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

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Methodical assessment of the differences between the QNSE and MYJ PBL schemes for stable conditions
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  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 Sukoriansky  c and Timo Vihma d a College of Marine Science, University of South Florida, St. Petersburg, FL, USA b  National Center for Atmospheric Research, Boulder, CO, USA c Department Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel  d Finnish 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 recent years. 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–Janji´c (MYJ) schemes,in the weather research and forecasting model. Our objectives are to isolate the effect of stability functions on the near-surface values and vertical profiles of virtual temperature,mixing ratio and wind speed. The results demonstrate that the QNSE stability functions yield 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 isthesensitivityofthemodelresultstothechoiceoftheturbulentPrandtlnumberforneutralstratification( Pr  t0 ) :intheMonin–Obukhovsimilarityfunctionforheat,thechoiceof  Pr  t0  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 energy subroutine and the stability functions, including the effect of   Pr  t0 . 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 Thepastdecadeofdevelopmentinnumericalweatherforecastinghas 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(SkamarockandKlemp,2008).Althoughstudiesof thiskindhavebeenvaluableindeterminingoptimalcombinationsfor model physics options and possible incompatibilities amongschemes,studiesoftheactualphysicswithintheparametrizationsarerare.Yet,understandingtheroleofvariousprocessesandtheirinteraction 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 coefficient K  . The  K  -formulation has a large effect on the model results. In1.5-level PBL schemes in the Mellor–Yamada hierarchy such astheMellor–Yamada–Janji´c(MYJ;Janji´c,2001)andquasi-normalscale elimination (QNSE; Sukoriansky   et al.,  2005),  K   takes thefollowing form: K   = α lS √  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 similarity functions, which are determined empirically. The validity of these log–linear expressions is supported by direct numericalsimulation (DNS) studies (e.g. Ansorge and Mellado, 2014).A substantial body of literature exists that addresses similarity functions: reviews have been given by e.g. H¨ogstr¨om (1996) andFoken(2006).Thefunctionalformsoftheseempiricalexpressionsdiverge 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 andgravitywaves,oftenfoundinstableconditions,furthercomplicatethe situation. Yet an accurate prediction of the evolving stably stratified 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–V¨ais¨al¨afrequency,verticalwindshearandseveralempiricallydeterminedcoefficients. The srcinal 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 Janji´c’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 srcinating inclose proximity to first principles based upon the Navier–Stokesequationshavebeenrare.Therelativelyrecentlydevelopedquasi-normal scale elimination (QNSE) theory of stably stratifiedturbulence (Sukoriansky   et al.,  2005, 2006) yields such a set of analytical functions for the vertical eddy viscosity andeddy diffusivity. Close to the surface in the approximation of the 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 stability functions on model simulations. Such comparisons are critical,however,notonlyduetotheusageoftheQNSEstabilityfunctionsin 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 wouldilluminatethe impactof 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 boundary layer 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 only significant 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  S M  and  S H  as afunction of gradient Richardson number. OthertestsrelatedtoMOSTfunctionsarecarriedoutaswell.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 stability functionsfromthemodellingperspective.TheoreticaltestsfortheQNSE 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 eddy viscosities and eddy diffusivities that account for the transportprocesses on the eliminated scales. This is not possible for RANSmodelsastheyarebaseduponensemble-averagingovertheentiredomain of fluctuating modes (Ferziger, 1993), a procedure thatlumps all scales together (Sukoriansky   et al.,  2005, 2006; Galperin and Sukoriansky, 2010; Sukoriansky and Galperin, 2013).The QNSE operates with the inverses of the stability functions, S M  = K  M / K  0  and  S H  = K  H / K  0 , where  K  0  is the eddy viscosity at  Ri g  = 0,  K  M  and  K  H  are the actual eddy viscosity and eddy diffusivity, respectively, all in the vertical direction. Furthermore, Ri g  refers to the gradient Richardson number, and  S M  and  S H  andare given in Figure 1 in terms of   Ri g  evaluated using adjacent gridpoints. Due to the breaking of internal gravity waves accountedfor in the QNSE, the eddy viscosity remains finite at high valuesof   Ri g . 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 S m,h ( Ri loc )  S m , h  TKE  , l  ,( ∂ u ∂ z   ) 2 ,  ∂ v  ∂ z   2 ,  ∂θ  v  ∂ z   l   If   N  2 ≥ 0, l  =  1 l  − 1 b  + l  − 1 s where  l  b  =  kz  1 + kz  λ , l  s  = 0 . 75 √  TKE  N   , and λ =  0 . 0063 u ∗  f  Preliminary   l  = l  0 kz  / ( kz  + l  0 ) l  0  = 0 . 1   hpbl  0  zq d z  /   hpbl  0  q d z q = (2 TKE  ) 0 . 5 l   adjusted to satisfy an equation for l  / q If   N  2 < 0,  l   as in MYJTKE  TKE  = TKE   l  , K   M   ∂ V  ∂ z   2 , K  H   N  2  Iterative procedure  TKE  = TKE   l  , TKE  , K   M   ∂ V  ∂ z   2 , K  H   N  2  Ri is the gradient Richardson number, k is the von K´arm´an constant, hpbl is theboundary layer height, K M  is the eddy viscosity, K H  is the eddy diffusivity, N isthe Brunt-V¨ais¨al¨a 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 of the stability functions ( S ), turbulence length scale ( l  ) andTKE. The formulations for the length scale in the QNSEscheme do not srcinate 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 eddy viscosity 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(Janji´c, 2001). Further details are given in Table 1. The stability functions in MYJ and QNSE are graphed as functions of   Ri g  inFigure1.TheMYJfunctionsfeatureacriticalRichardsonnumberof 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 of primary 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 boundary layer scheme and are derived from them in the constant flux layer approximation. Since the QNSE theory applies to stably stratified and weakly unstably stratified turbulence, its resultscannot be used in strongly unstable conditions. Therefore, theQNSE expressions for the weakly unstable regime are extendedintothestronglyunstableregimeusingPaulson’s(1970)similarity functions. Contrasting with the QNSE approach, the similarity functions 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: Sf  m  =  X  MYJ,QNSE + log   z z  o  , (3) Sf  h  = Pr  t0  Y  MYJ,QNSE + log    z z  T  , (4)where  Sf  m  and  Sf  h  are the integrated similarity functions formomentumandheat,respectively,and  X  and Y   representdifferent Table 2 .  Model configurations used in the experiments.Abbreviation TKE Stability function  l   Sfc layer  Pr  t0 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 1Q, QNSE; M, MYJ. expressions used in the QNSE and MYJ schemes. The roughnesslength is given by   z  o , the thermal roughness length is  z  T , and  z  denotes height above the ground level. The theoretically derivedvalue of   Pr  t0  in QNSE is 0.72, while in MYJ it is increased to1 by modifying one of the srcinal Mellor–Yamada constants(Janji´c, 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,  Pr  t0  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  Pr  t0 . Experiment MYJ Pr072 uses itsusual Holtslag and de Bruin and Paulson MOST functions, butwith Pr  t0  equaltotheQNSEvalueof0.72,whereastheexperimentMYJ Pr1 qnse Sf utilizes QNSE-based MOST functions alongwith  Pr  t0  from MYJ. These tests provide important informationabout the impact of the value of   Pr  t0  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  Pr  t0  was assigned the value 1 in both surface andboundary layer schemes.Note that in the QNSE surface layer scheme  z  T  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 stability functions,weusedtheQNSEexpressionforbothofthesevariablesin 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 HorizontalresolutionVerticallevelsSoutheasternCanada12 Feb 2003 1800 UTC to17 Feb 2003 0000 UTC– 50CASES97 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 UTC30 km20 May 1997 0000 UTC to21 May 1997 1800 UTCCASES99 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-exchangecoefficient( K  HS ),whereasMYJusesanexponentialformula based on surface temperature ( T  sfc ) and surface pressure( P  sfc ).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 flow over the inner domain, with no synoptic-scale fronts presentto complicate the near-surface validation. The srcinal 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.VerticalprofileswereexaminedinmodelrunsfortheCASES97and 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 betweenturbulencestrengthandwindspeedsinthemodel.Furtherdetailsof 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. Atthe 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.Table4providestheerrorstatisticsforthesoutheasternCanadaexperiments. The QNSE yields considerably smaller average bias( − 0.9K)forthe2mvirtualtemperature(T2)thanMYJ( − 2.3K).For the MYJ l TKE experiment, the bias is close to that of MYJ( − 2.2K).TheRMSEalsofeaturesaslightlylowervalueforQNSE(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  Pr  t0 in the surface-layer scheme is set to 0.72 instead of 1 given inthe srcinal MYJ MOST function, both the bias and root meansquareerror(RMSE)ofT2arereducedclosertotheQNSEvalues.The bias drops to − 1.7 K and the RMSE becomes 3.4 K. On theother hand, curiously, if the value  Pr  t0  = 1 is used in tandemwiththeQNSE-basedsimilarityfunctions(insteadoftheHoltslagand 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,mixingratio,andwindspeedisstudiedinthemodelexperiments for CASES97 and CASES99 (Table 3). As our focusis on stably stratified conditions, only soundings exhibiting stabletemperature profiles were selected. The layer bulk Richardsonnumber ( Ri b ) for the layer 2–150 m was between 0 and 2 in amajority of the profiles. In only eight cases was  Ri b  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 by MYJ 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 accurately reproduce the shapes and heights of the temperature inversions:outofthe46profiles,theinversiontopwasplacedatthesamelevelas 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)QNSEandMYJ Pr072and(iii)MYJ l TKE.QNSE and MYJ Pr072 show the largest bias and RMSE. It issignificant that the slight modification in the value of   Pr  t0  (from1 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   Pr  t0  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 asmallerbiasthantheQNSEbetween80and500m,butbelowthisthe absolute biases are similar in magnitude. The RMSEs for the c  2014 Royal Meteorological Society   Q. J. R. Meteorol. Soc.  (2015)
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