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

• Advantages:
  – Better than homogeneous assumption, since
    clouds can form before grids reach saturation
• Disadvantages:
  – 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
• Could add further predictors
• 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)
  advantages:
   – More accurate
     calculation of radiative                     qs       qt
     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
TASK 1: Specification of PDF

 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)
            TASK 1: Specification of PDF

                              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
        TASK 2: Specification of PDF
                 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
       TASK 2: Specification of PDF
                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)


• Disadvantage:
    – 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
  sinks are rather ad-hoc in
  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
• Advantages
  – 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
• Disadvantages
  – 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
                                  If assume fast adjustment,
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
for radiative transfer… etc.
               Variability in Clouds

Is this desirable to have so many independent tunable
                       parameters?

                                                    ‘N’ Metrics
  ‘M’ tuning                                        e.g. Z500 scores,
 parameters                                         TOA radiation
                                                    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
  Advantage of Statistical
         Scheme
Microphysics                            Convection Scheme

               Can use information in
                  Other schemes



                Statistical Cloud Scheme




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

                                          Knowing the PDF allows the
                                            IPA bias to be tacked

Use variance to calculate Effective
thickness Approximation                 G(qt)              ETA adjustment
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
         radiation calculations                                       Total
                                                   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