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Entrainment…..but what about detrainment?Some new views on lateral mixing in shallow cumulus convection
Multiscale PhysicsRegional Climate Division
A. Pier Siebesma, Wim de Rooy, Roel Neggers and Stephan de Roode
Faculty for Applied SciencesClimate Research
1. Importance detrainment vs entrainment2. A turbulent mixing view on entrainment and detrainment3. Thermodynamic constraints on cloud mixing4. How to put things at work
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Strong Dependency of convective activity on tropospheric humidity
Mass Flux
Derbyshire et al, QJRMS 2004
Not reproduced by any convection parameterization!!
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…….Also for shallow convectionStephan de Roode
v 10.61 q q
4 8 12 16
BOMEXExp 1Exp 2Exp 3Exp 4
total specific humidity q [g/kg]
q 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
0 0.005 0.01 0.015 0.02 0.0250
500
1000
1500BOMEX[m/s]Exp1Exp2Exp3Exp4
core mass-flux [m/s]
heig
ht a
bove
clou
d ba
se [m
]
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•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)
Led to many interesting studies……………..
Mainly concentrating on the role of entrainment MM
M
But …., what about detrainment?
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Entrainment and Detrainment
De Rooy and Siebesma MWR 2007
Detrainment varies much more than Entrainment
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A closer “turbulent” look at entrainment/detrainment
Lx
Ly
Ac
Lb
Average Budget Equation over (cloudy core) area Ac:Apply Gauss Theorem:
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•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.
convergence
divergence
Can we restore this?
Organized versus Turbulent entrainment/detrainment
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Asai Kasahari (1967) Revisited
Apply Reynolds decomposition on the cloud core interface: bbib
b
i
b
i uuuuuu ,
ecc
b
i wuu
econvergencuuifdivergenceuuif
ibeb
ibcb
00
Diffusivity approach for the turbulent term at the interface:
Upwind approximation at the interface:convergence
divergence
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Gives finally:
ltq , eccc
ccbbi
c
zwa
wauuH
Lz
1
,
divergenceifL
econvergencifz
MML
1
econvergencifL
divergenceifz
MML
1
Entrainment
Detrainment
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Shallow Convection: mostly divergence
L
z
MML1
So what determines the shape of the mass flux (or the organized detrainment)?
Organized detrainment
Turbulent entrainment/detrainment
(Tiedtke 1989)
The variation in organized detrainment from case to case explains the larger spread in detrainment
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The Kain-Fritsch Scheme 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 !
PC E
Courtesy: Stephan de Roode
c
c
Thermodynamic arguments: Kain-Fritch (1990)
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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
KAIN FRITSCH
.)(2 20
00 cuu MdpME
c
.)1()()1(2 20
1
0 cuu MdpMDc
uM0
)(p
)(p
0.5
.)1(
,2
0
20
c
c
use:MDME
c
Thermodynamic arguments: Kain-Fritch (1990)
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13De Rooy and Siebesma MWR 2007
)(
5.0*)(*
b
btb
zMzzzMm
Is the decrease of mass flux well correlated with c ?
Normalized mass flux in the middle of the cloud layer
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And how about relative humidity only……..?
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•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
utulu qwa ,, ,,,
How to put these ideas to work? Neggers JAS 2009
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Reconstruction of the cloud core fraction
Assume that the 2 parcels lie on a mixing line
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Example: Reconstruction of the cloud core fraction
c
•Determine c
•Calculate the core fraction ac
•Determine mass flux directly: M=ac wc
No explicit detrainment parameterization required anymore
zM
M1
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•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.
Conclusions
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•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
utulu qwa ,, ,,,
Remarks:•No closure at cloud base required.
•No convection triggering required.
•No detrainment parameterization required!
•Pdf used for cloud scheme and possible for radiation.
An reconstruct the flux:
uuuwaw________
zM
M1
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22 April 2023
Further new concepts: a bimodal statistical cloud scheme
Extension of EDMF into the representation of sub-grid clouds
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
updraft mode
passive mode
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Horizontal or vertical mixing? Lateral mixing
Adopted in cloud parameterizations:
Cloud-top mixing
Observations (e.g. Jensen
1985)
However: cloud top mixing needs substantial adiabatic cores within the clouds.
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adiabat
(SCMS Florida 1995)
No substantial adiabatic cores (>100m) 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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zinv
The (simplest) Mathematical Framework :
)(
)()1(
u
euuu
e
u
u
u
Mz
K
wawawaw
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22 April 2023
The flexible updraft area partitioning allows the representation of gradual transitions between different convective regimes:
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Overview
M2w2a2
subcloud
cloud
cloud base
Shallow Cumulus
+ K-diff.a1
inversion
M1
w1
M1w1a1PBL
inversion
dry PBL
+ K-diff.
stratocumulus
+ K-diff.a2 w2 M2
10%
+ K-diff.
subcloud
cloud base
inversion
10%
10%
M2: humidity supply for StCu clouds (coupling to surface)
Mass flux contribution acts like a more intelligent counter-gradient contribution
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Backtracing particles in LES: where does the air in the cloud come from?
Entrance level
Cloudtop
Cloudbase Cloudto
pMeasurement
level
Latera
l entr
ainmen
t
Cloudtop entrainment
Inflow from subcloud
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 included0 = F (wcore,z)
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2. Non-linear character of many cloud related processes
crlcrl qqHqqKA
With:ql : cloud liquid waterql : critical threshold
H : Heaviside functionA : Autoconversion rate
: Kessler Autoconversion Rate (Kessler 1969)
Example 1: Autoconversion of cloud water to precipitation in warm clouds
Autoconversion rate is a convex function:
_______
ll qAqA
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
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
updraft mode
passive mode
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Single column model & IFS results
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l
qt
qsat
Cloudfraction
Condensate
SCMLES
Tested for a large number of GCSS Cases………………..
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EDMF bimodal clouds: a closer look
The advective PDF captures convective (updraft) clouds, while the diffusive PDF picks up the more passive clouds
BOMEX ATEX
SCMLES
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Transient & steady state shallow cumulus
Continental: ARM SGP Marine: RICO
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PBL equilibration: response to a +1 g/kg perturbation in ML humidity
Moist convective inhibition effects
RICO
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A slow, but rewarding Working Strategy
Large Eddy Simulation (LES) Models
Cloud Resolving Models (CRM)Single Column Model
Versions of Climate Models
3d-Climate Models
NWP’s
Observations from
Field Campaigns
Global observational
Data sets
Development Testing Evaluation
See http://www.gewex.org/gcss.html
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Conclusions and Outlook
•Partly operational in ECMWF (fully later this year)•Implemented in ECHAM, RACMO, AROME (but coupled with a TKE scheme)
•Coupling with TKE-schemes•Initialisation from other layers than the surface layer•Microphysics•Extension to deep convection.•Momentum transport
Status:
Further research on:
•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!!)
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ccwaM
zM
M
1
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Early Plume models (1)
R
z
0
z
wAdlv plumeplume
L
Continuity Equation
Assume circular geometry: 022
zwR
Rv pr
Scaling Ansatz : cr wv
02 22
zwR
RvR pr
pwRM 2 02
z
MMwv
R p
r
1.0221
with
Ror
RzM
M
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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:
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
1410.4500
1.022 mxR
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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:
ecc
bc
cb
c
AL
zw
wuH
z
1)(
Dynamical entrainment Turbulent entrainment
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•Convective Mass Flux : M = 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:
Classic “Mechanistic” view on entrainment and detrainment
MMM
)( ecc
z
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45
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Cloud ensemble:
approximated by
1 effective cloud:
Clouds: use a bulk approach:
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and apply the mass flux approximation on ……
)()( cc
ceawwawaw 1
a
wc
aa
)( cM
,l tq
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aBwbz
wz
M
qz
cc
tcc
22
l
21
,for)(
•Simple Bulk Mass flux parameterization
zM
zw
tc
conv
)(
Requires only a parameterization for c and M :Tiedtke 1989, Betts 1974:R ~ 2 10-4 m-1
Based on entraining plume modelsWhere : fractional entrainment rate : fractional detrainment ratePlus boundary conditions at cloud base are required (I.e. mass
flux closure )
M
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Typical Tradewind Cumulus Case (BOMEX)
Data from LES: Pseudo Observations
,for)( l tcc q
z
Diagnose
through conditional sampling:
Total moisture (qt =qv +ql)
Entrainment factor
Measure of lateral mixing
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-13 m 1031 ~
Trade wind cumulus: BOMEX
LES
Observations
Cumulus over Florida: SCMS
(Neggers et al (2003) Q.J.M.S.)
Order of magnitude larger than
in operational models!!
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•Mass Flux
ccwaM
•Decreasing with height
•Also observed for other cases
• Obvious reason………..
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M cwca
•Due to decreasing cloud (core) cover
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53
zM
Diagnose detrainment from M and
~ 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.
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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
aBwbz
wz
MM
qz
cc
tcc
22
l
21
1
,for)(
M
The old working horse:
Entraining plume model:
Plus boundary conditionsat cloud base.
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Asai Kasahari Revisited
Need to make assumptions on boundary fields:
convergence
divergence
Remark: direct interaction with the environment assumed
e
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Final Result
So that:
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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Only the relativehumidity is varied !!
In the case of RH = 25%a low cloud top is expected !
But… things may vary
Mass Flux!!!
Derbyshire et al, QJRMS 2004
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Detrainment
ccwaM
zM
M
1
Remark 1: dependancy on the gradient of the cloud fraction affects only detrainmentRemark 2: If ac can be determined indepently no parameterization for detrainment is needed ( see later)
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Evaluation with LES (BOMEX)
LES
x
LES
x
=2/3 Simpson 1969
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LES
Including the “cloud mantle” : RICO
iba iba
LES
=0.9
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Evaluation with LES (ARM)
LES
x
LES
x
=2/3 Simpson 1969
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•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.
Conclusions
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65
Results for the Relative Humidity Sensitivity Test Case
• decreases as the relative humidity decreases !c
c
cross
zM
Looks qualitatively ok!!
De Rooy and Siebesma MWR 2007
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66
Large-eddy simulation -The BOMEX shallow cumulus case
300 305 3100
500
1000
1500
2000
2500
3000
BOMEXExp 1Exp 2Exp 3Exp 4
heig
ht [m
]
potential temperature [K]
v 10.61 q q
4 8 12 16
BOMEXExp 1Exp 2Exp 3Exp 4
total specific humidity q [g/kg]
q 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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67
Results for cloud core : mass flux
0 0.005 0.01 0.015 0.02 0.0250
500
1000
1500BOMEX[m/s]Exp1Exp2Exp3Exp4
core mass-flux [m/s]
heig
ht a
bove
clou
d ba
se [m
]
v
zM
Looks qualitatively ok
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68
Parameterization: = 0 2
Does it work? Check from LES results.
0 0.1 0.2 0.3 0.40
0.001
0.002
0.003
0.004BomexExp1Exp2Exp3Exp4
fract
ional
entra
inm
ent r
ate
[m-1
]di
agno
sed
from
LES
critical mixing fraction squared *2
core sampling
theory
0 = 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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70
Cumulus clouds are the condensed, visible parts of updrafts that are deeply rooted in the subcloud mixed layer (ML)
LeMone & Pennell (1976, MWR)
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Step 1 : Initialisation of updraft parcel near surface
1. Initialisation in the surface layer2. Use well-established surface layer similarity
theory to generate the varainces of of w, , q}
w, , q}
3. Assume Gaussian shape of pdf
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Step 2 : Parcel Ascents
Dry updraft
Moist updraft
K diffusionFlexible moist area fraction
Top 10 % of updrafts that is explicitly modelled
rising, entraining plume model for wi and i {qt ,l }I
Use this to:
1) Partition which part of the top 10% of the pdf will remain dry and which part will become moist.
2) Perform a dry updraft ascent.3) Perform 2 moist ascents.
Parcel entrainment i is sensitive to wi
zc
wii ,
1min
As a consequence, different updrafts have different profiles due to
i) different initialization
ii) different entrainment
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Traditionally it is implicitly assumed:
So that :
And the classic bulk mass flux models follow readily from the above equations.
Traditionally: