Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015) DOI:10.1002/qj.2503

Methodical assessment of the differences between the QNSE and MYJPBL schemes for stable conditions

Esa-Matti Tastula,a* Boris Galperin,a Jimy Dudhia,b Margaret A. LeMone,b Semion Sukorianskyc

and Timo Vihmad

aCollege of Marine Science, University of South Florida, St. Petersburg, FL, USAbNational Center for Atmospheric Research, Boulder, CO, USA

cDepartment Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, IsraeldFinnish Meteorological Institute, Helsinki, Finland

*Correspondence to: E.-M. Tastula, College of Marine Science, University of South Florida, 830 1st Street Southeast, St. Petersburg,FL 33701, USA. E-mail: tastulae@mail.usf.edu

The increasing number of physics parametrization schemes adopted in numerical weatherforecasting models has resulted in a proliferation of intercomparison studies in recentyears. Many of these studies concentrated on determining which parametrization yieldsresults closest to observations rather than analyzing the reasons underlying the differences.In this work, we study the performance of two 1.5-order boundary layer parameterizations,the quasi-normal scale elimination (QNSE) and Mellor–Yamada–Janjic (MYJ) schemes,in the weather research and forecasting model. Our objectives are to isolate the effect ofstability functions on the near-surface values and vertical profiles of virtual temperature,mixing ratio and wind speed. The results demonstrate that the QNSE stability functionsyield better error statistics for 2 m virtual temperature but higher up the errors related toQNSE are slightly larger for virtual temperature and mixing ratio. A surprising finding isthe sensitivity of the model results to the choice of the turbulent Prandtl number for neutralstratification (Prt0): in the Monin–Obukhov similarity function for heat, the choice of Prt0 issometimes more important than the functional form of the similarity function itself. Thereis a stability-related dependence to this sensitivity: with increasing near-surface stability,the relative importance of the functional form increases. In near-neutral conditions, QNSEexhibits too strong vertical mixing attributed to the applied turbulent kinetic energysubroutine and the stability functions, including the effect of Prt0.

Key Words: stable boundary layer; stability function; NWP system

Received 30 May 2014; Revised 3 December 2014; Accepted 11 December 2014; Published online in Wiley Online Library

1. Introduction

The past decade of development in numerical weather forecastinghas witnessed numerous studies in which various physicsparametrizations were compared with observations (e.g. Zhangand Zheng, 2004; Jankov et al., 2005; Tastula and Vihma, 2011;Draxl et al., 2014). In particular, the studies have been commonfor models that allow the choice of several different physicsconfigurations, such as the weather research and forecasting(WRF) model (Skamarock and Klemp, 2008). Although studies ofthis kind have been valuable in determining optimal combinationsfor model physics options and possible incompatibilities amongschemes, studies of the actual physics within the parametrizationsare rare. Yet, understanding the role of various processes and theirinteraction is an essential prerequisite for model developmentand may shed more light than the outputs from the proverbial‘black-box’ model studies.

A planetary boundary layer (PBL) parameterization providesestimates for turbulent fluxes of grid-scale variables. The mostcommon approach is the downgradient one

w′φ′ = −K∂φ

∂z, (1)

where the turbulent flux of a quantity φ is estimated via thevertical gradient of its mean profile and an exchange coefficientK. The K-formulation has a large effect on the model results. In1.5-level PBL schemes in the Mellor–Yamada hierarchy such asthe Mellor–Yamada–Janjic (MYJ; Janjic, 2001) and quasi-normalscale elimination (QNSE; Sukoriansky et al., 2005), K takes thefollowing form:

K = αlS√

TKE, (2)

c© 2014 Royal Meteorological Society

E.-M. Tastula et al.

where α is a non-dimensional coefficient, S is the inverse of astability function, l is a turbulence length scale, and TKE is theturbulence kinetic energy.

Close to the surface, parametrizations of turbulent fluxesof momentum, heat and moisture are typically based onthe Monin–Obukhov similarity theory (MOST), which relatesmean profiles of meteorological quantities to their respectivesurface fluxes (Monin and Obukhov, 1954). Under the MOSTassumptions, the stability functions in Eq. (2) become non-dimensional gradient functions, also known as similarityfunctions, which are determined empirically. The validity ofthese log–linear expressions is supported by direct numericalsimulation (DNS) studies (e.g. Ansorge and Mellado, 2014).A substantial body of literature exists that addresses similarityfunctions: reviews have been given by e.g. Hogstrom (1996) andFoken (2006). The functional forms of these empirical expressionsdiverge strongly in stable conditions (Grachev et al., 2007),whereas unstable conditions feature much closer agreement(Businger et al., 1971).

Turbulence in stable conditions is structurally complicated.The unsteadiness and intermittency of the stable PBL havebeen major obstacles in the way of gaining even qualitativeadvances (Mahrt, 2014). Low-level jets, terrain slope flows andgravity waves, often found in stable conditions, further complicatethe situation. Yet an accurate prediction of the evolving stablystratified PBLs is essential in many practical applications, such aspredictions related to air pollution (Mahrt, 1999).

A widely used formulation of the stability functions in stableconditions was developed by Mellor and Yamada (1982). Thestability functions given therein depend on the Brunt–Vaisalafrequency, vertical wind shear and several empirically determinedcoefficients. The original Mellor–Yamada 2.5-level model wasfound problematic in several studies and various correctionshave been suggested (e.g. Galperin et al., 1988; Helfand andLabraga, 1988; Deleersnijder, 1992). The coefficients were revisedin Janjic’s (2001) non-singular implementation. A revision of theclosure constants and turbulence length-scale formulation wasalso suggested by Nakanishi and Niino (2009).

Analytical expressions for the stability functions originating inclose proximity to first principles based upon the Navier–Stokesequations have been rare. The relatively recently developed quasi-normal scale elimination (QNSE) theory of stably stratifiedturbulence (Sukoriansky et al., 2005, 2006) yields such a setof analytical functions for the vertical eddy viscosity andeddy diffusivity. Close to the surface in the approximation ofthe constant flux layer, these functions constitute the MOSTsimilarity functions. Although numerical weather prediction(NWP) systems such as the WRF model have adopted theQNSE-based turbulence parametrization, no previous studieshave assessed the isolated effect of the analytical QNSE stabilityfunctions on model simulations. Such comparisons are critical,however, not only due to the usage of the QNSE stability functionsin NWP systems to compute near-surface turbulent exchange,but also due to the need for better theoretical understandingof turbulence in stably stratified conditions. Moreover, such anapproach would illuminate the impact of stability functions uponvertical profiles of different variables; an aspect usually missing inintercomparisons of boundary layer schemes. Having explainedthe approach used in this study, it is important to point outthat the lack of understanding of stably stratified turbulenceis not the only factor limiting predictive stable boundarylayer skills: radiative transport, representation of soil/vegetation,local topography/surface heterogeneity and wave–turbulenceinteractions are among other factors that challenge modellers.

The purpose of this study is to compare the performance of theQNSE and MYJ models by scrutinizing their stability functions.To accomplish this goal, we have modified the MYJ PBL andsurface layer codes in the WRF model in such a way that the onlysignificant difference between the QNSE and MYJ approaches isthe representation of stability functions (see section 4 for details).

Figure 1. The inverses of the QNSE and MYJ stability functions SM and SH as afunction of gradient Richardson number.

Other tests related to MOST functions are carried out as well. TheMYJ model was selected as a reference because it is a widely usedscheme in the WRF model and its code structure is similar to thatof QNSE.

The study compares the performance of these stabilityfunctions from the modelling perspective. Theoretical tests for theQNSE approach are addressed in Tastula et al.(2014). The modelsimulations test the model’s skill in reproducing observations of:(i) near-surface temperature, mixing ratio, and wind speed at 12stations in southeastern Canada in winter; (ii) vertical profilesof virtual temperature, mixing ratio, and wind speed during theCASES97 and CASES99 field campaigns in Kansas, USA; (iii)turbulence regimes during CASES99.

2. The QNSE theory

The feature that sets the QNSE theory apart from thetraditional Reynolds-averaged Navier–Stokes (RANS) modelsis the inclusion of waves and turbulence anisotropy. This isenabled by the spectral nature of QNSE, which is based uponsuccessive ensemble-averaging over infinitesimally thin spectralshells yielding scale-dependent horizontal and vertical eddyviscosities and eddy diffusivities that account for the transportprocesses on the eliminated scales. This is not possible for RANSmodels as they are based upon ensemble-averaging over the entiredomain of fluctuating modes (Ferziger, 1993), a procedure thatlumps all scales together (Sukoriansky et al., 2005, 2006; Galperinand Sukoriansky, 2010; Sukoriansky and Galperin, 2013).

The QNSE operates with the inverses of the stability functions,SM = KM/K0 and SH = KH/K0, where K0 is the eddy viscosityat Rig = 0, KM and KH are the actual eddy viscosity and eddydiffusivity, respectively, all in the vertical direction. Furthermore,Rig refers to the gradient Richardson number, and SM and SH andare given in Figure 1 in terms of Rig evaluated using adjacent gridpoints. Due to the breaking of internal gravity waves accountedfor in the QNSE, the eddy viscosity remains finite at high valuesof Rig.

3. Closure of TKE in MYJ and QNSE

A turbulence closure module that uses either the MYJ orQNSE schemes invokes the prognostic equations for TKE; theremaining equations are reduced to algebraic relationships. With

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

Stability Functions in QNSE and MYJ Models

Table 1. Formulations for calculating the inverse of stability functions (S), lengthscale (l) and TKE production/dissipation in the MYJ and QNSE schemes.

QNSE MYJ

S Sm,h(Riloc) Sm,h

[TKE, l, ( ∂u

∂z )2,(

∂v∂z

)2,(

∂θv∂z

) ]

l If N2 ≥ 0, l = 1

l−1b +l−1

s

where lb = kz

1+ kzλ

, ls =0.75

√TKEN , and

λ = 0.0063u∗f

Preliminary l = l0kz/(kz + l0)

l0 = 0.1∫ hpbl

0 zqdz/∫ hpbl

0 qdzq = (2TKE)0.5

l adjusted to satisfy an equation forl/q

If N2<0, l as in MYJ

TKE TKE =TKE

[l, KM

(∂V∂z

)2, KH N2

] Iterative procedure TKE =TKE

[l, TKE, KM

(∂V∂z

)2, KH N2

]

Ri is the gradient Richardson number, k is the von Karman constant, hpbl is theboundary layer height, KM is the eddy viscosity, KH is the eddy diffusivity, N isthe Brunt-Vaisala frequency, f is the Coriolis parameter, and u∗ is the frictionvelocity. The subscript loc refers to calculation on the basis differences betweenadjacent model levels.

its three variables, Eq. (2) encapsulates the three differencesbetween the QNSE and MYJ schemes, namely, the effect ofthe stability functions (S), turbulence length scale (l) andTKE. The formulations for the length scale in the QNSEscheme do not originate from the QNSE theory; rather, thelength scale for stable stratification is based on Detering andEtling (1985) and Sukoriansky and Galperin (2008). The TKEproduction/dissipation is a basic implementation of the TKEequation that, nevertheless, employs the QNSE-based eddyviscosity and eddy diffusivity, whereas MYJ uses a PBL-height-based expression for l. In addition to the traditional TKE balanceequation, it also uses an iterative approach to better adjust theturbulent fluxes to their values at the current computational step(Janjic, 2001). Further details are given in Table 1. The stabilityfunctions in MYJ and QNSE are graphed as functions of Rig inFigure 1. The MYJ functions feature a critical Richardson numberof 0.505 beyond which the flow becomes laminar. As the goalof the article is to validate the analytical stability functions fromthe QNSE theory, isolating the effect of stability functions is ofprimary importance. To accomplish this, the QNSE formulationsfor the turbulence length scale and TKE are transferred to theMYJ scheme.

As both MYJ and QNSE schemes employ MOST in the surfacelayer, the differences between the schemes in this layer aredictated by the similarity functions. The QNSE-based functionsare compatible with the stability functions used in the boundarylayer scheme and are derived from them in the constant fluxlayer approximation. Since the QNSE theory applies to stablystratified and weakly unstably stratified turbulence, its resultscannot be used in strongly unstable conditions. Therefore, theQNSE expressions for the weakly unstable regime are extendedinto the strongly unstable regime using Paulson’s (1970) similarityfunctions. Contrasting with the QNSE approach, the similarityfunctions in the MYJ surface layer scheme are not based on thestability functions in the MYJ boundary layer scheme. Instead,empirical formulations of Holtslag and De Bruin (1988) are usedin stable conditions and the aforementioned Paulson functionsare invoked in the case of unstable stratification.

The integrated similarity functions are introduced in the MYJand QNSE surface layer schemes as follows:

Sfm = XMYJ,QNSE + log

(z

zo

), (3)

Sfh = Prt0

[YMYJ,QNSE + log

(z

zT

)], (4)

where Sf m and Sf h are the integrated similarity functions formomentum and heat, respectively, and X and Y represent different

Table 2. Model configurations used in the experiments.

Abbreviation TKE Stability function l Sfc layer Prt0

MYJ M M M M 1MYJ Pr072 M M M M 0.72MYJ Pr1 qnse Sf M M M Q 1QNSE Q Q Q Q 0.72MYJ 1 TKE Q M Q M 1MYJ TKE Q M M M 1QNSE Pr1 Q Q Q Q 1

Q, QNSE; M, MYJ.

expressions used in the QNSE and MYJ schemes. The roughnesslength is given by zo, the thermal roughness length is zT, and zdenotes height above the ground level. The theoretically derivedvalue of Prt0 in QNSE is 0.72, while in MYJ it is increased to1 by modifying one of the original Mellor–Yamada constants(Janjic, 2001).

4. Model, set-up and experiments

The WRF model is a state-of-the-art NWP system used for bothresearch and operational applications. It is a community-basedmodel developed in collaboration among several institutions inthe United States. The WRF software framework (WSF) includesdynamic solvers, physics packages, initialization programs,and the WRF variational data assimilation system. The twodynamics solvers are the advanced research WRF (ARW) and thenonhydrostatic mesoscale model (NMM). The ARW is discussedbelow in more detail, as it is the solver utilized in this study.

The ARW solver features fully compressible, Euler non-hydrostatic equations, which are conservative for scalar variables.The top of the model domain is a constant-pressure surface, andthe applied vertical coordinate is based on a terrain-followinghybrid level approach. In the horizontal regime, Arakawa C-grid staggering is used. The time integration part of the modelcurrently uses a second- or third-order Runge–Kutta schemewith a smaller time step for acoustic and gravity wave modes.A full description of the solver is presented in Skamarock andKlemp (2008).

To study the issues presented in section 1, seven differenttypes of WRF model experiments were carried out (Table 2)for three different time periods and two different domainset-ups (Table 3). These include experiments with unmodifiedQNSE and MYJ schemes, and the MYJ scheme with the QNSEturbulent length-scale and TKE production/dissipation (labelledas MYJ l TKE). The comparison between MYJ l TKE and QNSEis the most important as the only differences between the two isthe representation of the stability functions and MOST functionsin the surface layer. In MYJ l TKE, Prt0 is 1 whereas QNSEuses the analytical result of 0.72 from spectral theory. Next, twoexperiments test the relative significance of the functional formof the MOST functions and Prt0. Experiment MYJ Pr072 uses itsusual Holtslag and de Bruin and Paulson MOST functions, butwith Prt0 equal to the QNSE value of 0.72, whereas the experimentMYJ Pr1 qnse Sf utilizes QNSE-based MOST functions alongwith Prt0 from MYJ. These tests provide important informationabout the impact of the value of Prt0 upon the MOST-basedsurface layer results and predictions in the bulk of the PBL.Experiment MYJ TKE refers to the MYJ scheme using the QNSETKE subroutine. Finally, QNSE Pr1 refers to a QNSE experimentin which Prt0 was assigned the value 1 in both surface andboundary layer schemes.

Note that in the QNSE surface layer scheme zT is given aslightly different expression than in MYJ. Other differences arethe computation of saturation mixing ratio at the surface overwater and potential temperature. To isolate the effect of stabilityfunctions, we used the QNSE expression for both of these variablesin MYJ l TKE. For water points, the surface saturation mixingratio in QNSE is related to surface moisture flux and the surface

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

E.-M. Tastula et al.

Table 3. Case studies.

Simulation period Horizontalresolution

Verticallevels

SoutheasternCanada

12 Feb 2003 1800 UTC to17 Feb 2003 0000 UTC

– 50

CASES97 28 Apr 1997 0000 UTC to29 Apr 1997 1800 UTC4 May 1997 0000 UTC to5 May 1997 1800 UTC Parent domain 5610 May 1997 0000 UTC to11 May 1997 1800 UTC

30 km

20 May 1997 0000 UTC to21 May 1997 1800 UTC

CASES99 4 Oct 1999 0000 UTC to8 Oct 1999 0000 UTC7 Oct 1999 0000 UTC to11 Oct 1999 0000 UTC10 Oct 1999 0000 UTC to14 Oct 1999 0000 UTC13 Oct 1999 0000 UTC to17 Oct 1999 0000 UTC Nest 6 km 5616 Oct 1999 0000 UTC to20 Oct 1999 0000 UTC19 Oct 1999 0000 UTC to23 Oct 1999 0000 UTC22 Oct 1999 0000 UTC to26 Oct 1999 0000 UTC25 Oct 1999 0000 UTC to29 Oct 1999 0000 UTC

heat-exchange coefficient (KHS), whereas MYJ uses an exponentialformula based on surface temperature (Tsfc) and surface pressure(Psfc).

The ability of the MYJ and QNSE schemes to reproduce thenear-surface observations at 12 stations in very cold conditionswas tested in the model runs featuring a cold air outbreak insoutheastern Canada (Figure 2(a)). During the period of thesimulation there was strong steady southsoutheastward flowover the inner domain, with no synoptic-scale fronts presentto complicate the near-surface validation. The original QNSEimplementation in WRF did not include the option for fractionalsea ice. For the case study in question, however, this option wascrucial as the Gulf of St Lawrence was partially covered by seaice during the simulation period. A sea-ice wrapper was thereforeadded for the QNSE surface-layer scheme in the surface drivermodule.

Vertical profiles were examined in model runs for the CASES97and CASES99 field campaigns in April and May 1997, and 5–29October 1999 (Figure 2(b)). A total of 46 radio soundings wasused to evaluate the model performance in the lowest 1000 mduring stable conditions. The CASES99 turbulence data fromSun et al. (2012) were used to validate the relationship betweenturbulence strength and wind speeds in the model. Further detailsof the model runs are listed in Table 3.

The physics options used in all experiments are the WRFsingle-moment three-class scheme for microphysics (Hong andLim, 2006), the rapid radiative transfer model (RRTM) for long-wave radiation (Mlawer et al., 1997), the Dudhia (1989) schemefor short-wave radiation, and the Noah land surface model forland surface (Chen and Dudhia, 2001a, 2001b; Ek et al., 2003).The initial and boundary conditions are from the ERA-Interimreanalysis (Dee et al., 2011).

5. Results

5.1. Near-surface quantities

The model experiments for southeastern Canada are from 12February 2003 at 1800 UTC to 17 February 2003 at 0000 UTC. At

the end of the 24 h model spin-up period, a surface low-pressurearea was centred close to 51◦N, 60◦W and moved slowly eastwardbringing a northerly flow from the Labrador Peninsula to theGulf of St Lawrence region before moving over the Atlantic. Tocheck the validity of the model simulation, surface observationsfrom 12 weather stations (inside the nest in Figure 2(a))were used.

Table 4 provides the error statistics for the southeastern Canadaexperiments. The QNSE yields considerably smaller average bias(−0.9 K) for the 2 m virtual temperature (T2) than MYJ (−2.3 K).For the MYJ l TKE experiment, the bias is close to that of MYJ(−2.2 K). The RMSE also features a slightly lower value for QNSE(3.4 K) than for MYJ (3.7 K) and MYJ l TKE (3.6 K). The mostsurprising details were found, however, in the comparison of theexperiments MYJ, MYJ Pr072 and MYJ Pr1 qnse Sf. When Prt0

in the surface-layer scheme is set to 0.72 instead of 1 given inthe original MYJ MOST function, both the bias and root meansquare error (RMSE) of T2 are reduced closer to the QNSE values.The bias drops to −1.7 K and the RMSE becomes 3.4 K. On theother hand, curiously, if the value Prt0 = 1 is used in tandemwith the QNSE-based similarity functions (instead of the Holtslagand de Bruin and Paulson functions), then the results are almostidentical to those from the MYJ experiment. The correlationcoefficient between observations and model predictions for T2decreases when the Prandtl number is reduced from 1 to 0.72:MYJ, MYJ l TKE and MYJ Pr1 qnse Sf all yield 0.7 whereasMYJ Pr072 and QNSE produce 0.6.

The error statistics for the 2 m mixing ratio (MR2) and 10 mwind speed reveal no advantage of QNSE over MYJ: the MR2RMSE and bias in MYJ l TKE (0.21 and 0.07 g kg−1) are close tothose for QNSE (0.20 and 0.09 g kg−1). For W10, QNSE yieldsa slightly smaller bias (2.7 m s−1) than MYJ l TKE (3.1 m s−1).The RMSE is higher for QNSE, however. The similarity betweenMYJ and MYJ Pr1 qnse Sf is also obvious for the MR2 and W10results.

5.2. Vertical profiles

The effect of stability functions on the vertical profiles of virtualtemperature, mixing ratio, and wind speed is studied in the modelexperiments for CASES97 and CASES99 (Table 3). As our focusis on stably stratified conditions, only soundings exhibiting stabletemperature profiles were selected. The layer bulk Richardsonnumber (Rib) for the layer 2–150 m was between 0 and 2 in amajority of the profiles. In only eight cases was Rib greater than2. The Coordinated Universal Time (UTC) for the profiles usedranges from 0000 UTC to 1230 UTC. For the error analysis, thesounding data were linearly interpolated to model levels.

Figure 3 displays the RMSE and bias for the selected variables.The modelled profiles for the first five CASES97 and CASES99experiments listed in Table 3 converge at the approximate heightof 800 m. The errors in the virtual temperature profile arecharacterized by slightly negative biases close to the surfaceand positive values higher aloft. The QNSE features higherRMSEs than MYJ l TKE between 30 and 200 m and largerbiases in the layer 10–150 m. Unlike in the southeastern Canadaexperiments, QNSE and MYJ l TKE yield almost identical 2 mvirtual temperature bias. Below 30 m, the errors produced byMYJ Pr072 are larger than those in any of the four otherexperiments. Consistent with Table 4, the differences betweenMYJ and MYJ Pr1 qnse Sf are very small.

Both PBL schemes under investigation struggle to accuratelyreproduce the shapes and heights of the temperature inversions:out of the 46 profiles, the inversion top was placed at the same levelas in observations in 13 (MYJ) and 12 (QNSE) cases. If we countonly the times when the virtual temperature bias was less than orequal to 1 K at that level, the numbers are reduced to seven (MYJ)and five (QNSE). These failures reflect difficulties of simulatingstable boundary layers. They stem from such phenomena as low-level jets (LLJ), the interaction of turbulence with internal gravity

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

Stability Functions in QNSE and MYJ Models

(a)

(b)

Figure 2. Domain set-ups for southeastern Canada experiments (a) and CASES97 and CASES99 experiments (b).

waves, turbulence intermittency and anisotropy (e.g. Mahrt,2014). All these factors are poorly represented in numericalmodels yet they affect the virtual temperature profile.

The mixing ratio error profiles in Figure 3 feature a very clearseparation between three groups of experiments: (i) MYJ andMYJ Pr1 qnse Sf, (ii) QNSE and MYJ Pr072 and (iii) MYJ l TKE.QNSE and MYJ Pr072 show the largest bias and RMSE. It issignificant that the slight modification in the value of Prt0 (from

1 to 0.72) in the surface-layer scheme makes the MYJ Pr072experiment fall within the QNSE group. This means that themoisture profile is sensitive to the value of Prt0 used in thesurface-layer scheme much more than the temperature profile.The stability functions applied have a profound effect on themixing ratio bias up to 500 m. Experiment MYJ l TKE yields asmaller bias than the QNSE between 80 and 500 m, but below thisthe absolute biases are similar in magnitude. The RMSEs for the

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

E.-M. Tastula et al.

Table 4. Near-surface error statistics from southeastern Canada experiments (13 February 2003 1800 UTC to 17 February 2003 0000 UTC) averaged over 12 stations.

T2 (K) MR2 (g kg−1) W10 (m s−1)

RMSE Bias r RMSE Bias r RMSE Bias r

MYJ 3.7 −2.3 0.7 0.21 0.07 0.6 4.6 3.0 0.5MYJ Pr072 3.4 −1.7 0.6 0.21 0.04 0.5 4.9 3.0 0.5MYJ Pr1 qnse Sf 3.7 −2.3 0.7 0.21 0.07 0.6 4.6 3.0 0.5QNSE 3.4 −0.9 0.6 0.20 0.09 0.6 4.9 2.7 0.4MYJ 1 TKE 3.6 −2.2 0.7 0.21 0.07 0.6 4.6 3.1 0.5

(a)

(c) (d)

(f)(e)

(b)

Figure 3. Error statistics for (a,b) virtual temperature, (c,d) mixing ratio and (e,f) wind speed based on 46 soundings from the CASES97 and CASES99 field campaigns.

mixing ratio also reflect the better performance of MYJ l TKE.The difference in the MR2 values between MYJ and MYJ l TKEin Table 3 are due to the different formulations for the saturationmixing ratio in the MYJ and QNSE PBL schemes: MYJ l TKEadopts the QNSE formulation. Closer to the surface, the divisioninto three groups in the RMSE becomes blurred. The mixing ratiobias, however, is clearly divided all the way from 2 to 400 m.

Wind speeds, on average, are underestimated below and slightlyoverestimated above 200 m. It is noteworthy that the profiles ofthe wind-speed bias and RMSE both display a local maximumat the height of about 100 m. This reflects the failure of themodel to correctly capture the LLJ, which is higher than thatobserved, with lower than observed shear from 50 to 100 m.Problems in modelling LLJs are not rare and have been discussed

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

Stability Functions in QNSE and MYJ Models

(a)

(c) (d)

(f)(e)

(b)

Figure 4. Vertical profiles of virtual temperature and exchange coefficient for heat from CASES97 on 5 May 1997 at (a,b) 0200, (c,d) 0500 and (e,f) 0800 UTC.

by e.g. Tastula et al. (2012). The fact that all model configurationsproduce similar wind-speed bias and RMSE profiles hints that thedifficulties could be related to the initial conditions rather thanthe model parametrizations, as suggested by the work of Van deWiel et al. (2010).

LeMone et al. (2014) found that during some near-neutralnights with high wind speed, the height of virtual temperaturemaxima in the QNSE was too low, despite well-reproduced wind-speed profiles. In this study, we scrutinize this finding in anattempt to understand what could cause such a behaviour. Thisanalysis utilizes profiles from CASES97: 5 May at 0200, 0500 and0800 UTC as shown in Figure 4.

The night in question was characterized by strong shear inthe boundary layer, which was captured almost identically byboth the QNSE and MYJ (not shown). The virtual temperatureprofiles from the two experiments were, however, drastically

different (Figure 4). The MYJ maintains a shape of temperatureprofile that is closer to that observed, although it exhibits toomuch mixing close to the surface by 0800 UTC. On the otherhand, the QNSE develops a less stable layer from 50 to 300 m.Experiment MYJ l TKE isolates the effect of stability functionsfor the observed differences in the profiles. Comparing the KH

profiles from the MYJ and MYJ l TKE reveals that a significantpart of the greater mixing observed in the QNSE is due to theturbulence length-scale and TKE subroutines used. We will showthat the remaining differences between the QNSE and MYJ l TKEare due to the stability functions and Prt0.

To assess the relative role of the turbulence length-scale andTKE subroutines in the differences between the KH profiles of theMYJ and MYJ l TKE, a new experiment was carried out (denotedas MYJ TKE) in which the TKE subroutine in the MYJ schemewas replaced with that used in the QNSE scheme. The KH profiles

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

E.-M. Tastula et al.

Figure 5. Vertical profile of TKE from CASES97 on 5 May 1997 at 0800 UTC.

obtained for 0500 and 0800 UTC (not shown) feature MYJ TKEyielding values much higher than the MYJ and comparable tothose from the MYJ l TKE. This result is consistent with the TKEsubroutine having a larger effect on the differences in KH than thelength-scale subroutine. Plotting the TKE profiles at 0800 UTC(Figure 5) shows higher TKE below 200 m in the QNSE andMYJ l TKE than in the MYJ, MYJ Pr072, or MYJ Pr1 qnse Sf.The largest TKE values are associated with the MYJ TKE, however,demonstrating that changing the TKE subroutine in the MYJ hasa profound effect on the TKE profile up to 750 m.

Table 1 and Eq. (2) provide a clue as to why such elevatedlevels in TKE can lead to large KH values: KH is affected not onlyby TKE via the TKE subroutine itself, but also via the turbulencelength-scale subroutine for which TKE is an input parameter.Therefore, the effect of overestimated TKE is amplified by the wayKH is calculated. Further evaluations of TKE against observationsare given in section 5.3.

Based on the curves for the QNSE and MYJ l TKE in Figure 4,the stability functions applied also significantly affect the verticalprofiles of virtual temperature and KH. With a strong shear andnear-neutral stratification (i.e. close to zero Rig: the left-hand sideof Figure 1), a faithful replication of the prevailing meteorologicalconditions during the night of 5 May depended on the choiceof a value of Prt0. In conditions very close to neutral, the MYJ-based SH approaches 1, whereas in the QNSE it attains the valueof 1.39. As a result, any mixing characteristic related to SH isaffected by the choice of Prt0. We can therefore hypothesize thatthe low value of Prt0 in the QNSE is also likely to contribute tothe excessive mixing of temperature on 4–5 May in CASES97.In order to quantify this hypothesis, Prt0 was changed to 1 inthe QNSE (denoted as QNSE Pr1). The corresponding curves inFigure 4 demonstrate that Prt0 indeed has a substantial effect onthe virtual temperature and KH profiles. It must be mentionedthat although Prt0 can be altered in the QNSE stability functions,the results obtained with this set-up no longer represent thespectral theory, as the value of Prt0 in the stability functions is afundamental result of the QNSE theory and is consistent with datacollected in small-scale neutral boundary layers (e.g. Sukorianskyet al., 2005).

The small difference between the results from the MYJ andMYJ Pr1 qnse Sf observed in Figures 3 and 4 requires furtheranalysis. To study the matter, the percentage differences in Sf h

(Eq. (4)), denoted by �Sfh% in Figure 6, were plotted forfour nights during the CASES97 experiments. The percentage

differences were determined out by subtracting Sf h of the MYJfrom that of MYJ Pr072 (dashed line), and Sf h of MYJ from thatof MYJ Pr1 qnse Sf (solid line), and dividing by Sf h of the MYJ.In addition, we plotted the Monin–Obukhov stability parameterζ = z/L from the MYJ Pr072 and MYJ Pr1 qnse Sf at the firstfull-model level (z) for the same four nights. The Obukhov

length is defined as L = −u3∗/[κ

(g

v0

)wθv0

], where −wθv0 is

the surface virtual temperature flux, u∗ is the friction velocity,κ is the von Karman constant, w and θv0 are the fluctuations invertical velocity and virtual potential temperature, respectively, gis the gravitational acceleration and v0 is the reference virtualpotential temperature.

The dominant effect of Prt0 on �Sfh% is obvious in Figure 6(a):decreasing Prt0 from 1 to 0.72 decreases Sf h by approximately28% when ζ stays below 0.3. At higher stabilities, the magnitude of�Sfh% decreases for the MYJ Pr072, but the behaviour of �Sfh%for the MYJ Pr1 qnse Sf is different: the percentage change is atits largest, −10%, and it is closely tied to the changes in ζ . Withincreasing stability, the functional form of Sf h becomes moreimportant, however, within the range of stabilities present in themodel grid point during the time period in question, the effect ofPrt0 dominates over the functional form of Sf h.

The value of Sfh directly affects KHS because the two are relatedby KHS = u∗κ/Sf h. Therefore, the time series of KHS for the samefour nights features the same characteristics between the MYJ,MYJ Pr072 and MYJ Pr1 qnse Sf schemes (Figure 7(c)): KHS

values from the MYJ and MYJ Pr1 qnse Sf are similar, whereasthose of MYJ Pr072 are higher. An interesting feature is thatthe results are quite different for both the surface sensible andlatent heat fluxes (Figure 7(a,b)). There is no clear distinctionbetween the behaviour of the MYJ Pr072 and MYJ Pr1 qnse Sf inrelation to the MYJ, even though KHS was used to determine both.To investigate the reason for this, the formulation for the heatfluxes in the model needed to be considered. As an example, thesensible heat flux is calculated as – KHSCpPsfc(T1 − Tsfc)/(RdT1),where Cp is the specific heat of dry air at constant pressure, T1 istemperature at the first-model level and Rd is the gas constant fordry air. The similarity functions affect the sensible heat flux notonly via KHS but also through the ratio (T1 − Tsfc)/T1, which wasinfluenced by the KHS at previous time steps. Based on Figure 3,an increase in KHS when changing from the MYJ to MYJ Pr072leads to colder temperatures at the first-model level, implyingthat more cold air from the surface is brought up than warmer airfrom above is brought down. The surface flux results reveal thatKHS does not have such a direct effect on the ratio (T1 − Tsfc)/T1.Moreover, the fact that the vertical profile of the mixing ratioexhibits similarity between the MYJ and MYJ Pr1 qnse Sf butdissimilarity between the MYJ and MYJ Pr072 suggests that theeffect of KHS dominates of over the surface latent heat flux whendetermining the lower boundary value of humidity in the MYJPBL scheme.

As the representation of the turbulent fluxes in the QNSEand MYJ schemes depends on both the vertical gradient of thequantity in question and the turbulent exchange coefficient, it isalso important to consider the fluxes obtained from the variousmodel experiments. Figure 8 displays the turbulent fluxes forvirtual temperature, water-vapour mixing ratio and momentumaveraged over the same radio sounding times as in Figure 3.All experiments yield similar virtual temperature flux below100 m, but above this altitude the QNSE features oscillatingbehaviour, which is strongest around the height of 500 m.The water-vapour mixing ratio flux features more variabilitybetween different experiments. The choice of length scale andTKE subroutines has a significant influence on the mixing ratioflux: the MYJ l TKE features higher values than the MYJ from 50to 500 m. The QNSE also yields higher values than the MYJ, butthe same type of oscillating effect as with the virtual temperatureflux is also observed with the mixing ratio flux. Again, theMYJ and MYJ Pr1 qnse Sf are more alike than the MYJ andMYJ Pr072.

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

Stability Functions in QNSE and MYJ Models

(a)

(b)

Figure 6. (a) Normalized differences in similarity functions and (b) modelled ζ during four nights in CASES97 experiments.

(a)

(b)

(c)

Figure 7. Time series of (a) surface sensible and (b) latent heat fluxes and (c) the modelled surface-heat exchange coefficient during four nights in CASES97experiments. The observed fluxes are averages over three grassland sites.

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

E.-M. Tastula et al.

(a) (b)

(d)(c)Virtual temperature flux (K m s-1)

Momentum flux u′w′ (m2 S–2) Momentum flux v′w′ (m2 S–2)

Mixing ratio flux 10–3 × (g kg–1 m s–1)

Figure 8. (a,b) Averaged turbulent fluxes of virtual temperature, mixing ratio and (c,d) momentum during the CASES97 and CASES99 field campaigns. The sameaveraging is applied as in Figure 3.

The momentum fluxes between 50 and 400 m for the u andv wind components exhibit both the QNSE and MYJ l TKEproducing more negative fluxes than the other experiments. Thereason for this is likely to be the aforementioned overestimationof TKE by the QNSE TKE subroutine.

In this study, it has been shown that the choice of Prt0 canprofoundly affect the vertical profiles of virtual temperature,mixing ratio, the respective fluxes profiles and the KH profile,but what is the observational record of the value of Prt0 reportedin the literature? Observations of Prt0 feature a large amount ofscatter under all conditions (Monti et al., 2002; Galperin et al.,2007), partially due to self-correlation (Anderson, 2009). Esauand Grachev (2007) in their Figure 3 provide a summary of Prt0

from DNS, observations and several studies, and show that thevalues from various sources range from about 0.6 to about 1.1.There is thus no indisputable evidence of what the value of Prt0

should be in parametrization schemes of the PBL, but gettingthis value right is important as demonstrated by the findings inthis study: changes in Prt0 in the surface layer alone can resultin substantial changes in the profiles higher up in the boundarylayer.

Another possible factor affecting the performance of the QNSEmay be a strong vertical shear: QNSE theory does not includethe direct dynamical effect of the shear on turbulent fluctuations

(Sukoriansky et al., 2005; Galperin and Sukoriansky, 2010) and sothe model performance may suffer in conditions of strong shear.

5.3. Turbulence regimes

Figure 9 displays a turbulence velocity scale (defined as thesquare root of TKE) as a function of wind speed for the fourlowest model levels: black lines denote a crude approximationfor CASES99 observations from Sun et al. (2012). The pointwhere the slope of the black lines change designates the thresholdbetween two turbulence regimes: weak turbulence generated bylocal shear (the region to the left of the threshold) and strongturbulence generated by the shear over the entire layer (theregion to the right). The physical mechanism behind this regimetransition was recently explored by Sun et al. (2012) and Vande Wiel et al. (2012). Any observations well above the black lineare grouped as the third regime created by top-down turbulentevents. The bin-averaged standard deviations of the observedturbulent velocity scale typically range from 0.1 to 0.5 m s−1

(Sun et al., 2012; the vertical error bars in their figure 1(a)).Bearing this in mind, three initial remarks can be made: (i) atheights of 10 m, models overestimate the strength of turbulence;(ii) for other heights, there is an overall agreement betweenthe models and observations except for QNSE and MYJ l TKE;

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

Stability Functions in QNSE and MYJ Models

(a) (b)

(d)(c)

Figure 9. Turbulent velocity scale as a function of wind speed at four different vertical levels (a–d). The black lines are estimates from CASES99 based on Sunet al. (2012).

(iii) models often underestimate the weak turbulence regime nearthe threshold wind.

The difference between model results at 10 m and the heightsabove is obvious: at 10 m, models do not show the differentturbulence regimes, but rather, a quasi-exponential relationshipbetween wind speed and turbulence velocity scale is displayed.This could be interpreted as a failure of the MOST to followregime changes. At higher altitudes, however, the models appearto produce a threshold wind speed. Note that the MYJ turbulentvelocity scale of 0.1 m s−1 corresponds to its prescribed minimumvalue.

Figure 9 provides evidence that the TKE subroutine used inthe QNSE scheme overestimates TKE at high wind speeds, aswas hinted in section 5.2 by the findings for a night with strongwinds. What then is the fundamental difference between thetwo TKE production/dissipation approaches? Although in bothsubroutines the same equation is solved, and the dissipationterms are identical, the MYJ TKE subroutine applies an iterativeprocedure as described in detail in Janjic (2001). Future research

should assess the need for implementing the iterative TKEapproach into the QNSE PBL scheme.

6. Conclusions

This study concentrated on two turbulence closure schemes,MYJ and QNSE, with the purpose of elucidating the effectof the stability functions on the accuracy of temperature,moisture, wind speed and turbulence predictions in the WRFmodel. The investigation utilized near-surface observationsfrom 12 stations in southeastern Canada and sounding datafrom two field campaigns, CASES97 and CASES99. The mainfindings are:

i) The QNSE stability functions provide the best near-surfaceresults for virtual temperature.

ii) The QNSE functions yield slightly larger biases and RMSEsfor virtual temperature between 30 and 200 m and formixing ratio between 80 and 500 m.

c© 2014 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2015)

E.-M. Tastula et al.

iii) The value of the neutral Prandtl number (Prt0) inthe Monin–Obukhov similarity functions for heat andmoisture appears more important than the functionalform of the similarity function itself. However, when thenear-surface stability increases, the relative importance ofthe functional form increases as well. Changing Prt0 in thesurface-layer scheme may affect the moisture profile up to500 m. A Prt0 value of 0.72 may lead to excessive scalarmixing in near-neutral conditions.

iv) The TKE subroutine used in the QNSE scheme yields toohigh turbulent kinetic energy at high wind speeds, a featurethat can significantly amplify KH.

The results demonstrate that the stability functions, as wellas the value of Prt0, may have a wide-ranging effect upon ver-tical profiles of many variables. The isolated influence of thesefunctions is often camouflaged in PBL scheme validation stud-ies when chosen parametrizations also employ different lengthscales, TKE production subroutines or surface-layer representa-tions. Other options may also play a role. This is well demonstratedin the present study by comparing the results from the QNSE,MYJ l TKE and MYJ model configurations. One of the mostimportant findings is that combining the results for both the vir-tual temperature and mixing ratio in the altitude range between,approximately, 30 and 500 m, the MYJ stability functions performbetter than those originating from the QNSE formulation. Onthe other hand, the performance of the QNSE with respect tothe near-surface virtual temperature is clearly superior. Similarresults for near-surface temperature were obtained by Dim-itrova et al. (2014) (online at http://www3.nd.edu/∼dynamics/materhorn/news.php) in their intercomparison between differ-ent boundary-layer schemes used in the WRF model and byTastula et al. (2014) in their one-dimensional WRF simulations.

To the authors’ knowledge, the relative importance of the Prt0

and the functional form of the MOST function for heat hasnot been addressed in previous studies. The results, however,point to the fact that the choice of Prt0 in the surface layermay bear significant repercussions for the entire boundary layer,especially in the case of humidity. Moreover, the effect of Prt0

may overwhelm the effect of the actual function itself. Thismeans that previous model validation studies that concentratedon the performance of different similarity functions and theircombinations may, in fact, have been reporting about the choiceof Prt0, in case Prt0 varied among the employed functions.Coosing the correct value for Prt0 is therefore crucial, but noconsensus on the matter exists, due to the spread of observedvalues. Most parametrization (such as the MYJ, YSU, Boulac) usePrt0 = 1, whereas the QNSE and the original Mellor–Yamadamodel use 0.72. Values between these two are also used: forinstance, in the HARMONIE model (Termonia et al., 2012)the value for Prt0 is approximately 0.88 (C. Fortelius, 2014;personal communication). A future work on this topic includeslooking into the possibility of using different mixing lengths fortemperature and momentum as a way to control the neutralPrt0 without interfering with the stability functions. All in all, asalso concluded by Sterk et al. (2013), the choice of the stabilityfunctions and the surface-layer parametrization affect not justnear-surface quantities but vertical profiles in the entire boundarylayer.

Acknowledgements

The participation of EMT was funded by a Paul L. GettingMemorial Fellowship in Marine Science and Elsie and WilliamKnight Fellowship in Marine Science. The work of TV wassupported by the Academy of Finland (contract 259537).LeMone and Dudhia are supported by the National Centerfor Atmospheric Research, which is sponsored by the NationalScience Foundation. We thank Dr Jielun Sun and the reviewersfor valuable comments. Environment Canada is acknowledged

for providing the near-surface weather data used in section 5.1.We would also like to acknowledge high-performance computingsupport from Yellowstone provided by NCAR’s Computationaland Information Systems Laboratory, sponsored by the NationalScience Foundation.

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