1. AN ECMWF LECTURER Numerical Weather Prediction Parametrization of diabatic processes The ECMWF cloud scheme tompkins@ecmwf.int

2. The ECMWF Scheme Approach 2 Additional prognostic variables: –Cloud mass (liquid + ice) –Cloud fraction Sources/Sinks from physical processes: –Deep and shallow convection –Turbulent mixing Cloud top turbulence, Horizontal Eddies –Diabatic or Adiabatic Cooling or Warming Radiation, dynamics... –Precipitation processes –Advection qtqt G(q t ) qsqs x y C Some (not all) of these are derived from a distribution approach

3. Basic assumptions Clouds fill the whole model layer in the vertical (fraction=cover) Clouds have the same thermal state as the environmental air rain water/snow falls out instantly but is subject to evaporation/sublimation and melting in lower levels cloud ice and water are distinguished only as a function of temperature - only one equation for condensate is necessary

4. Schematic of Source/Sink terms cloud 1. Convective Detrainment 1 2 2. (A)diabatic warming/cooling 3 3. Subgrid turbulent mixing 4 4. Precipitation generation 5 5. Precipitation evaporation/melting

5. ECMWF prognostic scheme - Equations Cloud liquid water/ice q l Cloud fraction C A: Large Scale Advection S: Source/Sink due to: S CV : Source/Sink due due to shallow convection (not post-29r1) S BL : Source/Sink Boundary Layer Processes (not post-29r1) c: Source due to Condensation e: Sink due to Evaporation G p : Precipitation sink D:Detrainment from deep convection Last term in liquid/ice : Cloud top entrainment (not post-29r1)

6. Linking clouds and convection Basic idea use detrained condensate as a source for cloud water/ice Examples Ose, 1993, Tiedtke 1993, DelGenio et al. 1996, Fowler et al. 1996 Source terms for cloud condensate and fraction can be derived using the mass-flux approach to convection parametrization.

7. Convective source terms - Water/Ice k k+1/2 k-1/2 (M u ql u ) k+1/2 (M u ql u ) k-1/2 (-M u ql) k-1/2 (-M u ql) k+1/2 D u ql u Standard equation for mass flux convection scheme ECHAM, ECMWF and many others... write in upstream mass-flux form

8. Convective source terms - Cloud Fraction k k+1/2 k-1/2 (M u C) k-1/2 (M u C) k+1/2 DuDu Can write similar equation for the cloud fraction Factor (1-C) assumes that the pre-existing cloud is randomly distributed and thus the probability of the convective event occurring in cloudy regions is equal to C

9. Stratiform cloud formation Local criterion for cloud formation: q > q s (T,p) Two ways to achieve this in an unsaturated parcel: increase q or decrease q s Processes that can increase q in a gridbox ConvectionCloud formation dealt with separately Turbulent MixingCloud formation dealt with separately Advection

10. Advection and cloud formation Advection does not mix air !!! It merely moves it around conserving its properties, including clouds. butThere are numerical problems in models u t t+  t Because of the non-linearity of q s (T), q 2 > q s (T 2 ) This is a numerical problem and should not be used as cloud producing process! Would also be preferable to advect moist conserved quantities instead of T and q

11. Stratiform cloud formation Postulate: The main (but not only) cloud production mechanisms for stratiform clouds are due to changes in q s. Hence we will link stratiform cloud formation to dq s /dt. = 

12. Stratiform cloud formation Existing clouds“New” clouds The cloud generation term is split into two components: And assumes a mixed ‘uniform-delta’ total water distribution 1-C qtqt G(q t ) C

13. Stratiform cloud formation: Existing Clouds Already existing clouds are assumed to be at saturation at the grid-mean temperature. Any change in q s will directly lead to condensation. qtqt G(q t ) C Note that this term would apply to a variety of PDFs for the cloudy air (e.g. uniform distribution)

14. Stratiform cloud formation Formation of new clouds RH crit = 80 % is used throughout most of the troposphere Due to lack of knowledge concerning the variance of water vapour in the clear sky regions we have to resort to the use of a critical relative humidity RH>RH crit 1-C qtqt G(q t ) C We know q e from similarly

15. 1-C qtqt G(q t ) C RH<RH crit Term inactive if RH<RH crit Perhaps for large cooling this is accurate? As we stated in the statistical scheme lecture: 1.With prognostic cloud water and here cover we can write source and sinks consistently with an underlying distribution function 2.But in overcast or clear sky conditions we have a loss of information. Hence the use of Rh crit in clear sky conditions for cloud formation

16. Evaporation of clouds Processes: e=e 1 +e 2 Large-scale descent and cumulus- induced subsidence Diabatic heating Turbulent mixing (E2) qtqt G(q t ) C GCM grid cell  x Note: Incorrectly assumes mixing only reduces cloud cover and liquid! no effect on cloud cover

17. Problem: Reversible Scheme? 1-C qtqt G(q t ) C Cooling: Increases cloud cover 1-C qtqt G(q t ) C Subsequent Warming of same magnitude: No effect on cloud cover Process not reversible

18. Precipitation generation Sundqvist, 1978 Collection Bergeron Process Mixed Phase and water clouds C 0 =10 -4 s -1 C 1 =100 C 2 =0.5 ql crit =0.3 g kg -1

19. Precipitation generation – Pre 25r3 Pure ice clouds (T<250 K) For IWC <100 - explicitly calculate ice settling flux Heymsfield and Donner (1990) Ice falling in cloud below is source for ice in that layer, ice falling into clear sky is converted into snow Two size classes of ice, boundary at 100  m McFarquhar and Heymsfield (1997) IWC >100 falls out as snow within one timestep

20. Precipitation generation – Post 25r3 Pure ice clouds (T<250 K) For IWC <100 - explicitly calculate ice settling flux Heymsfield and Donner (1990) Ice falling in cloud below is source for ice in that layer, ice falling into clear sky is converted into snow Two size classes of ice, boundary at 100  m McFarquhar and Heymsfield (1997) IWC >100 falls according to Heymsfield and Donner (1990) Small Ice has fixed fall speed of 0.15 m/s

21. Numerical Integration I n both prognostic equations for cloud cover and cloud water all source and sink terms are treated either explicitly or Implicitly. E.g: Solve analytically and exactly as ordinary differential equations (provided K j >0)

22. Numerical Integration How is this accomplished? If a term contains a n th power for example: The first order approximation is used: Beginning of timestep value

23. Numerical Aspects – Ice fall speeds Remember the original ice assumptions? “IWC >100 falls out as snow within one timestep” – V = CFL limit Solved implicitly to prevent noise Can calculate fall speed implied by CFL limit

24. Resulting Ice Fall Speed T511: pre-25r3 SMALL ICE LARGE ICE IN CLOUD ICE MIXING RATIO (KG/KG)

25. Resulting Ice Fall Speed T95: pre-25r3 (corrected at 25r3) SMALL ICE LARGE ICE IN CLOUD ICE MIXING RATIO (KG/KG)

26. Real advection Perfect advection Numerical Aspects - Vertical Advection Problem: we neglect vertical subgrid-scales w

27. Ice sedimentation: Further Numerical Problems Since we only allow ice to fall one layer in a timestep, and since the cloud fills the layer in the vertical, the cloud lower boundary is advected at the CFL rate of  z /  t Solution (not very satisfactory): Only allow sedimentation of cloud mass into existing cloudy regions, otherwise convert to snow – Results in higher effective sedimentation rates AND vertical resolution sensitivity Ice flux Snow flux

28. Numerical Aspects - Ice settling Problem : A “bad” assumption and numerics (always imperfect) lead to a realistic looking result. 1. We implicitly assume that all particles fall with the same velocity - wrong Start with single layer ice cloud in level k - expect that settling moves some of the ice into the next layer and that some stays in this layer due to size distribution. Assume  t =  z / v ice and ignore density Perfect numerics in this case - explicit scheme: 2. Use analytical integration Correct but not as expected as expected but incorrect

29. 100 vs 50 layer resolution Ice flux Snow flux Sink at level k depends on dz k k+1 This is correct, but if then converted to snow and “lost” will introduce a resolution sensivity which is which?

30. Planned Revision for 29r3 (sept 2005) To tackle ice settling deficiency Options: (i) Implicit numerics (ii) Time splitting (iii) Semi-Lagrangian fall speed Constant Explicit Source/Sink Advected Quantity (e.g. ice) Implicit Source/Sink (not required for short timesteps) Implicit: Upstream forward in time, j=vertical level n=time level Solution Note: For multiple X-bin bulk scheme (X=ice, graupel, snow…) with implicit (“D” above) cross-terms leads to set of X-linear algebraic equations what is short?

31. Improved Numerics in SCM Cirrus Case 29r1 Scheme 29r3 Scheme 100 vs 50 layer resolution time evolution of cloud cover and ice which is which? time cloud fraction cloud ice

32. Precipitation evaporation and melting Melting Occurs whenever T > 0 C Is limited such that cooling does not lead to T < 0 C Evaporation (Kessler 1969, Monogram) Newtonian Relaxation to saturated conditions

33. Precipitation Evaporation Jakob and Klein (QJRMS, 2000) Mimics radiation schemes by using a ‘2-column’ approach of recording both cloud and clear precipitation fluxes This allows a more accurate assessment of precipitation evaporation NUMERICS: Solved Implicitly to avoid numerical problems

34. Precipitation Evaporation Numerical “Limiters” have to be applied to prevent grid scale saturation Clear sky region Grid can not saturate Clear sky region Grid slowly saturates

35. Precipitation Evaporation Threshold for rainfall evaporation based on precipitation cover RH crit Clear sky Precipitation Fraction 70% 0 100% Critical relative humidity for precipitation evaporation Instead can use overlap with clear sky humidity distribution Humidity in most moist fraction used instead of clear sky mean Clear sky 1

36. Cirrus Clouds, new homogeneous nucleation form for 29r3 Want to represent super-saturation and homogeneous nucleation Include simple diagnostic parameterization in existing ECMWF cloud scheme Desires: –Supersaturated clear-sky states with respect to ice –Existence of ice crystals in locally subsaturated state Only possible with extra prognostic equation… GCM gridbox C Clear skyCloudy region

37. q v cld =?q v env =? cloudclear GCM gridbox Three items of information: q v, q i, C (vapour, cloud ice and cover) We know: q c occurs in the cloudy part of the gridbox We know: The mean in-cloud cloud ice What about the water vapour? In the bad-old-days was easy: Clouds: q v cld =q s Clear sky: q v env =(q v -Cq s )/(1-C) (rain evap and new cloud formation) C Unlike “parcel” models, or high resolution LES models, we have to deal with subgrid variability I F S

38. GCM gridbox The good-new-days: assume no supersaturation can exist C q v cld q v env (q v -Cq s )/(1-C)qsqs 1. q v +q i q v -q i (1-C)/C3. Klaus Gierens:Humidity in clear sky part equal to the mean total water 4.(q v -Cq s )/(1-C)qsqs My assumption: Hang on… Looks familiar??? qvqv qvqv 2. Lohmann and Karcher: Humidity uniform across gridcell The bad old days No supersaturation

39. GCM gridbox qvqv qvqv 2. Lohmann and Karcher: Humidity uniform across gridcell q v cld q v env q v cld : reduces as ice is formed Artificial flux of vapour from clear sky to cloudy regions!!! Critical S crit reached Assumption ignores fact that difference processes are occurring on the subgrid-scale q v env =q v cld uplifted box From Mesoscale Model 1 timestep q i : increases

40. GCM gridbox q v cld q v env q v cld : reduces to q s Critical S crit reached q i : increases q v env unchanged No artificial flux of vapour from clear sky from/to cloudy regions Assumption seems reasonable: BUT! Does not allow nucleation or sublimation timescales to be represented, due to hard adjustment q v env =unchanged q i : decreases uplifted box 4.(q v -Cq s )/(1-C)qsqs My assumption: Hang on… Looks familiar??? microphysics Difference to standard scheme is that environmental humidity must exceed S crit to form new cloud From GCM perspective

41. A C B RH PDF at 250hPa one month average Aircraft observations A: Numerics and interpolation for default model B: The RH=1 microphysics mode C: Drop due to GCM assumption of subgrid fluctuations in total water

42. Summary of Tiedtke Scheme Scheme introduces two prognostic equations for cloud water and cloud mass Sources and sinks for each physical process - Many derived using assumptions concerning subgrid-scale PDF for vapour and clouds J More simple to implement than a prognostic statistical scheme since “short cuts” are possible for some terms. Also nicer for assimilation since prognostic quantities are directly observable L Loss of information (no memory) in clear sky (a=0) or overcast conditions (a=1) (critical relative humidities necessary etc). Difficult to make reversible processes.  Nothing to stop solution diverging for cloud cover and cloud water. (e.g: liquid>0 while a=0). More unphysical “safety switches” necessary. Last Lecture: Validation!

43. Example Problem 1.Convection detrains into a clear-sky grid-box at a normalized rate of 1/3600 s -1. This is the highest level of convection so there is no subsidence from the layer above (see picture). How long does it take for the cloud cover to reach 50%? State the assumptions made. 2.The detrained air has a cloud liquid water mass mixing ratio of 3 g kg -1. If the conversion to precipitation is treated as a simple Newtonian relaxation (i.e. [dq l / dt] precip = q l / tau) with a timescale of 2 hours, what is the equilibrium cloud mass mixing ratio? GridboxM=1/3600 s -1 M=0 s -1 Answers: (1) 42 minutes (2) 2 g kg -1.