Stochastic parameterization of Convective Gravity Waves and effects in models

(1)

1) Motivation and Formalism

2) 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 [email protected]

(2)

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=1

where

The C'_ns generalised the intermittency coefficients of Alexander and Dunkerton (1995), and used in Beres et al. (2005).

1) Motivation and formalism

(3)

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^ê^z/^2Hêî^kⁿ^x^lⁿ ^y^−ⁿ^t^}

WKB passage from one level to the next with a small dissipation :

∣^w^n∣^≤^ws= ²

∣^^kn∣^N ^e

−z/2HS_c∣k^_n∣

k^∗ ,

m= N∣^k∣

 =−k⋅u

w_n , k_n, l_n,_n chosen randomly

Saturation:

w zz= wz ^m^^m^z^^^z^ ^z^^e⁻ⁱ^∫^z^z^z^^m^{z '−i}^^{ }^m³^^{dz '}

S_c, k^* , µ: Tunable parameters EP-flux:

Fzdz= k

∣^k∣^sign^{ }¹^sign^{  }^z2^^z^{⋅ }^z^^Min^∣^F^^^z^^∣^e⁻²^^^m³^^z^,^^r^∣²^^N³^∣^e^−^z^{z /}^H ^S^c²^∣^k^^k^∗2⁴^∣

Critical level EP theorem

with dissipation Breaking

(4)

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 _GWs^t^t⁼^^t^^−t ^t ^{∂ }^∂^ut _GWs^t ^M^^t t ∑_{n '}^M₌₁ _¹ ^{∂ }^Fⁿ

z

∂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!):

C_n²=^^t^^−t ^t M^^t t , F=∑_n=1^∞ ^Cⁿ² ^^Fⁿ

Lott et al. GRL 2012

(5)

Subgrid scale precipitation over the gridbox considered as a stochastic series

P '_r=∑_n=1^∞ ^Cⁿ^Pⁿ^' ^where P_n'=ℜ[^P^ⁿ^e^{i }^kⁿ^⋅^x^−ⁿ^t^] ^taking ^∣^P^ ⁿ^∣⁼^P^r

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 k_n's are chosen randomly between 0 and k*

F_nl=_r k_n

∣k^_n∣

∣k^_n∣²e^−mⁿ²^^z²

N_n³ G_uw^^{R L}^r ^{H c}^W ^p² ^P^r² ^mⁿ⁼^N^^∣^^kⁿⁿ^∣^,^ⁿ^=ⁿ^−^kⁿ^⋅^U

The ω_n's are chosen randomly between 0 and k_nC_max

Tuning parameter max phase speed

z tuning parameter or scale of the heating depth

(6)

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.

G_uw=2.4, S_c=0.25, k*=0.02km^-1,

µ=1kg/m/s

∆t=1day and M=8 Dz=1km (source depth~5km)

(7)

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

(8)

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

(9)

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

(10)

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

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

(11)

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)

(12)

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)

Lott and Guez, JGR13

(13)

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.

(14)

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