Consistent Two-Equation Closure Modelling for Atmospheric Research: Buoyancy and Vegetation Implementations

Transcription

1 Boundary-Layer Meteorol (2012) 145: DOI /s ARTICLE Consistent Two-Equation Closure Modelling for Atmospheric Research: Buoyancy and Vegetation Implementations Andrey Sogachev Mark Kelly Monique Y. Leclerc Received: 31 August 2011 / Accepted: 6 April 2012 / Published online: 10 May 2012 Springer Science+Business Media B.V Abstract A self-consistent two-equation closure treating buoyancy and plant drag effects has been developed, through consideration of the behaviour of the supplementary equation for the length-scale-determining variable in homogeneous turbulent flow. Being consistent with the canonical flow regimes of grid turbulence and wall-bounded flow, the closure is also valid for homogeneous shear flows commonly observed inside tall vegetative canopies and in non-neutral atmospheric conditions. Here we examine the most often used two-equation models, namely E ε and E ω (where ε is the dissipation rate of turbulent kinetic energy, E, andω = ε/e is the specific dissipation), comparing the suggested buoyancy-modified closure against Monin Obukhov similarity theory. Assessment of the closure implementing both buoyancy and plant drag together has been done, comparing the results of the two models against each other. It has been found that the E ω model gives a better reproduction of complex atmospheric boundary-layer flows, including less sensitivity to numerical artefacts, than does the E ε model. Re-derivation of the ε equation from the ω equation, however, leads to the E ε model implementation that produces results identical to the E ω model. Overall, numerical results show that the closure performs well, opening new possibilities for application of such models to tasks related to the atmospheric boundary layer where it is important to adequately account for the influences of both vegetation and atmospheric stability. Keywords Atmospheric boundary layer Atmospheric surface layer Canopy flow Monin Obukhov similarity theory Non-neutral stratification Turbulence closure Two-equation closure models A. Sogachev (B) M. Kelly Wind Energy Division, Technical University of Denmark, Risø Campus, Frederiksborgvej 399, Building 118, P.O. Box 49, 4000 Roskilde, Denmark anso@risoe.dtu.dk M. Kelly mkel@risoe.dtu.dk M. Y. Leclerc Laboratory for Environmental Physics, The University of Georgia, 1109 Experiment Street, Griffin, GA , USA mleclerc@uga.edu

2 308 A. Sogachev et al. 1 Introduction The two-equation closure approach, based on coupled transport equations for the turbulent kinetic energy (TKE), E, and a supplementary length-scale determining variable ϕ (typically the dissipation rate), provides the minimum level of complexity capable of simulating the effective turbulence length scale, l, as a dynamic variable the condition needed to adequately simulate airflow over complex terrain or through the diurnal cycle (e.g. Ayotte et al. 1999; Finnigan 2007) without additional speculation (e.g. LaunderandSpalding 1974; Kantha 2004). Being of relatively low computational cost, the approach appears to be the optimal choice for practical tasks where the interaction and joint effects of heterogeneities are more interesting than a highly detailed description of the turbulence field (Sogachev and Panferov 2006) and/or where uncertainties introduced by the underlying surface (e.g. forest properties) limit the accuracy of modelled wind statistics, regardless of the turbulence closure chosen (Pinard and Wilson 2001). This is in contrast to higher-order closure models (e.g. Rao et al. 1974; Launder et al. 1975) or large-eddy simulations (Deardorff 1972; Moeng 1984; Shaw and Schumann 1992), which are capable of effectively simulating flow statistics but with a greater computational cost. These are not without their own drawbacks; e.g. the former kind of models suffer from an abundance of required constants (Wichmann and Schaller 1986), and the latter typically have issues e.g. within transition regions of the diurnal cycle (Basu et al. 2008). In spite of the appealing properties mentioned above, the two-equation closure methodology receives limited attention in atmospheric research, as opposed to its use in computational fluid dynamics (CFD) research and industrial applications (see Hanjalić 2005, orhanjalić and Kenjereš 2008 for a review). Applications of the closure to atmospheric and oceanic flows have highlighted serious uncertainties in the treatment of buoyancy and vegetativedrag effects (Duynkerke 1988; Svensson and Häggkvist 1990; Apsley and Castro 1997; Wilson et al. 1998; Baumert and Peters 2000; Sanz 2003; Kantha 2004; Katul et al. 2004; Sogachev and Panferov 2006). These uncertainties raise concerns about correctly solving conventional black box supplementary equations (e.g. Wilson et al. 1998; Cheng et al. 2002; Wilson 2011), which considerably limits dissemination of this closure methodology and its subsequent application to atmospheric flows. Based on ideas and results of Sogachev and Panferov (2006), who suggested an approach to minimizing the uncertainties for vegetative canopy flows, Sogachev (2009) tried to generalize the closure method for two-equation models for any source/sink term as well as one associated with buoyancy. According to him any non-shear source/sink term (X i ) appearing in the TKE equation leads to a term of the form ) α i (Cϕ1 C ϕ2 (X i ) ϕ (1) E in the supplementary equation for ϕ. In the expression above Cϕ1 is a form of C ϕ1 updated for atmospheric boundary-layer (ABL) modelling (defined below as Eq. 15)(Apsley and Castro 1997); C ϕ1 and C ϕ2 are conventional coefficients of production and destruction terms in the equation for ϕ, respectively (e.g. Pope 2000; Wilcox 2002), and α i are newly introduced coefficients. Despite progress in the generalization of vegetative-drag effects, Sogachev (2009) did not reach a universal form for buoyancy effects: his suggested method proved to be suitable only with α i defined specifically for each model. This paper provides a consistent general form, correcting the extra terms appearing in the supplementary equations. Additionally an improved form for the modified shear-production coefficient Cϕ1, different from that in Apsley and Castro (1997)orSogachev (2009), is

3 Consistent Two-Equation Closure Modelling for Atmospheric Research 309 found more appropriate for ABL simulation. This in turn dictates modification of the other source/sink coefficients α i. We examine the behaviour of the improved E ε and E ω closures using the ABL model SCADIS (SCAlar DIStribution) based on Reynolds-averaged Navier Stokes (RANS) equations (Sogachev et al. 2002, 2005; Sogachev and Panferov 2006). Assessment of the performance of these two closures includes the comparison of model results to Monin Obukhov similarity theory (MOST) (Monin and Obukhov 1954) over flat rough surfaces, and evaluation of the results from the two models for (forested) canopy-flow cases. 2 Consistent Two-Equation Closure for Atmospheric Research 2.1 Background To obtain the eddy viscosity, K l E, and thus close the Reynolds-averaged momentum and energy equations based on the Boussinesq approximation for turbulent fluxes (Pope 2000; Pielke 2002), two differential transport equations must be solved. These equations, applied to atmospheric flow in one-dimensional form, are (Kantha 2004; Sogachev 2009) E ( K t z σ ϕ E ϕ t z E z ( K ϕ σ ϕ z ) = P ε + B + S p S d, (2) ) = ϕ E (C ϕ1 P C ϕ2 ε + C ϕ3 B + C ϕ4 S p C ϕ5 S d ), (3) where σ ϕ E and σ ϕ are the Schmidt numbers for E and ϕ, respectively (with σ ϕ E also depending on the ϕ equation used for closure), t is time, z is height, P is the rate of shear production, ε is the rate of dissipation of TKE, B is the buoyancy source/sink term (e.g. Pope 2000; Pielke 2002), while S p and S d are wake production and enhanced dissipation due to the surface drag of canopy elements, respectively (e.g. Raupach and Shaw 1982; Finnigan 2000; Finnigan and Shaw 2008). Further l and K are expressed in terms of E and ε as μ E 3/2 l = C3/4 ε E 2 K = C μ, (4) ε, (5) where ε can be obtained from ϕ (e.g. for the specific dissipation of TKE, ϕ = ω = ε/e), and the closure constant C μ equals the squared ratio of the equilibrium shear stress to TKE. More exhaustive descriptions of two-equation models can be found in Pope (2000) and Wilcox (2002). The modelling framework implicit in (3) is that each source/sink term in the E equation implies a corresponding term in the ϕ equation, with a number of model coefficients: C ϕ1, C ϕ2, C ϕ3, C ϕ4 and C ϕ5. The usual way of specifying the constants is by studying the behaviour of the equation for ϕ in various flows. C ϕ1 and C ϕ2 are model constants selected to be consistent with both the von Kármán constant, κ, in the logarithmic law region of nearwall flow and experimental observations for decaying homogeneous, isotropic turbulence (see Sect. 2.3 below for values). The specification of the coefficients C ϕ3, C ϕ4 and C ϕ5 is relatively unclear, lacking any consensus, and is a source of inconsistency in two-equation closures applied to real atmospheric flows (Sogachev 2009).

4 310 A. Sogachev et al. 2.2 Consistent Representation of Non-Shear Sources in Different Supplementary Equations Sogachev and Panferov (2006) showed that considering the behaviour of the transport equation for ϕ in a homogeneous turbulent shear flow is fruitful to remedying problems with the model coefficients for vegetation-related source/sink terms in Eqs. 2 and 3. The ratio between the production and dissipation rates of TKE is constant and the turbulence time scale τ = E/ε is also fixed in this regime (e.g. Pope 2000). For neutrally-stratified free flow, the model predicts that τ does not change with time for the corresponding neutral value of P/ε: P ε = C ϕ2 γ ϕ C ϕ1 γ ϕ, (6) where γ ϕ depends on the form of ϕ (for example, γ ϕ ={1, 0} for ϕ ={ε, ω}). Sogachev and Panferov (2006) argued that the model conventional constants (C ϕ1, C ϕ2 ) need to be adjusted for adequate reconstruction of the ratio between the production and dissipation rates of TKE in the vegetative canopy (homogeneous turbulent shear flow is typical here (e.g. Seginer et al. 1976)). The adjustment suggested for the coefficient C ϕ2, implying foliage-related terms, allowed provision of corresponding values of the coefficients C ϕ4 and C ϕ5. Using slightly different reasoning, Sogachev (2009) obtained the same values for the coefficients C ϕ4 and C ϕ5. He suggested that production and dissipation are influenced by additional sources/sinks with equal magnitude in such a manner as to hold the ratio between total production and total dissipation constant, equal to C ϕ2 /C ϕ1. Both methods proved to be identical, mostly due to the assumption S p = S d, i.e. removing vegetative effects from the TKE equation. Such methods are difficult to apply to the buoyancy term, which cannot be eliminated from the TKE equation; this defies partitioning of extra source/sink terms of the TKE equation into only total production and total dissipation contributions to the supplementary equation. To overcome such difficulties Sogachev (2009) assumed that non-shear terms cannot explicitly appear in any form of the supplementary equation. Thus only mutually compensating terms (±α i X i ) which affect production and dissipation proportionally to X i were taken into account in the supplementary equations, resulting in the term (1). As mentioned in the introduction, such a form proved to be non-universal. Applying the mutually-compensating source concept of Sogachev (2009) not only for buoyancy, but for all source/sink terms, we introduce into the TKE equation mutually compensating terms Y α P α ε ε + α B B + α p S p α d S d,whereα, α ε,α B,α p and α d are some model coefficients. Though the compensating terms for α P and α ε ε do not appear to have a non-shear nature, they can be considered in the present context as effects on P and ε due to the ABL s structure, not directly associated with shear. For homogeneous turbulent shear flow the closure equations become E = (P + Y ) (ε + Y ) + B + S p S d, (7) t ϕ ϕ = C ϕ1 t E (P + Y ) C ϕ ϕ2 E (ε + Y ) + C ϕ ϕ3 E B + C ϕ ϕ4 E S ϕ p C ϕ5 E S d. (8) Let us now explore what the values of P/ε would be for the E ε and E ω models. Invoking the constant turbulence time scale limit, for the E ε model one obtains E (dε/dt) = ε (de/dt), which leads to P ε + B + S p S d = C ε1 P C ε2 ε + C ε3 B + C ε4 S p C ε5 S d + (C ε1 C ε2 )(Y ). (9)

5 Consistent Two-Equation Closure Modelling for Atmospheric Research 311 Note that the right-most term in Eq. 9 appears due to our assumption about the additional effects of the source terms on production and dissipation. With this assumption, and with the non-shear coefficients taken to be C ε3 C ε4 C ε5 1, only the production and dissipation terms remain subsequently the non-shear source terms will not explicitly affect the dissipation equation or the ratio P/ε. Thus the unitary values of these three coefficients can be accepted as first estimates, and we have P ε = (C ε2 1) (C ε1 1), (10a) with ( ) α P C ε2 αε = C ε + α B B + α p S p α d S d ε2 (C ε1 C ε2 ). (10b) ε Invoking a procedure analogous to that which led to (9), but now for the E ω model, where ε t = ωe = ( ) P ε + B + S p S d ω t [ ω + C ω1 E (P + Y ) C ω ω2 E (ε + Y )+C ω ω3 E B + C ω ω4 E S p C ω5 ω E S d ] E, one finds that to retain only terms with explicit production and dissipation, the coefficients C ω3, C ω4 and C ω5 should all be zero. Then P ε = C ω2, (12a) C ω1 with ( ) α P C ω2 αε = C ε + α B B + α p S p α d S d ω2 (C ω1 C ω2 ), (12b) ε and the modification of P/ε is similar to that for E ε model as in (10). If model coefficients satisfy condition (6) the ratios are identical for the two models. Thus the respective supplementary closure equations for the E ε and E ω models, after re-arranging and retaining turbulent diffusion terms, are ε t ( ) K ε = ε z σ ε z E [C ε1 + (C ε1 C ε2 ) α] P [C ε2 + (C ε1 C ε2 ) α ε ] ε (11), [ + [(C ε1 C ε2 ) α B + 1] B + (Cε1 C }{{} ε2 ) α p + 1 ] S p [(C ε1 C ε2) α d + 1] S }{{}}{{} d C ε3 C ε4 C ε5 (13) ( ) K ω = ω z σ ω z E [C ω1 + (C ω1 C ω2 ) α] P [C ω2 + (C ω1 C ω2 ) α ε ] ω ω t +[(C ω1 C ω2 ) α B ] B + }{{} C ω3 [ ] (Cω1 C ω2 ) α p S p [(C ω1 C ω2) α d ] }{{}}{{} C ω4 C ω5 S d. (14)

6 312 A. Sogachev et al. It is worth noting that all non-shear coefficients (C ω3, C ω4 and C ω5 )inthee ω model are related to the analogous E ε model coefficients by C ωn = C εn 1, just as with the coefficients ( ) C ϕ1, C ϕ2 (see Eq. 6); this is consistent with the derivation of the ω equation implied by the E ε model (e.g. Pope 2000), a fact overlooked by Sogachev (2009). Note that two-equation closures based on Eq. 2 and either of Eqs.13 or 14 are consistent, independent of the parametrization for S p and S d used or the calibrated coefficients α B, α p and α d. Consistency here means that the closure is suitable for the description of three flow regimes generally considered for the definition of two-equation model constants grid turbulence, wall-bounded flow (α = α ε = 0), and homogeneous shear flow for both E ε and E ω models (and ideally other E ϕ models with different forms for ϕ). Based on previously derived results Eqs. 13 and 14 can be simplified. Choosing α = (l/l e ), we recover the updated form of C ϕ1 suggested by Apsley and Castro (1997)forthe ABL: Cϕ1 = C ϕ1 + (C ϕ2 C ϕ1 ) (l/l e ), (15) where l e is interpreted as a limiting or equilibrium size of turbulent eddies in the ABL. With l e l 0 = G/f (where G is the geostrophic wind speed and f is the Coriolis parameter (Blackadar 1962)), this modification, along with α ε = 0, allows a natural transition from the standard two-equation model while satisfying the logarithmic law velocity profile in the surface layer and providing a suitable model solution for the whole neutrally-stratified ABL over a bare surface. Other options to obtain the same result are to choose α = 0and α ε = (l/l e ), and thus Cϕ2 = C ϕ2 + (C ϕ1 C ϕ2 ) (l/l e ), (16) or to use some combination of both Cϕ1 (l/l e) and Cϕ2 (l/l e) such that Cϕ1 (l/l e)p = Cϕ2 (l/l e)ε for l = l e. Using Blackadar (1962) formulation for l e (l 0 ), however, does not reflect most of the variation in boundary-layer depth as affected by buoyancy (e.g. diurnal variations). Thus in this work we employ the Mellor and Yamada (1974) (M Y) length scale l e l M Y = z Edz, (17) Edz which accounts for the ABL-integrated effect of TKE. The choice of for the coefficient allows l M Y = l 0 for a neutrally stratified ABL, thus model solutions for this condition are identical. Another simplification, based on the findings of Sogachev and Panferov (2006) and Sogachev (2009), is S p = S d,α d = 1andα p = 0, with the expression for S d thoroughly validated for neutral canopy flows (Sogachev and Panferov 2006). By using Exp. 15 (i.e. α = (l/l e ), α ε = 0) and a simplification of the terms representing vegetation effects, we can then reduce the supplementary equations to ε t z ( K ε σ ε z ) ω t ( ) K ω z σ ω z =C ε1 0 ε E P C ε 2 ε2 E + [(C ε1 C ε2 ) α B + 1] ε E B (C ε1 C ε2 ) ε E S d, (18) =Cω1 ω E P C ω2ω 2 + [(C ω1 C ω2 ) α B ] ω E B (C ω1 C ω2 ) ω E S d. (19) Both equations are different from those suggested in Sogachev (2009) only in respect of the buoyancy terms. Aside from missing the value 1 added to C ε3 (square brackets in Eq. 18),

7 Consistent Two-Equation Closure Modelling for Atmospheric Research 313 the difference ( C ϕ1 C ϕ2 ) is now constant. Thus, due to the aforementioned corrections and new limiting length scale l e, the selection of α B = 1 verified by Sogachev (2009) against observations for the E ω model is no longer valid and needs to be recast. 2.3 Model Constants Specification of General Constants with Regard to ABL Modelling The RANS two-equation model constants, specified as (σ ε E,σ ε, C ε1, C ε2 ) = (1, 1.3, 1.44, 1.92) as in Launder and Spalding (1974) and(σ ω E,σ ω, C ω1, C ω2 ) = (2, 2, 0.52, 0.833) as in Wilcox (1988), were derived by providing better fits of model results for logarithmic law wind profiles in neutral flow. In such numerical experiments the von Kármán constant was assumedtobe0.43andtheengineeringvaluec μ = 0.09 was used. However, in atmospheric research the value of C μ can vary from (see Sogachev and Panferov 2006 for a discussion) and very often the value C μ = 0.03 is accepted (e.g. Duynkerke (1988); Katul et al. (2004)). Values of the von Kármán constant according to different observations also vary between (e.g. Högström 1985; Foken 2006). Such changes can be easily implemented in the model by way of the Schmidt number, σ ϕ, which can be updated according to other parameters via σ ϕ = C 1/2 μ κ 2 ( Cϕ2 C ϕ1 ). (20) The condition (20) ensures the model solution will satisfy the constant-stress logarithmic velocity law in the near-neutral surface layer, and follows from considering constant-flux flow with de/dz = 0(Pope 2000). As seen from (6) the original coefficients for the E ε model are far from giving the typical ratio P/ε observed for conditions of homogeneous shear turbulence. Some reported model constants represent a compromise between experimental data and model optimization, which is why the modelled values of P/ε do not always correspond with observed values (see Kantha et al for a discussion on model constants). In the following calculation, we use one of the alternative sets of constants, (C ε1, C ε2 ) = (1.52, 1.833) (Kantha et al. 2005), which provides the model ratio of P/ε close to observations. For the sake of comparison, we accept the value of the von Kármán constant recommended in analytical expressions, κ = 0.4 (Högström 1996). With C μ = 0.09, applying Eq. 20 we obtain other model constants (σ ε E,σ ε, C ε1, C ε2 ) = (1, 1.7, 1.52, 1.833) and (σ ω E,σ ω, C ω1, C ω2 ) = (1.7, 1.7, 0.52, 0.833). In these sets, we accept that traditionally σ ε E = 1, and σ ω E = σ ω, though there is no strict regulation on that Specification of α B For homogeneous shear flow in the constant time-scale limit, ignoring vegetation effects for ε andω, leads to a balance between shear-production, dissipation, and buoyant contributions (see Eqs. 10 and 12): [(C ϕ1 γ ϕ) + (C ϕ2 C ϕ1 )α B Ri f ] P = (C ϕ2 γ ϕ )ε, (21) where Ri f = B/P is the (local) flux Richardson number. This is equivalent to

8 314 A. Sogachev et al. P ε = ( Cϕ2 γ ϕ ) ( ), (22a) C ϕ1 γ ϕ with C ϕ1 C ϕ1 + ( C ϕ2 C ϕ1 ) αb Ri f = C ϕ1 + ( C ϕ2 C ϕ1 ) [l/le + α B Ri f ]. (22b) One can see that the effective coefficient of the production term in the supplementary equation can change depending on the degree of stratification, Ri f, which is one way of allowing for consistency with MOST (e.g. Freedman and Jacobson 2003). In neutral conditions α B Ri f = 0 so C ϕ1 = C ϕ1, and again C ϕ1 C ϕ2 as l l e. In the very stable limit P/ε 1soweexpectC ϕ1 C ϕ2, which from (22b) requires α B Ri f,cr = (1 l/l e ) for some critical Richardson number Ri f,cr. The bracketed term in (22b) should vary between 0 and 1 in the stable ABL as l ranges from 0 to l e, regardless of Ri f. Consistent with this and the form α = (l/l e ) implied by the Apsley and Castro (1997) expression (15) forcϕ1, for stable regimes we choose α B = (1 l/l e ) (implying Ri f,cr = 1). This is similar to considerations made by Freedman and Jacobson (2003) for the E ε closure, apart from the form parametrizing such a dependence (e.g. they used a non-linear function of Ri f for the bracketed term in Eq. 22b,lackingany{l, l e } dependence but with Ri f,cr = 0.213). A serious shortcoming of this study is that, although it permits some Ri f dependence, it ignores the buoyancy production term in the ε equation. Freedman and Jacobson (2003) showed, however, that if the effects of shear and buoyancy are separated, setting C ε3 = 1(asC ϕ3 = γ ϕ in this work) allows consistency with MOST. We note the above analysis can be repeated using a modified dissipation coefficient Cϕ2 (l/l e) instead of Cϕ1 (l/l e), but we forego such a development, and here use Cϕ1. To estimate α B in unstable conditions, we consider the limiting case of a balance between buoyant production and dissipation. According to the two-equation model framework (neglecting transport) the ratio ε B = α ( ) B Cϕ1 C ϕ2 (C ϕ1 γ ϕ )Rif 1 + (C ϕ3 γ ϕ ) (23) C ϕ2 γ ϕ should approach 1 in the convective limit when the inverse value of Ri f approaches zero (compare to e.g. Kantha 2004). With C ϕ3 = γ ϕ from (23) wehave ( ) α B (C ϕ1 C ϕ2 ) = (C ϕ2 γ ϕ ) + Cϕ1 γ ϕ Rif 1, (24) which in the convective limit becomes lim α B = ( )/( ) C ϕ2 γ ϕ Cϕ2 C ϕ1 ε/b 1 (25) or α B 2.66 for the constants quoted earlier. Thus for unstable conditions we choose α B = 1 [ 1 + ( ) ( )] C ϕ2 γ ϕ / Cϕ2 C ϕ1 (l/le ) (l/l e ) to satisfy the convective limit (25) ensuring α B 1 in the neutral-like l 0 limit, just as with the stable form. Noting, however, that free convection will never be fully reached in our RANS simulations (Rif 1 = 0), the coefficient 3.66 could be reduced somewhat, to enable optimum results in both near-neutral and strongly convective regimes.

9 Consistent Two-Equation Closure Modelling for Atmospheric Research SCADIS Model and Experimental Design To assess the robustness of the suggested closure, the present study uses the ABL RANS model SCADIS (Sogachev et al. 2002). The model includes a set of momentum equations, the continuity equation, equations for moisture and heat transport, and also allows a transport equation for any passive tracer of interest. SCADIS belongs to the 1.5-order closure class of models because the fluxes are expressed as a product of a turbulent diffusion coefficient and the gradient of the corresponding mean quantity; SCADIS can operate with E l closure as well as the closures described above. The time march method is used to solve the non-linear two-point boundary-value problem. Considering the vegetation as a multi-layer medium and implementing parametrizations for radiation transfer, drag forces on leaves, and stomatal conductance enables SCADIS to properly describe the exchange between the vegetative canopy and the atmosphere. The full description of the model, equations, and numerical details can be found in Sogachev et al. (2002, 2005). Only one additional change has been made to the model following introduction of the new formulation (17) for the length scale l e, namely the minimum value of ϕ is limited ( in such a way ) as to prohibit( l from exceeding ) l e for any given TKE values: ε = max ε, Cμ 3/4 E 3/2 /l e and ω = max ω,cμ 3/4 E 1/2 /l e for E ε and E ω closures, respectively. In the cases presented here, incident solar radiation drives the diurnal cycle of the surface heat budget, and as such the diurnal variations of meteorological variables in the ABL were computed for July 1 at 50 o latitude and clear-sky conditions according to Sogachev et al. (2002). The initial profile of temperature was given by a vertical gradient of K m 1 with a temperature of 267 K at the upper boundary of the model domain, which was set to 4 km. The initial values of air moisture were constant and equal to kg m 3. The geostrophic wind speed, G,wastakenas10ms 1. Evaporation and transpiration were calculated with properties typical for sandy loam soil and for a spruce forest, respectively (Sogachev et al. 2002). It was assumed that the model with these initial conditions approximates a typical variation of wind speed, turbulence and temperature over the course of a summer day at temperate latitudes. To properly resolve these variations, the model vertical grid consisting of 150 nodes was arranged with variable sizes and non-linear with height (about 0.06 m near the ground and 220 m near the upper boundary condition of the domain). 4 Monin Obukhov Similarity Theory For many years Monin Obukhov similarity theory (MOST) (see Garratt 1992 for a summary of the theory) has been used to assess and validate different turbulence parametrizations. According to MOST, the structure of the atmospheric surface layer (ASL) is defined through two key parameters: the friction velocity u = τ s,0 1/2 and the Obukhov length L = u 3 θ v,0/κgh 0,whereτ s,0 and H 0 are the near-surface values of kinematic Reynolds stress, τ s and the vertical turbulent heat flux H, respectively, g is the acceleration of gravity, and θ v,0 is the representative surface value of potential temperature θ v. Within the horizontally homogeneous surface layer, τ s and H are then considered to be practically independent of height. Wind speed, u, and temperature profiles normalized by the friction velocity u and temperature scale T = H 0 /u, respectively, are assumed to be unique functions of the dimensionless height ζ = z/l: i.e. u z ( ) κz = φ m (ζ ), (26) u

10 316 A. Sogachev et al. ( ) θ v κz = φ h (ζ ). (27) z T Despite numerous field experiments that have been carried out around the world in the decades since MOST was established, there still remain a few discrepancies in the literature regarding the exact forms of the functions φ m (ζ ) and φ h (ζ ). Widelyusedformsofφ m and φ h are (Businger et al. 1971; Dyer 1974): { 1 + δ φ m (ζ ) = m (ζ ) for L > 0 (1 γ m (ζ )) 1/4 (28) for L < 0, { χ + δ φ h (ζ ) = h (ζ ) for L > 0 χ (1 γ h (ζ )) 1/2 (29) for L < 0. In the literature the coefficients δ m and δ h range from 4 to 10, γ m and γ h range from 15 to 28 (see e.g. Högström 1996; Li et al for a review), and the coefficient χ ranges from 0.74 (Businger et al. 1971) to0.95(högström 1988). Figure 3 illustrates below several experimental datasets derived over ground covered by sparse sagebrush and grass (Li etal. 2008), over short grass (Klipp and Mahrt 2004) and over water (Vickers and Mahrt 1999) thatgive observed non-dimensional shear values φ m (ζ ) with yet a larger actual spread than is implied by the range of published δ m and γ m values. As a basis for assessment of the model closure below we consider the commonly-used Businger Dyer analytical expressions with δ m = δ h = 5,γ m = γ h = 15, and χ = 0.74 (Dyer and Hicks 1970). The last two choices define the approximation of the turbulent Prandtl number, Pr = φ h /φ m = K/K H (K H is the eddy conductivity), in the model as a function of the gradient Richardson number, Ri g : { 0.74 for Ri Pr = g (1 15Ri g ) 1/4 (30) for Ri g < 0. 5 Assessment of the Closure 5.1 Airflow Over Rough Bare Ground Using input information for a fair summer day, we calculated the ABL structure over a relatively low-roughness land surface (z 0 = 0.03 m). Figure 1 shows the key surface characteristics, u and L 1, along with l e normalized by the Blackadar scale, l 0, simulated by both the E ε and E ω models. Both models give the same daily course of l e but are slightly different for friction velocity and Obukhov length during daytime conditions, though this difference will be treated below. Simulated values of L and u can be used in analytical MOST expressions for reconstruction of the wind or temperature profiles in the atmospheric surface layer. These expressions provide a result resembling pure lines, but we note they have been derived from observations having scatter. Both φ m and φ h estimated from SCADIS-modelled profiles also exhibit scatter that complicates the presentation of results. To simplify our closure assessment, we clarify this issue with an example using data from the E ε model. Figure 2a presents the function φ m calculated from SCADIS results for the modelled day as described in the previous section, up to a height of 50 m and sampled every 10 minutes. Though the data have large scatter, much of the data fall along a line corresponding to MOST functions. The inset shows general agreement of model results with the similarity theory for

11 Consistent Two-Equation Closure Modelling for Atmospheric Research 317 Fig. 1 Evolution of surface characteristics over a day: u, inverse Obukhov length L 1, and normalized l e during fair weather over low-roughness land, calculated by E ε and E ω models. l e is normalized by the neutral (Blackadar) value l 0 ( 24.2m) Fig. 2 Effective φ m as function of stability (z/l) over a simulated day for heights below 50 m (crosses) calculated by the E ε model. In (a) all modelled data are presented; in (b) data for transition periods ( ) and ( ) are excluded. Circles are results for selected heights. Time and arrows in (a) indicate time sequence of data

12 318 A. Sogachev et al. Fig. 3 φ m from observations (open symbols), analytical expression (line)(eq.28 with δ m = 5andγ m = 15), and from the two-equation models (red E ε, blue E ω). The grey area indicates the range of analytical solutions (28)forφ m derived using different values of δ m and γ m (e.g. Högström 1988) conditions approaching neutral (L 1 0), which occur around 0500 and 1800 local time (seen in Fig. 1), and the inset evokes the height dependence of φ m : groups of φ m values (grey x s) corresponding to the same time proceed outward from the main line (cluster) with increasing height. A primary cause for the scatter of data is the inclusion of points outside the surface layer, i.e. larger values of z/l. Limiting the heights considered to the usual measurement levels over low-roughness surfaces (e.g. z 10 m, blue dots in the plot), i.e. within the surface layer, we find that the scatter decreases, especially for stable conditions where the surface layer can be significantly shallower than 50 m. Considering the scatter of modelled data (and deviation from MOST) for any specific height, we re-iterate that there is also a dependence on the time that the measurements have been made and how stationary (equilibrated) conditions were at that time. As seen in Fig. 2a, φ m (z/l) at each specific height exhibits a clear diurnal evolution, following a hysteretic counter-clockwise pattern (indicated by arrows). The simulated day begins at 0000 with relatively large φ m (e.g. z/l 2, φ m 15 cluster of white circles for z = 30 m), during a stable regime. As shown by Fig. 1 this is a steady regime, and subsequently these large φ m values fall on the implied similarity theory line. In the morning the local heat flux becomes positive and the ABL grows, with φ m and L 1 generally decreasing as φ m increasingly deviates from similarity theory and the main line of φ m (z/l). φ m then approaches a midday minimum, with a large number of points falling in a common cluster corresponding to the daytime regime of nearly constant L 1. Later ( ) the daytime ABL collapses, and φ m appears to approach an asymptotic value (akin to that described in Baas et al. 2008), as shown by the right-most (z/l > 2) points in Fig. 2a. One can see that the transition periods between both and are characterized by high rates of change of L 1 and u (Fig. 1), and the corresponding φ m (z/l) introduce scatter away from the main cluster of points that conform to MOST (Fig. 2a). It can be seen from Fig. 2a that during the transition periods after sunrise and following sunset, changing conditions at the surface particularly those represented by the Obukhov length do not exactly follow conditions in the atmosphere above. This is attributed to the

13 Consistent Two-Equation Closure Modelling for Atmospheric Research 319 Fig. 4 Wind profiles for different hours in the atmospheric surface layer during fair weather over low-roughness land from analytical MOST expressions (black lines) (Paulson 1970), and calculated via local SCADIS results using E ε (red lines) and E ω (blue lines) models low turbulence levels and the progressive, slow upward reversal of the temperature profile from an inversion to typical lapse conditions. A good indicator for the TKE development in the ABL is l e, and from Fig. 1 one sees that l e and L 1 indeed do not correlate well in some time periods; l e lags L due to the effective (thermal) inertia of the atmosphere. Excluding data for the transition periods we reduce scatter, making the trend clearer and enhancing conformity to MOST, as shown in Fig. 2b. Thus, unless otherwise noted, in the following plots we present modelled data that exclude transition periods, acknowledging that analytical φ m (z/l) functions were originally derived for steady conditions and corresponding surfacelayer fluxes. Figure 3 shows observed values of φ m obtained from various measurement campaigns, the analytical result of Eq. 28 (with δ m = 5andγ m = 15), as well as diagnosed SCADIS model values for both E ε and E ω closures. Li et al. (2008) analyzed a 10-day data series collected in May 2002 from five levels up to about 20 m above the ground covered by sparse sagebrush and grass, with no fully vegetated canopy present. The data of Vickers and Mahrt (1999) were derived at 10-m height over coastal waters from April through November Observations of Klipp and Mahrt (2004) are given for the non-dimensional shear at a height of 5 m above short grass derived in October 1999 for the nocturnal period, when conditions ranged from near-neutral with relatively strong turbulence to very stable with weak and/or intermittent turbulence. In Fig. 3 modelled data are again plotted for every 10 min up to 50 m height, but with transient condition points removed. The Figure illustrates that the models basically reproduce an analytical φ m function during non-transient conditions, but

14 320 A. Sogachev et al. also experience scatter in unstable cases as occurs with observations. Note that the E ω model gives better agreement with the analytical solution in unstable cases, but for all cases the modelled results fall within the range of observed scatter, with both closures producing a good match to MOST in near-neutral conditions (inset). Figure 4 compares the wind profiles produced using the E ε and E ω models to profiles obtained via Paulson (1970) analytical expressions for φ m (z/l) using δ m = 5and γ m = 15, for four different times during the modelled day. For each case the analytical wind profiles were estimated using u and L derived from corresponding model results. One can see that both models give results very close to analytical values, testifying that the suggested closure is consistent with regard to the buoyancy term. Better overlap of the model results and analytical approaches occur at 1800 local time, where (as seen from Fig. 1) the surface conditions are close to neutral. In convective conditions the wind profiles of both models are slightly shifted with respect to the diagnosed analytical functions, and the E ω model fares remarkably better than the E ε model (e.g. 1200). Again one can see that analytical MOST solutions have limited use in the morning transition period (0600 local time). Both models show that, although the winds are strongly influenced by buoyancy caused by radiation forcing in the lower ASL, the wind field still retains the structure of nighttime flow far from the surface. This is in good agreement with data presented in Fig. 1: the surface parameters u and L 1 are not in direct relation with the ABL-integrated limiting scale l e. Analytical solutions are typically inadequate for describing the flow during non-steady state conditions owing to the fact that the atmosphere becomes gradually unstable (vertically) as the boundary layer grows after sunrise, or that the surface cools and then progressively cools layers above. As inferred from results presented in Fig. 3, both models provide an ideal match with Paulson s solution in the steady stable case for the lower ASL (0000 local time) (Fig. 4c, d). But with increasing height (and stability z/l), the model and analytical MOST profiles diverge (Fig. 4a, b). A comparison of the dimensionless temperature gradient φ h (Fig. 5) also provides nonideal but reasonable agreement with the conventional analytical functions (Eq. 29). Note that the two E ϕ models behave very similarly, again with some advantage in using the E ω model. 5.2 Flow Through Vegetated Canopies Due to the roughness sublayer formed above plant canopies, the flux gradient relationship does not obey MOST (e.g. Harman and Finnigan 2007), causing difficulty in making any assessment of the models in a universal way. For assessment of the closure implementing both buoyancy and plant drag together, we begin by comparing the results from the two models against each other. Sogachev and Panferov (2006) showed that E ε and E ω models behave similarly in conditions of neutrally-stratified free flow. Concentrating on the universal modification of two-equation models for canopy flow, Sogachev and Panferov (2006) were satisfied with the derived results and did not explore apparent differences between models in detail, focusing on validation of the E ω canopy model. Thus, in our calculation, the E ω model is considered as a standard for comparison. Figure 6 shows profiles of basic modelled flow statistics inside and above two types of vegetation in a steady, neutrally-stratified ABL. The vegetation types have an identical leaf area index (LAI) of 3, but for illustrative proposes have different height and foliage distributions: one extends 15 m high with a vertically uniform leaf area density (A) and another is 25 m high but close to a real A distribution. Results presented in Fig. 6a reveal little difference in wind-speed profiles between models, though the discrepancies in TKE (Fig. 6b) and l

15 Consistent Two-Equation Closure Modelling for Atmospheric Research 321 Fig. 5 Dimensionless temperature gradient φ h derived analytically (black lines)(eq.29 with δ h = 5, γ h = 15, and χ = 0.74) and calculated using different models (crosses)forz < 50 m. The figure shows the full diurnal cycle with scatter (a) and with transition periods eliminated (b) Fig. 6 Flow statistics within and above two vegetation types of different heights and foliage distributions derived by E ε (red)ande ω (blue) models in a neutrally stratified ABL. In (a) grey areas indicate vertical profiles of leaf area density, A, of vegetation types, scaled by a factor of 10 for illustrative purposes. Dotted lines represent results for taller, non-uniform vegetation canopy; solid lines for lower uniform canopy (Fig. 6c) profiles are more remarkable. The E ε model overestimates values of TKE and l inside the canopy. Inclusion of buoyancy effects in the modelled canopy flow leads to more pronounced differences between the two models. One can see that, compared to the E ω model, the E ε model gives increasingly disparate values of wind speed (Fig. 7a) and mixing length (Fig. 7b) as conditions become more unstable.

16 322 A. Sogachev et al. Fig. 7 Flow statistics within and above vegetation canopy with vertically-varying leaf area density, A, derived by E ε (red) ande ω (blue) models for different hours in the lower ABL. In (a) thegrey area indicates the vertical profile of A, scaled by a factor of 10 for illustrative proposes 5.3 Consistency between Equations for ε and ω A general conclusion from the results presented in Sects. 5.1 and 5.2 is that the difference between the E ε and E ω models becomes more pronounced for large TKE values, which are usually observed in the convective ABL or above the forest even in neutral cases. Considering that the description of buoyancy and canopy effects, along with diffusion rates of ε and ω (applying the same Schmidt number), are similar in both models, the only reason for the difference would appear to be the different diffusion rates (transport) of E in the models. However, simply establishing σ ε E = σ ε in the E ε model in the same manner as in the E ω model does not improve the situation; a revision of the diffusion terms appears necessary, and is relatively straightforward given that both models can be derived from each other. Usually the E ω model is derived from E ε model, under the (often implicit) assumption that the E ε model is used as a reference model (Pope 2000; Wilcox 2002). However, as seen above, the E ω closure tends to give results that better match similarity theory than is the case with the E ε model (Figs. 3, 4, 5), and the E ω model was thoroughly verified for canopy flows (Sogachev and Panferov 2006). Furthermore, the E ε closure is more sensitive to the choice of constants, and susceptible to numerical instabilities, as well as being potentially less suited to canopy flow due to its less reliable behaviour in the presence of significant local pressure gradients (Wilcox 2002). Noting these factors, in order to obtain consistent results between the two closures we derive an ε equation from the E- andω-equations; i.e.(dε/dt) ω = E(dω/dt) + ω(de/dt), which in three dimensions leads to ( ) ( ε K ε t ω σ ω + 2 ] σe ε E E (C ε1 C ε2 ) ε E S d, ) = C ε1 [( 1 + C μ σe ε 1 ) ( 1 E 2 E σ ω σe ε + 1 ) E ε E σ ω ε ε E P C ε 2 ε2 E + [(C ε1 C ε2 ) α B + 1] ε E B (31) where is the gradient operator. In one dimension this is the same as (18), but with the Schmidt number σ ε in the original ε transport replaced by σ ω, plus extra diffusion-like terms:

17 Consistent Two-Equation Closure Modelling for Atmospheric Research 323 ( ) ε t + 2 σ ε E ( K ε ω z σ ω z ( ) ] E 2 = C ε ε1 z E P C ε2 ) [( 1 + C μ σe ε 1 ) ( E 2 E 1 σ ω z 2 σe ε + 1 ) E E ε σ ω ε z z ε 2 E + [(C ε1 C ε2 ) α B + 1] ε E B (C ε1 C ε2 ) ε E S d. (32) We note that the extra TKE diffusion term disappears when setting σ ε E = σ ω, and can be taken as partial justification for such a choice. Applying the E ε model with the corrective diffusion terms of Eq. 32 with σ ε E = σ ω demonstrates the full compatibility of E ε results with those of the E ω model (presented in Figs. 6, 7). Inclusion of the diffusion terms also causes the two models characteristic surface-layer parameters (e.g. u and L 1 presented in Fig. 1, and modelled φ m and φ h ) to be identical above a bare surface (not shown). 6 Discussion and Conclusions Two-equation closure is a pragmatic compromise between simple first-order and more complex higher-order closure schemes for modelling ABL flows using the Reynolds-averaged Navier Stokes equations. The problem of poorly-defined extra coefficients appearing in the supplementary equations inherent in earlier attempts to treat vegetative canopy and/or buoyancy effects had seriously limited the use of such closures in many applications. Here, we demonstrate the development of a consistent closure based on the well-known coefficients (C ϕ1, C ϕ2 ) of the production and destruction terms in the supplementary (ϕ) equation, respectively, and suitable for three canonical (asymptotic) flow regimes: grid turbulence, wall-bounded flow and homogeneous shear flow. To be applicable to atmospheric research the closure accommodates additional source/sink terms and associated coefficients, without affecting the consistency of the closure. Applying the previously defined coefficients for canopy effects (Sogachev and Panferov 2006; Sogachev 2009), we extend application of the closure to stratified flows both over bare surfaces and within forests, introducing only a single buoyancy-related coefficient α B. We apply Monin Obukhov similarity theory (MOST) and observations to assess the suggested closure implemented within the most popular twoequation schemes, for buoyancy-affected conditions occurring during a typical clear-sky diurnal cycle over bare land. Based on scaling arguments and confirmed by numerical experiments, α B = (1 l/l e ) appears to be the most suitable coefficient for stably-stratified conditions and α B = 1 [ 1 + ( ) ( )] C ϕ2 γ ϕ / Cϕ2 C ϕ1 (l/le ) in unstable regimes, with the Mellor and Yamada (1974) formulation for l e.though the closure approach has proven to be robust, several issues should be discussed. First, though the prescription α B = (1 l/l e ) works well in stable conditions, it is a simplified form based on the global (ABL mean) Mellor-Yamada length scale l e = l M Y. While use of l M Y is a great improvement over the Blackadar scale, which does not represent diurnal variations, the simplified nature of this non-local form for α B results in a compromise between matching MOST in mildly stable conditions versus the matching in strongly stratified regimes. We note that local similarity where the ABL scale l e is replaced by the local Obukhov length, subsequently implying (l/l e ) B/ε as in Apsley and Castro (1997) would result in an implicitly non-linear dependence upon the local heat flux when using α B (l/l e ). Such a potential numerical instability precludes practical use of a local scale for buoyancy, given our prescription for α B, and supports use of the Mellor Yamada

18 324 A. Sogachev et al. Fig. 8 As in Fig. 3 but for E ω model results with theoretically-derived α B and α B = 0 length scale. Note also that the simplest choice for α B is to set it to zero, i.e. ignore the contributions of buoyancy to the shear-production and dissipation components of the supplementary equation. With the use of the Mellor Yamada scale for l e, such a choice for α B can actually give reasonable results. But setting α B = 0 does not give the proper (theoretical) behaviour in the convective limit, disallowing a balance between buoyant production and dissipation; practically, however, the difference in model results was found to be minor (see Fig. 8). Using α B = 0 gives results that are worse at small z/l and better at larger z/l; at larger z/l the model simulation with the theoretically-based α B follows the convective regime in a consistent way. These numerical results also remind us that MOST does not apply for significant portions of a simple diurnal cycle, and its use becomes uncertain in some regimes above the ASL. More measurements are needed in stable cases at greater heights (e.g. > 80 m) over a distribution of Obukhov lengths (e.g. Kelly and Gryning 2010), and the selection of data for improving MOST formulations should be done carefully. Second, though we use a specific parametrization for canopy-related terms (Sogachev and Panferov 2006), our approach is not limited to using any particular forms existing in the literature; it demands only calibration of the parametrizations themselves, including the supplementary coefficient(s) associated with them. Applying a comprehensively tested canopy scheme, we find that both E ε and E ω models respond adequately to plant drag with the suggested closure. Nevertheless, the proper description of foliage-related terms is still questionable. Several different aspects, including the effect of the roughness sublayer (e.g. Harman and Finnigan 2007) or thermal stratification inside the canopy layer (e.g. Leclerc et al. 1990; Jacobs et al. 1994; Tóta et al. 2008), need to be involved for a robust solution of this problem. The parametrization for enhanced dissipation within plant canopies was derived by Sogachev and Panferov (2006) based on observations in neutrally stratified atmospheric and wind-tunnel flows, and involves only one fitted parameter. This process is probably not only shear dependent, but also depends on local thermodynamic conditions, and as such the parameter may again need to be adjusted. Other issues can be associated with the temperature and wind-speed dependence of the drag coefficient, usually accepted as a constant in models. The problem remains a challenging task, mostly due to the lack of

19 Consistent Two-Equation Closure Modelling for Atmospheric Research 325 observations inside and above vegetative canopies under different atmospheric conditions. It requires a separate study and is outside the scope of the present work. Third, we find that in stable regimes the E ω and the E ε models produce similar results, both above bare and forested surfaces. Above bare surfaces in convective cases, the behaviour of the E ω model corresponds closer to MOST. Based on the above we suggest that the relevant differences between the models especially above forested surfaces, for which the E ω model was verified are due to the transport description in the E ε model. Introducing corrective transport terms consistent with the E ω model in the ε equation, together with updating the Schmidt number for E (σ ε E = σ ε) in the E ε model, ensures identical models (given their identical numerical implementation). Nevertheless, because of the complex form of the extra effective diffusion term in the ε equation, some numerical issues can arise (particularly in the three-dimensional formulation), and so we continue to recommend the E ω closure over E ε schemes. In summary, the closure developed here represents a pragmatic means of modelling atmospheric flows; it is useful not only over homogeneous surfaces during non-steady atmospheric conditions, but it also allows treatment of flows over complex terrain and transient conditions where MOST becomes inapplicable, and where other closures are much more computationally prohibitive or uncertain. Acknowledgements The authors would like to thank the three anonymous reviewers for their constructive comments, which have led to further improvement of the manuscript, inspiring critical discussions along the way. AS and MK acknowledge partial support for this work by the Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence at DTU Wind Energy (formerly Risø DTU) under the Danish Council for Strategic Research, Grant no References Apsley DD, Castro IP (1997) A limited-length-scale k ε model for the neutral and stably-stratified atmospheric boundary layer. Boundary-Layer Meteorol 83:75 98 Ayotte KW, Finnigan JJ, Raupach MR (1999) A second-order closure for neutrally stratified vegetative canopy flows. Boundary-Layer Meteorol 90: Baas P, de Roode SR, Lenderink G (2008) The scaling behavior of a turbulent kinetic energy closure model for stable stratified conditions. Boundary-Layer Meteorol 127:17 36 Basu S, Vinuesa JF, Swift A (2008) Dynamic LES modeling of a diurnal cycle. J Appl Meteorol Climatol 47: Baumert H, Peters H (2000) Second-moment closures and length scales for weakly stratified turbulent shear flows. J Geophys Res 105: Blackadar AK (1962) The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J Geophys Res 67: Businger J, Wyngaard JC, Izumi Y, Bradley EF (1971) Flux profile relationships in the atmospheric surface layer. J Atmos Sci 28: Cheng Y, Canuto V, Howard A (2002) An improved model for the turbulent PBL. J Atmos Sci 59: Deardorff JW (1972) Numerical investigations of neutral and unstable planetary boundary layers. J Atmos Sci 18: Duynkerke PG (1988) Application of the E ε turbulence closure model to the neutral and stable atmospheric boundary layer. J Atmos Sci 45: Dyer AJ (1974) A review of flux profile relationships. Boundary-Layer Meteorol 7: Dyer AJ, Hicks BB (1970) Flux gradient relationships in the constant flux layer. Q J R Meteorol Soc 96: Finnigan JJ (2000) Turbulence in plant canopies. Annu Rev Fluid Mech 32: Finnigan JJ (2007) Turbulent flow in canopies on complex topography and the effects of stable stratification. In: Gayev YA, Hunt JCR (eds) Flow and transport processes with complex obstructions. Springer, Dordrecht pp