Presentation on theme: "Turbulence and surface-layer parameterizations for mesoscale models"— Presentation transcript:
1Turbulence and surface-layer parameterizations for mesoscale models Dmitrii V. MironovGerman Weather Service, Offenbach am Main, GermanyCroatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
2Outline Budget equations for the second-order turbulence moments Parameterizations (closure assumptions) of the dissipation, third-order transport, and pressure scramblingA hierarchy of truncated second-order closures – simplicity vs. physical realismThe surface layerEffects of water vapour and cloudsStably stratified PBL over temperature-heterogeneous surface – LES and prospects for improving parameterizationsConclusions and outlookCroatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
3ReferencesMironov, D. V., 2009: Turbulence in the lower troposphere: second-order closure and mass-flux modelling frameworks. Interdisciplinary Aspects of Turbulence, Lect. Notes Phys., 756, W. Hillebrandt and F. Kupka, Eds., Springer-Verlag, Berlin, Heidelberg, doi: / )Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
4Recall a Trivial Fact … Transport equation for a generic quantity f Split the sub-grid scale (SGS) flux divergenceConvection (quasi-organised)mass-flux closureTurbulence (quasi-random)ensemble-mean closure
5Energy Density Spectrum Quasi-organized motions(mass-flux schemes)ln(E)Quasi-random motions(turbulence closure schemes)Resolved scales(-1 is effectivelya mesh size)ViscousdissipationSub-grid scales-1-1ln(k)Cut-off at very high resolution(LES, DNS)
7Second-Moment Budget Equations (cont’d) Turbulence kinetic energy (TKE)(Monin and Yaglom 1971)
8Physical Meaning of Terms Time-rate-of-change,advection by mean velocityMean-gradient production/destructionBuoyancy production/destruction)Coriolis effectsThird-order transport (diffusion)PressurescramblingViscousdissipation
9Closure Assumptions: Dissipation Rates Transport equation for the TKE dissipation rateSimplified (heavily parameterized) ε-equation
10Closure Assumptions: Dissipation Rates (cont’d) Algebraic diagnostic formulations (Kolmogorov 1941)Closures are required for the dissipation time or length scales!
11Closure Assumptions: Third-Order Terms Numerous parameterizations, ranging from simple down-gradient formulations,to very sophisticated high-order closures.
12Closure Assumptions: Third-Order Terms (cont’d) An “advanced” model of third-order terms (e.g. Canuto et al. 1994)take transport equations for all (!) third-order moments involved,neglect /t and advection terms,use linear parameterizations for the dissipation and the pressure scrambling terms,use Millionshchikov (1941) quasi-Gaussian approximation for the forth-order moments,The results is a very complex model (set of sophisticated algebraic relations) that still has many shortcomings.
13Skewness-Dependent Parameterization of Third-Order Transport Down-gradient term(diffusion)Non-gradient term(advection)But there exists another possibility, namely, the possibility to account for non-local advective nature of the third-order transport due to the presence of coherent structures in convective flows.This can be done through introduction of an additional skewness-dependent non-gradient (ballistic) term into the formulation of the third-order transport term.Physically, skewness…In order to determine skewness… -> next slideAccounts for non-local transport due to coherent structures, e.g. convective plumes or rolls – mass-flux ideas!(Gryanik and Hartmann 2002)
14Skewness-Dependent Parameterization of Third-Order Transport (cont’d) Plume/roll scale“advection” velocityBut there exists another possibility, namely, the possibility to account for non-local advective nature of the third-order transport due to the presence of coherent structures in convective flows.This can be done through introduction of an additional skewness-dependent non-gradient (ballistic) term into the formulation of the third-order transport term.Physically, skewness…In order to determine skewness… -> next slide
15Analogies to Mass-Flux Approach A top-hat representation of a fluctuating quantityUpdraughtOnly coherent top-hat part of the signal is accounted forDowndraught(environment)After M. Köhler (2005)
16Closure Assumptions: Pressure Scrambling Transport equation for the Reynolds stressTransport equation for the temperature (heat) fluxFor later use we denote the above pressure terms by ij and i
17Temperature Flux Budget in Boundary-Layer Convection Pressure termFree convectionConvection with rotationBudget of <u’3’> in the surface buoyancy flux driven convective boundary layer that grows into a stably stratified fluid. The budget terms are estimated on the basis of LES data (Mironov 2001). Red – mean-gradient production/destruction <u’3’><>/x3, green – third-order transport –<u’3u’3’>/x3, black – buoyancy g3<’2>, blue – pressure gradient-temperature covariance <’ p’/x3>. The budget terms are made dimensionless with the Deardorff (1970) convective scales of depth, velocity and temperature.
18Linear Models of ij and i The simples return-to-isotropy parameterisation (Rotta 1951)Analogously, for the temperature flux (e.g. Zeman 1981)
20Linear Models of ij and i (cont’d) Equation for the temperature flux
21Linear Models of ij and i (cont’d) Poisson equation for the fluctuating pressureDecompositionContribution to p’ due to buoyancyNB! The volume of integration is the entire fluid domain.
22Linear Models of ij and i (cont’d) The buoyancy contribution to i is modelled asThe simplest (linear) representation… satisfying … we obtainCf. Table 1 of Umlauf and Burchard (2005):Cb = (1/3, 0.0, 0.2, 1/3, 1/3, 1/3, 1.3).NB! The best-fit estimate for convective boundary layer is 0.5.
23Linear Models of ij and i (cont’d) Similarly for the buoyancy contribution to ij (Reynolds stress equation)… satisfying … we obtainTable 1 of Umlauf and Burchard (2005):Cub = (0.5, 0.0, 0.0, 0.5, 0.4, 0.495, 0.5).3/10?
24Non-Linear Intrinsically Realisable TCL Model The buoyancy contribution to i is a non-linear function of departure-from-isotropy tensorThe representationRealisability. The two-component limit constraints (Craft et al. 1996)… together with the other constraints (symmetry, normalisation) … yields
25Models of i against data Buoyancy contribution to i in convective boundary-layer flows (Mironov 2001).Short-dashed – LES data,solid – linear model with Cb=0.5,long-dashed – non-linear TCL model (Craft et al. 1996).3 is scaled with the Deardorff (1970) convective scales of depth, velocity and temperature.TCL model (sophisticated and physically plausible) still does not perform well in some important regimes.
26Truncated Second-Order Closures Mellor and Yamada (1974) used “the degree of anisotropy” (the second invariant of departure-from-isotropy tensor) to scale and discard/retain the various terms in the second-moment budget equations and to develop a hierarchy of turbulence closure models for PBLs.
27Truncated Second-Order Closures (cont’d) The most complex model (level 4 of MY74)prognostic transport equations (including third-order transport terms) for all second-order moments are carried.Simple models (levels 1 and 2 of MY74)all second-moment equations are reduced to the diagnostic down-gradient formulations.The most simple algebraic modelconsists of isotropic down-gradient formulations for fluxes,and production-dissipation equilibrium relations for the TKE and the scalar variances.
28Two-Equation TKE-Scalar Variance Model (MY74 level 3) Transport equations for the TKE and for the scalar variance(s)Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.
29One-Equation TKE Model (MY74 level 2.5) Transport equation for the TKEDiagnostic formulation(s) for the scalar variance(s)Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g.
30Comparison of 1-Eq and 2-Eq Models Equation for <’2>Production = Dissipation (implicit in all models that carry the TKE equation only).Equation for <w’’>Summarizing, the main differences between the one-equation and two-equation model formulations are the following…Then, if we want to use the two-equation model, two topics appear – how to model this term and this one.No counter-gradient term (cf. turbulence models using “counter-gradient corrections” heuristically).1-Eq Models are Draft Horses of Geophysical Turbulence Modelling
31Importance of Scalar Variance Prognostic equations for <ui’2> (kinetic energy of SGS motions) and for <’2> (potential energy of SGS motions).Convection/stable stratification =Potential Energy Kinetic Energy.No reason to prefer one form of energy over the other!The TKE equationThe <’2> equation
32Exercise Given transport equation for the temperature flux, make simplifications and invoke closure assumptions to derive a down-gradient approximation for the temperature flux,(Hint: the dimensions of Kθ is m2/s.)
33The Surface LayerThe now classical Monin-Obukhov surface-layer similarity theory (Monin and Obukhov 1952, Obukhov 1946).The surface-layer flux-profile relationshipsMOST breaks down in conditions of vanishing mean velocity (free convection, strong static stability).
34The Surface Layer (cont’d) The MO flux-profile relationships are consistent with the second-moment budget equations. In essence, they represent the second-moment budgets truncated under the surface-layer similarity-theory assumptionsturbulence is continuous, stationary and horizontally-homogeneous,third-order turbulent transport is negligible, andchanges of fluxes over the surface layer are small as compared to their changes over the entire PBL.
35Effects of Water Vapour and Clouds Quasi-conservative variablesVirtual potential temperature is defined with due regard for the water loading
36Turbulence and Clouds qt qt x x Δx Δx Account for humidity Neglect SGS fluctuations of temperature and humidity, all-or-nothing schemeqtqtno clouds, C = 0C = 1xxΔxΔxAccount for humidityfluctuations onlyAccount for temperatureand humidity fluctuationsqtqtxxCloud cover 0<C<1, although the grid box is unsaturated in the mean
37Turbulence and Clouds (cont’d) If PDF of s is known, thencloud cover, cloud condensate =integral over supersaturated part of PDFcloud covercloud condensateHowever, PDF is generally not known!SGS statistical cloud schemesassume a functional form of PDFwith a small number of parameters.Input parameters (moments predicted by turbulence scheme) →Assumed PDF → Diagnostic estimates of C, , etc.after Tompkins (2002)
38Turbulence and Clouds (cont’d) Buoyancy flux (a source of TKE),is expressed through quasi-conservative variables,where Aθ and Aq are functionsof mean state and cloud coverAθ = Aθ (C, mean state)Aq = Aq (C, mean state)functional form depends on assumed PDFAq is of order 200 for cloud-free air, but ≈ 800 ÷ 1000 within clouds!Clouds-turbulence coupling: clouds affect buoyancy production of TKE, turbulence affect fractional cloud cover (where accurate prediction of scalar variances is particularly important).
39LES of Stably Stratified PBL (SBL) Traditional PBL (surface layer) models do not account for many SBL features (static stability increases turbulence is quenched sensible and latent heat fluxes are zero radiation equilibrium at the surface too low surface temperature)No comprehensive account of second-moment budgets in SBLPoor understanding of the role of horizontal heterogeneity in maintenance of turbulent fluxes (hence no physically sound parameterization)LES of SBL over horizontally-homogeneous vs. horizontally-heterogeneous surface [the surface cooling rate varies sinusoidally in the streamwise direction such that the horizontal-mean surface temperature is the same as in the homogeneous cases, cf. GABLS, Stoll and Porté-Agel (2009)]Mean fields, second-order and third-order momentsBudgets of velocity and temperature variance and of temperature flux with due regard for SGS contributions (important in SBL even at high resolution)(Mironov and Sullivan 2010, 2012)
40in Homogeneous and Heterogeneous Cases Surface Temperaturein Homogeneous and Heterogeneous Casesss1s =(s1+ s2)homogeneous caseheterogeneous cases2time8hsampling9.75hys1+ cold stripesxwarm stripe s2
41Mean Potential Temperature cf. Stoll and Porté-Agel (2009)Blue – homogeneous SBL,red – heterogeneous SBL.
42TKE and Temperature Variance LargeBlue – homogeneous SBL, red – heterogeneous SBL.
43TKE BudgetDecreased inmagnitudeLeft panel – homogeneous SBL, right panel – heterogeneous SBL.Red – shear production, blue – dissipation, black – buoyancy destruction, green – third-order transport,thin dotted black – tendency .
44Temperature Variance Budget Net sourceLeft panel – homogeneous SBL, right panel – heterogeneous SBL.Red – mean-gradient production/destruction, blue – dissipation, green – third-order transport, black (thin dotted) – tendency .
45Key Point: Third-Order Transport of Temperature Variance LES estimate of <w’’2> (resolved plus SGS)In heterogeneous SBL,the third-order transportof temperature variance isnon-zero at the surfaceSurface temperature variations modulate local static stability and hence the surface heat flux net production/destruction of <’2> due to divergence of third-order transport term!
46Third-Order Transport of Temperature Variance xs2zzs1as2a
47Enhanced Mixing in Horizontally-Heterogeneous SBL An Explanation increased <’2> near the surface reduced magnitude of downward heat flux less work against the gravity increased TKE stronger mixingDecreased(in magnitude)IncreasedIncreasedКак смоделировать этот эффект? --- Переход к следующему слайду.downwardupward
48Can We Improve SBL Parameterisations? In order to describe enhanced mixing in heterogeneous SBL,an increased <’2> at the surface should be accounted for.Elegant way: modify the surface-layer flux-profile relationships. Difficult – not for nothing are the Monin-Obukhov surface-layer similarity relations used for more than 1/2 a century without any noticeable modification!Less elegant way: use a tile approach, where several parts with different surface temperatures are considered within an atmospheric model grid box.
49Tiled TKE-Temperature Variance Model: Results Blue – homogeneous SBL,red – heterogeneous SBL.(Mironov and Machulskaya 2012, unpublished)
50Conclusions and Outlook Only a small fraction of what is currently known about geophysical turbulence is actually used in applications … but we can do betterBeware of limits of applicability!TKE-Scalar Variance turbulence scheme offers considerable prospects (IMHO)Improved models of pressure termsInteraction of clouds with skewed and anisotropic turbulencePBLs over heterogeneous surfacesCroatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
51Thanks for your attention! Acknowledgements: Peter Bechtold, Vittorio Canuto, Sergey Danilov, Stephan de Roode, Evgeni Fedorovich, Jean-François Geleyn, Andrey Grachev, Vladimir Gryanik, Erdmann Heise, Friedrich Kupka, Cara-Lyn Lappen, Donald Lenschow, Vasily Lykossov, Ekaterina Machulskaya, Pedro Miranda, Chin-Hoh Moeng, Ned Patton, Jean-Marcel Piriou, David Randall, Matthias Raschendorfer, Bodo Ritter, Axel Seifert, Pier Siebesma, Pedro Soares, Peter Sullivan, Joao Teixeira, Jeffrey Weil, Jun-Ichi Yano, Sergej Zilitinkevich.The work was partially supported by the NCAR Geophysical Turbulence Program and by the European Commission through the COST Action ES0905.Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
52Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.
53Exercise: derive down-gradient approximation for fluxes from the second-moment equations Transport equation for the temperature flux(!) Using Rotta-type return-to-isotropy parameterisation of the pressure gradient-temperature covariancethen neglecting anisotropyyields the down-gradient formulation