1. On the vertical structure of the
tropospheric eddy momentum flux
Farid Ait Chaalal(1,3) and Tapio Schneider(2,3)
(1)Brown University, Providence, USA (2)ETH, Zurich, Switzerland
(3)Caltech, Pasadena, USA
19th Conference on Atmospheric and Oceanic Fluid Dynamics
17–21 June 2013, Newport, Rhode Island
2. Eddy momentum flux maximum in the upper troposphere
Held, 2000: upper level zonal mean flow favors linear wave
meridional propagation
Upper level enhancement not captured without nonlinear
saturation (Simmons and Hoskins, 1978; Merlis and Schneider,
2009)
10 6
m s 2
Motivation
Eddy momentum flux
convergence in the
atmosphere (colors), zonal
wind (contours, m/s) and
tropopause (grey line).
ERA 40 1980-2001
Latitude
Sigma
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10
3030
−10
−60 −30 0 30 60
0.2
0.8
−50
0
50
10
20
20
0
0
3. Surface friction?
Large scale eddy-eddy interactions?
Outline
Why is eddy momentum flux concentrated in
the upper troposphere?
4. An idealized dry GCM
GFDL pseudospectral dynamical core
Radiation: Newtonian relaxation toward temperature
profile
Convection: relaxation of vertical lapse rate toward
0.7 ⨉ (dry adiabatic)
Uniform surface, no seasonal cycle
Run at T85 with 30 vertical sigma-levels
600 days average after 1400 days spin-up
(Schneider andWalker, 2006)
5. The role of surface friction
Colors: eddy momentum flux divergence
Contours and dashed lines: zonal mean flow (m/s)
Dotted line: potential temperature (K)
Grey line: tropopause (2K/km lapse rate)
Simulation with positive 90 K
pole-to-equator temperature
contrast ∆h.
Simulation with
negative ∆h = -90 K.
Poles heated, tropics cooled.
10 6
m s 2
LatitudeSigma
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−30
−40 −40
300
320
350
−60 −30 0 30 60
0.2
0.8
−10
−5
0
5
10
Latitude
Sigma
30 30
10
10
20 20
300
320
350
−60 −30 0 30 60
0.2
0.8
−50
0
50
10 6
m s 2
6. Removal of eddy-eddy interactions
Quasi-linear (QL) model
(O’Gorman and Schneider, 2007)
Retained:
Wave-mean flow interaction (mean flow = zonal average)
Interaction of waves of opposite zonal wavenumber
(Reynolds stress)
Neglected:
All other eddy-eddy interactions
(Statistics equivalent to 2nd order cumulant expansion,
see Brad Marston’s talk)
8. Eddy Momentum Flux Divergence
Latitude
Sigma
30 30
10
10
20 20
300
320
350
−60 −30 0 30 60
0.2
0.8
−50
0
50
Colors:
eddy momentum
flux convergence
Contours:
zonal mean flow
(m/s)
Dotted lines:
potential
temperature (K)
Grey line:
tropopause
Eddy momentum flux convergence
Full
No
eddy-eddy
10 6
m s 2
9. Restoring barotropic triads
Latitude
Sigma
10
10
1010
40 40
300
320
350
−60 −30 0 30 60
0.2
0.8
−50
0
50
Latitude
Sigma
30 30
10
10
20 20
300
320
350
−60 −30 0 30 60
0.2
0.8
−50
0
50
10 6
m s 2
Colors:
eddy momentum
flux convergence
Contours:
zonal mean flow
(m/s)
Dotted lines:
potential
temperature (K)
Grey line:
tropopause
Full
With
barotropic
triads
10. Baroclinic wave lifecycle experiments
Initialize a zonal wavenumber 6 perturbation in the
zonally averaged circulation (fully non-linear model)
Experiments run with full model, no eddy-eddy model,
and model with only barotropic triads
Why eddy momentum flux not maximum in the
upper troposphere in the QL model ?
Why enhancement captured when barotropic triads
are restored?
Also: eddy kinetic energy larger in the QL model
usually, larger momentum flux expected
Removal of eddy-eddy interactions
11. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5 full model
Days
W/kg
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
Days
W/kg
only barotropic eddy−eddy
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
Days
W/kg
no eddy−eddy
Baroclinic conversion.
Eddy available potential energy
to eddy kinetic energy
Barotropic conversion.
Zonal kinetic energy to
eddy kinetic energy
Full
No eddy-eddy Only barotropic eddy-eddy
Energy conversion
12. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
Days
W/kg
only barotropic eddy−eddy
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
DaysW/kg
no eddy−eddy
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5 full model
Days
W/kg
Colors: QG potential vorticity flux
Contours: zonal mean flow (m/s)
Arrows: QG Eliassen-Palm vector
Full No eddy-eddy Only barotropic eddy-eddy
10 5
m s 2
Latitude Latitude Latitude
Sigma
Maximum
of barotropic conversion
13. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
Days
W/kg
only barotropic eddy−eddy
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5
DaysW/kg
no eddy−eddy
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
−5
0
5
10
x 10
−5 full model
Days
W/kg
Sigma
Latitude Latitude Latitude
Minimum
of baroclinic conversion
Colors: QG potential vorticity flux
Contours: zonal mean flow (m/s)
Arrows: QG Elliassen-Palm vector
Full No eddy-eddy Only barotropic eddy-eddy
10 5
m s 2
14. Potential vorticity rearrangement and mixing
Days
W/kg
15 20 25 30 35
−5
0
5
10
x 10
Days
W/kg
15 20 25 30 35
−5
0
5
10
x 10
For a theoretical study of critical layers (SWW solution) in the QL approximation:
Haynes and McIntyre, 1987
Full
(350 K
isentrope)
No
eddy-eddy
(320 K
isentrope)
Day 26 Day 30 Day 35
Day 17 Day 22 Day 30
PV fields on isentropes
210
PVU
15. Potential vorticity rearrangement and mixing
Days
W/kg
15 20 25 30 35
−5
0
5
10
x 10
15 20 25 30 35
−5
0
5
10
x 10
−5
Days
W/kg
only barotropic eddy−eddy
No
eddy-eddy
(320 K
isentrope)
Only
barotropic
eddy-eddy
(320 K
isentrope)
Day 17 Day 22 Day 30
Day 25 Day 28 Day 32
PV fields on isentropes
210
PVU
16. Conclusions
Surface friction not an important factor
Wave packets generated in LT propagate upward until
they reach UT and tropopause, where horizontal
propagation is favored
Vorticity rearragement and mixing through eddy-eddy
interaction near critical layers essential
Keeping only barotropic triads captures the enhancement
Understanding nonlinear barotropic critical layer
dynamics might suffice
Why is eddy momentum flux concentrated in
the upper troposphere?
17. Conclusions
What still favors meridional propagation of waves in the upper-
troposphere? (tropopause as a wave guide? ...)
Index of refraction
for zonal wavenumber 6 baroclinic waves.
Latitude
Sigma
30
30
1010
−60 −30 0 30 60
0.2
0.8
0
5
10
15
20
Why is eddy momentum flux concentrated in the upper
troposphere?
Full
No
eddy-eddy
18. Dry GCM without eddy-eddy interactions
Removal of the eddy-eddy interactions (O’Gorman and Schneider,
2007).
Advection of a quantity a = a + a0 by the meridional flow v = v + v0
(zonal mean/eddy decomposition):
@a
@t
= v
@a
@y
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0 @a0
@y
transformed into
@a
@t
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0
@a0
@y
Statistics of such a model are equivalent to a second order cumulant
expansion (third order cumulants set to 0 in the second order
equations).
Farid Ait-Chaalal (Caltech) Second-Order Atm. Circulation June 26, 2012 4 / 19
@a
@t
= v
@a
@y
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0
@a0
@y
@a
@t
= v
@a
@y
= v
@a
@y
v
@a0
@y
v0 @a
@y
v0 @a0
@y
Removal of the eddy-eddy interactions
19. Latitude
Sigma
0
0
−5
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−30
−40
−40
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320
350
−60 −30 0 30 60
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10
The role of surface friction
Latitude
Sigma
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−10
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−40
−40
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320
350
−60 −30 0 30 60
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Simulation with ∆h = -90 K
Latitude
Sigma
10
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10
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−50
−50
−40
−30
−10
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320
350
−60 −30 0 30 60
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Colors: Eddy momentum fluxes ()
Dashed lines: winds in m/s
Dotted line: potential temperature in K
Grey line: tropopause
Friction/10 Friction Friction * 10
20. The role of surface friction
Colors: Eddy momentum fluxes divergence
Dashed lines: winds in m/s
Dotted line: potential temperature in K
Grey line: tropopause
Simulation with positive ∆h = 90 K Surface friction balances upper level
momentum fluxes divergence (EMFD)
Long been recognized effect on
midlatitudes eddies amplitude, jet streams
strength and location, energy conversions
(James, 1987; Robinson 1997; Chen et al.,
2007; etc...)
Can it explain upper level EMDF
e n h a n c e m e n t ? S o m e t i m e
suggested in text books (e.g.
Vallis, 2006)Latitude
Sigma
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5.10 6
m s 2
