1)
Motivation and Formalism2) Off-line test with ERAI and GPCP 3) Online results with LMDz
4) Summary and perspectives
Stochastic parameterization of Convective Gravity Waves and effects in models
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 concept (even though they are probably related to the model gridscales x δy). 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 :
∣wn∣≤ws= 2
∣kn∣N e
−z/2HSc∣kn∣
k∗ ,
m= N∣k∣
=−k⋅u
w
n, k
n, l
n,
n chosen randomlySaturation:
w zz= wz
mmzz ze−i∫zzz
mz '−i m3
dz 'S
c, k
*, µ :
Tunable parameters EP-flux:1) Motivation and formalism
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
Few waves (say M=8) are launched at each physical time step (δt=30mn), but their effect is redistributed over a longer time scale (∆t=1day), so around 400 waves are active at the same time:
This excellent spectral resolution is the major benefit of the method.
M and ∆t are two extra tunable parameters (Could be random numbers as well)
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 Fn1) Motivation and formalism
Lott et al. GRL 2012
Subgrid scale precipitation over the gridbox considered as a stochastic series
P 'r=
∑
n=1∞ CnPn' where Pn'=ℜ[
Pnei kn⋅x−nt]
taking∣
P n∣
=PrThe 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
Nn3 Guw
R Lr H cW p
2 Pr2 mn=N∣knn∣,n=n−kn⋅UThe ωn's are chosen randomly between 0 and knCmax
Tuning parameter max phase speed
z tuning parameter or scale of the heating depth
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)
Precipitations and Gravity wave drag averaged over 1week (1-7 January 2000) GPCP datas and ERAI
The CGWs drag is much larger
Than the drag due to Hines (1997) in the QBO region but
smaller in the mesosphere.
Launching few large GWs than a more uniform ensemble of smaller waves make that the non-orographic
GWs Scheme act at lower level
then when they are not keyed To convection.
Hines (1997) (CI: 5m/s/d)
CGWs (CI: 2m/s/d)
QBO region (zoom) CI=0.1m/s/dayCI=0.02m/s/day
Wind (contour), non-orog GWDs (colors)
2) Offline tests with ERAI and GPCP
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
Model set-up:
The model vertical resolution is increased up to 80 levels compared to the 50 stratospheric level version documented in Lott et al. 2005, or to the 39 levels used for CMIP 5 (Maury et al. 2011) Horizontal resolution, 96x95 grid.
GWs setup:
Launch altitude of the waves 500hPa,
/home/flott_local/lmdz/TEST_DISVERT/Deltaz_50lev.jpg
3) On-line results with LMDz
Results for the Equatorial winds:
QBO of irregular period around 26month, 20% too small amplitude
Westerly phase lacks of connection with the stratopause SAO
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
Relatively good spread of the periods taking into account that it is a forced
simulation with climatological SST (no ENSO)
Periods related to the annual cycle (multiples of 6 months) are not favoured:
probably related to the weak relations with the SAO
Histogram of QBO periods
3) On-line results with LMDz
Lott and Guez, JGR13
MERRA
Easterlies at 20hPa Westerlies at 20hPa
LMDz
3) On-line results with LMDz
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 withou 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: Very high spectral resolution, which is good for the treatments of critical levels (an important aspect of the QBO dynamics).
Very cheap cost (but about the same cost as the Hines (1997) Parameterization schemes).
Easy to relate to the convective sources, seems benefitial to do so
More fundamental: 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 likely to have a Fourier representation that resemble to the Spectra 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.
4) Summary and perspectives
Reconcile the “multiwaves” techniques and the “globally spectal” techniques, via evaluation of spectra issued from the multiwaves techniques, compare of the stresses produced by
both, test in 1D models with input from GPCP datasets.
How to relate the subgrid-scale precipitations to the gridscale ones: analyse the spectra in high resolution runs and used them to relate the large-scale to the small scale? Build
surrogate statistic processes that mimics the subgrid-scale precipitations and Fourier transform them!
Impact on the QBO, off having the sources and when the climate change (2xC02 experiments) Can these replace the non-orographic GWs schemes that impose a uniform background
everywhere? LMDz still needs Hines (1997) but can not we get rid of it by specifying sources from fronts?
Test at lower vertical resolution to be compatible with CMIP's type of configurations
4) Summary and perspectives