16th ams blt 2004 conzemius & fedorovichl

8/2/2019 16th AMS BLT 2004 Conzemius & Fedorovichl

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* Corresponding author address: Robert J Conzemius,School of Meteorology, University of Oklahoma, 100 EastBoyd, Norman, OK 73019-1013;e-mail: [email protected]

5.6 NUMERICAL MODELS OF ENTRAINMENT INTO SHEARED CONVECTIVE BOUNDARY LAYERSEVALUATED THROUGH LARGE EDDY SIMULATIONS

Robert Conzemius* and Evgeni FedorovichSchool of Meteorology, University of Oklahoma, Norman, Oklahoma

1. INTRODUCTION

Atmospheric convective boundary layer (CBL)entrainment is the downward mixing of free atmosphericair into the CBL as the CBL grows. Primarily, the CBLentrainment is driven by the buoyancy production ofturbulence kinetic energy (TKE) due to heating of thelower surface or cooling at the CBL top. The effects ofshear-generated turbulence in the CBL are secondary.However, cases of a purely buoyancy-driven CBL arerare, and in many situations, the flux of buoyancy isrelatively weak, allowing the effects of shear-generatedturbulence to become roughly equivalent to those ofbuoyancy-generated turbulence

Large eddy simulation (LES), which can resolvemost of the energy-containing motions in the CBL, hasbecome a standard tool for fundamental studies of theCBL dynamics. However, the spatial and temporalresolution needed in the LES for an adequatereproduction of the CBL turbulence properties is stillbeyond the limits of numerical models commonlyemployed for weather prediction purposes, as well as inapplied climate and air pollution research. In the presentstudy, we use LES to evaluate predictions ofentrainment into the sheared CBL by numerical modelsof two types: (i) models with turbulence closureschemes based on Reynolds averaging, which resolve

some vertical structure of the CBL and are commonlyapplied in numerical weather prediction (NWP) and (ii)models based on vertically integrated momentum,buoyancy, and TKE balance equations assuming aparameterized CBL vertical structure (the integralbudget approach). The latter approach is widely used topredict integral CBL parameters (e.g., depth ofconvectively mixed layer) in applied atmosphericdispersion studies.

Previously, Moeng and Wyngaard (1989) haveevaluated NWP turbulence closures for parameterizingthe turbulence structure of shear-free CBLs, and Ayotteet al. (1996) have done the same for both sheared andshear-free CBLs.

2. REYNOLDS AVERAGING-BASED TURBULENCECLOSURE SCHEMES

Two turbulence closure schemes, based upon

Reynolds averaging (RANS), such as used in NWP, areevaluated here. The two schemes are both 1.5 order, e-l closure schemes and therefore, both have the samebasic formulations of the TKE equation, eddydiffusivities of heat and momentum, and dissipation.The differences between these two schemes are in thespecification of the length scale used to close theproblem and in constants in the expressions for theeddy diffusivities and TKE dissipation.

In the actual NWP grids used at the time of thiswriting, the grid cell size is becoming small enough thatthe implied horizontal averaging of the non-resolvedturbulence within the grid cell may differ considerablyfrom an ensemble average. As such, the assumptionsused in the RANS-based NWP turbulence closures maynot hold. This is a particularly important problem incontemporary NWP. However, in this evaluation, it isassumed that the grid cell sizes are large enough thatthe implied horizontal averaging can be consideredrepresentative of an ensemble average, and theassumptions of a Reynolds averaging-based schemewill still be valid.

In such a case, the TKE equation has the followingform:

0 0

1,

e U Vwu wv

t x x

gw we wp

z

(1)

where e is the TKE, U and V are the total horizontalcomponents of the flow, u, v, and w are the turbulent

components, p and are the deviations of pressureand potential temperature from their horizontal

averages, g is the acceleration of gravity, 0 is the

reference value of potential temperature, is

dissipation of TKE, and brackets denote averagedquantities.

The turbulent fluxes are parameterized according tothe following hypotheses:

0

12 m

ewe wp K

z, (2)

m

Uwu K

z, (3)

m

Vwv K

z, and (4)

Preprints, 16th Symp. On Boundary Layers and Turbulence,

Amer. Meteor. Soc., 9-13 August, Portland, Maine, USA, CD-

ROM, 5.6.


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hw Kz

, (5)

where mK and hK are the eddy diffusivities of

momentum and heat, respectively. The eddy diffusivityof momentum has the following general form:

1/ 2m KK e l (6)

where K is a constant, and l is a turbulence length

scale. The eddy diffusivity for heat is related to the eddydiffusivity for momentum through the turbulent Prandtlnumber, which generally ranges from one third to oneunder unstable conditions. Dissipation is parameterizedby the expression

3 / 2e

l, (7)

where is a constant. The values of K , , l, andthe turbulent Prandtl number are scheme-dependent.

The first scheme evaluated in this study is that ofXue et al. (2001). The turbulence length scale in thisclosure is dependent on the hydrostatic stability. Instable conditions, the length scale is

1/2 10.76sl e N , (8)

where N is the Brunt-Visl frequency. In neutral orunstable conditions (the boundary layer portion of theflow), the length scale is

0

1 exp 4 /

1.8 0.0003 exp 8 /

i

u i

i

z z

l l z z z (9)

where z is the height, iz is the boundary layer depth,

and 0 0.25l . The boundary layer depth is defined as

the level at which a parcel, lifted from the lowest modelgrid level above ground, becomes neutrally buoyant.

Because ul doesnt become small until iz z , where

the atmosphere is stable, the maximum ofs

l andu

l is

taken. The turbulent Prandtl number is

1

1 2Pr max , 1

3t

v

l, (10)

wherev

is the vertical dimension of grid cell. The

constants in (6) and (7) are 0.1K and 3.9 at

the first grid level above ground and 0.93 at all

other grid points.The scheme of Fiedler and Kong (2003), hereafter

referred to as F&K, does not depend on thedetermination of the CBL depth. Rather, it defines the

length scale as the geometric mean of an upward lengthscale and a downward length scale. The upward anddownward length scales are defined as the verticaldistance an air parcel would travel from its originalheight, working against buoyancy forces, until all its TKEwere expended. Mathematically, these length scalesare defined through the following integrals:

( )

*( ) ( ) ( , )( )

upz z

v v

z v

ge z z z z dz

z, and (11)

( )

*( ) ( ) ( , )( )

downz z

v v

z v

ge z z z z dz

z, (12)

where up and down are the vertical distances of parcel

travel and *v is the virtual potential temperature a

parcel would have if it ascended or descended from its

starting level zto a new level z . Once up and down

are determined, the mixing length is 1/2

up downl .The specifications for the constants are

0.25 2 0.35K , 0.5 , and Pr 1t .

3. CLOSURE SCHEMES BASED ON INTEGRALBUDGETS

To simplify the numerical complexity andcomputational cost and to develop a conceptualunderstanding of CBL growth, the horizontally averagedprofiles of buoyancy, momentum, and TKE (Eq. 13) canbe integrated over the depth of the CBL to form theintegral budgets of those quantities. In order to simplifythe integration, the boundary layer structure is

parameterized. The simplest parameterization is knownas the zero order model (ZOM), developed by Lilly(1968) and others following. Figure 1 shows the ZOMprofiles for buoyancy. In the ZOM, the buoyancy istaken to be constant within the CBL and has a finitediscontinuity at the CBL top, where it jumps to its freeatmospheric value. Above the CBL, the buoyancy is alinear function of height. Integrating the buoyancy

balance equation to some arbitrary leveli

z z , one can

obtain the buoyancy flux, which varies linearly from apositive value at the surface to a negative value justbelow the interface at the CBL top (a fraction of thenegative of the surface value), where it jumps back tozero. Originally, the ZOM was only applied to situationsin which the shear contribution of TKE in the CBL couldbe neglected, but later authors [Zeman and Tennekes(1977), Tennekes and Driedonks (1981), Driedonks(1982), Boers et al. (1984), Fairall (1984), Batchvarovaand Gryning (1990, 1994), Fedorovich (1995), Pino etal. (2003) extended it to CBL types with shearcontribution to the generation of TKE. The momentumprofiles in the ZOM are prescribed in a similar manner tothe buoyancy profile, with a constant value in the mixedlayer, a jump at the top of the CBL, and a linear rate ofchange in the free atmosphere. Fedorovich (1995)


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derived the complete ZOM equations for the CBL withwind shear contributions to the TKE but did not proposesolutions to the equations.

For the shear-free case, analytic solutions to theentrainment equation are possible when certainassumptions are made regarding the profiles of TKEand its dissipation rate. In particular, the Deardorff

(1970) and Zilitinkevich and Deardorff (1974) scalinghypotheses are used, and the TKE and dissipationprofiles in the CBL are assumed to be self-similar.Given such assumptions, the TKE and dissipation, when

scaled by the convective velocity scale 1/ 3

* s iw B z ,

integrate to constants, and the system of equationsdescribing CBL entrainment is closed. Fedorovich et al.(2004) have compared ZOM solutions with LES of theshear-free CBL and found good agreement as long asthe parameters of entrainment are retrieved from LES ina manner that is consistent with their definitions in theZOM.

0

z z

zi

Bs

Bi

0

zizi /2

b

b

bs

db/dz=N2

Bi

zil

ziu

Figure 1. Profiles of buoyancy in the zero order model(ZOM) compared to the actual horizontally averagedprofiles in the CBL.

Similar scaling assumptions regarding the integrals ofTKE and dissipation can be applied to the CBL withshear, except that the appropriate velocity scale may bedifferent from the shear-free case. In the ZOM, theshear generation of TKE within the CBL is zero becausethe momentum is assumed to be well-mixed there. This

means that shear only contributes to the generation ofTKE at the CBL top or at the surface. Most authorsconsider the surface shear and adjust the velocity scale

upward by including the friction velocity *u .

At the CBL top, the shear contribution is modeled interms of the velocity jumps, and the assumption is madethat a certain fraction of the shear-generated energy isavailable for entrainment, and the other portion isdissipated.

The derivations of the entrainment equations arefound in the papers listed above, and the equations areonly listed in their final form here. Some authorspresent their solutions in terms of the CBL growth rate,

/idz dt , and others show the entrainment flux ratio,

which is defined as the negative of the buoyancy flux of

entrainment to the surface buoyancy flux ( /i sB B ). In

terms of the latter quantity, the following general form iscommon to most of the equations:

3

3 2 22*

*1

i

i m F

s s

T P

i i

dzb

B w Cdt

B B w u vwC C

bz bz

(13)

where u and v are velocity jumps across the

entrainment zone and b is the buoyancy jump.

0( / )i iB g w ( /ibdz dt in the ZOM) is the

buoyancy flux of entrainment, 1/3

* s iw B z is theDeardorff (1970) convective velocity scale and FC , TC ,

and PC are constants describing the relative

contributions of the buoyancy generation, spin-up, andshear generation of TKE, respectively. The parameter

FC represents the portion of buoyancy-generated TKE

that is not dissipated before being used for entrainment,

and PC represents the portion ofshear-generated TKE

that is available for entrainment. Finally, the adjusted

velocity scale is 3 3 3* *mw w Au , where *u is the friction

velocity, except Zeman and Tennekes (1977) and

Tennekes and Driedonks (1981) use 2 2 2* *mw w Au .

One limitation of (13) is immediately clear: if theshear across the CBL top becomes sufficiently large,the denominator of the entrainment equation can go tozero, and the entrainment ratio will become infinite.While this result is expected if the surface buoyancy fluxapproaches zero, it is not clear from the expression thatthe right hand side will behave in such a fashion. Someof this has to do with the limitations of the assumptionsthat are made, particularly with the spin-up term. Thespin-up of TKE is scaled only by the convective velocityscale, implying that the shear-generated TKE at theCBL top is not transported into the CBL where it isstored. If the shear is included in the velocity scale,then the spin-up term in the denominator will becomelarger and compensate for a larger shear term.

The entrainment parameterizations of Zeman andTennekes (1977), Tennekes and Driedonks (1981),Boers et al. (1984), and Pino et al. (2003) follow (13)reasonably closely, except that Zeman and Tennekes(1977) were unable to include the entrainment zoneshear term because they did not know what portion ofthe shear-generated TKE is available for entrainment.Table 1 describes the values of the constants that wereused in the entrainment expressions.


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Table 1. Values of Constants in Entrainment EquationsAuthor A FC TC PC

Tennekes(1973)

12.5 0.2 0 0

Zeman andTennekes

(1977)

4.6* 0.5 0.024i

m

Nz

w 3.55 0

Tennekes andDriedonks(1981)

4* 0.6 0.03i

m

Nz

w 4.3 0.7

Driedonks(1982)

25 0.2 0 0

Boers (1984) 23 0.32 0.75 1

Batchvarovaand Gryning(1990,1994)

12.5 0.2 0 0

Pino et al.(2003)

8 0.2 4 0.7

Other studies [Tennekes (1973), Driedonks (1982),Batchvarova and Gryning (1990,1994)] consider onlythe surface shear or parameterize the total shearcontribution. In this case, (13) is reduced to

3 3*/ /i s mB B w w . Batchvarova and Gryning (1990,

1994) use the formula

(1 )

1 1 2i

s

B X X

B X Y X (14)

where 3 3 3 3* * */ (1 / )m FX w w C Au w and

2 2 2* / iY Bu N z , with 8B .

The entrainment equation of Stull (1976a,b)partially follows the ZOM approach but adds a

parameterization of the entrainment zone thickness overwhich the velocity jump occurs. Although his approachis somewhat inconsistent with ZOM methodology, theproblems of the unboundedness of (13) are avoided.Stulls entrainment equation is

2 3*

1 2 33 3* *

mi i i

s

u uB z z uA A A

B w w, (15)

where is the entrainment zone thickness, mu is the

mixed layer velocity, and 1 0.1A , 2 0.05A , and

3 0.001A .

More recently, Sorbjan (2004) developed aparameterization specifically for the heat flux at thesheared CBL top. This parameterization takes intoaccount the Richardson number of the entrainment zoneand therefore requires a finite entrainment zonethickness. In the ZOM, the entrainment zoneRichardson number is zero because the entrainmentzone depth is infinitesimally small while the velocity

jump remains finite. Thus, shear becomes infinitelylarge and although the buoyancy gradient is also large,if one assumes a linear variation of buoyancy and

momentum across the entrainment zone and takes thelimit as the entrainment zone depth goes to zero, onewill find that the Richardson number tends to zero. TheSorbjan parameterization is

22* 1/ 2

1 /

1 1/i H i

c RiB c w N

Ri(16)

with 0.015Hc and 2 1.5c . The Richardson number

is the interfacial Richardson number

2 2/Ri b u v , where the velocity andbuoyancy jumps are interpreted as their changes across

the full layer . This approach is most consistent with

the first order model (FOM) of entrainment (Betts 1974),in which the buoyancy profile is like the ZOM profile inthe mixed layer and the free atmosphere, but the

entrainment zone has a finite thickness , and the

buoyancy and velocity profiles there are linear.Other FOM models [Mahrt and Lenschow (1976),

Kim (2001)] have been considered for this study, but themathematical formulation of the entrainment equation isnot much different from (13) other than being a bit morecomplex. Additionally, the entrainment zone thicknessis an additional dependent variable in the FOMequations that requires additional parameterizations orassumptions to close the problem and presents moreopportunities for the developed equations to haveundesirable mathematical consequences. Fairall (1984)also developed a shear parameterization for CBLentrainment, but the expression was rathercumbersome to evaluate from LES data and did notprovide additional insight beyond what was provided bythe simpler, ZOM-based entrainment equations.

4. EVALUATION METHODOLOGY

The following cases of CBL with wind shears werestudied with LES:

1. No mean shear (NS case), which was thereference case.

2. Height-constant geostrophic wind of the 20 m/smagnitude throughout the whole simulationdomain (GC case).

3. Geostrophic wind with the magnitude that linearlyincreased from zero at the surface to 20 m/s atthe domain top (GS case).

In the GC and GS cases, geostrophic wind had only thelongitudinal (x) component ug, so the ycomponent of thegeostrophic wind, vg, was set equal to zero. For all

simulated cases, the surface roughness length,geographic latitude, and reference temperature wereprescribed to be 0.01 m, 40 N, and 300 K, respectively.

In the initial-flow configurations (see Fig. 2), virtualpotential temperature changed vertically at a constantrate of 0.001, 0.003, or 0.010 K/m throughout the entiredomain starting from the surface. The initial windvelocity in the domain was geostrophic (zero in the NScase), with the vertical velocity component set equal tozero. The surface heat flux had values of 0.03, 0.10, or


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0.30 K m/s and was kept constant with time throughoutthe run.

0

z

NS, GS, GC

N2

=(g/0)(d

/dz)=const

0 20ug(m/s)0

z GS

GC

NS

Figure 2. Initial profiles of the virtual potentialtemperature andxcomponent of the geostrophic windvelocity, ug, for the simulated CBL cases.The LES grid domain was 5.125.121.6 km with gridcells of 20 meters in all dimensions. Considering allpossible combinations of shear, stratification, andsurface heat flux, a total of 24 LES runs wereconducted. The combinations with the strongest

stratification (0.010 K/m) and weakest heat flux (0.03Km/s) were not conducted due to the excessive timenecessary for these cases to run to completion. TheLES were allowed to continue until the CBL depthreached approximately 1000 meters, at which point itwas possible for the entrainment zone to impinge uponthe sponge layer, so the run was stopped.

Turbulence statistics were calculated every 100seconds. The averaging was carried out over thehorizontal planes only in order to avoid uncertaintiesassociated with complementary time averaging. The

CBL depth iz was determined from the minimum of

kinematic heat flux 'w (resolved + subgrid).

The evaluation of the RANS-based NWPturbulence closures was carried out in a similar mannerto Moeng and Wyngaard (1989) and Ayotte et al.(1996). The LES code was reduced to a one-

dimensional column model, and the turbulence closureschemes described in Section 2 were used. The CBLdepth was determined in the same manner as in LES,and the CBL depth versus time determinations werecompared with those of LES.

The evaluation of the various schemes within theintegral budget approach was done by retrieving the

parameters sB , iz , b , u , and v from LES data

and using them in the respective formulations (13), (14),or (15) with the constants indicated in Table 1.Unfortunately, the evaluation of the above parameterswas not very straightforward. The definitions of theparameters of entrainment were dependent on theauthor evaluating them, regardless of whether theevaluation was made based on atmospheric data orLES results. For example, in the ZOM, the buoyancy

jump across the entrainment zone can be evaluated inthe strict sense of the ZOM, in which case the linear freeatmospheric profile of buoyancy is extrapolateddownward from the upper edge of the entrainment zone

to the defined CBL depth iz . However, many authors

took the full change across the entire depth of theentrainment zone. Unfortunately, in most cases, theactual value of the parameterized heat flux at theinversion is highly sensitive to the size of the jumpsused. Additionally, there is usually a difference between

the actual entrainment heat flux, iB (the minimum of

heat flux in the entrainment zone), and the ZOM-

parameterized heat flux, /ibdz dt . Some authors havetuned their parameterizations for the former, and othersfor the latter. To the extent possible, this study tries touse the parameterizations in a manner consistent withthe way they were originally evaluated by theirrespective authors.

Because of the large number of comparisonsperformed, only the most important and representativeresults are shown in the following sections.

5. RESULTS

5.1 RANS-Based Closures in NWP

The cases with a heat flux of 0.03 Km/s and a free

atmospheric stratification of 0.003 K/m arerepresentative of the differences between the variousnumerical models of entrainment and LES. With thisparticular combination of surface heat flux and freeatmospheric stratification, the shear promotes asignificant deepening of the CBL in the GS and GCcases, relative to the shear-free (NS) case. The CBLdepth versus time for the ARPS e-lclosure is comparedwith LES in Fig. 3. The CBL depth for the NS casegrows more slowly with the ARPS e-l model than with


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LES. For the GC case, and particularly in the GS case,the relationship is reversed, and the e-lmodel predictsfaster CBL growth than is seen in LES.

0 10000 20000 30000 40000 50000

Time (s)

0

400

800

1200

zi

(m)

NS, ARPS

NS, LES

GS, ARPS

GS, LES

GC, ARPS

GC, LES

Figure 3. Comparison of the CBL depth ( iz ) versus

time predictions of ARPS e-lclosure versus LES

Figure 4 shows the heat flux profiles in LES versus theARPS e-lclosure. In the LES, the heat flux profiles aregenerally smoother than in the ARPS closure, and it isobvious that, in the cases with shear, the RANS-basedentrainment heat flux is much larger than in LES. Figure5 shows the overall effect on the potential temperatureprofiles. If a different definition of the CBL depth, suchas the height of the maximum temperature gradient,were used, the results would still be the same.

-0.04 -0.02 0 0.02 0.04

w'(Km/s)

0

400

800

1200

z(m)

NS, ARPS

NS, LES

GS, ARPS

GS, LES

GC, ARPS

GC, LES

Figure 4. Comparison of the heat flux profiles of ARPSe-lclosure versus LES at t=10000s.

296 297 298 299

(K)

0

400

800

1200

z(m)

NS, ARPS

NS, LES

GS, ARPS

GS, LES

GC, ARPS

GC, LES

Figure 5. Comparison of the potential temperatureprofiles of ARPS e-lclosure versus LES at t=10000s.

The RANS-based closures predict much faster CBLgrowth compared to LES when shear is present. Figure6 shows the TKE profiles. Surprisingly, the TKE in the

ARPS e-l closure is less than in LES, despite thegreater entrainment of heat. Looking at the profile forthe GS case more carefully, it can be seen that the TKEat the top of the CBL is greater in the ARPS e-lclosurethan in LES. For the GC case, the LES has greaterenergy, but the energy calculations in LES do notdistinguish between truly turbulent motions associatedwith entrainment and wave-like motions that do not mix

heat.

0 0.5 1 1.5 2 2.5

TKE(m2/s2)

0

400

800

1200

z(m)

NS, ARPS

NS, LES

GS, ARPS

GS, LES

GC, ARPS

GC, LES

Figure 6. Comparison of the TKE profiles of ARPS e-lclosure versus LES at t=10000s.


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0 10000 20000 30000 40000 50000Time (s)

0

400

800

1200

z(m)

NS, F&K

NS, LES

GS, F&K

GS, LES

GC, F&K

GC, LES

Figure 7. Comparison of the CBL depth iz predictions

with F&K e-lclosure and LES.

The F&K closure predictions of the CBL depth arecompared with LES data in Fig. 7. Basically, the results,in a relative sense, are similar to those in the ARPScase. The CBL depths agree very closely with LES forthe NS case, but in the GS and GC cases, theentrainment is remarkably faster than in LES.

-0.16 -0.12 -0.08 -0.04 0 0.04

w''(Km/s)

0

400

800

1200

z(m)

NS, F&K

NS, LES

GS, F&K

GS, LES

GC, F&K

GC, LES

Figure 8. Comparison of the heat flux profiles obtainedwith F&K e-lclosure and LES at t=10000s.

The heat flux profiles are shown in Fig. 8. If themaximum temperature gradient height is used for thedefinition of the CBL depth, the results are very similar,as can be seen in Fig. 9. In Fig. 10, the TKE is shown.Unlike the ARPS closure, the F&K closure showsgreater TKE in the cases with shear when compared toLES, and the TKE in the NS case is very comparable to

TKE from the LES. The TKE values in the interior of theCBL are actually rather similar to the LES predictions,but at the top of the CBL (where shear is fairly large),there is much more TKE predicted by F&K than by LES.

296 297 298 299

(K)

0

400

800

1200

z(m)

NS, F&K

NS, LES

GS, F&K

GS, LES

GC, F&K

GC, LES

Figure 9. The potential temperature profiles from F&Ke-land LES at t=10000s.

0 0.5 1 1.5 2 2.5

TKE(m2/s2)

0

400

800

1200

z(m)

Figure 10. The TKE profiles from F&K e-l and LES at

t=10000s.

Finally, the momentum profiles in ARPS, F&K, and LESare compared in Fig. 11. Between the two e-lmodels,the momentum gradients from F&K more closely matchthe LES momentum gradients than do those from

ARPS, but the values of momentum are greater in theF&K data for the GS case because of the enhancedentrainment of momentum. The GC momentum fromF&K is less than predicted by LES momentum for theCBL interior, perhaps due to the enhanced upward


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mixing of weaker momentum from the surface. InARPS, the TKE is much smaller in the lowest model gridlevel because of the greater dissipation there. Thiscauses less frictional slowing of the momentum, andmomentum throughout most of the CBL in ARPS isgreater than the LES momentum for the GC case.

0 4 8 12 16 20

U(m/s)

0

400

800

1200

z(m)

GS, LESGC, LES

GS, ARPS

GC, ARPS

GS, F&K

GC, F&K

Figure 11. The momentum profiles from F&K e-l, ARPS,and LES at t=10000s.

The primary conclusion regarding performance of twoconsidered e-lclosures is that for cases with shear, theypredict faster entrainment than LES does. Thisoverestimation is not as large with ARPS as it is withF&K, but the ARPS-predicted CBL growth in the NScase is also slower than in LES. Another common

feature of the two closures is the larger relativedifference in CBL entrainment between shear-free andsheared CBL cases than is predicted by LES. Onemore disadvantage of e-l closures, earlier noted inMoeng and Wyngaard (1989), is their inability to directlyaccount for counter-gradient turbulent flux in the upperportion of the CBL.

5.2 Integral Budget-Based Closure

The results of the integral budget methods aregrouped according to the type of method. Theparameterizations of Tennekes (1973), Driedonks(1982), and Batchvarova and Gryning (1990, 1994) takeinto account only the surface shear through their frictionvelocity terms and are presented together as Group I.The parameterizations of Zeman and Tennekes (1977),Tennekes and Driedonks (1981), Boers et al. (1984),and Pino et al. (2003) take into account more than justthe surface shear. They also include either the spin-upterm, velocity jump across the entrainment zone(although some may use the full jumps rather than theZOM jumps), or both and are thus assigned to Group II.Finally, the parameterizations of Stull (1976a,b) and

Sorbjan (2004), which differ somewhat from the typicalZOM methodology, constitute Group III.

Figure 12 shows results obtained with theparameterizations of Group I for the GS case. Theparameters of entrainment were retrieved from LESdata. The black dots denote the entrainment flux ratiodefined from the ZOM parameterization for the

entrainment zone heat flux, /i iB bdz dt , and theblue dots denote the entrainment flux ratio defined fromthe actual minimum of buoyancy flux in the LESentrainment zone. Since the parameterizations in Fig.12 take into account only the surface shear, which isinitially zero, they underestimate the entrainment fluxratio in the GS case and predict values consistent withcommonly accepted shear-free value of about 0.2.

0 4000 8000 12000 16000 20000 24000Time (s)

0

0.2

0.4

0.6

0.8

1

-Bi

/Bs

LES bdzi/dt

LES -Bi/Bs

Tennekes (1973)

Driedonks (1982)

Batchvarova & Gryning (1991,1994)

Figure 12. Entrainment flux ratio predictions of Group Iclosures compared with LES for the GS case with

/ 0.003d dz K/m and 0.03sQ K m/s.

Figure 13 shows results for parameterizations fromGroup II. The entrainment parameterization of Zemanand Tennekes (1977) does not take entrainment zoneshear into account and fortuitously performs better thanthe others when compared to LES data. Among thosethat do take the entrainment zone shear into account,the parameterization of Pino et al. (2003) performs thebest, but all suffer from problems with the denominator

going to zero, when the entrainment rate becomesunbounded.

Figure 14 shows the comparisons with the Group IIIparameterizations. Since these parameterizations donot include negative-sign shear terms in thedenominator, the parameterized entrainment flux ratiodoes not become unbounded. The Stull (1976a,b)parameterization predicts an entrainment flux ratio thatclimbs steadily with time, while LES data show a fairly


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constant value. The Sorbjan (2004) entrainment fluxratio also increases with time.

0 4000 8000 12000 16000 20000 24000Time (s)

0

0.2

0.4

0.6

0.8

1

-Bi/

Bs

LES bdzi/dt

LES -Bi/Bs

Zeman & Tennekes (1977)

Tennekes & Driedonks (1981)

Boers (1984)

Pino et al (2003)

Figure 13. Entrainment flux ratio predictions by Group IIclosures compared to LES for the GS case with

/ 0.003d dz K/m and 0.03sQ K m/s.

0 4000 8000 12000 16000 20000 24000Time (s)

0

0.2

0.4

0.6

0.8

1

-Bi/

Bs

LES bdzi/dt

LES -Bi/BsStull (1976)

Sorbjan (2004)

Figure 14. Entrainment flux ratio predictions byGroup III closures compared to LES for the GS case

with / 0.003d dz K/m and 0.03sQ K m/s.

Figure 15 shows the entrainment flux ratios by theGroup I parameterizations for the GC case. All methodspredict higher entrainment flux ratios than LES. Thepredicted ratios decrease with time, and the LES datashow this decrease to some extent as well.

The initial value of surface shear is very large, andso the shear-generated TKE at the surface should be

very large as well. As friction decreases the momentumin the CBL, the surface shear-generated TKE alsodecreases, resulting in a decrease in the predictedentrainment flux ratio.

0 5000 10000 15000 20000 25000Time (s)

0

0.4

0.8

1.2

1.6

2

-Bi/

Bs

LES bdzi/dt

LES -Bi/Bs

Tennekes (1973)

Driedonks (1982)

Batchvarova & Gryning (1994)

Figure 15. Entrainment flux ratio predictions of Group Iclosures compared to LES for the GC case with

/ 0.003d dz K/m and 0.03sQ K m/s.

Figure 16 shows entrainment predictions for the GCcase by the Group II parameterizations. Again, all ofthem overpredict the entrainment flux ratio, but the Pino(2003) parameterization performs the best. Thedenominator goes to zero in these expressions for theGC case just like it does for the GS case.

Figure 17 shows performance of the Group IIIparameterizations for the GC case. These perform thebest of all the groups of parameterizations. The

parameterization of Stull (1976a,b), which takes bothsurface and elevated shears into account, performsbetter than the Sorbjan (2004) parameterization, whichonly uses entrainment zone shear. This does notnecessarily mean that the surface shear is equallyimportant to the entrainment zone shear. It merelymeans that the Stull parameterization is better tuned forthis particular case. In fact, LES results indicate that itis primarily the entrainment zone shear that drives theenhancement of entrainment relative to the shear-freecases. Even the GC cases, which start without


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entrainment zone shear, eventually develop similarshear at the CBL top when compared to the GS cases.

0 5000 10000 15000 20000 25000Time (s)

0

0.4

0.8

1.2

1.6

2

-Bi/

Bs

LES bdzi/dt

LES -Bi/Bs



Boers (1984)

Pino et al (2003)

Figure 16. Entrainment flux ratio predictions of Group IIclosures compared to LES for the GC case with

d/dz=0.003 K/m and surface heat flux of 0.03 K m/s.

0 5000 10000 15000 20000 25000Time (s)

0

0.4

0.8

1.2

1.6

2

-Bi/

Bs

LES bdzi/dt

LES -Bi/Bs

Stull (1976)

Sorbjan (2004)

Figure 17. Entrainment flux ratio predictions of Group IIIclosures compared to LES for the GC case with

d/dz=0.003 K/m and surface heat flux of 0.03 K m/s.

In order to illustrate a situation, in which theparameterizations perform reasonably well, we show inFig. 18 the Group II parameterizations for the GS case

with a surface kinematic heat flux of 0.10 K m/s and afree atmospheric stratification of 0.010 K/m.

All parameterizations perform reasonably well, butthe entrainment flux ratio is close to the shear-free valueof 0.2 anyway. Of those methods that take entrainmentzone shear into account, the Pino et al. (2003)parameterization is closest to the LES entrainment flux

ratio. Overall, the Zeman and Tennekes (1977)parameterization is the closest to the LES results, but itdoes not specifically take the entrainment zone shearinto account.

0 10000 20000 30000Time (s)

0

0.2

0.4

0.6

0.8

1

-Bi/

Bs

LES bdzi/dt

LES -Bi/Bs



Boers (1984)

Pino et al (2003)

Figure 18. Entrainment flux ratio predictions by the

Group II parameterizations compared to LES for the GScase with d/dz=0.010 K/m and surface heat flux of 0.10K m/s.

5. SUMMARY AND CONCLUSIONS

The e-l closures of Xue et al. (2001) and Fielderand Kong (2003) both predict a greater difference inentrainment between shear-free CBLs and shearedCBLs, with entrainment predicted for sheared CBLsbeing too large. At the same time, the F&K closurepredicts CBL growth very close to the LES predictionsfor the NS case. The more rapid entrainment in caseswith shear seems to occur regardless of the TKE valuesin the parameterized CBL. The ARPS scheme

produces less TKE than LES but still predicts too fastentrainment when shear is present. The F&K schemegenerally produces TKE values that are comparable tothose of LES, but the TKE in the entrainment zone,where significant shear is present, is significantlygreater than the LES-predicted TKE. It is possible thatthe shear production of TKE in such schemes isoverestimated, perhaps because the appropriate lengthscales for shear-generated turbulence in the CBL aredifferent from the length scales for buoyancy-generatedturbulence. It is possible that the formulation of the


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master length scale l for CBL turbulence needs to berevised to account for the effects of shear.

The analysis in the present paper is somewhatlimited by the fact that the initial momentum profiles donot necessarily reflect real atmospheric profiles. Inparticular, the surface winds in the GC case wouldprobably never be equal to their geostrophic value, and

so any NWP model initialization would havesubgeostrophic winds at the surface and thereforeweaker wind shears there. Likewise, the GS case hasvery large geostrophic shear, probably three or fourtimes greater than its typical atmospheric value.Nevertheless, similar magnitudes of total shear(between 10 meters above ground level and a kilometeror so above ground level) are not unheard of. In thisway, the cases with strong shear serve as a critical testof how an e-l scheme in NWP might perform withrespect to the development of the CBL. It is particularlyimportant for NWP and air quality model schemes tocorrectly predict boundary layer depth, because thisaffects the concentration of moisture in the loweratmosphere as well as the concentration of air

pollutants.The tests of the integral budget-based entrainment

schemes suggest that they overestimate entrainment insituations when strong shear is present. Again, it mustbe stated that the shear in the LES cases is ratherstrong, but the cases were designed in this manner sothat the relative effects of shear could be more easilyseen. Adhering to the strict mathematical form of theZOM provides entrainment equations that contain theentrainment zone shear as a negative sign term in thedenominator. This has the unfortunate result of causingthe denominator to drop to zero and the entrainment fluxratio to become unbounded well before the surface heatflux goes to zero. Entrainment equations by Stull(1976a,b) and Sorbjan (2004), which deviate from the

traditional ZOM methodology in this respect, avoid thisproblem and therefore produce more realistic results forthe strong shear conditions. The success of these twoparameterizations suggests the importance ofaccounting for the finiteness of the entrainment zonethickness in any entrainment equation for sheared CBL.Given that Kelvin-Helmholtz instabilities appear to be aninherent feature of convective entrainment in thepresence of wind shears, as our and other LES show(see, e.g., Kim et al. 2003), any Ri-limited entrainmentequation would seem to be most suited to model thegrowth dynamics of sheared CBL.

Acknowledgement: Authors gratefully acknowledgesupport by the National Science Foundation (grant

ATM-0124068).

Ayotte, K. W., P. P. Sullivan, A. Andren, Scott C. Doney,A. A. M. Holtslag, W. G. Large, J. C. McWilliams,C.-H. Moeng, M. J. Otte, J. J. Tribbia, and J. C.Wyngaard, 1996: An evaluation of neutral andconvective planetary boundary-layerparameterizations relative to large eddysimulations. Bound. Layer Meteorol.,79, 131-175.

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Atmos. Sci., 27, 1211-1213.Driedonks, A. G. M., 1982: Models and observations of

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