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

siebesma@knmi.nl

Faculty for Applied SciencesClimate Research

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

Strong Dependency of convective activity on tropospheric humidity

Mass Flux

Derbyshire et al, QJRMS 2004

Not reproduced by any convection parameterization!!

…….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

1500

BOMEX[m/s]Exp1Exp2Exp3Exp4

core mass-flux [m/s]

heig

ht

ab

ove c

lou

d b

ase

[m

]

•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?

Entrainment and Detrainment

De Rooy and Siebesma MWR 2007

Detrainment varies much more than Entrainment

A closer “turbulent” look at entrainment/detrainment

Lx

Ly

Ac

Lb

Average Budget Equation over (cloudy core) area Ac:Apply Gauss Theorem:

•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

Asai Kasahari (1967) Revisited

Apply Reynolds decomposition on the cloud core interface:

bbib

b

i

b

i uuuuuu ,

ecc

b

i wuu

econvergencuuif

divergenceuuif

ibeb

ibcb

0

0

Diffusivity approach for the turbulent term at the interface:

Upwind approximation at the interface:convergence

divergence

Gives finally:

ltq , eccc

ccbbi

c

z

wa

wauuH

Lz

1

,

divergenceifL

econvergencifz

M

ML

1

econvergencifL

divergenceifz

M

ML

1

Entrainment

Detrainment

Shallow Convection: mostly divergence

L

z

M

ML

1

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

11

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)

12

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

0

0 cuu MdpMEc

.)1()()1(2 20

1

0 cuu MdpMDc

uM0

)(p

)(p

0.5

.)1(

,2

0

20

c

c

use:MD

ME

c

Thermodynamic arguments: Kain-Fritch (1990)

13De Rooy and Siebesma MWR 2007

)(

5.0*)(*

b

btb

zM

zzzMm

Is the decrease of mass flux well correlated with c ?

Normalized mass flux in the middle of the cloud layer

14

And how about relative humidity only……..?

•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

16

Reconstruction of the cloud core fraction

Assume that the 2 parcels lie on a mixing line

17

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

z

M

M

1

z

M

M

1

18

•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

•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________

z

M

M

1

z

M

M

1

19 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

21

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.

22

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.

23

zinv

The (simplest) Mathematical Framework :

)(

)()1(

u

euuu

e

u

u

u

Mz

K

wawawaw

19 April 2023

The flexible updraft area partitioning allows the representation of gradual transitions between different convective regimes:

25

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

26

Backtracing particles in LES: where does the air in the cloud come from?

En

tran

ce le

vel

Cloudtop

Cloudbase Cloudto

p

Measurement level

Late

ral e

ntra

inm

ent

Cloudtop entrainment

Inflow from subcloud

Courtesy Thijs Heus

27

Height vs. Source level

Virtually all cloudy air comes from below the observational level!!

28

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)

2. Non-linear character of many cloud related processes

crlcrl qqHqqKA

With:

ql : cloud liquid water

ql : critical threshold

H : Heaviside function

A : 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

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

Single column model & IFS results

l

qt

qsat

Cloudfraction

Condensate

SCMLES

Tested for a large number of GCSS Cases………………..

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

Transient & steady state shallow cumulus

Continental: ARM SGP Marine: RICO

PBL equilibration: response to a +1 g/kg perturbation in ML humidity

Moist convective inhibition effects

RICO

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

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!!)

ccwaM

z

M

M

1

ccwaM

z

M

M

1

Early Plume models (1)

R

z

0

z

wAdlv plumeplume

L

Continuity Equation

Assume circular geometry: 022

z

wRRv p

r

Scaling Ansatz : cr wv

02

22

z

wR

R

vR pr

pwRM 2 02

z

MM

w

v

R p

r

1.0221

withR

orRz

M

M

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 mx

R

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:

ecc

bc

cb

c

A

L

z

w

wuH

z

1

)(

Dynamical entrainment Turbulent entrainment

•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

45

Cloud ensemble:

approximated by

1 effective cloud:

Clouds: use a bulk approach:

and apply the mass flux approximation on ……

)()( cc

ceawwawaw 1 )()( cc

ceawwawaw 1

a

wc

aa

)( cM )( cM

,l tq

aBwbz

w

z

M

qz

cc

tcc

22

l

2

1

,for)(

aBwbz

w

z

M

qz

cc

tcc

22

l

2

1

,for)(

•Simple Bulk Mass flux parameterization•Simple Bulk Mass flux parameterization

z

M

z

w

tc

conv

)(

z

M

z

w

tc

conv

)(

Requires only a parameterization for c and M :Tiedtke 1989, Betts 1974:

R ~ 2 10-4 m-1

Based on entraining plume models

Where : fractional entrainment rate

: fractional detrainment ratePlus boundary conditions at cloud base are required (I.e. mass flux closure )

M

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

-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!!

•Mass Flux•Mass Flux

ccwaM ccwaM

•Decreasing with height

•Also observed for other cases

• Obvious reason………..

•Decreasing with height

•Also observed for other cases

• Obvious reason………..

MM cwcwca

ca

•Due to decreasing cloud (core) cover

53

z

M

z

M

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.

Derivation of Budget Equations (2)

Average Budget Equation over area Ac:

Use Leibniz:

Apply Gauss Theorem:

56

Classic Bulk Mass Flux Model

aBwbz

w

z

M

M

qz

cc

tcc

22

l

2

1

1

,for)(

aBwbz

w

z

M

M

qz

cc

tcc

22

l

2

1

1

,for)(

M

The old working horse:

Entraining plume model:

Plus boundary conditions

at cloud base.

Asai Kasahari Revisited

Need to make assumptions on boundary fields:

convergence

divergence

Remark: direct interaction with the environment assumed

e

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

59

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

Detrainment

ccwaM

z

M

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)

Evaluation with LES (BOMEX)

LES

x

LES

x

=2/3 Simpson 1969

LES

Including the “cloud mantle” : RICO

iba iba

LES

=0.9

Evaluation with LES (ARM)

LES

x

LES

x

=2/3 Simpson 1969

•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

65

Results for the Relative Humidity Sensitivity Test Case

• decreases as the relative humidity decreases !c

c

cross

z

M

z

M

Looks qualitatively ok!!

De Rooy and Siebesma MWR 2007

66

Large-eddy simulation -

The BOMEX shallow cumulus case

300 305 310

0

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

67

Results for cloud core : mass flux

0 0.005 0.01 0.015 0.02 0.0250

500

1000

1500

BOMEX[m/s]Exp1Exp2Exp3Exp4

core mass-flux [m/s]

heig

ht

ab

ove c

lou

d b

ase

[m

]

v

z

M

z

M

Looks qualitatively ok

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

0.004

BomexExp1Exp2Exp3Exp4

fracti

on

al en

train

men

t ra

te

[

m-1

]

dia

gn

ose

d f

rom

LE

S

critical mixing fraction squared *

2

core sampling

theory

0 = 5e-3 m-1 Do not use for entrainment!!

How to make better use of

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

70

Cumulus clouds are the condensed, visible parts of updrafts that are deeply rooted in the subcloud mixed layer (ML)

LeMone & Pennell (1976, MWR)

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}

pdf

w, , q}

3. Assume Gaussian shape of pdf

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 ,

1

min

As a consequence, different updrafts have different profiles due to

i) different initialization

ii) different entrainment

Traditionally it is implicitly assumed:

So that :

And the classic bulk mass flux models follow readily from the above equations.

Traditionally: