1. Further steps towards a scale separated turbulence scheme: Matthias Raschendorfer DWD Aim: General valid (consistent) description of sub grid scale (SGS) processes Problem:Closure assumptions are constraints additional to the only valid first principals  General valid closure assumptions can’t exist  SGS turbulence and e.g. convection can’t be described by one set of 2-nd order equations Solution:Scale separation  A system of closure equations for each scale separated process with specific closure assumptions wake vortices by SSO (sub grid scale orography) blocking horizontal shear vortices surface induced density flow patterns shallow and deep convection patterns Complication:Larger scale SGS processes are interacting with turbulence!!  Scale interaction terms Validation: COSMO Rome 2011  Operational verification  Statistical procession with the package TMOS using ACARS turbulence data

2.  Turbulence closure is only valid for scales not larger than - the smallest peak wave length L p of inertial sub range spectra from samples in any direction ( ) - the largest (horizontal) dimension D g of the control volume  Scale separation by - averaging these budgets along the whole control volume ( double averaging ) Consistent partial solution for turbulence by spectral separation: Turbulence is that class of sub grid scale structures being in agreement with turbulence closure assumptions! - considering budgets with respect to the separation scale generalized turbulent budgets including additional scale interaction terms Matthias Raschendorfer Oberpfaffenhofen: 10-11.05.2010 Filter is a moving volume average with infinitesimal vertical extension and horizontal dimension L. WakeNet workshop

3. Physical meaning of the scale interaction terms:  Budgets for the non turbulent SGS structures (SGS circulations) : T he scale interaction term is shifting Co-Variance (e.g. S ub grid scale K inetic E nergy) form the circulation part of the spectrum ( CKE ) to the turbulent part ( TKE ) by virtue of shear generated by the non turbulent SGS flow patterns. CKE TKE production terms dependent on: specific length scales and specific velocity scales (= ) production terms depend on: single turbulent length scale and single turbulent velocity scale (= ) circulation -scale turbulence -scale statistical moments Matthias Raschendorfer and other We need to consider additional length scales besides the turbulent length scale! COSMO Rome 2011 DWD source term scale interaction sink

4. Separated semi parameterized TKE equation (neglecting laminar shear and transport): buoyancy production eddy- dissipation rate (EDR) labil: neutral: stabil: time tendency transport (advection + diffusion) shear production by sub grid scale circulations expressed by turbulent flux gradient solution to be parameterized by a non turbulent approach shear production by the mean flow : with respect to the separation scale L buoyant part of buoyant and wake part of mean (horizontal) shear production of circulations, according Kolmogorov Matthias Raschendorfer : correction factor in case of sloped model layers COSMO Rome 2011 DWD

5.  Separated horizontal shear production term: effective mixing length of diffusion by horizontal shear eddies velocity scale of the separated horizontal shear mode scaling parameter  Equilibrium of production and scale transfer towards turbulence: scaling parameter horizontal shear eddy isotropic turbulence horizontal grid plane TKE-production by separated horizontal shear modes: grid scale ……….effective scaling parameter separated horizontal shear additional TKE source term Matthias Raschendorfer DWD COSMO Rome 2011

6. out_usa_shs_rlme_a_shsr_0.2 Pot. Temperature [K] SN 06.02.2008 00UTC + 06h -92 E out_usa_shs_rlme_a_shsr_1.0 Matthias Raschendorfer = (dissipation) 1/3 frontal zone Oberpfaffenhofen: 10-11.05.2010WakeNet workshop

7.  SSO-term in filtered momentum budget: blocking term TKE-production by separated wake modes due to SSO: currently Lott und Miller (1997)  Pressure term in kinetic energy budget: wake source sources of mean kinetic energy MKE buoyancy production sources of sub grid scale kinetic energy SKE pressure transport expansion production from inner energy DWD Matthias RaschendorferCOSMO Rome 2011 Equilibrium of production and loss by scale transfer

8. moderate light SN 06.02.2008 00UTC + 06h -77 E mountain ridge SSO-effect in TKE budget out_usa_rlme_tkesso out_usa_rlme_sso out_usa_rlme_tkesso – out_usa_rlme_sso MIN = 0.00104324 MAX = 10.3641 AVE = 0.126079 SIG = 0.604423MIN = 0. 00109619 MAX = 10.3689 AVE = 0.127089 SIG = 0.804444 MIN = -0.10315 MAX = 0.391851 AVE = 0.00100152 SIG = 0.00946089 = (dissipation) 1/3 Increased due to separated wake terms Matthias Raschendorfer DWD COSMO Rome 2011

9. 10X10 GP above Appalachian mountains out_usa_shs_rlme_sso out_usa_shs_rlme_a_shsr_0.2 COSMO user seminarOffenbach: 09-11.03.2009 Matthias Raschendorfer

12. virtual potential temperature of ascending air circulation scale temperature variance ~ circulation scale buoyant heat flux circulation term TKE-Production by convection (thermal circulations):  Circulation scale 2-nd order budgets with proper approximations valid for thermals: convective thermals virtual potential temperature of descending air Matthias Raschendorfer vertical velocity scale of circulation can be derived directly form current mass flux convection scheme COSMO Rome 2011 DWD Equilibrium of production and loss by scale transfer

13. Matthias Raschendorfer COSMO Rome 2011 DWD

14. Matthias Raschendorfer reference including horizontal shear – and SSO- production including horizontal shear –, SSO- and convective production pot. temperature [K] COSMO Rome 2011 DWD

15. Turbulence index = 1 (light)Turbulence index = 4 (moderate) Turbulence index = 5 (severe) Colours for measurement height in [m] Matthias Raschendorfer COSMO Rome 2011 DWD

16. Matthias Raschendorfer COSMO Rome 2011 DWD

17. Matthias Raschendorfer Distribution between Model- and ARCAS-EDR: -Prediction-pedictor correlation: 0.44 COSMO Rome 2011 DWD

18. Matthias Raschendorfer Final distribution after successive regression: -21 predictors -most effective besides edr: p, dt_tke_(con, sso, hsh) -Successive cubic regression of residuals -Prediction-pedictor correlation: 0.627 -Variance reduction: 39.9 % COSMO Rome 2011 DWD

19. Conclusion:  A double filter approach formally generates a system of 2-nd order equations valid for turbulence closure approximations  It differs form the usual single filter approach (according to the grid scale) only by additional scale interaction terms  They describe the source of turbulent 2nd order moments by the action of shear from non turbulent (larger scale) sub grid scale flow structures  Those are  Horizontal shear eddies  Wake eddies by SSO  Convective vertical flow circulations  For them exist specific closure assumptions and they generate their own larger scale diffusion (e.g. by coherent mass flux transport)  Scale interaction is able to generate a needed larger amount of EDR compared to measurements  However, the used ACARS EDR data seem to be biased by  The domination of either low-level or high level measurements  The avoiding of strong turbulence events except in low levels near air ports  The influence of aircrafts ahead during the low level flights  Uncertainties of altitude registration  Flight activities  Thus for the time being, simply pressure or altitude is a significant predictor for EDR Matthias Raschendorfer COSMO Rome 2011 DWD

20.  Correction of ACARS data and considering other data sources including LES-data  Some revisions concerning the solution of TKE equation and implicit formulation of vertical diffusion  Reformulation of the surface induced density flow term (original circulation term) in the current scheme to become a thermal SSO production dependent on SSO parameters  Investigation of adoptions regarding the turbulent length scale above the boundary layer  Generating a consistent ensemble of sub grid scale parameterizations by expressing the non turbulent ones scale dependent, containing the scale interaction terms as sink terms.  A revised formulation of mass flux convection has already started (talk in Moskow)  Adoption of the sub grid scale cloud description in the framework of scale separation  Expression of sub grid scale transport by SSO eddies and horizontal shear eddies Next steps: Matthias Raschendorfer COSMO Rome 2011 DWD

21. turbulent peak wavelength ln [wave number k] frequency of aircraft oscillations aircraft velocity with respect to mean wind TKECKE model resolution turbulence attenuation function and velocity of the aircraft spectrum of vertical oscillations inertial sub range spectrum of atmosphere EDR by regression of the Kolmogorov spectrum Aircraft measurements of EDR (from ACARS data base): T urbulent C irculation Oberpfaffenhofen: 10-11.05.2010 Matthias Raschendorfer E nergy K inetic WakeNet workshop

22. Effect of the density flow driven circulation term for stabile stratification: Even for vanishing mean wind and negative turbulent buoyancy there remains a positive definite source term TKE will not vanishSolution even for strong stability horizontal scale of a grid box turbulent buoyancy flux circulation buoyancy flux Matthias Raschendorfer DWD CLM-Training Course