Numerical Weather Prediction Parametrization of diabatic proce

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```					Cloud Parametrization 2:
Cloud Cover

tompkins@ictp.it
Cloud Cover: The problem

Most schemes presume
GCM Grid cell: 40-400km    cloud fills GCM box in
vertical
Still need to represent
horizontal cloud cover, a

a
GCM Grid
First: Some assumptions!
qv = water vapour mixing ratio
qc = cloud water (liquid/ice) mixing ratio
qs = saturation mixing ratio = F(T,p)
qt = total water (vapour+cloud) mixing ratio
RH = relative humidity = qv/qs

(#1) Local criterion for formation of cloud: qt > qs
This assumes that no supersaturation can exist
(#2) Condensation process is fast (cf. GCM timestep)
qv = q s , q c = q t – q s
!!Both of these assumptions are suspect in ice clouds!!
Partial cloud cover
Homogeneous
Distribution of water
vapour and temperature:
q
Note in the                                      qs ,1
second
case the                                        q
relative
humidity=1                                       qs ,2
from our
assumptions
x
One Grid-cell
Partial coverage of a grid-box with clouds is only possible if
there is a inhomogeneous distribution of temperature
and/or humidity.
Heterogeneous distribution of T and q

cloudy=
q

q
qs
RH=1                RH<1
x

Another implication of the above is that clouds must exist
before the grid-mean relative humidity reaches 1.
The interpretation does not change much if
we only consider humidity variability
cloudy
qt

qs
qt

RH=1                   RH<1
x

Throughout this talk I will neglect temperature variability

In fact : Analysis of observations and model data indicates
humidity fluctuations are more important
Simple diagnostic schemes: RH-based schemes

qt                 RH=60%

qs

qt

x
Take a grid cell with a certain (fixed)   1
distribution of total water.
C
At low mean RH, the cloud cover is
zero, since even the moistest part of     0                            RH
the grid cell is subsaturated            60        80       100
Simple diagnostic schemes: RH-based schemes

qt                RH=80%

qs
qt

x
1
Add water vapour to the gridcell,
the moistest part of the cell    C
become saturated and cloud
forms. The cloud cover is low.     0                            RH
60        80       100
Simple diagnostic schemes: RH-based schemes

qt               RH=90%

qs
qt

x
1

Further increases in RH          C
increase the cloud cover
0                            RH
60        80       100
Simple diagnostic schemes: RH-based schemes

RH=100%
qt

qt
qs

x
1
The grid cell becomes
C
overcast when RH=100%,
due to lack of supersaturation       0                            RH
60        80       100
Simple Diagnostic Schemes:
Relative Humidity Schemes
• Many schemes, from the                       1

1970s onwards, based cloud                   C
cover on the relative humidity
(RH)                                         0
60         80   100
RH

• e.g. Sundqvist et al. MWR
1989:
C=1−
    1− RH
1− RH crit

RHcrit = critical relative humidity at which
cloud assumed to form
(function of height, typical value is 60-80%)
Diagnostic Relative Humidity
Schemes
• Since these schemes form cloud when
RH<100%, they implicitly assume subgrid-
scale variability for total water, qt, (and/or
temperature, T) exists
• However, the actual PDF (the shape) for
these quantities and their variance (width)
are often not known
• “Given a RH of X% in nature, the mean
distribution of qt is such that, on average, we
expect a cloud cover of Y%”
Diagnostic Relative Humidity
Schemes

– Better than homogeneous assumption, since
clouds can form before grids reach saturation
– Cloud cover not well coupled to other processes
– In reality, different cloud types with different
coverage can exist with same relative humidity.
This can not be represented
• Can we do better?
Diagnostic Relative Humidity
Schemes
• E.g: Xu and Randall (1996)
sampled cloud scenes from a 2D
cloud resolving model to derive an
empirical relationship with two
predictors:

C=F RH ,q c 
• More predictors, more degrees of freedom=flexible
• But still do not know the form of the PDF. (is model valid?)
• Can we do better?
Diagnostic Relative Humidity
Schemes

• Another example is the scheme of Slingo, operational at
ECMWF until 1995.
• This scheme also adds dependence on vertical velocities
• use different empirical relations for different cloud types, e.g.,
middle level clouds:

{
0                ω≥0                                        2
C m = C m ω /ω crit
¿

C   ¿
m
ω crit ≤ω 0
ωω crit
C = max
m
¿
[ 
RH −RH crit
1−RH crit
,0
]
Relationships seem Ad-hoc? Can we do better?
Statistical Schemes
• These explicitly specify           qt
the probability density
function (PDF) for the                                       q
total water qt (and                                          qs

sometimes also
x
temperature)

Cloud cover is
PDF(qt)
∞
integral under
C=∫ PDF q t dq t
supersaturated
q
∞s                                              part of PDF
q c =∫ q t −q s PDF q t dqt                                 qt
qs                                      qs
Statistical Schemes

• Others form variable ‘s’ that also takes temperature
variability into account, which affects qs              L
T L =T − q L
'      '                             Cp
s=a L q t −α T L                           LIQUID WATER TEMPERATURE
conserved during changes of state
−1
Cloud mass if T
variation is
neglected
qs
∂ qs
α L=  T L 
∂T
             [   L
aL = 1 α L
Cp     ]
S                                         ∞
qt
C=∫ G s ds
T                        ss
TL

S is simply the ‘distance’ from the linearized        INCREASES COMPLEXITY
saturation vapour pressure curve                  OF IMPLEMENTATION
Statistical Schemes

• Knowing the PDF has

PDF(qt)
– More accurate
fluxes                                            C
– Unbiased calculation of
microphysical
processes                  y
• However, location of
clouds within gridcell                      x
unknown
e.g.
microphysics
bias
Statistical schemes
• Two tasks: Specification of the:
(1) PDF shape
(2) PDF moments
• Shape: Unimodal? bimodal? How many
parameters?

• Moments: How do we set those parameters?
TASK 1: Specification of the PDF

• Lack of observations to determine qt PDF
– Aircraft data
• limited coverage
modis image from NASA website
– Tethered balloon
• boundary layer
– Satellite
• difficulties resolving in vertical
• no qt observations
• poor horizontal resolution
– Raman Lidar
• only PDF of water vapour

• Cloud Resolving models have also been used
• realism of microphysical parameterisation?
Wood and field
JAS 2000
Aircraft
observations low
clouds < 2km

Aircraft
Observe
d PDFs
Height

Heymsfield and
PDF(qt)          McFarquhar
JAS 96
qt   Aircraft IWC obs
during CEPEX
PDF             Data

More examples
from Larson et al.
JAS 01/02

Note significant
error that can
occur if PDF is
unimodal

Conclusion: PDFs are mostly approximated by uni or bi-
modal distributions, describable by a few parameters

Many function forms have been used
symmetrical distributions:
PDF( qt)

qt                     qt
Uniform:             Triangular:
Letreut and Li (91)        Smith QJRMS (90)

qt                      qt
Gaussian:            s4 polynomial:
Mellor JAS (77)   Lohmann et al. J. Clim (99)

skewed distributions:
PDF( qt)

qt                      qt                          qt

Exponential:            Lognormal:                  Gamma:
Sommeria and Deardorff             Bony & Emanuel          Barker et al. JAS (96)
JAS (77)                        JAS (01)

qt                             qt
Beta:            Double Normal/Gaussian:
Tompkins JAS (02)      Lewellen and Yoh JAS (93), Golaz et al.
JAS 2002
moments
Need also to determine the
moments of the                                                  saturation
distribution:

PDF(qt)
– Variance (Symmetrical                                             cloud forms?
PDFs)
– Skewness (Higher                                                              qt

order PDFs)                                 e.g. HOW WIDE?
– Kurtosis (4-parameter
PDFs)
Skewness                         Kurtosis
Moment 1=MEAN                                                          positive

e
negative
itive

tiv
Moment 2=VARIANCE

ga
Moment 3=SKEWNESS
po s

ne
Moment 4=KURTOSIS
moments
(1-RHcrit)qs

• Some schemes fix the

G(qt)
moments (e.g. Smith                        1-C          C
1990) based on critical
RH at which clouds                                              qt
assumed to form                            q q
q            s
• If moments (variance,                      e
t
skewness) are fixed,                  q v =Cq s 1−C  qe
                   
then statistical schemes
are identically          where q e =q s 1− 1− RH crit 1−C 

equivalent to a RH                    qv
                         2
formulation                   RH= =1− 1− RH crit 1−C 
qs
• e.g. uniform qt
distribution = Sundqvist
form
∴ C =1−

Sundqvist formulation!!!
1− RH
1 −RH crit
Clouds in GCMs
Processes that can affect distribution
moments

convection

microphysics
turbulence

dynamics
Example: Turbulence
In presence of vertical gradient of total water,
turbulent mixing can increase horizontal variability

dry air

moist air
'
dq2          d qt
t        '   '
=−2 w q t
dt            dz
Example: Turbulence
In presence of vertical gradient of total water,
turbulent mixing can increase horizontal variability

dry air

moist air

while mixing in the horizontal     dq2
'        '
q2
plane naturally reduces the          t
=−
t

horizontal variability     dt         τ
Specification of PDF moments
If a process is fast                                                       turbulence
compared to a GCM timestep,
an equilibrium can be
assumed, e.g. Turbulence
dq2
'
d q t qt 2
'
local
'
d qt '   '
t      ' '
=−2 w q t      −                      equilibrium    q 2=−τ2 w q t
dt             dz    τ                                    t             dz
Source       dissipation

Example: Ricard and Royer, Ann Geophy, (93), Lohmann et al. J. Clim (99)

– Can give good estimate in boundary layer, but above, other
processes will determine variability, that evolve on slower timescales
Prognostic Statistical Scheme

• I previously introduced a
prognostic statistical
scheme into ECHAM5
climate GCM
• Prognostic equations are
introduced for the                    convective
detrainment
variance and skewness
of the total water PDF
precipitation
• Some of the sources and               generation
their derivation!                     mixing
qs
Scheme in action

Minimum
Maximum
qsat
Scheme in action

Turbulence breaks up cloud

Minimum
Maximum
qsat
Scheme in action

Turbulence breaks up cloud             Turbulence creates cloud

Minimum
Maximum
qsat
Production of variance

Courtesy of
Steve Klein:

Change due to difference in
means
Transport
Change due to difference in variance

Also equivalent terms due to entrainment
Microphysics
• Change in variance
' ' Where P is the
P=Kq n
P qt      precipitation
l
−     ql

generation rate, e.g:                               qlcrit
P=Kq l 1−e                              
• However, the tractability depends on the PDF form
for the subgrid fluctuations of q, given by G:
!!   G
s ty ! n d
q t _max                                                na r P a
ty
p ret m fo
∫         ' '
P qt G qt dq t                   et for
n g on
Ca ing
en
d
qt =qsat                          D   ep
But quickly can
get untractable

• E.g: Semi-Lagrangian
ice sedimentation
• Source of variance is
far from simple, also
depends on overlap
assumptions
• In reality of course
wish also to retain the
sub-flux variability too
Summary of statistical
schemes
– Information concerning subgrid fluctuations of humidity
and cloud water is available
– It is possible to link the sources and sinks explicitly to
physical processes
– Use of underlying PDF means cloud variables are
always self-consistent
– Deriving these sources and sinks rigorously is hard,
especially for higher order moments needed for more
complex PDFs!
– If variance and skewness are used instead of cloud
water and humidity, conservation of the latter is not
ensured
Issues for GCMs

If we assume a 2-parameter PDF for total water,
and we prognose the mean and variance such
that the distribution is well specified (and the cloud
water and liquid can be separately derived
assuming no supersaturation) what is the potential
difficulty?
Which prognostic equations?
Take a 2 parameter distribution & Partially cloudy conditions

Can specify distribution with          Can specify distribution with
(b) Mean                               (b) Water vapour
(c) Variance                           (c) Cloud water
of total water                         mass mixing ratio

qt qsat                                  qsat

Cloud                                  Cloud

qv       ql+i
Variance
Which prognostic equations?
qsat
• Cloud water budget conserved
(a) Water vapour                         • Avoids Detrainment term
(b) Cloud water                          • Avoids Microphysics terms (almost)
mass mixing ratio

qv          ql+i

But problems arise in
Overcast
conditions
qsat                 (…convection +
microphysics)
Clear sky                                                      (al la Tiedtke)
conditions
(turbulence)

qv                      qv+ql+i
Which prognostic equations?
Take a 2 parameter distribution & Partially cloudy conditions
qsat
(a) Mean
(b) Variance
of total water

• “Cleaner solution”
• But conservation of liquid water compromised
due to PDF
• Need to parametrize those tricky
microphysics terms!
qcloud
qs        qcrit   If supersaturation allowed, equation
for cloud-ice no longer holds
PDF(qt)

∞

q i ≠∫ q t −q s PDF q t  dqt
qs
qs        qcrit    qt
PDF(qt)

derivation is straightforward
∞

q i = ∫ q t −q s  PDF  q t dq t
q cloud
qs        qcrit    qt

Much more difficult if want to
PDF(qt)

integrate nucleation equation
explicitly throughout cloud

qt                 qi = ???
Issues for GCMs

What is the advantage of knowing the total
water distribution (PDF)?
You might be tempted to say…

Hurrah! We now have cloud
variability in our models, where
before there was none!

WRONG! Nearly all components of GCMs
contain implicit/explicit assumptions
concerning subgrid fluctuations
e.g: RH-based cloud cover, thresholds for
precipitation evaporation, convective
triggering, plane parallel bias corrections
Variability in Clouds

Is this desirable to have so many independent tunable
parameters?

‘N’ Metrics
‘M’ tuning                                        e.g. Z500 scores,
fluxes

With large M, task of reducing error in N metrics
Becomes easier, but not necessarily for the right reasons

Solution: Increase N, or reduce M
Scheme
Microphysics                            Convection Scheme

Can use information in
Other schemes

Statistical Cloud Scheme

Thus ‘complexity’ is not
synonymous with ‘M’: the
Use in other Schemes (II)
IPA Biases

Knowing the PDF allows the
IPA bias to be tacked

Use variance to calculate Effective
Effective Liquid = K x Liquid
Cahalan et al 1994                                                     Total
qsat               Water

Use Moments to find best-fit
Gamma Function   Gamma Distribution PDF and
G(qt)
apply Gamma weighted TSA
Total
Barker (1996)
qsat           Water
Thick
Cloud etc.
G(qt)
Split PDF and perform N
qsat               Water
Cloud Inhomogeneity and
microphysics biases
qs
qs  q L Most current microphysical

G(qt)

schemes use the grid-mean
cloud mass (I.e: neglect
qt              variability)

cloud

cloud range
Result is not equal in the two cases since                       precip

mean
microphysical processes are non-linear                          generation
Example on right: Autoconversion based on Kessler
Grid mean cloud less than threshold and gives zero
precipitation formation
qL0       qL

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