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.