KNMI

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Entrainment..but what about detrainment? Some
new views on lateral mixing in shallow cumulus
convection
A. Pier Siebesma, Wim de Rooy, Roel Neggers and
Stephan de Roode siebesma_at_knmi.nl

1. Importance detrainment vs entrainment
2. A turbulent mixing view on entrainment and
   detrainment
3. Thermodynamic constraints on cloud mixing
4. How to put things at work

Faculty for Applied Sciences
Climate Research
Multiscale Physics
Regional Climate Division
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Strong Dependency of convective activity on
tropospheric humidity
Derbyshire et al, QJRMS 2004
Mass Flux
Not reproduced by any convection
parameterization!!
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.Also for shallow convection
Stephan de Roode
dq???? dq ?g/kg?
BOMEX
Exp 1 0.04 -0.2
Exp 2 0.07 -0.4
Exp 3 -0.07 0.4
Exp 4 -0.13 0.7
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Led to many interesting studies..

• Grabowski et al. (2006) Need entrainment rate to
  decrease with time of day
• Kuang and Bretherton (2006) Weaker entrainment
  rates for deep than for shallow convection -
  increasing parcel size as cold pools form?
• Khairoutdinov and Randall (2006) Demonstration
  of downdraft/cold pool role in transition from
  shallow to deep convection
• Bechtold et al. (2008) Explicit parameterization
  of entrainment rate as f(1-RH)

Mainly concentrating on the role of entrainment
?M
?M
M
But ., what about detrainment?
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Entrainment and Detrainment
Detrainment varies much more than Entrainment
De Rooy and Siebesma MWR 2007
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A closer turbulent look at entrainment/detrainme
nt
Average Budget Equation over (cloudy core) area
Ac
Apply Gauss Theorem
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Traditionally
Traditionally it is implicitly assumed
So that
And the classic bulk mass flux models follow
readily from the above equations.
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Organized versus Turbulent entrainment/detrainment

• However in these last steps entrainment/detrainmen
  t is treated rather advective (as opposed to a
  turbulent mixing process.

• The very old notion that there is a distinction
  between dynamical and turbulent entrainment (i.e.
  Houghton and Cramer 1951) has gone.

Can we restore this?
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Asai Kasahari (1967) Revisited
Apply Reynolds decomposition on the cloud core
interface
divergence
Diffusivity approach for the turbulent term at
the interface
convergence
Upwind approximation at the interface
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Gives finally
Entrainment
Detrainment
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Shallow Convection mostly divergence
Organized detrainment
(Tiedtke 1989)
Turbulent entrainment/detrainment
The variation in organized detrainment from case
to case explains the larger spread in detrainment
So what determines the shape of the mass flux (or
the organized detrainment)?
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The Kain-Fritsch Scheme
Thermodynamic arguments Kain-Fritch (1990)

• The periphery of a cloud consists of air parcels
  that have distinct fractions environmental air
  and cloudy air 1-
• is the mixed fraction at which the mixed
  parcel is neutrally buoyant.
• Positive (negative) buoyant mixtures are
  entrained (detrained).
• A greater yields a greater entrainment and
  smaller detrainment !

Courtesy Stephan de Roode
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KAIN FRITSCH
Thermodynamic arguments Kain-Fritch (1990)

• The mixed parcels have distinct probabilities of
  occurrence.
• Ascribe a PDF to the mixed parcels in
  order to determine the expectation values of the
  mass of the entrained and detrained air.
• Specify an inflow rate in order to
  set the upper bounds of entrainment and
  detrainment.
• dictates the vertical gradient of the updraft
  mass flux

use
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Is the decrease of mass flux well correlated with
cc ?
Normalized mass flux in the middle of the cloud
layer
De Rooy and Siebesma MWR 2007
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And how about relative humidity only..?
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How to put these ideas to work? Neggers JAS 2009

• Assume a Gaussian joint PDF(ql,qt,w) shape for
  the cloudy updraft.
• Mean and width determined by the multiple
  updrafts
• Determine everything consistently from this joint
  PDF

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Reconstruction of the cloud core fraction
Assume that the 2 parcels lie on a mixing line
Buoyancy along the mixing line at a specified
height
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Example Reconstruction of the cloud core fraction

• Determine cc
• Calculate the core fraction ac
• Determine mass flux directly Mac wc

?c
No explicit detrainment parameterization required
anymore
Buoyancy along the mixing line at a specified
height
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Conclusions

• In shallow cumulus it is detrainment rather than
  entrainment that regulates the shape of the mass
  flux and hence the moistening of the cloud layer.
• This shape is regulated the zero buoyancy point
  on the mixing line cc strong decrease of the
  mass flux is promoted by low CAPE but also
  through low RH.
• The physical relationship is made explicit in the
  Dual Eddy Diffusivity Mass Flux framework in
  which the cloud core fraction can be directly
  related to cc
• This allows a direct determination of the mass
  flux which makes an explicit detrainment
  parameterization obsolete.

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• Assume a Gaussian joint PDF(ql,qt,w) shape for
  the cloudy updraft.
• Mean and width determined by the multiple
  updrafts
• Determine everything consistently from this joint
  PDF

An reconstruct the flux

• Remarks
• No closure at cloud base required.
• No convection triggering required.
• No detrainment parameterization required!
• Pdf used for cloud scheme and possible for
  radiation.

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Further new concepts a bimodal statistical cloud
scheme
Extension of EDMF into the representation of
sub-grid clouds
updraft mode
passive mode
The observed turbulent PDF in shallow cumulus has
a clear bimodal structure 1 updraft mode, 1
passive (diffusive) mode This decomposition
conceptually matches that defining EDMF -gt
favours an integrated treatment of transport and
clouds within the PBL
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Horizontal or vertical mixing?
Cloud-top mixing
Observations (e.g. Jensen 1985)
However cloud top mixing needs substantial
adiabatic cores within the clouds.
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(SCMS Florida 1995)
No substantial adiabatic cores (gt100m) found
during SCMS except near cloud base. (Gerber)
Does not completely justify the entraining plume
model but It does disqualify a substantial
number of other cloud mixing models.
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The (simplest) Mathematical Framework
zinv
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The flexible updraft area partitioning allows
the representation of gradual transitions between
different convective regimes
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Overview
dry PBL
Mass flux contribution acts like a more
intelligent counter-gradient contribution
inversion
M1
a1
K-diff.
w1
PBL
Shallow Cumulus
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inversion
M2
w2
a2
cloud
cloud base
stratocumulus
a1
w1
subcloud
K-diff.
M1
10
inversion
K-diff.
cloud base
a2
w2
M2
K-diff.
subcloud
M2 humidity supply for StCu clouds (coupling to
surface)
10
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Backtracing particles in LES where does the air
in the cloud come from?
Courtesy Thijs Heus
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Height vs. Source level
Virtually all cloudy air comes from below the
observational level!!
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Conclusions

• Kain Fritsch looks reasonable at first sight.
• Thermodynamic considerations alone is not enough
  to parameterize lateral mixing and the mass flux
• Kinematic ingredients need to be included

e0 F (wcore,z)
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2. Non-linear character of many cloud related
processes
Example 1 Autoconversion of cloud water to
precipitation in warm clouds
Kessler Autoconversion Rate (Kessler 1969)
With ql cloud liquid water ql critical
threshold H Heaviside function A
Autoconversion rate
Autoconversion rate is a convex function
Larson et al. JAS 2001
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Further new concepts a bimodal statistical cloud
scheme
Extension of EDMF into the representation of
sub-grid clouds
updraft mode
passive mode
The observed turbulent PDF in shallow cumulus has
a clear bimodal structure 1 updraft mode, 1
passive (diffusive) mode This decomposition
conceptually matches that defining EDMF -gt
favours an integrated treatment of transport and
clouds within the PBL
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Single column model IFS results
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Tested for a large number of GCSS Cases..
?l
qsat
qt
Cloud fraction
Condensate
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EDMF bimodal clouds a closer look
BOMEX ATEX
The advective PDF captures convective (updraft)
clouds, while the diffusive PDF picks up the more
passive clouds
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Transient steady state shallow cumulus
Continental ARM SGP
Marine RICO
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Moist convective inhibition effects
PBL equilibration response to a 1 g/kg
perturbation in ML humidity
RICO
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A slow, but rewarding Working Strategy
See http//www.gewex.org/gcss.html
Large Eddy Simulation (LES) Models Cloud
Resolving Models (CRM)
Single Column Model Versions of Climate Models
3d-Climate Models NWPs
Global observational Data sets
Observations from Field Campaigns
Development
Testing
Evaluation
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Conclusions and Outlook

• EDMF framework is explained, that presently
  extend its range of applicability to
  conditionally
• unstable cloud layers (shallow cumulus)
• Just enough complexity is added to enable gradual
  transitions to and from shallow cumulus
  convection
• Attaching a bimodal statistical cloud scheme to
  the EDMF framework makes the treatment of
  transport and cloud consistent throughout the PBL
  scheme
• The double PDF allows representation of complex
  cloud structures, such as cumulus rising into
  stratocumulus
• Scheme is calibrated against independent
  datasets (LES), and tested for a broad range of
  different PBL scenarios (GCSS!!)

Status

• Partly operational in ECMWF (fully later this
  year)
• Implemented in ECHAM, RACMO, AROME (but coupled
  with a TKE scheme)

Further research on

• Coupling with TKE-schemes
• Initialisation from other layers than the surface
  layer
• Microphysics
• Extension to deep convection.
• Momentum transport

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Early Plume models (1)
L
Continuity Equation
R
Assume circular geometry
z
Scaling Ansatz
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Early Plume models (2)
Plume models have proven extremely succesful for
plumes but

• Can not straightforwardly be translated to clouds
  because
• Plume-environment mixing is essentially a
  dilution process, hence plume width grows with z.
  With clouds phase transition come into play that
  calls for detrainment process as well.
• Plume entrainment rate gives estimates an order
  of magnitude smaller than for entrainment in
  clouds.
• In parameterization there is a need for an
  entrainment rate for cloud ensembles rather than
  for individual clouds (bulk model vs spectral
  model

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Also for shallow convection (ARM case)
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Also for shallow convection (ARM case)
De Rooy and Siebesma MWR 2007
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Asai Kasahari Revisited
Intermezzo
Steady state model with no gradient in fraction
and with mass flux appr for conserved variables
Dynamical entrainment
Turbulent entrainment
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Classic Mechanistic view on entrainment and
detrainment

• Convective Mass Flux M r ac wc
• Crucial parameter in parameterizing convective
  transport in large scale models
• Shape and Magnitude determined by the inflow
  (entrainment) and the outflow (detrainment)
• Entrainment determined (by conditional sampling)
  using simplified budget equations
• Detrainment as a residual of the continuity
  equation

?M
?M
M
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Clouds use a bulk approach
Cloud ensemble approximated by
1 effective cloud
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and apply the mass flux approximation on

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• Simple Bulk Mass flux parameterization

e
d
M
Requires only a parameterization for fc and M

• Tiedtke 1989, Betts 1974
• d 0.2/R 2 10-4 m-1
• Based on entraining plume models
• Where e fractional entrainment rate
• d fractional detrainment rate

Plus boundary conditions at cloud base are
required (I.e. mass flux closure )
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Diagnose
through conditional sampling
Entrainment factor Measure of lateral mixing
Typical Tradewind Cumulus Case (BOMEX) Data from
LES Pseudo Observations
Total moisture (qt qv ql)
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Trade wind cumulus BOMEX
LES
Order of magnitude larger than in operational
models!!
Observations
(Neggers et al (2003) Q.J.M.S.)
Cumulus over Florida SCMS
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• Mass Flux

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

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• Due to decreasing cloud (core) cover

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Diagnose detrainment from M and e
e 2 10-3 m-1 and d 3 10-3 m-1

• Entrainment and detrainment order of magnitude
  larger than previously assumed
• Detrainment systematically larger than
  entrainment
• Mass flux decreasing with height
• Due to larger entrainment a lower cloud top is
  diagnosed.

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Simplest conceptual entrainment model for
entrainment in clouds
Siebesma 1997 Bretherton and Grenier JAS 2003
Shallow convection hc 1000m e 10-3 m-1 !!
Neggers et al 2001 JAS Cheinet 2003 JAS
Alternative
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Derivation of Budget Equations (2)
Average Budget Equation over area Ac
Use Leibniz
Apply Gauss Theorem
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Classic Bulk Mass Flux Model
The old working horse
Entraining plume model
Plus boundary conditions at cloud base.
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Asai Kasahari Revisited
Need to make assumptions on boundary fields
e
Remark direct interaction with the environment
assumed
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Final Result
g
So that
g
Remark Gregory 2001 and Nordeng 1994 are special
cases of these results!
Remark no dependancy on the gradient of the
cloud fraction
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But things may vary

• Only the relative
• humidity is varied !!
• In the case of RH 25
• a low cloud top is expected !

Mass Flux!!!
Derbyshire et al, QJRMS 2004
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Detrainment
g
Remark 1 dependancy on the gradient of the cloud
fraction affects only detrainment
Remark 2 If ac can be determined indepently no
parameterization for detrainment is needed ( see
later)
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Evaluation with LES (BOMEX)
ex
dx
dLES
eLES
g
g
g2/3 Simpson 1969
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Including the cloud mantle RICO
eiba
diba
eLES
dLES
g
g
g0.9
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Evaluation with LES (ARM)
ex
dx
dLES
eLES
g
g
g2/3 Simpson 1969
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Conclusions

• Proposed relations not a ready to use as
  parameterization but..
• Expressions derived from first principles
• Provides insight in the mechanisms of entrainment
  and detrainment
• The gradient of core fraction appears only in the
  detrainment and is responsible for the fact that
  detrainment is a much strongly varying quantity
  from case to case.

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Results for the Relative Humidity Sensitivity
Test Case

• decreases as the relative humidity
  decreases !

Looks qualitatively ok!!
De Rooy and Siebesma MWR 2007
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Large-eddy simulation -The BOMEX shallow cumulus
case
dq???? dq ?g/kg?
BOMEX
Exp 1 0.04 -0.2
Exp 2 0.07 -0.4
Exp 3 -0.07 0.4
Exp 4 -0.13 0.7
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Results for cloud core mass flux
DTv
c
Looks qualitatively ok
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Parameterization e e0 ?2Does it work? Check
from LES results.
theory
e0 5e-3 m-1
Do not use for entrainment!!
How to make better use of
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Eddy Diffusivity Mass Flux Parameterization

• Siebesma and Teixeira An advection-diffusion
  scheme for the convective boundary layer
  description and 1d results. AMS proceedings 2000
• Siebesma, Soares and Teixeira A combined eddy
  diffusivity Mass flux approach for the convective
  boundary layer. JAS 64, (2007)
• Soares, Miranda, Siebesma and Teixeira An eddy
  diffusivity/ mass flux parameterizaiton for dry
  and shallow cumulus convection. QJRMS 130 (2004)
• De Rooy and Siebesma MWR 2007
• Neggers Kohler and Beljaars A dual mass flux
  framework for boundary layer convection. Part 1
  Transport Accepted for JAS
• Neggers A dual mass flux framework for boundary
  layer convection. Part ii Clouds. Accepted for
  JAS

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• LeMone Pennell (1976, MWR)

Cumulus clouds are the condensed, visible parts
of updrafts that are deeply rooted in the
subcloud mixed layer (ML)
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Step 1 Initialisation of updraft parcel near
surface

• Initialisation in the surface layer
• Use well-established surface layer similarity
  theory to generate the varainces of of fw, q,
  q

3. Assume Gaussian shape of pdf
pdf
fw, q, q
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Step 2 Parcel Ascents
rising, entraining plume model for wi and ?i
? qt ,?l I

• Use this to
• Partition which part of the top 10 of the pdf
  will remain dry and which part will become moist.
• Perform a dry updraft ascent.
• Perform 2 moist ascents.

Parcel entrainment ?i is sensitive to wi
As a consequence, different updrafts have
different profiles due to i) different
initialization ii) different entrainment