Mass Flux Concepts:A. Pier Siebesma (siebesma@knmi.nl)

KNMI, De Bilt

The Netherlands

� Mass flux approximations� Entraining Plume model� Mixing mechanisms� Transition from shallow to deep

convection� Mass flux in the PBL and Boundary

Layer

)()( φφφφφ −+′′+′′−=′′ cc

ceawwawaw 1

a

wc

aa

)( φφ −≈ cM

What is the mass flux concept?Estimating (co)variances through smart conditional sampling of joint pdf’s

Why does it work so well for cumulus?Horizontal Variability

Bimodal joint pdf due to :�Buoyancy production by condensation in clouds (w)�Non-local transport (qt)

Courtesy : Bjorn Stevens

Cumulus: Typically 80~90% repesented for moist conserved variables by mass flux appr.

Courtesy : Bjorn Stevens

How well does it work for “dry” convection?

Assuming Joint Gaussian pdf and using updraft/downdraft decomposition:

______

6.0)( φφφ ′′≈− wM du

Wyngaard&Moeng BLM 1992

Remarks:• Scu gives similar results for fluxes as dry convection (50~60%) when using updraft/downdraft.(de Laat and Duynkerke BLM 1998)

•Deep Convection: Additional decomposition of cloudy downdrafts is advisable.

•Applying the same technique on variances and higher moments gives progressively worse results.

Health warning: Be careful by applying mass flux concept on variances, skewness and beyond.

How to estimate updraft fields and mass flux?

{ }

aBwbz

w

z

M

M

qz

cc

tcc

+−=∂∂

−=∂∂

∈−−=∂∂

22

l

2

1

1

,for)(

ε

δε

θφφφεφ

M

εδ

The old working horse:Entraining plume model:

Plus boundary conditionsat cloud base.

Isn’t the plume model in conflict with other proposed mixing mechanisms in cumulus?

Pakuch 1979 JASEmanuel 1991 JASBlyth 80’s

However: all these other concepts need substantial adiabatic cores within the clouds.

Two-point mixingEpisodic mixing, etc…….

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.

What is simplest entraining plume based parameterization?

Simply use diagnosed typical values for ε and δ based on LES and observations and suitable boundary conditions at cloud base (closure)

-13 m 1031 −

= ~ε

Trade wind cumulus: BOMEX

LES

Observations

Cumulus over Florida: SCMS

(Neggers et al (2003) Q.J.R..M.S.)

•Mass Flux•Mass Flux

δε −=∂∂

z

M

M

1

ccwaM =

δ = ε + 0.5 10-3

Diagnose δ using M and ε

Works reasonably well for shallow cumulus:BOMEX: Siebesma et al JAS 2003ARM: Brown et al QJRMS 2002SCMS: Neggers et al QJRMS 2003ATEX: Stevens et al JAS 2001

Steady State Mass flux profiles of CRM’s!!

Derbyshire et al, QJRMS 2004 EUROCS special issueBut does it work for deep(er) convection

4 cases: RH=25%,50%,70% 90% Same θ−profile use nuding

ε and δ change with environmental conditions so a more flexible approach is required!!

Simplest conceptual model for entrainment

{ }

τφφφ

φθφ

)(

:,

eccc

mixingc

tl

zw

Fdt

d

qfor

−−=∂∂

=

∈

clt,c θq ,

cw

elt,e θq ,

ew

hc 1

1

where)(cc

ecc

hwz≈=−−=

∂∂

τεφφεφ

Shallow convection: hc ~ 1000mε ~10-3 m-1 !!

Siebesma 1997 Bretherton and Grenier JAS 2003

w

1

1

c0

∝≈τ

εcw

Alternative:Neggers et al 2001 JASCheinet 2003 JAS

But how about detrainment? (or, the mass flux?)Kain-Fritsch buoyancy sorting scheme (1991) is the only scheme to my knowledge that makes a prediction of ε and δ

PC E

�The periphery of a cloud consists of air parcels that have distinct fractions environmental air χ and cloudy air 1-χ

Courtesy: Stephan de Roode

Buoyancy sorting mixing principle� The mixed parcels have distinct probabilities of occurrence P(χ)� Specify an inflow rate ε0 M � Assume the simplest PDF (Bretherton and Grenier (2003):

�

.)(2 20

0

0 cuu MdpMEc

χεχχχεχ

== ∫.)1()()1(2 2

0

1

0 cuu MdpMDc

χεχχχεχ

−=−= ∫)(χp

χ

0.5

.)1(

,2

0

20

c

c

χεδχεε

−=

=use:

MD

ME

δε=

=

Remark: if χc < 0.5 then : E<D => dM/dz < 0if χc > 0.5 then : E>D => dM/dz > 0

Parameterization: ε = ε0 χ2

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

frac

tiona

l ent

rain

men

t rat

e

ε [m

-1]

diag

nose

d fr

om L

ES

critical mixing fraction squared χ*

2

core sampling

theory

ε0 = 5e-3 m-1

εLES

χ2c

Another Avenue to derive Mass flux M:

∑=≡i

iccc wN

waM ,

1 Multiple Mass Flux ParameterizationArakawa Schubert JAS 1974Neggers et al. JAS 2001Cheinet JAS 2003

Works good for shallow convection but……only gives decreasing mass flux

Can mass flux parameterize transition from shallow to deep convection?� Deep Convection mass flux parameterization in the tropics starts to precipitate hours too early within the diurnal cycle� CRM studies suggest this is related to a slow transition from shallow to deep convection

Thanks to Jon Petch ( Met office) (QJRMS 2004 EUROCS special issue)

Is this the end of the classic Entraining Plume Model?

During early stage of the day: small scale structures in the pbl that trigger small strongly entraining cumulus clouds with limited vertical extendDuring later stage of the day: larger scale structures in the pbltrigger larger cloud structures that entrain less and reach higher cloud tops.

•Double counting of processes•Interface problems•Problems with transitions between different regimes

This unwanted situation has led to:

( ) Swzt

+′′∂∂−≅

∂∂ φφ

Standard parameterization approach:

zKw∂∂−≅′′ φφ )( φφφ −≅′′ uMw

Attemps to unify convective transport inclear, subcloud and cloud layer

Three roadmaps:

1. Only Eddy Diffusivity (Bechtold et al. JAS 52 1995)

2. Only Mass Flux (Cheinet JAS 2003)

3. Both Mass Flux and Eddy Diffusivity(Siebesma and Teixeira : AMS Proceedings 2000)

(Lappen and Randall. JAS 58 2001)

(Soares et al., QJRMS, 130 2004)

(Siebesma et al. submitted to JAS)

Extending the plume model to the subcloud layer.

zinv

The Idea :•Nonlocal (Skewed) transport through strong updrafts in clear and cloudy boundary layer by advective Mass Flux(MF) approach•Remaining (Gaussian) transport done by an Eddy Diffusivity (ED) approach

Barbados Oceanographic and Meteorological Experiment (BOMEX)

Strongest updrafts: are always cloudyroot deeply into the subcloud layer

LES results on shallow cumulus

Motivation

zinv

Advantages :

•One updraft model for : dry convective BL, subcloud layer, cloud layer.•No trigger function/closure for moist convection needed

•No switching required between moist and dry convection needed

•Easy transition to neutral and stable PBL

zinv

The Parameterization of the

Eddy Diffusivity-Mass Flux (ED-MF) approach

)( φφφφ −+∂∂−≅′′ uM

zKw

Entrainment

Steady State Updraft Equations

( )θθεθ

−−=∂∂

uu

z

Entraining updraft parcel:

PBw

z

pgw

z

ww

uw

vuvuwu

u

−+−=∂∂−−+−=

∂∂

2

,0

2 1)(

ερθθθε

KKK

ε: Fractional entrainment rate:

advection

entrainment buoyancy

pressure

Vertical velocity eq. of updraft parcel:

wu, θu

Initialisation of updraft eq.?

h (km)

x(km)

0

5

1

Use LES to derive updraft model in clear boundary layer.

0

Updraft at height z composed

of those grid points

that contain the highest p%

of the vertical velocities:

p=1%,3%,5%:

Use LES results to diagnose entrainment ε

( )lulul

zθθ

θε −

∂∂

−= /

From LES

τ

ε

u

ie

w

zzzc

/1

11

≅

−+=Fit:

Remark : ε high near interfaces and low in the middle of the p bl

Remark:. 1/ ε has similar behaviour has length scale formulations in TKE scheme

1/εzi

Use LES results to diagnose vertical velocity budget

PBwz

wuw

u −+−=∂∂ 2

2

2

1ε

5.0

15.0

____2

____2

≈≈

≈∂

∂−≅∂

∂−=

bb

z

w

z

wP

w

u

εε

µµ Bwz

wuw

u +−=∂∂− 2

2

)21(2

1 εµ

wu-budget: B

P

E

Adv

Similar format as Simpson (1969)

0.1 0.2 0.3 0.4

0.04

0.06

0.08

0.10

0.12

r = 0.97926

Y = 0.29758 X

∆θv (

K)

(w'θv')

S/ e1/2 (K)

10 m 30 m 50 m

Initialisation of the updraft

θθθ ∆+=uInitialise updraft at lowest model layer:

2~1

______

≈′′

=∆ ασθ

αθw

srfwWith an excess that scales with the Surface flux (Troen and Mahrt 1986):

LES

Initial buoyancy

w

srfw

σθ

______

′′

obs

z/zinv

0

1

0.1

K w * /zinv

23/1

3*

3* 139

−

+≡

ii z

zz

z

zkwukK

Mass flux:

uu waM ≡

Holtslag 1998

Eddy Diffusivity:

K-profile:

au =0.03

Single Column Model tests for convective BL

γθθ Kz

Kw +∂∂−=′′

)( θθθθ −+∂∂−=′′ uM

zKw

zKw

∂∂−=′′ θθOnly Diffusion: ED

Diffusion + Mass Flux: ED-MF

Diffusion + Counter-Gradient: ED-CG

z

w

t ∂′′∂−=

∂∂ θθSolve with implicit solver (Teixeira

2000)

Mean profiles after 10 hours

Mean profilePBL height growth

ED : Unstable Profiles : Too aggressive top-entra inment : too fast pbl -growth

Counter-gradient: Hardly any top-entrainm ent : too slow pbl-growth. Howcome??

EDMF

ED-CG

ED

ED-CG

ED

BL-growth

EDMF

Breakdown of the flux into an eddy diffusivity and a countergradient contribution

γθθ Kz

Kw +∂∂−=′′

No entrainment flux since the countergradient (CG) term is balancing the ED-term!!

______

θ ′′w

EDCG

total

Conclusions

ED MF provides good frame work for integral turbule nt mixing in CT -PBL

•Correct internal structure

•Correct ventilation (top-entrainment) for free atmo sphere

•Easy to couple to the cumulus topped BL

Countergradient approach

•Correct internal structure but…..

•Underestimation of ventilation to free atmosphere

•Cannot be extended to cloudy boundary layer

Two flavours of ED-MF now implemented in operational models

1. ECMWF : K-profile ED + Mass Flux

Impact on low cloud cover

Roel Neggers and Martin Kohler

2. Meso-NH Model : TKE-closure based ED + Mass FluxPedro Soares and coworkers ( QJRMS, 130 2004) (special EUROCS issue)

Matching with convection scheme through Mb = au wu (similar to Grant closure)

Other important untouched Topics

•Momentum transport (Brown 1998 QJRMS, Lappen and Randall submitted

•Triggering (Bechtold )

•Transition scu � cu•Equilibrium Solutions (Stevens. Grant)

A statistical mass flux framework for organized updrafts

Each updraft corresponds to a certain fraction of the joint PDF

Model each fraction’s averaged internal statistics

vertical integration of updraft budget equations

updraft initialization depends on its position in the PDF

The moist area fraction is a free variable

closure is done on the area fraction instead of on the mass flux as a whole

a flexible area fraction can act as a ‘soft trigger function’ (dry moist transition)

PDF of {w, qt ,θl }Dry updrafts

Moist updrafts

Top % of updrafts that is explicitly modelled

K diffusion

iii waM ≡

� Estimate cloud convective properties on the basis of observations in sub-cloud layer

Properties:

- Presence of BL clouds

- Cloud base(/top) height

- In-cloud properties

(LWC, vertical velocity)Surface fluxes

Mean profiles T, q

cloud base

Courtesy Manfred Wendisch, IfT Leipzig

Extensively tested against observations in Cabauw (Marijn de Hay)

Sensitivity study (1)Sensitivity to dilution

� 30-60 m bias in Vaisala CT75 cloud base (red to green)

� Lateral mixing:- suppresses cloud presence- causes higher cloud base

� UND run undershoots cloud base, in some cases better with REF but great variability

10

11

21

zinv

The Idea :

•Nonlocal (Skewed) transport through strong updrafts in clear and cloudy boundary layer by advective Mass Flux (MF) approach•Remaining (Gaussian) transport done by an Eddy Diffusivity (ED) approach

Divide subcloud updrafts into only two groups (N=2):

i) those which will stop at cloud base (i=1)

ii) those which will become clouds (i=2)

N=2: two degrees of freedom

M1subcloud

cloud

cloud base

∑=

−+∂∂−=

N

iiiPBL M

zKw

1

)('' φφφφ

Explicitly model vertical advective transport by strongest multiple updrafts:

M2

M2

inversion

Mtotal

A statistical mass flux framework for organized updrafts

the relevant population statistics are represented without ‘shooting’ too many parcels

Vertical velocity and entrainment

ii w

1

τε =

ii wz

c

1

τε +=

2 SCM runs:

Moist

Dry

BOMEX

LES

SCM

zinv

The Mathematical Framework :

)(

)()1(

φφφφφφφφ

−+∂∂−≅

−+′′−+′′=′′

u

euuu

e

u

u

u

Mz

K

wawawaw

Courtesy : Dave Stevens ; Lawrence Livermore National Laboratory

•ATEX :Marine CumulusToppedWith Scu

Buoyancy reversal -

Dependency on mean vertical gradients Γ

Γθ =∂θ∂z

, Γq = ∂q

∂z

0.0

1.0

De Roode, 2004

Conclusions

• Buoyancy reversal does always occur in cumulus

• More active (positively buoyant) cloud updrafts if:

- More CAPE

- More moisture

• Improve parameterizations that only consider CAPE!

Roode, S.R. de, Buoyancy reversal in cumulus clouds, submitted to the J. Atmos. Sci.

(http://www.phys.uu.nl/~roode/publications.html)

Open Problems:

•Mass flux Closure

•Detrainment (or the mass flux)

•Vertical velocity equation

•Momentum transport

•Connection/Interaction with the cloud scheme

•Connection/Interaction with the dry convection

•Transition to other regimes (Scu / Deep Convection)

•Role of precipitation (RICO)

Results for cloud core :

vertical velocity and core fraction

0 1 2 3 4 5 60

500

1000

1500

BOMEXExp1Exp2Exp3Exp4

core vertical velocity [m/s]

heig

ht a

bove

clo

ud b

ase

[m]

0 0.01 0.02 0.03 0.04

core fraction

Results for cloud core :

fractional entrainment and detrainment

0 0.001 0.002 0.003 0.0040

200

400

600

800

1000

BOMEXExp1Exp2Exp3Exp4

fractional entrainment rate ε [m -1]

heig

ht a

bove

clo

ud b

ase

[m]

0 0.005 0.01 0.015

fractional entrainment rate δ [m-1

]

χ* from LES

0 0.1 0.2 0.3 0.4 0.5 0.60

200

400

600

800

1000

BOMEXExp1Exp2Exp3Exp4

χ*

heig

ht a

bove

clo

ud b

ase

[m]

Parameterization: ε = ε0 χ2

Does it work? Check from LES results.

0 0.001 0.002 0.003 0.0040

200

400

600

800

1000

BOMEXExp1Exp2Exp3Exp4

fractional entrainment rate ε [m -1]

heig

ht a

bove

clo

ud b

ase

[m]

0 0.1 0.2 0.3 0.4 0.5 0.60

200

400

600

800

1000

BOMEXExp1Exp2Exp3Exp4

χ*

heig

ht a

bove

clo

ud b

ase

[m]

•Boundary layer equilibrium

•subcloud velocity closure

•CAPE closure: based on

srfbcb qwqqM ′′=− )( ,

∗= cwM b

τ

CAPE

t

CAPE ≈∂

∂

Data provided by: S. Rodts, Delft University, thesis available from:http://www.phys.uu.nl/~www.imau/ShalCumDyn/Rodts.html

Mixing between Clouds and Environment

(SCMS Florida 1995)

4.0~3.0,=

adl

lq

q

adiabat

Due to entrainment between the clouds and

the environment.

4. Evaluation

of the

Shallow Convection Parametrization

Single Column Model

versionfull 3d model

•Methodology:

Difficult to isolate and assess the performance of one model component……

Use a well documented case and prescribe the large scale forcing!!

0subgridscale

large≈

∂∂+

∂∂=∂

∂ttt

φφφBOMEX ship array

•No observations of turbulent fluxes and mass flux, but..

•Large scale tendencies measured by a radiosonde array

•No observations of turbulent fluxes and mass flux, but..

•Large scale tendencies measured by a radiosonde array

observed observed

To be simulated by SCM

BOMEX Trade Wind Cumulus Experiment: 1969.(Siebesma and Holtslag JAS 1996)

•ECMWF SCM model 21r3

•Initial profiles

•Large scale forcings prescribed

•24 hours of simulation

•ECMWF SCM model 21r3

•Initial profiles

•Large scale forcings prescribed

•24 hours of simulation

Is SCM capable of reproducing the steady state?Is SCM capable of reproducing the steady state?

Too vigorous vertical mixing of qt and θl .

What to do……..?

)(

)(

,,

,,

tctct

lclcl

qqz

q

z

−−=∂∂

−−=∂∂

ε

θθεθ

1. Updraft Calculation in conserved variables:

{ } { },,,,, lvvtl qqq θθθ ⇒

2. Reconstruct non-conserved variables:

( )vcvv

gB θθ

θ−= ,

3. Check on Buoyancy:

B>0

continue Stop (= cloud top height)

Implementation simple bulk model:

Typical Tradewind Cumulus Case (BOMEX)

Data from LES: Pseudo Observations

{ } ,for)( l tcc q

zθφφφφε ∈−

∂∂−=Diagnose

through conditional sampling:

Total moisture (qt =qv +ql)

Entrainment factor

Measure of lateral mixing

=M cwca

•Due to decreasing cloud (core) cover

×

δε −=∂∂

z

M

Diagnose detrainment from M and ε :

ε ~ 2 10-3 m-1 and δ = 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.

Results for the Relative Humidity Sensitivity Test Case

• decreases as the relative humidity decreases !cχ

cχ

crossχ

δε −=∂∂

z

M

M

1

Looks qualitatively ok!!

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 θ [K]

θv = θ +δθ( )1+ 0.61 q+ δq( )[ ]

4 8 12 16

BOMEXExp 1Exp 2Exp 3Exp 4

total specific humidity q [g/kg]

0.7-0.13Exp 4

0.4-0.07Exp 3

-0.40.07Exp 2

-0.20.04Exp 1

BOMEX

δq [g/kg]δθ [Κ]

Thanks to: Stephan de Roode

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 a

bove

clo

ud b

ase

[m]

χ

∆Τv

δε −=∂∂

z

M

M

1

Looks qualitatively ok

χc

Conclusions:

•Buoyancy Sorting Mechanism looks “qualitatively ok”.•Thermodynamic considerations alone is not enough to parameterize lateral mixing and hence the mass flux•Kinematic ingredients need to be included

ε0 = F (wcore,z)