Atmospheric flows are governed by the equations of fluid dynamics. These equations are nonlinear. But because atmospheric flows are inhomogeneous and anisotropic, the nonlinearity may manifest itself only weakly through interactions of non-trivial mean flows with disturbances or eddies. In such situations, the quasi-linear (QL) approximation, that retains eddy-mean flow interactions but neglect eddy-eddy interactions, hold promise in resolving large-scale atmospheric dynamics. The statistics of the QL system corresponds to closing the hierarchy of statistical moments at the second order. Hence, exploring QL dynamics paves the way for the development of direct statistical simulations of geophysical flows. Using a hierarchy of idealized general circulation models, we identify when the QL approximation captures large-scale dynamics. We show that the QL dynamics fails to capture the flow when the dissipation of large-scale eddies occurs through strongly nonlinear eddy-eddy interactions in upper tropospheric surf zones, as it is often the case on Earth. But we demonstrate that the QL approximation captures eddy absorption when it arises from the shearing by the mean flow, for example when the eddy amplitude is small enough or the planetary rotation rate is large enough. These results illustrate different classes of nonlinear processes that can control wave dissipation in the upper troposphere and show that in some parameter regimes the QL approximation is accurate to resolve large-scale dynamics.

1. Quasi-linear approaches to
large-scale atmospheric flows
(or: how turbulent is the atmosphere?)
Farid Ait-Chaalal(1),
in collaboration with:
Tapio Schneider(1,3) and Brad Marston(2)
(1)ETH, Zurich, Switzerland, (2)Brown University, Providence, USA
(3)Caltech, Pasadena, USA

2. The general circulation
Superposition of a mean flow and turbulent eddies
Source: EUMETSAT, https://www.youtube.com/watch?v=m2Gy8V0Dv78
March 2013 brightness temperature (clouds)

3. Relative vorticity (s-1) at 725 hPa in an idealized dry GCM
The general circulation

4. FMS GFDL pseudospectral dynamical core
Radiation: Newtonian relaxation of temperatures toward a fixed
profile
Convection: Relaxation of the vertical lapse rate toward
0.7 ⨉ (dry adiabatic)
Uniform surface, no seasonal cycle
Run at T85 (256 x 128 in physical space) with 30 vertical sigma-
levels
600 days average after 1400 days spin-up
(Held and Suarez, 1994; Schneider andWalker, 2006)
An idealized dry general circulation model (GCM)
Convenient to play with:We can change rotation rate, pole-to-
equator temperature contrast, surface friction, convection, etc….

5. Contours:
Zonal flow
(m/s)
Green line:
Tropopause
Sigma
30
20
10
a
−60 −30 0 30 60
0.2
0.8
−1
−0.5
0
0.5Sigma
4020
10
10
b
0.2
0.8
−0.5
0
0.5
Latitude
Mid-latitude jet
Surface westerlies
Surface easterlies
(trade winds)
An idealized dry GCM:The mean zonal flow

6. Sigma
30
30
20
10
20
10
−10
295
320
350
a
−60 −30 0 30 60
0.2
0.8
−30
−20
−10
0
10
20
30
Colors:
Eddy momentum
flux (EMF)
convergence
Contours:
Zonal flow
(m s-1)
Dotted lines:
Potential
temperature (K)
Green line:
Tropopause
Eddy momentum
flux (EMF)
Friction on surface westerlies
balances vertically averaged
convergence of momentum
Friction on easterlies (trade winds)
balances vertically averaged
divergence of momentum
(Held 2000, Schneider 2006)
u0v0 cos
EMFconvergence(10-6ms-2)
Eddy zonal
wind
Eddy meridional
wind
Overbar:
zonal-time mean
Eddy momentum flux
An idealized dry GCM:The mean zonal flow
a = a + a0

7. Sigma
53
1
3
1
−5
−3
−1
−3
−1
a
−60 −30 0 30 60
0.2
0.8
−30
−20
−10
0
10
20
30
Colors:
Eddy momentum
flux (EMF)
convergence
(10-6 m s-2)
Contours:
Mass stream
function
(1010 kg s-1)
Dotted lines:
Potential
temperature (K)
Green line:
Tropopause
Ferrel cell
(Coriolis torque on the upper branch balances locally
EMF convergence)
Hadley cell
(Coriolis torque on the upper branch balances locally
EMF divergence)
(Held 2000, Schneider 2006,Walker and Schneider 2006, Korty and Schneider 2007, Levine and Schneider 2015, etc…)
An idealized dry GCM:The mean meridional flow
Streamfunction(1010kgs-1)
Eddy momentum flux

8. Heating the poles and cooling the equator
Warm pole
Cold tropics
Near surface
temperature
Near surface
relative vorticity
Westerlies
Easterlies
(Ait-Chaalal and Schneider, 2015)

9. Heating the poles and cooling the equator
Reversed insolation
Latitude
Sigma
2
2
−2
−10
−20
−40 −40
−60 −30 0 30 60
0.2
0.8
−10
−5
0
5
10
Latitude
Sigma
295
320
350
e
−60 −30 0 30 60
0.2
0.8
−1
0
1
Earth-Like
EMF(m2s-2)Streamfunction(1010kgs-1)
Latitude
Sigma
30
20
10
5
−5
−5
−60 −30 0 30 60
0.2
0.8
−40
−30
−20
−10
0
10
20
30
40
Latitude
Sigma
295
320
350
f
−60 −30 0 30 60
0.2
0.8
−6
0
6
Contours: Zonal mean flow (m/s) Dotted lines: Potential temperature (K) Green line:Tropopause
(Ait-Chaalal and Schneider, 2015)
EMF(m2s-2)Streamfunction(1010kgs-1)

10. Large-scale eddies and the general circulation
Large-scale motion in the atmosphere is controlled by eddy—
mean-flow interactions (e.g., Held 2000, Schneider 2006).
Atmospheric flows look linear from macroturbulent
scalings and do not exhibit nonlinear cascades of energy over a
wide range of parameters (Schneider and Walker 2006, Schneider
andWalker 2008, Chai andVallis 2014)
What happens if we retain eddy-mean flow
interactions and neglect eddy-eddy interactions, in
other words if we make a quasi-linear (QL)
approximation?

11. Why is the QL approximation interesting?
QL dynamics ~ closing the equations for statistical moments
at the second order
Is it possible to build statistical models to “solve climate”
based on QL dynamics as a closure strategy?
"More than any other theoretical procedure, numerical integration is also
subject to the criticism that it yields little insight into the problem. The
computed numbers are not only processed like data but they look like data,
and a study of them may be no more enlightening than a study of real
meteorological observations. An alternative procedure which does not
suffer this disadvantage consists of deriving a new system of equations
whose unknowns are the statistics themselves...."
Edward Lorenz, The Nature and Theory of the General Circulation of the
Atmosphere (1967)

12. The QL approximation
Take for example the meridional advection of a scalar (zonal mean/
eddy decomposition)
a = a + a0
@a
@t
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0 @a0
@y
@a
@t
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0
@a0
@y
becomes
Equation for the mean flow:
Equation for the eddies:
@a0
@t
= ¯v
@a0
@y
v0 @¯a
@y
(v0 @a0
@y
v0
@a0
@y
).
QL
@¯a
@t
= ¯v
@¯a
@y
v0
@a0
@y
.
Removing eddy-eddy interactions in the GCM:
Eddy-eddy interactions
(O’Gorman and Schneider 2007; Ait-Chaalal et al., 2015)
@a
@t
= v
@a
@y
= ¯v
@¯a
@y
¯v
@a0
@y
v0 @¯a
@y
v0 @a0
@y

13. The QL approximation conserves invariants consistent with the order of
truncation, for example zonal momentum and energy (Marston et al., 2014).
In the literature
Stochastic structural stability (S3T) theory to study coherent structures in
stable flows: Farrell, Ioannou, Bakas, Krommes, Parker, etc…
Cumulant expansions of second order (CE2): Marston, Srinivasan,Young, etc…
Some attempts to recover atmospheric statistics from linearized GCMs with a
stochastic forcing: Whitaker and Sardeshmuck, 1998; Zhang and Held 1999; Delsole
2001
Here: we look at unstable planetary baroclinic flows with large-scale forcing
and dissipation.
The QL approximation

14. Full
The QL approximation: Mean zonal flow
Contours:
Zonal flow (m/s)
Green line:
Tropopause
Sigma
30
20
10
a
−60 −30 0 30 60
0.2
0.8
−1
−0.5
0
0.5
Latitude
Sigma
40
20
10
10
b
−60 −30 0 30 60
0.2
0.8
−1
−0.5
0
0.5
(O’gorman and
Schneider, 2007)
QL

15. Eddy Momentum Flux Divergence
Colors:
Eddy momentum
flux (EMF)
Contours:
Zonal flow (m/s)
Dotted lines:
Potential
temperature (K)
Green line:
Tropopause
The QL approximation:The eddy momentum flux
EMF(m2s-2)EMF(m2s-2)
Full
Sigma
30
2010
a
−60 −30 0 30 60
0.2
0.8
−50
0
50
Latitude
Sigma
40
10
10
b
−60 −30 0 30 60
0.2
0.8
−20
−10
0
10
20
(Ait-Chaalal and
Schneider, 2015)
QL

16. Eddy Momentum Flux Divergence
Colors:
Eddy kinetic
energy (EKE)
Contours:
Zonal mean flow
(m/s)
Dotted lines:
Potential
temperature (K)
Green line:
Tropopause
EKE(m2s-2)EKE(m2s-2)
Full
Sigma
30
20
10
a
−60 −30 0 30 60
0.2
0.8
100
200
300
Latitude
Sigma
10
10
40
b
−60 −30 0 30 60
0.2
0.8
150
250
350
(Ait-Chaalal and
Schneider, 2015)
QL
0.5 (u02 + v02)
The QL approximation:The eddy kinetic energy

17. How is large-scale eddy decay captured in the QL
model?
Why is the eddy momentum flux not maximum in the upper
troposphere in the QL model ?
Why are weak momentum fluxes associated with high EKE in the
QL model?
The QL approximation: Summary

18. 5/29/13 7:28 PMMac App Store - GCM
MacMac
Screenshots
Description
Idealized General Circulation Models (GCMs) of planetary atmospheres, solved by a variety of methods.
GCM Support
What's New in Version 1.0.4
New wave lifecycle model, better organized menu. Bug fixes to CE3 (now conserves 3rd Casimir) and the calculation
of the eddy diffusivity.
Free
Category: Education
Updated: May 23, 2013
Version: 1.0.4
Size: 1.4 MB
Language: English
Seller: Brad Marston
© 2013 M3 Research
Rated 4+
Requirements: OS X 10.8.3 or
later, 64-bit processor
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GCM
By Brad Marston
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GCM, by Brad Marston
Solves one-layer and two-layers models of the atmosphere in spectral space
and on the geodesic grid
Solves for averages and equal-time two-point correlations (direct statistical
simulations, CE2 at the second order, CE3 at the their order)
Length nondimensionalized
with planet radius
Time nondimensionalized
with day length

19. A prototype model for the upper troposphere
Two-dimensional flow (barotropic)
Wavenumber 6 perturbation in a westerly jet
Initial value problem: how does the perturbation decay when eddy-
eddy interactions are suppressed?
Relative vorticity field
Vorticity of the eddies about 6 times larger than that of the mean flow.
Rossby number of order 0.2 in mid-latitudes.
Jet relative vorticity Jet + eddies relative vorticity

20. “Earth-like” parameters, large-amplitude eddies
An prototype model for the upper troposphere
Relative vorticity field

21. “Earth-like” parameters, large-amplitude eddies
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-0.01
-0.001
0
0.001
0.01
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-0.01
-0.001
0
0.001
0.01
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-0.01
-0.001
0
0.001
0.01
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-0.01
-0.001
0
0.001
0.01
x10-3
10
1
0
-1
-10Eddy kinetic energy Eddy kinetic energy
Eddy momentum flux convergence Eddy momentum flux convergence
x10-3
10
0
-10
x10-3
10
0
-10
x10-3
10
1
0
-1
-10
Time Time
Time Time
(Ait-Chaalal et al., 2015)
Full QL (CE2)
An prototype model for the upper troposphere

22. The QL dynamics
d
T = 1.2 T = 4.0
T = 5.9 T = 17.5
a b
c e
V
10
0
-1
-10T = 7.5
X
X
1
Relativevorticity
Relative vorticity field evolution in the QL approximation

23. The fully nonlinear dynamics
Day 1.2 Day 4.0
Day 7.5 Day 17.5
a b
d e
Day 5.9c
7
0.7
0
-0.7
-7
X
X X X
T = 1.2 T = 4.0
T = 5.9 T = 7.5 T = 17.5
10
1
0
-1
-10-10
Relativevorticity
Relative vorticity field evolution in the fully nonlinear dynamics
(for some theory, seeWarn andWarn 1978 or Stewartson 1978)

24. Vorticity - streamfunction relationship:
Flow - streamfunction relationship:
Mean-flow and eddy vorticity equations:
Shear Eddy-eddy interactions Beta-term
“Rossby number”, ratio of the mean flow vorticity to the planetary rotation rate
Relative amplitude of the eddies to the mean flow (need not to be small !!)
A prototype model for the upper troposphere

25. Decreasing the amplitude of the eddies (by a factor 3)
Relative vorticity field
A prototype model for the upper troposphere

26. EQ
30N
60N
30S
60S
0 10 20
-0.001
-0.0001
0
0.0001
0.001
EQ
30N
60N
30S
60S
0 10 20
-0.001
-0.000
0
0.0001
0.001
EQ
30N
60N
30S
60S
0 10 20
-0.001
-0.0001
0
0.0001
0.001
EQ
30N
60N
30S
60S
0 10 20
-0.001
-0.0001
0
0.0001
0.001
x10-3
10
1
0
-1
-10Eddy kinetic energy Eddy kinetic energy
Eddy momentum flux convergence Eddy momentum flux convergence
x10-3
10
0
-10
x10-3
10
0
-10
x10-3
10
1
0
-1
-10
Time Time
Time Time
(Ait-Chaalal et al., 2015)
Full QL (CE2)
Decreasing the amplitude of the eddies (by a factor 3)
A prototype model for the upper troposphere

27. Mean-flow and eddy vorticity equations:
Shear Eddy-eddy interactions Beta-term
“Rossby number”, ratio of the mean flow vorticity to the planetary rotation rate
Relative amplitude of the eddies to the mean flow (need not to be small !!)
A prototype model for the upper troposphere

28. EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
EQ
30N
60N
30S
60S
0 10 20 30 40
-10
-1
0
1
10
50
EQ
30N
60N
30S
60S
0 10 20 30 40 50
-10
-1
0
1
10
Full CE2
Ro=
0.06
Ro=
0.04
Ro=
0.03
Ro=
0.02
x10-3 x10-3
x10-3 x10-3
x10-4 x10-4
x10-4 x10-4
Time Time
Decreasing the Rossby number
(= increasing the rotation rate or decreasing both the mean flow and the eddies)
Relative vorticity (full) Eddy kinetic energy
(Ait-Chaalal et al., 2015)
A prototype model for the upper troposphere

29. Eddy absorption can be linear or nonlinear, QL captures the later
but not for the former (in which case eddies are “reemitted”
from the surf zone).
Eddies need not to be “small” for linear absorption. Smaller is the
Rossby number, larger are the eddies that can be absorbed
linearly. A theory that would describe the transition is missing.
Is this relevant to a baroclinic atmosphere?
A prototype model for the upper troposphere
How is large-scale eddy decay captured in the QL
model?

30. Baroclinic wave lifecycle experiments
Initialize a zonal wavenumber 6 perturbation in the
zonally averaged circulation (fully nonlinear model)
Let it evolve without forcing and dissipation
Experiments run with the full model and the QL
model
Back to the (baroclinic) GCM
(Simmons and Hoskins, 1978; Thorncroft et al., 1993; etc…)

31. Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
EAPE > EKE
ZKE > EKE
Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
Baroclinic conversion: eddy
available potential energy
(EAPE) to eddy kinetic energy
(EKE).
Barotropic conversion: Zonal
kinetic energy (ZKE) to eddy
kinetic energy (EKE).
Back to the (baroclinic) GCM
Baroclinic wave lifecycle experiments
(Ait-Chaalal and Schneider, 2015)

32. Baroclinic wave lifecycle experiments
Day 42
Sigma
0 30 60
0.2
0.8 −4
0
4
Day 23
Sigma
0 30 60
0.2
0.8
−1
0
1
Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
EAPE > EKE
ZKE > EKE
A2 B2
21
PVU
0
A1 B1
Full QL
a
b
c
QGPVFlux(10-5ms-2)
Latitude Latitude
Sigma
Grey arrows: Eliassen-Palm flux
(~ baroclinic equivalent of the
barotropic momentum flux)
Colors: Potential vorticity flux
(~ baroclinic equivalent of
the barotropic momentum flux
convergence)
Potential vorticity on the 300K isentrope
@¯u
@t
= r · F = (
@A
@t
)
r · F = v0q0
F = R cos
0
@
u0v0
f v0✓0/@p
¯✓
1
A
(Ait-Chaalal and Schneider, 2015)

33. Baroclinic wave lifecycle experiments
Day 46
Sigma
0 30 60
0.2
0.8 −4
0
4
Day 29
Sigma
0 30 60
0.2
0.8
−1
0
1
21
PVU
0
Full QL
a
b
c
Latitude Latitude
Sigma
QGPVFlux(10-5ms-2)
Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
Time (days)
Conversion(m2
s−3
)
0 25 50 75 100
−1
0
1
x 10
−4
EAPE > EKE
ZKE > EKE
Potential vorticity on the 300K isentrope
Grey arrows: Eliassen-Palm flux
(~ baroclinic equivalent of the
barotropic momentum flux)
@¯u
@t
= r · F = (
@A
@t
)
r · F = v0q0
F = R cos
0
@
u0v0
f v0✓0/@p
¯✓
1
A
Colors: Potential vorticity flux
(~ baroclinic equivalent of
the barotropic momentum flux
convergence)
(Ait-Chaalal and Schneider, 2015)

34. Back to the (baroclinic) GCM
Sigma
30
2010
a
−60 −30 0 30 60
0.2
0.8
−50
0
50
Latitude
Sigma
40
10
10
b
−60 −30 0 30 60
0.2
0.8
−20
−10
0
10
20
Sigma
30
20
10
a
−60 −30 0 30 60
0.2
0.8
100
200
300
Latitude
Sigma
10
10
40
b
−60 −30 0 30 60
0.2
0.8
150
250
350
Full
QL
Eddy momentum flux Eddy kinetic energy

35. Example of a baroclinic flow in which QL works
Latitude
Sigma
40
40
−10
−60 −30 0 30 60
0.2
0.8
Latitude
Sigma
40
40
−60 −30 0 30 60
0.2
0.8
Latitude
Sigma
10
10
20
20
−60 −30 0 30 60
0.2
0.8
Latitude
Sigma
30 30
−60 −30 0 30 60
0.2
0.8
Full
QL
Earth-like Reduced surface friction
Also works in many other situations (e.g., the reversed insolation
experiment)

36. Conclusive remarks
Eddy-eddy interactions do matter for eddy absorption in the
upper troposphere.They have to be parametrized in some
way to achieve direct statistical simulations.
Eddy absorption can be linear in some regimes (without the
requirement of small-amplitude waves). In what case QL
dynamics and the second order cumulant expansion capture
the dynamics.
QL maybe more promising for giant plants, e.g. to study the
long-term evolution of jets.