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Turbulence and surface-layer parameterizations for mesoscale models Dmitrii V. Mironov German Weather Service, Offenbach am Main, Germany Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

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Budget equations for the second-order turbulence moments Parameterizations (closure assumptions) of the dissipation, third-order transport, and pressure scrambling A hierarchy of truncated second-order closures – simplicity vs. physical realism The surface layer Effects of water vapour and clouds Stably stratified PBL over temperature-heterogeneous surface – LES and prospects for improving parameterizations Conclusions and outlook Outline Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

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References Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June Mironov, 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: / )

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Recall a Trivial Fact … Transport equation for a generic quantity f Split the sub-grid scale (SGS) flux divergence Convection (quasi-organised) mass-flux closure Turbulence (quasi-random) ensemble-mean closure

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Energy Density Spectrum ln(E) ln(k) Resolved scales ( -1 is effectively a mesh size) Quasi-random motions (turbulence closure schemes) Quasi-organized motions (mass-flux schemes) Sub-grid scales Viscous dissipation Cut-off at very high resolution (LES, DNS)

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Temperature (heat) flux Reynolds stress Second-Moment Budget Equations Temperature variance

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Turbulence kinetic energy (TKE) Second-Moment Budget Equations (contd) (Monin and Yaglom 1971)

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Time-rate-of-change, advection by mean velocity Physical Meaning of Terms Mean-gradient production/destruction Buoyancy production/destruction) Coriolis effects Third-order transport (diffusion) Pressure scrambling Viscous dissipation

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Closure Assumptions: Dissipation Rates Transport equation for the TKE dissipation rate Simplified (heavily parameterized) ε -equation

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Closure Assumptions: Dissipation Rates (contd) Algebraic diagnostic formulations (Kolmogorov 1941) Closures are required for the dissipation time or length scales!

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Closure Assumptions: Third-Order Terms Numerous parameterizations, ranging from simple down- gradient formulations, to very sophisticated high-order closures.

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Closure Assumptions: Third-Order Terms (contd) 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, An advanced model of third-order terms (e.g. Canuto et al. 1994) The results is a very complex model (set of sophisticated algebraic relations) that still has many shortcomings.

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Accounts for non-local transport due to coherent structures, e.g. convective plumes or rolls – mass-flux ideas! (Gryanik and Hartmann 2002) Skewness-Dependent Parameterization of Third-Order Transport Down-gradient term (diffusion) Non-gradient term (advection)

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Skewness-Dependent Parameterization of Third-Order Transport (contd) Plume/roll scale advection velocity

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Analogies to Mass-Flux Approach A top-hat representation of a fluctuating quantity After M. Köhler (2005) Updraught Downdraught (environment) Only coherent top-hat part of the signal is accounted for

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Transport equation for the temperature (heat) flux Transport equation for the Reynolds stress Closure Assumptions: Pressure Scrambling For later use we denote the above pressure terms by ij and i

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Budget of 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 / x 3, green – third-order transport – / x 3, black – buoyancy g 3, blue – pressure gradient-temperature covariance. The budget terms are made dimensionless with the Deardorff (1970) convective scales of depth, velocity and temperature. Temperature Flux Budget in Boundary-Layer Convection Pressure term Free convection Convection with rotation

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Linear Models of ij and i The simples return-to-isotropy parameterisation (Rotta 1951) Analogously, for the temperature flux (e.g. Zeman 1981)

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Linear Models of ij and i (contd)

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Equation for the temperature flux Linear Models of ij and i (contd)

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Poisson equation for the fluctuating pressure Decomposition Contribution to p due to buoyancy Linear Models of ij and i (contd) NB! The volume of integration is the entire fluid domain.

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The buoyancy contribution to i is modelled as The simplest (linear) representation … satisfying … we obtain Cf. Table 1 of Umlauf and Burchard (2005): C b = (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. Linear Models of ij and i (contd)

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Similarly for the buoyancy contribution to ij (Reynolds stress equation) … satisfying … we obtain Table 1 of Umlauf and Burchard (2005): C u b = (0.5, 0.0, 0.0, 0.5, 0.4, 0.495, 0.5). 3/10? Linear Models of ij and i (contd)

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Non-Linear Intrinsically Realisable TCL Model The buoyancy contribution to i is a non-linear function of departure-from-isotropy tensor The representation … together with the other constraints (symmetry, normalisation) … yields Realisability. The two-component limit constraints (Craft et al. 1996)

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Buoyancy contribution to i in convective boundary-layer flows (Mironov 2001). Short-dashed – LES data, solid – linear model with C b =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. Models of i against data TCL model (sophisticated and physically plausible) still does not perform well in some important regimes.

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Truncated 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.

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Truncated Second-Order Closures (contd) 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 model consists of isotropic down-gradient formulations for fluxes, and production-dissipation equilibrium relations for the TKE and the scalar variances.

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Two-Equation TKE-Scalar Variance Model (MY74 level 3) Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g. Transport equations for the TKE and for the scalar variance(s)

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One-Equation TKE Model (MY74 level 2.5) Algebraic formulations for the Reynolds stress components and for the scalar fluxes, e.g. Transport equation for the TKE Diagnostic formulation(s) for the scalar variance(s)

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Comparison of 1-Eq and 2-Eq Models Equation for Production = Dissipation (implicit in all models that carry the TKE equation only). Equation for No counter-gradient term (cf. turbulence models using counter- gradient corrections heuristically). 1-Eq Models are Draft Horses of Geophysical Turbulence Modelling

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Importance of Scalar Variance The TKE equation The equation Prognostic equations for (kinetic energy of SGS motions) and for (potential energy of SGS motions). Convection/stable stratification = Potential Energy Kinetic Energy. No reason to prefer one form of energy over the other!

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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 K θ is m 2 /s.) Exercise

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The Surface Layer The now classical Monin-Obukhov surface-layer similarity theory (Monin and Obukhov 1952, Obukhov 1946). The surface-layer flux-profile relationships MOST breaks down in conditions of vanishing mean velocity (free convection, strong static stability).

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The Surface Layer (contd) 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 assumptions (i) turbulence is continuous, stationary and horizontally-homogeneous, (ii) third-order turbulent transport is negligible, and (iii) changes of fluxes over the surface layer are small as compared to their changes over the entire PBL.

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Effects of Water Vapour and Clouds Quasi-conservative variables Virtual potential temperature is defined with due regard for the water loading

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qtqt qtqt qtqt x ΔxΔx x ΔxΔx qtqt xx Neglect SGS fluctuations of temperature and humidity, all-or-nothing scheme Account for humidity fluctuations only Account for temperature and humidity fluctuations no clouds, C = 0 C = 1 Cloud cover 0

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after Tompkins (2002) cloud cover, cloud condensate = integral over supersaturated part of PDF If PDF of s is known, then However, PDF is generally not known! SGS statistical cloud schemes assume a functional form of PDF with a small number of parameters. Input parameters (moments predicted by turbulence scheme) Assumed PDF Diagnostic estimates of C,, etc. cloud cover cloud condensate Turbulence and Clouds (contd)

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Buoyancy flux (a source of TKE), is expressed through quasi-conservative variables, where A θ and A q are functions of mean state and cloud cover A q is of order 200 for cloud-free air, but 800 ÷ 1000 within clouds! Turbulence and Clouds (contd) functional form depends on assumed PDF A θ = A θ (C, mean state) A q = A q (C, mean state) Clouds-turbulence coupling: clouds affect buoyancy production of TKE, turbulence affect fractional cloud cover (where accurate prediction of scalar variances is particularly important).

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LES 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 SBL Poor 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 moments Budgets 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)

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s Surface Temperature in Homogeneous and Heterogeneous Cases time 8h9.75hsampling s = ( s1 + s2 ) s1 s2 homogeneous case heterogeneous case x s2 s1 s y warm stripe cold stripe +

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Mean Potential Temperature Blue – homogeneous SBL, red – heterogeneous SBL. cf. Stoll and Porté-Agel (2009)

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TKE and Temperature Variance Blue – homogeneous SBL, red – heterogeneous SBL. Large

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TKE Budget Left panel – homogeneous SBL, right panel – heterogeneous SBL. Red – shear production, blue – dissipation, black – buoyancy destruction, green – third-order transport, thin dotted black – tendency. Decreased in magnitude

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Temperature Variance Budget Left panel – homogeneous SBL, right panel – heterogeneous SBL. Red – mean-gradient production/destruction, blue – dissipation, green – third-order transport, black (thin dotted) – tendency. Net source

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Key Point: Third-Order Transport of Temperature Variance LES estimate of (resolved plus SGS) In heterogeneous SBL, the third-order transport of temperature variance is non-zero at the surface Surface temperature variations modulate local static stability and hence the surface heat flux net production/destruction of due to divergence of third-order transport term!

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a a s2 s1 s s2 x zz Third-Order Transport of Temperature Variance

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Enhanced Mixing in Horizontally-Heterogeneous SBL An Explanation increased near the surface reduced magnitude of downward heat flux less work against the gravity increased TKE stronger mixing Increased Decreased (in magnitude) downwardupward

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In order to describe enhanced mixing in heterogeneous SBL, an increased 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. Can We Improve SBL Parameterisations?

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Tiled TKE-Temperature Variance Model: Results Blue – homogeneous SBL, red – heterogeneous SBL. (Mironov and Machulskaya 2012, unpublished)

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Conclusions and Outlook Only a small fraction of what is currently known about geophysical turbulence is actually used in applications … but we can do better Beware of limits of applicability! TKE-Scalar Variance turbulence scheme offers considerable prospects (IMHO) Improved models of pressure terms Interaction of clouds with skewed and anisotropic turbulence PBLs over heterogeneous surfaces Croatian - USA Workshop on Mesometeorology, Ekopark Kraš Resort near Zagreb, Croatia June 2012.

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Thanks 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.

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Transport equation for the temperature flux then neglecting anisotropy (!) Using Rotta-type return-to-isotropy parameterisation of the pressure gradient- temperature covariance yields the down-gradient formulation Exercise: derive down-gradient approximation for fluxes from the second-moment equations

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