STATISTICS OF SURFACE-LAYER TURBULENCE OVER TERRAIN
            WITH METRE-SCALE HETEROGENEITY

         EDGAR L ANDREAS1 ...
380                           EDGAR L ANDREAS ET AL.

narrow the error bars on the Monin–Obukhov similarity functions and,...
STATISTICS OF SURFACE-LAYER TURBULENCE                      381
     tq =  (z=L):
    tq tq                               ...
382                              EDGAR L ANDREAS ET AL.

   Because u and, thus, t and q lose their significance gradually ...
STATISTICS OF SURFACE-LAYER TURBULENCE                    383
In the asymptotic limit of free convection (i.e., large    )...
384                                  EDGAR L ANDREAS ET AL.




Figure 1. Patchy vegetation characterizes the Sevilleta. T...
STATISTICS OF SURFACE-LAYER TURBULENCE                      385
In the late summer of 1991, vegetation covered 37.7% of th...
386                            EDGAR L ANDREAS ET AL.

three-axis sonic. Statistics from these instruments included w , t ...
STATISTICS OF SURFACE-LAYER TURBULENCE                       387
4.2. QUALITY CONTROL

The three-axis ATI sonic anemometer...
388                                 EDGAR L ANDREAS ET AL.




Figure 2. Three-axis sonic wind direction histograms for ty...
STATISTICS OF SURFACE-LAYER TURBULENCE                    389
or, more specifically, the drag coefficient at neutral stabili...
390                               EDGAR L ANDREAS ET AL.




                                                             ...
STATISTICS OF SURFACE-LAYER TURBULENCE                     391
for     4 0:1,
   w =u = 1:20(0:70 3:0 )1=3;               ...
392                                EDGAR L ANDREAS ET AL.




Figure 5. Nondimensional standard deviation in longitudinal ...
STATISTICS OF SURFACE-LAYER TURBULENCE                                393




Figure 6. Nondimensional standard deviations...
394                               EDGAR L ANDREAS ET AL.




Figure 7. Nondimensional standard deviations in specific humid...
STATISTICS OF SURFACE-LAYER TURBULENCE                          395
homogeneous surfaces, Ariel and Nadezhina (1976) concl...
396                                EDGAR L ANDREAS ET AL.




Figure 8. Nondimensional temperature-humidity covariance as ...
STATISTICS OF SURFACE-LAYER TURBULENCE                        397
    By writing (5.11b) we do not mean to suggest any fun...
398                                  EDGAR L ANDREAS ET AL.




Figure 9. The t q correlation as a function of local time....
STATISTICS OF SURFACE-LAYER TURBULENCE                                   399




Figure 10. The t q correlation versus the...
400                                 EDGAR L ANDREAS ET AL.




Figure 11. Temperature skewness versus stability. All these...
STATISTICS OF SURFACE-LAYER TURBULENCE                        401
error of about 27%. The scatter in the data in Figure 11...
402                             EDGAR L ANDREAS ET AL.

tq=jt qj has the same stability dependence as t2=t2 but is 24% sma...
STATISTICS OF SURFACE-LAYER TURBULENCE               403
where upper-case letters denote averages and unhatted lower-case ...
404                                           EDGAR L ANDREAS ET AL.
   
       q   2

                 =
                ...
0:2:
DBo
(A14)

Consequently, the bracketed term in (A13) is nearly 1, and the equation reduces to
   
       ts 2 =  t 2 :
       ...
STATISTICS OF SURFACE-LAYER TURBULENCE                       405
   Let us define some additional Monin-Obukhov similarity ...
406                                 EDGAR L ANDREAS ET AL.

In unstable conditions, we expect ttt , ttq , tqq , and qqq to...
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  1. 1. STATISTICS OF SURFACE-LAYER TURBULENCE OVER TERRAIN WITH METRE-SCALE HETEROGENEITY EDGAR L ANDREAS1 , REGINALD J. HILL2 , JAMES R. GOSZ3 , DOUGLAS I. MOORE3 , WILLIAM D. OTTO2 and ACHANTA D. SARMA4 1 U.S. Army Cold Regions Research and Engineering Laboratory, 72 Lyme Road, Hanover, New Hampshire 03755-1290, U.S.A. 2 National Oceanic and Atmospheric Administration, Environmental Technology Laboratory, 325 Broadway, Boulder, Colorado 80303-3328, U.S.A. 3 Biology Department, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A. 4 R. & T. Unit for Navigational Electronics, Osmania University, Hyderabad – 500007, India (Received in final form 23 June, 1997) Abstract. The Sevilleta National Wildlife Refuge has patchy vegetation in sandy soil. During midday and at night, the surface sources and sinks for heat and moisture may thus be different. Although the Sevilleta is broad and level, its metre-scale heterogeneity could therefore violate an assumption on which Monin-Obukhov similarity theory (MOST) relies. To test the applicability of MOST in such a setting, we measured the standard deviations of vertical (w ) and longitudinal velocity (u ), temperature (t ), and humidity (q ), the temperature-humidity covariance (tq ), and the temperature skewness (St ). Dividing the former five quantities by the appropriate flux scales (u , t , and q ) j j j j j j yielded the nondimensional statistics w =u , u =u , t = t , q = q , and tq=t q . w =u , t = t , and St have magnitudes and variations with stability similar to those reported in the literature and, thus, seem to obey MOST. Though u =u is often presumed not to obey MOST, our u =u data also j j agree with MOST scaling arguments. While q = q has the same dependence on stability as t = t , j j its magnitude is 28% larger. When we ignore tq=t q values measured during sunrise and sunset transitions – when MOST is not expected to apply – this statistic has essentially the same magnitude and stability dependence as t =t 2 . In a flow that truly obeys MOST, t =t 2 , q =q 2 , and tq=t q should all have the same functional form. That q =q 2 differs from the other two suggests that the Sevilleta has an interesting surface not compatible with MOST. The sources of humidity reflect the patchiness while, despite the patchiness, the sources of heat seem uniformly distributed. Key words: Bowen ratio, Heterogeneous terrain, Monin–Obukhov similarity, Skewness of temper- ature, Sonic anemometer/thermometer, Statistics of turbulence 1. Introduction Monin–Obukhov similarity theory (MOST) has been the most important develop- ment in boundary-layer meteorology in the last 50 years. By unifying the inter- pretation of diverse observations, it provided the theoretical foundation on which boundary-layer meteorology has risen as a discipline. Yet, despite the success of MOST in unifying theory and observations, disturb- ing uncertainties persist in some of the universal functions that it predicts should exist. Compare, for example, the recent summaries by Panofsky and Dutton (1984), Sorbjan (1989), and Kaimal and Finnigan (1994). In light of this persistent uncer- tainty, it is still important to report high-quality turbulence data that may help Boundary-Layer Meteorology 86: 379–408, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
  2. 2. 380 EDGAR L ANDREAS ET AL. narrow the error bars on the Monin–Obukhov similarity functions and, indeed, answer questions about MOST’s applicability. Although MOST is founded on the assumption of horizontal homogeneity, it really is the only conceptual framework we have for treating near-surface turbulence in the atmospheric boundary layer. Consequently, much current research focuses on extending MOST to heterogeneous surfaces (e.g., Beljaars and Holtslag, 1991; Roth and Oke, 1993; Roth, 1993; Katul et al., 1995). Here our objective is also to use MOST to investigate turbulence statistics over a heterogeneous surface – but one that is heterogeneous only at scales from tens of centimetres to several metres. At larger scales, our site is homogeneous. We did our work at the Sevilleta National Wildlife Refuge, a semi-arid grassland between Albuquerque and Socorro, New Mexico. The Sevilleta’s vegetation is patchy, with bare ground between clumps of plants. It is easy to imagine that, during daytime, the bare ground is a heat source in the late summer, and the plants are water vapour sources. Because of this source heterogeneity, the statistics of temperature and humidity and, especially, their covariance might not follow the same similarity relations, as MOST predicts they should (Hill, 1989). Our results, however, suggest that the surface heat sources are not as heteroge- neous as the plant cover. The measured nondimensional temperature variance and temperature skewness values follow Monin–Obukhov similarity functions similar to those already reported in the literature. The nondimensional humidity variance, on the other hand, follows a similarity relation that is 60% above the temperature relation. But then the nondimensional temperature-humidity covariance coinciden- tally follows practically the same similarity relation as does temperature variance. This latter result is possible because the temperature-humidity correlation coeffi- cient – even during stationary periods – typically has an absolute value of 0.8 or less. We conclude that this behaviour of the temperature-humidity covariance is evidence of how the Sevilleta’s metre-scale heterogeneity leads to violations of MOST. In a flow that strictly obeys MOST, the correlation coefficient between any two conservative scalars must be 1 (Hill, 1989). 2. Theoretical Background Monin–Obukhov similarity theory predicts the following relationships (e.g., Wyn- gaard, 1973; Sorbjan, 1989, p. 69 ff.; Hill, 1989): w = (z=L); u 33 (2.1) t = (z=L); jt j tt (2.2) q jqj = qq (z=L); (2.3)
  3. 3. STATISTICS OF SURFACE-LAYER TURBULENCE 381 tq = (z=L): tq tq (2.4) Here, w , t , and q are the standard deviations in vertical velocity, temperature, and specific humidity, and tq is the temperature-humidity covariance, where t and q are, respectively, the turbulent fluctuations in temperature and specific humidity, and the overbar denotes a time average. u , t , and q are flux scales such that u2 = uw, ut = wt, and uq = wq are, respectively, the kinematic surface stress, temperature flux, and specific humidity flux. In addition, z is the measurement height, and L is the Obukhov length, L 1 = gkwt3v ; Tv u (2.5) where g is the acceleration of gravity, k (= 0.4) is the von K´ rm´ n constant, a a Tv = T (1 + 0:61Q) (2.6) is a representative virtual temperature of the atmospheric surface layer (ASL), and wtv = wt(1 + 0:61Q) + 0:61Twq (2.7) is the virtual temperature flux. Also in (2.6) and (2.7), T and Q are representative surface-layer values of air temperature and specific humidity. In Equations (2.1)–(2.4), the crux of MOST is the functions; these are nondi- mensional – presumably universal – functions of the stability parameter z=L. Although MOST predicts that universal forms for should exist, the actual func- tions must be found experimentally. MOST does, however, provide insights into the functional forms of the s for very unstable and very stable stratification. Because in very unstable or free-convection conditions, u loses its significance as the appropriate velocity scale, it is common to define a free-convection velocity scale (e.g., Hess, 1992) = !1 3 uf = zgwtv : Tv (2.8) This scale, in turn, lets us define new temperature and humidity scales, = !1 3 wt = Tv wt3 tf = u ; f zgwtv (2.9) = !1 3 qf = wq = Tv wq 3 uf zgwtv : (2.10)
  4. 4. 382 EDGAR L ANDREAS ET AL. Because u and, thus, t and q lose their significance gradually as the ASL approaches free convection, we can recast Equations (2.8)–(2.10) for large as uf = k 1=3 ( )1=3; u (2.11) tf = qf = k1=3( ) 1=3 : t q (2.12) The three scales uf , tf , and qf , with (2.11) and (2.12), lead immediately to the well-known predictions for the asymptotic behaviours of 33 , tt , and qq in free convection (e.g., Wyngaard, 1973; Sorbjan, 1989, p. 71 ff.), 33 ( ) = A3u ( )1=3; (2.13) tt ( ) = Atu( ) 1=3 ; (2.14) qq ( ) = Aqu( ) 1=3; (2.15) where A3u , Atu , and Aqu are constants. In very stable conditions, the turbulent eddies are small and often never interact with the surface. Consequently, the height of the observation, z , has no significance. This is z -less stratification (Wyngaard, 1973; Dias et al., 1995). MOST suggests that the asymptotic behaviours of the functions in the limit of very stable stratification (i.e., large ) are 33 ( ) = A3s; (2.16) tt ( ) = Ats; (2.17) qq ( ) = Aqs; (2.18) where A3s , Ats , and Aqs are constants. We also study the temperature skewness, t3 St 3 ; (2.19) t where t3 is the third moment of temperature. Businger (1973) stated that it is not possible to use MOST to predict the asymptotic behaviour of St , as has been done for 33 , tt , and qq . He now, however, agrees that the following MOST arguments are accurate (J. A. Businger, 1995, personal communication). We can rewrite St as t3 tf 3 jtj 3 : St = t3 t f t (2.20)
  5. 5. STATISTICS OF SURFACE-LAYER TURBULENCE 383 In the asymptotic limit of free convection (i.e., large ), t3 = Bu; t3 f (2.21) a constant. Thus, substituting (2.12), (2.14), and (2.21) in (2.20) yields St = Bu[k( ) 1 ][Atu3 ( )] = kBu=A3 ; tu (2.22) a constant. That is, in the free-convection limit, MOST predicts that St becomes a constant. In very stable conditions (i.e., for large ), on the other hand, MOST suggests t3 = B ; t3 s (2.23) yet another constant. But we can also rewrite the skewness as St = t3 j j : t3 t 3 (2.24) t Thus, with (2.17) and (2.23), (2.24) becomes St = Bs=A3 ; ts (2.25) which is again a constant. That is, in z -less stratification also, St approaches a constant. Monin and Yaglom (1971, p. 462) also give these two asymptotic predictions for temperature skewness, but without proof. The same arguments also hold for humidity skewness. 3. The Sevilleta The Sevilleta National Wildlife Refuge is in the rift valley of the Rio Grande River; our research site was in the area known as McKenzie Flats (34 210 5.1500 N, 106 410 9.4700 W). Although mountains rise 1000 m above the valley floor to the east and to the far west, McKenzie Flats itself is a large (100 km2 ) grassland area that is relatively level and fairly homogeneous at kilometre scales. Otto et al. (1995) give other details of the site, including maps. On a scale of tens of centimetres to several metres, however, the Sevilleta’s vegetation is patchy (see Figure 1). Vegetation transect studies are done annually in this area. These consist of 400-m-line-intercept measurements at 1 cm resolution that yield percentages covered by bare ground, litter, and various plant species.
  6. 6. 384 EDGAR L ANDREAS ET AL. Figure 1. Patchy vegetation characterizes the Sevilleta. The patchiness, however, is visible only in the foreground here because the vegetation occludes the line-of-sight in the background. In the insert, the scale is in centimetres.
  7. 7. STATISTICS OF SURFACE-LAYER TURBULENCE 385 In the late summer of 1991, vegetation covered 37.7% of the area; bare ground, 33.3%; and litter, 29.0%. The dominant plant species (see Figure 1) were black grama (Bouteloua eripoda), with 18.0% coverage, and blue grama (Bouteloua gracilis), with 8.8% coverage. These values also correspond closely to the leaf area index because of the low stature of these plants. During our experiment, plant pedestals, typically, were 5 cm above the surface, leaves reached 30 cm, and seed stalks reached 60 cm. The spacing between clumps of plants was on the order 20–30 cm. Turner et al. (1991) and Gosz (1993, 1995) give additional details of the Sevilleta’s vegetation. We collected the data described here on 4–16 August 1991. August is the rainy season in New Mexico. The Sevilleta had, at least, light rain on 10 of the 13 days of our experiment. Some storms yielded heavy rain, with one log entry showing 1 cm of rain in an hour. Dew was common in the morning. As a consequence, the turbulent sensible and latent heat fluxes were, generally, of comparable magnitude. 4. The Data 4.1. DATA COLLECTION AND PROCESSING A three-axis ‘Kaimal’ sonic anemometer/thermometer made by Applied Technolo- gies, Inc. (ATI; Boulder, Colorado) was our primary turbulence instrument. This was positioned at the top of a thin beam with the center of the w-anemometer path 4 m above the ground. A sampling height of 4 m is sufficient to allay concern over the loss of high-frequncy response because of the path-averaging in this instrument (Kaimal and Finnigan, 1994, p. 219). The ATI sonic provides digital values of the three velocity components and temperature 10 times per second. We logged these data through the communications port of a personal computer. Each three-axis sonic run started on the hour and continued for 40.96 minutes. We made 69 such runs. After the experiment, we computed the turbulence statistics for each sonic run. This analysis included routines to remove spikes and detrend the time series. We also rotated the statistics into a coordinate frame in which the mean transverse and vertical velocity components and the mean transverse turbulent stresses were all zero. The statistics thus computed included run averages of uw , wt, u , t , L, w , u, t , and t3 , where u is the standard deviation in longitudinal velocity. Fifty metres southeast of the three-axis sonic tower was our so-called eddy- correlation tower that held turbulence instruments made by Campbell Scientific, Inc. (Logan, Utah). Here, again 4 m above the ground, were a vertically oriented single-axis sonic anemometer, a 76-m chromel-constantan thermocouple, and a krypton hygrometer. A Campbell data logger sampled these instruments at 10 Hz and automatically computed averages from the top of the hour to 40 minutes after the hour; these eddy-correlation statistics, thus, coincided with those from the
  8. 8. 386 EDGAR L ANDREAS ET AL. three-axis sonic. Statistics from these instruments included w , t , q , wt, wq , and tq. We have 118 runs with these data. Using u from the three-axis sonic, we could also compute t and q from these data. There are 30 of these coincident runs. Midway between the sonic and eddy-correlation towers was a Campbell Bowen ratio station. This measured the Bowen ratio, defined as Bo = Lpwt = Lptq ; c wq c (4.1) v v by measuring temperature and dew point at two heights. That is, the Bowen ratio station approximated the Bowen ratio as Bo = L p(Q2 T1 )) : c (T v 2 Q1 (4.2) In (4.1) and (4.2), cp is the specific heat of air at constant pressure, and Lv is the latent heat of vaporization of water. On this Bowen-ratio tower, matched 76-m- diameter chromel-constantan thermocouples placed at heights of 0.87 and 2.83 m measured the vertical temperature gradient. At the same two heights, air intakes led to a single cooled-mirror dew-point hygrometer. At 2-minute intervals, a pump and valve alternately sent air from one intake and then the other to the hygrometer. The dew points at the two heights were thus measured. A second Campbell data logger, again synchronized to sample for 40 minutes starting on the hour, collected these temperature and dew-point data, calculated the vertical temperature (T2 T1) and specific humidity (Q2 Q1 ) differences, and output 40-minute averages of the Bowen ratio estimated according to (4.2). Although the Bowen ratio station yielded other data, which we mention briefly later, here we primarily use these Bowen ratios (in the Appendix). Schotanus et al. (1983) and Kaimal and Gaynor (1991) explain that the tem- perature measured by a sonic thermometer is not a true temperature; it contains humidity information also. If ts is the instantaneous temperature measured by a e sonic thermometer, tes = t(1 + 0:51q); ~ ~ (4.3) where t and q are the instantaneous temperature and specific humidity in the sonic ~ ~ path. Notice, (4.3) is not very different from the virtual temperature, (2.6). Kaimal and Gaynor demonstrate that the sonic temperature can, therefore, be used directly for computing the Obukhov length [see (2.5)] without humidity corrections. Since we also use the sonic temperature to compute t and t3 , we worry that these might be biased by humidity. In the Appendix, however, we demonstrate that, for the range of Bowen ratios we encountered, the statistics of interest here, t=jt j and t3=t3 , can be computed directly from the sonic temperature without corrections for humidity.
  9. 9. STATISTICS OF SURFACE-LAYER TURBULENCE 387 4.2. QUALITY CONTROL The three-axis ATI sonic anemometer/thermometer and the fast-responding tem- perature and humidity sensors on the eddy-correlation tower were fixed in place to accept southerly winds coming up the Rio Grande Valley. Of course, the winds were not always head-on to the ATI sonic. But because of the thoughtful design of this instrument, it can measure the wind vector accurately even in flows that are not head-on. Kaimal et al. (1990) show that this anemometer has no directional bias for winds 45 from head-on. We, however, interpret their data as suggesting that there is no bias for winds to almost 90 from head-on. We graded our sonic runs on the basis of mean wind direction and the variability in the direction. We label runs with an average wind direction within 90 of head- on to the sonic and with small directional variability our ‘best’ data, and those with a mean wind direction within 90 of head-on but with higher directional variability ‘questionable’ data. We reject runs with winds coming predominantly from the backside of the sonic or with high directional variability. Figure 2 shows wind- direction histograms for typical runs that we identified as ‘best’, ‘questionable’, and ‘rejected’. Of the 69 original runs, we judged 31 as ‘best’ and 11 as ‘questionable’ and rejected 27 for use in our analysis. As will be seen later, most of the questionable runs occurred in lighter winds and were therefore associated with moderately unstable stratification. 5. Turbulence Statistics 5.1. DISPLACEMENT HEIGHT Over surfaces with vegetation, it is often necessary to account for the displacement height d in the scaling. Then in Equations (2.1)–(2.4), for example, the correct height scale would not be the height above ground z but rather z d (e.g., Lloyd et al., 1991; Roth, 1993; Roth and Oke, 1993). Typically, d is 60–70% of the height h of the vegetation if the vegetation is dense (Stanhill, 1969; Monteith, 1980; Wieringa, 1993). With the patchy vegetation of the Sevilleta, however, we suspect that d will be a smaller percentage of h (Wieringa, 1993). Since d is interpreted as the average height within a canopy where the momentum is absorbed (Raupach, 1992; Wieringa, 1993), h at the Sevilleta would be the height below which the plants are most dense. That is, h should be roughly what we earlier called the leaf height, 30 cm. Hence, d may have been as large as 20 cm during our experiment but likely was smaller. Since d is defined in the context of momentum exchange, we can investigate its importance to us by considering the wind speed profile u ln z d U (z) = k z d z0 m L (5.1)
  10. 10. 388 EDGAR L ANDREAS ET AL. Figure 2. Three-axis sonic wind direction histograms for typical runs that we judged to yield the best data, questionable data, and data that we rejected. 0 is head-on to the sonic.
  11. 11. STATISTICS OF SURFACE-LAYER TURBULENCE 389 or, more specifically, the drag coefficient at neutral stability, evaluated for a refer- ence height of 10 m (e.g., Andreas and Murphy, 1986), CDN 10 = k2 : (5.2) [kCDz=2 1 ln[(z d)=10] + m [(z d)=L]]2 In (5.1), z0 is the roughness length, which is monotonically related to CDN 10 by CDN 10 = [ln[(10 k d)=z ]]2 ; 2 (5.3) 0 where d and z0 must both be in metres. In (5.1) and (5.2), m is a stability correction. For unstable stratification, we used the Businger-Dyer formulation for m (Andreas and Murphy, 1986); for stable stratification, we used m [(z d)=L] = 5[(z d)=L]: (5.4) Also in (5.2), CDz = [u =U (z)]2 : (5.5) All the quantities in (5.2) necessary to compute CDN 10 – namely, U (z ), u , and L – come directly from the three-axis sonic anemometer/thermometer. We thus computed CDN 10 for two possible displacement heights: the maximum likely value, 20 cm, and the minimum, 0 cm. Figure 3 shows CDN 10 plotted as a function of stability with d = 0 cm and only for three-axis sonic runs for which j j 0:2. Confining our analysis to this stability range minimizes the importance of the stability correction necessary in (5.2). In Figure 3, the mean value of CDN 10 is 5.36 10 3 , and the standard deviation of the mean is 0.08 10 3. From (5.3), the corresponding value of the roughness length, z0 , is 4.2 cm. We also computed CDN 10 values using d = 20 cm for the same runs depicted in Figure 3. The mean of these CDN 10 values is 5.30 10 3 , and the standard deviation of this mean is again 0.08 10 3 . Consequently, on the basis of a Student’s t-test, we can reject the hypothesis that the means of the two CDN 10 distributions (one with d = 0 cm, and one with d = 20 cm) are the same only at the 37% significance level. In other words, the means are not statistically different, and we can henceforth exclude any concern for the displacement height in our analysis. Evidently, the displacement height is less than 20 cm, as we supposed. 5.2. VARIANCE AND COVARIANCE STATISTICS Figure 4 shows w =u plotted versus . The stability range that these data cover is fairly wide, 4 1. The solid line in the figure is:
  12. 12. 390 EDGAR L ANDREAS ET AL. j j Figure 3. Neutral-stability, 10-m drag coefficients for three-axis sonic runs for which z=L 0:2. Here the displacement height d is taken as 0 cm. Figure 4. Nondimensional standard deviation in vertical velocity as a function of stability. The vertical velocity data came from the three-axis ATI sonic anemometer/thermometer and from the single-axis Campbell sonic anemometer. For both data sets, u and L came from the ATI sonic. The line represents Equation (5.6).
  13. 13. STATISTICS OF SURFACE-LAYER TURBULENCE 391 for 4 0:1, w =u = 1:20(0:70 3:0 )1=3; (5.6a) for 0:1 0, w =u = 1:20; (5.6b) for 0 1, u=u = 1:20(1 + 0:2 ): (5.6c) The formulation on the unstable side [(5.6a) and (5.6b)] is our own since w =u seems to be constant for near-neutral stability. Andreas and Paulson (1979) and H¨ gstr¨ m (1990) also report that w =u is independent of for 0:1 0. o o Kader and Yaglom (1990) justify the existence of this constant region theoretically and call it the dynamic sublayer. In the free-convection limit, (5.6a) becomes w =u = 1:73( )1=3 , as (2.13) predicts. The multiplicative constant here is within the range of previously reported values (e.g., Panofsky and Dutton, 1984, p. 161; Sorbjan, 1989, p. 75; Hedde and Durand, 1994). On the stable side of Figure 4, (5.6c) has Kaimal and Finnigan’s (1994, p. 16) stability dependence with a slightly smaller multiplicative constant: 1.20 instead of 1.25. Our value of w =u at neutral stability, 1.20, is within the range of previously reported values (e.g., Panofsky and Dutton, 1984, p. 160 ff.; Hedde and Durand, 1994). Andreas and Paulson (1979) and H¨ gstr¨ m (1990), however, suggest that o o the value of w =u at neutral stability may vary; it depends on the measurement height and probably other variables. H¨ gstr¨ m proposes that the relation o o w =u j0 = 0:12 ln(zf=u ) + 1:99 (5.7) predicts w =u at neutral stability, where f is the absolute magnitude of the Coriolis parameter. For the Sevilleta data (with f based on 34 north latitude, and u 0:4 m s 1 for our near-neutral runs), (5.7) predicts w =u 1:14 at neutral stability, in fair agreement with our fitted result, 1.20. Compared to reports of w =u , the literature contains relatively few plots of u=u . This may be because u=u is commonly presumed not to obey MOST (e.g., Panofsky, 1973; Panofsky and Dutton, 1984, p. 165; Sorbjan, 1989, p. 77) because large-scale motions, which do not scale with z , influence u . Nevertheless, Bradley and Antonia (1979), Kader and Yaglom (1990), and Hedde and Durand (1994), among others, treat u =u as a MOST statistic. In fact, Kader and Yaglom disparage the u results of Panofsky et al. (1977) – which are the primary evidence for the u dependence on the inversion height, zi, in unstable stratification – stating that their ‘conclusions : : : do not seem to be very reliable’ because their measurements
  14. 14. 392 EDGAR L ANDREAS ET AL. Figure 5. Nondimensional standard deviation in longitudinal velocity as a function of stability. All these data came from the three-axis ATI sonic anemometer/thermometer. The line represents Equation (5.8). were ‘at relatively large heights’. Kader and Yaglom, thus, conclude that ‘there have been no reliable measurements of’ u for z=L 0:1 in the atmospheric surface layer. In light of this perceived deficiency, we present Figure 5 with our u =u data plotted versus z=L. These are true atmospheric surface-layer statistics because, our measurement height was 4 m. The line in the figure is: for 4 0:1, u=u = 5:49( )1=3; (5.8a) for 0:1 0, u=u = 2:55; (5.8b) for 0 1, u=u = 2:55(1 + 0:8 ): (5.8c) MOST does seem useful in organizing our u data. On the unstable side of Figure 5, u =u is constant for 0:1 0, as is w =u in Figure 4. This stability region corresponds to what Kader and Yaglom (1990) call the dynamic sublayer, where MOST predicts that both w =u and u =u should be constant. As increases, u=u becomes proportional to ( )1=3, as MOST predicts following
  15. 15. STATISTICS OF SURFACE-LAYER TURBULENCE 393 Figure 6. Nondimensional standard deviations in temperature as a function of stability. The data derive from the three-axis ATI sonic anemometer/thermometer or from the Campbell eddy-correlation instruments, as noted. For all data, u (necessary for computing t ) and L came from the ATI sonic. The line represents Equation (5.9) with C = 3.2. the same arguments that led to (2.13). In near-neutral stability, u =u = 2:55, a value that agrees very well with most other observations of this quantity in neutral stratification (e.g., Ariel and Nadezhina, 1976; Stull, 1988, p. 366; Sorbjan, 1989, p. 69 ff.; Kader and Yaglom, 1990). In conclusion, for z=zi 1, MOST seems to be a useful context for organizing u data. Figures 6 and 7 show, respectively, the nondimensional standard deviations for temperature and humidity. We fitted the data in both of these figures with lines of the same form: for 4 0, s=js j = C (1 28:4 ) 1=3; (5.9a) for 0 1, s=js j = C ; (5.9b) where s is the scalar standard deviation and s is the corresponding flux scale. For temperature (Figure 6), C = 3.2; for humidity (Figure 7), C = 4.1. The temperature data from the eddy-correlation tower plotted in Figure 6 (the squares) are more scattered than the data from the three-axis sonic (the circles) and, on the unstable side, even appear to be biased somewhat high. On the stable side of Figure 6, there is no obvious bias. This scatter is not unexpected because of the way we had to compute t =jt j. For the three-axis sonic points in Figure 6,
  16. 16. 394 EDGAR L ANDREAS ET AL. Figure 7. Nondimensional standard deviations in specific humidity as a function of stability. All these data came from the Campbell eddy-correlation instruments, but u (necessary for computing q ) and L came from the ATI sonic. The line represents Equation (5.9) with C = 4.1. u, t , and wt (which yielded t = wt=u ) all came from the same instrument. Thus random scatter in t was likely mitigated by coincident scatter in wt. For the eddy-correlation points in Figure 6, however, only t and wt came from the eddy-correlation instruments; u (which yielded t = wt=u ) and z=L still came from the three-axis sonic. Thus, the fact that another instrument, 50 m away, was necessary for deriving the eddy-correlation t =jt j values makes the scatter reasonable. With the exception of the ( ) 1=3 behaviour in the free-convection limit, there is little consensus as to the form of (5.9a). Our function has the stability dependence recommended by De Bruin et al. (1993). There is even less guidance as to the behaviour of tt ( ) and qq ( ) on the stable side of Figures 6 and 7 (e.g., Panofsky and Dutton, 1984, p. 169 ff.; Sorbjan, 1989, p. 75). Our data, though, definitely do not follow Kaimal and Finnigan’s (1994, p. 16) suggestion that tt decreases with increasing . On the basis of (2.17) and (2.18), we interpret Figures 6 and 7 as suggesting that tt ( ) and qq ( ) are independent of throughout the stable region [see (5.9b)]. Weaver (1990) reaches the same conclusion. In the atmospheric surface layer above horizontally homogeneous surfaces, tem- perature and humidity statistics are often presumed to differ little (e.g., Brutsaert, 1982, p. 67 ff; Panofsky and Dutton, 1984, p. 170 ff.). Ohtaki (1985) and Hedde and Durand (1994) confirm that over homogeneous surfaces, such as dense vegetation or the ocean, this is true. Likewise, on reviewing many data sets collected over fairly
  17. 17. STATISTICS OF SURFACE-LAYER TURBULENCE 395 homogeneous surfaces, Ariel and Nadezhina (1976) conclude that temperature and humidity statistics ‘have similar characteristics’. But for more complex surfaces, Smedman-H¨ gstr¨ m (1973) and Beljaars et al. (1983) find that nondimensional o o temperature and humidity standard deviations have different values at the same stability, as we have found. Katul et al. (1995) suggest that, even over homogeneous surfaces, temperature and humidity statistics could differ because temperature is an active scalar conta- minant while moisture is generally not. Such an argument logically implies then that t =jt j and q =jq j would be relatively alike in near-neutral stability but would diverge as j j increases and temperature assumes a more active role in the dynam- ics. That is, the shapes of plots of t =jt j and q =jq j versus would be different. In Figures 6 and 7, however, our t =jt j and q =jq j data exhibit the same shape – the same stability dependence – for 4 1. This hypothesis that temperature is an active scalar also implies that the propor- tionality of t =jt j to ( ) 1=3 should break down with increasing atmospheric instability, since this prediction relies strictly on MOST and, thus, takes no account of temperature’s active role in flow dynamics. But, as we mentioned, the propor- tionality of t =jt j to ( ) 1=3 is the most robust feature of this statistic. Therefore, if temperature does behave as an active scalar, evidence of this does not have any obvious manifestation in its variance statistics. As a result, temperature’s presumed active role in the dynamics does not seem to explain the difference in the t =jt j and q =jq j levels in Figures 6 and 7. De Bruin et al. (1993) recommend that C = 2.9 for tt in (5.9a). This is not much different from our value of 3.2. Notice, with this constant, that (5.9) implies that t =jt j approaches 3.2 at neutral stability (see Figure 6). Most investigations, however, find values of t =jt j between 2 and 3 at neutral stability (e.g., Tillman, 1972; Ohtaki, 1985; H¨ gstr¨ m, 1990; Kader and Yaglom, 1990; Kaimal and Finni- o o gan, 1994, p. 16), but few had as many data for 0:1 0:1 as we have. Beljaars et al. (1983) and Wang and Mitsuta (1991) do report that t =jt j is 3.5 and 3.0, respectively, at neutral stability. There are not as many observations of q =jq j in the literature. With C = 4.1 in (5.9), we suggest that q =jq j = 4:1 at neutral stability. Ohtaki (1985), on the other hand, suggests that t =jt j = q =jq j = 2:5 at neutral stability. Beljaars et al. (1983) likewise suggest that q =jq j = 2:5 at neutral stability; but unlike our and Ohtaki’s results, this value is much less than their value for t =jt j at neutral stability, 3.5. Hedde and Durand (1994) report that t =jt j = q =jq j in the free- convection region but do not have enough data near neutral stability to infer values here. Thus, the humidity data lead to no consensus. The fact that both nondimensional temperature and humidity standard deviations have the same dependence on stability in (5.9) and that this dependence has been reported elsewhere (i.e., De Bruin et al., 1993) supports MOST. The fact that, at the Sevilleta, the magnitudes of these two statistics are different is contrary to MOST. Another test of MOST over the Sevilleta, where we expect that during daytime the
  18. 18. 396 EDGAR L ANDREAS ET AL. Figure 8. Nondimensional temperature-humidity covariance as a function of stability. All the tq data came from the Campbell eddy-correlation instruments. The u (necessary for computing t and q ) and L values came from the ATI sonic. The line represents Equation (5.10). sources of heat and moisture are different, is to look at the temperature-humidity covariance. Figure 8 shows tq=t q as a function of stability. For each data point in Figure 8, we needed measurements of wt, wq , and tq from the eddy-correlation tower and simultaneous measurements of u and L from the three-axis sonic. As we mentioned earlier, there are 30 of these coincident runs. The line in Figure 8 is: for 4 0, tq=tq = 10(1 28:4 ) 2=3 ; (5.10a) for 0 1, tq=tq = 10: (5.10b) Using (2.3), (2.4), and (5.9), we can also write this as tq tq t q tq = t q jtj jqj = 0:76tt ( )qq ( ); (5.11a) 2 ( ): tt (5.11b) This implies that the t q correlation coefficient, tq=t q , has a typical magnitude of 0.76.
  19. 19. STATISTICS OF SURFACE-LAYER TURBULENCE 397 By writing (5.11b) we do not mean to suggest any fundamental relation- ship. This near-equality is probably just coincidence since tt ( ) 6= qq ( ) and tq=tq 6= 2 ( ). qq Equation (5.11a) is a generalization of MOST as applied to tq=t q above hetero- geneous surfaces. If a flow truly obeys MOST, then tt = qq and jtq=t q j = 1 (e.g., Hill, 1989). Consequently, instead of (5.11a), we would have tq=t q = tt ( )qq ( ). But Hill (1989) emphasizes that scalar-scalar correlations, such as tq, are sensitive indicators of deviations from MOST. Consequently, Figure 8 and our fitting its data with (5.11a) confirms that some feature of the Sevilleta violates the conditions on which MOST relies. Among the 30 points available for plotting in Figure 8 are some that fell far from (5.10). On scrutinizing these points, we found that all came from the sunrise and sunset transitions – periods when the steady-state assumption on which MOST relies is invalid. Therefore, to further investigate MOST as it applies to tq , we constructed Figure 9. Here we plot the t q correlation as a function of local time. Remember that each run started on the hour and lasted 40 minutes. Thus, the times plotted in Figure 9 are the starting times for the runs. These data show a clear diurnal cycle. The t q correlation is high and positive from mid-morning until late afternoon; during the night the correlation is negative. Consequently, there are two transitions during which tq crosses zero – one around sunrise and the other around sunset. The wild outliers in Figure 8 came from these transition periods; the well-behaved points in Figure 8 came from daytime measurements for negative and from nighttime measurements for positive. Typical magnitudes of the t q correlation for these well-behaved points are as we suggest in (5.11a), 0.76. The fact that jtq=t q j is never 1 in Figure 9 confirms that the Sevilleta data vio- late MOST. Priestley and Hill (1985) speculate that jtq=t q j may not be perfectly 1 as a consequence of entrainment from heights where the gradients of potential temperature and specific humidity differ from their near-surface averages. De Bruin et al. (1993) offer a similar explanation for imperfect t q correlation but with an added constraint. For their data, the surface sensible heat flux was large; conse- quently, large boundary-layer eddies – for example, through “top-down” diffusion (Wyngaard and Brost, 1984) – had a small effect on near-surface temperature fluc- tuations. On the other hand, their surface latent heat flux was small, so the large eddies affected the humidity fluctuations much more than the temperature fluctu- ations. For such conditions, t and q would be poorly correlated, and q would be only weakly related to q . Figure 7, however, shows that, for our Sevilleta data, q and q are closely related. Figure 10 provides further insight into the hypothesis by De Bruin et al. (1993). Here we plot tq=t q versus the Bowen ratio, Bo, where Bo came from (4.1) with wt and wq values measured on the eddy-correlation tower. In making this plot, we excluded t q and Bowen ratio pairs collected during nonstationary periods as indicated by high variability in simultaneous scintillometer data (Otto et al., 1995).
  20. 20. 398 EDGAR L ANDREAS ET AL. Figure 9. The t q correlation as a function of local time. All of these data came from the Campbell eddy-correlation instruments. The 2s indicate that there are two data points with the same coordinates. The stability values (i.e., z=L) in Figure 10 came from wind speed and heat flux data available from the Campbell Bowen ratio station (Otto et al., 1996; Hill et al., 1997). The point Figure 10 makes is that, generally, neither the sensible nor the latent heat flux was negligible in comparison to the other at the Sevilleta: The magnitude of Bo is typically 1, and, during the day at least, both sensible and latent heat fluxes were usually 100–200 W m 2 . Thus, the points in the upper right quadrant in Figure 10 are daytime values, when both sensible and latent heat fluxes were positive. The points in the lower left quadrant are nighttime values, when the latent heat flux was still usually positive, but the sensible heat flux was negative. Our conclusion on seeing Figure 10 is that the explanation for jtq=t q j values less than 1 suggested by De Bruin et al. (1993) is not applicable for the Sevilleta, at least during daytime. The latent heat flux was not so small that large boundary- layer eddies could have dominated the variability in surface-level q values. We thus reiterate our hypothesis that the metre-scale surface heterogeneity leads to distributed heat and moisture sources that cannot produce temperature and humidity fluctuations with perfect correlation or anticorrelation. In summary, turbulence in the surface layer over the Sevilleta violates MOST, but not violently. The behaviour of tt ( ) is consistent with other reports in the literature. That the stability dependence in qq ( ) and tq ( ) is like that for tt ( ) is another result consistent with MOST. But the fact that qq does not have the same value as tt for all values is a breakdown in MOST. Because qq is sig- nificantly larger than tt though the magnitude of the Bowen ratio is typically 1,
  21. 21. STATISTICS OF SURFACE-LAYER TURBULENCE 399 Figure 10. The t q correlation versus the Bowen ratio. The data points came from the Campbell eddy-correlation instruments; we used data from the Campbell Bowen ratio station to assign z=L values (Otto et al., 1996; Hill et al., 1997). The data have also been screened to exclude nonstationary periods as judged by high variability in simultaneous scintillometer measurements (Otto et al., 1995). this breakdown seems to result because the surface moisture sources are hetero- geneously distributed while the temperature sources are more homogeneous. The evidence in (5.11a) and Figures 9 and 10 that temperature and humidity do not have perfect positive or negative correlation also argues against MOST and corrob- orates our conclusion that its breakdown results because the surface temperature and moisture sources are not similarly distributed. 5.3. TEMPERATURE SKEWNESS Businger (1973) suggests that one possible use of the temperature skewness is for estimating u and wt. Figure 11 shows our temperature skewness data as a function of stability. Because the data came from only the ATI sonic, there are fewer points than in some of the other plots; but the stability range covered is still 4 1. The solid line in Figure 11 is our least-squares fit to the data:
  22. 22. 400 EDGAR L ANDREAS ET AL. Figure 11. Temperature skewness versus stability. All these data came from the ATI sonic thermome- ter. The solid line represents Equation (5.12); the dashed line is from Tillman (1972); the dotted line shows the asymptotic free-convection limit, relation (5.13). for 4 0:01, St = 0:255 ln( ) + 1:044; (5.12a) for 0.01 1, St = 0:15: (5.12b) The only similar relations between skewness data and stability that we know of are those by Tillman (1972; or see Businger, 1973) and Antonia et al. (1981). Ohtaki (1985) also shows plots of temperature, humidity, and carbon dioxide skewness; but his absolute values are smaller than ours and those in the Tillman and Antonia sets and too scattered to fit with stability relations. Wyngaard and Sundararajan (1979) also plot temperature skewness data but do not fit a line. The dashed line in Figure 11, Tillman’s result, is essentially identical to our (5.12a). Antonia et al. likewise compare Tillman’s fit with two data sets they had and find negligible differences. Sreenivasan et al. (1978) suggest that measuring temperature skewness requires very long averaging times. Their suggested relation between the averaging time for skewness, Tsk , and the mean-square error in the measured skewness, 2 , is Tsk U=z = 180=2, where U is the mean wind speed at height z. Since our averaging time was 40.96 minutes and since U=z was typically 1 s 1 for our data, this relation implies that our skewness data should typically have a root-mean-square
  23. 23. STATISTICS OF SURFACE-LAYER TURBULENCE 401 error of about 27%. The scatter in the data in Figure 11 is roughly in line with this assessment; and these data are more scattered than those in Figures 4–7, the variance statistics, which require much shorter averaging times to yield comparable accuracy (Sreenivasan et al., 1978). Nevertheless, the similarity between our skewness data and the fits reported by Tillman (1972) and Antonia et al. (1981) suggests that we have enough data to capture the general trend in skewness with stability and also corroborates our conclusion in the Appendix that, for the Sevilleta data, the sonic-temperature skewness is essentially equal to the true-temperature skewness. Lastly, the similarity of these three data sets confirms our conclusion in the last section that, over the Sevilleta, temperature statistics obey MOST. Earlier we showed that, in very unstable and very stable stratification, the temperature skewness should approach constants. In Figure 11, St appears to be constant throughout the stable region [see (5.11b)]. On the unstable side of Figure 11, St likewise seems to be constant for large . The dotted line in the figure is St = 0:82 (5.13) for 0:2. Tillman’s (1972) data do not show this asymptotic limit in free convection, but he has only six data points for 0:2 and only one point with 0:7. Surprisingly, Antonia et al. (1981) do not find this asymptotic limit in their temperature skewness data either, although they have roughly 40 data points for which 0:2. On the other hand, the 13 temperature skewness values for which 0:8 that Wyngaard and Sundararajan (1979) plot tend to a constant value of approximately 1, in good agreement with (5.13). Ohtaki’s (1985) temperature, humidity, and carbon dioxide skewness data also all seem to be nearly constant for 0:2; but the absolute values of his constants are about half as large as ours, 0.82, and the value implied by Wyngaard and Sundararajan’s data. Clearly, there is still work to do on the similarity behaviour of scalar skewness. 6. Conclusions The Sevilleta’s metre-scale heterogeneity provides an interesting surface over which to evaluate turbulence statistics. Despite the heterogeneity, some turbulence statistics appear to obey Monin-Obukhov similarity theory, while others deviate mildly. For example, w =u , t =jt j, and t3 =t , in general, depend on stability for 3 4 1 in ways that have been reported before and, thus, follow MOST. If anything, w =u is the most deviant of this group. We find that w =u is constant for 0:1 0, as Kader and Yaglom (1990) recommend, while most others suggest that, for unstable stratification, w =u increases monotonically with . Statistics involving humidity, on the other hand, do not truly follow MOST; q =jq j has the same stability dependence as t =jt j but is 28% larger. Likewise,
  24. 24. 402 EDGAR L ANDREAS ET AL. tq=jt qj has the same stability dependence as t2=t2 but is 24% smaller than tq =t q. We believe these inequalities result because, at the Sevilleta, the surface sources of heat and moisture are not the same. The surface heat sources seem to be homogeneously distributed – at least, in part, because of the uniformity of the radiative forcing (Katul et al., 1995) – and, thus, lead to temperature statistics similar to those found over more ideal surfaces. The surface moisture sources, in contrast, seem to be heterogeneously distributed, as is the vegetation. Thus, the turbulent humidity fluctuations (parameterized as q ) are relatively large when compared to the total moisture flux. In other words, the heterogeneity fosters unusually large q =jq j values. Yet another way of saying this is, because of the heterogeneity, the t q correlation is not exactly +1 or 1. During midday and late night, when temperature and humidity have high positive or negative correlation over homogeneous surfaces, we find the magnitude of the t q correlation over the Sevilleta to be, typically, 0.76. As a consequence, the Sevilleta does not support MOST when the statistic of interest involves humidity. A corollary is that other photosynthetic gases, such as carbon dioxide, will not obey MOST over such a surface either. Acknowledgements We thank Greg Shore and Yorgos Marinakis (both at the University of New Mexico) and Jim Wilson (from the Environmental Technology Laboratory) for help with the data collection. Rod Frehlich, Markus Furger, Ken Gage, and two anonymous reviewers offered helpful comments on the manuscript. The U.S. Department of the Army supported this work through Project 4A161102AT24; the U.S. National Science Foundation supported it with grant BSR-89-18216. This is contribution 110 to the Sevilleta Long-Term Ecological Research Program. Appendix: Temperature Statistics from a Sonic Thermometer The instantaneous temperature measured by a sonic thermometer (ts ) is related to e the actual instantaneous temperature (t) and specific humidity (q ) by (Schotanus et ~ ~ al., 1983; Kaimal and Gaynor, 1991) tes = t(1 + 0:51q): ~ ~ (A1) By using the Reynolds decompositions, tes = Ts + ts; (A2a) t = T + t; ~ (A2b) q = Q + q; ~ (A2c)
  25. 25. STATISTICS OF SURFACE-LAYER TURBULENCE 403 where upper-case letters denote averages and unhatted lower-case letters denote turbulent fluctuations, we can derive (to first order) Ts = T (1 + 0:51Q); (A3) ts = t(1 + 0:51Q) + 0:51Tq: (A4) The turbulent vertical sonic-temperature flux is then wts u ts = wt(1 + 0:51Q) + 0:51Twq; (A5) where w is the turbulent vertical velocity fluctuation. From (4.1), we realize that we can write this in terms of the Bowen ratio, :51Q) + 0:51T ; uts = wt (1 + 0 DBo (A6) where D Lv =cp is roughly 2500 K. In turn, we can also define the sonic- temperature flux scale, :51Q) + 0:51T : ts = t (1 + 0 DBo (A7) From (A4), the sonic-temperature variance is ts = t2(1 + 0:51Q)2 + 2(0:51T )(1 + 0:51Q)tq + (0:51T )2q : 2 2 (A8) Thus, from (A7) and (A8), the nondimensional temperature variance statistic that we compute from the sonic thermometer is # tq 1 + 2 2 + 2 q 2 ts 2 = t 2 t t # ts t ; (A9) 2 2 1+ DBo + DBo where = 1 +:5151Q : 0 T 0: (A10) The experimental findings that we report in Section 5 are that (t =t )2 = 1:6(q =q )2 and that tq=t q (t =t )2 . Hence, in the numerator of (A9) tq = tq t 2 q = 1 ; t2 tq t t DBo (A11)
  26. 26. 404 EDGAR L ANDREAS ET AL. q 2 = q 2 t 2 q 2 = 1:6 : t q t t (DBo)2 (A12) Thus, (A9) becomes 2 2 3 0:6 ts = t 61 + 2 2 6 DBo 7 : 7 ts t 4 6 1+ 27 5 (A13) DBo During our experiment at the Sevilleta, 150 K, D 2500 K, and the Bowen ratio rarely was measured to have an absolute value less than 0.3. Therefore,
  27. 27. 0:2:
  28. 28. DBo
  29. 29. (A14) Consequently, the bracketed term in (A13) is nearly 1, and the equation reduces to ts 2 = t 2 : ts t (A15) The unadjusted, nondimensional sonic-temperature variance is, to second order, equal to the true nondimensional temperature variance. We likewise consider the skewness measured by the sonic thermometer. The third moment of sonic temperature is, from (A4), ! 3t2 q 32 tq 2 3 q3 : t s t 3 = 3 (1 + 0:51Q) 3 1+ + + (A16) t3 t3 t3 From (A8), (A11), and (A12), we see that 2 1=2 2 3 0:6 6 ts = t(1 + 0:51Q) 1 + DBo 61 + DBo 2 7 ; 6 7 (A17) 4 7 5 1+ DBo where we have already argued that the term to the 1/2 power is virtually 1. Thus, the sonic-temperature skewness, Sts = t3 =ts , is related to the true temperature s 3 skewness, St = t3 =t , by 3 ! 3t2 q 32 tq 2 3 q3 St 1+ + + Sts = t3 t3 t3 : (A18) 3 1+ DBo
  30. 30. STATISTICS OF SURFACE-LAYER TURBULENCE 405 Let us define some additional Monin-Obukhov similarity functions: t3 ttt ( ) t3 ; (A19) ttq ( ) tt2 qq ; 2 (A20) tqq ( ) ttqq2 ; 2 (A21) qqq ( ) q3 : 3 q (A22) With these, the terms in the numerator of (A18) become t2q = t2 q t3 q = ttq 1 ; t 3 t2 q t3 t ttt DBo (A23) tq2 = tq2 t3 q 2 = tqq 1 ; t 3 tq t3 t 2 ttt (DBo)2 (A24) q3 = q3 t3 q 3 = qqq 1 : t 3 q t3 t 3 ttt (DBo)3 (A25) Substituting (A23)–(A25) in (A18) and doing a binomial expansion of the denominator yields # ttq + 3 2 tqq + 3 qqq 3 Sts = St 1 + DBo DBo ttt DBo ttt ttt # 3 2 1 DBo + 6 DBo : (A26) Using the binomial expansion is valid because, at the Sevilleta, j=DBoj 0:2. For the Sevilleta measurements, we expect, jttq j jttt j, jtqq j jttt j, and jttt j jqqq j. Hence, we can safely ignore third-order terms in (A26). On multiplying the right-hand side of (A26) out, we finally get # Sts = St 1 + DBo ttq 3 2 9 ttq + 3 tqq + 6 : ttt 1 + DBo ttt ttt (A27)
  31. 31. 406 EDGAR L ANDREAS ET AL. In unstable conditions, we expect ttt , ttq , tqq , and qqq to all be negative. In stable conditions, they should all be positive. Thus, the ratios of similarity functions in (A27) should always be positive. Consequently, in a surface layer that truly obeys MOST (i.e., the temperature and humidity fluctuations are highly correlated), the bracketed term in (A27) is nearly 1. Over the Sevilleta, where the t q correlation was not perfect, the ratios of third-order similarity functions in (A27) will likely be less than 1. But from our observations of tt , qq , and tq , we expect that these ratios of third-order functions will not be far from 1. Thus, (A27) suggests that, to second order, the sonic-temperature skewness equals the true temperature skewness. References Andreas, E. L. and Murphy, B.: 1986, ‘Bulk Transfer Coefficients for Heat and Momentum over Leads and Polynyas’, J. Phys. Oceanogr. 16, 1875–1883. Andreas, E. L. and Paulson, C. A.: 1979, ‘Velocity Spectra and Cospectra and Integral Statistics over Arctic Leads’, Quart. J. Roy. Meteorol. Soc. 105, 1053–1070. Antonia, R. A., Chambers, A. J., and Bradley, E. F.: 1981, ‘Temperature Structure in the Atmospheric Surface Layer: II. The Budget of Mean Cube Fluctuations’, Boundary-Layer Meteorol. 20, 293– 307. Ariel, N. Z. and Nadezhina, Ye. D.: 1976, ‘Dimensionless Turbulence Characteristics under Various Stratification Conditions’, Izv., Atmos. Oceanic Phys. 12, 492–497. Beljaars, A. C. M. and Holtslag, A. A. M.: 1991, ‘Flux Parameterization over Land Surfaces for Atmospheric Models’, J. Appl. Meteorol. 30, 327–341. Beljaars, A. C. M., Schotanus, P., and Nieuwstadt, F. T. M.: 1983, ‘Surface Layer Similarity under Nonuniform Fetch Conditions’, J. Clim. Appl. Meteorol. 22, 1800–1810. Bradley, E. F. and Antonia, R. A.: 1979, ‘Structure Parameters in the Atmospheric Surface Layer’, Quart. J. Roy. Meteorol. Soc. 105, 695–705. Brutsaert, W. A.: 1982, Evaporation into the Atmosphere: Theory, History, and Applications, D. Reidel, Dordrecht, 299 pp. Businger, J. A.: 1973, ‘Turbulent Transfer in the Atmospheric Surface Layer’, in D. A. Haugen (ed.), Workshop on Micrometeorology, American Meteorological Society, Boston, pp. 67–100. De Bruin, H. A. R., Kohsiek, W., and Van Den Hurk, B. J. J. M.: 1993, ‘A Verification of Some Methods to Determine the Fluxes of Momentum, Sensible Heat, and Water Vapour Using Stan- dard Deviation and Structure Parameter of Scalar Meteorological Quantities’, Boundary-Layer Meteorol. 63, 231–257. Dias, N. L., Brutsaert, W., and Wesely, M. L.: 1995, ‘Z -less Stratification under Stable Conditions’, Boundary-Layer Meteorol. 75, 175–187. Gosz, J. R.: 1993, ‘Ecotone Hierarchies’, Ecol. Appl. 3, 369–376. Gosz, J. R.: 1995, ‘Edges and Natural Resource Management: Future Directions’, Ecol. Int. 22, 17–34. Hedde, T. and Durand, P.: 1994, ‘Turbulence Intensities and Bulk Coefficients in the Surface Layer above the Sea’, Boundary-Layer Meteorol. 71, 415–432. Hess, G. D.: 1992, ‘Observations and Scaling of the Atmospheric Boundary Layer’, Aust. Meteorol. Mag. 41, 79–99. Hill, R. J.: 1989, ‘Implications of Monin-Obukhov Similarity Theory for Scalar Quantities’, J. Atmos. Sci. 46, 2236–2244. Hill, R. J., Otto, W. D., Sarma, A. D., Wilson, J. J., Andreas, E. L., Gosz, J. R., and Moore, D. I.: 1997, ‘An Evaluation of the Scintillation Method for Obtaining Fluxes of Momentum and Heat’, NOAA Tech. Memo. ERL ETL-275, Environmental Technology Laboratory, Boulder, Colo., 55 pp.
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