15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013 Modelling Scalar Skewness: an Attempt to Improve the Representation of Clouds and Mixing Using a Double-Gaussian Based Statistical Cloud Scheme Dmitrii Mironov 1, Ekaterina Machulskaya 1, Ann Kristin Naumann 2, Axel Seifert 1,2, and Juan Pedro Mellado 2 1) German Weather Service, Offenbach am Main, Germany 2) Max Plank Institute for Meteorology, Hamburg, Germany 2 September 2013

15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013 Outline  Motivation  3-parameter double-Gaussian PDF of linearized saturation deficit, a priori testing  Transport equation for the skewness: derivation, closure assumptions, and coupling with the TKE- Scalar Variance mixing scheme  Results from single-column numerical experiments with the TKESV-S-3PDG coupled system  Conclusions and outlook

15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013 From PP UTCS Final Status Report (Lugano, 13 September 2012): a Look into the Future  Development of a three-moment (mean, variance, and skewness) statistical cloud scheme that accounts for non- Gaussian effects. Co-operation with Axel Seifert and Ann Kristin Naumann, Hans Ertel Centre on Cloud and Convection (HErZ), Hamburg.  Further development and testing of transport equations for the skewness of scalar quantities, coupling the skewness equations with the three-moment statistical cloud scheme.

Motivation (Bougeault, 1982) Two-parameter PDF is insufficient – mean saturation deficit normalized by its variance In cumulus regime, the PDF asymmetry (incl. non-Gaussian tails) should be accounted for. With the same mean and variance, cloud cover and the amount of cloud condensate vary as function of skewness.

Motivation (cont‘d) Assumed PDFs used in cloud schemes (e.g. Larson et al. 2001) One delta function (no variability), uniform (unfavorable shape) – poor Gaussian, triangular – insufficient (symmetric) Two delta – insufficient (ignore small-scale fluctuations) Gamma, log-normal – insufficient (allow only positive skewness) Beta – good, however unimodal 5-parameter double-Gaussian – remarkably good (very flexible, etc.), but complex and too computationally expensive 3-parameter double-Gaussian – good (flexible enough, etc.), likely an optimal choice

Double-Gaussian PDF Five parameters, viz., a, s 1, s 2, σ 1, and σ 2 should be determined to specify a double-Gaussian PDF. To these end, five PDF moments should be predicted, e.g. the first five moments. Too complex and too computationally expensive!

3-Parameter Double-Gaussian PDF Using LES data, Naumann et al. (2013) proposed Cf. Larson et al. (2001) Now σ 1 and σ 2 are functions of S and only 3 moments are needed to specify PDF.

Double-Gaussian PDF: Relation between a and S The relation f(a, σ 1, σ 2, S) = 0 follows from the definition of double-Gaussian PDF moments and is independent of any assumptions about σ 1 and σ 2.

Relation between a and S If σ 1 and σ 2 are functions of S only (3-parameter PDF), the relation f(a, σ 1, σ 2, S) = 0 reduces to F(a, S) = 0. F(a, S) is the 7th-order polynomial with respect to a. Numerical solution (Naumann et al. 2013) 0 ≤ a ≤ 1 as S → +∞, a → 0 as S → –∞, a → 1

Relation between a and S (cont’d) Asymptotic analysis suggests from cubic equation

Relation between a and S (cont’d) Power-law interpolation between the two asymptotics ← selects the smallest value Similarly for S < 0

Buoyancy term in the TKE equation In terms of quasi-conservative variables, where A and B are functions of mean state and cloud cover B is ≈ 180 for cloud-free air, but ≈ 800 ÷ 1000 within clouds! Formulations of A and B are required to model the TKE production Effect of Clouds on the Buoyancy Production of TKE

R is equal to the cloud fraction C for Gaussian PDF R=C does not hold in many situations, e.g. for cumulus clouds (C is small but is dominated by ) Non-Gaussian correction is required to compute the buoyancy flux! can be obtained without further assumptions for clear-sky (“dry”) and overcast (“wet”) grid boxes for “wet”, is used Interpolation: Effect of Clouds on the Buoyancy Production of TKE (cont’d)

In terms of liquid water flux Approximation of F with due regard for skewness S Formulation of Naumann et al. (2013)

A Priori Testing of Cloud Schemes using assumed PDF, compute C,... compare C,, etc., with observations and/or LES (DNS) PDF parameters are computed by a turbulence model PDF parameters are determined using observational and/or numerical (LES, DNS) data (“ideal input”)

A Priori Testing: Cloud Fraction and Cloud Water

A Priori Testing: Expression for Buoyancy Flux (cloud fraction and cloud water are from LES) LES data

A Priori Testing: Expression for Buoyancy Flux (cloud fraction and cloud water are diagnosed from assumed PDF) LES data

Coupling with Turbulence Scheme using assumed PDF, compute C,... compare C,, etc., with observations and/or LES (DNS) PDF parameters are computed by a turbulence model PDF parameters are determined using observational and/or numerical (LES, DNS) data (“ideal input”)

First- and Second-Order Moments The statistical cloud scheme requires first three moments of the distribution of linearized saturation deficit First-order moment is provided by the grid-scale equations Second-order moment should be provided by turbulence scheme TKESV scheme carries transport equations for

Recall: linearized saturation deficit is defined as Third-Order Moment Simplification: neglect the time rate-of-change (and advection) of Q and Π Strictly speaking, four third-order moments are required to determine

The equation for the third-order moment of s Closure is required for dissipation rate, third-order moment, and fourth-order moment Transport Equation for Dissipation rate:,, l is the length scale

Third-order moment (Mironov et al. 1999, Gryanik and Hartmann 2002) Equation for : Closure Assumptions small-scale random fluctuations PBL-scale coherent structures (≈ Gaussian) (two-delta function = mass-flux) As the resolution is refined, the SGS motions are (expected to be) increasingly Gaussian. Then, S  0 and the parameterization of the third-order transport term reduces to the down-gradient diffusion approximation. interpolation

Equation for : Closure Assumptions (cont’d) No need for equations of higher order! Fourth-order moment (Gryanik and Hartmann, 2002) Gaussian formulation two-delta function (=mass-flux) interpolation

TKESV + New Cloud Scheme: Skewness of s BOMEX shallow cumulus test case ( Profiles are computed by means of averaging over last 3 hours of integration (hours 4 through 6). LES data are from Heinze (2013).

TKESV + New Cloud Scheme: Cloud Fraction and Cloud Water

TKESV + New Cloud Scheme: Variances of Temperature and Humidity

TKESV + New Cloud Scheme: TKE and Buoyancy Flux

15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013 Conclusion and Outlook  Transport equation for the skewness of linearized saturation deficit is developed and coupled to the TKESV mixing scheme and the statistical cloud scheme based on a 3- parameter double-Gaussian PDF  First results from single-column numerical experiments look promising  Comprehensive testing (stratus clouds, etc.)  Numerical issues, implementation into COSMO and ICON  Effect of microphysics on the scalar variance and skewness (in co-operation with the HErZ-CC team)

15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013 Acknowledgements: Peter Bechtold, Rieke Heinze, Bob Plant, Siegfried Raasch, Pier Siebesma, Jun-Ichi Yano Thank you for your kind attention!

UNUSED

Equation for a and S Unfortunately, it may be not so easy to find the solution numerically in some cases: May we approximate the solution without solving the equation?

15th COSMO General Meeting, Sibiu, Romania, 2-6 September 2013

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