Theories of Mixing in Cumulus Convection

1. Theories of Mixing in Cumulus Convection A. Pier Siebesma Royal Netherlands Meteorological Institute (KNMI) De Bilt The Netherlands 1. Motivation 2. Essential Thermodynamics 3. Phenomenology and Observations 4. Cloud Mixing Models 5. Parameterizations 6. Remaining Problems. Many thanks to: Roel Neggers (KNMI) Harm Jonker, Stephaan Rodts (TU Delft)

2. References • A.P. Siebesma and J.W.M. Cuijpers, Evaluation of parametric assumptions for shallow cumulus convection, J. Atmos. Sci., 52, 650-666, 1995 • A.P. Siebesma and A.A.M. Holtslag, Model impacts of entrainment and detrainment rates in shallow cumulus convection, J. Atmos. Sci., 53, 2354-2364, 1996 • A.P. Siebesma , ‘’Shallow Cumulus Convection” published in: Buoyant Convection in Geophysical Flows, p441-486. Edited by: E.J. Plate and E.E. Fedorovich and X.V Viegas and J.C. Wyngaard. Kluwer Academic Publishers. • R.A.J. Neggers,A.P. Siebesma and H.J.J. Jonker. A multiparcel method for shallow cumulus convection. Accepted for J. of Atm Sci. 2002. • R.A.J. Neggers,A.P. Siebesma and H.J.J. Jonker. Size statistics of cumulus cloud populations in large-eddy simulations. Submitted to J. of Atm Sci. 2002. See also: http://www.knmi.nl/~siebesma theories of cloud mixing

3. Motivation and Objectives theories of cloud mixing

4. Tropopause 10km Subsidence ~0.5 cm/s inversion 10 m/s Cloud base ~500m Cartoon of Hadley Circulation Shallow Convective Clouds No precipitation Vertical turbulent transport No net latent heat production Fuel Supply Hadley Circulation Stratocumulus Interaction with radiation Deep Convective Clouds Precipitation Vertical turbulent transport Net latent heat production Engine Hadley Circulation theories of cloud mixing

5. 50 km Shallow cumulus not resolved by “state of the art” global atmospheric models. theories of cloud mixing

6. Large scale advection Large scale subsidence Vertical turbulent transport Net Condensation Rate Grid Averaged Budget Equations theories of cloud mixing

7. Schematically: Objectives • Understand Cumulus Convection…. • Design Models….. • But ultimately design parameterizations of: theories of cloud mixing

8. 2. Thermodynamics theories of cloud mixing

9. 2.1 Moisture Variables qv :Specific Humidity ql :Liquid Water qt = qv + ql :Total water specific humidity (Conserved for phase changes theories of cloud mixing

10. qs(p,T) qt ql qv T Remark 1: In thermodynamic equilibrium: qt = qv if qv < qsat: undersaturation qt = qsat + ql if qv > qsat: oversaturation Remark 2: qsat = qsat (p,T) is a state function (Clausius-Clayperon) theories of cloud mixing

11. 2.2 Used Temperature Variables Potential Temperature Conserved for dry adiabatic changes Liquid Water Potential Temperature Conserved for moist adiabatic changes Virtual Potential Temperature Directly proportional to the density Measure for buoyancy theories of cloud mixing

12. Grid averaged equations for conserved variables: Parameterization issue reduced to a turbulent mixing problem! theories of cloud mixing

13. 3. Phenomenology and Observations theories of cloud mixing

14. Typical Tradewind Cumulus Strong horizontal variability ! theories of cloud mixing

15. Horizontal Variability and Correlation Mean profile height theories of cloud mixing

16. Mean profile “Level of zero kinetic energy” Inversion Level of neutral buoyancy (LNB) non-well mixed layer height Level of free convection (LFC) Lifting condensation level (LCL) well mixed layer Poor man’s cloud model: adiabatic ascent theories of cloud mixing

17. Schematic picture of cumulus convection: Cumulus convection: • more intermittant • more organized than Dry Convection. theories of cloud mixing

18. adiabat Mixing between Clouds and Environment (SCMS Florida 1995) Due to entraiment! Data provided by: S. Rodts, Delft University, thesis available from:http://www.phys.uu.nl/~www.imau/ShalCumDyn/Rodts.html theories of cloud mixing

19. Liquid water potential temperature Virtual potential temperature Entrainment Influences: • Vertical transport • Cloud top height theories of cloud mixing

20. 4. Cloud Mixing Models theories of cloud mixing

21. hc 4.1 lateral mixing bulkmodel Fractional entrainment rate theories of cloud mixing

22. Diagnose through conditional sampling: Typical Tradewind Cumulus Case (BOMEX) Data from LES: Pseudo Observations theories of cloud mixing

23. Trade wind cumulus: BOMEX LES Observations Cumulus over Florida: SCMS theories of cloud mixing

24. Updraft Calculation in conserved variables: 3. Check on Buoyancy: 2. Reconstruct non-conserved variables: Implementation simple bulk model: continue Stop (= cloud top height) B>0 theories of cloud mixing

25. Criticism: • No correct simultaneous prediction of cloud top height (=zero buoyancy level) and cloud fields (Warner paradox) Due to: Bulk model theories of cloud mixing

26. 4.2 Multiparcel Mixing Models Ensemble of parcels (cloud elements) Each parcel has a different mixing fraction with environment Are send to their zero buoyancy level Spectral mass flux models: (Arakawa Schubert 1974) Stochastic versions: Raymond, Blyth, Emanuel) theories of cloud mixing

27. 4.3 Example: Lateral mixing multiparcel model Ensemble of parcels (cloud elements) Parcels are send to their zero vertical velocity level. All parcels obey the same dynamical equations. All parcels only interact with a background (mean) field. theories of cloud mixing

28. 1. Parcel equations: 2. Fractional Entrainment rate e theories of cloud mixing

29. Test: Test their properties in the cloud layer Initialise core parcels at cloud base BOMEX, LES data theories of cloud mixing

30. Other results: ql, qt, and ql in the cloud core: theories of cloud mixing

31. Other results: vertical velocity cloud core cover entrainment theories of cloud mixing

32. Other results: variance of qt, ql theories of cloud mixing

33. 5. Turbulent Flux Parameterizations theories of cloud mixing

34. wc a a a 5.1 Mass Flux Approximation theories of cloud mixing

35. No observations of turbulent fluxes and mass flux • Use Large Eddy Simulation (LES) based on observations BOMEX ship array observed observed To be modeled by LES theories of cloud mixing

36. 10 different LES models Initial profiles Large scale forcings prescribed 6 hours of simulation Is LES capable of reproducing the steady state? theories of cloud mixing

37. Mean profiles after 6 hours Use the last 4 simulation hours for analysis of ……. theories of cloud mixing

38. Cloud cover theories of cloud mixing

39. Turbulent Fluxes theories of cloud mixing

40. Mass Flux Decreasing with height Also observed for other cases Obvious reason……….. theories of cloud mixing

41. Conditional Sampling of: • Total water qt • Liquid water potential temperature ql liquid water virtual pot. temp. theories of cloud mixing

42. Test of Mass flux approximation theories of cloud mixing

43. Simple Bulk Mass flux parameterization Where: “Empty” equation detrainment theories of cloud mixing

44. 6. Open Problems theories of cloud mixing

45. 6.1 Issues within the mass flux parameterization Entrainment Formulation (relatively easy) Mass Flux Formulation (hard) Closure Problem, i.e. boundary values at cloud base theories of cloud mixing

46. Boundary layer equilibrium • subcloud velocity closure • CAPE closure: based on theories of cloud mixing

47. 6.2 Issues beyond the mass flux parameterization theories of cloud mixing

48. K-diffusion versus Mass flux theories of cloud mixing

49. K-diffusion theories of cloud mixing

50. OPTIONS • Do all mixing processes with K-diffusion • Do all mixing with mass flux (Randall and coworkers) • Design a blend between mass flux and K-diffusion (two-scale approach) theories of cloud mixing