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Similarity Equations for Wind and Temperature Profiles in the Radix Layer, at the Bottom of the Convective… Santoso, Edi; Stull, Roland B. Jun 30, 2001

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1446 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S q 2001 American Meteorological Society Similarity Equations for Wind and Temperature Profiles in the Radix Layer, at the Bottom of the Convective Boundary Layer EDI SANTOSO AND ROLAND STULL Atmospheric Science Programme, Department of Earth and Ocean Sciences, The University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 13 January 1999, in final form 17 May 2000) ABSTRACT In the middle of the convective boundary layer, also known as the mixed layer, is a relatively thick region where wind speed and potential temperature are nearly uniform with height. Below this uniform layer (UL), wind speed decreases to zero at the ground, and potential temperature increases to the surface skin value. This whole region below the UL is called the radix layer (RxL), and is of order hundreds of meters thick. Within the bottom of the RxL lies the classical surface layer (order of tens of meters thick) that obeys traditional Monin- Obukhov similarity theory. The RxL depth is shown to depend on friction velocity, Deardorff velocity, and boundary layer depth. The wind RxL is usually thicker than the temperature RxL. Using RxL depth, UL wind speed, and UL potential temperature as length, velocity, and temperature scales, respectively, one can form dimensionless heights, ve- locities, and temperatures. When observations obtained within the RxL are plotted in this dimensionless frame- work, the data collapse into similarity curves. This data collapse is tightly packed for data collected over single- location homogeneous surfaces, and shows more scatter for data collected along 72-km flight tracks over het- erogeneous surfaces. Empirical profile equations are proposed to describe this RxL similarity. When these profile equations are combined with the flux equations from convective transport theory, the results are new flux–profile equations for a deep region within the bottom of the convective boundary layer. These RxL profile similarity equations are calibrated using data from four sites with different roughnesses: Minnesota, BLX96-Lamont, BLX96-Meeker, and BLX96-Winfield. The empirical parameters are found to be invariant from site to site, except for the profile shape parameter for wind speed. This parameter is found to depend on standard deviation of terrain elevation, rather than on the aerodynamic roughness length. The resulting parameter values are compared with independent data from a forested fifth site, Koorin, and it is found that displacement height must be subtracted from all the heights in the RxL profile equations. The resulting profile equations could be useful for calculating wind loading on bridges, wind turbine power estimation, air pollutant transport, or other applications where wind speeds or temperatures are needed over the bottom hundreds of meters of the convective boundary layer. 1. Introduction The term mixed layer (ML) is used in this paper to represent the whole convective boundary layer that is nonlocally statically unstable (Stull 1991), and which is undergoing vigorous convective overturning associated with coherent thermals rising from the warm underlying surface. Within this ML, winds are zero near the ground (Fig. 1a) and smoothly increase with height until finally becoming tangent to a vertically uniform, subgeostroph- ic, wind speed layer in the mid-ML (Santoso and Stull 1998a). Across the top of the ML, the winds increase to their nearly geostrophic magnitudes above. Analo- gous profiles exist for potential temperature (Fig. 1b). Corresponding author address: Roland Stull, Atmospheric Science Programme, Dept. of Earth and Ocean Sciences, The University of British Columbia, 6339 Stores Rd., Vancouver, BC V6T 1Z4, Canada. E-mail: rstull@eos.ubc.ca a. Subdomains of the ML One can identify subdomains of the ML that have different similarity scalings. In the middle is the uniform layer (UL) as described above. Convective ML scales such as Deardorff’s (1970) convective velocity (w * , which will be referred to as the ‘‘Deardorff velocity’’ here) and average ML depth (zi) apply here, and are associated with the large thermal circulations that trans- port heat upward and momentum downward. The Dear- dorff velocity is defined as 1/3 g w* 5 z w9u9 , (1)i y s1 2[ ]Ty where g is gravitational acceleration, Ty is average ab- solute virtual temperature, and , is a surface eddy-w9u9y s covariance value representing kinematic vertical flux of virtual potential temperature (a measure of buoyancy flux). 1 JUNE 2001 1447S A N T O S O A N D S T U L L FIG. 1. Idealized vertical profiles of (a) wind speed M and (b) potential temperature u in the atmospheric boundary layer. Here G represents the geostrophic wind speed, uskin is the potential temper- ature of the surface skin, and zi is the mixed layer depth. The sub- scripts are UL for the uniform layer, R for the radix layer, S for the surface layer, u for potential temperature, and M for wind speed. Above the UL is the entrainment zone, a transition layer between the UL and the nearly geostrophic free atmosphere above. Within the entrainment zone are sub- adiabatic temperature profiles, overshooting thermals, intermittent turbulence, and wind shear (Deardorff et al. 1980). Both free convection and entrainment scales are important here (Sorbjan 1999). Because of the inter- mittency, the average top (zi) of the ML is near the middle of the entrainment zone. At the very bottom of the ML is the surface layer (SL), the nearly constant flux region where Monin-Obu- khov (MO) similarity theory applies (Businger et al. 1971; Dyer 1974). In this layer the wind profile is nearly logarithmic with height (z) and is dominated by me- chanically generated small-eddy turbulence within the wall shear flow (Stull 1997a). The dominant SL scales required for similarity of mean profiles (rather than for similarity of mean gradients) are aerodynamic rough- ness length (z0), friction velocity (u*), and Obukhovlength (L): 33u* 1 u* L 5 2 5 2 z , (2)i1 2g k w* k w9u9y sTV where k ù 0.4 is the von Kármán constant. There is a region or gap between the top of the SL and the bottom of the UL where SL similarity theory was not designed to work, and indeed where it has been shown (Santoso and Stull 1998a) to give rather poor results. In this region, one might expect that both SL and ML scales should be important. b. The radix layer To better explain the portion of the boundary layer below the UL, Santoso and Stull (1998a) analyzed data from the 1973 Minnesota field experiment (Izumi and Caughey 1976), and identified a radix layer (RxL) that obeys a similarity scaling different than MO. The word ‘‘radix’’ means ‘‘origin’’ or ‘‘root’’ in Latin, because the roots of convective thermals are in this layer. The new RxL scaling was found to apply to the whole region between the surface and the bottom of the UL, and thus includes the traditional SL as a subdomain. Typical depths of the RxL are on the order hundreds of meters for wind profiles and tens of meters for temperature profiles. Based on the definitions above there is superposition of layers; namely, the ML contains the RxL as a sub- domain, and the RxL contains the SL as a subdomain (Fig. 1). At the top of the RxL, wind speed (M) and potential temperature (u) become tangent to the UL, allowing one to define the top of the RxL as the lowest altitude where ]M/]z 5 0 or ]u/]z 5 0. Within the RxL, one finds that SL scales decrease in importance with increasing altitude, while ML scales increase. Using data collected in the Minnesota field experi- ment, Santoso and Stull (1998a) proposed new similar- ity equations for wind and potential temperature profiles within and above the RxL. When the dimensionless ratio of wind speed or potential temperature divided by their UL values were plotted against dimensionless ratio of height divided by RxL depth, the data points collapsed quite tightly around the proposed similarity curves. There was also evidence that some of the parameters in these similarity equations were universal. While these results showed promise for universal sim- 1448 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S ilarity profiles in the RxL, there were two drawbacks. First, the success of the theory depended on knowledge of the RxL depth, but this depth was difficult to pinpoint from the observations because of the gradual blending of the wind and potential temperature profiles into the UL. Second, the mechanisms that control the RxL depth had not yet been identified in that first paper (Santoso and Stull 1998a). c. Goals The purpose of this work is to improve the data- analysis methodology, thereby allowing development of well-defined similarity profiles for the RxL. We utilize Buckingham Pi dimensional analysis to identify the rel- evant dimensionless groups of variables, and to help find the parameters that control RxL depth. We also employ convective transport theory (CTT; Stull 1994) to relate surface fluxes to ML scales, and combine these with RxL profiles to create new flux–profile relation- ships. As motivation for our work, we review in section 2 basic premises and limitations underlying SL similarity theory. Then in section 3 we develop parameterizations for the RxL depths and improve RxL similarity theory for wind and potential temperature profiles. In section 4 we again use data from the Minnesota field experiment to recalibrate our new parameterizations and similarity equations. In section 5, data from Boundary Layer Ex- periment 1996 (BLX96) is used to determine whether RxL profiles depend on surface conditions such as roughness. In section 6 we compare our results against independent data collected during the Koorin field cam- paign. A flux–profile relationship is proposed in section 7, followed by summary and recommendations in sec- tion 8. Tables of field experiment data and error analyses are included in the appendixes. 2. Basic premises for SL similarity theory Monin-Obukhov similarity theory has been the fa- vored tool in deriving SL profile equations. The theory is built on the following premises. First, turbulent ver- tical fluxes are assumed approximately uniform with height in the SL. This simplifies the theory by allowing flux variations to be neglected. It also constrains its applicability to the bottom 10% of the ML, assuming that turbulent fluxes vary roughly linearly with height during free convection. For real MLs, the SL has been observed to be as small as 1% of zi during free con- vection (Dyer 1967). Second, turbulence is assumed to be generated pre- dominantly by mechanical shear flow near the ground. Such turbulence is idealized as consisting of small ed- dies, causing downgradient local transport. These char- acteristics lead to the widely known ‘‘law-of-the-wall’’ for statically neutral flow, which has a solid theoretical basis and strong support from laboratory experiments. For slightly nonneutral conditions, small eddies can still dominate at heights less than | L | , which is often used to define the top of the SL within which MO theory is valid. A third premise, although rarely acknowledged, is that feedback exists between the mean flow and tur- bulence within the SL. Namely, 1) turbulence transports momentum, 2) momentum-flux divergence alters the mean-wind profile, and 3) shear in the mean-wind pro- file generates the small-eddy turbulence that feeds back to (1). Thus, this feedback loop is closed for shear- driven SLs; namely, at heights less than | L | . However, there is no reason why MO similarity the- ory should work for the strongly convective ML at heights above | L | , because small-size eddies are less important there. Also, the feedback loop is broken for that situation (Stull 1997a). Namely, 1) turbulence still transports momentum and 2) momentum-flux diver- gence modifies the mean wind profile. However, the mean wind profile does not generate large-eddy tur- bulence. Instead, 3) at heights above | L | , turbulence is driven by convective instabilities caused by the warm underlying surface. In the limit of strong surface heat flux and very weak mean winds (i.e., as u * /w * becomes very small), the Obukhov length becomes very small (see appendixes A and B), resulting in an extremely shallow SL. For the BLX96 field experiment described later in this paper, we find that SL similarity theory is valid only for the bottom 0.2%–2% of the ML, in agree- ment with Dyer’s (1967) findings. While MO similarity theory identifies the dimen- sionless groups that are relevant in the SL (e.g., z/L, z/ z0, and M/u*), no form of dimensional analysis is ablegive the relationships between these groups. Such re- lationships must be estimated from field experiments. For convective conditions, empirical SL relationships have been suggested by Swinbank (1968), Zilitinkevich and Chalikov (1968), Businger et al. (1971), Dyer (1974), Dyer and Bradley (1982), Foken and Skeib (1983), Högström (1988), Kader and Perepelkin (1989), Kader and Yaglom (1990), Frenzen and Vogel (1992), Brutsaert (1999), and others. All the premises listed earlier in this subsection imply that the MO similarity theory is strongly dependent on surface characteristics, but is virtually independent of factors higher in the ML. For this reason one cannot expect the empirical SL equations to merge smoothly into the UL, because no information about the UL is considered in those equations. As pointed out by Pa- nofsky (1978), convective-matching-layer and free-con- vective-layer formulations (Priestley 1955; Kaimal et al. 1976) fail near the bottom of the UL, where the shear and potential-temperature gradient approach zero. The limited altitude range of applicability of tradi- tional MO SL similarity was illustrated by Santoso and Stull (1998a), who tested the Businger et al. (1971)– Dyer (1974) similarity equations against Minnesota field experiment data. In the SL, all the data collapsed to a 1 JUNE 2001 1449S A N T O S O A N D S T U L L single curve when plotted as dimensionless groups based on SL similarity. However, at higher altitudes the data scattered away from a universal curve. Namely, MO similarity theory works well in the SL, but is less successful higher in the RxL and in the UL. It is ob- viously desirable to find a new similarity theory that can be applied over a greater depth. One approach is to include zi/L into MO SL profile theories, such as de- scribed by Khanna and Brasseur (1997, 1998) and Jo- hansson et al. (2001). An alternative approach, taken here, is to refine a RxL similarity theory that is not based on MO SL similarity. 3. RxL theory a. Review of previous work 1) RXL PROFILE EQUATIONS The first step in any similarity analysis is to hypoth- esize which variables are relevant to the physics. Be- cause thermal structures exist within the whole ML in- cluding the SL, we can infer that similarity theory for RxL should depend on both SL and ML parameters. For wind and potential temperature profiles, Santoso and Stull (1998a) hypothesized that RxL depths (zRM and zRu, for wind and potential temperature, respectively) are the relevant height scales, and that the winds and potential temperature in the UL ( UL and UL) are theM u relevant velocity and temperature scales. Additional constraints are that the partial derivatives of the mean variables with respect to height are zero at the top of (and above) the RxL, and mean variables in the UL are constant with height. Based on Minnesota data, the following empirical relations (Santoso and Stull 1998a) were suggested to describe the vertical profiles of mean wind speed M and potential temperature within and above theu RxL  A1z zM exp A 1 2 for z # zUL 1 RM1 2 1 2[ ]M(z) 5  z zRM RM (3a) M for z . z UL RM  A2z z(u 2 u ) 1 2 exp A 1 2 for z # z0 UL 2 Ru5 1 2 1 2 6[ ]u(z) 2 u 5  z zUL Ru Ru (3b) 0 for z . z , Ru where 0 is potential temperature near the surface; ov-u erbars represent a horizontal-average ergodic approxi- mation to the ensemble average; and A1 and A2 are em- pirical constants. Both sets of equations above satisfy the desired zero gradient at the top of (and above) the RxL (Arya 1999; Santoso and Stull 1999a). Both sets also show that the mean profiles and the vertical gradients are continuous and smoothly merge at the top of RxL. Namely, the RxL profiles are tangent to the UL at finite height as ob- served, rather than asymptotically approaching the UL at infinite height, or rather than crossing the UL at an arbitrary matching height (Panofsky 1978). A nonlinear regression (Bevington 1969; Press et al. 1992) was used to determine best-fit parameters UL,M zRM, and A1 for wind; and 0, UL, zRu, and A2 for po-u u tential temperature, for each dataset from the Minnesota experiment. An iterative process was used to minimize the sum of squared deviations between the regression equation (RxL and UL taken together) and the data. Among best-fit parameters, UL and UL were found toM u the most statistically robust, having least root-mean- squared (rms) errors. The RxL depths, zRM and zRu cal- culated individually using this method (with zero ver- tical gradient as the desired constraint) were very sen- sitive to even small errors in measured data. With such high uncertainties, the attempted parameterizations of the RxL depths as a function of SL and ML variables were less successful. 2) CONVECTIVE TRANSPORT THEORY (CTT) Thermal diameters are sufficiently large (order of the mixed layer depth of 1–2 km) that there is a very large central core in each thermal that is protected from small- eddy lateral entrainment (Crum et al. 1987). Within this core, air parcels from near the surface are moved to the middle and the top of the convective boundary layer (BL) with virtually no dilution (Stull and Eloranta 1985). Because the buoyant thermals are anisotropic with more energy in the vertical, they can efficiently transport heat, momentum, and moisture vertically away from the surface. Stull (1994, 1997b) utilized these characteristics of thermals to show that during free-convective conditions the surface vertical turbulent flux s of any meanw9c9 variable is proportional to w * times D , the mag-c c 1450 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S nitude of the difference of between the surface skinc and the UL: w9c9 5 C*w*Dc , (4)s where C * is an empirical mixed layer transport coef- ficient. In the notation of this paper D 5 skin 2 ULu u u and D 5 skin 2 UL 5 2 UL, because skin 5 0M M M M M by definition. The CTT concept was found applicable for many different sites (Kustas et al. 1996; Greischer and Stull 1999), particularly when one accounts for the variation of C * (Santoso and Stull 1998b). b. New RxL depth equations To overcome the sensitivity problem of RxL depths, we now reverse our steps. First we parameterize the RxL depths as functions of both ML and SL scales, because of the superposition of large buoyantly driven and small shear-driven eddies in the RxL. The large eddies scale to zi and w* (Deardorff 1970, 1972; Dear-dorff et al. 1980; Wyngaard et al. 1971; Kaimal et al. 1976; Sorbjan 1986; Stull 1988). The small eddies scale to the RxL depth (zRM and zRu) and u*. The small-eddylength scale of RxL depth was chosen because at the bottom of the ML the shear needed to generate shear- driven eddies exists only within the RxL. That is, mean- shear-driven eddies exist everywhere within the RxL, they dominate only in the SL, and they do not exist in the UL. Using Buckingham Pi analysis (Stull 1988) with this set of scales (zR, zi, u*, w*) yields RxL dimensionlessgroups of zR/zi and u*/w*, where we use zR as a genericRxL depth. The relationship between these groups is not given by Buckingham Pi analysis; it must be found empirically. The relationship that was found to work best for wind is BMu* z 5 E z , (5a)RM M i1 2w* where BM and EM are empirical constants, and the sub- script M denotes wind speed. For potential temperature (subscript u) the corresponding relationship is Buu* z 5 E z , (5b)Ru u i1 2w* where Bu and Eu are empirical constants that are not necessarily the same as those for momentum. Relation- ships (5) agree with the suggestions of Plate (1997) that the altitude of the base of the UL should be proportional to u * /w * . The adequacy of (5) will be tested with Min- nesota data in section 4. These parameterizations of the RxL depths have a functional form similar to that of the Obukhov length (2), except that the exponents might be found to differ. We will show in section 4 that there is a simple rela- tionship between RxL depth, Obukhov length, and ML depth. c. Revised RxL profile equations Based on the arguments above we propose new em- pirical equations for wind and potential temperature pro- files in the RxL (z # zR) and UL (z . zR): M(z) 5 F(z) (6a) MUL u(z) 2 uUL 5 1 2 F(z), (6b) u 2 uskin UL where F(z) is a function of the form A D D z z exp A 1 2 for z # zR51 2 6 5 1 2 6[ ]F(z) 5 z zR R 1 for z . z , R (7) and A and D are empirical profile-curvature parameters. Based on CTT, we use skin in (6) instead of 0 fromu u (3). We use two shape parameters (A, D) in (7) instead of only one (A) in (3) because we found the profile shape to depend on both meteorological and surface characteristics. The formulation of (7) is designed to allow separation of these two effects, with D depending on surface characteristics and A on meteorological char- acteristics. The net result is that the exponential portion of (7) contributes more than it did in (3), relative to the ‘‘power law’’ portions of those equations. Substituting (5) into (7) yields a new functional equa- tion for the wind and potential temperature profiles D A D(z* ) exp[A(1 2 z* )] in the RxL F(z) 5 (8)51 in the UL where the dimensionless height is B1 z w* z* [ (9)1 2E z u*i and where (zR, A, B, D, E, z*) 5 (ZRM, AM, BM, DM, EM, z *M ) for momentum, and 5 (zRu, Au, Bu, Du, Eu, z*u)for potential temperature. As will be shown in section 6, all heights including the RxL depth should be mea- sured from a datum defined as surface elevation plus displacement distance. That is, over dense forest can- opies and other regions of significant displacement dis- tance zd where the effective aerodynamic surface is above the physical surface, replace all heights z in Eqs. (5)–(9) with zp 2 zd, where zp is height above the phys- ical surface. These revised profile equations [(6), (8), (9)] iden- tically satisfy the zero-gradient condition at the top of the RxL for any values of the empirical parameters. The resulting mean profiles and vertical gradients in the RxL are continuous, and smoothly merge into the overlying UL. Also, the new profile equation is not an explicit 1 JUNE 2001 1451S A N T O S O A N D S T U L L function of RxL depth, thereby eliminating a problem area of the previous parameterization. As will be shown later, an iterative-graphical procedure can be used to simultaneously solve for those B, E, and zR values that allow the field-experiment data to collapse about a com- mon profile curve with minimum scatter. The equations for the RxL depth (5) and the dimen- sionless similarity equations for mean profiles [(6), (8), (9)] are the basis for the RxL theory as used in the remainder of this paper. d. Comparison with traditional power laws The traditional surface-layer power-law (SLPL) wind profile is (M/M10) 5 (z/z10)p, where M10 is the wind speed at reference height z10 (z10 5 10 m for standard surface observations), and p is the power (Pasquill and Smith 1983; Panofsky and Dutton 1984; Arya 1988). The empirical parameters are M10, z10, and p. This profile equation and its variations have been used extensively in fluid mechanics as well as in air pollution meteorol- ogy and wind engineering. Based on observed wind profiles, p is usually found to be a function of static stability and surface roughness. Such SLPLs have been found to represent the observed wind profiles over a greater depth than the Monin–Obukhov SL similarity profiles (Arya 1999). However, these SLPLs have not had the strong theoretical underpinning associated with the MO similarity theory. There are four main differences between traditional SLPL and the new RxL profiles. First, the RxL profile includes the product of an exponential function times the power function of height. The exponential contri- bution is required to make the profile become tangent to the UL at the top of the RxL, as is observed (Santoso and Stull 1999a). Second, the RxL reference height is at the base of the UL (in the mid ML), rather than at 10 m in the SL. Thus, the RxL approach incorporates ML forcings into the parameterization. Third, RxL in- cludes surface skin effects, because surface forcings are another important contributor to profile processes. Fourth, a limitation of RxL is that it is designed only for the statically unstable boundary layer, while tradi- tional power laws have been designed for a wide range of static stabilities. Finally, both approaches include the effect of aerodynamic roughness, within the parameter p for SLPLs, or within the parameter C * for RxL profiles (Santoso and Stull 1999a). 4. Calibration of empirical parameters The parameters in any similarity theory must be de- termined empirically. As discussed in detail in the first RxL paper (Santoso and Stull 1998a), calibration of RxL parameters requires statistically robust wind and poten- tial temperature data that are consistent and contiguous over a large range of heights from near the surface through the interior of the ML. One cannot use data that have profile gaps and mismatches, such as are typical of field experiments where instantaneous rawinsonde observations in the middle of the ML are combined with time-averaged observations from instrumented towers in the SL. We use data from three field sites that satisfy the required statistical robustness. Data from one of the sites, Minnesota, is used in this section to find (i.e., calibrate) the RxL empirical parameters of Eqs. (5), (6), (8), and (9). In later sections, we determine the terrain roughness dependence of the parameters using BLX96 field data, and finally we compare the parameters with independent data from the Koorin field campaign. Al- though not used in the present study, valid data for testing RxL theory could also come from very tall (.300 m) towers, such as the Boulder Atmospheric Ob- servatory in the USA, and the Cabauw tower in the Netherlands. a. The Minnesota site The Minnesota site was located at 488349N, and 958519W with elevation of 255 m above mean sea level (MSL). It was a flat, recently harvested, and plowed farm ‘‘square-mile section’’ (1.609 km on each side) with no aerodynamically significant vegetation close by. A uniform fetch of 10 km existed to the north, which was the predominant wind direction. Details about the Minnesota field experiment, including a description of the site, instruments, and experimental procedures, can be found in Izumi and Caughey (1976) and Kaimal et al. (1976). This ideal site is virtually horizontally ho- mogeneous. Profile robustness was achieved in the Minnesota field experiment by locating sensors at fixed heights on both a 32-m tower and along the tether of a tethered balloon and then averaging over 75-min at each height. The resulting profiles were statistically consistent, smooth, and contiguous. There were 11 runs that were obtained during the Minnesota experiments. Table A1 in appen- dix A lists key dates, times, and the scales from the Minnesota experiment that are important for the RxL analysis. There were no measurements of surface skin tem- peratures during the Minnesota experiment. However, we can estimate them indirectly using the improved CTT equation for surface heat flux s as proposed by San-w9u9 toso and Stull (1998b): w9u9 5 w9u9 1 C* w*Du , (10)s 0 H where the heat-flux intercept parameter is 0 5 0.022w9u9 km s21 and the ML transport coefficient for heat is C *H 5 0.0039. One can rearrange (10) to solve for D 5u skin 2 UL, which can be used directly in the denom-u u inator of (6b). 1452 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S FIG. 2. (a, b) Wind speed M and (c, d) potential temperature u profiles for the Minnesota field experiment, plotted in dimensionless form using radix layer similarity theory. (a) and (c) Linear axes to focus on the UL and top of the RxL; (b) and (d) Semilog axes to focus on the middle and bottom of the RxL. TABLE 1. Best-fit radix-layer parameter values for the homogeneous terrain of Minnesota and Koorin, with the 6 errors given as standard deviations. For the heterogeneous terrain of BLX96, the 6 errors are roughly 5 to 10 times larger for wind parameters, and 2 times larger for temperature parameters (see error values given in section 5c). The standard deviation of terrain elevation is sz. Radix parameter (dimensionless) A B D E Potential temperature Wind speed 0.50 (60.01) 0.25 (60.017) 0.75 (60.005) 0.75 (60.007) 0.20 (60.038) DM 5 a 1 b · sz 0.143 (60.036) 0.50 (60.061) For a 5 0.35 (60.10) and b 5 0.018 (60.004) m21. b. Methodology—Separation of parameters Parameters A and D appear only in the profile shape equation (8), while parameters B and E appear in the dimensionless height z * definition (9). This separation of parameters allows one to empirically solve for the best-fit values of B and E first, independent of the as- yet-unknown values of A and D. Such an approach over- comes the sensitivity problem in calculating the RxL depths that plagued the original RxL paper. To get B and E for momentum, first estimate the val- ues of wind speed in the UL. Then, plot the observed data in the form of dimensionless wind (z)/ ULM M against dimensionless height z * . The result will be a cloud of points clustered along a vertical profile. An iterative approach is then used to vary parameters B and E until the cloud of points exhibits the tightest packing. A similar approach using ( (z) 2 UL)/( skin 2 UL)u u u u versus z * allows one to find B and E for temperature. The end result is shown by the data points in Fig. 2 for all 11 runs of the Minnesota experiment (ignore the solid line for now). The figure also includes plots in semi-log form to show more detail near the surface. The resulting best-fit parameter values (Table 1) are EM 5 1/2 (60.061) and BM 5 3/4 (60.007) for wind, and Eu 5 1/7 (60.036) and Bu 5 3/4 (60.005) for potential temperature, with 6 error given as standard deviations as discussed in appendix C. Because B for momentum and temperature are identical and equal to 3/4, we will use the symbol B 5 3/4 without subscript, and will test later whether this value for B can be considered ‘‘uni- versal.’’ Next, knowing constants E and B, the RxL depths for wind and temperature can be calculated using (5) for each Minnesota run. Once the RxL depth is known, then those data points above the RxL depth can be averaged to refine the values of mean characteristics in the UL: UL and UL, for each run. If necessary, these revisedM u values of UL wind and potential temperature can be used to reestimate B and E as described in the previous two paragraphs. The final values are listed in Table A2 in appendix A for the Minnesota field experiment. To get A and D, a nonlinear regression is applied to minimize the sum of squared errors between the re- gression equations (6), (8), and (9) and the data, with the RxL and UL taken together when calculating squared errors. In Fig. 2, this corresponds to finding the solid line that best fits the cloud of data points. This procedure is done separately for wind and temperature. Using all 11 Minnesota runs, one finds that AM 5 1/4 (60.017) and DM 5 1/2(60.06) for the wind profile, and Au 5 1/2(60.01) and Du 5 1/5(60.038) for the potential temperature profile. The lines plotted in Fig. 2 show the resulting best-fit profile equations [(6), (8), (9)]. In Fig. 2 the potential temperature data points are packed around the best-fit curves quite tightly. The wind speed data points also pack tightly around the best-fit 1 JUNE 2001 1453S A N T O S O A N D S T U L L curves, except near the bend in the line where the best- fit curve is slightly to the right of the data in the top half of the RxL. Namely, the RxL similarity winds are too fast by 3% for heights of 0.5 , z *M , 1. This range of heights is roughly 3–50 times the value of | L | , a region where MO similarity theory gives a much poorer fit to the data, as demonstrated by Santoso and Stull (1998a). For the Koorin field experiment data shown later, the best-fit curve is slightly off in the other di- rection (4% too slow) in the top half of the RxL. Further improvements to the curve fit in this region might be possible with different profile equations (8), or better estimates of displacement distance (see section 6). The statistical spreads of the data points from their similarity curves, measured by the standard deviations of nondimensional wind and potential temperature, are sM/MUL 5 0.028 (dimensionless), and 5s(u2u )/(u 2u )UL skin UL 0.013 (dimensionless), respectively. These statistics were used for error propagation calculations in appendix C to estimate the error bounds of the similarity param- eters A, B, D, and E that were presented in Table 1. The Minnesota field experiment data do not allow us to test whether the RxL profiles vary with surface rough- ness length because all the observations were made at one location. Surface-roughness effects will be exam- ined in section 5 using data from the BLX96 experiment, which was conducted over three sites with different land surface characteristics. c. Relationship between RxL depth, ML depth, and Obukhov length Because the Obukhov length and RxL depth are both boundary layer length scales, it is instructive to compare them. Combining the definition for Obukhov length (2) with the definitions for RxL depth (5), and using the best-fit parameters found above, one finds the following relationship between RxL depth and Obukhov length for the statically unstable boundary layer: 3 1/4z 5 E[k(2L)z ] ,R i (11a) where (zR, E) is (zRM, EM) for wind speed, and (zRu, Eu) for potential temperature. The equation above shows that RxL depth is more strongly related to zi than to L. The equation can also be rewritten in nondimensional form zR/zi to show the relationship with the convective boundary layer stability parameter 2zi/L: 21/4 z zR i1/45 E · k 2 . (11b)1 2z Li The Obukhov length is more strongly dependent on the ratio of u * /w * than the RxL depth. Similar to Obu- khov length, the RxL depths are indirectly related to surface roughness length z0 via the friction velocity u*.Table A2 in appendix A includes the RxL depth, the Obukhov length L, and 0.1zi for comparison of the var- ious length scales. Plots of Eq. (11b) for each experi- mental run are given in Fig. A1 in appendix A, to il- lustrate the robustness of this relationship. 5. Dependence on terrain roughness To determine whether surface conditions such as aero- dynamic roughness z0 or topography influence the RxL profiles and depths, data from Boundary-Layer Exper- iment 1996 (BLX96) is analyzed next. Components of the BLX96 experiment were designed specifically to examine this issue. a. Site characteristics During BLX96 the University of Wyoming King Air aircraft instrumented with boundary layer/turbulence sensors was flown over three different tracks in Oklahoma and Kansas having different land use and surface roughness. The three flight tracks were named after nearby villages: Lamont and Meeker (in Oklahoma) and Winfield (in Kansas). Relatively flat to- pography, large areas of uniform land use, and frequent fair weather motivated the selection of the field site. A total of 12 good flights were made between 15 July and 13 August 1996. The Lamont flight track was predominantly over crop land. Vegetation coverage consisted of 60%–80% wheat fields, 40%–20% pasture, and a small number of trees less than 10 m tall. Roughly 40% of the cultivated fields were recently plowed at the time of the experiment, leaving reddish-brown bare soil. Terrain under the track was quite flat, but gently rising to the west, with ele- vations ranging from 320 to 425 m MSL. An estimate of average aerodynamic roughness based on averaged direct calculations using Monin-Obukhov profile equa- tions and tables of roughness classification (Smedman- Högström and Högström 1978; Stull 1988; Wieringa 1980, 1986) was z0 5 0.1 m. The Meeker track was over a region mainly covered by forest. Vegetation coverage consisted of 40%–50% pasture, 40%–60% wooded areas with trees less than 10-m tall, and 10%–30% cropland. The track had some small rolling hills that ranged from 40 to 60 m in height. The remaining terrain was relatively flat, with elevations from east to west ranging from 250 to 280 m MSL. The average aerodynamic roughness length was z0 5 1.4 m. The Winfield track was primarily over pasture land. Vegetation coverage consisted of 30%–60% pasture, 10%–50% forested areas, with trees less than 10 m tall and more wooded at west end, and the remaining area was cultivated. Terrain was rising to the west, with el- evations ranging from 250 to 400 m MSL. Small hills near the center of the track ranged from 70 to 100 m in height. The average aerodynamic roughness length was z0 5 0.9 m. A more detailed description of the goals, sites, in- struments, and experimental procedures of BLX96 can 1454 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S FIG. 3. Example of (a) wind speed M and (b) potential temperature u data for one flight leg (SS) during BLX96 at the Meeker site on 2 Aug 1996, obtained from aircraft slant ascent–descent flights, and sorted and averaged into nonoverlapping vertical bins of 2-m depth. Each plotted data point at bin center is based on an average of at least 50 observations. be found in Stull et al. (1997) and Berg et al. (1997). Tables of BLX96 surface fluxes, boundary layer depth, and scaling parameters are given by Santoso and Stull (1998b). b. Field experiment procedure and instrumentation To investigate the RxL, a vertical zigzag flight pattern was flown over a 70-km horizontal distance to provide vertical profiles of horizontal mean wind, temperature, and humidity between altitudes of about 10 and 700 m above ground level (AGL). This spans the SL, the RxL, and the bottom of the UL. This zigzag flight pattern was selected as being the most likely to give statistically robust data in a heterogeneous, nonstationary boundary layer, based on earlier trials made by flying a virtual aircraft through synthetic boundary-layer turbulence (Santoso and Stull 1999b). To get vertical profiles, the data are sorted by altitude into nonoverlapping bins of 2-m vertical depth. The average value within each bin is assigned to a height at the bin center. The resulting wind and potential tem- perature data show some scatter, but the large number of data points helps improve the statistical robustness of the sample. As an example, Fig. 3 shows plots of mean wind and potential temperature profiles for a sin- gle zigzag flight labeled as leg SS, obtained at the Meek- er site on 2 August 1996. Each plotted data point at bin center is based on an average of at least 50 observations. The lowest safe altitude that was reached during the flights was dependent on terrain conditions and obstruc- tions near the surface. For the Lamont track, which was relatively flat with bare soil, most vertical zigzag flights reached altitudes lower than 10 m AGL. For the Meeker and Winfield tracks, which had more complicated terrain conditions and obstructions, the lowest altitudes were a bit higher than 10 m AGL, which was just above canopy heights. A downward-looking Heiman pyranometer on the air- craft was used to get surface skin temperatures (Tskin) averaged along the flight track during zigzag legs. An adjustment of 0.1 K per 100 m between aircraft and surface was added to the measured Tskin value to com- pensate for partial absorption of surface radiation by moisture in the atmosphere, and contamination by at- mospheric reemission at a lower air temperature (Perry and Moran 1994; Greischar and Stull 1999). Because the other potential temperatures were defined with re- spect to a reference pressure of 100 kPa, Tskin was also converted to a potential temperature (uskin) for the same reference pressure. To find the ML depth zi, higher ascent/descent sound- ings were flown at the beginning, middle, and end of each of the 4-h flights. These were interpolated to the midtimes of the zigzag flights to account for the ML nonstationarity. Heat flux, buoyancy flux, and Deardorff velocity were calculated iteratively using (1), (10), and the following buoyancy flux equation ø (1w9u9 w9u9y s s 1 0.61 r) 1 0.61 u · where r is mixing ratio andw9r9 ,s s is surface kinematic moisture flux.w9 r9 Momentum flux magnitude (equal to the square of the friction velocity u * ) was estimated using CTT: 2u 5 C w M ,D UL* * * (12) where the ML transport coefficients for momentum flux C *D at Lamont, Winfield, and Meeker are 0.019, 0.028, and 0.040, respectively, as reported by Santoso and Stull (1998b) for BLX96. These coefficients were found to be dependent on surface roughness length. Table B1 in appendix B lists flight tracks, dates, midtimes, and the boundary layer scales for all the vertical zigzag legs. c. RxL parameters for BLX96 Using information listed in Table B1 in appendix B, the depths of the RxL for wind speed and potential temperature are estimated using (5) for each leg. Be- cause these RxL depths define the location of the bottom of the UL, the mean wind speed and potential temper- ature in the UL are determined next. Finally, these are all used to give dimensionless profiles of mean wind speed and potential temperature: (z)/ UL versus z*MM M and ( (z) 2 UL)/( skin 2 UL) versus z*u. Wind datau u u ufrom the Meeker 28 July leg AA were excluded from the Meeker plot, because wind was so slow (about 1.5 m s21) that UL was of the same order as w*, resultingMin excessive sampling noise. Namely, the mean flow is effectively calm when UL , w*.MThe dimensionless profiles for the remaining 18 cases are plotted in Figs. 4–6 for the Lamont, Meeker, and Winfield sites. The height parameters (z, zRM, and zRu) in these plots have been corrected by the displacement height zd due to pasture, cropland, and forest. As rec- ommended by Garratt (1978), the displacement height 1 JUNE 2001 1455S A N T O S O A N D S T U L L FIG. 4. Vertical profiles of dimensionless (a, b) wind speed and (c, d) potential temperature for the BLX96-Lamont site. Data points are from all seven flights at the Lamont site. The solid line is the RxL profile using the best nonlinear regression parameters for this data, while the dashed line is the RxL profile using parameters from the Minnesota field experiment. For potential temperature, the dashed and solid lines coincide. Again, (a) and (c) are in linear–linear co- ordinates, while (b) and (d) are vertically logarithmic to show more detail in the bottom of the RxL. FIG. 5. Same as in Fig. 4 but for six flights at the BLX96-Meeker site (excluding wind data from flight leg AA of 28 Jul 96, because of nearly calm wind). zd is taken as 64% of the estimated roughness-element height average. The displacement heights for individual vegetation types (trees, pasture, etc.) were weighted by the relative coverage of those types under each flight path footprint to give flight-leg averaged values. The resulting displacement heights for Lamont, Meeker, and Winfield are 0.3, 2.7, and 1.8 m, respectively. Next, the parameters B and E are calculated to de- termine if they are significantly different from those for Minnesota. As shown in Table 1 and appendix C, error bounds on the B and E parameters for the heterogeneous surfaces under the 72-km-long flight tracks of BLX96 are 5 to 10 times greater than for the homogeneous terrain of Minnesota. Within the scatter of these BLX96 datasets, we are unable to discern a significant difference from the Minnesota values. We hypothesize that B and E might be universal constants rather than parameters. Table B2 in appendix B lists calculated RxL depths for wind and potential temperature for each leg of BLX96. Again, we include the traditional length scales of Obu- khov length L and 0.1zi for comparison to the calculated RxL depths. Because parameters B and E for the three BLX96 sites do not vary significantly from those of Minnesota, we wondered if parameters A and D are also constant. The RxL wind profile equations using the Minnesota parameter values of A and D are plotted in Figs. 4–6 as the dashed lines, along with the BLX96 data points. The solid lines in these figures represent the RxL profile using parameters that are the best fit for the BLX96 data using direct nonlinear regression. For potential temper- ature, the best-fit RxL parameters are Au 5 0.50 (60.02), 0.51 (60.02), and 0.49 (60.01) for the Lamont, Meeker, and Winfield BLX96 sites, compared to Au 5 0.50 (60.01) for Minnesota. Similarly, Du 5 0.199 (60.051), 0.198 (60.060), and 0.203 (60.035) for BLX96, com- pared to Du 5 0.20 (60.038) for Minnesota. All of these BLX96 parameters yield profile curves that are virtually identical to those using the Minnesota parameters; hence 1456 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S FIG. 6. Same as in Fig. 4 but for all six flights at the BLX96- Winfield site. FIG. 7. Terrain elevation under the BLX96 flight paths at (a) La- mont, (b) Meeker, and (c) Winfield. Horizontal (approximately east– west) distance x is relative to a datum at the east end of the Lamont track. the dashed line is hidden behind the solid line for the potential temperature figures. Our tentative conclusion based on these two datasets is that Au and Du are con- stant. The scatter of the data points about the best-fit wind and temperature lines for BLX96 have standard deviations of 5 0.095, 0.095, and 0.147 (dimen-sM/MUL sionless) and , and 0.011s 5 0.020, 0.025(u2u )/(u 2u )UL skin UL(dimensionless) for the Lamont, Meeker, and Winfield sites. For wind, the Minnesota parameter values do not pro- vide the best fit to the data for all sites, as exhibited by the difference between dashed and solid lines. Further analysis revealed that these site-to-site profile differ- ences of wind could not be explained by parameter AM. We found AM 5 0.254 (60.062), 0.248 (60.062), and 0.245 (60.017) for the Lamont, Meeker, and Winfield BLX96 sites, compared to AM 5 0.25 (60.017) for Min- nesota. Namely, the best-fit AM value for BLX96 was negligibly different from the best-fit Minnesota value given the error bounds for each experiment, and will be considered a constant here. For this reason, our investigation focused on the var- iation of DM with terrain characteristics. When holding AM constant, the nonlinear regression best-fit values of DM for BLX96 are DM 5 0.62, 0.73, and 0.89 (60.06) for Lamont, Meeker, and Winfield, which are the solid lines plotted in Figs. 4–6. When these values of DM were plotted against aerodynamic roughness length z0, no significant correlation was found (correlation coef- ficient r 5 0.24). When plotted against the displacement height zd, again no significant correlation was found. Based on this limited dataset, we conclude that neither aerodynamic roughness nor displacement height can ex- plain the site-to-site variation of profile curvature (i.e., of parameter DM). So there must be some other char- acteristic that causes DM to vary from site to site. Next we investigated whether larger-scale topograph- ic variability could explain the differences between the sites. Figure 7 shows the plots of surface elevations under the Lamont, Meeker, and Winfield flight tracks, which were flown approximately east–west. By eye, the Lamont surface topography (Fig. 7a) appears relatively smooth and flat and is related to the smallest DM, while Winfield (Fig. 7c) has the roughest topography and is related to the largest DM. These plots suggest that re- solvable topographic characteristics might have caused the variation of DM. To better quantify such a relationship, we analyze the discrete variance (energy) spectrum of the surface to- pography for each site under the flight track. From hor- izontal low-level aircraft flights, measurements of air- craft pressure altitudes and radar altitudes AGL sampled at 1 Hz, are used to estimate surface elevations MSL under the aircraft. Taylor’s hypothesis is used with mean aircraft speed (order of 100 m s21) to convert the re- 1 JUNE 2001 1457S A N T O S O A N D S T U L L FIG. 8. Variation of the wind-profile shape parameter DM with respect to standard deviation of terrain elevation sz for BLX96 (C) and Minnesota (3), and the corresponding linear regression (solid line). sulting Fourier spectrum from frequency (after being demeaned and detrended) to wavenumber. To allow better comparison of the discrete variance (energy) spectrum between sites, we truncate the time series from all sites to be the same length as the shortest time series (still corresponding to roughly 70-km hor- izontal flight distance) for each flight track. We flew many low-level flights over each site following virtually the same track based on GPS navigation. Our results show that the total spectral energy density (e.g., total variance) from flight to flight over the same site varies over only a small percentage (about 6%) of the total variance, supporting our above assumption. We found the standard deviation of topography elevations for the Lamont, Meeker, and Winfield tracks were sz 5 12.9, 16.8, and 30.2 m, respectively. These are plotted as the (C) data points in Fig. 8. Also in Fig. 8 is the corresponding data point (3) for the Minnesota field experiment. This experiment was over a single location rather than being along a flight track, and no terrain spectra were provided in the Min- nesota data book. Therefore, we performed a spectral analysis of current digital elevation data (from the U.S. Geological Survey web site: 1-m contour accuracy, 30-s interval distance) for the Minnesota field site, assuming that topography has not changed significantly since 1973 (in contrast to aerodynamic roughness, which usually does change as vegetation and snow cover varies). We found the standard deviation of topography elevation sz 5 12.2 m. Even though the crop characteristics and recent plowing of neighboring farm fields at Minnesota were similar to those at Lamont (and therefore might be expected to have nearly equal values of aerodynamic roughness length), the Minnesota site has slightly smoother resolvable surface topography. Thus, the Min- nesota site has smaller standard deviation of elevation than any the BLX96 sites, and indeed corresponds with the smaller best-fit value of DM. The variations of parameter DM were found to be correlated (with correlation coefficient r 5 0.92) to the standard deviation topography elevation sz: D 5 a 1 bs ,M z (13) where a 5 0.35 (60.10) is dimensionless and b 5 0.018 (60.004) m21. This relation is shown in Fig. 8 as the straight line. The conclusion is that rougher terrain-el- evation variations cause greater curvature in the wind speed profile, as indicated by large values of DM. Unlike the ML transport coefficient C *D for mo- mentum fluxes that was found to be dependent on sur- face roughness length z0 (Santoso and Stull 1998b), the curvature parameter DM for RxL wind profiles is de- pendent on standard deviation sz of surface topogra- phy. These different dependencies are probably due to differences in their measurement heights. The mo- mentum fluxes were measured close to the ground sur- face, therefore, local surface elements and obstacles that create the aerodynamic roughness have more in- fluence on these near-surface (z 5 tens of meters) mea- surements. For RxL wind profiles that span heights of order hundreds of meters, the local surface elements or obstacles are felt only by the near-surface part of the RxL wind profiles. The higher part of the wind profiles is not influenced by local surface elements, but are more likely influenced by much larger footprint area (e.g., surface topography). Thus, in hindsight it is not surprising we found that the RxL wind profiles are dependent on standard deviation of surface topog- raphy via the curvature parameter DM. An obvious question is why the RxL momentum pro- file apparently depends on terrain characteristics, while the temperature profile does not. One explanation is that the wind profile has surface skin values that are always zero, regardless of wind speed in the UL. Thus, the magnitude of wind shear and of surface stress must always reflect the frictional drag at the surface. Contrast this with potential temperature, where the surface skin temperature can increase as the UL layer warms during the day, in order to maintain the temperature difference necessary to drive surface heat fluxes sufficient to re- spond to solar forcings. A similar explanation also ap- plies to CTT (Stull 1994), which was also found to have heat flux parameters independent of surface roughness, but momentum flux parameters that do depend on rough- ness (Santoso and Stull 1998b). 6. Comparison with Koorin data From Fig. 8 in the previous section, data from both Minnesota and BLX96 were used to find a relationship between the standard deviation of resolvable terrain roughness and the parameter DM. The other RxL pro- file parameters were suggested to be constants, based on those same field experiments. We will now com- pare these parameters in the RxL profile equations 1458 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S FIG. 9. Same as in Fig. 4 but for the Koorin site. (a) and (b) Wind speed is using DM 5 0.59 [based on Eq. (13)] and using the aero- dynamic displacement distance. (c) and (d) Potential temperature us- ing ground-level skin average temperature and zero displacement distance. against independent data from the Koorin field ex- periment. The Koorin experiment was conducted in northern Australia at a site with with small hills that ranged from 50 to 70 m in height [see Figs. 1.1 and 5.3 in Clarke and Brook (1979)]. Within 50-km radius from the ob- servation site the surface topography was quite similar to Winfield in general. Vegetation coverage in that re- gion consisted of a forest of well-spaced and uniformly distributed eucalyptus and acacia trees of height 5–10 m, with sparse grass beneath. The vegetation coverage over the Koorin site might be quite similar to that over Meeker. The roughness length at Koorin was z0 5 0.4 m and the displacement distance was zd 5 5.1 m. Based on the resolvable terrain roughness (calculated from a 1:100 000 scale topographic map, with contour interval of 20 m), the standard deviation of terrain elevation is approximately sz 5 13.0 m. Measurements were taken at Koorin between 15 July and 13 August 1974, during the austral winter. Vertical fluxes of momentum, heat, and moisture were deter- mined using the eddy correlation technique. Mean wind speeds were measured at heights of 11.55, 15.65, 22.15, 32.15, and 48.65 m on a tower, and temperature sensors were 0.63 m lower. All were above canopy-top heights. There were 118 experimental runs made during 30 days of convective conditions, giving a total of nearly 600 wind-profile data points. For temperature, there were only 23 experimental runs that can be used because of missing surface skin temperature measurements on the other days. In addition to the tower measurements, there were less-frequent radiosonde soundings to greater height. Observation periods for the mean winds, heat, mois- ture, and momentum fluxes were identical, but the av- eraging time for mean wind speed was an hour while for the fluxes it was half an hour. We will assume here that the measured fluxes are representative of 1-h fluxes. The long averaging time of the tower data in Koorin was fortuitous, because the resulting wind profiles have reduced sampling error. The ML depths zi were esti- mated from soundings, and were used in the calculation of the Deardorff velocity. RxL depths are then estimated using (5). Unfortunately the radiosonde soundings cannot be used to give UL characteristics, because the soundings suffer from the lack of statistical robustness described earlier (the difficulty of merging instantaneous sound- ings with time-average SL and RxL winds). Instead, UL and UL must be inferred from RxL profiles. Equa-M u tions (6), (8), and (9) can be used to solve for UL andM UL values for each profile. Because the data are usedu to help determine one of the parameters ( UL and UL)M u in each profile equation, it means that we cannot in- dependently test the quality of the proposed RxL profile equations and parameters. This is a very unfortunate limitation of the Koorin dataset. However, the available data do allow us to compare the shape or curvature of the profile equations [(6), (8), (9)] to the shape of the plotted data, because these shapes are not affected by the value of wind speed and potential temperature in the UL. The RxL wind parameter values are taken as constants as previously discussed: AM 5 1/4, BM 5 3/4, and EM 5 1/2, except that DM 5 0.59 was found from Fig. 8 [or Eq. (13)] based on the resolvable terrain roughness of sz 5 13.0 m at Koorin. The resulting RxL profile curves for wind are plotted in Figs. 9a and 9b, and fit the data quite well. For potential temperature, it is not obvious what is the appropriate displacement height and what are rep- resentative skin temperatures. From photographs in the Koorin data book (Clarke and Brook 1979), it appears that the trees were sparsely leaved, and had large dis- tances between neighboring trees, as would be ex- pected in a semiarid region. Those photographs show most of the sunlight reaching the ground and the sparse grass, with relatively little intercepted by the trees. This 1 JUNE 2001 1459S A N T O S O A N D S T U L L resulted in substantial differences between radiometric ‘‘skin’’ temperatures measured on tree leaves, grass, and bare soil (see plots and discussion by Stull 1994). In fact, leaf skin temperatures were cooler than the air temperature just above, in spite of the fact that there was vigorous convection associated with strong solar heating. Since this implies that most of the thermals were rising from the ground rather that from the sparse canopy top, it would be appropriate to set the tem- perature displacement distance to zero, and to use an area-average uskin that ‘‘sees’’ the ground and grass, which is listed in the Koorin data book as the ‘‘black ball’’ method. The result of this approach is plotted in Figs. 9c and 9d, using the previously defined profile constants of Au 5 1/2, Bu 5 3/4, Du 5 1/5, and Eu 5 1/7. Given the limitations of the comparison described above, at best we can conclude that the shape of the RxL wind and potential temperature profile equations agrees well with the shape of the data when the pre- viously proposed ‘‘universal’’ constants are used. Also, the data collapse tightly into a single similarity curve after properly accounting for displacement distance. 7. Flux–profile relationships The RxL profile equations are a function of the mean difference of wind speed or potential temperature be- tween the surface skin and the UL. Similarly CTT (Stull 1994) gives surface fluxes as a function of the same wind or temperature difference. By combining both the- ories, one can relate fluxes to profiles, thereby giving new flux–profile relationships for the RxL. Substituting CTT equations for momentum (12) and heat (10) fluxes into (6) yields flux–profile relationships for mean wind speed and potential temperature in the RxL: 2u* M(z) 5 F(z, z , A, D) (14a)RC* w*D (w9u9 2 w9u9 )s 0u(z) 2 u 5UL C* w*H 3 [1 2 F(z, z , A, D)], (14b)R where F has been defined by (8) and (9), and where the appropriate A or D parameters for wind or temperature must be used. Also, z and zR should be interpreted as distances above the displacement height, when the equa- tions are applied to forest or urban canopy regions. The flux–profile relationship for wind speed (14a) is dependent on a wide range of scales of terrain rough- ness. First, the ML transport coefficient for momentum flux C *D depends on small-scale roughness elements that affect the aerodynamic roughness length z0. Second, parameter DM depends on resolvable-scale topographic variations that affect the standard deviation of terrain elevation sz. Such dependence over the wide range of scales should be expected because the RxL profile equa- tions were designed and calibrated as the average over a heterogeneous region (e.g., by using 72-km flight legs in BLX96), rather than being for one column over a single land use. Equation (14) is easy to use to diagnose wind and temperature profiles given measurements of surface heat flux, momentum flux, and ML depth. However, it is more difficult to go in the other direction: to get the fluxes from the profiles. The reason is that the profile function F depends on the RxL depth zR, which itself is a function of the surface fluxes. For comparison, sim- ilar difficulties are encountered when trying to get the fluxes from the classical Businger-Dyer–profile rela- tionships for the unstable surface layer, which are of the form u* z 2 d M 5 ln5 1 2k z0 1/41 1 z 2 d 2 z02 2 ln 1 1 2 151 1 22[ ]2 2 L 1/21 1 z 2 d 2 z02 ln 1 1 2 151 1 22[ ]2 2 L 1/4 z 2 d 2 z p0211 2 tan 1 2 15 2 ,1 1 22 6[ ]L 2 (15) where the fluxes are hidden in each term containing Obukhov length L. Nonetheless, there are many non- linear regression packages available (Press et al. 1992) that can easily solve (14) for the fluxes, given obser- vations of the mean wind and potential temperature pro- files in the RxL. For situations where the fine vertical structure of the ML and SL are not well resolved, such as using satellite radiance estimates of ML temperature or global climate models with coarse vertical grid spac- ing, the solution of (14) for surface fluxes could be quite a valuable approach. 8. Summary and recommendations The radix layer (RxL) is identified as the region be- tween the surface and the base of the uniform layer (UL), in the bottom of the convective boundary layer. The classical surface layer is the bottom subdomain of the RxL. The depth of the RxL is given by (5), and depends both on mixed layer characteristics such as ML depth, and on surface characteristics such as heat and momentum fluxes. These new depth equations eliminate the uncertainty in RxL depth that was discussed in a previous paper (Santoso and Stull 1998a). Similarity shapes of RxL wind and potential-temper- ature profiles are given by (6), (8), and (9). These equa- tions are designed to give zero wind speed at the ground 1460 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S (or at the displacement height for forest canopies), and to become tangent to the UL wind speed at the top of the RxL. These apply only to free convective conditions where thermals are the dominant turbulence process. They have been calibrated to work in mesoscale (order of 50 km) regions over heterogeneous surfaces. These equations are not applicable in the blending layer and the roughness sublayer, in the immediate vicinity of the individual roughness elements. The RxL depth parameterization includes two param- eters: B and E. The profile equations include two more parameters A and D. Based on the five field sites ex- amined here (Minnesota, BLX96-Lamont, BLX96- Meeker, BLX96-Winfield, Koorin), all but one of the parameters are constant, suggesting that they might be universal. Only the D parameter for wind depends on standard deviation of surface terrain elevation sz, and surprisingly does not depend on traditional aerodynamic roughness length z0. These parameter values are sum- marized in Table 1. For wind speed the terrain elevation parameters are a 5 0.35 (dimensionless) and b 5 0.018 m21. As shown in (11) there is a simple relationship be- tween RxL depth and Obukhov length, although the RxL depth also includes a stronger dependence on ML depth. The RxL depth varies from day to day analogous to the variability of Obukhov length, depending on varying external forcings. The RxL depth for wind is in general greater than that for potential temperature. The param- eterizations show that the ratio zRM:zRu is roughly 7:2. Flux–profile equations for the convective RxL were suggested in (14), which extends over a greater depth than traditional surface-layer Monin-Obukhov flux–pro- file relationships. The dimensionless form of these flux– profile relationships is in terms of the mean profiles of wind and temperature, rather than of their gradients. As was discussed in a previous paper (Santoso and Stull 1998a), nondimensional gradient forms of SL flux–pro- file equations tend to hide or disguise errors and give false pictures of the accuracy of the relationships. Finally, the data from Minnesota and Koorin show much less scatter than the data from BLX96. There are two reasons. One, the Minnesota and Koorin data are measured over single points on the earth’s surface hav- ing relatively homogeneous surroundings, compared to the BLX96 data, which was measured by aircraft over a 72-km mesoscale distance having surface heteroge- neity. Second, the long time averages from multiple sensors at different heights during Minnesota and Koor- in provided more robust statistics than the sequentially sampled heights by aircraft flying ascent–descent zigzag patterns during BLX96. For further tests of the equations presented in the paper, it would be very appropriate for other investigators to analyze meteorological data from very tall instrumented towers such as the Boulder At- mospheric Observatory or Cabauw, which would likely produce statistically robust data. Potential applications for wind speed diagnoses using the RxL equations include air-pollutant transport, wind loading on bridges and other tall structures, and for calculating power output from wind turbines during con- vective conditions. The RxL potential temperature pro- file could be used to help determine chemical reaction rates of air pollutants within the RxL. Both profile equa- tions could be used in global climate models to estimate vertically unresolved effects associated with relative coarse vertical grid spacing, but only during convective conditions. Acknowledgments. This research was funded by the U.S. National Science Foundation (NSF) under Grant ATM-9411467. The University of Wyoming King Air aircraft is also sponsored by NSF. The Canadian Natural Science and Engineering Research Council and the En- vironment Canada Atmospheric Environment Service also provided significant grant support. The U.S. De- partment of Energy is gratefully acknowledged for their Grant DE-FG02-92ER61361, as well as for their data and field support at the Southern Great Plains Atmo- spheric Radiation Measurement program. Josh Hacker provided excellent daily forecasts that were used for flight planning and served as an airborne scientist, and Larry Berg also served as one of the airborne scientists. APPENDIX A Minnesota Measurements FIG. A1. Lines show the theoretical relationship [Eq. (11b)] be- tween radix-layer depth zR (made dimensionless with average mixed layer depth zi) vs convective boundary layer stability parameter zi / L, where L is the surface-layer Obukhov length. Data points represent every run reported in appendixes A and B, with circles for wind, squares for temperature, open for Minnesota, and solid for BLX96. Solid line uses E 5 1/2 for wind; dashed line uses E 5 1/7 for temperature. 1 JUNE 2001 1461S A N T O S O A N D S T U L L TABLE A1. Dates, times, and boundary layer scaling variables for the Minnesota field experiment. Friction velocity is u*, surface kinematic heat flux , Deardorff velocity is w*, mixed layer depth is zi, Obukhov length is L. Central Daylight Time (CDT) 5 UTC 2 5 h.w9u9s Run Time (CDT) Date (1973) u* (m s21) w9u9s (K m s21) w* (m s21) zi (m) 2L (m) 2A1 2A2 3A1 3A2 5A1 6A1 6A2 6B1 7C1 7C2 7D1 12:17–13:32 13:32–14:47 15:10–16:25 16:25–17:40 16:22–17:37 14:01–15:16 15:16–16:31 16:52–18:07 14:15–15:30 15:30–16:45 16:50–18:05 9/10 9/10 9/11 9/11 9/15 9/17 9/17 9/17 9/19 9/19 9/19 0.461 0.454 0.371 0.320 0.194 0.241 0.228 0.265 0.282 0.301 0.249 0.196 0.209 0.186 0.116 0.069 0.210 0.162 0.072 0.221 0.181 0.099 2.00 2.23 2.41 2.06 1.35 2.43 2.21 1.77 1.95 1.89 1.58 1250 1615 2310 2300 1085 2095 2035 2360 1020 1140 1225 38.23 34.10 20.96 21.54 8.16 5.08 5.64 19.87 7.74 11.49 11.92 TABLE A2. Estimates of radix-layer parameters for the Minnesota field experiment: MUL is wind speed in the uniform layer, zRM is radix- layer depth for wind, uUL is potential temperature in the uniform layer, Du is potential temperature difference between the surface skin and the uniform layer, zru is radix-layer depth for temperature. Also shown for comparison are the Obukhov length L and 10% of the ML depth zi, which are often used as depth scales for the classical surface layer. Run Wind MUL (m s21) zRM (m) Potential temperature uUL (K) Du (K) zRu (m) Other depth scales 2L (m) 0.1 zi 2A1 2A2 3A1 3A2 5A1 6A1 6A2 6B1 7C1 7C2 7D1 11.7 11.8 9.8 8.9 no UL 7.5 7.6 7.8 6.8 7.0 6.2 208 245 284 284 no UL 185 185 284 120 144 153 295.9 296.7 296.2 296.3 285.5 292.5 293.0 293.5 284.3 285.0 285.3 22.1 21.3 17.3 11.6 8.8 19.6 16.0 7.1 25.9 21.4 10.8 59.4 69.9 81.1 81.3 36.2 52.8 53.0 81.1 34.2 41.0 43.8 38.2 34.1 21.0 21.5 8.2 5.1 5.6 19.9 7.7 11.5 11.9 125 162 231 230 109 210 204 236 102 114 123 APPENDIX B BLX96 Measurements TABLE B1. The same as Table A1 but for the BLX96 field experiment. Track Leg Date (1996) Midtime (CDT) m (m s21) w9u9s (K m s21) w* (m s21) zi (m) 2L (m) Lamont AA SS AA SS AA AA SS 7/23 7/23 7/27 7/27 8/4 8/13 8/13 13:51 14:66 12:78 14:54 13:93 14:68 15:20 0.309 0.310 0.280 0.298 0.519 0.410 0.406 0.086 0.068 0.049 0.054 0.032 0.079 0.095 1.484 1.460 1.108 1.308 1.070 1.556 1.644 1010 1194 680 984 903 1323 1325 22.8 28.6 27.5 29.1 257.8 60.4 49.7 Meeker AA SS AA SS AA SS 7/16 7/16 7/28 7/28 8/2 8/2 12:57 14:27 13:15 14:91 12:78 14:55 0.609 0.678 0.284 0.429 0.382 0.423 0.052 0.060 0.041 0.040 0.038 0.039 1.251 1.408 1.286 1.504 1.151 1.253 1074 1379 1119 1794 867 1090 309.6 385.3 30.1 104.0 79.1 104.8 Winfield AA SS AA SS AA SS 7/15 7/15 7/25 7/25 7/31 7/31 12:01 13:79 12:89 14:69 13:03 14:79 0.452 0.532 0.461 0.460 0.433 0.504 0.094 0.107 0.127 0.113 0.126 0.126 1.653 1.949 2.005 2.124 1.878 2.054 1379 1965 1781 2382 1506 1952 70.7 99.7 54.0 60.6 46.1 72.3 1462 VOLUME 58J O U R N A L O F A T M O S P H E R I C S C I E N C E S TABLE B2. The same as Table A2 but for the BLX96 field experiment. Site abbreviations are Lmt 5 Lamont, Mkr 5 Meeker, and Wfd 5 Winfield. Track Leg Date (1996) Wind MUL (m s21) zRM (m) Potential temperature uUL (K) Du (K) zRu (m) Other depths 2L (m) 0.1 zi Lmt AA SS AA SS AA AA SS 7/23 7/23 7/27 7/27 8/4 8/13 8/13 3.4 3.4 3.7 3.6 13.2 5.6 5.2 156 187 121 162 262 243 232 303.0 303.4 299.3 301.0 307.2 301.7 301.8 11.0 8.1 6.2 6.2 2.3 9.4 11.2 44.5 53.3 34.6 46.4 75.0 69.8 66.3 22.8 28.6 27.5 29.1 257.8 60.4 49.7 100 119 68 98 90 132 133 Mkr AA SS AA SS AA SS 7/16 7/16 7/28 7/28 8/2 8/2 7.4 8.2 1.6 3.1 3.2 3.6 313 399 180 350 189 241 303.4 304.6 303.0 304.4 302.6 304.2 6.2 6.9 3.8 3.0 3.6 3.4 89.4 113.9 51.5 100.0 54.1 69.0 309.6 385.3 30.1 104.0 79.1 104.8 107 138 112 179 87 109 Wfd AA SS AA SS AA SS 7/15 7/15 7/25 7/25 7/31 7/31 4.5 5.2 3.8 3.6 3.6 4.5 261 371 296 378 250 341 302.3 303.4 302.7 303.9 301.9 303.7 11.0 11.0 13.3 10.9 14.0 12.9 74.6 106.0 84.5 108.1 71.6 97.3 70.7 99.7 54.0 60.6 46.1 72.3 138 197 178 238 151 195 TABLE C1. Propagation of error (as a standard deviation s) into the dimensionless profile parameters A, B, D, and E from errors in dimensionless wind or temperature observations sobs, where subscript obs 5 M/MUL for wind and obs 5 (u 2 uUL)/(uskin 2 uUL) for tem- perature. Observa- tion error sobs Radix parameter value errors sA sB sD sE 0.01 0.05 0.10 0.15 0.007 0.029 0.065 0.072 0.005 0.009 0.020 0.028 0.033 0.105 0.164 0.253 0.031 0.098 0.152 0.236 APPENDIX C Error Propagation Analysis For this analysis, we assume that the greatest uncer- tainties (i.e., variances) are associated with nondimen- sional profile data [ (z)/ UL and ( (z) 2 UL)/( skin 2M M u u u UL)], and not in height (z), ML depth (zi), surface fric-u tion velocity (u * ), and Deardorff velocity (w * ). Follow- ing Bevington (1969), the square of the probable error in a parameter p depends on the uncertainties (var-2(s )p iance) in nondimensional data by2(s )c 2N ]p 2 2s 5 s , (C1)Op c 1 2]cj51 j where c is the nondimensional wind or potential tem- perature and p is a parameter such as A, B, D, or E. The partial derivative ]p/]c is simply (]c/]p)21. Unfortu- nately, one cannot use this analytical method directly because the partial derivatives of the parameters A, B, D, and E with respect to the nondimensional profile variables contain a singularity (divide by zero) at the top of the RxL and within the UL. These subdomains cannot be neglected, because the best-fit parameters (A, B, D, and E) were obtained by fitting the whole profile equation to data both below and above the top of the RxL, specifically for the purpose of reducing sampling uncertainty. To overcome this problem, synthetic data were cre- ated by introducing random perturbation errors into the nondimensional ideal profiles. Four experiments are conducted, where the random errors are set to yield standard deviations of sc 5 0.01, 0.05, 0.10, and 0.15 (dimensionless). These error standard deviations span the range of observed scatter in the data points for the nondimensional Minnesota and BLX96 profiles, as re- ported in the body of this paper. For each experiment, 10 different trials (i.e., different random perturbations from the ideal profiles) are used. For each trial of each experiment, nonlinear regression is used to find the best-fit parameters to the perturbed profiles. From the set of 10 values for each parameter for each experiment, the standard deviation of each pa- rameter is computed as a measure of the parameter sen- sitivity to scatter in the data points. The results of sen- sitivity analyses for the parameters A, B, D, and E are shown in Table C1 and Fig. C1. It can be seen that the uncertainties in the parameters A, B, D, and E increase as the uncertainties in (z)/ UL or (u(z) 2 uUL)/(uskinM M 2 uUL) increase. Finally, we found that parameters A and D vary op- positely in a special way. If A is fixed, then the uncer- tainty in D is relatively small, as given above. If D is fixed, then the uncertainty in A is relatively small, as given above. However, if both are allowed to vary, one finds a locus of sets of A and D, which give profile 1 JUNE 2001 1463S A N T O S O A N D S T U L L FIG. C1. Propagation of errors (as a standard deviation s) into the dimensionless RxL profile parameters (a) A, (b) B, (c) E, and (d) D, from errors in dimensionless wind or temperature observations sobs, where subscript obs 5 M/MUL for wind, and obs 5 (u 2 uUL)/(uskin 2 uUL) for temperature. curves with nearly identical goodness of fit. 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