Parameterization in large-scale atmospheric modelling

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Parameterization in large-scale atmospheric
modelling

• General parameterization problem
• Evaluation of terms involving averaged
  quadratic and higher order products of
  (unresolved) deviations from large-scale
  variables and effects of unresolved processes in
  large-scale models (e.g. NWP models and GCMs).
• Examples
• (a) Turbulent transfer in the boundary layer
• (b) Effects of unresolved wave motions (e.g.
  gravity-wave drag)
• (c) Cumulus parameterization
• Other kinds of parameterization problems
  radiative transfer, cloud microphysical
  processes, chemical processes

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Large-scale variables and equations Let an
overbar denote the result of an averaging or
filtering operation which suppresses
fluctuations with temporal and spatial scales
smaller than pre-defined limits. e.g. for some
appropriately smooth and bounded variable after
averaging
We refer to this as the large-scale variable and
assume that our model has sufficient spatial and
temporal resolution to represent the variation of
this variable once we have determined the
equations governing it and an appropriate
solution methodology.
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Typically, if the variable, has the
following governing equation

And the mass continuity equation is
Then applying the averaging operation gives,
approximately
if
In cases to be considered (e.g. cumulus
parameterization)
Determining this term is a goal of the
parameterization
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Atmospheric Equations
Motion
Mass continuity
Thermodynamic

Or
Vapour
Condensed water
Equation of State (ideal gas)
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Energy Conservation (e.g., Gill, 1982, ch. 4)
(kinetic energy)
(moist static energy)
Molecular dynamic and kinematic viscosity
For air
at 15C , 100hPa
Kolmogorov scales (for which viscosity and
dissipation are independent parameters)
These are small for the atmosphere ( 1mm, .1
m/s) . Therefore it is permissible to neglect
viscous terms for parameterization purposes but
not to ignore effects/processes that lead to
dissipation and associated heating
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• Quasi-anelastic approximations for AGCM
  (Atmospheric GCM) parameterization
• Background state
• hydrostatically balanced
• slowly varying (on the smaller, unresolved
  horizontal and temporal scales - e.g. that of
  quasi-balanced planetary scale circulation
  regime).
• deviations from it are small enough to allow
  linearization of the equation of state (ideal gas
  law) to determine relationships between key
  thermodynamic variables

gt
gt
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Using these results leads to the following
Terms in curly brackets negligible for the
parameterized scales but not for the resolved
scales
Terms involving will also be neglected
compared to unity. The approximate mass
continuity equation which will be used is
Upon using this continuity equation to develop
the flux-form equations and averaging
other such (horizontal) terms
other

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Parameterization of the effects of Moist
Convection in GCMs

• Mass flux schemes
• Basic concepts and quantities
• Quasi-steady Entraining/detraining plumes
  (ArakawaSchubert and similar approaches)
• Buoyancy sorting
• Raymond-Blythe, Emanuel
• Kain-Fritsch
• Closure Conditions, Triggering
• Adjustment Schemes
• Manabe
• Betts-Miller

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Traditional Assumptions for Cumulus
Parameterization
1. Quasi-steady assumption effects of averaging
over a cumulus life-cycle can be represented in
terms of steady-state convective elements
. Transient (cloud life-cycle) formulations
Kuo (1964, 1974) Fraedrich(1974), Betts(1975),
Cho(1977), von SalzenMcFarlane (2002). 2.
Pressure perturbations and effects on momentum
ignored Some of these effects have been
reintroduced in more recent work, but not
necessarily in an energetically consistent manner
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Parameterization of Moist Convection
Starting equations (neglect terms in curly and
other small terms brackets and assume implicitly
that the background state is slowly varying on
the parameterized scales)
plus similar equations for vapour, condensed
water, and other scalar quantities
For the traditional formulation ignore
crossed-out terms
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Spatial Averages
For a generic scalar variable,
Large-scale average
Convective-scale average (for a singlecumulus
up/downdraft)
Environment average (single convective element)
Where
Vertical velocity
Ensemble of cumulus clouds
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Cumulus effects on the larger-scales
Start with a general conservation equation for
Plus the assumption
(similar to using anelastic assumption for
convective-scale motions)
(i) Average over the large-scale area (assuming
fixed boundaries)
Mass flux (positive for updrafts)

Also
Top hat assumption
In practice (e.g. in a GCM) the prognostic
variables are also implicitly time averages over
convective cloud life-cycles
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(ii) Apply cumulus scale sub-average to the
general conservation equation, accounting for
temporally and spatially varying boundaries
(Leibnitz rule)
Mass continuity gives
the outward directed normal flow
velocity (relative to the cloud
boundary)
Entrainment (inflow)/detrainment (outflow)
Define


Top hat
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Summary for a generic scalar, c (steady and top
hat in cloud drafts neglect crossed-out terms)
When both updrafts and downdrafts are present,
both entraining environmental air
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Basic cumulus updraft equations (top-hat,
traditional)
Dry static energy sCpTgz Moist static
energy hsLq
mass conservation
dry Static Energy
vapour
condensate
moist Static Energy

(virtual temperature)

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Entrainment/Detrainment
Traditional organized (e.g.plume) entrainment
assumption
(draft perimeter)
Arakawa Schubert (1974) (and descendants, e.g.
RAS, Z-M) - l is a constant for each updraft
saturated homogeneous (top-hat) entraining
plumes - detrainment is confined to a narrow
region near the top of the updraft, which is
located at the level of zero buoyancy (determines
l )
Kain Fritsch (1990) (and descendants, e.g.
Bretherton et al, 2004 ) - Rc is specified
(constant or varying with height) for a given
cumulus - entrainment/detrainment controlled
by bouyancy sorting (i.e. the effective value of
is constrained by buoyancy sorting)

• Episodic Entrainment and non-homogeneous mixing
• (RaymondBlythe, Emanuel, EmanuelZivkovic-Rothman
  )
• Not based on organized entrainment/detrainment
• entrainment at a given level gives rise to an
  ensemble of mixtures of undiluted and
• environmental air which ascend/descend to levels
  of neutral buoyancy and detrain

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zt
zb
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Determining fractional entrainment rates (e.g.
when at the top of an updraft)
Note that since updrafts are saturated with
respect to water vapour above the LCL
This determines the updraft temperature and w.v.
mixing ratio given its mse.
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Fractional entrainment rates for updraft ensembles
(a) Single ensemble member detraining at zzt
Detrainment over a finite depth
(b) Discrete ensemble based on a range of tops
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Buoyancy Sorting Entrainment produces mixtures
of a fraction, f, of environmental air and (1-f)
of cloudy (saturated cumulus updraft) air. Some
of the mixtures may be positively buoyant with
respect to the environment, some negegatively
buoyant, some saturated with respect to water,
some unsaturated
saturated (cloudy)
positively buoyant
1
0
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Kain-Fritsch (1990) (see also Bretherton et al,
2003) Suppose that entrainment into a cumulus
updraft in a layer of thickness dz leads to
mixing of lMcdz of environmental air with an
equal amount of cloudy air. K-F assumed that all
of the negatively buoyant mixtures (fgtfc) will be
rejected from the updraft immediately while
positively buoyant mixtures will be incorporated
into the updraft. Let P(f) be the pdf of mixing
fractions. Then
This assumes that negatively buoyant air detrains
back to the environment without requiring it to
descend to a level of nuetral bouyancy first).
Emanuel Mixtures are all combinations of
environement air and undiluted cloud-base air.
Each mixture ascends(positively buoyant)/descends
(negatively buoyant), typically without further
mixing to a level of nuetral buoyancy where it
detrains.
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Closure and Triggering

• Triggering
• It is frequently observed that moist convection
  does not occur even when there is a positive
  amount of CAPE. Processes which overcome
  convective inhibition must also occur.
• Closure
• The simple cloud models used in mass flux schemes
  do not fully determine the mass flux. Typically
  an additional constraint is needed to close the
  formulation.
• The closure problem is currently still poorly
  constrained by theory.

Both may involve stochastic processes
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Closure Schemes In Use (typically to determine
the net mass flux at thebase of the convective
layer)

• Moisture convergence Precipitation (Kuo, 1974-
  for deep precipitating convection)
• Quasi-equilibrium Arakawa and Schubert, 1974 and
  descendants (RAS, Z-M, ZhangMu, 2005)
• Prognostic mass-flux closures (Pan Randall,
  1998ScinoccaMcFarlane, 2004)
• Closures based on boundary-layer forcing
  (EmanuelZivkovic-Rothman, 1998 Bretherton et
  al., 2004)
• Stochastic closures may combine one of the above
  with a stochastic formulation for cumulus
  ensemble properties (e.g. CraigCohen papers,
  PlantCraig)

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Zonally averaged variance of latent heating for
different convective closures and downdraft
evaporation efficiency parameters
(Scinocca McFarlane, 2004)
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Lecture 2

• Cumulus Friction and Energetics
• Parameterization of Boundary Layer Processes

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Cumulus friction and energetics
(usually parameterized)
Take the dot product of with the momentum
eq. It can be shown that
Total energy
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The dissipation heating term is intrinsically
positive. Choose it as
Assume , consistent with top-hat that
is negligible
All parameterized
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From the mean equations (ignoring, for
simplicity, non-cumulus contributions to prime
terms)
(1)
(2)
(for top-hat profiles)
Kinetic energy
cumulus k.e. eq.
(3)
parameterized
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Combine (2) (3)
The R.H.S. should be in flux form. QR is the
radiative flux divergence. Dissipational heating
should be positive. Suggests
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In summary, assuming top-hat cumulus profiles
(a)
(b)
(c)
(d)
(e)
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The cumulus pgf term must be parameterized, e.g.
Gregory et al, 1997 propose the following for the
horizontal component associated with updrafts
For the vertical component, the pgf is often
assumed to partially offset the buoyancy and
enhance the drag effect of entrainment. Since
Let
(Siebesma et al, 2003)
Typical choice
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Parameterization of Boundary Layer Processes in
AGCMs
- The atmospheric boundary layer (ABL) is the
region adjacent to the surface of the earth
within which the exchange of momentum, heat,
moisture, and other constituents between the
atmosphere and the surface takes place mainly by
turbulent processes. - Within a sub-layer near
the surface vertical fluxes of momentum, heat,
and moisture are almost independent of height.
- Within the remainder of the ABL quantities
that are typically conserved under adiabatic
motion are found to be nearly uniform with height
(well mixed)(e.g. potential temperature and
specific humidity for cloud-free conditions or
equivalent potential temperature and total water
mixing ratio in cloudy conditions).
Cartoon of typical structure for a cloud-free
convectively Active ABL
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Cloud-free ABL - neglect effects of water
vapour condensation - ignore (for simplicity)
virtual temperature effects (i.e. water vapour
is passive)
Basic equations for the large (resolved) scale
(1)
(2)
(buoyancy)
The depth of the ABL (and of turbulent regions in
the free atmosphere) is typically small compared
to a density scale-height (e.g.
). Therefore vertical variations in the
background density are often ignored in ABL
modelling.
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Potential Temperature vs Static Energy
If departures from hydrostatic conditions are
small
and
It can also be shown that
Also
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Therefore the R.H.S. of (2) is approximtely
Usual current approach combine a turbulent
kinetic energy (tke) equation with an eddy
diffusivity formulation. Get a tke equation by
forming an equation for and averaging.
Turbulent Kinetic Energy Equation
Approximate tke ( ) equation (e.g. Stull,
1988)
Eddy diffusion approximation for second moments
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Diffusivities
Traditional approach
Dissipation
Physical and dimensional considerations suggest
Specifying the lengths and
coefficients is a closure issue. Large
literature on this topic. Several hypotheses have
been explored in recent work (e.g.
SanchezCuxart, 2004, LenderinkHoltslag, 2004,
and references therein)
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Boundary conditions and constraints
Matching to the surface layer Monin - Obukhov
similarity requires that
where
(Monin-Obukov length)
k von Karmen constant, Pr turbulent
Prandtl number, UL, qL wind speed, potential
temperature at reference level ( )
roughness heights (where surface values
apply).
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The functions are derived from
field campaign observations (e.g. Dyer, 1974) .
Moisture and other tracers treated similarly.
Bulk exchange formulae (resulting from fits to
non-linear solutions)
(Bulk Richardson Number)
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• Limitation
• Dependence of vertical fluxes on local mean
  gradients does not account
• for heat transfer in the convectively active ABL
  where mean gradients are small
• (slightly stable) but upward heat flux is
  positive.
• Requires introduction of non-local effects.
• For a scalar quantity,

• Approaches
• Introduce prognostic equations for second moments
  with associated closure
• Assumptions to derive the nonlocal effects
• (e.g. Deardorf, 1966, Mellor Yamada, 1974,
  Cuijpers Holtslag , 1993,
• Abdella McFarlane, 1997, GryanikHartmann,
  2002, .).
• Simplest formulations give

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(b) Represent non-local transfer effects as being
associated with plume-like Eddies (e.g. Siebesma
et al, 2007)
(From Siebesma et al)
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