Improving non-local parameterization of the convective boundary layer

Transcription

1 Boundary-Layer Meteorol DOI ORIGINAL PAPER Improving non-local parameterization of the convective boundary layer Zbigniew Sorbjan 1, 2, 3 Submitted on 21 April 2008 and revised on October 17, 2008 Abstract This paper proposes improvements in the "non-local" parameterization scheme of the convective boundary layer. The counter-gradient terms for components of momentum fluxes are introduced in a form analogous to those for other scalars. The scheme also includes explicit expressions for entrainment fluxes of momentum, temperature, and humidity. The simplified procedure for calculating the boundary-layer height is proposed. It consists of two steps: the evaluation of the convection level followed by the assessment of the depth of the interfaciallayer. Keywords: Boundary-layer parameterization, Convective boundary layer, Entrainment fluxes, Non-local parameterization scheme Z.Sorbjan, sorbjanz@mu.edu 1. Department of Physics, Marquette University, Milwaukee, WI , USA, 2. Faculty of Environmental Engineering, Warsaw University of Technology, Warsaw, Poland, 3. Geophysical Institute of Polish Academy of Sciences, Warsaw, Poland. 1

2 1 Introduction Mass and energy transfer through the atmospheric boundary layer (ABL) regulates a broad variety of processes in the entire atmosphere. As a result, weather and climate models show appreciable sensitivity to the ABL formulation, especially in long-term integrations over lands and oceans. In recent years, weather forecasting and climate simulations have progressed significantly, thereby allowing unprecedented improvements in spatial resolution at an affordable computing cost. Finer grids require further refining of parameterizations in order to minimize biases caused by inaccurate representation of unresolved physical processes. The development of a refined parameterization is the subject of this paper. Effects of the ABL in weather and climate models are usually included by calculating tendencies of basic meteorological parameters due to divergence of turbulent fluxes: $ u ' & ) " & v ) "t &# ) & ) % q ( turb $ R x ' = * " & ) & R y ) "z & H ) & ) % Q ( (1) where u and v are the horizontal components of the wind velocity, Θ is the virtual potential temperature, q is the specific humidity, R x, R y, H, and Q are the turbulent fluxes. The tendencies are evaluated in a vertical domain, defined by the boundary layer height h. Generally, the fluxes can be assumed linear, as in bulk models, expressed in terms of the K theory, or higher-order closures (e.g., Ayotte et al 1996; Zhang and Zheng 2004). One of the parameterization schemes widely used in current operational practice was originally developed by Troen and Mahrt (1986). Subsequently, it has been augmented and modified by a number of researchers (e.g., Holtslag et al 1990; Vogelezang and Holtslag 1996; Holtslag et al 1995; Brown and Grant 1997; Noh et al 2003; Brown et al 2006, Brown et al 2008). The scheme is applicable in both stable and convective conditions and is simple and robust (Holtslag and Boville 1993; Giorgi et al. 1993; Alpaty et al. 1996; Hong and Pan 1996; Noh et al. 2003; Hong et al. 2006, Collins et al 2006). 2

3 Despite its advantages, the scheme includes several deficiencies. For example, entrainment fluxes are not explicitly evaluated, and are therefore inaccurate (Noh et al., 2003). Effects of non-local mixing for momentum are misrepresented, causing the surface winds to be too weak (e.g., Brown et al., 2006). The ABL growth oversimplifies the processes in the interfacial layer. The paper has the following structure. Properties of the convective boundary layer are briefly reviewed in Section 2. The procedure for calculation of the boundary-layer height is discussed in Section 3. Computation of turbulent fluxes is considered in Sections 4 and 5. Final remarks are presented in Section 6. 2 Convective boundary layer The atmospheric boundary layer constitutes a complex system, which is under permanent transition, enforced by a variety of internal and external factors. Its brief description based on the large-eddy simulations (LES) of Sorbjan (2005; 2006) is presented below, in order to identify the parameters for an appropriate parameterization of the convective boundary layer. A typical distribution of the potential virtual temperature and its flux in a mid-day (quasisteady) boundary layer during shear-less conditions (free-convection) is shown in Figure 1. Note that the vertical coordinate in the figure is scaled by parameter z i, defined as the height of the minimum heat flux. Two cases are considered in that figure. The first one was obtained for a weak interfacial inversion (case W). The second case was generated for a strong interfacial inversion (case S), characterized by a sharp, nearly 15 K temperature jump above the mixed layer. The formation of large temperature jumps in interfacial layers occurs mainly because of subsidence above the boundary layer and mixing in the boundary layer (Sorbjan, 2007). Figure 1 indicates that the mean value of the temperature gradient " i in the interfacial layer affects the temperature flux at the top of the mixed layer. For typical values of " i, the ratio of the entrainment and surface fluxes H i /H o is about 0.2 during shear-less conditions (it is larger during sheared convection). It follows from the figure that when the temperature gradient in the interfacial layer increases or decreases, the entrainment temperature flux also increases or decreases. For example, for a very small temperature gradient, as during free encroachment, the 3

4 entrainment flux is expected to also be very small (Sorbjan, 1996). In a case, when the temperature gradient is large, the entrainment flux is also large, because very warm air is entrained into the mixed layer. The described trend can be explained in terms of the K theory. A typical distribution of the humidity and humidity flux, obtained in the same case of a mid-day (quasi-steady) boundary layer, during shear-less and barotropic convection, is shown in Figure 2. The two cases considered in Figure 2 will be referred to as drying and moistening. They were obtained in cases (W) and (S), depicted in Figure 1. It is worth stressing that the same value of the surface humidity flux Q o was used in both simulations. In the moistening case, the humidity flux decreases with height. In the drying case, the humidity flux increases with height. Figure 2 indicates a strong sensitivity of humidity fields to the temperature gradient in the interfacial region. A simple condition for the presence of moistening or drying regimes in the mixed layer was derived by Sorbjan (2005). If q * < "0.022g i w * /N i the mixed layer will be dried, otherwise it will be moistened, where g i is the mean humidity gradient in the interfacial layer, N i = interfacial Brunt-Vaisala frequency, β is the buoyancy parameter, w * = ("z i H o ) 1/ 3 "# i is is the convective velocity scale, q * = Q o /w * is the convective humidity scale, H o and Q o are the surface temperature and humidity fluxes. The ratio w * /N i can be assumed to be proportional to the depth of the interfacial layer (Sorbjan, 2007). Consequently, the product g i w * /N i is the measure of the humidity jump in the interfacial layer. A typical distribution of wind velocity components and momentum flux in the convective boundary layer is shown in Figure 3. It was obtained for mid-day (quasi-steady), barotropic conditions (forced convection), for weak (W) and strong (S) interfacial inversion layers, and for three values of the geostrophic wind G = 5, 10, and 15 m s -1. The geostrophic wind vector was oriented along the x-axis. The figure indicates that wind velocity is nearly uniform in the mixed layer. The v- component of wind is very small. Consequently, the cross-isobar angle in the convective case is also very small. The x-component of the momentum flux linearly decreases with height, while the y-component is near zero. The momentum flux at the top of the mixed layer is proportional to wind shear. Note that the x-component of the momentum flux is the largest in the case where G 4

5 = 15 m s -1. It is larger in case W (weak interfacial inversion, w * /N i = 83.6 m), and smaller in case S (strong interfacial inversion, w * /N i = 17.9 m). This implies that the momentum flux at the top of the mixed layer is also proportional to the parameter w * /N i. Figures 1 3 refer to the barotropic boundary layer, when the horizontal temperature gradient is zero and the geostrophic wind is independent of height. In the baroclinic case, the geostrophic wind varies with height, and the horizontal temperature gradient is non-zero. The effects of the resulting wind shear at the top of the mixed layer can be parameterized as in the case of the barotropic forced convection (Sorbjan, 2004). 3 Depth of the convective boundary layer The height of the convective boundary layer h can be defined in various ways. It can be associated with the height z i of the minimum heat flux H i, with the height Z i of the maximum positive temperature gradient above the mixed layer, or defined as Z T = z i + Δh, where Δh is the depth of the interfacial layer. The height z = z i approximately indicates the top of the mixed layer. The level z = Z i is located in the interfacial layer. The level z = Z T marks the top of the boundary layer. Above this level, the turbulent heat flux is negligible. In stable conditions, Troen and Mahrt (1986) defined h as the value which satisfies the following equation, based on the definition of the critical Richardson number: "h # h $# s U h 2 = Ri cr (2) where β = g/t o is the buoyancy parameter, Θ h is the potential temperature at z = h, Θ s is surface temperature, U h = u h 2 + v h 2 is the modulus of the wind velocity at z = h, and Ri cr is the 5

6 critical Richardson number, usually assumed to be equal to 1/4. For a constant temperature gradient γ in the stable boundary layer, the solution to Equation (2) is in the form: h = Ri cr U h N (3) where N = "# is the Brunt-Vaisala frequency. If the temperature gradient is not constant with height, the value of h in (2) can be found iteratively. Troen and Mahrt (1986) assumed that (2) can also be used to evaluate the height of the convective boundary layer. This ad hoc approach is unphysical in convective conditions. The Richardson number in the mixed layer is near zero, and therefore below its critical value. Let us rewrite (2) in the form: "(h) = " s + Ri c U h 2 #h (4) where h " Z T is the height of the convective boundary layer. Some authors increase the value of the critical Richardson number to Ri cr /a, where a is equal to about 1/3 (e.g., Noh et al, 2003). In such a formulation, (4) can be rewritten as: " h = " s + Ri cr a U h 2 #h (5) We will assume that the parameter a can be interpreted as a relative depth of the interfacial layer: a = (h " z c ) / h, where z c is the level of convection, for which: "(z c ) = " s (6) The level of convection z c can be evaluated iteratively by solving (6). The resulting value is expected to be approximately equal to the mixed layer height z i. For this reason, based on Figure 6

7 1, one can conclude that the value of " s in (6) should be calculated (using Monin-Obukhov similarity) as the temperature Θ(z 1 ) at the level z 1 " 0.05 z i. Based on (6), equation (5) can be rewritten in the form: " h #" c = Ri cr U h 2 $(h # z c ) (7) Equation (7) refers to the stably stratified interfacial layer on the top of the mixed layer, and therefore is more pertinent than (2). The equation implies that in the interfacial layer there exists equilibrium between turbulence production by wind shear and the stabilizing effects of buoyancy. The interfacial layer is turbulent, and the resulting Richardson number is equal to the critical value. Following Vogelzang and Holtslag (1996), we can further modify the above equation and replace U 2 2 h in (7) by U h "U c = u h " u c [( ) 2 + ( v h " v c ) 2 ]. As a result, we will obtain: U " h # " c = Ri h #U c cr $(h # z c ) 2 (8) If the temperature gradient " i above the mixed layer is assumed constant, then the solution of (8) is: h " z c = Ri c U h "U c N i (9) which is analogous to (3). In the case of free convection, the shear term in (10) disappears, which indicates that the depth of the interfacial layer h " z c = 0. This result disagrees with the observations, which show that h " z c > 0. This flaw in the scheme can be avoided by assuming that: h " z c = c h z c (10) where the constant c h is in the range from 0.15 to 0.2, as seen in Figures

8 The above discussion implies that the procedure for evaluating the height h of the boundary layer in daily conditions can be improved by executing it in two steps. During the first step, the level of convection level z c can be evaluated iteratively, by solving (6). During the second step, the height h can be obtained from (10). 4 Turbulent fluxes For a specified height of the boundary layer, Troen and Mahrt (1986) evaluated the turbulent fluxes based on the K theory. Specifically, the momentum fluxes were expressed in terms of local gradients: R x = "K m du dz (11a) R y = "K m dv dz (11b) where K m is the eddy viscosity. The modified K theory was applied for scalars: H = "K h ( d# dz " g $ ) (12a) Q = "K q ( dq dz " g q ) (12b) where K h, K q are the eddy diffusivities, g θ and g q are the counter-gradient terms. The eddy viscosity was assumed to be in the form: % "u * h '# m (z /L) K m = & ' "u * h (' # m (0.1h /L) z h (1$ z /h)2 for z < 0.1h z h (1$ z /h)2 for z > 0.1h (13) 8

9 where L = "u 3 * /(#$H vo ) is the Monin-Obuhkov length, H vo = ( q)H o T o Q o is the surface flux of the virtual temperature, H o and Q o are the surface fluxes of the potential temperature and humidity, u * is the friction velocity, T o is a reference temperature, and " m (z / L) = (1# 7z / L) #1/ 3 is the Monin-Obukhov similarity function for wind. During free convection, when u * 0, u * /" m (0.1h /L) = (u 3 * w 3 * ) 1/ 3 # w s $ 0.65w *. The choice of the power coefficient in (13) is quite arbitrary. In the surface layer, where z/h is small, the coefficient K m assumes a shape predicted by the surface similarity theory. The Prandtl numbers, Pr θ = K m /K h and Pr q = K m /K q, are used to prescribe the eddy diffusivities for temperature and humidity. Troen and Mahrt (1986) assumed that the Prandtl numbers were constant above the level of 0.1h. According to Holtslag and Boville (1993) the Prandtl numbers vary from 1 in neutral conditions to 0.6 in very unstable cases. The counter-gradient terms are defined for z/h > 0.1 as: g " = C " H o w s h (14a) g q = C q Q o w s h (14b) According to Troen and Mahrt (1986), C θ " 5 and C q " 10. One might notice two major drawbacks in the above scheme. The first one is related to the values of the entrainment fluxes at the top of the mixed layer. Using (12) and (13) for temperature, we can obtain that at z = z c, the flux entrainment H i ~ K h γ i ~ w s z c γ i. In free convection, when u * " 0, this yields H i " w * z c γ i,. One would rather expect that the flux entrainment H i ~ w * 2 Ni /β ~ w * 2 " i /# (Sorbjan, 2004; 2005; 2006). The second shortcoming is due to the fact that (11) incorrectly describes the momentum fluxes in the mixed layer. Referring to Figure 3a, we see that the velocity gradients in the mixed layer are nearly negligible. As a result, (11) implies that the momentum fluxes will vanish, 9

10 which disagrees with Figure 3b. 5 Modified turbulent fluxes To improve the scheme described in the previous Section, we propose the following expressions: $ "K m ( du dz " g ) + R z x xc for z /z c #1 & z c R x = % & h " z R xc for 1< z /z c < h /z c '& h " z c (15a) $ "K m ( dv dz " g z y) + R yc for z /z c #1 & z R y = c % & h " z R yc for 1< z /z c < h /z c '& h " z c (15b) & "K h ( d# dz " g z $ ) + H c for z /z c %1 ( z H = c ' ( h " z H c for 1< z /z c < h /z c )( h " z c (15c) $ "K q ( dq dz " g z q) + Q c for z /z c #1 & z Q = c % & h " z Q c for 1< z /z c < h /z c '& h " z c (15d) where R xc, R yc, H c, Q c are the entrainment fluxes. In the mixed layer (z/z c 1), the fluxes consist of three components. The local terms are proportional to local gradients. The effects of large eddies are expressed by counter-gradient. The effects of entrainment are included in the entrainment terms. We will assume that counter-gradient parameters in (15) can be expressed as: 10

11 g x = C r (z c / L) R xo w s z c (16a) g y = C r (z c / L) R y o w s z c (16b) H o g " = C " w s z c (16c) g q = C q Q o w s z c (16d) where R xo = "u * 2 cos# o, R yo = "u * 2 sin# o, α o = arctg (v o /u o ) is the cross isobar angle, u o, v o are the components of the surface layer wind, and 3 z C r (z i / L) = c i / L r 1+ z i / L = c w * r u 3 * + w (17) 3 * 3 3 During convective conditions, u * << w*, and Cr (z i / L) = c r. On the other hand, in nearly neutral conditions w * " 0, and C r (z i / L) " 0. Consequently, the counter-gradient terms g x, and g y disappear during the transition between the convective and stable states. Note that (16c) and (16d) coincide with (14a) and (14 b). We will assume that c r =11, and C θ = C q = 5.5. The eddy viscosity will be rewritten as: %' K m = "u z (1# 7z * /L)1/ 3 (1# z /z c ) 2 for z /z c $ 0.1 & (' "w s z (1# z /z c ) 2 for 0.1< z /z c <1 (18) Note that the eddy diffusivity in (18) is not defined above the convection layer z c. 11

12 Based on the gradient-based similarity in the interfacial layer, entrainment fluxes for scalars take on the following form (Sorbjan 2006, 2008): H c = "c H # i w * 2 N i f H (R i ) (19a) Q c = "c Q q i w * 2 N i f Q (R i ) (19b) where f H, f Q are the stability functions of the Richardson number of the form f H = f Q = (1+c/R i )/(1+1/R i ) 0.5, with c = 8. The functions were evaluated based on the LES model (Sorbjan, 2006). Note that 1/R i = 0 in free convection, and 1/R i > 0 in shear convection. We will assume that c H = 0.012, and c Q = Similarly, it can be assumed that: # 2 w R x c = " c * R 2 N S f (R ) & % i i i ( s x i (20a) $ i ' # 2 w R y c = " c * R 2 N S f (R ) & % i i i ( s y i (20b) $ i ' f i (R i ) is the stability functions of the interfacial Richardson number, and s xi = (u h " u c ) /(h " z c ), s yi = (v h " v c ) /(h " z c ), and S i = s xi 2 + s yi 2. Based on the results of the LES model, we will assume that f i =1 1/R i, and c R = 0.2. Note that the entrainment fluxes in (20) are in a form implied by the K theory, with the eddy viscosity proportional to a product of the squared length scale w * /N i, the stability function f i (R i ) of the Richardson number, and the wind shear S i. When the ratio S i /N i is incorporated into the stability function, (19) and (20) have an analogous form. The above parameterization includes several elements improving the original scheme of 12

13 Troen and Mahrt, namely: (i) the counter-gradient terms for all scalars, (ii) the directly specified entrainment fluxes for all scalars, (iii) the simplified procedure for evaluating the height h of the boundary layer. The parameterization differs from the approach of Frech and Mahrt (1995) and Brown and Grant (1997), who modified the scheme of Troen and Mahrt (1986) by adding the non-local components of the momentum flux into (11 a, b). The added components were defined as products of a specified, non-linear function of height and the relative wind shear components in the mixed layer. Brown et al (2006) and Brown et al (2008) neglected the effects of the relative wind shear and slightly modified the stability coefficient. As a result, their non-local components of the momentum flux are similar to the counter-gradient terms K m g u and K m g v in (15), but differ slightly with respect to the stability coefficients. Contrary to our approach, Frech and Mahrt (1995), and Brown and Grant (1997), Brown et al (2006), and Brown et al (2008), did not specify entrainment fluxes. The negative effects of such a simplification for momentum are expected to be observed in strong baroclinic conditions. The most complete parameterization of the convective boundary layer was presented by Noh et al (2003). The authors included the non-local terms for temperature and momentum in the mixed layer, and also in the interfacial layer, and directly specified entrainment fluxes for these parameters. The entrainment fluxes were evaluated based on the bulk-modeling approach. The fluxes in the interfacial layer were calculated by using an ad hoc profile of the eddy diffusivity. The difference between the parameterization Noh et al (2003) and the presented approach is in the details. The local and counter-gradient terms in (15) are evaluated for all scalars in the mixed layer (z < z c ). The entrainment fluxes (19)-(20) are directly specified on the top of the mixed layer, based on gradient-based similarity (Sorbjan, 2004; 2005; 2006). Therefore, they have appropriate asymptotes and depend on the interfacial Richardson number. The result of the proposed approach for the interfacial fluxes is depicted in Figure 4. Figure 4a displays the values of dimensionless entrainment humidity flux Q i /Q o, obtained from a LES model in the barotropic, quasi-steady free convection cases, when f Q (R i ) = 1, as described by Sorbjan (2005), versus the estimated values based on Equation (19b). Figure 4b shows the 2 dimensionless momentum fluxes R xi /w * and R yi /w 2 *, obtained from a LES model of Sorbjan (2006) in the barotropic, quasi-steady cases, versus the estimated values based on equations (20 13

14 a, b). Figure 5a displays the values of a dimensionless humidity flux H/H o (the black continuous line) obtained from Eqs. (15c), (16c), (18), and (19a). The local, entrainment and counter-gradient contributions to the flux, indicated as (LOC), (ENT), and (CTG), are marked by red, blue, and green lines. The LES results (Sorbjan 2005) in the barotropic, quasi-steady free convection case A (h = 950 m, γ i = 7.33 K km -1 ) are represented by a dashed line. Figure 5b 2 shows the dimensionless momentum flux R xi /u * (the black continuous line), obtained from Equations (15a), (16a), (17), (18), and (20a). The LES result of Sorbjan (2006)), obtained in the barotropic, quasi-steady forced convection case W15 (h = 850 m, α o = 2.66 o, γ i = 7.25 K km -1 ), is represented by a dashed line. The entrainment fluxes (19) and (20) crucially depend on the accuracy in evaluating interfacial temperature, humidity and wind velocity gradients. Consequently, a fine vertical resolution is required for their appropriate assessment. The vertical grid increment Δz should be smaller than a fraction of the interfacial layer depth (h z c ). Presently, the vertical resolution of global and regional climate models is significantly coarser. When a fine grid resolution cannot be applied, entrainment fluxes for scalars (potential temperature, humidity) can also be obtained by integrating (1) with respect to height (Sorbjan, 1995). Specifically for the potential temperature Θ, it can be derived that: z c H c = H o " % #$ dz (21) 0 #t where H c = (H c 2 + H c1 ) /2, H o = (H o2 + H o1 ) /2, "# /"t = (# 2 $ # 1 ) /(t 2 $ t 1 ), while the subscripts 1 and 2 indicate the subsequent instants of time. Analogous expressions for the momentum fluxes would be incorrect, because the velocity tendencies are balanced not only by divergence of the momentum fluxes, but also by ageostrophic wind components. 14

15 6 Final remarks The "non-local" parameterization scheme in convective conditions has been discussed in this paper. Several improvements have been put forward which do not require significant alterations in the original algorithm. A modified procedure for calculating the height h of the boundary layer has been suggested. It consists of two steps: the evaluation of the level of convection and then the assessment of the depth of the interfacial layer. The counter-gradient terms have been introduced for the components of the momentum fluxes in a form analogous to those for scalars. Direct inclusion of the entrainment fluxes, R xi, R yi, H i, Q i for momentum, temperature, and passive scalars, has been proposed through the use of the gradient-based similarity in the interfacial layer. Since the interfacial fluxes within the introduced scheme are based on temperature, humidity, and wind velocity gradients in the interfacial layer, an increased vertical resolution is required for improved performance. A simplified approach has been also proposed for temperature and humidity in the case when a fine grid resolution cannot be applied. Acknowledgments Preliminary tests of the scheme were performed in May 2008, using the RegCM model at the International Center for Theoretical Physics in Trieste (Italy). Helpful discussions with Dr. Filippo Giorgi are gratefully acknowledged. The work has been partly supported by the Central and Eastern Europe Climate Change Impact and Vulnerability Assessment Project (CECILIA), financed by UE 6.FP, Contract GOCE to Warsaw University of Technology, Warsaw, Poland. References Alapaty K, Pleim JE, Raman S, Niyogi DS, Byun DW (1997) Simulation of atmospheric boundary layer processes using local- and nonlocal-closure schemes. J Appl Meteorol 36: Ayotte, KW., Sullivan PP, Andren, Doney ASC, Holtslag A, Large WG, McWilliams JC, Moeng C-H, Otte M, Tribbia J, Wyngaard JC (1996) An evaluation of neutral and convective planetary boundary layer parameterizations relative to large eddy simulation. 15

16 Boundary-Layer Meteorol 79: Brown AR, Grant LM (1997) Non-local mixing of momentum in the convective boundary layer. Boundary-Layer Meteorol 84: 1-22 Brown AR, Beljaars ACM, Hersbach H (2006) Errors in parameterizations of convective boundary layer turbulent momentum mixing. Quart J Roy Meteorol Soc 132: Brown AR, Beare RJ, Edwards JM, Lock AP, Keogh SJ. Milton SF, Walters DN (2008) Upgrades to the boundary layer scheme in the Met Office Numerical weather prediction model. Boundary-Layer Meteorol 128: Collins WD, Bitz CM, Blackmon ML, Bonan GB, Bretherton CS, Carton JA, Chang P, Doney SC, Hack JJ, Henderson TB, Kiehl JT, Large WG, McKenna DS, Santer BD, Smith RD (2006) The Community Climate System Model Version 3 (CCSM3). J Climate 19 (11): Frech M, Mahrt L (1995) A two scale mixing formulation for atmospheric boundary layer. Boundary-Layer Meteorol 73: Giorgi F, Marinucci MR, Bates GT (1993) Development of a Second-Generation Regional Climate Model (RegCM2). Part I: Boundary-Layer and Radiative Transfer Processes. Mon Wea Rev 121: Holtslag, AAM, de Bruijn EIF, Pan H-L (1990) A high resolution air mass transformation model for short-range weather forecasting. Mon Wea Rev 118: Holtslag, AAM, B. Boville, 1993 Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate Model. J Climate 6: Holtslag AAM, Van Meijgaard E, De Rooy WC (1995) A comparison of boundary layer diffusion schemes in unstable conditions over land. Boundary-Layer Meteorol 76 (1-2): Hong S-Y, Pan H-L (1996) Nonlocal boundary layer vertical diffusion in a medium-range forecast model. Mon Wea Rev 124: Hong S-Y, Noh Y, Dudhia J (2006) New Vertical Diffusion Package with an Explicit Treatment of Entrainment Processes. Mon Wea Rev 134, 9: Noh Y, Cheon WG, Hong SY (2003) Improvement of the K-profile Model for the Planetary Boundary Layer based on Large Eddy Simulation Data. Boundary-Layer Meteorol 107:

17 Sorbjan Z (1995) Toward evaluation of heat fluxes in the convective boundary layer. J Appl Meteor 34, 5: Sorbjan Z (1996) Effects caused by varying strength of the capping inversion based on a largeeddy simulation of the shear-free convective boundary layer. J. Atmos. Sci. 53: Sorbjan Z (2004) Large-eddy simulations of the baroclinic boundary layer. Boundary-Layer Meteorol 112: Sorbjan Z (2005) Statistics of scalar fields in the atmospheric boundary layer based on largeeddy simulations. Part I: Free convection. Boundary-Layer Meteorol 116 (3): Sorbjan Z (2006) Statistics of scalar fields in the atmospheric boundary layer based on largeeddy simulations. Part II: Forced convection. Boundary-Layer Meteorol 119 (1): Sorbjan Z (2008) Gradient-based similarity in the atmospheric boundary layer. Acta Geophys 56 (1): Troen I, Mahrt L (1986) A simple model of the atmospheric boundary layer: Sensitivity to surface evaporation. Boundary-Layer Meteorol 37: Vogelezang DHP, Holtslag AAM (1996) Evaluation and model impacts of alternative boundarylayer height formulations. Boundary-Layer Meteorol 81: Zhang D-L, Zheng W-Z (20004) Diurnal cycles of surface winds and temperatures as simulated by five boundary layer parameterizations. J Appl Meteorol 43:

18 Figure Captions Figure 1. A typical distribution of: (a) the potential temperature, and (b) the potential temperature flux in the boundary layer during quasi-steady and shear-less convection for weak (W) and strong (S) interfacial inversion, obtained from a LES model (runs A and B) of Sorbjan (2005). Figure 2. A typical distribution of: (a) the humidity, and (b) the humidity flux in the boundary layer for shear-less convection cases shown in Figure 1, based on a LES model (runs A and B) of Sorbjan (2005). The initial humidity profile is marked by a red line. Figure 3. A typical distribution of: (a) the wind velocity components u and v for weak (W) and 2 strong (S) interfacial inversion, (b) the dimensionless momentum fluxes R x /w * (solid black 2 lines) and R y /w * (red lines) in the boundary layer in case (W). The momentum flux, obtained in case (S) for the geostrophic wind of 15 m s -1, is indicated by a broken line. Both plots were obtained based on a LES model (Sorbjan, 2006) for the geostrophic wind G = 5, 10 and 15 m s -1. Figure 4. Dimensionless values of: (a) entrainment humidity fluxes Q i /Q o, obtained from the LES model in the barotropic, quasi-steady free convection cases A, B, C, and D, described by Sorbjan (2005), versus the estimated values based on Eq. (21b), (b) The entrainment momentum 2 2 fluxes R xi /w * (negative values) and Ryi /w * (positive values), obtained from the LES model of Sorbjan (2006) for six cases of barotropic, quasi-steady, forced convection, versus the estimated values based on Eq. (22a) and (22b). Figure 5. Values of: (a) the dimensionless heat flux H/H o, obtained from Eq. 15c (solid line), and from the LES model (dashed line) in the barotropic, quasi-steady free convection case A, 2 described by Sorbjan (2005), and (b) the dimensionless momentum flux R xi /u *, obtained from Eqs. 15a (solid line), and from the LES model (dashed line) for the barotropic, quasi-steady 18

19 forced convection case W15, described by Sorbjan (2006). The local, counter-gradient and entrainment contributions to the fluxes are indicated by letters LOC, CTG, and ENT. 19

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