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 Machulskaya1, Ann Kristin Naumann2,

Axel Seifert 1,2, and Juan Pedro Mellado2

1) German Weather Service, Offenbach am Main, Germany

2) Max Plank Institute for Meteorology, Hamburg, Germany

(dmitrii.mironov@dwd.de)

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

s

lst TqqQ

)(

1

– 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, s1, s2 , σ1, and σ2 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 σ1 and σ2 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, σ1, σ2, S) = 0 follows from the definition of double-Gaussian PDF moments and is independent of any assumptions about σ1 and σ2 .

Relation between a and S

If σ1 and σ2 are functions of S only (3-parameter PDF), the relation f(a, σ1, σ2, 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

ltlv qwDqwBwAg

wg ***

In terms of quasi-conservative variables

, where A and B are functions of mean state and cloud cover

tlv qwBwAw

B is ≈ 180 for cloud-free air, but ≈ 800 ÷ 1000 within clouds!

Formulations of A and B are requiredto model the TKE production

Effect of Clouds on the Buoyancy Production of TKE

wetv

dryvv wRwRw )1(

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

)( lstl Tqqq for “wet”,

is used

tdryldrydry

v qwBwAw

twetlwetwet

v qwBwAw

Interpolation:

Effect of Clouds on the Buoyancy Production of TKE (cont’d)

vw

vw wet

vw

In terms of liquid water flux

swFCqw l

Approximation of F with due regard for skewness S

Formulation of Naumann et al. (2013)

s

lst TqqQ

QSQF

QSQSQF

)(

where

,0as1,

0as25.0exp5.11,

1

11

12

11

A Priori Testing of Cloud Schemes

using assumed PDF,compute C, ...

compare C, , etc.,with observationsand/or LES (DNS)

lq

lq

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 observationsand/or LES (DNS)

lq

lq

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 MomentsThe 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, S0 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 (http://www.knmi.nl/~siebesma/BLCWG/#case5) .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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