On Modelling the One-Dimensional Atmospheric Boundary Layer · Keywords: Atmospheric boundary layer, Depth of the ABL, Diurnal cycle, Neutral/stable ABL, Turbulence closure, Turbulence

ON MODELLING THE ONE-DIMENSIONAL ATMOSPHERICBOUNDARY LAYER

WENSONG WENG� and PETER A. TAYLOR��

Department of Earth and Atmospheric Science, York University, 4700 Keele Street, North York,Ontario, Canada M3J 1P3

(Received in final form 23 July 2002)

Abstract. Several commonly used turbulence closure schemes for the atmospheric boundary layer(ABL) are applied to simulate neutral, nocturnal and diurnal cycle situations in a one-dimensionalABL. Results obtained with the different schemes, E − �, E − ε and its modified versions, and twoversions of the q2� Level 2.5, are compared and discussed.

Keywords: Atmospheric boundary layer, Depth of the ABL, Diurnal cycle, Neutral/stable ABL,Turbulence closure, Turbulence length scale.

1. Introduction

The atmospheric boundary layer (ABL) plays a vital role in the transfers of mo-mentum, heat and mass between land or ocean surfaces and the atmosphere, whichare critical for accurate numerical weather prediction and climate modelling. Muchprogress has been made in our understanding, modelling and parameterization ofthe ABL in the last two or three decades. One approach to include the effects ofthe ABL in a large-scale model is to resolve the boundary-layer structure expli-citly and effectively by including several computational levels. To model thoseturbulent fluxes of momentum, heat and mass, the Reynolds averaged equationsare often used. To specify how to complete the system of Reynolds averaged equa-tions is referred to as the turbulence closure problem. A frequently used form ofturbulence closure in boundary-layer modelling is the second-order, single-pointclosure model (Mellor and Yamada, 1974; Launder et al., 1975). In this typeof closure, equations for the second moments are derived by applying Reynoldsdecomposition and averaging to the momentum, heat and scalar equations. Theresult is a set of differential equations for the ensemble-averaged second moments.Although the derivation of these equations is straightforward, a number of terms inthe equations require parameterization, specifically third-order moments, pressurestrain correlations and dissipation terms. With suitable parameterization for theseterms a closed set of equations results.

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Boundary-Layer Meteorology 107: 371–400, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

372 WENSONG WENG AND PETER A. TAYLOR

In principle, equations for turbulent moments of any order can be derived. Al-though models containing prognostic third-order moments have been used (Andréet al., 1978; Briere, 1981; Moeng and Randall, 1984), these are far from practicalfor use in the boundary layer of large-scale models. Even at second-order, thesemodels are often too complex for inclusion in a large-scale model due to com-putational cost. Additionally, it is questionable if the advantage of using a morephysically complete, but potentially less robust, closure can be realized at verticaland horizontal resolutions similar to those of most large-scale models. For thesereasons, most implementations of single-point closure in large-scale models arebased on a truncation of the second-moment equations to some manageable form.

In this paper we study some of the closure schemes commonly used in ABLmodelling. They are all 11

2 -order schemes in which the equations for the turbulentkinetic energy (TKE) and a turbulence length scale are used. The turbulent lengthscale equation can either be diagnostic (E − �, E − ε − � and q2� Model I) orprognostic (E − ε, or its modification and q2� Model II).

2. Descriptions of the Model

The model context is a one-dimensional, horizontally homogeneous, dry boundarylayer, whose mean structure only depends on time t and the vertical coordinatez. The primary meteorological variables we shall consider are the velocity com-ponents U and V , and the potential temperature . The pressure P is determinedby the hydrostatic relationship and ∂P/∂x and ∂P/∂y are independent of height(i.e., the boundary layer is barotropic) and represented by the geostrophic windcomponents, i.e., (∂P/∂x, ∂P/∂y) = ρf

(Vg, −Ug

), where ρ is the air density,

f is the Coriolis parameter and Ug and Vg are components of the geostrophic wind.Moisture could have been included as a passive scalar but we wish to avoid phasechanges and latent heat release. For moist, but unsaturated, air, one can replace

by v, the virtual potential temperature, in the analysis below.

2.1. THE MODEL EQUATIONS

In an idealised, horizontally homogeneous ABL and in the absence of radiative fluxdivergence and moisture, the Reynolds averaged equations describing the dynamicsof the ABL can be written as

∂U

∂t= f (V − Vg) − ∂ 〈uw〉

∂z, (1)

∂V

∂t= f (Ug − U) − ∂ 〈vw〉

∂z, (2)

∂

∂t= −∂ 〈wθ〉

∂z, (3)

ONE-DIMENSIONAL ATMOSPHERIC BOUNDARY-LAYER MODELLING 373

where u, v and w are horizontal and vertical components of wind velocity per-turbation, is the potential temperature (any passive scalar can be treated in thesame way), −〈uw〉 and −〈vw〉 are the (kinematic) shear stress components inthe x and y-directions respectively, 〈wθ〉 is the vertical kinematic heat flux (posit-ive upwards) and angle brackets (〈〉) represents an ensemble/short-time averagedmean.

2.2. TURBULENCE CLOSURE

To close the above set of Equations (1)–(3), a model for the turbulent fluxes isrequired. There are several ways to model turbulence effects in the Reynoldsaveraged equations. The simplest is to represent the turbulent flux term in eachmean-field equation through an eddy diffusivity. Thus, to close Equations (1)–(3),the turbulent fluxes are commonly modelled as

−〈uw〉 = Km

∂U

∂z, (4)

−〈vw〉 = Km

∂V

∂z, (5)

−〈wθ〉 = Kh

∂

∂z, (6)

where Km is the eddy viscosity and Kh is the eddy diffusivity for heat. The essentialpremise of this kind of closure is that turbulent fluxes are locally related to meanvertical gradients by an eddy diffusivity K, which is a property of the turbulentflow. All the complexities of turbulence must be accounted by this K. The problemof the closure is then effectively reduced to the problem of specifying K.

For low-order turbulence closures, the eddy diffusivities are often expressed as(Ayotte et al., 1996; Xu and Taylor, 1997a)

Km = �m (αE)1/2 , (7)

Kh = �m (αE)1/2/ P r , (8)

where �m is a turbulent mixing length, E is the turbulent kinetic energy (TKE), theconstant α is the ratio of the surface shear stress to TKE and P r is the turbulentPrandtl number (defined as P r = Km/Kh, the value of 0.74 is used here). One ap-proach to model �m is simply to formulate a diagnostic equation for the length scalethat relates it to the distance from the surface and the stability of the atmosphere;another uses the prognostic equation for the dissipation rate of TKE, ε. One canthen formulate a length scale from E and ε, i.e.,

�d = (αE)3/2/

ε , (9)


where �d is often called a dissipation length scale (Xu and Taylor, 1997a). With theassumption, �m = �d , the eddy diffusivities (7) and (8) can be expressed as

Km = (αE)2/

ε , (10)

Kh = (αE)2/

(εP r) = Km

/P r . (11)

Using the ε-equation is analogous to having a prognostic equation for the turbulentlength scale.

Both approaches need an equation for the TKE, which, in one-dimensionalform, is

∂E

∂t= Ps + Pb − ε + ∂

∂z

(Km

∂E

∂z

), (12)

where Ps and Pb are TKE production terms by the shear and buoyancy respectively

Ps ≡ −〈uw〉 ∂U

∂z− 〈vw〉 ∂V

∂zwith Pb ≡ βg 〈wθ〉 .

Here β is the coefficient of thermal expansion, g is the acceleration due to gravity.The last term of Equation (12) represents turbulent diffusion of TKE in the vertical.

In the following, a brief description of commonly used simple closures is given.

2.2.1. E − � ClosureAs the name of the closure suggests this closure uses the TKE Equation (12) and adiagnostic equation for the turbulent length scale. There are quite a few formulaein the literature, see for example, the review by Holt and Raman (1988), but herewe look at two of them. The first is based on André et al.’s (1978) model (to bereferred as AMLTV78), which is also used by Duynkerke and Driedonks (1987).In the model, the assumption �m = �d = � is used. The turbulent length scale, � ischosen as,

� ={

�nc , ∂/∂z ≤ 0 ;min(�nc, �s) , ∂/∂z > 0 ,

(13)

where �nc is the characteristic length for neutral and unstable stratification while �s

is the characteristic length scale for stable stratification. They are modelled as

1

�nc

= φm

κ(z + z0)+ 1

�0, (14)

with �0 = 0.0027|Ug|f −1 and

�s = cs

[E

/(βg∂/∂z)

]1/2, (15)


where κ is the von Kármán constant, z0 is the surface roughness length, |Ug| =(U 2

g + V 2g

)1/2and cs is a constant (a value of 0.75 was used in André et al. (1978)

while Duynkerke and Driedonks (1987) used 0.36, which is employed here). φm is astability function for the non-dimensional vertical gradient of wind and a generallyaccepted form of φm is

φm ={

(1 − γ1z/Lo)1/4 , for z/Lo ≤ 01 + βcz/Lo , for z/Lo > 0

(16)

where constants γ1 = 15, βc = 4.7. Lo (Businger, 1973) is the Obukhov length anddefined by Lo = −u3∗

/(κβg 〈wθ〉) . Note Equation (14) is based on Blackadar’s

(1962) formula and was used in Xu and Taylor’s (1997a) study for neutral ABLflow with φm = 1.

The second model is based on Delage’s (1974) model. To model the nocturnalboundary layer, Delage used the following formulae for the turbulent length scales

1

�m

= 1

κ(z + z0)+ 1

�0+ βc

κLo, (17)

1

�d

= 1

κ(z + z0)+ 1

�0+ (βc − 1)

κLo. (18)

Equation (17) is based on (14) and (16b), while Equation (18) is deduced fromthe TKE Equation (12) with the assumption of a constant flux layer. The secondE − � model (D74) uses these turbulent length scales for the stable situation andthe formula (14) for neutral and unstable flows.

2.2.2. E − ε and its Modified ClosuresIn these closures, the equations for the turbulent fluxes and the TKE are the sameas in the E − � scheme. The eddy diffusivities are calculated from Equations (10)–(11). In the E − ε closure scheme the rate of dissipation of TKE, ε, is governed bythe prognostic equation,

∂ε

∂t= ε

ECε1 (Ps + Pb) − Cε2

ε2

E+ Cε

∂

∂z

(Km

∂ε

∂z

), (19)

where the three terms on the RHS are loosely labelled as the production, dissipationor destruction and diffusion of ε respectively. The often used constants are Cε1 =1.44, Cε2 = 1.92 and Cε = α(Cε2 − Cε1)/κ2.

Previous applications of this ε-equation to the completely neutral ABL yiel-ded too deep a boundary layer. There is however a concern that most ‘neutral’ABLs do in reality have a stable cap preventing them from becoming too deep(see Zilitinkevich and Esau, 2002). Various investigators proposed modification byadjusting between production and destruction in the standard ε-equation. Detering


and Etling (1985) chose to increase the production term and changed the originalconstant Cε1 to the variable C ′

ε1 defined by

C ′ε1 = Cε1

�

hc

, (20)

where � is given by Equation (9) and hc is a characteristic scale of the ABL andproportional to the ABL height, h. They compared the results of their modifiedE − ε model with different hc to observations (Leipzig wind profile). The optimumform of hc, chosen as given the best fit between model results and observations,was found with

hc = c1u∗/f and c1 = 0.0015 . (21)

Here we shall refer to this version of the E − ε model as E − ε (DE85). Notethat the length scale used in Equation (20) is hc and not the ABL height itself,h, which was claimed to be used in their studies by Holt and Raman (1988) andHurley (1996). A limited-length-scale E − ε model was proposed by Apsley andCastro (1997). In their model, a global maximum length scale is imposed by thefinite boundary-layer depth or by the Obukhov length in stably stratified condition.

In their study of the ABL using the E−ε and Launder et al.’s (1975) full second-order closures, Xu and Taylor (1997, 1997b) modified the standard ε-equation byre-parameterizing the production term and referred to the new scheme as E − ε −�

closure. The modified equation reads

∂ε

∂t= Cε1

α3E2

�2ε

− Cε2ε2

E+ Cε

∂

∂z

(Km

∂ε

∂z

), (22)

where �ε is a characteristic dissipation length scale. For simplicity, Equation (14)was used for �ε in neutrally stratified flow. Here we shall apply this model to otherconditions by using Equations (14) and (17) for �ε . This model is referred as E −ε − �.

2.2.3. q2� Closure, Level 2.5This scheme is derived from the second-order closures described by Mellor andYamada (1974, 1982) in which the equations are simplified (truncated) based onan anisotropic scaling. The model is classified into different levels according to itscomplexity, ranging from most sophisticated, full second-order closure, Level 4 tothe simplest Level-1 closure. The intermediate Level-2.5 model is most commonlyused in studying the evolution of the ABL, see for example, Yamada and Mellor(1975); Yamada (1983); Andrén (1990); Haberle et al. (1993). The model uses anequation for twice the TKE (q2 ≡ 2E) in combination with either an algebraicexpression for the master turbulent length scale � or an equation for q2 multipliedby � (q2�). The equations are

∂q2

∂t= 2

(Ps + Pb − q3

B1�

)+ ∂

∂z

(�qSq

∂q2

∂z

)(23)


and either

1

�= 1

κ(z + z0)+ 1

�0with �0 = C�

∫ ∞

0z q dz∫ ∞

0q dz

, (24)

or

∂q2�

∂t= E1�

(Ps + Pb

)− q3

B1

[1 + E2

(�

�R

)2]

+ ∂

∂z

(�qS�

∂q2�

∂z

), (25)

where the closure constants used here are B1 = 16.6, Sq = S� = 0.2, E1 =1.8, E2 = 1.33 (Mellor and Yamada, 1982). Various values of C� were used byprevious investigators, i.e., 0.1 (Mellor and Yamada, 1974; Yamada and Mellor,1975; Andrén, 1990); 0.2 (Ayotte et al., 1996) or 0.25 (Duynkerke and Driedonks,1987; Holt and Raman, 1988). The impact of different values of C� is discussed, seebelow; �R is a measure of the distance away from the surface (Mellor and Yamada,1982). In the following, we refer to the q2� closure Level 2.5 with the diagnosticEquation (24) as q2� Model I and the q2� closure Level 2.5 with the prognosticEquation (25) as q2� Model II.

Following Yamada (1983), the turbulent fluxes are calculated from (4)–(6).Eddy diffusivities in these models may be written

Km = �qSm , (26)

Kh = �qSs = �qSmCα , (27)

where Sm and Cα are functions of stability. They are given as,

Sm = 1.96

(0.1912 − Rif )(0.2341 − Rif )

(1 − Rif )(0.2231 − Rif ), Rif < 0.16 ;

0.085 , Rif ≥ 0.16 .

(28)

Cα = 1.318

0.2231 − Rif

0.2341 − Rif

, Rif < 0.16 ;1.12 , Rif ≥ 0.16 ,

(29)

where Rif is the flux Richardson number, defined as

Rif = βg 〈wθ〉〈uw〉 (∂U/∂z) + 〈vw〉 (∂V /∂z)

. (30)

It can be derived from the gradient Richardson number, Ri, defined as

Ri = βg (∂/∂z)

(∂U/∂z)2 + (∂V /∂z)2 , (31)


using the following conversion,

Rif ={

0.6588[Ri + 0.1776 − (Ri2 − 0.3221Ri + 0.03156)1/2], Ri < Ric;Rif c, Ri ≥ Ric,

(32)

where Rif c(= 0.191) and Ric(= 0.195) are critical flux and gradient Richardsonnumbers. For neutrally stratified flow, 〈wθ〉 = 0 and Rif = 0. From Equations(28) and (29), we then have Sm = 0.3933 and Cα = 1.2559.

For q2� Level-2.5 closure, q is computed from the prognostic equation while thenon-dimensional eddy diffusivities Sm and Ss are calculated from the diagnosticequations that depend only on the local gradient Richardson number. In essencethis means that the q2� Level-2.5 closures, as used here (q2� Models I and II), areanalogous to the E − � and E − ε models common in the engineering community.

3. Numerical Scheme and Boundary Conditions

As in other studies, the model uses a stretched vertical coordinate to ensuresufficient resolution near the surface and to resolve strong gradients. We set

Z = lnz + z0

z0+ z

b0, (33)

where b0 is a constant (67.5 m is used in these calculations). Equations are trans-formed into the new coordinate system before they are discretized into their finitedifference equivalents. Flow variables are stored on a staggered grid, where meanvariables (U , V and T ) are at layer midpoints and turbulent quantities (E andturbulence fluxes) at layer lower boundary levels and zt (the top of the computationdomain). The numerical scheme employed for time integration is Crank–Nicolson.The resulting set of difference equations is solved using a block LU factorizationalgorithm (Karpik, 1988).

The surface boundary conditions used are a non-slip condition for velocity(U = V = 0), a specified time dependent temperature or heat flux if required,and the assumption that production balances the dissipation of TKE (P = ε). Atthe upper boundary, we specify (U, V ) = (Ug, Vg), = g (constant) and set thevertical derivatives of TKE, ε, shear stresses and other turbulent fluxes to zero.

It was found that the results of the standard E − ε model and q2� Model II aresensitive to the height of the computational domain, see the discussion below. Twosets of grid point number (Ng) and computational domain heights (zt ) are usedwith Ng = (75, 141) and zt = (3000, 6000) m. All the simulations presentedhere were obtained with 141 vertical grid points and the computational domain topis set to zt = 6000 m unless stated otherwise.


TABLE I

Parameters of the neutral ABL with different turbulence closures,zt = 6000 m. The boundary-layer height, h, is calculated as the heightwhere E is 5% of the surface value, E0; α0 is the surface cross-isobarangle. c1 = 0.0045 and C� = 0.055 are used in E − ε (DE85) and q2�

Model I calculations respectively. Other parameters are discussed in thetext.

Turbulence closure E0 (m2 s−2) u∗ (m s−1) h (m) α0 (deg)

E − � 0.44 0.36 750 25.9

E − ε 0.59 0.42 3584 14.3

E − ε (DE85) 0.49 0.38 772 25.2

E − ε − � 0.46 0.37 812 24.9

q2� Model I 0.43 0.36 771 25.6

q2� Model II 0.61 0.43 4635 13.9

4. Results and Discussions

We first look at the behaviour of various models with different turbulence closuresin modelling the one-dimensional, neutrally stratified, planetary boundary layerover horizontally homogeneous terrain. Secondly, a nocturnal boundary layer isstudied by applying constant cooling at the surface. Finally the models are run in adiurnal cycle study based loosely on experimental data.

In all calculations, a value of α = 0.3 is used in E − �, E − ε and their variedforms so that they are close to the constants used in the q2� models (α can becalculated from the constant B1 used in q2� models, i.e., α = 2/B

2/31 ≈ 0.31 with

B1 = 16.6, see Xu and Taylor, 1997b). All the results plotted in the figures onlyshow the lower portion of the computational domain so that the main features canbe easily identified.

4.1. NEUTRAL STRATIFICATION

Although a truly steady state, neutrally stratified, barotropic ABL above a horizont-ally homogeneous surface is an idealization rarely, if ever, achieved in nature, it hasbeen a starting point, and a reference for many of the theories and models of theboundary layer. We consider a typical situation with 〈wθ〉 = 0, (Ug, Vg) = (10, 0)

m s−1, f = 10−4 s−1, and the surface roughness length z0 = 0.1 m, which leadsto a surface Rossby number, Ro = 106. Following Xu and Taylor (1997a), theintegration starts with an Ekman-like solution at t = 0 (see the lines in Figure 1)and ends at t = 8π/f ≈ 70 hr (4 inertial cycles) in an approximately steady state.Note that the results do not depend on the initial conditions.


Figure 1. Initial profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) and results fromE − � closure under neutral stratification.

Figures 1–3 show the vertical profiles of wind (U, V ), shear stress (〈uw〉 , 〈vw〉)and TKE (E) from the models with different turbulence closures under neutralstratification. Some of the calculated ABL parameters are listed in Table I.

It has been argued that the model with E − � closure works well, at least in thissimple situation (Xu and Taylor, 1997a; Hess and Garratt, 2002). From Figure 1, wecan see that the ABL then has finite depth, the turbulent quantities (TKE and shearstresses) decrease appreciably over the depth of the boundary layer and U ≈ Ug atupper levels. There also exits a supergeostrophic wind around z = 500 m and thewind vector has Ekman spiral behaviour. The surface value of TKE (E0) is about0.44 m2 s−2, and the surface friction velocity, u∗ = (〈uw〉2 + 〈vw〉2

)1/4 ≈ 0.36 ms−1. One of the important variables of the ABL is the depth of the boundary layer,h. In neutral stratification, the commonly used estimation formula for h is

h = cnu∗/f , (34)

where cn is an empirical constant. If we define h as the height where the TKE isabout 5% of its surface value, h ≈ 750 m, this compares favourably with 720 mcalculated according to the formula with cn = 0.2 (van Ulden and Holtslag, 1985).However, it is lower than the 1080 m obtained with cn = 0.3 (Arya, 1985, p.148).


Figure 2. Vertical profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) from standardE − ε and E − ε − � closures under neutral stratification.

In contrast to the results of the E − � model, there are no supergeostrophic U

component speeds in model results obtained with the standard E − ε closure andq2� Model II. The surface values of TKE and u∗ are about 35% and 18% largerthan those with E − � closure. These models predicted very large values of theturbulent length scale (see Figure 6), which lead to excessive turbulent mixing andvery deep boundary layers. In fact, h is 3584 m for the E−ε model and 4635 m forq2� Model II if one uses the definition that h is the height where the TKE is 5% ofits surface value. Note the larger difference in the upper part of the computationaldomain when zt = 3000 m is used instead of 6000 m. If we impose the upperboundary condition at zt = 3000 m, h is not defined (E > 0.05E0 in the wholecomputational domain). In fact, E (z = 3000 m) is still about 20% of its surfacevalue for E − ε closure and about 28% of E0 for q2� Model II when zt = 3000 mis used, see Figures 3 and 4.

The deep ABL height with the standard E − ε model is associated with theε-equation (19). With proper modifications, such as those proposed by Deteringand Etling (1985) or Xu and Taylor (1997a), one can obtain similar results to thoseof E − � closure. In our use of the DE85 model, further modification is needed.Close to the surface, the first two terms on the RHS of ε-equation (19) are almost


Figure 3. Vertical profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) from q2� ModelsI and II under neutral stratification.

in balance and � can be much smaller then hc. If we directly use Equation (20), itleads to model failure due to the excessive decrease of ε production. To overcomethis problem, we have replaced Equation (20) by

C ′ε1 ≡ Cε1 max (1, �/hc) so that C ′

ε1 ≥ Cε1 . (35)

This modification is in line with DE85’s idea that the ε production term is increasedin the upper part of the boundary layer. In the calculations, we also find that themodel results are dependent on the coefficient, c1, used in Equation (21). Goodagreement with results of the E − � model can be achieved with c1 = 0.0045.Figure 4 shows the vertical distribution of E from different models. Results of theE − ε − � closure are almost identical to those of the E − � closure. For the neutralABL, more detailed results of E − ε − � closure can be found in Xu and Taylor(1997a).

The results from q2� Model I compare well with those of E − � closure providethat an appropriate value of C� is used. There is a supergeostrophic U componentwind, and the predicted surface values of E0 and u∗, and the ABL height, aresimilar to those of the E − � model, see Table I. Note that the model results aresensitive to the actual C� value used in Equation (24), especially the estimated


Figure 4. Vertical profiles of TKE (E) from E − �, standard E − ε and its modified closures underneutral stratification.

boundary-layer height. Figure 5 shows the variations of h and the surface valuesof TKE and u∗ with C�. By using a least squares fit, the model results of h can beapproximated by h = 10745.4C� + 183.594, in m. Generally speaking, the largerthe value of C�, the higher the boundary layer and more turbulence is produced.For a realistic neutral ABL, we think C� should be less than 0.1 and with C� = 0.2the boundary layer is probably too deep (h ≈ 2378 m). To match the results of theE − � closure model, C� = 0.055 is appropriate, which we use in the rest of ourcalculations.

The differences in model results with different turbulence closures are primarilycaused by the way the turbulent length scales are modelled. Figure 6 shows verticalprofiles of the turbulent length scale. As expected, the E −�, the modified versionsof E − ε closure and q2� Model I predict similar distributions, with � being almosta constant in the upper part of the ABL. The � profile of q2� Model I is almost


Figure 5. Variation of the boundary-layer height, surface values of TKE and shear stress with differentvalues of C� from q2� Model I under neutral stratification.

identical to that of E − � closure with C� = 0.055, while we believe that thestandard E − ε closure and q2� Model II predict far too large values for � abovethe surface layer.

The surface cross-isobar angles (α0) predicted by the modified E−ε models arein good agreement with those of E − � closure and q2� Model I, while the modelswith a prognostic length scale equation (the standard E − ε model and q2� ModelII) predict much smaller values of α0, see Table I. These results are consistent withthose of Xu and Taylor (1997a) and Koo and Reible (1995).

4.2. STABLY STRATIFIED FLOW

Here we look at the behaviour of various models in stably stratified conditions.From their neutral stratification state with uniform potential temperature, = 285K, the models are run for 12 hr under the cooling rate of 1 K per hr at the surface.Due to the net loss of heat to the ground and no compensating heat flux at thetop, the boundary layer as a whole must cool. A quasi-steady state stable boundarylayer is achieved if the turbulent fluxes are independent of time (Nieuwstadt, 1985).Provided that the surface cooling is not too extreme, the ABL may reach a quasi-steady state stable boundary layer with the relaxation time of O

(f −1

).


Figure 6. Vertical profiles of the turbulent length scale from E − �, standard E − ε and its modifiedversion closures and q2� Models I and II under neutral stratification.

Figure 7 shows the evolution with time of the surface friction velocity (u∗0),the surface kinematic heat flux (〈wθ〉0) and the Obukhov length (Lo) from modelswith different turbulence closures. There are fairly rapid decreases of u∗ within thefirst few hours after the transition, which then level off after about 3 hr for E − �

(D74), E − ε − � and q2� (Models I and II) closures, and after 4 hr for the standardE − ε closure. For E − � (AMLTV78) closure, u∗0 decreases slowly after trans-ition, decreases quickly about 2 hr later and then levels off again around 4 hr aftertransition. This behaviour is due to the formulation of the turbulent length scale, �,(13). The effect of the stable stratification on � only comes into play when ∂/∂z

is large enough so that �s from Equation (15) is used instead of �nc from Equation(14). For E − ε (DE85) closure, u∗0 decreases slowly during the whole coolingperiod and also the magnitude of the change is the smallest. We can see from thefigure that the behaviours of u∗0 and 〈wθ〉0 are very similar from the two closuresof E −� (D74) and E −ε −�. Also the results are similar from q2� Models I and II


TABLE II

Some parameters of the nocturnal ABL with different turbulence closures. The resultsshown use zt = 6000 m and are at t = 12 hr. The boundary-layer heights, h1, h2 andh3, are calculated where E is 5% of the surface value, E0, and from Equation (36) withc = 0.37 and 0.4 respectively. c1 = 0.0045 is used in E − ε (DE85) and C� = 0.055is used in q2� Model I. The nocturnal jet strength (Sjet) is measured by the strongest

wind speed, |U| =√

U2 + V 2 and the height (Hjet) is where it occurs.

E0 u∗ h1 h2 h3 Sjet Hjet

Turbulence closure (m2 s−2) (m s−1) (m) (m) (m) (m s−1) (m)

E − � (D74) 0.15 0.21 103 92 99 13.1 122

E − � (AMLTV78) 0.28 0.29 177 127 136 12.7 189

E − ε 0.35 0.32 162 125 135 12.3 342

E − ε (DE85) 0.34 0.32 162 125 135 12.1 302

E − ε − � 0.16 0.22 106 95 101 13.1 122

q2� Model I 0.19 0.24 169 90 96 11.3 225

q2� Model II 0.20 0.25 176 95 101 11.9 263

and the results of E − ε (DE85) are almost identical to those of E − ε. The surfaceheat flux 〈wθ〉0 and the Obukhov length Lo generally decrease for all turbulenceclosures but for E − � (AMLTV78) closure the switch in formula for � leads toa period of increasing surface heat flux between 2 1

2 and 313 hr (∂ 〈wθ〉0 /∂t > 0).

The downward heat fluxes predicted by E − � (D74) and E − ε − � closures arethe smallest, while E − ε and E − ε (DE85) closures produce the largest and allthe others closures in between. All models predict similar evolutions of Lo. After12 hr, q2� Models I and II predict the strongest stratification effects (lower Lo).

Figures 8–12 show the vertical profiles of mean wind, potential temperature andturbulent quantities, after 12 hr of cooling, from the various closure schemes for thelower portion of the computational domain while some of the parameters are listedin Table II. E−� (D74) and E−ε−� closures predict the shallowest boundary layerwhereas E − ε and E − ε (DE85) closures show the deepest. Some broad featuresof wind speed profiles are similar for all the closures – the supergeostrophic windor nocturnal jet is apparent at low level, wind speed and direction assume theirgeostrophic values at around 400 m, and turbulence levels are lower and fall offmore steeply with height compared with those of their respective neutral ABLs.There are, however, marked differences in the structure of the TKE. For all theE−� and E−ε closure models, E decreases steadily with height from its maximumsurface value. For q2� Model I and II, we see from Figure 10 that, with increasingheight, E initially increases from its surface value, reaches a maximum value aloft(≈ 15 m) and then decreases with height.


Figure 7. Evolution of the surface friction velocity (u∗0), kinematic heat flux (〈wθ〉0) and Obukhovlength (LO ) under a cooling rate of 1 K h−1 from different turbulence closures.

The E − � (D74) and E − ε − � closures predict the strongest nocturnal jetswhile q2� Models I & II have the weakest. The jet location predicted with E − �

(D74) and E − ε − � closures are very close to the surface compared with thoseobtained using the standard E − ε and the modified version (DE85). From Figures8 to 11 it can been seen clearly that the E − ε closure predicts one of the deepest


Figure 8. Vertical profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) from E−� closurewith two different formulation of � after 12 hr cooling.

nocturnal boundary layers and the largest surface values of TKE and turbulentfluxes (−〈uw〉 , −〈vw〉 , −〈wθ〉). Vertical distributions of eddy diffusivity for mo-mentum are plotted in Figure 12. There are large differences between the models,with the standard E − ε closure producing the largest value of Km, calculated fromEquation (10). This larger value of Km leads to a larger downward turbulent heatflux, see Figure 11. Note that although there are large differences in neutral ABLpredictions, the results of the q2� Model I and II are quite similar for the nocturnalboundary layer, as for results from the E − ε closure and its modified version(DE85), see Figures 7 and 9–12.

Based on similarity arguments, Zilitinkevich (1972) proposed that the height ofthe quasi-steady state stable boundary layer can be estimated from

h = c (u∗Lo/f )1/2 , (36)

where c is a constant. From his local scaling model, Nieuwstadt (1985) found thatc2 = √

3κRif . Based on the Cabauw experiment, he suggested the use of Rif ≈0.2 which leads to c ≈ 0.37, slightly smaller than the value c ≈ 0.4 given byGarratt (1982). Table II shows nocturnal ABL parameters from the different closureschemes, including the boundary-layer heights from the above equation. Generallyspeaking, the boundary-layer heights obtained from the equation are smaller thanthe heights where E is 5% of their surface values, calculated from the variousmodels. E − � (D74) and E − ε − � closures show the best agreement.


Figure 9. Vertical profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) from E − ε andits modified closures after 12 hr cooling.

For a quasi-stationary stable boundary layer, Nieuwstadt’s (1985) local scalingmodel predicts that the vertical profiles of turbulent fluxes obey

|τ |/u2∗0 =

(1 − z

h

)3/2, (37)

〈wθ〉/〈wθ〉0 =(

1 − z

h

). (38)

There was good agreement between this model and the observations taken on themeteorological mast at Cabauw (Nieuwstadt, 1985). Our model results with E − �

(D74 and AMLTV78) and E−ε−� closures show very good agreement with Equa-tions (37) and (38) except very close to the top of the boundary layer, as Nieuwstadt(1985) suggested, see Figure 13. The results of other turbulence closures can alsobe approximated by variations of Equations (37) and (38) with additional scalingand offset coefficients, see Figure 13.


Figure 10. Vertical profiles of wind (U, V ), TKE (E) and shear stress (〈uw〉 , 〈vw〉) from q2� closureafter 12 hr cooling.

4.3. DIURNAL CYCLE

Our diurnal cycle studies are based on running the models by specifying the diurnalvariation of the surface potential temperature (0) or the surface heat flux (〈wθ〉0).Several forms were tried including the sinusoidal variations of 0 and an empiricalformulation of 〈wθ〉0 (Stull, 1988, Section 7.3). Here, to construct a typical diurnalvariation of 0, we use the measured screen height (z = 2 m) temperature fromDay 33 of the Wangara Experiment (Clarke et al., 1971). We take their measure-ments of hourly potential temperature, i (i = 0, 1, . . . , 23 representing differenthours), at screen height and then set the surface potential temperature, 0 at 1900hours, equal to 19 and estimate 0 at other hours by 0 = 19+1.1×(i −19),where i = 0, 1, . . . , 23 (The factor 1.1 is an attempt to extrapolate the screen heighttemperature to the surface). A cubic spline interpolation was used for 0 at timesbetween hours, see Figure 14a. The initial potential temperature profile for all themodel runs is given by (z, 0) = 19 = 281.1 K for 0 ≤ z ≤ 1000 m, and (z, 0)

= 281.1 + 0.0035 (z − 1000) for z > 1000 m. The initial velocity field is a steadystate, neutral ABL with (Ug, Vg) = (10, 0) m s−1, f = 10−4 s−1 and z0 = 0.1 m.Thereafter the surface potential temperature is allowed to vary according to Figure14a with the calculation starting at 1900 local time (LT).


Figure 11. Vertical profiles of potential temperature () and kinematic heat flux (〈wθ〉) from differentturbulence closures after 12 hr cooling.

The simulations are carried out for 10 days with the surface potential temper-ature repeating cyclically every 24 hr. We should stress that we are not attemptingto match the Wangara observations, just using the surface temperature to give areasonable diurnal cycle. In future work, we will couple our ABL model to a landsurface scheme. Seeking a periodic diurnal cycle is somewhat artificial, imposing arequirement of no net heat flux over a day. This leads to a relatively warm boundarylayer and accounts for a late development of the deep convective boundary layer.The averaged mixed layer temperature m (between 1000–2000 m) for the tenthday of the simulation is 285.9 K for E − � (D74) closure, 286.4 K for the standardE − ε closure and q2� Model I and 287.1 K for the q2� Model II. There is onlya short period of time (1200 to 1800 LT) where 0 � m, see Figure 14a. In thefollowing, we shall look at the model results from the last day of the simulations.

Figures 14b–c show the resultant surface friction velocity and kinematic heatflux for the tenth cycle with different turbulence closures. Generally speaking, thediurnal cycles of u∗0 and 〈wθ〉0 closely reflect the variation of the surface potentialtemperature, 0. They increase rapidly between 0800–1000 LT in the morning anddecrease quickly in late afternoon from 1700–1900 LT. u∗0 reaches its maximumvalue around 1700 while 〈wθ〉0 obtains its maximum upward flux around 1000


Figure 12. Vertical profiles of the eddy diffusivity for momentum (Km) from different turbulenceclosures after 12 hr cooling.

LT. For u∗0, the magnitudes of the daily changes are smallest from E − ε closure(DE85) while q2� Model II predicts the largest; and for 〈wθ〉0, the E − ε closurepredicts the largest daily variation while E − � (D74) and E − ε − � closures givethe smallest daily changes.

There are qualitative differences in the time variation of the surface frictionvelocity, associated with surface heating and thermal stratification effects. Theseare sensitive to the imposed time variation of the surface temperature, and theresponse of the models to changes in thermal stratification. The results of E − �

(D74) and E − ε − � models indicate there is a local minimum while all the othermodel results show a local maximum in u∗0 around 1000 LT.

In Figures 15 to 17, we have plotted the diurnal cycles of the mean wind com-ponent, U , potential temperature, and the TKE, E. Note in these figures, two


Figure 13. Turbulent fluxes normalised by their surface values as a function of non-dimensionalheight, z/h (here h is calculated where E is 0.05E0 from the model), from different turbulenceclosures after 12 hr cooling.

days are plotted for better visualization and data for day 2 are the same as thosefor day 1. The maximum wind speed, the so-called nocturnal jet, occurs aroundmidnight. E − � (D74) closure predicts the strongest jet while the E − ε closureproduces the weakest. The height of the jet is around 200–300 m and decreasesslightly toward morning, see Figure 15.

It can be seen from Figure 16 that there are deep constant potential temperatureregions above which is the initial state stratification. The top of the ABL growsfrom the original 1000 m to 2230, 3000, 2565 and 2800 m for E − � (D74), thestandard E − ε closures and q2� Models I and II respectively. The largest changesare confined below 500 m, and even lower with E − � (D74) closure. Note alsothe slightly backward tilted vertical profile of in the upper part of the ABL fromE − � (D74) closure while others are more well mixed. A surface based inversiondevelops over the lowest 300 m (150 m for E − � (D74) closure) between 1800and 0800 LT. It is subsequently destroyed, from the surface upward, by heatingbetween 0800 and 1800 LT. Later in the afternoon between 1500 and 1800 LT, thewhole layer is essentially neutral.

Figure 17 shows the contours and vertical profiles of TKE for different tur-bulence closures. As expected, stronger turbulence occurs in the afternoon and


Figure 14. Surface variables changing with time. a) Potential temperature (0), b) friction velocity(u∗0) and c) kinematic heat flux (〈wθ〉0).


Figure 15. Mean wind component, U (m s−1), from different turbulence closures. From top to bottomthe closures used are E − � (D74), E − ε (the standard), q2� Models I and II.


Figure 16. Potential temperature, (K), from different turbulence closures. From top to bottom theclosures used are E − � (D74), E − ε (the standard), q2� Models I and II.


Figure 17. The TKE, E (m2 s−2), from different turbulence closures. From top to bottom the closuresused are E − � (D74), E − ε (the standard), q2� Models I and II. Note the different contour intervalsand different x-axis scales in profile plots.


reaches about 2000 m depth. The maximum values of E from the standard E − ε

and q2� Model II are about twice as large as those from E − � (D74) closure andq2� Model I. Note the slight forward tilting of the vertical distribution of E andthe fact that there are turbulence maxima aloft, for some closures, especially E − �

(D74). With E − � (D74) closure and q2� Model I there is significant (up to 0.05m2 s−2) turbulence aloft at midnight, and later with E − � (D74) closure.

5. Conclusions

Several different 112 -order turbulence closure models are used to simulate the at-

mospheric boundary layer. The eddy viscosity concept is employed to model theturbulent fluxes. All models use the turbulent kinetic energy equation together witheither a diagnostic expression or a prognostic equation to represent the turbulentlength scale. The effects of stability are realized via the turbulent length scale andthe TKE in E − � closure, the TKE and the dissipation rate of TKE in E − ε

closure and its modified versions and the turbulent length scale, the TKE and astability function in q2� Models I and II.

The turbulence length scale plays a very important role in any turbulence closuremodels. In the simple situations studied here, the models with TKE and a diagnosticequation for � are quite good, while the closures with TKE and a prognostic equa-tion for � do not guarantee success although these models do carry more physicalprocesses.

Full second-order closure scheme models, which can predict all components ofthe Reynolds stress, have significant advantages for some flow situations, e.g., flowover hills. Ayotte and Taylor (1995) used the LRR second-order scheme for theirneutral, linear ABL model of flow over topography but limited the time integrationof the background flow calculation to 4 hr in order to avoid having too deep aboundary layer. The problem with the ε- or q2�-equation, and prediction of a verydeep ABL and large values for the length scale in this flow situation, need to beresolved. However a model including prognostic equations for the shear stress isrequired for flows that are rapidly distorted by topography.

For an idealized neutral ABL, the results of q2� Model I, with an appropriatevalue of C�, agree well with that of E − � closure while the standard E − ε closuremodel and q2� Model II predict too much turbulent mixing and ABL heights thatare too large. In q2� Model I, the modelled ABL depth is sensitive to the constantC�, which is used in the formulation of the turbulent length scale, Equation (24).To closely match the ABL height of E − � closure, C� = 0.055 can be used in q2�

Model I.With modification to the ε-equation, one can obtain similar results from E − ε

closure (E − ε − � and DE85) to those of E − � closure although some furthermodification is needed to the model proposed by Detering and Etling (1985).


For the nocturnal ABL, the results of E − ε − � closure are similar to those ofthe E−� closure; the results of the E−ε (DE85) closure are similar to those of thestandard E − ε closure and the two q2� models show similar results. The standardE − ε closure model predicts the largest nocturnal boundary-layer depth and thestrongest turbulence; the E − � model estimates the lowest h and weak turbulence,while the results of q2� model are in between. In diurnal cycle simulations, againthe standard E − ε closure and q2� Model II predict the deepest ABL and strongerturbulence; the TKE maxima are about twice as large as those from E − � (D74)closure and q2� Model I.

The present study has been an attempt to re-evaluate commonly used ABLmodels using high vertical resolutions in simple atmospheric conditions. Futureresearch will incorporate a soil model or land surface scheme coupled to the sur-face energy budget, and make comparisons with the large-eddy simulations andexperimental data. Applications to the Martian boundary layer are also planned.

Acknowledgements

This research has been partially funded by the MATADOR project (Optech Ltd,Canadian Space Agency) and partially by grants from the Natural Science andEngineering Research Council of Canada.

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On Modelling the One-Dimensional Atmospheric Boundary Layer · Keywords: Atmospheric boundary layer, Depth of the ABL, Diurnal cycle, Neutral/stable ABL, Turbulence closure, Turbulence