Mass Flux Concepts - atmos.washington.edu
Transcript of Mass Flux Concepts - atmos.washington.edu
Mass Flux Concepts:A. Pier Siebesma ([email protected])
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)