1. transport equations for the scalar variances
Development and testing of the moist second-order turbulence-convection model including
transport equations for the scalar variances
Ekaterina Machulskaya1 and Dmitrii Mironov2
1 Hydrometeorological Centre of Russian Federation, Moscow, Russia
2 German Weather Service, Offenbach am Main, Germany
COSMO General Meeting, Offenbach am Main, Germany
7-11 September 2009

2. Outline Moist TKE-Scalar Variance closure model Outline of test cases
DYCOMS-II: Nocturnal stratocumuli
BOMEX: Shallow cumuli
Conclusions and outlook

3. Moist TKE-Scalar Variance Closure Model
Prognostic equations for TKE and for variances of scalars (liquid water potential temperature, total water specific humidity)
Diagnostic equations for the scalar fluxes and for the Reynolds-stress components
Algebraic expression for turbulence length (time) scale
Statistical SGS cloud scheme, either Gaussian (Sommeria and Deardorff 1977), or with exponential tail to account for the effect of cumulus clouds (Bechtold et al. 1995)
Optionally, prognostic equations for temperature and humidity triple correlation (to determine temperature and humidity skewness)

4. Outline of Test Cases DYCOMS-II
Nocturnal marine stratocumuli (LES and some observational data)
Prescribed surface fluxes of heat and moisture, prescribed large-scale subsidence, prescribed radiation heating
Fractional cloud cover of order 80%, profiles of some second-order moments available (LES), no well-documented second-moment budgets
BOMEX
Trade-wind shallow cumuli (LES and some observational data)
Prescribed surface fluxes of heat and moisture, prescribed large-scale subsidence, prescribed radiation heating, prescribed large-scale drying rate of the sub-cloud layer, prescribed z-dependent geostrophic wind (baroclinicity)
Fractional cloud cover of order 10%, profiles of some second-order moments available, some LES-based second-moment budgets available for similar cases

5. Sc Case Liquid water specific humidity.
Left: TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).
Right: LES data (Stevens et al. 2005)

6. Sc Case Vertical virtual potential temperature (buoyancy) flux.
Left: TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).

7. Data suggest cloud fraction of order 0.8
Sc Case (cont’d)
Data suggest cloud fraction of order 0.8
Fractional cloud cover simulated by the TKE model (red) and by the TKE-Scalar Variance (TKE-TPE) model (blue).

8. Sc Case (summary)
TKE-Scalar Variance model performs well and slightly better than TKE model
However, vertical resolution is more critical
LES:
cloud fraction = 0.8
Fractional cloud cover simulated by the TKE model (red) and by the TKE-Scalar Variance (TKE-TPE) model (blue) at high vertical resolution (left) and low vertical resolution (right).

9. Cu Case Variance of the total water specific humidity.
Left: TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).
Right: LES data (Cuijpers et al. 1996)

10. Data suggest cloud fraction of order 0.1
Cu Case (cont’d)
Data suggest cloud fraction of order 0.1
Fractional cloud cover with Gaussian SGS statistical cloud scheme.
TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).

11. Cu Case (cont’d) Data suggest cloud fraction of order 0.1
Gaussian SGS cloud scheme gives no clouds!
Fractional cloud cover with the l and qt profiles taken from LES.
Curves shows cloud cover with non-Gaussian SGS statistical cloud scheme (includes exponential tail to account for the effect of cumuli).

12. Cu Case (cont’d) 1 0.5 Strong mean undersaturation =
Cloud fraction 1
Exponential tail
0.5
Linear approximation
of the error function
(Gaussian distribution)
Normalised saturation deficit
Strong mean undersaturation =
shallow cumuli case

13. Cu Case (cont’d)
Fractional cloud cover simulated by the TKE-Scalar Variance model using Gaussian (blue) and non-Gaussian (green) SGS statistical cloud scheme.

14. Cu Case (summary)
SGS cloud-cover parameterisation is crucial for successful simulation of Cu
Not only fractional cloud cover should be parameterised realistically, but also the effect of cumuli on the buoyancy production of TKE must be accounted for (fractional coverage of Cu is small, but the effect on fluxes is large – no way to describe this effect with Gaussian schemes!)
Models (both TKE and TKE-Scalar Variance) have insufficient mixing in the upper part of the Cu cloud layer

15. Conclusions and Outlook
Moist TKE-Scalar Variance model shows reasonable performance (better than TKE model, but there is room for improvement)
The use of prognostic equations for the scalar triple correlations causes numerical problems (small time step is required); expert assistance is requested
A comprehensive data set on second-moment budgets (from LES) is very desirable (otherwise component model testing is very difficult)
Consolidation of the TKE-Scalar Variance model
Formulation of the best-compromise version in terms physics, numerical stability, and computational efficiency
Implementation into the COSMO model (effort should be co-ordinated with WG2 and WG6)
In order to obtain a linear algebraic system of equation with respect to all second-order moments, pressure correlation terms are parameterized linearly.

16. Thank you for your attention!
Thanks are due to Pier Siebesma (KNMI) for discussions and helpful suggestions.

17. Stuff Unused

18. Sc Case (cont’d)
Vertical virtual potential temperature (buoyancy) flux.
Left: TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).
Right: LES data (Steven et al. 2005)

19. Data suggest cloud fraction of order 0.1
Cu Case (cont’d)
Data suggest cloud fraction of order 0.1
Fractional cloud cover with Gaussian SGS statistical cloud scheme.
TKE model (red) vs. TKE-Scalar Variance (TKE-TPE) model (blue).

20. Cu Case (cont’d)
Liquid water potential temperature (left) and total water specific humidity (right) simulated by the TKE-Scalar Variance model using Gaussian (blue) and non-Gaussian (green) SGS statistical cloud scheme.