# how turbulent is the atmosphere at large scales

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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

5/29/13 7:28 PMMac App Store - GCM

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