Transcript Document

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
microphysical processes*, chemical processes
transfer,
cloud
Ae  (1   i ) AL
( Ac )i   i ( z, t ) AL
AL
i
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:
1
 ( x, y , z , t ) 
Lx Ly Lz
t  2 x  Lx 2 y  L y 2 z  Lz 2
 

 dzdydxdt 
t  2 x  Lx 2 y  L y 2 z  Lz 2
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.

Typically, if the variable,

has the following governing equation:

D


 V  
;  Q,
Dt
t
And the mass continuity equation is:
    
 


   V  0;
t
Then applying the averaging operation gives, approximately:
 
1
 V      ( V  )  Q
t

if

 1

In cases to be considered (e.g. cumulus parameterization)    V   
Determining this term is a goal of the parameterization
(  w )
z
Atmospheric Equations



DV
1
2
ˆ
ˆ
 fk V   P  gk  V
Dt



   V  0
t
Motion
Thermodynamic

DT 1 Dp
cp

 Lc  e     Frad  k 2c pT  D
Dt  Dt
DS
1 Dp
 gw 
 Lc  e  k 2T  D;
Dt
 Dt
Vapour
Condensed
water
Equation of
State (ideal gas)

V  iˆu  ˆjv  kˆw
 
Mass continuity
Or:
D
 
[
  V  ]
Dt t
S  c pT  
Dq v
 ec
Dt
Dq w
 c  e  Pr
Dt
P  RTv
Tv  T (1  .61qv  qw )
D
 gw
Dt
Energy Conservation (e.g., Gill, 1982, ch. 4)
 
E  V V / 2
h  c pT  Lqv  

(kinetic energy)
(moist static energy)
D
E  h   p   2E  k 2 (c pT )    Frad
Dt
t




[  (E  h)  p]    [ V (E  h)  E  k(c pT )  Frad ]  0
t
 V 2 V 2 V 2 

D  


 x
y
z 


For air
  1.4 105 m2 / s
, 
Molecular dynamic and kinematic viscosity
at 15C , 100hPa
Kolmogorov scales (for which viscosity and
dissipation are independent parameters):

LK   3 D

1/ 4
;U K  D
1/ 4
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
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 quasibalanced 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:
p  RTv
p
  g 
z
 1  1 Tv 

p

  z Tv z 
p   Tv
 
p  Tv
=>
1 
g  R Tv 
g
1 
  

 z
RTv  g z 
RTv
=>
Tv
  p 
 p

 
 g         g
z   
Tv
 z


 p  
Tv
 
 g      g
Tv
   z
Using these results leads to the following:


   p 
Tv 
DV
ˆ
ˆ


 (1    )
  H  p   k     g   fk V   H p
Dt
Tv 
 z   
Terms in curly brackets:



negligible for the parameterized scales
but not for the resolved scales



{  (1     ) }     (1     )V  0
t
 T    p   p 
Dh  p 



 V   H p    w g v       V   H p   Q
Dt  t


 Tv z     t
 (1     )
(h  c pT  Lv q  )
Terms involving    will also be neglected compared to unity. The approximate mass continuity
equation which will be used is:
 


{  }    V  0
t
Upon using this continuity equation to develop the flux-form equations and averaging:
 






 VH
  wV 
   V VH  fkˆ V  H p 
t
z




+ other such (horizontal) terms


h 
  wh 
 p 

   V h    V   H p   Q 
 (wp)   wB 
t
z
z
 t

 
Q  QR  D
B  gTv Tv
+ other…
Parameterization of the effects of Moist
Convection in GCMs
• Mass flux schemes
– Basic concepts and quantities
– Quasi-steady Entraining/detraining plumes (Arakawa&Schubert
and similar approaches)
– Buoyancy sorting
• Raymond-Blythe, Emanuel
• Kain-Fritsch
– Closure Conditions, Triggering
• Adjustment Schemes
– Manabe
– Betts-Miller
Traditional Assumptions for Cumulus Parameterization:
1. Quasi-steady assumption: effects of averaging over a cumulus lifecycle 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 Salzen&McFarlane
(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]
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):


   p 
Tv 
( V )
ˆ
    VV   H  p  k     g 
t
Tv 
 z   


 w
  (  w)

  H  V 
 0   H V 
t
z
z
  p 
 Tv   p 
 h

    hV  
 V   H p    w g     Q
t
 t

 Tv z   


plus similar equations for vapour, condensed water, and other scalar quantities
For the traditional formulation ignore crossed-out terms
Ae  (1   i ) AL
( Ac )i   i ( z, t ) AL
AL
i
Spatial Averages
For a generic scalar variable,
Large-scale average:
:
 1 

  
dA


 AL  A
L
Convective-scale average
(for a singlecumulus up/downdraft) :

1
 e  
 AL  Ac
Environment average
(single convective element):
Where
 1 
 c     dA
 Ac  Ac
 c ,e /   O (1)
  Ac AL  1
   c  (1   ) e
Vertical velocity:
        *  (1   ) ˆ
w  wc  (1   )we
Ensemble of cumulus clouds:

  dA
 Ae
   i
i
wc  w , we
   i i  (1   )  e
i
Cumulus effects on the larger-scales
Start with a general conservation equation for


 (  )
 ( w )
  H  ( V ) 
 Q
t
z
Plus the assumption:

(similar to using anelastic assumption for convective-scale motions)
(i) Average over the large-scale area (assuming fixed boundaries):

(  wc (  c   )) (   ( w*  * ) c )
(  )
( w  )
   ( V  ) 
 Q 

 others
t
z
z
z
Mass flux (positive for updrafts): M c  wc
Also: Q   (Q )c  (1   )(Q )e ; “Top hat” assumption:
(w*  * )c  0
In practice (e.g. in a GCM) the prognostic variables are also implicitly time
averages over convective cloud life-cycles
(ii) Apply cumulus scale sub-average to the general conservation equation,
accounting for temporally and spatially varying boundaries (Leibnitz rule):
(  c ) 
(  [wc c  (w*  * )c ])

vn b dl 
  (Q )c

t
Ac 
z
Mass continuity gives:
( awc )
 

v
dl

 0 ; vn
n

t
Ac 
z

the outward directed normal flow velocity
(relative to the cloud boundary)
Entrainment (inflow)/detrainment (outflow):
E

Ac
 v [1  H (v )]dl
n
Define:  E 
Top hat:
D
n

EAc

Ac
n
 D  c
n
D 
c
;
{
 v H (v )dl
 vn  b [1  H (vn )]dl
 E  e  
1; f  0
H( f ) 
0; f  0
;

DAc
 v  H (v )dl
n
c
b
n
Summary for a generic scalar,  :
(steady and top hat in cloud drafts: neglect crossed-out terms)

 


[ M c (  c   )    w*  * c ]
   
  w  
   V  

 Q  other
t
z
z
 M c 
   
DE
0
t
z
   c 
 M c  c    ( w*  * ) c
 D D  E 
  Q c
t
z


 

   
  w  


   V  
 Mc
 D D     (1   )Q e  other
t
z
z
When both updrafts and downdrafts are present, both entraining environmental air:
M c   wc  M u  M d ; E  Eu  Ed ; D  Du  Dd
M c  c  M u  u  M d  d ; D c  Du  u  Dd  d
Basic cumulus updraft equations (top-hat, traditional)
{Dry static energy: s=CpT+gz; Moist static energy : h=s+Lq; M u  wu }
M u
0
z
 M u su 
Du su  Eu s 
 Lcu
z
 M u qu 
Du qu  Eu q 
 cu
z
 M u lu 
Du lu 
 cu  Pu
z
 M u hu 
Du hu  Eh 
0
z
Du  Eu 
v  Tv ( po p) ;   R / c p ; T  T (1  .608q  l )
v
mass conservation
dry Static Energy
vapour
condensate
moist Static Energy
(virtual temperature)
Entrainment/Detrainment
Traditional organized (e.g.plume) entrainment assumption:
E    vn [1  H (vn )]dl  vin  Pc  wc Pc
Pc   dl (draft perimeter)
c
c
 P 
2
E   c   wc  M c 
wc
Rc
 Ac 
Arakawa & Schubert (1974) (and descendants, e.g. RAS, Z-M):
-  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  )
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
(Raymond&Blythe, Emanuel, Emanuel&Zivkovic-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
zt
zb
Determining fractional entrainment rates (e.g. when
hi
 i h  hi 
z
at the top of an updraft)
hi ((zt )i , i )  h* ((zt )i )
hi ( zb )  h ( zb )
h* (( zt )i )  h ( zb ) exp[ i (( zt )i  zb )]  i
Tc  Te
( zt ) i
 h ( z) exp[  ( z  ( z ) )]dz
i
t i
zb
Note that since updrafts are saturated with respect to water vapour above the LCL:
*
L
hi  h*
L

q
 Ti  T   (qi  q * (T , p))  (Ti  T )(1 
)  O(Ti  T ) 2
c 
cp
c p T
 p
This determines the updraft temperature and w.v. mixing ratio given its mse.
Fractional entrainment rates for updraft ensembles
(a) Single ensemble member detraining at z=zt
zb  z  zt 
Eu   ( zt )M u ; Du  0
M u  M b exp ( zt )(z  zb )
Detrainment over a finite depth
zt : D( zt )  M u ( zt ) / zt
(b) Discrete ensemble based on a range of tops
M u   M ( z, i ); i   ((zt )i )
i
Eu   i M i
i
M u hu   M i hi
M i 
Du  
i i zt
i
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
v saturated (cloudy)
v c
positively
buoyant
v e
fc
0
f*
1
f
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 Mcdz of environmental air with an equal amount of cloudy air. K-F
assumed that all of the negatively buoyant mixtures (f>fc) 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:
fc
E  2o M u  fP( f )df
0
1
D  20 M u  (1  f ) P( f )df
fc
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.
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
Closure Schemes In Use
(typically to determine the net mass flux at the
base of the convective layer)
• Moisture convergence~ Precipitation (Kuo, 1974- for deep
precipitating convection)
• Quasi-equilibrium [Arakawa and Schubert, 1974 and descendants
(RAS, Z-M, Zhang&Mu, 2005)]
• Prognostic mass-flux closures (Pan & Randall,
1998;Scinocca&McFarlane, 2004)
• Closures based on boundary-layer forcing (Emanuel&ZivkovicRothman, 1998; Bretherton et al., 2004)
• Stochastic closures may combine one of the above with a stochastic
formulation for cumulus ensemble properties (e.g. Craig&Cohen
papers, Plant&Craig)
Zonally averaged variance of
latent heating for different
convective closures and
downdraft evaporation
efficiency parameters
(Scinocca & McFarlane, 2004)
Lecture 2
• Cumulus Friction and Energetics
• Parameterization of Boundary Layer
Processes
Cumulus friction and energetics
M c  wc
M c
DE 0
z
(usually parameterized)




   p    ˆ
( M cVc )
 DVc  EVE    H pc     Bc       k
z
 z    c 


 M c hc 
  p  
 Dhc  EhE   M c Bc   V   H p   w     Dc
z
z     c

Take the dot product of

Vc with the momentum eq. It can be shown that
M c K c 
 DKc  EKE  E
z
 
Vc  VE
2
2

  p  
 Vc    H p  kˆ     M c Bc
z    c

Total energy:

M c Kc  hc   DK  hD  E( K  h) E  E
z
 
Vc  VE
2
2
   
  p   
ˆ



  (V  Vc )    H p  k     Dc

z     c


The dissipation heating term is intrinsically positive. Choose it as
Dc  E


Vc  V E
2
2
Assume , consistent with top-hat that
   
  p   
ˆ

  V  Vc    H p   k   
z     c




is negligible
 
M c hc 
  p  

 Dhc  EhE  M c Bc  Vc    H p  kˆ     E
z
z    c

All parameterized
 
Vc  VE
2
2
From the mean equations (ignoring, for simplicity, non-cumulus contributions
to prime terms):
 


 



 M c (Vc  V ) H
 VH
   V VH  fkˆ V  H p 
t
z





(1)

M c hc   h      
h 
  p  
 p 

    V h    V  H p   Q 
  V  V    H p  kˆ     M c Bc (2)
t
z
z    
 t



c



Q  QR  D



   

 





p
 V  V    H p  kˆ      Vc  V
z    


c
 
ˆ   p   (for top-hat profiles)

p


k
H
z    c


Kinetic energy: VH  eq(1) + cumulus k.e. eq.


 K H 

    V K H  V   H p  M c ( K c  K H ) M c Bc  D
t
z
  
   p   
   Vc  V   H pc   wc     
(3)

 z    c 




parameterized
 
Vc  V
2
2
H
E
 
Vc  V
2
2
Combine (2) +(3):



 M c hc  K c   h  K H  
  (h  K H ) 
 p 
    V (h  K H )     QR 
t
z
 t 
  2



2
Vc  V
w  w   


H
 D  ( D  E )
E c

2
2


 




The R.H.S. should be in flux form. QR is the radiative flux divergence.
Dissipational heating should be positive. Suggests :
 

Vc  V

D  ( D  E )
2


2
H

w  w    D  D
E c
c
E

2

2


 M c hc  h 
  h 
 p

  V h 
V  H p   Q 

t
z
 t




  p  

   M c Bc
 Vc  VH    H p   kˆ
z     c

 


In summary, assuming top-hat cumulus profiles
M c
DE 0
z
(a)
 
M c hc 
  p  


 Dhc  EhE  M c Bc  Vc    H p       E
z
z    c

 
Vc  VE
2
2




 ( M cVc )
  p  
ˆ


 DVc  EVE     H p   k      Bc kˆ
z
z     c

(c)
(b)
(hE  h )


(VE  V )

  
 M c hc  h 
  h 
  p  
 p 

    V h    V   H p   QR 
  Vc  VH    H p   kˆ   
t
z
z     c
 t


 2
  2
Vc  V
Vc  VH
(d)
H
 M c Bc  D
E
2
2

 



 



 M c (Vc  V ) H
 VH
   V VH  fkˆ V  H p 
t
z




(e)
The cumulus pgf term must be parameterized, e.g. Gregory et al, 1997
propose the following for the horizontal component associated with updrafts:

V
   H pu  M u
z
  .7
For the vertical component, the pgf is often assumed to partially offset
the buoyancy and enhance the drag effect of entrainment.
Since wE  wc

 

M c wc   Dwc       p     Bc
z
 z     c
   p  
wc
  wc
 Ewc           Bc
z
 z     c
Let
   p  
    aEw c   Bc / 1   
 z     c
  
Typical choice:
a 1
0 2
(Siebesma et al, 2003)
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
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:
 






 VH
  wV 
   V VH  fkˆ V  H p 
t
z



(1)


S 
  wS  
 p 



    V S     V  H p   Q 
 (wp)   wB 

  t
t
z
z

B  gTv Tv  gT  / T
(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. h / H  .1 ). Therefore
abl
vertical variations in the background density are often ignored in ABL modelling.
Potential Temperature vs Static Energy
  T  p0 / p 
(  R / c p )
S  c pT  
If departures from hydrostatic conditions are small:


  p0   T g  1  p0  S
  gz and
 

  
z  p   z c p  c p  p  z
It can also be shown that



d  p0   dT 1 dp   p0   dS
1  p




   



wB




(
V
p
)

V


p
H

dt  p   dt  dt   p   dt
  t

Also:

p  
p 
c p    0   c pT   

 p 
 p    wT   wp

w 
 c p w    0  c p


g
z
z
z 

 p 

 w   p0   wS   wp

 cp
   

 wB 
z
z

 p   z


Therefore the R.H.S. of (2) is approximtely Q  c p T
 w  z
Usual current approach: combine a turbulent kinetic energy (tke) equation
with an eddy diffusivity formulation. Get a tke equation by forming an equation for
 
V   V  and averaging.
Turbulent Kinetic Energy Equation:
Approximate tke (

e ) equation: (e.g. Stull, 1988)

 V 
 
e
 wV  

wV   V  / 2  wp /   g w  /   Dk
t
z z


 
 
E   V  V 2  V   V  / 2   E  e 

Eddy diffusion approximation for second moments:


wV    K m V z
w    K H  z
 
wV   V  / 2  wp /    K e e / z
Diffusivities:
Traditional approach
K m, H ,e  lm, H ,ecm, H ,e e
Dissipation:
Physical and dimensional considerations suggest
D  e /  d  cd e 3/ 2 / ld
Specifying the lengths lm, H ,e,d and coefficients is a closure issue.
Large literature on this topic. Several hypotheses have been explored in recent
work (e.g. Sanchez&Cuxart, 2004, Lenderink&Holtslag, 2004, and references therein)
Boundary conditions and constraints
Matching to the surface layer:
Monin - Obukhov similarity requires that:

2 1/ 2
 /   [(u ' w' )  (v' w' ) ]
2
 u*2
( w' ')  u**
where
u* 
kU L
ln(z L z0 )  m ( z L L)  m ( z0 L)
* 
(k / Pr)( L   s )
ln(z L zt )  H ( z L L)  H ( zt / L)
vu*3
L
kg(w'v ') s
(Monin-Obukov length)
k : von Karmen constant,
Pr : turbulent Prandtl number,
UL, L: wind speed, potential temperature at reference level ( z L )
z0 , zt roughness heights (where surface values apply).
The functions  m , H 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):
RiB  gzL [(v ) L  (v ) s ] [(v ) s U L2 ]
(Bulk Richardson Number)
u*2  C D Fm ( z L z 0 , RiB )U L2
u* *  C H FH ( z L z 0 , z L / z t , RiB )U L ( L   s )
u* q*  C Q FQ ( z L / z 0 , z L / z q , RiB )U L (q L  q s )
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,

:



w    K 
 w  nl
z
Approaches:
(a) 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, Gryanik&Hartmann, 2002, ….).
Simplest formulations give:
(w ) nl  K H  cg
(b) Represent non-local transfer effects as being associated with plume-like
Eddies (e.g. Siebesma et al, 2007)
(From Siebesma et al)
 w nl  M (u  )
M
 (  d ) M
z
 u
  ( u )
z