Parameterization of convective momentum transport and energy
conservation in GCMs
N. McFarlane
Canadian Centre for Climate Modelling and
Analysis (CCCma)
Parameterization in larger-scale atmospheric modeling
General parameterization problem: Evaluation of terms involving averaged products of (unresolved) deviations from (resolved) larger-scale variables and effects of unresolved processes in larger-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) on larger scales
(c) Cumulus parameterization
Other kinds of parameterization problems: radiative transfer, cloud microphysical processes, chemical processes
** The parameterization scheme(s) should preserve integral constraints of the basic equations – e.g. conservation of energy and momentum
kgPVkfDt
VD ˆ1ˆ
radp FecLDt
Dp
Dt
DTc
1
radFecLDt
Dpgw
Dt
DS
1 TcS p; gw
Dt
D
ceDt
Dqv
P ecDt
Dqw
vRTP )61.1( wvv qqTT
Basic Equations
0
Vt
wkvjuiV ˆˆˆ
][
VtDt
D
Motion
Mass continuity
Thermodynamic
Or:
Vapour
Condensedwater
Equation ofState (ideal gas)
2/VV
K (kinetic energy)
vvvp LqpTcLqTch (moist static energy)
222
z
V
y
V
x
V
D
Energy Conservation (e.g., Gill, 1982, ch. 4)
0])([])([
radFhVpht
KK
For air sm /105.1 25 at 15C , 100hPa
Kolmogorov scales 4/14/13 ; KK UL
These are small for the atmosphere (~ 1mm, .1 m/s) => Permissibleto neglect viscous terms for modeling/parameterization purposes but not to ignore effects/processes that lead to dissipation and associated heating
(k.e. dissipation rate)
Energy conservation:
Hydrostatic primitive equations and parameterization
(GCM resolved) Background state:
-hydrostatically balanced - slowly varying (on the smaller, unresolved horizontal and temporal scales). - deviations from it are small enough to allow linearization of the equation of state (ideal gas law) to determine relationships between key thermodynamic variables:
vTRp
z
T
Tzpg
z
p v
v
11
=>v
v
v TR
g
z
T
g
R
TR
g
z
11
v
v
T
T
p
p
=>
v
v
T
Tgg
z
p
v
v
T
Tg
p
zg
z
p
z
VwpVkfVV
t
VHH
H
ˆ
Bwpwzz
hwQpV
t
phV
t
hRH
)(D
/phqLTc HvvH KKEvv TTgB
GCM equations [hydrostatic p.e + parameterized terms]
0
Vt
2HHH VV
K
z
VVwVVw
zpVV
tH
HHHH
)()()(
KK
Bwpwzz
VVw
z
hwVVw
zQp-V
tH
HR
)()()]([(
DEE )
(+ other terms)
Boville and Bretherton (2003) in the context of turbulent transfer (PBL) in CAM2:
2
z
VK
z
VVw H
mH
D
Neglect the terms Bwpwz
)(
Include a tke equation (Stull,1988; Lendrick & Holtslag,2004)
D
Bwz
VVwpwVVw
zDt
D
2/e
dd lc /e 2/3D
2e VV
emm lK
AL
Liic AtzA ),()(
Li
ie AA )1(
Cumulus effects on the larger-scales
Qz
wV
t H
)(
)()(
Start with basic equations, e.g. for a quantity
(i) Average over the larger-scale area (assuming fixed boundaries):
othersz
w
z
wQ
z
wV
tccc
))(())(()()(
)( **
Mass flux (positive for updrafts): cc wM
ec QQQ ))(1()( “Top hat” assumption: 0)( ** cw
Quasi-steady assumption: effects of averaging over a cumulus life-cycle can be represented in terms of steady-state convective elements .
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]
ec )1( 1 cww
0
ED
z
M ccc wM
kp
zBpVEVD
z
VM
c
ccHEDc ˆ)()(
c
c
HcEDc p
zwpVMBEhDh
z
MhD
)(
(usually parameterized)
(2) Average over convective areas (updrafts/downdrafts). Note:
cc
c
HEDc wB
p
zwpVE)D
z
MD
KKK
()(
(K.E.1)
wM
cccc VwVMVM
)( cccccccc MVMVMMM KKKK
)(
2/ccc VV
K 2/ VV
K 2/)(2/ DDDD VVVV
K
top-hat: neglect 2
2
Ec VVE
cD
2
ˆ2
Ec
c
HcccEccc
VVE
p
zkpVBMEhDh
z
hM
K.E. equation from the cumulus momentum equation:
22
22
EcDcED
ccccVV
EVV
DEKDKz
VVw
z
VVwKM
cDc
ccc BMKDz
VVMKMKM
z
c
HcDcEc p
zkpV
VVD
VVE
ˆ*22
22 D
cc
c
Hc BMp
zkpV
ˆ
*()
(K.E.2)
(K.E.1 – K.E.2):
and cD VV
=> (top hat)
From the mean equations:
z
VVMpVkfVV
t
V HccHH
H
)(ˆ
cc
c
Hcc
RH BMp
zkpVV
z
hhMQpV
t
phV
t
h
ˆ
D
Kinetic energy:
c
ccHc
cH
c
ccHccHHH
p
zwpVV
VVE
VVDBMKKM
zpVKV
t
K
22)(
22
)1(eqVH
+ cumulus k.e. eq.
c
Hc
c
H
p
zkpVV
p
zkpVV
ˆˆ
(for top-hat profiles)
(1)
22
22)(
)()(
wwE
VVED
z
KhKhMQKhV
t
pKh
cHc
HcccRH
H
D
Total energy::
The R.H.S. should be in flux form. QR is the radiative flux divergence.
2
2
22)(
wwE
VVED cH
c
D
cc
c
HHc
ccRH
BMp
zkpVV
z
hhMQpV
t
phV
t
h
ˆ
D
In summary, assuming top-hat cumulus profiles
2
2
Ec
c
HcccEccc
VVE
p
zpVBMEhDh
z
hM
22
ˆ
22
HcH
c
cc
c
HHccc
RH
VVE
VVDBM
p
zkpVV
z
hhMQpV
t
phV
t
h
0
ED
z
M c
kBp
zkpVEVD
z
VMc
c
HEccc ˆˆ)(
z
VVMpVkfVV
t
V HccHH
H
)(ˆ
(a)
(b)
(c)
(d)
(e)
)( VVE
)( hhE
Parameterization of convective (horizontal) momentum transport in GCMs
Schneider & Lindzen, 1976 (SL76) - ignored the PGF
Helfand, 1979 – Implemented SL76 in the GLAS GCM
Zhang & Cho, 1991 (ZC91), Wu&Yanai, 1993– parameterized PGF using an idealized cloud model
Zhang & McFarlane, 1995 (JGR) – ZC91 in CCC GCMII
Gregory et al., 1997 (G97) – parameterized the PGF on the basis of CRM results (implemented in the UKMO GCM)
Grubisic&Moncrieff, 2000; Zhang & Wu, 2003 – evaluated G97 and other possibilities from CRM results
Richter & Rasch, 2008 – effects of SL76 and G97 using NCAR CAM3
The cumulus pgf term must be parameterized, e.g. Gregory et al, 1997propose the following for the horizontal component associated with updrafts:
z
VMp uuH
7.
For the vertical component, the pgf is often assumed to partially offsetthe buoyancy and enhance the drag effect of entrainment. Since cE ww
c
c
ccc Bp
zDwwM
z
c
c
cc
c Bp
zEw
z
ww
Let
1/cc
c
BaEwp
z
Typical choice: 1a 20 (Siebesma et al, 2003)
(H)
(V)
Need to ensure that (H) and (V) are consistent with eachother.
Gregory et al. parameterization for updraft PGF in
)()1( VVDz
VM
t
Vuuu
u
(Zhang&Wu, 2003: )
Grubisic & Moncrieff alternative - adddependence on detrainment for PGF:
z
VuMVuVD
z
VuVuM
u
)1()()(
<=>
Zhang & McFarlane, 1995
(JJA)
ZM95
(Richter&Rasch, 2008; CAM3, Z-M) SL: = 0; GR: = .7
ZM95
Control(no CMF)
=.2
Summary
1. Momentum transport by convective clouds has important effects in GCM simulations of the large-scale atmospheric circulation
2. Introduction of parameterized momentum transport by cumulus clouds may result in an unbalanced total energy budget if associated dissipational heating effects areignored
