1)
Motivation and Formalism2) Off-line test with ERAI and GPCP 3) Online results with LMDz
4) Summary and perspectives
A stochastic parameterization of the Gravity Waves due to convection and and impact on the equatorial
stratosphere
F. Lott,
Laboratoire de Météorologie Dynamique, Paris flott@lmd.ens.fr
Classical arguments: see Palmer et al. 2005, Shutts and Palmer 2007, for the GWs: Piani et al. (2005, globally spectral scheme) and Eckeman (2011,
multiwaves scheme)
1) The spatial steps ∆x and ∆y of the unresolved waves is not a well defined. The time scale of the GWs life cycle t is certainly larger than the time step (t) of the model, and is also not well defined.
2) The mesoscale dynamics producing GWs is not well predictable (for the mountain gravity waves see Doyle et al. MWR 11).
These calls for an extension of the concept of triple Fourier series, which is at the basis of the subgrid scale waves parameterization to that of stochastic series
:
w '=
∑
n=1∞ Cnw 'n∑
n∞=1 Cn 2=1where
The
C'
ns generalised the intermittency coefficients of Alexander and Dunkerton (1995), and used in Beres et al. (2005).1) Motivation and formalism
For the w'n we next use the linear WKB theory of hydrostatic monochromatic waves, and treat their breaking as if each w'n was doing the entire wave field (using Lindzen (1982)'s criteria for instance): they are viewed as independent realizations.
w '
n=ℜ { w
n z e
z/2He
iknxln y−nt}
WKB passage from one level to the next with a small dissipation :
m= N∣k∣
=−k⋅u
w
n, k
n, l
n,
n chosen randomlyS
c, k
*, µ :
Tunable parameters EP-flux:Fzdz= k
∣k∣sign
1sign z2z⋅ z
Min
∣Fz∣e−2m3z,r∣2N3∣e−zz /H Sc2∣kk∗24∣
Critical level EP theorem
with dissipation Breaking
Stochastic parameterization of GWs due to convection and impact on the Eq. Strato.
1) Motivation and formalism
Few waves (say M=8) are launched each δt=30mn, but their effect is redistributed over
∆t=1day: around 400 waves are active at the same time!
This excellent spectral resolution is a benefit of the method.
The redistribution is done via an AR-1 protocol which forces to keep the GWs tendency, e.g. 2 other 3D fields that will be needed at the re-start of the model:
∂∂ut
GWstt=t−t t
∂ ∂ut
GWst Mt t ∑n 'M=1 1 ∂ Fnz
∂z
At each time we promote M new waves and degrade the probability of all the others by the AR-1 factor (∆t-δt)/∆t (they loose their AAA!):
Cn2=
t−t t
Mt t , F=∑n=1∞ Cn2 FnLott et al. GRL 2012
1) Motivation and formalism
Subgrid scale precipitation over the gridbox considered as a stochastic series
P 'r=
∑
n=1∞ CnPn' where Pn'=ℜ[
Pnei kn⋅x−nt]
We take
∣
P n∣
=Pr The subgrid scale standard deviation of the precipitation equals the gridscale mean: White noise hypothesis!Distributing the related diabatic forcing over the vertical via a Gaussian function yields the EP-flux at the launch level (see also Beres et al.~(2004), Song and Chun (2005):
G
uw tuning parameter of the CGWs amplitude The kn's are chosen randomly between 0 andk*
Fnl=r kn
∣kn∣
∣kn∣2e−mn2z2
N n3 Guw
R Lr H cWp
2Pr2 mn=N∣kn∣n
,n=n−kn⋅U
The ωn's are chosen randomly between 0 and knCmax
Tuning parameter max phase speed
z tuning parameter or scale of the heating depth
Stochastic parameterization of GWs due to convection and impact on the Eq. Strato.
1) Motivation and formalism
a) Precipitation Kg.s-1.day-1
b) Surface Stress amplitude (mPa)
30S 30N Eq 60N 90N
60S 90S
30S 30N Eq 60N 90N
60S
90S0 60E 120E 180 120W 60W 0
0
180 120W 60W
120E 60E
0
30S 30N Eq 60N 90N
60S 90S
30S 30N Eq 60N 90N
60S 90S
7 6 5 4 3 2 1 0
8 10 4 2
0 6
mPa Kg.s-1.day-1
30 26 22 16 12 8 4
42 36 30 24 18 12 6
Precipitations and surface stresses averaged over 1week (1-7 January 2000) Results for GPCP datas and ERAI
2) Offline tests with ERAI and GPCP
Why precipitations instead of heatings?
a) Models are tuned to produced realistic precips b) global datasets exist covering long periods, d) there are known uncertainties on model vertical profiles of heatings
The CGWs stress is now well distributed along where there is strong precipitations
It is stronger on average in the tropical regions, but quite significant in the
midlatitudes.
The zonal mean stress comes from very large values issued from quite
few regions.
Guw=2.4, Sc=0.25, k*=0.02km-1,
µ=1kg/m/s
∆t=1day and M=8 Dz=1km (source depth~5km)
On the benefit of having few large GWs rather than a large ensemble of small ones:
Offline it happens that the scheme can be used taking for the precip the zonal and temporal mean values.
Here are only shown the stress and tendencies of the waves with positive phase speed.
2) Offline tests with ERAI and GPCP
Lott and Guez, JGR 2013
CGWs stress
CGWs drag
Same zonal mean stress
Real precip. Stress amplitude (CI=2mPa) Uniformized precip. Stress amplitude (CI=2mPa)
Eq 30N 60N 90N
30S 60S
90S0 60E 120E 180E 60W 120W 0 60E 120E 180E 60W 120W
Real precip. du/dt *e(-z/2H), CI= 0.1 m/s/d Uniformized precip. du/dt *e(-z/2H), CI= 0.1 m/s/d
Eq 30N 60N 30S
60S 60S 30S Eq 30N 60N
More drag near and above stratopause Slightly less drag in the QBO region
50 60
40 30 20 10
50 60
40 30 20 10
0.15 0.25 0.35 0.45
0.05 0.05 0.15 0.25 0.35 0.45
LMDz version with 80 levels, dz<1km In the stratosphere
QBO of irregular period with mean around 26month, 20% too small amplitude
Westerly phase lacks of connection with the stratopause SAO
Also, no negative impacts on the SAO, subtropical winds,
ect...
(even slightly positive improvement of the SAO phase, and of the subtropical summer eastearlies
in the mesosphere)
3) On-line results with LMDz
Lott and Guez, JGR13
a) LMDz with convective GWs LMDz+CGWs
b) MERRA
1000 100
10 20 1 0.1 1000 100
10 20 1 0.1
1990 1992 1994 1996 1998
2 4 6 8
No negative impacts on other climatological aspects of the model SAO:
January zonal mean zonal wind
CGWs improve the phase at the stratopause
CGWs reduce easterly biases in the subtropics summer mesosphere
(1) LMDz+CGWs
MERRA
(2) LMDz without CGWs
LMDz+CGWs
MERRA
LMDz without CGWs
Impact: (1) – (2)
100 10 1 0.1
100 10 1 0.1
100 10 1 0.1
100 10 1 0.1
1000
100 10 1 0.1
1000
100 10 1 0.1
1000 100
10 1 0.1
Eq 30N 60N 1000 30S
60S
90S 90N 90S 60S 30S Eq 30N 60N 90N
Eq 30N 60N 30S
60S
90S 90N
Eq 30N 60N 30S
60S
90S 90N
FEB
1979 APR JUN AUG OCT DEC
FEB 1979
APR JUN AUG OCT DEC
FEB 1979
APR JUN AUG OCT DEC
20 60 85
-10 -35 -50 -75
-90 5 45
20 60 85
-10 -35 -50 -75
-90 5 45 -90 -75 -50 -35 -10 5 20 45 60 85
20 60 85
-10 -35 -50 -75
-90 5 45
15 35 45 -5
-15 -25 -35
-45 5 25 55
-55
3) On-line results with LMDz
Periodicities and sensitivity tests
We change the dissipation on the vorticity only
Effect of dissipation Eq. wind at 20hPa
MERRA, High dissip, low dissip In its set-up for LMDz,
Hines decreases the QBO period By around 2-3 months
Guw=2.2 (instead of 2.4) increases the QBO period of 2 months (no effect on the histogram)
3) On-line results with LMDz
Lott and Guez, JGR13
ERAI 21, 11 cases
LMDz+CGWs 10 cases
LMDz without CGWs 10 cases 20S
20N
Eq
20S 20N
Eq
20S 20N
Eq
80E 0
80W 40W 40E
80E 0
80W 40W 40E
80E 0
80W 40W 40E
Composite of Rossby-gravity waves with s=4-8 Temp (CI=0.1K) and Wind at 50hPa & lag = 0day
Equatorial waves:
Remember also that when you start to have positive zonal winds, the planetary scale Yanai wave
is much improved
(the composite method is described
in Lott et al. 2009)
3) On-line results with LMDz
Lott and Guez, JGR13
Advantages of stochastic multiwaves methods:
Very high spectral resolution, which is good for the treatments of critical levels; Very cheap cost; Easy to relate to the convective sources.
More fundamental concerning the stochastic approach:
there is no reason to treat the mesoscale dynamics as predictable from the large-scale flow and using few tunable parameters.
Defect: What is true for critical levels is not for the waves breaking far from them, linear theory is not adapted to describe it. In this sense, the globally spectral methods (Hines (1997), Warner and McIntyre (1997)) are may be more adapted.
But: Imposing spectral shapes at all time is also quite incorrect since:
Spectra are the superposition of individual periodograms, each realisation
is not supposed torealize the entire spectra, this becomes a problem when we relate to sources.
Observations with constant level balloons aften show that the waves in the stratosphere have quite narrowbanded spectra (Hertzog et al. 2008), and are highly intermittent.
Prospective: Reconcile the “multiwaves” techniques and the “globally spectal” techniques, via evaluation of spectra issued from the multiwaves techniques.