13B.3 1

STRUCTURE OF A DAYTIME CONVECTIVE BOUNDARY LAYER REVEALED BY AVIRTUAL RADAR BASED ON LARGE EDDY SIMULATION

D. E. Scipion1 ,2 ,∗, R. D. Palmer2 , E. Fedorovich2 , P. B. Chilson2 , A. M. Botnick2

1 School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma, USA2 School of Meteorology, University of Oklahoma, Norman, Oklahoma, USA

Abstract

The daytime atmospheric convective boundary layer(CBL) is characterized by strong turbulence that is pri-marily caused by buoyancy forced from the heated un-derlying surface. The present study considers a com-bination of radar and large eddy simulation (LES) tech-niques to characterize the CBL. Data representative ofthe daytime CBL with wind shear have been generatedby LES, and used in the virtual boundary layer radar(BLR) with multiple vertical and off-vertical beams. Amultiple radar experiment (MRE) was conducted usingfive virtual BLRs located equidistant within the LES, andpointing into the same resolution volumes at many al-titudes. Three-dimensional wind fields were retrievedfrom the radar and compared with the LES output. Fi-nally, the three-dimensional wind fields obtained fromthe MRE are used to estimate the turbulent kineticenergy and the turbulent kinematic momentum fluxes.These data are then compared with LES statistics for atime average vertical profile at the center of the domain.

1. INTRODUCTION

Turbulence in the daytime atmospheric convectiveboundary layer (CBL) is primarily forced by heatingof the surface, radiational cooling from clouds at theCBL top, or by both mechanisms. The CBL is consid-ered clear when no clouds are present [Holtslag andDuynkerke, 1998], as in this study. In this case, the mainforcing mechanism in the CBL is heating of the surface.

Turbulent convective motions in the CBL transport heatupward in the form of convective plumes or thermals.These rising motions and associated downdrafts effec-tively mix momentum and potential temperature fields inthe middle portion of the CBL [Zilitinkevich, 1991]. Theresulting mixed layer is typically the thickest sublayerwithin the CBL. The CBL is topped by the entrainmentzone, with relatively large vertical gradients of averaged

∗ Corresponding author address: Danny E. Scipion,University of Oklahoma, School of Meteorology, 120David L. Boren Blvd., Rm 5900, Norman, OK 73072-7307;e-mail: dscipion@ou.edu

(in time or over horizontal planes) meteorological fields.The entrainment zone is often called the interfacial orcapping inversion layer, as it is collocated with the re-gion of maximum gradients in the potential temperatureprofile.

A widely used instrument for the study and monitoringof the lower atmosphere is the boundary layer radar(BLR). The term BLR is generally applied to a class ofpulsed Doppler radar that transmits radio waves verti-cally, or nearly vertically, and receives Bragg backscat-tered signals from refractive index fluctuations of the op-tically clear atmosphere. The operating frequency of thistype of radar is typically near 1 GHz. Therefore, theBragg scale is such that BLRs are sensitive to turbu-lent structures, which have a spatial scales near 15 cm.Enhanced refractive index variations are often associ-ated with the entrainment zone just above the CBL,which can be detected by clear-air radar. There havebeen many studies in which BLRs are used to estimatethe height of the atmospheric boundary layer (ABL)and thickness of the entrainment layer [e.g., Angevineet al., 1994; Angevine, 1999; Cohn and Angevine, 2000;Grimsdell and Angevine, 2002]. Profiles of the windvector directly above the instrument are obtained usingthe Doppler beam swinging (DBS) method [Balsley andGage, 1982]. BLRs are also sensitive to Rayleigh scat-ter from hydrometeors and are used to study clouds andprecipitation [Gage et al., 1994; Ecklund et al., 1995].Thus, the BLR can be used to study the boundary layerunder a wide variety of meteorological conditions, andhas been proven invaluable for such investigations [e.g.,Rogers et al., 1993; Angevine et al., 1994; Wilczak et al.,1996; Dabberdt et al., 2004].

Complementary to field observations of the CBL by in-situ and remote sensing measurement methods, numer-ical simulation approaches – specifically, the large eddysimulation (LES) technique – are widely employed tostudy physical processes in the atmospheric CBL.

The LES method is based on the numerical integra-tion of filtered equations of flow dynamics and ther-modynamics that resolve most of the energy-containingscales of turbulent transport. Any motions that are not

13B.3 2

resolvable are assumed to carry only a small fractionof the total energy of the flow, and are parameterizedwith a subgrid (or subfilter) closure scheme. In the LESof the atmospheric CBL, the environmental parameterssuch as surface heating, stratification, and shear can beprecisely controlled.

In addition to adopting LES and radar methodologiesas is conventionally done, in the present investigation,these two approaches have been synthesized throughthe development and implementation of an enhancedradar simulator based on the work of Muschinski et al.[1999]. High resolution three-dimensional wind andthermodynamic fields representing a clear CBL are gen-erated using LES. These data are then probed using avirtual ultra high frequency (UHF) BLR, similar to thoseconventionally used in numerous atmospheric studiesand field campaigns.

As mentioned, the time-series data for the virtual BLRare generated following the approach developed inMuschinski et al. [1999], which are compiled by sum-ming the contribution from each point within the radarresolution volume. This is determined by the radar pulseand beam width. The virtual BLR output data are thenemployed to estimate CBL characteristics, which arecompared to the “ground-truth” (reference) LES data.Comparison between the components of the velocityvector from the BLR and LES, as well as, turbulent ki-netic energy and turbulent kinematic momentum fluxescomputed over an hour of simulation time, are presentedin this work.

2. ESTIMATION OF THE STRUCTURE FUNCTIONPARAMETER OF REFRACTIVITY FROM THE LES

The LES code employed in the present study hasbeen developed along the lines described in Nieuwstadt[1990]; Fedorovich et al. [2001, 2004]. With respect tomany of its features, the code was specifically designedto simulate CBL-type flows characterized by the pres-ence of large-scale turbulent structures transporting thedominant portion of the kinetic and thermal energy ofthe flow. Fields of atmospheric parameters generatedby LES are used as input fields for the BLR simulator.

The simulation run for the present study has beenperformed in a rectangular domain composed of10 m grid cells. The domain size is X×Y×Z =5120×5120×2000 m3. Correspondingly, there are512×512×200 grid points. The time discretization is1 s. The following external parameters were assignedfor the run; the free-atmosphere horizontal wind was setto 5 m s−1 in the x direction and 0 m s−1 in the y direc-

tion; the free-atmosphere potential temperature gradientwas 0.004 K m−1; the surface kinematic heat flux, sur-face kinematic moisture flux, and the surface roughnesslength were 0.2 K m s−1, 10−4 m s−1, and 0.01 m, re-spectively.

A subset of the LES output was used for theradar simulator. The radar sub-domain size was1060 m ≤ x ≤ 4060 m, 1060 m ≤ y ≤ 4060 m,and 0 m ≤ z ≤ 1600 m. The output included resolved(in the LES sense) three-dimensional fields of potentialtemperature Θ, specific humidity q, flow velocity compo-nents u, v and w, and subgrid turbulence kinetic energyE, as presented in Fig. 1, and summarized in Table 1.See Conzemius and Fedorovich [2006] for additional in-formation about the numerical setup.

Table 1: Description of the LES sub-domain used by theradar simulator

Property SpecificationNumber of grid points 300×300×160Spatial resolution 10 mTime step 1 sDim. of the sub-domain 3000×3000×1600 m3

Variables u, v, w,E, q,Θ

According to the radar equation, the backscattered sig-nal power from a collection of distributed targets is di-rectly proportional to the radar reflectivity η. The valueof η represents the cumulative effect of the radar crosssections for the individual targets. For the case of scatterresulting from turbulent variations in the refractive index(clear-air scatter), the echo power is also related to theradar reflectivity; however, η is now given by the wellestablished theoretical relationship

η = 0.379C2nλ−1/3, (1)

where C2n is the structure function parameter of the re-

fractive index, and λ is the radar wavelength [Tatarskii,1961; Ottersten, 1969]. It is assumed that the turbu-lence is isotropic and in the inertial subrange. Further,the radar resolution volume is assumed to be uniformlyfilled with turbulence.

The structure function parameter of refractivity, C2n, is by

definition derived from the refractive index field using

C2n =

⟨[n(r + δ)− n(r)]2

⟩r

|δ|2/3, (2)

where n is the refractive index and < · >r, as men-tioned in Muschinski et al. [1999]. This denotes the spa-

13B.3 3

y(m)

Ran

ge (

m)

Zonal Wind

−1000 −500 0 500 1000 1500

500

1000

1500

−10

−5

0

5

10

Cross Section at X = 0 m − Time: 6306 s

y(m)

Ran

ge (

m)

Meridional Wind

−1000 −500 0 500 1000 1500

500

1000

1500

−10

−5

0

5

10

Cross Section at X = 0 m − Time: 6306 s

y(m)

Ran

ge (

m)

Vertical Wind

−1000 −500 0 500 1000 1500

500

1000

1500

−5

0

5

Cross Section at X = 0 m − Time: 6306 s

y(m)

Ran

ge (

m)

Potential Temperature

−1000 −500 0 500 1000 1500

500

1000

1500

294

295

296

297

298

299

Cross Section at X = 0 m − Time: 6306 s

y(m)

Ran

ge (

m)

Specific Humidity

−1000 −500 0 500 1000 1500

500

1000

1500

4.5

5

5.5

6x 10−3

Cross Section at X = 0 m − Time: 6306 s

y(m)

Ran

ge (

m)

Subgrid Kinetic Energy

−1000 −500 0 500 1000 1500

500

1000

1500

0

0.05

0.1

0.15

0.2

0.25

Cross Section at X = 0 m − Time: 6306 s

Figure 1: Examples of LES output fields in the sub-domain of the radar simulator. Top-left: zonal wind. Top-right: meridionalwind. Middle-left: vertical velocity. Middle-right: potential temperature. Bottom-left: specific humidity. Bottom-right: subgrid kineticenergy. All data presented refer to the same single realization in time (one LES time step).

13B.3 4

tial average over a volume within which the n irregulari-ties are assumed to be statistically isotropic and homo-geneous. Here, r represents the position vector, and δdenotes the spatial separation.

The refractive index is related to the refractivity, N ,through n = 1 +N ×10−6. If we assume that the back-ground atmospheric pressure profile is hydrostatic, thenthe refractivity is found directly from the simulation out-put parameters through the following equations [Beanand Dutton, 1966; Holton, 2004]:

N =77.6T

(P + 4811

e

T

), (3)

dlnP = − g

RTdz, (4)

T = Θ(P

P0

)0.286

, (5)

e =qP

0.622 + q, (6)

where P is the atmospheric pressure (hPa), e(q, T ) isthe partial pressure of water vapor (hPa), g is the gravi-tational acceleration (9.81 m s−1), R is the gas constantfor dry air (287 J kg −1 K −1), and T is the absolutetemperature (K).

To calculate C2n within each grid cell of the LES domain,

Eq. 2 was applied along a line (beam) connecting theposition of the virtual radar and the center of the grid cellwhere C2

n needed to be estimated (Fig. 2: left). Then,the refractive index n was calculated in the center of thegrid cell at a level z with respect to the ground and attwo levels displaced from z in height by an increment∆z (Fig. 2: right). That is, the points at z, z −∆z, andz+∆z are considered. A bilinear interpolation was thenapplied in order to obtain an estimate of n between theclosest 4 points on the upper and lower planes, therebycalculating values of n along the beam:

nxy = (1− dx)(1− dy)nx0y0 +(dx)(1− dy)nx1y0 + (1− dx)(dy)nx0y1 +(dx)(dy)nx1y1, (7)

dx =x− x0

x1 − x0, dy =

y − y0y1 − y0

,

where nxy represents the refractive index n at point(x, y), which does not match any of the grid points, andnxiyj denotes refractive index values at xi,yj ; i, j =0, 1.

Since the radar is sensitive to variations in the refractiveindex on the order of the Bragg scale (λ/2 ∼16 cm fora frequency of 915 MHz), which is much smaller thanthe LES grid cell size (10 m), one may assume the sim-ulated turbulence to be approximately isotropic on the

scales sensed by the radar; however, the estimation ofC2n needs to be rescaled by λ/2 to accurately repre-

sents the Bragg scale. This modification along with theaverage over two consecutive layers constitute a sig-nificant refinement from the work of Muschinski et al.[1999]:

C2n(x, y, z, t) =

12

[(n1−nδ · λ2

)2(λ2

)2/3 +

(n−n2δ · λ2

)2(λ2

)2/3], (8)

where n1, n, and n2 represents the refractive index atlevels z + ∆z, z, and z −∆z respectively, as depictedin Fig. 2: right, and λ is the wavelength of the radar. Anexample vertical profile of the specific humidity q, fromthe LES output together with the corresponding calcu-lated profiles of refractivity N , and the horizontal aver-aged structure function parameter of refractivity, C2

n arepresented in Fig. 3 for a single realization in time.

3. RADAR SIMULATOR

Various approaches can be found in the literature to beused in generating time-seres data for radar simulations.One that was studied by Sheppard and Larsen [1992];Holdsworth and Reid [1995]; Yu [2000]; Cheong et al.[2004] consists of creating a sampling domain popu-lated with scattering points. These points move withinthe domain according to the field of instantaneous windvector. Another method considers the grid cells of themodel as a scattering center, and the phase of the radarsignal from the scattering center is modulated by the lo-cal instantaneous velocity field [Muschinski et al., 1999].Therefore, by varying the phase, without actually mov-ing the scatterer, the expected Doppler velocity can begenerated. Some implementation advantages exist withsuch a method, especially related to scatterer positionupdate.

The radar simulator for this study was developed fol-lowing the Eulerian frame approach of Muschinski et al.[1999]. The signal amplitude in the simulator after timeτ is proportional to C2

n and inversely proportional to r20 ,which is the range of the center of the sampling volume.The phase difference is proportional to the velocity vec-

13B.3 5

z

z + ∆z

z - ∆z

dx

dx

dy

dy

n1

n

n2

Figure 2: Left: scheme that represents an off-vertical pointing beam. The dotted lines represent the distance from the location ofthe radar to the grid points of the LES within the resolution volume. Along these dotted lines, C2

n is calculated, and later weightedin range (Wr) and beam width (Wb). Right: the dotted line represents the axis along which C2

n needs to be estimated. First thevalue of n at level z is obtained from the LES matrix, however at levels z + ∆z and z −∆z the dotted line does not match anyof the LES grid points. In order to obtain an estimate at those heights, a linear interpolation is made using the four closest points.Finally an average of the two gradients of n (from z and z + ∆z, and z and z −∆z) is computed.

4.8 5 5.2 5.4 5.6 5.8

x 10−3

200

300

400

500

600

700

800

900

1000

1100

1200

Ran

ge(m

.)

Specific Humidity (kg/kg)

270 275 280 285 290 295 300200

300

400

500

600

700

800

900

1000

1100

1200

Ran

ge(m

.)

Refractivity (N)

10−17 10−16 10−15 10−14200

300

400

500

600

700

800

900

1000

1100

1200

Cn2 , θ = 0o, φ = 0o

Ran

ge (

m.)

m−2/3

Figure 3: Left: vertical profile of specific humidity characteristic for the CBL. Center: vertical profile of the refractivity calculatedfrom the LES fields. Right: vertical profile of the horizontally-averaged structure function parameter of refractivity, C2

n. All datawere calculated from a single realization in time.

13B.3 6

tor according to

V (t0 + τ) = A′N∑p=1

√C2n(t0 + τ)(p)W (p)

r W(p)b ·

exp[−j(ϕ(p)0 + kB · r(p) +

k(p)B · v

(p)(t0 + τ)τ)], (9)

A′ =G

λr20

√0.0330k−11/6

B ,

where p represents each individual grid point of the Npoints contained within the resolution volume, G is aconstant proportional to the power transmitted and gainof the transmitter and receiver, ϕ(p)

0 is a random initialphase, kB is the Bragg wavenumber (kB = 4π

λ ), k(p)B

is the Bragg wave vector that is directed from the centerof the antenna to the center of the pth LES grid cell andhas the magnitude of kB , r(p) is the distance from thecenter of the antenna to the center of the cell, v(p) is theinstantaneous radial velocity. Wr represents the rangeweighting function and is described by Holdsworth andReid [1995] as

Wr = exp[

(r − r0)2

2σ2r

], (10)

where r represents the projection of the range of eachgrid point (

√x2 + y2 + z2) over the pointing direction,

r0 is the range of the center of the scattered volume, andthe variance σr = 0.35cτp/2, where c is the speed oflight and τp is the pulse width [Doviak and Zrnic, 1993].Finally, the beam-pattern weighting function (Wb) is de-fined by the following equation [Yu, 2000; Cheong et al.,2004]:

Wb = exp[− (θx − θx)2

2σ2x

− (θy − θy)2

2σ2y

], (11)

θx = tan−1(xz

), θy = tan−1

(yz

),

where θx and θx describe the antenna beam pointing indegrees and σx = σy = θ1/2.36 are proportional to thebeam width (θ1) in degrees. Significant improvementsover the work presented by Muschinski et al. [1999] are:the values ofC2

n, that are calculated over oblique beamsalong the radial, and velocity are now being interpolatedat each time τ , the range weighting function (Wr) is nowincluded as part of the simulator, and a range term (kB ·r(p)) which is important for further multiple frequenciesapplications is now incorporated in the phase term.

Since the LES generates data for variables at a timestep of one second, and the radar inter-pulse period(IPP) is on the order of milliseconds, an estimation ofany of the variables X at t0 + τ is needed. In order

to achieve this, the desired value is calculated using alinear interpolation scheme presented by Cheong et al.[2004]:

X(t0 + τ) = (1− τ)X(t0) + τX(t1), (12)

where t0 and t1 are two consecutive time steps of theLES τ is the intermediate value taken from 0 to 1.

3.1. Radar Setup

The virtual radar in this study is patterned after a VaisalaUHF BLR (LAP3000) operating at a central frequency of915 MHz, with a half-power beam width of 9◦. It is pos-sible to direct the radar beam vertically or, electronicallysteered 23◦ off vertical along 4 different azimuth angles:0◦, 90◦, 180◦, and 270◦. In the present study, differentoff-vertical positions and range resolutions were chosento validate the virtual BLR. The parameters chosen forthe virtual BLR are presented in Table 2.

Table 2: Virtual BLR - Specifications

Quantity ValueFrequency 915 MHzWavelength 32.8 mFull half-power beam width 9◦

Inter-Pulse Period 5 msResolution 50 mBeam inclination variable

In order to produce realistic results from the virtual UHFBLR, it is necessary to introduce additive white noiseto our generated time-series data. First, the maximumpower of the signal is estimated from the time-seriesdata; then, the variance of the background is computedbased on a desired Signal-to-Noise Ratio (SNR). Thecomplex white noise is generated based on the assump-tion that it has a Gaussian distribution described by thepreviously calculated variance. This procedure is inde-pendently applied to the real and imaginary componentsof the time-series data. The complex Gaussian noise issimply added to the original time-series data. An ex-ample of time-series data corresponding to a verticallypointing beam is presented in Fig. 4. The additive com-plex Gaussian noise shown was calculated for an SNRof 55 dB.

13B.3 7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1.5

−1

−0.5

0

0.5

1

1.5x 104

time (s)

V(t

)

Simulated wind profiler signal without noise (r = 475m.)

IQ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1.5

−1

−0.5

0

0.5

1

1.5x 104

time (s)

V(t

)

Simulated wind profiler signal with noise (r = 475m.)

IQ

Figure 4: time-series data from the simulated BLR Data V (t) obtained at a range of 475 m. The solid line represents the realpart (in phase) of V (t) and the dotted line is the imaginary part (quadrature) of V (t). Top: without additive noise. Bottom: withadditive Gaussian noise (SNR = 55 dB).

3.2. Spectral Analysis

A conventional spectral processing procedure was usedto calculate the Doppler moments from the resultingcomplex time-series data (including the additive noise).First, the Doppler spectra were found, then the noiselevel was estimated using the algorithm described inHildebrand and Sekhon [1974]. After the spectral lev-els were reduced to compensate for the estimated noiselevel, the first three moments were estimated (power,mean radial velocity, and spectrum width) followingDoviak and Zrnic [1993].

An example of the spectral analysis is provided in Fig. 5for a virtual radar located at x = 0 m, y = 0 m, andz = 0 m with respect to the center of the LES volume.The simulated radar beam has a width of 9◦ and is di-rected vertically. The range resolution was 50 m, andthe profile shown was calculated with 2 coherent inte-grations and 20 sec of incoherent integration. Overallthe results shown in each of the panels in Fig. 5 com-pare well with actual BLR data. Comparing the meanradial velocities estimated from the virtual BLR and thesimulated “truth” from the LES (single profile pointing tothe same direction), it can be seen that there is goodagreement between the two datasets. Some differences

are to be expected simply due to the disparate sizesof the sampling volumes resulting from a measurementalong a single radial versus a 9◦ beam width. Althoughnot shown here, additional radar simulator results weregenerated using a narrower beam width. Not surpris-ingly, the agreement between the estimated radial veloc-ity and the simulated “truth” from the LES was improved,for this case.

We now consider the estimated signal power from theradar simulator. The center-right panel shows the rangecorrected power from the virtual BLR and the beamweighted C2

n (Wb ·C2n) obtained from the LES. The max-

imum of the range corrected power represents the top ofthe CBL, also known as the inversion layer, and its evo-lution over one hour of the simulation period can be ob-served in the bottom panel. Again, there exists a markedagreement between the simulated radar and LES fields.

4. STATISTICAL ANALYSIS

4.1. Experiment Setup

In order to perform a statistical analysis, a special exper-iment was designed. The so-called multiple radar exper-

13B.3 8

−5 0 5

200

400

600

800

1000

1200

1400

Ran

ge (

m)

Radial Velocity (m/s)

Mean Radial Velocity vs. LES Radial Velocity

−8 −6 −4 −2 0 2 4 6 80

200

400

600

800

1000

1200

1400

1600

Ran

ge (

m)

Radial Velocity (m/s)

pos = (0,0), θ = 0o, φ = 0o, BW = 9o, ∆r = 50m.

75 85 95 105 115

200

400

600

800

1000

1200

1400R

ange

(m

)

Range Corrected Power (dB)

Channel A − time = 9606s

Radial Velocity (m/s)

Ran

ge (

m)

−5 0 50

200

400

600

800

1000

1200

1400

1600

15

20

25

30

35

40

45

50Mean Vel

rLES Vel

r

−17 −16.5 −16 −15.5 −15 −14.5 −14 −13.5 −13

200

400

600

800

1000

1200

1400

log10

(Cn2)

Time (s)

Ran

ge (

m)

Range Corrected Power (dB)

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

85

90

95

100

105

Figure 5: Spectral analysis of the time-series data at 9606 s. Top-left: intensity spectrum with additive white noise. Thecontinuous line represents the mean radial velocity and the error bars the spectrum width. Top-right: comparison between themean radial velocity estimated from the Doppler moments and the true radial velocity from the LES. Center-left: normalizedstacked spectra. Center-right: comparison between the range corrected power estimated form the virtual radar and the beamweighted C2

n from the LES. The peak corresponds to the CBL top (inversion layer). Bottom: Range Corrected Power (dB)estimated form the LES for the whole simulation period.

13B.3 9

iment (MRE) was designed using five different radarslocated equidistant from one another as illustrated inFig. 6. The center radar within the LES domain useda vertically oriented beam. The other four radar beamswere directed off-vertical (20◦) but pointed towards thevertical beam of the first radar, at heights starting at100 m up to 1500 m spaced every 100 m.

A procedure to validate the virtual radars is to comparethe retrieved wind velocity vector from the virtual BLRto the “truth” from the LES. The mean radial velocity forthe five radars was estimated following the proceduredescribed in Section 3. The radial velocity is a functionof the three dimensional wind field components u, v, w,the azimuth angle θ, and zenith angle φ as given by

vr(θ, φ) = usin(θ)sin(φ) + vcos(θ)sin(φ) +wcos(φ). (13)

After combining the 5 radars with Doppler Beam Swing-ing (DBS) techniques the three-dimensional wind com-ponents can be retrieved. The obtained winds at eachheight can be directly compared with the ones from theLES as presented in Fig. 7, as can be observed, there isgood agreement between estimates from the MRE andthe “truth” from the LES.

4.2. Second-order Turbulence Statistics

The retrieved values from the MRE are used to esti-mate the turbulence kinetic energy (TKE), and the turbu-lent momentum fluxes over one hour of the simulation,these values are compared with data from the LES. Theprocedure involves estimating the deviations from themean winds (u′, v′, and w′), which are used to estimatethe variances and covariances used in the calculations.The variances from the LES are composed of two terms(Eq. 14). The first term represents the resolved vari-ance obtained directly from the mean wind components,while the second term represents the subgrid compo-nent, which is calculated from the time-averaged sub-grid kinetic energy (Ei). The total TKE (ETOTAL) isobtained from the half-sum of the variances.

σ2ui

(LES) = u′iu′i +

23Ei, (14)

ETOTAL(LES) =12(σ2u + σ2

v + σ2w

),

σ2uir

(Radar) = u′iru′ir, (15)

ETOTAL(Radar) =12(σ2ur + σ2

vr + σ2wr

),

where u′i and Ei are the i component of the velocityvector and subgrid kinetic energy respectively.

The obtained estimates of TKE are presented in Fig. 8.Both resolved and subgrid components of the TKE fromthe LES are shown. It can be observed that subgrid en-ergy contributes to less than 10% of the variance in theLES case. The variances from the radar calculated over100 m tend to underestimate the results from the LES(calculated over 10 m). The same behavior is observedin the total TKE.

As in the calculation of the turbulent kinetic energy, thevertical components of momentum flux from LES arecomposed of two terms (Eq. 16). The first term rep-resents the co-variance obtained directly from the re-solved wind components, while the second term repre-sents the subgrid component.

Kmi = 0.12∆√Ei,

τxz(LES) = u′w′ +Kmi∂ui∂z

,

τyz(LES) = v′w′ +Kmi∂vi∂z

, (16)

τxz(Radar) = u′rw′r,

τyz(Radar) = v′rw′r, (17)

where τiz represents the component of the vertical kine-matic momentum flux; Kmi the subgrid-scale turbulentexchange coefficient; and ∆ = 3

√∆x×∆y ×∆z is the

effective grid spacing, which in this case is 10 m.

In Fig. 9 the estimates of the vertical kinematic mo-mentum flux components are presented. The compo-nents are presented in separate lines with their resolvedand subgrid components, similar to TKE. As can be ob-served, the subgrid contribution represents less than 2%of the co-variance in the LES case. The flux compo-nents from the radar, even when calculated with 1/10 theLES resolution, generally follows the trend of the LESstatistics.

5. CONCLUSIONS

Based on the work of Muschinski et al. [1999], an en-hanced radar simulator has been presented for stud-ies of the atmospheric convective boundary layer. Thissimulator is capable of producing realistic time-seriesdata for idealized CBLs and has sufficient flexibility fora wide range of virtual radar experiments. For example,one can select the beam direction, beam width, pulsewidth, and simultaneously deploy a large number of vir-tual radars; as in the case of the MRE, which consti-tutes an inexpensive way to test various theories. Goodagreement is observed from the wind retrieval obtained

13B.3 10

A (1000,0)

B (0,1000)

D (0,-1000)

C(-1000,0)E (0,0)

3000 m.

~ 1000 m.

~ 1500 m.

1000 m.XY

Z

45º

34º

Figure 6: Multiple radar experiment Setup. The experiment was conducted using five radars pointing approximately towards thesame resolution volume. Left: location of the radars (four oblique and one vertical pointing beams). Right: geometry of the radarspointing at only two different heights (∼1000 m and ∼1500 m), the idea is to have equidistant spacing (100 m) starting at 100 mup to 1500 m, which will cover almost all the LES vertical sub-domain.

Time (s)

Hei

ght (

m)

Zonal Wind (u) +E − Radar

0 500 1000 1500 2000 2500 3000

500

1000

1500

−10

−5

0

5

10

Time (s)

Hei

ght (

m)

Zonal Wind (u) +E − LES

0 500 1000 1500 2000 2500 3000

200400600800

100012001400

−10

−5

0

5

10

Time (s)

Hei

ght (

m)

Meridional Wind (v) +N − Radar

0 500 1000 1500 2000 2500 3000

500

1000

1500

−5

0

5

Time (s)

Hei

ght (

m)

Meridional Wind (v) +N − LES

0 500 1000 1500 2000 2500 3000

200400600800

100012001400

−5

0

5

Time (s)

Hei

ght (

m)

Vertical Wind (w) − Radar

0 500 1000 1500 2000 2500 3000

500

1000

1500

−5

0

5

Time (s)

Hei

ght (

m)

Vertical Wind (w) − LES

0 500 1000 1500 2000 2500 3000

200400600800

100012001400

−5

0

5

Figure 7: Comparison of the three-dimensional velocity field retrieved from the multiple radar arrangement using DBS (left) tothe field obtained from LES at the center of the subdomain (right).

13B.3 11

0 0.5 1 1.5 2

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

σu2

RadarResolvedSubgridLES

0 0.5 1 1.5 2

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

σv2

RadarResolvedSubgridLES

0 0.5 1 1.5 2

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

σw2

RadarResolvedSubgridLES

0 0.5 1 1.5 2

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

Turbulence Kinetic Energy (TKE)

RadarLES

Figure 8: Variances of the velocity components and total es-timates of TKE. The variances from the LES are being decom-posed in resolved and subgrid components. Even though theradar estimates are calculated at about 1/10 the LES resolu-tion, they follow the tendencies of the LES statistics.

−0.2 −0.1 0 0.1 0.2 0.3

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

X component − Kinematic Flux − u′w′

ResolvedSubgridLESRadar

−0.2 −0.1 0 0.1 0.2 0.3

200

400

600

800

1000

1200

1400

m2s−2

Hei

ght (

m)

Y component − Kinematic Flux − v′w′

ResolvedSubgridLESRadar

Figure 9: Vertical kinematic momentum flux components (Xand Y). The LES fluxes are decomposed in resolved and sub-grid parts. Again the subgrid component is very small, and theradar estimates fairly trail the ones from the LES.

from DBS from the 5 radars at each height with the pointmeasurements of the LES. This agreement is also ob-served in the estimates of the TKE and the vertical com-ponents of the kinematic momentum fluxes.

ACKNOWLEDGMENTS

Support for this work was provided by the National Sci-ence Foundation under Grant No. 0553345.

References

Angevine, W. M., 1999: Entraintment results with advec-tion and case studies from Flatland boundary layerexperiments. J. Geophys. Res., 104, 30947–30963.

Angevine, W. M., A. B. White, and S. K. Avery, 1994:Boundary-layer depth and entrainment zone charac-terization with a boundary-layer profiler. Boundary-Layer Meteor., 68, 375–385.

Balsley, B. B., and K. Gage, 1982: On the use of radarsfor operational profiling. Bull. Amer. Meteor. Soc., 63,1009–1018.

Bean, B. R., and E. J. Dutton, 1966: Radio Meteorology.Vol. 92 of Natl. Bur. Stand. Monogr. Supt. Doc. U.S.Govt. Printing Office, Washington D.C.

Cheong, B. L., M. W. Hoffman, and R. D. Palmer, 2004:Efficient atmospheric simulation for high-resolutionradar imaging application. J. Atmos. Oceanic Tech-nol., 21, 374–378.

Cohn, S. A., and W. M. Angevine, 2000: Boundary levelheight and entrainment zone thickness measured bylidars and wind-profiling radars. J. Appl. Meteorol., 39,1233–1247.

Conzemius, R. J., and E. Fedorovich, 2006: Dynam-ics of sheared convective boundary layer entrain-ment. Part I: Meteorological background and large-eddy simulations. J. Atmos. Sci., 63, 1151–1178.

Dabberdt, W. F., G. L. Frederick, R. M. Hardesty, W. C.Lee, and K. Underwood, 2004: Advances in meteoro-logical instrumentation for air quality and emergencyresponse. Meteor. Atm. Phys., 87, 57–88.

Doviak, R. J., and D. S. Zrnic, 1993: Doppler Radar andWeather Observations. Academic Press, San Diego,CA, second edition.

Ecklund, W. L., K. S. Gage, and C. R. Williams, 1995:Tropical precipitation studies using a 915-MHz windprofiler. Radio Sci., 30, 1055–1064.

13B.3 12

Fedorovich, E., R. Conzemius, I. Esau, F. K. Chow,D. Lewellen, C.-H. Moeng, D. Pino, P. Sullivan, andJ. V.-G. de Arellano, 2004: Entrainment into shearedconvective boundary layers as predicted by differentlarge eddy simulation codes. in Preprints, 16th Symp.on Boundary Layers and Turbulence, Amer. Meteor.Soc., 9-13 August, Portland, Maine, USA, pp. CD–ROM, P4.7.

Fedorovich, E., F. T. M. Nieuwstadt, and R. Kaiser,2001: Numerical and laboratory study of horizontallyevolving convective boundary layer. Part I: Transitionregimes and development of the mixed layer. J. At-mos. Sci., 58, 70–86.

Gage, K. S., C. R. Williams, and W. L. Ecklund, 1994:UHF wind profilers: A new tool for diagnosing andclassifying tropical cloud systems. Bull. Amer. Meteor.Soc., 75, 2289–2294.

Grimsdell, A. W., and W. M. Angevine, 2002: Obser-vation of the afternoon transition of the convectiveboundary layer. J. Appl. Meteorol., 41, 3–11.

Hildebrand, P. H., and R. S. Sekhon, 1974: Objectivedetermination of the noise level in doppler spectra. J.Appl. Meteorol., 13, 808–811.

Holdsworth, D. A., and I. M. Reid, 1995: A simple modelof atmospheric radar backscatter: Description andapplication to the full correlation analysis of spacedantenna data. Radio Sci., 30, 1263–1280.

Holton, J. R., 2004: An Introduction to Dynamic Meteo-rology. Elsevier Academic Press, fourth edition.

Holtslag, A. A. M., and P. G. Duynkerke, 1998: Clearand cloudy boundary layers. in Royal NetherlandsAcademy of Arts and Sciences, Amsterdam, p. 372.

Muschinski, A. P., P. P. Sullivan, R. J. Hill, S. A. Cohn,D. H. Lenschow, and R. J. Doviak, 1999: First synthe-sis of wind-profiler signal on the basis of large-eddysimulation data. Radio Sci., 34, 1437–1459.

Nieuwstadt, F. T. M., 1990: Direct and large-eddy sim-ulation of free convection. in Proc. 9th Internat. HeatTransfer Conference, pp. 37–47.

Ottersten, H., 1969: Mean vertical gradient of potentialrefractive index in turbulent mixing and radar detec-tion of CAT. Radio Sci., 4(12), 1247–1249.

Rogers, R. R., W. L. Ecklund, D. A. Carter, K. S. Gage,and S. A. Ethier, 1993: Research applications of aboundary-layer wind profiler. Bull. Amer. Meteor. Soc.,74(4), 567–580.

Sheppard, E. L., and M. F. Larsen, 1992: Analysis ofmodel simulations of spaced antenna/radar interfer-ometer measurements. Radio Sci., 27, 759–768.

Tatarskii, V. I., 1961: Wave Propagation in a TurbulentMedium. McGraw-Hill, New York.

Wilczak, J. M., E. E. Gossard, W. D. Neff, and W. L.Eberhard, 1996: Ground-based remote sensing of theatmospheric boundary layer: 25 years of progress.Boundary-Layer Meteor., 78, 321–349.

Yu, T.-Y., 2000: Radar studies of the atmosphere usingspatial and frequency diversity. Ph.D. thesis, Univer-sity of Nebraska, Lincoln, Nebraska.

Zilitinkevich, S. S., 1991: Turbulent penetrative convec-tion. in Avebury Technical, Aldershot, p. 179.