-  -0.3  0  I  I  0.1  0.2  0.3  (r 4/3_ro4/3)/(z _d)4/3 Figure 8. Comparisonof the lateral decay of w-T covariance observed by Heikinheimo [1986] (open squares, z d = 3.5 m; solid squares,z - d = 8 m) over an alfalfa field with that calculated from equations (21) and (32) (solid line) usinga value of Ri z = -0.2.  LEE AND BLACK: EDDY CORRELATION SENSIBLE HEAT FLUX  0.20  I  I  18,553  Hill, R. J., Implications of Monin-Obukhov similarity theory for  I  scalar quantities, J. Atmos. Sci., 46, 2236-2251, 1989. Hinze, J. O., Turbulence, 2nd ed., McGraw-Hill, New York, 1975. Jarvis, P. G., G. B. James, and J. J. Landsberg, Coniferous forest, in Vegetationand the Atmosphere,vol. 2, Case Studies,edited by J. L. Monteith, pp. 171-240, Academic, San Diego, Calif., 1976. Kaimal, J. C., J. C. Wyngaard, Y. Izumi, and O. R. Cote, Spectral characteristicsof surface layer turbulence, Q. J. R. Meteorol.  0.15  0.10  Soc., 98, 563-589, 1972.  Koprov, B. M., and D. Y. Sokolov, Spatial correlation functions of velocity and temperature componentsin the surface layer of the atmosphere,Izv. Acad. Sci. USSR Atmos. Oceanic Phys. Engl.  0.05  Transl., 9, 178-182, 1973.  Kristensen,L., On longitudinalspectralcoherence,BoundaryLayer I  0  -0.5  I  -1.0  Meteorol., 16, 145-153, 1979.  I  -1.5  -2.0  (z - d) /L  Kristensen, L., and D. R. Fitzjarrald, The effect of line averagingon scalar flux  measurements  with  a sonic anemometer  near  the  surface, J. Atmos. Oceanic Technol., 1, 138-146, 1984. Kristensen, L., and N. O. Jensen, Lateral coherence in isotropic turbulence in the natural wind, Boundary Layer Meteorol., 17,  Figure 9. Dependence of the maximum allowable longitu353-373, 1979. dinal (solid curve) and lateral (dotted curve) separation distance (/'max)on the ratio of height (z - d) to Monin- Lee, X., Atmospheric turbulence within and above a coniferous forest, Ph.D. thesis, Appendix C, Univ. of B.C., Canada, 1992. Obukhov length scale (L) in the unstablesurfacelayer.  ing that 3% is an acceptableflux underestimation,a conservative estimate is that the ratio of separationdistanceto the height above the displacement plane should not exceed about 5%.  In principle, the model can be extended to stable conditions by replacing the similarity functions for sensibleheat and turbulent kinetic energy with those for stable air. The experimental verification of this practice is challengingbecauseof the difficulty in measuringfluxes of small magnitude and the very strict requirement of fetch for the site. This issue will be addressed in the future.  Lee, X., and T. A. Black, Atmospheric turbulence within and above a Douglas-fir stand, II, Eddy fluxes of sensible heat and water vapour, Boundary Layer Meteorol., 64, 364-389, 1993. Leuning, R., O. T. Denmead, A. R. G. Lang, and E. Ohtaki, Effects of heat and water vapor transport on eddy covariance measurement of CO2 fluxes, Boundary Layer Meteorol., 23, 209-222, 1982.  Moore, C. J., Frequency responsecorrectionsfor eddy correlation systems, Boundary Layer Meteorol., 37, 17-35, 1986.  Pandolfo, J.P., Wind and temperature for constantflux boundary layers in lapse conditions with a variable eddy conductivity to eddy viscosity ratio, J. Atmos. Sci., 23,495-502, 1966. Panofsky, H. A., and J. A. Dutton, Atmospheric Turbulence: Models and Methods for EngineeringApplications, John Wiley, New York, 1984.  Schmitt, K. F., C. A. Friehe, and C. H. Gibson, Structure of marine surface layer turbulence, J. Atmos. Sci., 36, 602-618, 1979.  Sellers,W. D., PhysicalClimatology,Universityof ChicagoPress, Chicago, Ill., 1965.  Silverman, B. A., The effect of spatial averaging on spectrum Acknowledgments. This researchwas supportedby an operating grant from the Natural Scienceand EngineeringResearchCouncil of Canada. Richard Nistuk provided valuable help in the field. We gratefully acknowledgeHugh and Stan Reynoldsfor allowingus to use their crop fields. We also thank the two reviewers for their comments  on this work.  estimation, J. Appl. Meteorol., 7, 168-172, 1968.  Tatarskii, V. I., The Effects of the TurbulentAtmosphereon Wave Propagation, Kefer, Jerusalem, 1971. Wyngaard, J. C., and S. F. Clifford, Estimating momentum, heat and moisturefluxesfrom structureparameters,J. Atmos. Sci., 35, 1204-1211, 1978.  Wyngaard, J. C., and O. R. Cote, Cospectral similarity in the atmospheric surface layer, Q. J. R. Meteorol. Soc., 98, 590-603,  References Arya, S. P., Finite-differenceerrorsin estimationof gradientsin the atmospheric surface layer, J. Appl. Meteorol., 30, 251-253, 1991. Businger,J. A., Transfer of momentumand heat in the planetary boundarylayer, in Proceedingsof the Arctic Symposiumon Heat Budget and AtmosphericCirculation, pp. 305-322, Rand Corpo-  1972.  Wyngaard, J. C., and M. A. LeMone, Behavior of the refractive index structure parameter in the entraining convective boundary layer, J. Atmos. Sci., 37, 1573-1585, 1980. Wyngaard, J. C., Y. Izumi, and S. A. Collins, Behavior of the refractive-index-structure parameter near the ground, J. Opt. Soc. Am., 61, 1646-1650, 1971.  ration, Santa Monica, Calif., 1966.  Garratt, J. R., Transfer characteristicsfor a heterogeneoussurface of large aerodynamicroughness,Q. J. R. Meteorol. Soc., 104, 491-502, 1978.  Heikinheimo, M. J., Techniquesfor determiningcarbon dioxide and water vapour transport above a vegetated surface using the eddy-correlationmethod,Ph.D. thesis,Univ. of Guelph,Guelph, Ontario, Canada, 1986.  T. A. Black, Department of Soil Science, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z4. X. Lee, School of Forestry and Environmental Studies, Yale University, New Haven, CT 06511. (Received April 21, 1993' revised December 3, 1993; accepted April 5, 1994.)  


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