Some thoughts on statistical cloud schemes

1. ADRIAN Some thoughts on statistical cloud schemes a recipe for seafood laksa soup Adrian Tompkins, ECMWF With Contributions from: Rob Pincus (NOAA-CIRES Climate Diagnostics Center) Georg Bauml (Max Plank Institut fur Meteorologie, Hamburg) …and my cloud “mentor” Steve Klein Plus material stolen from Jean-Christoph Golaz and Vince Larson…

2. So just what will I speak about? • Some background things (preparing the ingredients) • Statistical cloud schemes (the laksa paste) • My scheme (so hands off !) (the soup) • Level of complexity required? (noodles) • Linking with other model components (combining the ingredients) • Conclusions and future developments? (presentation)

3. Some Background things Preparing the ingredients…Clean fish, slice garlic and chilli finely, grate ginger and palm sugar, toast and grind cumin and coriander seeds

4. ~500m ~100km Issues of Parameterization HORIZONTAL COVERAGE, C Most of this talk will concentrate on this issue

5. ~500m ~100km Issues of Parameterization VERTICAL COVERAGE Most models assume that this is 1 This can be a poor assumption with coarse vertical grids.

6. Issues of Parameterization Vertical overlap of cloud Important for Radiation and Microphysics Interaction ~500m ~100km

7. cloud inhomogeneity ~500m ~100km Issues of Parameterization

8. ~500m ~100km Issues of Parameterization Just these issues can become very complex!!!

9. q x One Grid-cell Horizontal Cloud Cover: The Problem Partial coverage of a grid-box with clouds is only possible if there is a inhomogeneous distribution of temperature and/or humidity Homogeneous distribution of water vapour and temperature: Cloud cover zero or one Local criterion for existence of cloud: qtotal > qsat (T,p) This assumes that no supersaturation can exist (poor assumption for ice clouds)

10. q x Horizontal Cloud Cover: The Problem Heterogeneous distribution of T and q… …clouds exist before the grid-mean relative humidity reaches 100%

11. 1 CC (cloud fraction) 0 0 RH0 1 Relative humidity Diagnostic Relative Humidity Schemes • Many schemes, from the mid 1970s onwards, based cloud cover on the relative humidity (RH): • E.g. Sundqvist et al. MWR 1989 RH0 = critical relative humidity at which cloud assumed to form, (function of height, typical value above BL is 60%)

12. Diagnostic Relative Humidity Schemes • Since these schemes form cloud when RH<100%, they implicitly assume subgrid-scale variability for total water, qt, (and/or temperature, T) exists • However, the actual PDF (the shape) for these quantities and their variance (width) are often not known • “Given a RH of X% in nature, the mean distribution of qt is such that, on average, we expect a cloud cover of Y%”

13. Diagnostic Relative Humidity Schemes • Advantages: • Better than homogeneous assumption, since clouds can form before grids reach saturation • Disadvantages: • Cloud cover not well coupled to other processes • In reality, different cloud types with different coverage can exist with same relative humidity. This can not be represented

14. Aside:

15. Statistical Cloud SchemesLaksa PasteSquish garlic, chilli, coriander and cumin, shrimp paste, coconut crème, tumeric, lemongrass, palm sugar in food processor

16. qv ql+qi Cloudy Region G(qt) Total Water qt= vapour + liquid + ice qsat Statistical Schemes for Cloud Parameterization Horizontal GCM grid points 1. The PDF form G(qt) 2. The moments of the distribution 3. The mean saturation mixing ratio qsat Cloud cover can be derived

17. Statistical schemes • Based on Mellor JAS 77 and Sommeria and Deardorff JAS 77, and much development by Bougeault, 81, 82… • Two tasks: Specification of the: (1) PDF (2) PDF moments • PDF of what?

18. G(qt) qt C y qs x Statistical Schemes • Some schemes just take variability of total water into account, and assume temperature is homogeneous • Examples: Le Treut and Li, Clim Dyn 91, Bony and Emanuel, JAS 01, Tompkins JAS 02 PROBABLY REASONABLE ABOVE THE PBL, especially in the tropics where gravity waves remove temperature perturbations efficiently

19. LIQUID WATER TEMPERATURE conserved during changes of state Cloud mass if T variation is neglected qs S qt T Statistical Schemes • Others form variable ‘s’ that also takes temperature variability into account, which affects qs S is simply the ‘distance’ from the linearized saturation vapour pressure curve INCREASES COMPLEXITY OF IMPLEMENTATION

20. Role of temperature variability Previous slide implicitly neglected temperature variability Reasonable but not robust assumption as this ARM SGP site data shows Tompkins 2003, QJRMS

21. TASK 1: Specification of the PDF • Lack of observations to determine PDF • Aircraft data • Limited coverage • e.g. Ek and Mahrt (91), Wood and Field JAS (00), Larson et al. JAS (01) • Tethered balloon • Boundary layer • e.g. Price QJ (01) • Satellite • Difficulties resolving in vertical • e.g. Barker et al. JAS (96) • Cloud Resolving models have also been used • Realism of microphysical parameterisation? • e.g. Bougeault JAS(82), Lewellen and Yoh JAS (93), Xu and Randall JAS (96), Tompkins JAS (02)

22. CRM data qsat Tompkins JAS 02 CRM data 90x90x21km 3D domain 350m horizontal resolution Instantaneous scene Fairly Unimodal Smooth PBL negative skewness Detrainment from downdraughts Above PBL positive skewness - Detrainment from updraughts G(qt) qt

23. Balloon and Satellite PDFs Price QJRMS 2001 Example Balloon data PBL humidity Barker et al. JAS 1996 PDF of Liquid Water Path (LWP)28km horizontal resolution - Large Cloud Cover scenes: Smooth Unimodal Greater sampling possibilities

24. Aircraft Observed PDFs Hemysfield and McFarlane JAS 98 Aircraft IWC obs during CEPEX Wood and field JAS 2000 Aircraft observations low clouds < 2km Height G(qt) qt qt ql T

25. One of the most comprehensive investigations of suitable PDFsJAS 2001 They used aircraft data to test many different PDF forms:

26. q-sat Cloudcover Problems with using assumed PDF Take 3 pieces of information 1. Cloud water 2. humidity 3. total water variance Throw one away 1. total water mean 2. total water variance Use this info to fit assumed PDF Extract humidity and cloud water Different Values!!! 1. Cloud water 2. humidity

27. Found multi parameter multimodal PDF gave smallest errorThe (many) defining parameters have to be specified in a scheme

28. They went further to examine joint PDFs: J. Atmos. Sci. 2002 They used aircraft and LES data to test 5 different PDF forms:

29. Examined examples of joint PDFs

30. Tested errors for each form Lewellen and Yoh best but requires iteration Note: these figures are not absolute but depend on closure assumptions

31. G(s or qt) s or qt s or qt Uniform: Letreut and Li (91) Triangular: Smith QJRMS (90) s or qt s or qt Gaussian: Mellor JAS (77) s4 polynomial: Lohmann et al. J. Clim (90) TASK 1: Specification of PDF Many function forms have been used Symmetrical distributions:

32. G(s or qt) s or qt s or qt s or qt Lognormal: Bony & Emanuel JAS (01) Gamma: Barker et al. JAS (96) Exponential: Sommeria and Deardorff JAS (77) s or qt s or qt Beta: Tompkins JAS (02) Double Normal: Lewellen and Yoh JAS (93) Golaz bimodal Gaussian TASK 1: Specification of PDF skewed distributions:

33. saturation G(s or qt) cloud forms? s or qt e.g. HOW WIDE? TASK 2: Specification of PDF moments • Need also to determine the moments of the distribution: • Variance (Symmetrical PDFs) • Skewness (Higher order PDFs)

34. (1-RH0)qs C G(qt) 1-C qt TASK 2: Specification of PDF moments • Some schemes fix the moments (e.g. Smith 1990) based on critical RH at which clouds assumed to form • If moments (variance, skewness) are fixed, then statistical schemes can be reduced to a RH formulation where Sundqvist formulation!!! Solution for Smith scheme more complex

35. TASK 2: Specification of PDF moments • Some schemes use the turbulence parameterisation to determine T’L and q’t , their correlation and thus also s’. Example: Ricard and Royer, Ann Geophy, (93), Lohmann et al. J. Clim (99) T’L , q’t • Disadvantage: • Can give good estimate in boundary layer, but above, other processes will determine variability Lohmann et al. (who used this turbulence approach to set the PDF variance) found it necessary to impose a minimum variance above the planetary boundary layer

36. Golaz scheme chooses double Gaussian 1 • 15 defining parameters, reduced to 10 through closures • E.g. skewness of T is zero, skewness of qtot fixed ratio of w • 10 prognostic equations for moments • E.g. w,q,T, q’T’, T’T’ etc… Creation of variance by subgrid motions Advection by large-scale flow Subgrid turbulence Dissipation

37. convection turbulence microphysics dynamics Clouds in GCMs - Processes that can affect distribution moments

38. Production of variance by unresolved motions in the presence of a vertical gradient This includes both convective and turbulent motions Change due to microphysical processes Includes advection by the resolved flow Transport of variance by unresolved motions Dissipation term Golaz et al. were able to neglect some terms

39. For convective motions? Production of variance Depends on definition of variance, above relevant if variance of whole grid box is considered If it is the environmental variance then only affected by detrainment al la Tiedtke. Steve Klein outlines how to tackle this with a mass flux convection scheme in a GCSS preprint from 2003

40. Production of variance Courtesy of Steve Klein: Change due to difference in means Transport Change due to difference in variance Also equivalent terms due to entrainment

41. Variance of updraughts Requires closure of variance of updraft air (implicitly zero in the Tiedtke approach) Simply carry variance as a passive tracer in mass flux scheme

42. Steve Klein diagnosed these terms from a 2D CRM model Equal? Skewness terms can be derived in similar way

43. Microphysics • Change in variance • However, the tractability depends on the PDF form for the subgrid fluctuations of q, given by G: Where P is the precipitation generation rate, e.g: Can get pretty nasty!!! Depending on form for P and G

44. Autoconversion loss from one level is straightforward, provided G(qt) and P are simple Explicit integration of microphysics Vs integration of simple variance term

45. But quickly can get untractable • E.g: Semi-Langrangian ice sedimention • Source of variance in bottom layer is tricky • In reality of course wish also to retain the sub-flux variability too

46. My SchemeThe soup Fry laksa paste for 3 minutes, then add coconut milk and stock, simmer for 15 minutes

47. Prognostic Statistical Scheme • Tompkins JAS (02) introduced a scheme for GCM (ECHAM5) • Prognostic equations are introduced for the variance and skewness of the total water PDF • These, specify both the shape and width of the PDF, allowing the cloud cover to be diagnosed. • A full treatment of the variance equation is included, with the third order skewness treated in an simplified way • AND IF YOU BELIEVE THAT, YOU WILL BELIEVE ANYTHING!!! • In fact, a short cut was taken to circumnavigate the microphysics and convective detrainment terms • (And ok, you got me, I admit it, the skewness budget was ‘botched’)

48. G(qt) qt0 qt qt1 Prognostic Statistical Scheme • Scheme uses the Beta distribution for qtvariability • Justified from use of CRM data (observations agree-ish) • Temperature variations are ignored. • POSITIVE / NEGATIVE SKEWNESS POSSIBLE, also SYMMETRIC • LIMITED FUNCTION • UNIMODAL Stupid assumption No1 – Fix p=constant

49. Which prognostic equations? Take a 2 parameter distribution & Partially cloudy conditions qsat Cloudcover • Can specify distribution with • Mean • Variance • of total water qsat Cloudcover • Can specify distribution with • Water vapour • Cloud water • mass mixing ratio qv ql+i

50. qv qv+ql+i Which prognostic equations? qsat • Water vapour • Cloud water • mass mixing ratio • Cloud water budget conserved • Avoids Detrainment term • Avoids Microphysics terms (almost) qv ql+i But problems arise in qsat Overcast conditions (…convection + microphysics) (al la Tiedtke) Clear sky conditions (turbulence)