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Wind and Temperature Profiles in the Radix Layer: The Bottom Fifth of the Convective Boundary Layer. Santoso, Edi; Stull, Roland B. 1998

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VOLUME 37 JUNE 1998JOURNAL OF APPLIED METEOROLOGYq 1998 American Meteorological Society 545Wind and Temperature Profiles in the Radix Layer: The Bottom Fifth of theConvective Boundary LayerEDI SANTOSO AND ROLAND STULLAtmospheric Science Programme, Department of Geography, University of British Columbia, Vancouver, British Columbia, Canada(Manuscript received 8 January 1997, in final form 13 November 1997)ABSTRACTIn the middle of the convective atmospheric boundary layer is often a deep layer of vertically uniform windspeed (MUL), wind direction, and potential temperature (uUL). A radix layer is identified as the whole regionbelow this uniform layer, which includes the classic surface layer as a shallower subdomain. An empirical windspeed (M) equation with an apparently universal shape exponent (A) is shown to cause observations from the1973 Minnesota field experiment to collapse into a single similarity profile, with a correlation coefficient ofroughly 0.99. This relationship is M/MUL5 F(z/zR), where F is the profile function, z is height above ground,and zRis depth of the radix layer. The profile function is F 5 (z/zR)Aexp[A(1 2 z/zR)] in the radix layer (z/zR# 1), and F 5 1 in the uniform layer (zR, z , 0.7zi). The radix-layer equations might be of value for calculationof wind power generation, wind loading on buildings and bridges, and air pollutant transport.The same similarity function F with a different radix-layer depth and shape exponent is shown to describethe potential temperature (u) profile: (u 2 uUL)/(u02 uUL) 5 1 2 F(z/zR), where u0is the potential temperatureof the air near the surface. These profile equations are applicable from 1 m above ground level to the midmixedlayer and include the little-studied region above the surface layer but below the uniform layer. It is recommendedthat similarity profiles be formulated as mean wind or potential temperature versus height, rather than as shearsor gradients versus height because shear expressions disguise errors that are revealed when the shear is integratedto get the speed profile.1. Structure of the convective mixed layerThe goal of this research is to describe the mean windspeed (M) and potential temperature (u) profiles withinthe bottom fifth of the convective boundary layer. Figure1a identifies layers in the convective mixed layer (ML),using wind speed as an example.Starting from the top, the free atmosphere is wheresurface friction is not felt, and the actual wind speed isnearly equal to the geostrophic or gradient wind speedG. This layer is above the boundary layer. Next is theentrainment zone, a region of subadiabatic temperatureprofiles, overshooting thermals, intermittent turbulence,and wind shear (Deardorff et al. 1980).Further down is a region where wind speed and di-rection are nearly uniform with height, z. This is theuniform layer (UL), where the wind speed is MULandthe potential temperature is uUL. The wind is subgeo-strophic because thermals communicate surface drag in-formation via nonlocal transport. Traditionally, the term‘‘mixed layer’’ is reserved for the whole turbulent regionbetween the surface and the average ML top, zi; hence,Corresponding author address: Dr. Roland Stull, Atmospheric Sci-ence Programme, Dept. of Geography, University of British Colum-bia, 1984 West Mall, Vancouver, BC, V6T 1Z2, Canada.E-mail: rstull@geog.ubc.cawe distinguish between the ML and the UL to avoidconfusion.Between the surface and the UL is a radix layer (RxL)of depth zRM. Like the roots of a tree, the radix (Latinfor ‘‘root’’) layer is where smaller plumes merge intolarge-diameter ML thermals. It contains the classic sur-face layer as a subdomain (Fig 1a), as will be explainedbelow. Winds in this layer are zero near the ground, andsmoothly increase to become tangent to MULat the RxLtop.Within the bottom of the RxL is the classic surfacelayer (SL), the nearly constant flux region whereMonin–Obukhov similarity theory applies. The SL isthe region where predominantly mechanically generatedturbulence within the wall shear flow produces a nearlylogarithmic wind profile. Again, we distinguish betweenthe RxL and the SL to avoid confusion because we retainthe traditional definition that the SL is that region whereMonin–Obukhov similarity applies. Although not thesubject of this paper, Zilitinkevich (1994) further sub-divides the traditional surface layer into (from bottomup) a mechanical turbulence layer, an alternative tur-bulence layer, and a free convection layer, where thetop two layers combine to form his convective and me-chanical layer.Radix-layer definitions also can be made for potentialtemperature, as sketched in Fig. 1b. The depth of the546 VOLUME 37JOURNAL OF APPLIED METEOROLOGYFIG. 1. Idealized profiles showing components of the convectivemixed layer for (a) wind M and (b) potential temperature u. Here, Gis geostrophic wind speed.SL and RxL for temperature can be different than formomentum. For potential temperature, we denote thesedepths as zSuand zRu, respectively.Based on these traditional and new definitions, wesee that the ML (order of a couple kilometers thick)contains the RxL (hundreds of meters thick) as a sub-domain, and the RxL contains the SL (tens of metersthick) as a subdomain. Hence, there is a superpositionof layers in the bottom fifth of the convective boundarylayer. This is an alternative view to the three-layer modelof Garratt et al. (1982), where their surface layer mergesdirectly into the uniform layer.Because the proposed profile equations will be ex-pressed as a dimensionless similarity theory, section 2starts by reviewing similarity hypotheses for the bottomof the boundary layer. In section 3 similarity equationsfor the radix layer are proposed and calibrated againstfield data. In section 4 there is a first attempt to identifyparameters that control the radix-layer depth. The valueof similarity wind profiles over similarity shear profilesis discussed in section 5.2. Similarity hypothesesa. Surface layerMost SL similarity theories are based on the followingpremises:1) flux is approximately uniform with height (constantflux 6 10%);2) turbulence consist of ‘‘small eddies,’’ causing localtransport;3) turbulence is predominantly generated mechanicallyby shear flow near the ground, with minor modifi-cations for static stability; and4) feedback exists between the mean flow and the dom-inant eddies.The first premise not only simplifies the theory by al-lowing flux variations to be neglected but it constrainsthe depth of applicability to the bottom 10% of the ML,assuming heat flux decreases roughly linearly withheight during near-free convection. The second premisesuggests that ML depth zishould not be relevant. Thethird premise implies that surface roughness length z0is important. The fourth premise is discussed later.Most classic SL similarity equations are strongly de-pendent on the surface but are virtually independent offactors higher in the ML. Typically lacking is depen-dence on ML depth, temperature within the UL, windswithin the UL, and geostrophic wind speed. For thisreason, we cannot expect the SL equations to mergesmoothly into the UL because no information about theUL is included in those equations. Panofsky (1978)points out that convective-matching-layer and free-con-vection-layer formulations (Priestley 1955; Kaimal etal. 1976) fail near the bottom of the UL, where the shearand potential-temperature gradient approach zero.This situation is illustrated in Fig. 2, where the ab-scissa has been normalized according to the Businger(1971)–Dyer (1974) similarity theory. In this normal-ization, all the data will collapse to a single curve re-gardless of static stability in those regions where SLsimilarity theory is valid. While Fig. 2a shows that SLsimilarity works well in the bottom 40 m of the ML forthe Minnesota dataset (to be described in more detaillater), Fig. 2b shows that SL theory is less successfulhigher in the RxL and in the UL. Namely, the classicJUNE 1998 547SANTOSO AND STULLFIG. 2. Wind profiles for all 11 runs of the Minnesota campaign. Abscissa is normalized using surface-layer similarity, where z0isaerodynamic roughness length, k 5 0.4 is von Ka´rma´n’s constant, and cmis the integrated wind profile stability–correction function ofBusinger and Dyer: (a) within the surface layer and (b) within the radix and uniform layers.surface layer does not extend up to the base of the ULfor these data, resulting in data points that do not col-lapse onto a single curve.Monin–Obukhov similarity theory has been the fa-vored tool for finding wind and temperature profiles inthe SL. Within this theory, dimensionless wind shear asa function of dimensionless height z/L is defined aszkz]Mf [ , (1)m12Lu]z*where L is the Obukhov length (negative for a staticallyunstable BL), k is von Ka´rma´n’s constant (approxi-mately 0.4), and u*is the friction velocity. This ex-pression can be integrated to yield the mean wind pro-file,zuz9*M(z) 5 f dz9, (2)E m12kz9 Lz95z0where z0is the aerodynamic roughness length and z9 isa dummy of integration. The function fmvaries awayfrom unity as the static stability varies from neutral;however, the functional form of fmis not known fromfirst principles.Many empirical estimates of the functional form offmhave been suggested for the statically unstable sur-face layer. These forms can be broadly classified intotwo groups. One group is the modified logarithmic:dzzf 5 ab2 g , (3)m12 1 2LLwhere a, b, g, and d are arbitrary empirical constantsthat usually differ from author to author (e.g., Busingeret al. 1971; Dyer 1974; Dyer and Bradley 1982; Ho¨g-stro¨m 1988; Frenzen and Vogel 1992).The other group is the power law:czzf 5 a 2b , (4)m12 1 2LLwhere a, b, and c are arbitrary constants that also differfrom author to author (e.g., Swinbank 1968; Zilitink-evich and Chalikov 1968; Foken and Skeib 1983; Kaderand Yaglom 1990). Both broad groups are a function ofz/L, which implies that both tacitly assume that the dom-548 VOLUME 37JOURNAL OF APPLIED METEOROLOGYFIG. 3. Wind observations from the Minnesota field experimentrun 7C1 compared to three surface-layer models [KP 5 Kader andPerepelkin 1989; ZS 5 Sorbjan 1986; BD 5 Businger–Dyer (Bus-inger et al. 1971)]: (a) above and (b) within the traditional SL. Seeappendix C for plots of the other runs.FIG. 4. Potential temperature (left) and wind (right) superpositionof transport processes (center) imposed on the background states (a)and (d), by large thermals (b) and (e) in the uniform layer, and smallershear-driven eddies (c) and (f) in the radix layer (after Stull 1994).inant generation of turbulence is mechanical, formed bywall shear (Stull 1997).Some SL similarity equations have been proposed toapply higher into the ML. For example, Fig. 3 comparesthe SL similarity relationships proposed by Kader andPerepelkin (1989) and Sorbjan (1986, hereafter ZS) tothe Minnesota data of run 7C1. Also plotted is an ex-tension of the Businger–Dyer profile equations abovethe surface layer for comparison. Of the three relation-ships plotted in Fig. 3, the proposal by Sorbjan appearsto work the best above the top of the SL; however, evenit has substantial errors as shown in appendix C for allMinnesota runs. The Sorbjan relationship will be dis-cussed in more detail in section 5. The difficulties ofmost SL theories at heights above the top of the tra-ditional SL provide motivation for the definition of anRxL and suggest that classic profile-matching methods(e.g., Rossby similarity) linking SL profiles with ULprofiles are probably not justified.In closing this brief review of SL similarity, we returnto the fourth premise listed at the start of this section.That feedback premise has the following interpretationfor flow very near the bottom boundary. Turbulencetransports momentum, momentum-flux divergence al-ters the mean-wind profile, and shear in the mean-windprofile generates small-eddy turbulence. The feedbackJUNE 1998 549SANTOSO AND STULLFIG. 5. Raw (a) wind speeds and (b) potential temperatures for all Minnesota runs, showing the range of conditions studied here.TABLE 1. Best-fit estimates of parameters in the dimensionless wind, Eq. (7), for A15 0.0959, and in the dimensionless potential temperature,Eq. (8), for A25 0.101. Also shown for comparison with the radix-layer depths for momentum zRMand temperature zRuare the Obukhovlength L, and 10% of the mixed-layer depth zi, which are often used as depth scales for the classic surface layer. Here, UL is the uniformlayer, and MULand uULare the wind speed and potential temperature in the UL. The surface potential temperature based on the best-fit profileis u0.RunWindMUL(m s21) zRM(m)Potential temperatureuUL(K) u0(K) zRu(m)Other depth scales2L (m) 0.1zi2A12A23A13A25A111.711.89.78.9(No UL)134.5111.8226.7216.5(No UL)295.95296.74296.23296.33285.51304.81305.52304.78302.71290.3932.2230.2753.0762.1726.6438. is closed, at least for shear-driven surface layers.This is a fundamental, but infrequently discussed, prem-ise underlying SL similarity theory.b. Radix layerIn the nearly free-convective ML, such feedback isbroken. Turbulence still transports momentum, and mo-mentum-flux divergence alters the mean wind profile.However, the mean wind profile does not generate large-eddy turbulence. Instead, surface heating generates thelarge, coherent, thermal structures. Because the thermalstructures have a length scale proportional to the MLdepth, we can infer that the wind profile in the radixlayer must also be a function of zi.According to convective transport theory (CTT) for550 VOLUME 37JOURNAL OF APPLIED METEOROLOGYFIG. 6. Data (open circles) from the 10 wind datasets collapse toa common similarity wind profile when radix-layer depth zRMis usedto make height z dimensionless, and uniform-layer wind MULis usedto make wind speed M dimensionless. Empirical Eq. (7) is plottedas the solid line, using a ‘‘universal’’ value of A15 0.0959. Notethat the wind speed Eq. (7) does become zero at the surface, eventhough there are no data low enough to show it.FIG. 7. Data (open circles) from the 11 potential-temperature da-tasets collapse to a common similarity profile when radix-layer depthzRuis used to make height z dimensionless, and temperature differencebetween the surface skin and the uniform layer (u02 uUL) is usedto make potential temperature u dimensionless. Empirical Eq. (8) isplotted as the solid line, using a universal value of A25 0.101.FIG. 8. Linear regression of dimensionless RxL depth zRMfor windagainst a dimensionless group (wB/MUL) that is proportional to thesquare root of the mixed-layer Richardson number (Stull 1994), wherez0is the aerodynamic roughness length, wBis the buoyancy velocity,and MULis the wind speed in the uniform layer.surface fluxes (Stull 1994), the UL is independent ofthe surface layer. Namely, the UL interacts with thesurface directly via the large thermal structures, makingit independent of the small-eddy SL and of the rough-ness length (Fig. 4, after Stull 1994). However, the SLin that previous paper, here better identified as the radixlayer, must depend on both roughness length and ULcharacteristics if it is to become tangent to the UL pro-file. This also suggests that the RxL is a function of zi.CTT identifies a buoyancy velocity scale, wB5[(g/Ty)Duyszi]1/2, that is valid for free convection. It isproportional to the Deardorff velocity, w*5where g is gravitational acceleration,1/3[(g/T )w9u9 z ],yys isubscript s denotes a surface value, Tyis average virtualtemperature, Duysis the virtual potential temperaturedifference between the surface skin and UL, andis the surface value of vertical flux of virtual po-w9u9ystential temperature, which is similar to a buoyancy fluxor a kinematic heat flux. The buoyancy velocity will beutilized later in the paper.3. Radix-layer wind and temperature profilesa. Profile equations in the radix layerFor the purpose of the similarity analysis in the RxL/ML system, we hypothesize here that RxL depths (zRMand zRu) are the relevant height scales and that the windsand temperatures in the UL (MULand uUL) are the rel-evant velocity and temperature scales. After testing sev-eral candidate functions, the following empirical profileequations were selected for further study based on theirclose agreement with the data.JUNE 1998 551SANTOSO AND STULLFIG. 9. Dimensionless potential temperature RxL depth zRushowslittle correlation when plotted against the RxL dimensionless groupthat was successful for wind.FIG. 10. Comparison of wind shear profiles from Sorbjan (ZS) andfrom Eq. (10, radix) for typical conditions at Minnesota (zi5 2000m, zRM5 200 m, L 5220 m, MUL5 10ms21, and w*5 2ms21).The differences are subtle but significant when integrated over heightto get wind speed.Equations for wind and potential temperature in theRxL are A1zzM exp A 1 2 , z # z ,UL 1 RM12 2[]RM RMM 5M , z . z UL RM(5)and A2zz(u 2 u )12 exp A 1 2 ,0UL 2125 12 6[] Ru Ruu 2 u 5 z # z ,UL Ru0, z . z , Ru(6)where A1and A2are empirical constants and u0is thepotential temperature of the air near the surface.Note that both sets of equations above satisfy thedesired constraint that the partial derivative of the pa-rameters (M or u) with respect to z is zero at the top ofthe radix layer, zR. Both sets of the equations also showthat the mean profiles and the vertical gradients are con-tinuous and smoothly merge at the top of the RxL.Namely, the RxL profiles are tangent to the UL at afinite height, as observed, rather than asymptoticallyapproaching the UL at infinite height, or rather thancrossing the UL at an arbitrary matching height (Pa-nofsky 1978).b. Calibration against Minnesota dataTo locate the top of the RxL, vertically contiguousmean (time averaged) wind speed and temperature dataare needed from near the surface through the interior ofthe ML. Many field experiment datasets do not satisfythis requirement because of the artificial discontinuitycreated when time-averaged surface-layer data frommasts or towers are combined with instantaneous ra-winsonde observations above (Clarke and Brooke1979). Also, many wind profiler systems do not giverobust wind-speed measurements below 100–200 m.However, published data from the 1973 Minnesotafield experiment (Izumi and Caughey 1976) are satis-factory. The instrument platforms included a combi-nation of a 32-m tower and sensors at fixed locationson the cable of a tethered balloon. Similar averagingtimes of 75 min were used at all heights from near thesurface through the midmixed layer, yielding profiles ofwind and temperature that are self-consistent, smooth,and contiguous. There were 11 datasets (‘‘runs’’) thatwere obtained during the Minnesota experiment. Thesite was a flat, recently harvested, and plowed farmsquare-mile section (1.609 km on a side) with no veg-etation close by, a roughness length of z0ł 2.4 mm,elevation of 255 m above sea level, at location 488349N,968519W. A uniform fetch of 10 km existed to the north,which was the predominant wind direction. A more de-tailed description of the site, instruments, and experi-mental procedures can be found in Izumi and Caughey(1976) and Kaimal et al. (1976). Appendix A (TableA1) lists key dates, times, and scales for the 11 runs.Figure 5 shows the range of wind speeds and potentialtemperature profiles present during the runs.A nonlinear regression (Bevington 1969; Press et al.1992) is used to determine best-fit parameters MUL, zRM,and A1for wind; and u0, uUL, zRu, and A2for potentialtemperature, for each dataset from the Minnesota ex-periment. An iterative process is used to minimize thesum of squared deviations between the regression equa-tion (RxL and UL taken together) and the data. Best-552 VOLUME 37JOURNAL OF APPLIED METEOROLOGYFIG. 11. Wind speed M profiles observed (data points) and foundby integrating the Sorbjan (1986) profile relationship (curves) up anddown from each data point.fit estimates of the parameters are listed in appendix Bfor each of the 11 datasets.c. Universal constantsUsing all the Minnesota runs, parameter A15 0.0956 0.011 (mean plus or minus standard deviation) forwind speed, and A25 0.104 6 0.009 for potential tem-perature (see appendix B). One wind dataset (run 5A1)did not have a UL in wind speed and, hence, was ex-cluded from the wind speed analysis of this study be-cause no top to the RxL could be found. The relativelysmall variations in A over the wide range of wind speeds(Fig. 5a), potential temperatures (Fig. 5b), and MLdepths suggest that they might be universal. We willmake the assumption of universality. Instead of aver-aging the previously calculated A values to find the uni-versal value, we rearrange the profile equations into adimensionless form and apply the nonlinear regressionto the full set of 10 or 11 runs grouped together as onerun. This allows us to find the best-fit A values for asuperset of data consisting of the whole Minnesota ex-periment. This is a more stringent test because it de-mands similarity simultaneously over a wide variety ofwind and depth regimes. That is, while any single da-taset might cluster tightly around the curve, the supersetis more likely to have scatter (i.e., unexplained variance)if the theory was poor.First, reframe the wind and temperature profile equa-tions in nondimensional form: A1zzexp A 1 2 ,112 2[] RM RMM5 z # z , (7)RMMUL1, z . z , RMand A2zz1 2 exp A 1 2 ,212 2[] Ru Ruu 2 uUL5 z # z , (8)Ruu 2 u0UL0, z . z . RuThe ‘‘universal’’ A values are computed using non-linear least squares analysis, as before, between M/MULversus z/zRMfor the 10 valid wind runs taken togetheras one large dataset. Similarly, nonlinear regression isused for (u 2 uUL)/(u02 uUL) versus z/zRuusing datafrom all 11 temperature runs taken together as one largedataset. The A values are determined such that M/MULapproaches one as z/zRMapproaches one, and (u 2 uUL)/(u02 uUL) approaches zero as z/zRuapproaches one.The resulting best-fit values are A15 0.0959 6 0.011(mean plus or minus standard deviation) for wind andA25 0.101 6 0.009 for temperature. The correlationcoefficient between observed and computed nondimen-sional equations is r 5 0.992 for wind speed and r 50.986 for potential temperature. Given these fixed valuesfor A1and A2, the other parameters in the wind andtemperature equations found using nonlinear regressionare listed in Table 1 for each of the Minnesota runs.Plots of dimensionless wind and temperature profilesJUNE 1998 553SANTOSO AND STULLusing these universal A values are shown in Figs. 6and 7.The parameters listed in Table 1 are not expected tobe universal, but they can be functions of external forc-ings such as geostrophic wind speed G, boundary effectssuch as ziand z0, and scaling variables such as wB. Mostsimilarity theories incorporate such locally varying non-universal parameters, such as the way the Obukhovlength is incorporated in SL similarity theory. Note fromTable 1 that wind speeds varied by roughly a factor of2, and RxL depths by a factor of 3, yet the data fromeach of the 10 wind datasets collapsed into a singlesimilarity curve in Fig. 6. Analogous similarity behavioris evident in Fig. 7 for the 11 temperature datasets. Forboth wind and temperature, the RxL depth exhibits verylittle correlation with either classic measures of surface-layer thickness, namely, the Obukhov length L, or 10%of the mixed-layer depth zi(see Table 1). This latterresult differs from the findings of Brutsaert and Sugita(1992) for a different dataset, although their results arefor the surface-layer depth rather than the RxL depth.There are noticeable differences between RxL depthsbetween Table 1 and the tables in appendix B. Becauseof the gradual approach of the RxL profile to the ULprofile, the RxL height is somewhat sensitive to the Aparameters and to scatter in the data. Because of thissensitivity, we looked for a better depth scale, such asan integral scale or a half-width height. To date, a betterdepth scale has not been found. However, the shape ofthe profile is relatively insensitive to the RxL depth,which means the former sensitivity is not very impor-tant.4. Similarity scalingWhat controls RxL depth? From Table 1 it is obviousthat RxL depth varies from run to run. It probably alsovaries from location to location, although the Minnesotacase study alone is insufficient to address dependencyon roughness length.In this section we attempt to parameterize RxL depthas a function of constraints and external forcings for theMinnesota case study. These factors include both SLvariables such as z0and surface buoyancy flux, but theyalso include ML variables such as zi. A new field pro-gram called Boundary Layer Experiment 1996 (BLX96)was recently conducted to study roughness dependencyof RxL characteristics (Stull et al. 1997). Results fromthis new study will be reported later in a separate paper.After regressing RxL depth against a wide range ofcandidate SL and ML variables, we found the followingempirical expression to provide the best fit for the Min-nesota data:3/2zwRM B5 c , (9)12zM0ULwhere the buoyancy velocity wBwas as defined earlierand c is an empirical constant. An expression equivalentto (9) is possible using the Deardorff velocity w*insteadof the buoyancy velocity wBbecause the two are linearlyrelated by w*ł 0.08wB(Stull 1994). The ratio wB/MULis the square root of the ‘‘mixed-layer Richardson num-ber’’ (Stull 1994). The roughness-length dependence in(9) is tentative, based on theoretical expectations.Figure 8 shows this regression. The dimensionlessslope is c 5 21.4 3 103and the correlation is r 5 0.545,which means that only about 30% of the variance in thedata is explained by the regression line. It is clear thatadditional work needs to be done to identify the factorsthat control the RxL depth.The RxL depth for temperature, zRu, shows even morescatter when plotted (Fig. 9) against the same parametersas for zRM. Also, there was very little correlation (r 50.145) between zRMand zRu. This might be expectedbecause the wind speed is very highly constrained be-tween zero at the surface and the geostrophic wind aloft,but the temperature profile floats as both the surfaceskin temperature and the UL temperature increase dur-ing the day.5. Wind profiles versus shear profilesDuring this work, it became apparent that inaccura-cies of wind profile relationships are hidden when thoseprofiles are expressed in terms of wind shear. However,when those profiles are integrated with height to get theactual wind speeds, the errors are revealed and can ac-cumulate to cause substantial discrepancies between theobserved wind speed and the parameterized profile. Thisis unfortunate because for many practical purposes, suchas wind loads on structures, pollutant transport, andwind power generation, it is the speed and not the shearthat is needed.As an example, Fig. 10 compares the wind shear usingthe Sorbjan (ZS 1986) relationship [his Eq. (38)], andusing the radix ralationship, which isA21]Mz z z zRM5 A 1 2 exp A 1 2 ,111212 12[]]zM z z zUL RM RM RMz # zRM5 0, z . z . (10)RMThe difference between these two curves is subtle; it isnot easy to discriminate between the two relationships.However, when integrated over height to get wind speed,the deficiencies of the ZS relationship compared withthe RxL formulation become apparent and significant(Fig. 3 and Fig. C1). The radix-layer wind speeds aremore accurate over a wider range of heights than theZS speeds.One might argue that this is an unfair test becauseintegrating up from a small wind near the surface, suchas from zero wind at the roughness height, might am-plify small initial errors. To examine this argument, one554 VOLUME 37JOURNAL OF APPLIED METEOROLOGYcan recompute the height integration, but starting at dif-ferent heights. This process is repeated for each datapoint in the observed wind profile, generating a set ofcurves such that each curve exactly passes through onepoint. If the profile similarity theory is valid, then allof the curves should lie nearly on top of each other.Examples of the integrated ZS wind-speed profilesare shown in Fig. 11, for runs 3A1 and 6A1. Most ofthe curves do not lie on top of each other. Furthermore,the direction of the error is not consistent: run 3A1 hasless shear than the integrated ZS curves, while run 6A1has more shear. We repeated this exercise for each ofthe Minnesota runs and found magnitudes of ZS wind-speed errors of roughly 63.0ms21for runs 2A1, 2A2;61.0 m s21for runs 3A1 and 3A2; 60.7ms21for runs6A1, 6A2, 6B1, and 7C2; and 60.3 m s21for runs 7C1and 7D1. This compares to wind-speed errors of 60.3ms21or less for all runs using RxL Eq. (5), as alreadyplotted in Fig. 6.Thus, it appears that wind speed gives a more sen-sitive test of the accuracy of a similarity relationshipthan does shear. We recommend that future proposalsfor similarity relationships be tested in their integratedform, such as wind speed profiles.6. Summary and recommendationsEquations (7) and (8) describe similarity relationshipsfor wind speed and potential temperature within thewhole radix layer. The radix layer is identified as theregion between the surface and the base of the uniformportion of the convective boundary layer. The top of theradix layer (i.e., the base of the uniform layer) is usuallywell above the top of the classic surface layer.Using data from the 1973 Minnesota experiment, theresulting best-fit value for the exponents in Eqs. (7) and(8) are A15 0.0959 6 0.011 for wind and A25 0.1016 0.009 for temperature. These exponents are relativelyconstant for a wide variety of wind speeds and surfaceheating, suggesting that they might be universal. Thecorrelation coefficient between observed and parame-terized profiles is r 5 0.992 for wind speed and r 50.986 for potential temperature. This suggestion of uni-versality of equations and exponents should be consid-ered tentative until independent verification tests arepublished and wind-speed ranges of validity are iden-tified.The radix-layer depth differs from run to run, anal-ogous to run-to-run variations of surface-layer scalessuch as the Obukhov length. The radix-layer depth forwind is greater than that for temperature, and there islittle correlation between the two. An attempt was madeto parameterize the radix-layer depth for wind as a func-tion of the mixed-layer Richardson number; however,the fit was poor, suggesting that more work is needed.The profile equations proposed here are applicable inthe range of 1 m , z , 0.7 zi. Thus, they span thepreviously little-studied region above the top of the sur-face layer and below the base of the uniform layer, whereclassic Monin–Obukhov similarity theory performspoorly. Potential applications of better wind speed pro-files include wind turbine electrical generation, windloads on buildings and bridges, and air pollutant trans-port.We recommend that profile similarity relationships bepresented and tested as wind-speed profiles, rather thanas shear profiles. The latter tend to hide or disguiseerrors and give a false picture of the accuracy of therelationship.Datasets for independent validation are scarce be-cause of the requirements for time- or space-averaged,not instantaneous, observations from the surface intothe midmixed layer. To test the above equations, a newfield program (Boundary Layer Experiment 1996) wasconducted in July and August 1996 in the central UnitedStates (Stull et al. 1997). Special vertical zigzag flightpatterns in the University of Wyoming King Air aircraftwere flown to measure the mean profiles of wind speedand potential temperature. Results from this new vali-dation will be forthcoming.Acknowledgments. This work was initiated while bothauthors were at the University of Wisconsin. The fol-lowing funding agencies are gratefully acknowledgedfor their support: U.S. National Science Foundation(NSF ATM-9411467), the U.S. Department of Energy(ARM Grant DE-FG02-9ZER61361), the Canadian At-mospheric Environment Service (AES subventions),and the Canadian Natural Sciences and Engineering Re-search Council (NSERC operating grants).JUNE 1998 555SANTOSO AND STULLAPPENDIX AScaling Variables for Individual Minnesota RunsTABLE A1. Dates, times, and boundary layer scaling variables for the Minnesota field experiment. Friction velocity is u*, Deardorff velocityis w*, buoyancy velocity is wB, mixed-layer depth is zi, Obukhov length is L, surface kinematic heat flux is Q0, surface-layer temperaturescale is T*52Q0/u*, and mixed-layer temperature scale is u*5 Q0/w*. Central Daylight Time (CDT) 5 UTC 2 5 h. The buoyancyvelocity wBwas calculated from its definition (see section 2b) using Duysequal to uyUL2 uy0and was not calculated from w*. Also L wascalculated using a von Ka´rma´n constant of 0.4.RunTime(CDT)Date(1973)u*(m s21)w*(m s21)wB(m s21)zi(m)2L(m)Q0(K m s21)T*(8C)u*(8C)2A12A23A13A25A1*1217–13321332–14471510–16251625–17401622–173710 Sep10 Sep11 Sep11 Sep15 Sep0.460.450.370.320.*1250161523102300108538.2334.1020.9621.548.160.1960.2090.1860.1160.06920.4220.4620.5020.3620.380.–15161516–16311652–18071415–15301530–164517 Sep17 Sep17 Sep19 Sep19 Sep0. 1650–1805 19 Sep 0.25 1.58 15.2 1225 11.92 0.099 20.40 0.06* No uniform layer on this day.APPENDIX BNonlinear Regression Parameters for Individual Minnesota RunsTABLE B1. Best-fit estimates of the parameters in Eq. (5) for windspeed in the combined radix and uniform layers. Here, MULis windspeed in the uniform layer, zRMis the radix-layer depth for wind, andA1is the exponent parameter.Run MUL(m s21) zRM(m) A12A12A23A13A25A16A111.8011.759.728.865.437.42115.8186.02226.73219.17610.00248.670.1070.1070.0960.0950.0960.0846A26B17C17C27D17.727.806.796.976.21894.13593.42102.07212.08236.900.0750.0820.1050.1010.096TABLE B2. Best-fit estimates of the parameters in Eq. (6) for po-tential temperature in the combined radix and uniform layers. Here,uULand u0are the potential temperatures in the uniform layer and inthe air adjacent to the surface, respectively; zRuis the radix–layerdepth for potential temperature; and A2is the exponent parameter forpotential temperature.Run uUL(K) u0(K) zRu(m) A22A12A23A13A25A16A16A26B17C17C27D1296.05296.75296.23296.32285.50292.52293.02293.52284.29284.93285.32304.92305.52304.78302.71290.39301.16301.43299.39293.16292.38291.1221.0223.7854.1370.4230.7629.5224.5719.0659.3865.1653.140.1180.1150.1000.0970.0940.1150.1060.0960.1000.1070.093556 VOLUME 37JOURNAL OF APPLIED METEOROLOGYAPPENDIX CComparison of Several Wind-SpeedProfile Relationships Against Each Individual Minnesota RunFIG. C1. Wind observations from each of the Minnesota field experiment runs, compared tothree surface-layer models [KP 5 Kader and Perepelkin 1989; ZS 5 Sorbjan 1986; BD 5Businger–Dyer (Businger et al. 1971)] and the RxL model.JUNE 1998 557SANTOSO AND STULLFIG. C1. (Continued)558 VOLUME 37JOURNAL OF APPLIED METEOROLOGYREFERENCESBevington, E. H., 1969: Data Reduction and Error Analysis for thePhysical Sciences. McGraw-Hill, 336 pp.Brutsaert, W., and M. Sugita, 1992: The extent of the unstable Monin–Obukhov layer for temperature and humidity above complexhilly grassland. Bound.-Layer Meteor., 51, 383–400.Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971:Flux-profile relationships in the atmospheric surface layer. J.Atmos. Sci., 28, 181–189.Clark, R. H., and R. R. Brook, 1979: The Koorin Expedition, At-mospheric Boundary Layer Data over Tropical Savannah Land.Bureau of Meteorology, Australian Government Publishing Ser-vice, 359 pp.Deardorff, J. W., G. E. Willis, and B. H. Stockton, 1980: Laboratorystudies of the entrainment zone of a convectively mixed layer,part 1. J. Fluid Mech., 100, 41–64.Dyer, J. A., 1974: A review of flux-profile relationship. Bound.-LayerMeteor., 7, 363–372., and E. F. Bradley, 1982: An alternative analysis of flux-gradientrelationships at the 1976 ITCE. Bound.-Layer Meteor., 22, 3–19.Foken, T., and G. Skeib, 1983: Profile measurements in the atmo-spheric near-surface layer and the use of suitable universal func-tions for the determination of the turbulent energy exchange.Bound.-Layer Meteor., 25, 55–62.Frenzen, P., and C. A. Vogel, 1992: The turbulent kinetic energybudget in the atmospheric surface layer: A review and experi-mental reexamination in the field. Bound.-Layer Meteor., 60,49–76.Garratt, J. R., J. C. Wyngaard, and R. J. Francey, 1982: Winds in theatmospheric boundary layer—Prediction and observation. J. At-mos. Sci., 39, 1307–1316.Ho¨gstro¨m, U., 1988: Non-dimensional wind and temperature profilesin the atmospheric surface layer: A re-evaluation. Bound.-LayerMeteor., 42, 55–78.Izumi, Y., and J. S. Caughey, 1976: Minnesota 1973 atmosphericboundary layer experiment data report. AFCRL-TR-76-0038, 28pp. [Available from Atmospheric Science Programme, Dept. ofGeography, University of British Columbia, 1984 West Mall,Vancouver, BC V6T 1Z2, Canada.]Kader, B. A., and V. G. Perepelkin, 1989: Effect of unstable strati-fication on the wind speed and temperature profiles in the surfacelayer. Atmos. Oceanic Phys., 25, 583–588., and A. M. Yaglom, 1990: Mean fields and fluctuation momentsin unstably stratified turbulent boundary layer. J. Fluid Mech.,212, 637–667.Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, O. R. Cote, and Y.Izumi, 1976: Turbulence structure in the convective boundarylayer. J. Atmos. Sci., 33, 2152–2169.Panofsky, H. A., 1978: Matching in the convective planetary bound-ary layer. J. Atmos. Sci., 35, 272–276.Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,1992: Numerical Recipes in FORTRAN: The Art of ScientificComputing. 2d ed., Cambridge University Press, 963 pp.Priestley, C. H. B., 1955: Free and forced convection in the atmo-sphere near the ground. Quart. J. Roy. Meteor. Soc., 81, 139–143.Sorbjan, Z., 1986: On similarity in the atmospheric boundary layer.Bound.-Layer Meteor., 34, 377–397.Stull, R. B., 1994: A convective transport theory for surface fluxes.J. Atmos. Sci., 51, 3–22., 1997: Reply. J. Atmos. Sci., 54, 579., E. Santoso, L. Berg, and J. Hacker, 1997: Boundary layerexperiment 1996 (BLX96). Bull. Amer. Meteor. Soc., 78, 1149–1158.Swinbank, W. C., 1968: A comparison between predictions of di-mensional analysis for the constant-flux layer and observationsin unstable conditions. Quart. J. Roy. Meteor. Soc., 94, 460–467.Zilitinkevich, S., 1994: A generalized scaling for convective shearflows. Bound.-Layer Meteor., 70, 51–78., and D. V. Chalikov, 1968: On the determination of the universalwind and temperature profiles in the surface layer of the at-mosphere. Atmos. Oceanic Phys., 4, 294–302.


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