Theories of Mixing in Cumulus Convection

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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)
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• 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

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• Motivation and Objectives

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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

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• Shallow cumulus not resolved by state of the
  art global atmospheric models.

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Grid Averaged Budget Equations
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Schematically

• Objectives
• Understand Cumulus Convection.
• Design Models..
• But ultimately design parameterizations of

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2. Thermodynamics
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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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Remark 1 In thermodynamic equilibrium qt
qv if qv lt qsat undersaturation qt qsat
ql if qv gt qsat oversaturation
Remark 2 qsat qsat (p,T) is a state function
(Clausius-Clayperon)
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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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Grid averaged equations for conserved variables
Parameterization issue reduced to a turbulent
mixing problem!
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3. Phenomenology and Observations
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Typical Tradewind Cumulus
Strong horizontal variability !
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Poor mans cloud model adiabatic ascent
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• Schematic picture of cumulus convection

• Cumulus convection
• more intermittant
• more organized
• than
• Dry Convection.

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Mixing between Clouds and Environment
(SCMS
Florida 1995)
Due to entraiment!
Data provided by S. Rodts, Delft University,
thesis available fromhttp//www.phys.uu.nl/www.i
mau/ShalCumDyn/Rodts.html
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Liquid water potential temperature
Virtual potential temperature

• Entrainment Influences
• Vertical transport
• Cloud top height

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4. Cloud Mixing Models
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4.1 lateral mixing bulk model
Fractional entrainment rate
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Diagnose
through conditional sampling
Typical Tradewind Cumulus Case (BOMEX) Data from
LES Pseudo Observations
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Trade wind cumulus BOMEX
LES
Observations
Cumulus over Florida SCMS
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Implementation simple bulk model
continue
Stop ( cloud top height)
Bgt0
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• 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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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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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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1. Parcel equations
2. Fractional Entrainment rate e
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Test
BOMEX, LES data
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Other results ql, qt, and ql in the cloud core
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Other results vertical velocity cloud core
cover entrainment
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Other results variance of qt, ql
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5. Turbulent Flux Parameterizations
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5.1 Mass Flux Approximation
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• 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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• 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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• Mean profiles after 6 hours

• Use the last 4 simulation hours for analysis of
  .

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• Cloud cover

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• Turbulent Fluxes

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• Mass Flux

• Decreasing with height
• Also observed for other cases
• Obvious reason..

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• Conditional Sampling of
• Total water qt
• Liquid water potential temperature ql

• liquid water
• virtual pot. temp.

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• Test of Mass flux approximation

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• Simple Bulk Mass flux parameterization

Where
Empty equation
detrainment
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6. Open Problems
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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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• Boundary layer equilibrium
• subcloud velocity closure
• CAPE closure based on

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6.2 Issues beyond the mass flux parameterization
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• K-diffusion versus Mass flux

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• K-diffusion

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• 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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• To be done

• Find equilibrium solutions for fql,qt

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• Cloud Boundaries
• Cloud size distributions

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Bulk means
Cloud ensemble approximated by
1 effective cloud
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Determination of the relaxation time

• Use LES
• Determine for each cloud cloud height h and
  vert. vel. w
• Estimate t by t1/wh

Conclusion Relaxation time constant
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• Due to decreasing cloud (core) cover

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Virtual potential temperature qv
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• Cloud Liquid water

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• Prescribe non-dimensionalised mass flux profile

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Case studies - The GEWEX Cloud System Study
CRMs
GCSS WG4 TOGA/COARE 6-day average cloud cover
SCMs
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