1. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 1 Some thoughts on statistical cloud schemes 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… ADRIAN a recipe for seafood laksa soup

2. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 4 ~500m ~100km Issues of Parameterization HORIZONTAL COVERAGE, C Most of this talk will concentrate on this issue

5. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 6 ~500m ~100km Issues of Parameterization Vertical overlap of cloud Important for Radiation and Microphysics Interaction

7. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 7 Issues of Parameterization ~500m ~100km cloud inhomogeneity

8. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 8 ~500m ~100km Issues of Parameterization Just these issues can become very complex!!!

9. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 9 Horizontal Cloud Cover: The Problem Local criterion for existence of cloud: q total > q sat (T,p) This assumes that no supersaturation can exist (poor assumption for ice clouds) 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 q x One Grid-cell

10. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 10 Horizontal Cloud Cover: The Problem Heterogeneous distribution of T and q… q x …clouds exist before the grid-mean relative humidity reaches 100%

11. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 11 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 RH 0 = critical relative humidity at which cloud assumed to form, (function of height, typical value above BL is 60%) CC (cloud fraction) 1 Relative humidity RH 0 1 0 0

12. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 12 Diagnostic Relative Humidity Schemes  Since these schemes form cloud when RH<100%, they implicitly assume subgrid-scale variability for total water, q t, (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 q t is such that, on average, we expect a cloud cover of Y%”

13. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 14 Aside:

15. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 15 Statistical Cloud Schemes Laksa Paste Squish garlic, chilli, coriander and cumin, shrimp paste, coconut crème, tumeric, lemongrass, palm sugar in food processor

16. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 16 Total Water q t = vapour + liquid + ice q sat G(q t ) Cloudy Region qvqv q l +q i 1. The PDF form G(q t ) 2. The moments of the distribution 3. The mean saturation mixing ratio q sat Cloud cover can be derived Horizontal GCM grid points Statistical Schemes for Cloud Parameterization

17. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 18 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 qtqt G(q t ) qsqs x y C PROBABLY REASONABLE ABOVE THE PBL, especially in the tropics where gravity waves remove temperature perturbations efficiently

19. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 19 Statistical Schemes  Others form variable ‘s’ that also takes temperature variability into account, which affects q s LIQUID WATER TEMPERATURE conserved during changes of state S is simply the ‘distance’ from the linearized saturation vapour pressure curve qsqs T qtqt S INCREASES COMPLEXITY OF IMPLEMENTATION Cloud mass if T variation is neglected

20. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 22 CRM data qtqt G(q t ) 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 q sat

23. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 24 Aircraft Observed PDFs Wood and field JAS 2000 Aircraft observations low clouds < 2km qtqt G(q t ) Height qtqt qlql T Hemysfield and McFarlane JAS 98 Aircraft IWC obs during CEPEX

25. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 25 They used aircraft data to test many different PDF forms: One of the most comprehensive investigations of suitable PDFs JAS 2001

26. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 26 Problems with using assumed PDF 1. Cloud water 2. humidity 3. total water variance Take 3 pieces of information Throw one away 1. total water mean 2. total water variance Use this info to fit assumed PDF Extract humidity and cloud water q-sat Cloud cover 1. Cloud water 2. humidity Different Values!!!

27. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 27 Found multi parameter multimodal PDF gave smallest error The (many) defining parameters have to be specified in a scheme

28. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 28 They used aircraft and LES data to test 5 different PDF forms: They went further to examine joint PDFs: J. Atmos. Sci. 2002

29. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 29 Examined examples of joint PDFs

30. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 31 TASK 1: Specification of PDF Many function forms have been used Symmetrical distributions: Triangular: Smith QJRMS (90) s or q t Gaussian: Mellor JAS (77) s or q t G(s or q t ) Uniform: Letreut and Li (91) s or q t s 4 polynomial: Lohmann et al. J. Clim (90)

32. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 32 TASK 1: Specification of PDF skewed distributions: s or q t G(s or q t ) Exponential: Sommeria and Deardorff JAS (77) Lognormal : Bony & Emanuel JAS (01) s or q t Gamma: Barker et al. JAS (96) s or q t Beta: Tompkins JAS (02) s or q t Double Normal: Lewellen and Yoh JAS (93) Golaz bimodal Gaussian

33. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 33 TASK 2: Specification of PDF moments  Need also to determine the moments of the distribution:  Variance (Symmetrical PDFs)  Skewness (Higher order PDFs) s or q t G(s or q t ) e.g. HOW WIDE? saturation cloud forms?

34. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 34 C 1-C 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 (1-RH 0 )q s qtqt G(q t ) Sundqvist formulation!!! where Solution for Smith scheme more complex

35. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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’. T’ L, q’ t Example: Ricard and Royer, Ann Geophy, (93), Lohmann et al. J. Clim (99)  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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 36  Golaz scheme chooses double Gaussian 1  15 defining parameters, reduced to 10 through closures  E.g. skewness of T is zero, skewness of q tot fixed ratio of w  10 prognostic equations for moments  E.g. w,q,T, q’T’, T’T’ etc… Advection by large-scale flow Subgrid turbulence Creation of variance by subgrid motions Dissipation

37. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 37 Clouds in GCMs - Processes that can affect distribution moments convection turbulence dynamics microphysics

38. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 38 Golaz et al. were able to neglect some terms Production of variance by unresolved motions in the presence of a vertical gradient This includes both convective and turbulent motions Dissipation term Includes advection by the resolved flow Transport of variance by unresolved motions Change due to microphysical processes

39. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 39 Production of variance For convective motions? 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 40 Production of variance Courtesy of Steve Klein: Change due to difference in variance Change due to difference in means Transport Also equivalent terms due to entrainment

41. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 42 Steve Klein diagnosed these terms from a 2D CRM model Equal? Skewness terms can be derived in similar way

43. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 46 My Scheme The soup Fry laksa paste for 3 minutes, then add coconut milk and stock, simmer for 15 minutes

47. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 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. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 48 POSITIVE / NEGATIVE SKEWNESS POSSIBLE, also SYMMETRIC LIMITED FUNCTION UNIMODAL Prognostic Statistical Scheme Scheme uses the Beta distribution for q t variability Justified from use of CRM data (observations agree-ish) Temperature variations are ignored. qtqt q t0 q t1 G(q t ) Stupid assumption No1 – Fix p=constant

49. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 49 Which prognostic equations? Take a 2 parameter distribution & Partially cloudy conditions q sat Cloud cover q sat Cloud cover Can specify distribution with (a)Mean (b)Variance of total water Can specify distribution with (a)Water vapour (b)Cloud water mass mixing ratio qvqv q l+i

50. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 50 Which prognostic equations? q sat (a)Water vapour (b)Cloud water mass mixing ratio qvqv q l+i q sat qvqv q v +q l+i Cloud water budget conserved Avoids Detrainment term Avoids Microphysics terms (almost) But problems arise in Clear sky conditions (turbulence) Overcast conditions (…convection + microphysics) (al la Tiedtke)

51. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 51 Which prognostic equations? Take a 2 parameter distribution & Partially cloudy conditions q sat (a)Mean (b)Variance of total water “Cleaner solution” But conservation of liquid water compromised due to PDF Need to parametrize those tricky microphysics terms!

52. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 52 Governing equations must be chosen with care…  Combine the advantages of the above two by using a flexible PDF that involves one extra degree of freedom in partially cloudy conditions. :o) Cloud budget ensured :o) Consistent treatment of Variance Equation :o) Allows BIMODALITY for example in partially cloudy conditions :o( PDF more complex :o( Requires parameterization of those tricky variance terms  PDF form devised (double Beta distribution) on beach in Gianova Lido, near Pescara, in summer 2002 under a yellow and red striped umbrelloni But…  Tested with ARM aircraft data  Failure… since cloud/vapour is a variance type – PDF was not flexible enough to find a solution. (failure rate>90%)

53. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 53 The scheme has five governing prognostic equations  Prognostic equations for:  * Cloud liquid water  * Cloud ice  * Water vapour  Variance of q tot  * “Skewness” of qtot  * = truly prognostic and used to fit PDF in partially cloudy conditions, otherwise variance is used  Processes represented:  Deep convective detrainment  Microphysics  Turbulence  Advection  Dissipation

54. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 54 Partially cloudy * q l * q v * q’ t 3 q’ t 2 * q l * q v * q’ t 3 q’ t 2 Time=iTime=i+1 Slaving of variance Subset process (turbulence) * = used to define PDF giving cloud cover

55. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 55 Clear sky or Overcast qlql qvqv * q’ t 3 * q’ t 2 qlql qvqv * q’ t 3 * q’ t 2 Time=iTime=i+1 Subset process (turbulence) * = used to define PDF giving cloud cover * *

56. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 56 Variance equation of Tompkins(2002) Production of variance by unresolved motions in the presence of a vertical gradient This includes turbulent motions only Dissipation term Includes advection by the resolved flow Transport of variance by unresolved motions Change due to microphysical processes EFFECT OF VARIANCE CHANGES ADDED TO CLOUD AND VAPOUR BUDGETS

57. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 57 Production of variance Vertical gradient of q t is known For turbulent motions the flux term is already calculated in the turbulence scheme

58. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 58 Transport of Variance For convective motions, simple advective term: For turbulent motions we parameterize the transport in terms of e (TKE) as in a typical 1.5 order closure Eddy length scale

59. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 59 Dissipation of Variance Dissipation defined as Newtonian relaxation, with the timescale resulting from two processes: Process 1: 3D Turbulence After Garrat, Atmospheric Boundary Layer, 1992, who suggests  =7.44 Implicit in this is the assumption that the eddy length scale  also describes the length-scale of the total water perturbations. This is not a good assumption above the PBL, where q t perturbations will occur on much larger scales MESOSCALE FLUCTUATIONS Thus, only applied in PBL - Results in a short time-scale (~10 minutes), and this is solved implicitly to avoid numerical instabilities

60. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 60 Aside: Mesoscale fluctuations  Fluctuations occur on length scales comparible to grid length.  Upscale cascade from turbulence  Gravity waves  Subgrid instabilities (cloud top instabilities)……  It is clear that modals are deficient in representing these  E.g: Lohmann shows inadequacy of ECMWF model, and how enhancement of turbulence activity can produce improved spectra From haag and Kaerchner ECMWF resolved motions only aircraft

61. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 61 Dissipation of Variance Process 2: Horizontal Mixing GCM grid cell  x Large-scale wind shear Horizontal wind shear will cause mixing (destruction of q t variance) even if stable temperature stratification suppresses vertical turbulence Use simple order 1 closure Assume the scale of the moisture variability occurs on gridscale - The internal horizontal moisture flux which destroys variance is proportion to the moisture gradient which is approximated by, where  x is the grid length. Subgrid-scale q t flux

62. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 62 But this variance change then needs to be translated into equivalent changes in cloud and vapour variables, (also discussed by Larson 2004) Implied Source/Sink of ice/liquid cloud water taken into account From Larson

63. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 63 The generation of cloud by cooling or warming also has to be consistent with the PDF qvqv q l+i q sat qvqv q l+i q sat

64. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 64 Skewness Budget: Convection convection Convective updraught detrainment increases the skewness of total water by introducing air with high water contents Change of skewness linearly related to detrained cloud mass flux at the moment 3D CRM Results Long Distribution Tail from convective detrainment

65. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 65 It gets worse: Skewness budget  Skewness less important than variance since it is a higher order moment (less effect on cloud cover). Microphysics term treated simply in both variance and skewness equations Dissipation time-scales identical to variance equation Tunable constant Detrainment term from cumulus parameterization scheme Embarrassing!!! I should have read some of David Randall’s work AND, instead of skewness I simply used the shape parameters

66. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 66 Actually do you know what the most embarrassing thing in that paper was? THIS!!!

67. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 67 Schematic: How the scheme works TIME Convection detrains cloud condensate at a certain level, BUT, also increases the distribution skewness and width Turbulent processes reduce width, as do microphysical processes Return to clear sky conditions Start with clear sky From the distribution the cloud fraction can be diagnosed

68. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 68 Scheme in action… Deep Convective Regime Deep convection: causes distribution to widen considerably and skewness increases (skewness also advected from neighbouring regions) min max q sat q t

69. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 69 Scheme in action Minimum Maximum q sat Turbulence breaks up cloud Turbulence creates cloud

70. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 70 Column model results for deep convective case

71. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 71 Shoddy things about my scheme  Skewness budget  Variance budget “implicit”  Microphysics still needed to be tackled for overcast conditions  Convection detrainment implicit  Working on ECMWF scheme to tackle these (the other nightmare)

72. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 72 Complexity required? Boil noodles

73. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 73 How much complexity? To start in ECMWF model have built the simplest statistical scheme possible qtqt G(q t ) qsqs Cloud water Vapour Simply use mean vapour and cloud water to fit a symmetrical PDF (Beta distribution) and then diagnose cloud cover

74. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 74 How much complexity? Results of a 4 month total cloud cover climatology at T95 resolution …simplicity aids data assimilation dilemma…

75. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 75 Links with other model components add fish to soup, simmer 5 minutes, add noodles

76. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 76 You might be tempted to say… Hurrah! We now have cloud variability in our models, where before there was none! WRONG! Nearly all components of GCMs contain implicit/explicit assumptions concerning subgrid fluctuations e.g: RH-based cloud cover, thresholds for precipitation evaporation, convective triggering, plane parallel bias corrections for radiative transfer… etc.

77. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 77 Variability in Clouds Is this desirable to have so many independent tunable parameters? ‘M’ tuning parameters ‘N’ Metrics e.g. Z500 scores, TOA radiation fluxes With large M, task of reducing error in N metrics Becomes easier, but not necessarily for the right reasons Solution: Increase N, or reduce M

78. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 78 Advantage of Statistical Scheme Statistical Cloud Scheme Radiation MicrophysicsConvection Scheme Thus ‘complexity’ is not synonymous with ‘M’: the tunable parameter space Can use information in Other schemes

79. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 79 Use in other Schemes (II) IPA Biases Knowing the PDF allows the IPA bias to be tacked Split PDF and perform N radiation calculations Total Water q sat G(q t ) Thick Cloud etc. Use Moments to find best-fit Gamma Distribution PDF and apply Gamma weighted TSA Barker (1996) Total Water q sat G(q t ) Gamma Function Total Water q sat G(q t ) Use variance to calculate Effective thickness Approximation Effective Liquid = K x Liquid Cahalan et al 1994 ETA adjustment

80. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 80 Variability in other Schemes: Radiation Beta Weighted Two Stream Approximation Effective Thickness Approximation = 0.7 (Cahalan et al. 1994) Seasonal Average Albedo differences Georg Bauml 2003, PhD Thesis

81. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 81 Rob Pincus McICA

82. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 82

83. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 83 McICA in the ECMWF model

84. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 84 Result is not equal in the two cases since microphysical processes are non-linear Example on right: Autoconversion nonlinear function. Grid mean cloud less than threshold and gives zero precipitation formation. Threshold often “tuned” to counter the neglect of variability. Cloud inhomogeneity and microphysics biases qLqL q L0 cloud range mean precip generation Most current microphysical schemes use the grid-mean cloud mass (i.e: neglect variability) cloud RealityModel

85. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 85 Cloud Variability and Microphysics Work with Rob Pincus  Used the ECWMF model modified so cloud cover is parameterized using the simple statistical scheme  With the column model, conducted two experiments  Exp1: Default microphysics  Exp 2: Divide cloud into 100 sub- columns and calculate microphysics taking subgrid fluctuations into account (using Rob’s cloud generator) 12435…

86. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 86 Liquid Ice Default 100 columns 0.05 0.4 0.1 Time Deep convection case

87. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 87 Clouds also become longer lived Default 100 columns

88. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 88 What is going on? Sundqvist Scheme is not strongly nonlinear when above critical threshold Creation of snow ‘side-effect’ of ice sedimentation. (Ice falling into clear sky is treated as snow) A~q 0.16 Non-diagonal Jacobians - Ice processes dominate in deep convective situations through accretion processes REALISM?

89. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 89 Conclusions and future directions? Garnish with fresh coriander leaves and serve

90. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 90 Conclusions  Move towards statistical schemes is a move towards greater coherency and self- consistency in models.  Information concerning microphysics, especially in ice phase crucial for further improvement, both in terms of means and spatial and temporal fluctuations.

91. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 91 Future  Intention at ECMWF is to move towards a statistical scheme framework to replace the existing prognostic cloud fraction  This will be accomplished in stages and will be combined with a simple treatment of the cloud ice variable that will allow super saturation with respect to ice  Statistical information to provide greater link to convection (sources!), boundary scheme, radiation  As always there is the issue of data assimilation

92. Thoughts on statistical cloud schemes Denver 2004 A.Tompkins 92 And finally… In terms of clouds… We have come a long way… But we can still improve