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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
The general circulation
Superposition of a mean flow and turbulent eddies
Source: EUMETSAT, https://www.youtube.com/watch?v=m2Gy8V0Dv78March 2013 brightness temperature (clouds)
https://www.youtube.com/watch?v=m2Gy8V0Dv78
Relative vorticity (s-1) at 725 hPa in an idealized dry GCM
The general circulation
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 and Walker, 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.
Contours: Zonal flow (m/s)
Green line: Tropopause
Sigm
a30
2010
a
60 30 0 30 60
0.2
0.8
1
0.5
0
0.5
Latitude
Sigm
a
40
20
10
10
b
60 30 0 30 60
0.2
0.8
1
0.5
0
0.5
Latitude
Mid-latitude jet
Surface westerlies
Surface easterlies(trade winds)
An idealized dry GCM: The mean zonal flow
Sigm
a
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
30Colors: 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 cosEM
F co
nver
genc
e (1
0-6 m
s-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
Sigm
a
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
Stre
amfu
ncti
on (
1010
kg
s-1 )
Eddy momentum flux
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)
Heating the poles and cooling the equator
Reversed insolation
Latitude
Sigm
a
22 2
10
20
40 40
60 30 0 30 60
0.2
0.810
5
0
5
10
Latitude
Sigm
a
295
320
350
e
60 30 0 30 60
0.2
0.81
0
1
e
Earth-Like
EMF
(m2 s-
2 )St
ream
func
tion
(10
10 k
g s-
1 )
Latitude
Sigm
a
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
Sigm
a
295
320
350
f
60 30 0 30 60
0.2
0.86
0
6
Contours: Zonal mean flow (m/s) Dotted lines: Potential temperature (K) Green line: Tropopause
(Ait-Chaalal and Schneider, 2015)
EMF
(m2 s-
2 )St
ream
func
tion
(10
10 k
g s-
1 )
Large-scale eddies and the general circulation
Large-scale motion in the atmosphere is controlled by eddymean-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 and Walker 2008, Chai and Vallis 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?
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)
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@a
0
@y v0 @a
@y v0 @a
0
@y@a
@t= v@a
@y v@a
0
@y v0 @a
@y v0 @a
0
@ybecomes
Equation for the mean flow:
Equation for the eddies: @a0
@t= v@a
0
@y v0 @a
@y (v0 @a
0
@y v0 @a
0
@y).
QL
@a
@t= v@a
@y v0 @a
0
@y.
Removing eddy-eddy interactions in the GCM:
Eddy-eddy interactions
(OGorman and Schneider 2007; Ait-Chaalal et al., 2015)
@a
@t= v@a
@y= v@a
@y v@a
0
@y v0 @a
@y v0 @a
0
@y
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 2001Here: we look at unstable planetary baroclinic flows with large-scale forcing and dissipation.
The QL approximation
Full
The QL approximation: Mean zonal flow
Contours: Zonal flow (m/s)
Green line: Tropopause
Sigm
a
30
2010
a
60 30 0 30 60
0.2
0.8
1
0.5
0
0.5
Latitude
Sigm
a
40
20
10
10
b
60 30 0 30 60
0.2
0.8
1
0.5
0
0.5
(Ogorman and Schneider, 2007)
QL
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
(m2 s-
2 )EM
F (m
2 s-
2 )
Full
Sigm
a
30
2010
a
60 30 0 30 60
0.2
0.850
0
50
Latitude
Sigm
a
40
10
10
b
60 30 0 30 60
0.2
0.8 20
10
0
10
20
(Ait-Chaalal and Schneider, 2015)
QL
Eddy Momentum Flux Divergence
Colors: Eddy kineticenergy (EKE)
Contours: Zonal mean flow (m/s)
Dotted lines: Potential temperature (K)
Green line: Tropopause
EKE
(m2 s-
2 )EK
E (m
2 s-
2 )
Full
Sigm
a
30
20
10
a
60 30 0 30 60
0.2
0.8 100
200
300
Latitude
Sigm
a
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
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
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GCMBy Brad MarstonOpen the Mac App Store to buy and download apps.
GCM, by Brad Marston
Solves one-layer and two-layers models of the atmosphere in spectral space and on the geodesic gridSolves 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
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 fieldVorticity 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
Earth-like parameters, large-amplitude eddies
An prototype model for the upper troposphere
Relative vorticity field
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
-10 Eddy 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
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
Rel
ativ
e vo
rtic
ity
Relative vorticity field evolution in the QL approximation
The fully nonlinear dynamics
Day 1.2 Day 4.0
Day 7.5 Day 17.5
a b
d eDay 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
Rel
ativ
e vo
rtic
ity
Relative vorticity field evolution in the fully nonlinear dynamics
(for some theory, see Warn and Warn 1978 or Stewartson 1978)
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
Decreasing the amplitude of the eddies (by a factor 3)
Relative vorticity field
A prototype model for the upper troposphere
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
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
-10 Eddy 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
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
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
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?
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)
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
EAPE > EKEZKE > EKE
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
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)
Baroclinic wave lifecycle experiments
Day 42
Sigm
a
0 30 60
0.2
0.8 4
0
4
Day 23
Sigm
a
0 30 60
0.2
0.81
0
1
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
EAPE > EKEZKE > EKE
A2 B2
21
PVU
0
A1 B1
Full QL
a
b
c
QG
PV F
lux
(10-
5 m
s-2 )
Latitude Latitude
Sigm
a
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 v00/@p
1
A
(Ait-Chaalal and Schneider, 2015)
Baroclinic wave lifecycle experiments
Day 46
Sigm
a
0 30 60
0.2
0.8 4
0
4
Day 29
Sigm
a
0 30 60
0.2
0.81
0
1
21
PVU
0
Full QL
a
b
cLatitude Latitude
Sigm
a
QG
PV F
lux
(10-
5 m
s-2 )
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
Time (days)
Con
vers
ion
(m2
s3 )
0 25 50 75 100
1
0
1
x 104
EAPE > EKEZKE > 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 v00/@p
1
A
Colors: Potential vorticity flux(~ baroclinic equivalent of the barotropic momentum flux convergence)
(Ait-Chaalal and Schneider, 2015)
Back to the (baroclinic) GCM
Sigm
a
30
2010
a
60 30 0 30 60
0.2
0.850
0
50
Latitude
Sigm
a
40
10
10
b
60 30 0 30 60
0.2
0.8 20
10
0
10
20
Sigm
a
30
20
10
a
60 30 0 30 60
0.2
0.8 100
200
300
Latitude
Sigm
a
10
10
40
b
60 30 0 30 60
0.2
0.8 150
250
350
Full
QL
Eddy momentum flux Eddy kinetic energy
Example of a baroclinic flow in which QL works
Latitude
Sigm
a
40 4010
60 30 0 30 60
0.2
0.8
Latitude
Sigm
a
40
40
60 30 0 30 60
0.2
0.8
Latitude
Sigm
a
10
10
2020
60 30 0 30 60
0.2
0.8
Latitude
Sigm
a
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)
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.