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. See also: http://www.knmi.nl/~siebesma
References
A.P. Siebesma and J.W.M. Cuijpers, Evaluation of parametric assumptions for shallow cumulus convection, J. Atmos. Sci., 52, , 1995
A.P. Siebesma and A.A.M. Holtslag, Model impacts of entrainment and detrainment rates in shallow cumulus convection, J. Atmos. Sci., 53, , 1996
A.P. Siebesma , ‘’Shallow Cumulus Convection” published in: Buoyant Convection in Geophysical Flows, p 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
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
See also:
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3. Motivation and Objectives
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4. Cartoon of Hadley Circulation
Subsidence
~0.5 cm/s
10 m/s
inversion
Cloud base
~500m
Tropopause 10km
Deep Convective Clouds
Precipitation
Vertical turbulent transport
Net latent heat production
Engine Hadley Circulation
Shallow Convective Clouds
No precipitation
Vertical turbulent transport
No net latent heat production
Fuel Supply Hadley Circulation
Stratocumulus
Interaction with radiation
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5. theories of cloud mixing
50 km
Shallow cumulus not resolved by “state of the art” global atmospheric models.
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6. Grid Averaged Budget Equations
Large scale
advection
Large scale
subsidence
Vertical turbulent transport
Net Condensation Rate
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7. theories of cloud mixing
Schematically:
Objectives
Understand Cumulus Convection….
Design Models…..
But ultimately design parameterizations of:
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8. theories of cloud mixing
2. Thermodynamics
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9. theories of cloud mixing
2.1 Moisture Variables
qv :Specific Humidity
ql :Liquid Water
qt = qv + ql :Total water specific humidity
(Conserved for phase changes
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10. theories of cloud mixing
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) T qt
qs(p,T) ql qv
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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
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12. Grid averaged equations for conserved variables:
Parameterization issue reduced to a turbulent mixing problem!
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13. theories of cloud mixing
3. Phenomenology
and
Observations
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14. Typical Tradewind Cumulus
Strong horizontal variability !
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15. Horizontal Variability and Correlation
Mean profile
height
Horizontal Variability and Correlation
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16. Poor man’s cloud model: adiabatic ascent
Mean profile
height
“Level of zero kinetic energy”
Level of neutral buoyancy (LNB)
Level of free convection (LFC)
well mixed layer
Inversion
non-well
mixed
layer
Lifting condensation level (LCL)
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17. Schematic picture of cumulus convection:
more intermittant
more organized
than
Dry Convection.
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18. Mixing between Clouds and Environment
(SCMS Florida 1995)
adiabat
Due to entraiment!
Data provided by: S. Rodts, Delft University, thesis available from:
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19. Virtual potential temperature Entrainment Influences:
Liquid water potential temperature
Virtual potential temperature
Entrainment Influences:
Vertical transport
Cloud top height
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20. theories of cloud mixing
4. Cloud Mixing Models
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21. 4.1 lateral mixing bulk model Fractional entrainment ratehc
Fractional entrainment rate
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22. Diagnose through conditional sampling:
Typical Tradewind Cumulus Case (BOMEX)
Data from LES: Pseudo Observations
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23. Trade wind cumulus: BOMEX Cumulus over Florida: SCMS
LES
Observations
Cumulus over Florida: SCMS
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24. Implementation simple bulk model:
Updraft Calculation in conserved variables:
continue
Stop (= cloud top height)
B>0
3. Check on Buoyancy:
2. Reconstruct non-conserved variables:
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25. theories of cloud mixing
Criticism:
No correct simultaneous prediction of cloud top height (=zero buoyancy level) and cloud fields (Warner paradox)
Due to: Bulk model
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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)
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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.
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28. 2. Fractional Entrainment rate e
1. Parcel equations:
2. Fractional Entrainment rate e
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29. Test: Test their properties in the cloud layer
BOMEX, LES data
Initialise core parcels at cloud base
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30. Other results: ql, qt, and ql in the cloud core:
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31. Other results: vertical velocity cloud core cover entrainment
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32. Other results: variance of qt, ql
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33. 5. Turbulent Flux Parameterizations
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34. 5.1 Mass Flux Approximationwc
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35. theories of cloud mixing
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
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36. theories of cloud mixing
10 different LES models
Initial profiles
Large scale forcings prescribed
6 hours of simulation
Is LES capable of reproducing the steady state?
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37. Mean profiles after 6 hours
Use the last 4 simulation hours for analysis of …….
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38. theories of cloud mixing
Cloud cover
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39. theories of cloud mixing
Turbulent Fluxes
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40. theories of cloud mixing
Mass Flux
Decreasing with height
Also observed for other cases
Obvious reason………..
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41. theories of cloud mixing
Conditional Sampling of:
Total water qt
Liquid water potential temperature ql
liquid water
virtual pot. temp.
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42. Test of Mass flux approximation
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43. Simple Bulk Mass flux parameterization
Where:
“Empty” equation
detrainment
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44. theories of cloud mixing
6. Open Problems
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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
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46. theories of cloud mixing
Boundary layer equilibrium
subcloud velocity closure
CAPE closure: based on
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47. 6.2 Issues beyond the mass flux parameterization
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48. K-diffusion versus Mass flux
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49. theories of cloud mixing
K-diffusion
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50. theories of cloud mixing
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)
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51. theories of cloud mixing
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52. Find equilibrium solutions for f={ql,qt}
To be done:
Find equilibrium solutions for f={ql,qt}
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53. theories of cloud mixing
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54. theories of cloud mixing
Cloud Boundaries
Cloud size distributions
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55. theories of cloud mixing
Bulk means:
Cloud ensemble:
approximated by
1 effective cloud:
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56. Determination of the relaxation time:w
Use LES
Determine for each cloud: cloud height h and vert. vel. w
Estimate t by t=1/wh
h(m)
t(sec)
Conclusion:
Relaxation time ~ constant
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57. Due to decreasing cloud (core) cover
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58. Virtual potential temperature: qv
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59. theories of cloud mixing
Cloud Liquid water
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60. Prescribe non-dimensionalised mass flux profile
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61. Case studies - The GEWEX Cloud System Study
CRMs
GCSS WG4
TOGA/COARE 6-day average cloud cover
SCMs
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62. theories of cloud mixing
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63. theories of cloud mixing
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64. theories of cloud mixing
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65. theories of cloud mixing
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