Transcript KNMI - University of British Columbia

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
[email protected]
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
Climate Research
Regional Climate Division
Faculty for Applied Sciences
Multiscale Physics
Strong Dependency of convective activity on tropospheric humidity
Derbyshire et al, QJRMS 2004
Mass Flux
Not reproduced by any convection parameterization!!
…….Also for shallow convection
Stephan de Roode
4
8
12
q g/kg
Exp 1
0.04
-0.2
Exp 2
0.07
-0.4
Exp 3
-0.07
0.4
Exp 4
-0.13
0.7
BOMEX
16
tot al specif ic humidit y
q [ g/ kg]
v    1  0.61q  q 
1500
height above cloud base [m]
BOMEX
Exp 1
Exp 2
Exp 3
Exp 4

BOMEX[m/s]
Exp1
Exp2
Exp3
Exp4
1000
500
0
0
0.005
0.01
0.015
core mass-f lux [m/s]
0.02
0.025
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
But …., what about detrainment?
M
M
M
Entrainment and Detrainment
Detrainment varies much more than Entrainment
De Rooy and Siebesma MWR 2007
A closer “turbulent” look at entrainment/detrainment
Ac
Ly
Lb
Average Budget Equation over (cloudy core) area Ac:
Apply Gauss Theorem:
Lx
Organized versus Turbulent entrainment/detrainment
•Traditionally entrainment/detrainment 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.
divergence
Can we restore this?
convergence
Asai Kasahari (1967) Revisited
Apply Reynolds decomposition on the cloud core interface:
u  ui 
b
 u  ui    ub  ui ,b b
b
divergence
Diffusivity approach for the turbulent term at the interface:
u  ui  
b
  wc c  e 
Upwind approximation at the interface:
b  c if
b  e if
ub  ui  0 divergence
ub  ui  0 convergence
convergence
Gives finally:
 
c
1 ac wc 
c  e 
 
 H ui ,b  ub 

z
ac wc z 
 L
  qt ,l 
 
1 M 
  

if convergence

 L M z 


if divergence
Entrainment
 
1 M 
  

if divergence

 L M z 


if convergence
Detrainment
L
L
Shallow Convection: mostly divergence


L
 
1 M 
  


 L M z 
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)?
Thermodynamic
The Kain-Fritsch
Scheme arguments: Kain-Fritch (1990)

The periphery of a cloud consists of air parcels that have distinct
fractions environmental air  and cloudy air 1- 
C
P
E
Courtesy: Stephan de Roode



 c is the mixed fraction at which the mixed parcel is neutrally buoyant.
Positive (negative) buoyant mixtures are entrained (detrained).
A greater  c yields a greater entrainment and smaller detrainment !
11
Thermodynamic arguments: Kain-Fritch (1990)
KAIN FRITSCH



The mixed parcels have distinct probabilities of occurrence.
Ascribe a PDF p (  ) to the mixed parcels in order to determine the
expectation values of the mass of the entrained and detrained air.
Specify an inflow rate  0 M u in order to set the upper bounds of
entrainment and detrainment.
c
E  2 0 M u  p(  )d   0 M u  c2 .
0.5
0
1
D  2 0 M u  (1   ) p (  )d   0 M u (1   c ) 2 .
p(  )
c
use:
E  M
D  M

   0  c2 ,
   0 (1   c ) 2 .


 c dictates the vertical gradient of the updraft mass flux
12
Is the decrease of mass flux well correlated with c ?
m* 
M zb  ( zt  zb ) * 0.5
M ( zb )
Normalized mass flux in the middle
of the cloud layer
De Rooy and Siebesma MWR 2007
13
And how about relative humidity only……..?
14
How to put these ideas to work?
Neggers JAS 2009
•Assume a Gaussian joint PDF(l,qt,w) shape for the
cloudy updraft.
•Mean and width determined by the multiple updrafts
•Determine everything consistently from this joint PDF
a, wu ,l ,u , qt ,u
Reconstruction of the cloud core fraction
Assume that the 2 parcels lie on a mixing line
16
Example: Reconstruction of the cloud core fraction
•Determine c
•Calculate the core fraction ac
•Determine mass flux directly: M=ac wc
c
No explicit detrainment parameterization required anymore
1 M
  
M z
17
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 c : 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 c
•This allows a direct determination of the mass flux which makes an explicit
detrainment parameterization obsolete.
18
•Assume a Gaussian joint PDF(l,qt,w) shape for the
cloudy updraft.
•Mean and width determined by the multiple updrafts
•Determine everything consistently from this joint PDF
a, wu ,l ,u , qt ,u
An reconstruct the flux:
w   au wu  u  
________
Remarks:
•No closure at cloud base required.
•No convection triggering required.
•No detrainment parameterization required!
•Pdf used for cloud scheme and possible for radiation.
1 M
  
M z
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
-> favours an integrated treatment of transport and clouds within the PBL

Latera
l
Adopted in cloud parameterizations:
Horizontal or vertical
mixing?

mixin
g
Cloud-top
mixing
Observations
(e.g. Jensen 1985)
However: cloud top mixing needs
substantial adiabatic cores within the clouds.
21
(SCMS Florida 1995)
No substantial adiabatic
cores (>100m) found
during SCMS except near
cloud base. (Gerber)
adiabat
Does not completely
justify the entraining
plume model but………
It does disqualify a
substantial number of
other cloud mixing models.
22
The (simplest) Mathematical Framework :
u
e
w   au w   (1  au ) w   au wu (u   e )


K

z
M (u   )
zinv
23
The flexible updraft area partitioning allows the representation of
gradual transitions between different convective regimes:
Overview
dry PBL
Mass flux contribution acts like a more
intelligent counter-gradient contribution
inversion
PBL
a1
M1
w1
+ K-diff.
Shallow Cumulus
10%
inversion
cloud
a2
w2
M2
cloud base
stratocumulus
subcloud
a1
w1
M1
+ K-diff.
10%
inversion
+ K-diff.
cloud base
subcloud
a2
w2
10%
M2
+ K-diff.
25
M2: humidity supply for StCu clouds (coupling to surface)
Backtracing particles in LES: where does the air in the
cloud come from?
Cloudtop
Cloudtop entrainment

Entrance
level
Cloudbase
Inflow from subcloud
Measurement level
Cloudtop
Courtesy Thijs Heus
26
Height vs. Source level
Virtually all cloudy air comes from below the
observational level!!
27
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
0 = F (w
core,z)
28
2. Non-linear character of many cloud related processes
Example 1: Autoconversion of cloud water to precipitation in warm clouds
A  K ql  qcr H ql  qcr 
: 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:
_______
Aql   Aql 
Larson et al. JAS 2001
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
-> favours an integrated treatment of transport and clouds within the PBL
Single column model & IFS results
Tested for a large number of GCSS Cases………………..
l
qsat
qt
Cloud
fraction
Condensate
SCM
LES
EDMF bimodal clouds: a closer look
SCM
LES
BOMEX
The advective PDF captures convective
(updraft) clouds, while the diffusive PDF
picks up the more passive clouds
ATEX
Transient & steady state shallow cumulus
Continental: ARM SGP
Marine: RICO
Moist convective inhibition effects
PBL equilibration: response to a +1 g/kg perturbation in ML humidity
RICO
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
3d-Climate Models
Versions of Climate Models
NWP’s
Global observational
Observations from
Data sets
Field Campaigns
Development
Testing
Evaluation
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
1 M
  
M z
M  ac wc
Early Plume models (1)
L
Continuity Equation
 vdl 
Aplume w plume
z
0
R
Assume circular geometry:
2Rvr 
R 2 wp
z
0
2R 2vr R wp

0
R
z
2
M  R2 wp
Scaling Ansatz :
z
2 vr
M
M
0
R wp
z
vr   wc
1 M
2

M z
R
or  
2
R
with   0.1
Early Plume models (2)
Plume models have proven extremely succesful for plumes but……
Can not straightforwardly be translated to clouds because:
1.
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.
2.
Plume entrainment rate gives estimates an order of magnitude smaller
than for entrainment in clouds.
3.
In parameterization there is a need for an entrainment rate for cloud
ensembles rather than for individual clouds (bulk model vs spectral
model

2 2 x0.1

 4 .10  4 m 1
R
500
Also for shallow convection (ARM case)
Also for shallow convection (ARM case)
De Rooy and Siebesma MWR 2007
Asai Kasahari Revisited
Intermezzo:
Steady state model with no gradient in fraction and with mass flux appr
for conserved variables:

c
L 
1 wc
  H (ub )
  b c  e 
z
wc z
Ac 

Dynamical entrainment
Turbulent entrainment
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:
M
M
M
c
  (c  e )
z
•Detrainment as a residual of the continuity equation:
45
Clouds: use a bulk approach:
Cloud ensemble:
approximated by
1 effective cloud:
and apply the mass flux approximation on
   l , qt 
……
e
 M (c   )
c
w   (1  a) w   a w   awc (c   )
wc
a
a
a
•Simple Bulk Mass flux parameterization
 

 t

M (c   )
 w 



z
z
 conv


M
Requires only a parameterization for c and M
:
 c
  ( c   ) for    l , qt 
Tiedtke 1989, Betts 1974:
z
-4
-1
0.2/R ~ 2 10 m
M
  
Based on entraining plume models
z
Where  : fractional entrainment
1 wc2
 b wc2  aB
rate
2 z
 : fractional detrainment rate
Plus boundary conditions at cloud base are required (I.e. mass flux
closure )
Diagnose
c
 
(c   ) for    l , qt 
z
Entrainment factor
through conditional sampling:
Measure of lateral mixing
Typical Tradewind Cumulus Case (BOMEX)
Data from LES: Pseudo Observations
Total moisture (qt =qv +ql)
Trade wind cumulus: BOMEX
LES
  1 ~ 3 103 m-1
(Neggers et al (2003)
Q.J.M.S.)
Order of
magnitude
larger than in
operational
models!!
Observations
Cumulus over Florida: SCMS
•Mass Flux
•Decreasing with height
•Also observed for other cases
• Obvious reason………..
M  ac wc
M
ac

wc
•Due to decreasing cloud (core) cover
Diagnose detrainment from M and 
:
M
  
z
 ~ 2 10-3 m-1 and  = 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.
53
Derivation of Budget Equations (2)
Average Budget Equation over area Ac:
Use Leibniz:
Apply Gauss Theorem:
Classic Bulk Mass Flux Model
The old working horse:
Entraining plume model:
c
  (c   ) for    l , qt 
z
1 M
  
M z
1 wc2
 b wc2  aB
2 z


M
Plus boundary conditions
at cloud base.
56
Asai Kasahari Revisited
Need to make assumptions on boundary fields:
e
divergence
convergence
Remark: direct interaction with the environment assumed
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
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
59
Detrainment
1 M
  
M z
M  ac wc
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)
Evaluation with LES (BOMEX)
x
x
LES
LES
g
g
g=2/3
Simpson 1969
Including the “cloud mantle” : RICO
iba
iba
LES
LES
g
g
g=0.9
Evaluation with LES (ARM)
x
x
LES
LES
g
g
g=2/3
Simpson 1969
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.
Results for the Relative Humidity Sensitivity Test Case
 cross
c
•  c decreases as the relative humidity decreases !
M
  
z
Looks qualitatively ok!!
De Rooy and Siebesma MWR 2007
65
Large-eddy simulation The BOMEX shallow cumulus case
v    1  0.61q  q 
3000
BOMEX
Exp 1
Exp 2
Exp 3
Exp 4
2500
BOMEX
Exp 1
Exp 2
Exp 3
Exp 4

q g/kg
Exp 1
0.04
-0.2
Exp 2
0.07
-0.4
Exp 3
-0.07
0.4
Exp 4
-0.13
0.7
height [m]
2000
BOMEX
1500
1000
500
0
300
305
pot ent ial t emperature
 [K]
310
4
8
12
16
tot al specif ic humidit y
q [ g/ kg]
66
Results for cloud core : mass flux
DTv

height above cloud base [m]
1500
BOMEX[m/s]
Exp1
Exp2
Exp3
Exp4
1000
500
0
0
M
  
z
0.005
0.01
0.015
0.02
0.025
core mass-f lux [m/s]
Looks qualitatively ok
67
Parameterization:  = 0 2
Does it work? Check from LES results.
core s am pling
-1
]
0.004
 [m
Bomex
Exp1
Exp2
Exp3
Exp4
fractional entrainment rate
diagnosed from LES
0.003
0.002
theory
0.001
0
0
0.1
0.2
critical mixing f raction squared
0 = 5e-3 m-1
0.3
0.4

2
*
Do not use for entrainment!!
How to make better use of
68
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

LeMone & Pennell (1976, MWR)
Cumulus clouds are the condensed, visible parts of updrafts
that are deeply rooted in the subcloud mixed layer (ML)
70
Step 1 : Initialisation of updraft parcel near surface
1.
Initialisation in the surface layer
2.
Use well-established surface layer similarity theory to
generate the varainces of of w, , q}
3.
Assume Gaussian shape of pdf
pdf
w, , q}
Step 2 : Parcel Ascents
rising, entraining plume model for wi and i  {qt ,l }I
Moist updraft
Use this to:
1)
2)
3)
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.
Dry updraft
K diffusion
Parcel entrainment i is sensitive to wi
 1 c
, 
  wi z 
 i  min
As a consequence, different updrafts have different profiles due to
i) different initialization
ii) different entrainment
Flexible moist area fraction
Top 10 % of updrafts that is explicitly modelled
Traditionally:
Traditionally it is implicitly assumed:
So that :
And the classic bulk mass flux models follow readily from the above equations.