Transcript Geostrophic Turbulence Atmospheric Energy & Enstrophy Cascades

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2 October, 2006
The Energy Spectrum
of the
Atmosphere
Peter Lynch
University College Dublin
Geometric & Multi-scale Methods for
Geophysical Fluid Dynamics
Lorentz Centre, University of Leiden
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Background
“Big whirls have little whirls … ”
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Figure from Davidson: Turbulence
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The Problem
• A complete understanding of the
atmospheric energy spectrum remains
elusive.
• Attempts using 2D and 3D and QuasiGeostrophic turbulence theory to explain
the spectrum have not been wholly
satisfactory.
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Quasi-Geostrophic Turbulence
• The characteristic aspect ratio of the
atmosphere is 100:1
L/H ~ 100
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Quasi-Geostrophic Turbulence
• The characteristic aspect ratio of the
atmosphere is 100:1
L/H ~ 100
• Is quasi-geostrophic turbulence more
like 2D or 3D turbulence?
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2D Vorticity Equation
• In 2D flows, the vorticity is a scalar:
v u
 
x y
• For non-divergent, non-rotating flow:

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

 d
u
v

0
t
x
y dt
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2D Vorticity Equation
• If we introduce a stream function , we
can write the vorticity equation as
 2
2
  J,  0

t
• The velocity is

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   
(u,v)   , 
 y x 
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Quasi-Geostrophic Potential
Vorticity
• In the QG formulation we seek to augment
the 2D picture in two ways:
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Quasi-Geostrophic Potential
Vorticity
• In the QG formulation we seek to augment
the 2D picture in two ways:
– We include the effect of the Earth’s
rotation.
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Quasi-Geostrophic Potential
Vorticity
• In the QG formulation we seek to augment
the 2D picture in two ways:
– We include the effect of the Earth’s
rotation.
– We allow for horizontal divergence.
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Quasi-Geostrophic Potential
Vorticity
• The equation of Conservation of Potential Vorticity is:
d   f 
0


dt  h 
 relative vorticity
– f - planetary vorticity
– h - fluid height

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Quasi-Geostrophic Potential
Vorticity
• To derive a single equation for a single variable, we
assume geostrophic balance:
g
V  k  h  k  
f
• This allows us to relate the mass and wind fields.

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QGPV Equation
• The Barotropic Quasi-Geostrophic Potential Vorticity
Equation is:
 2

2
   F J,  
0

t
x
f 02
where F 
gH
.


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Digression on Resonant Triads
(and the swinging spring … maybe … )
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2D versus QG
• 2D Case:
 2
2
  J,  0

t
• QG Case:
 2

2
   F J,  
0


t
x
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QG Turbulence: 2D or 3D?
• 2D Turbulence
– Energy & Enstrophy conserved
– No vortex stretching
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QG Turbulence: 2D or 3D?
• 2D Turbulence
– Energy & Enstrophy conserved
– No vortex stretching
• 3D Turbulence
– Enstrophy not conserved
– Vortex stretching present
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QG Turbulence: 2D or 3D?
• 2D Turbulence
– Energy & Enstrophy conserved
– No vortex stretching
• 3D Turbulence
– Enstrophy not conserved
– Vortex stretching present
• QG Turbulence
– Energy & Enstrophy conserved (like 2D)
– Vortex stretching present (like 3D)
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QG Turbulence: 2D or 3D?
• The prevailing view has been that QG
turbulence is more like 2D turbulence.
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QG Turbulence: 2D or 3D?
• The prevailing view has been that QG
turbulence is more like 2D turbulence.
• The mathematical similarity of 2D and
QG flows prompted Charney (1971) to
conclude that an energy cascade to
small-scales is impossible in QG
turbulence.
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Inverse cascade to largest scales
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Inverse cascade to largest scales
Inverse cascade to isolated vortices
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Inverse Energy Cascade
matlab examples
(Demo-01: QG01, QG24)
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Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is
injected at an intermediate scale, more
energy flows to larger scales.
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Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is
injected at an intermediate scale, more
energy flows to larger scales.
• Charney (1971) used Fjørtoft’s proofs to
derive the conservation laws for QG
turbulence.
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Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is
injected at an intermediate scale, more
energy flows to larger scales.
• Charney (1971) used Fjørtoft’s proofs to
derive the conservation laws for QG
turbulence.
• The proof used is really just a convergence
requirement for a spectral representation of
enstrophy (Tung & Orlando, 2003).
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2D Turbulence
• Standard 2D turbulence theory predicts:
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2D Turbulence
• Standard 2D turbulence theory predicts:
– Upscale energy cascade from the point of
energy injection (spectral slope –5/3)
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2D Turbulence
• Standard 2D turbulence theory predicts:
– Upscale energy cascade from the point of
energy injection (spectral slope –5/3)
– Downscale enstrophy cascade to smaller
scales (spectral slope –3)
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Decaying turbulence
Some results for a
1024x1024 grid
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E/E(1)
S/S(1)
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-3
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2D Turbulence
• Inverse Energy
Cascade
• Forward Enstrophy
Cascade
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2
E (k )   k
3
2
5
3
E(k)   k
3 3
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2D Turbulence
• Inverse Energy
Cascade
• Forward Enstrophy
Cascade
2
E (k )   k
3
2
5
3
E(k)   k
3 3
What observational evidence do we have?
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Two Mexican physicists,
José Luis Aragón and
Gerardo Naumis, have
examined the patterns
in van Gogh’s
Starry Night
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Two Mexican physicists,
José Luis Aragón and
Gerardo Naumis, have
examined the patterns
in van Gogh’s
Starry Night
They found that the PDF of luminosity
follows a Kolmogorov -5/3 scaling law.
See Plus e-zine for more information.
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Observational Evidence
• The primary source of observational
evidence of the atmospheric spectrum
remains (over 20 years later!) the study
undertaken by Nastrom and Gage (1985)
[but see also the MOZAIC dataset].
• They examined data collated by nearly 7,000
commercial flights between 1975 and 1979.
• 80% of the data was taken between 30º and
55ºN.
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The Nastrom & Gage Spectrum
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Observational Evidence
• No evidence of a broad mesoscale
“energy gap”.
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Observational Evidence
• No evidence of a broad mesoscale
“energy gap”.
• Velocity and Temperature spectra have
nearly the same shape.
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Observational Evidence
• No evidence of a broad mesoscale
“energy gap”.
• Velocity and Temperature spectra have
nearly the same shape.
• Little seasonal or latitudinal variation.
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2 October, 2006
Observed Power-Law
Behaviour
• Two power laws were evident:
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Observed Power-Law
Behaviour
• Two power laws were evident:
• The spectrum has slope close to –(5/3)
for the range of scales up to 600 km.
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Observed Power-Law
Behaviour
• Two power laws were evident:
• The spectrum has slope close to –(5/3)
for the range of scales up to 600 km.
• At larger scales, the spectrum steepens
considerably to a slope close to –3.
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The Nastrom & Gage Spectrum (again)
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The Spectral “Kink”
• The observational evidence outlined
above showed a kink at around 600 km
– Surely too large for isotropic 3D effects?
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The Spectral “Kink”
• The observational evidence outlined
above showed a kink at around 600 km
– Surely too large for isotropic 3D effects?
• Nastrom & Gage (1986) suggested the
shortwave –5/3 slope could be
explained by another inverse energy
cascade, from convective storm scales
(after Larsen, 1982)
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Larsen’s Suggested Spectrum
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The Spectral “Kink” (cont.)
• Lindborg & Cho (2001), however, could
find no support for an inverse energy
cascade at the mesoscales.
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The Spectral “Kink” (cont.)
• Lindborg & Cho (2001), however, could
find no support for an inverse energy
cascade at the mesoscales.
• Tung and Orlando (2002) suggested
that the shortwave k^(-5/3) behaviour
was due to a small downscale energy
cascade from the synoptic scales.
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The Spectral Kink
• Tung and Orlando reproduced the N&G
spectrum using QG dynamics alone.
20
(They employed  sub-grid diffusion.)

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The Spectral Kink
• Tung and Orlando reproduced the N&G
spectrum using QG dynamics alone.
20
(They employed  sub-grid diffusion.)
• The NMM model also reproduces the
spectral kink at the mesoscales when

physics is included (Janjic, EGU 2006)
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Where is the small scale energy in the observed spectrum coming from?
Atlantic case, NMM-B, 15 km, 32 Levels
No Physics
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With Physics
(Thanks to Zavisa Janjic for this slide)
2 October, 2006
An Additive Spectrum?
• Charney (1973) noted the possibility of
an additive spectrum:
5
3
E(k)  Ak  Bk
3
 Ck
2
• Tung & Gkioulekas (2005) proposed a
similar form:
3
5
E(k)  Ak  Bk
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3
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Current View of Spectrum
• Energy is injected at scales associated
with baroclinic instability.
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Current View of Spectrum
• Energy is injected at scales associated
with baroclinic instability.
• Most injected energy inversely cascades
to larger scales (-5/3 spectral slope)
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2 October, 2006
Current View of Spectrum
• Energy is injected at scales associated
with baroclinic instability.
• Most injected energy inversely cascades
to larger scales (-5/3 spectral slope)
• Large-scale energy is lost through
radiative dissipation & Ekman damping.
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2 October, 2006
Current Picture (cont.)
• It is likely that a small portion of the
injected energy cascades to smaller
scales.
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Current Picture (cont.)
• It is likely that a small portion of the
injected energy cascades to smaller
scales.
• At synoptic scales, the downscale
energy cascade is spectrally dominated
by the k^(-3) enstrophy cascade.
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Current Picture (cont.)
• Below about 600 km, the downscale energy
cascade begins to dominate the energy
spectrum.
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2 October, 2006
Current Picture (cont.)
• Below about 600 km, the downscale energy
cascade begins to dominate the energy
spectrum.
53
• The k
slope is evident at scales smaller
than this.

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Current Picture (cont.)
• Below about 600 km, the downscale energy
cascade begins to dominate the energy
spectrum.
53
• The k
slope is evident at scales smaller
than this.
53
• The k
slope is probably augmented by an
 inverse energy cascade from convective
scales.

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2 October, 2006
Inverse Enstrophy Cascade?
• It is possible that a small portion of the
enstrophy inversely cascades from
synoptic to planetary scales.
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2 October, 2006
Inverse Enstrophy Cascade?
• It is possible that a small portion of the
enstrophy inversely cascades from
synoptic to planetary scales.
• We are unlikely, however, to find
3
evidence of large-scale k behaviour:
– The Earth’s circumference dictates the
size of the largest scale.

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2 October, 2006
ECMWF Model Output
53
• The k
“kink” at mesoscales is not
evident in the ECMWF model output.

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ECMWF Model Output
53
• The k
“kink” at mesoscales is not
evident in the ECMWF model output.
• Excessive damping of energy is likely
to be the cause.
(Thanks to Tim Palmer & Glenn Shutts for the following figures)
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2 October, 2006
Energy spectrum in T799
run
k 3
E(n)
k
3
k 5 / 3
n = spherical
harmonic order
k 5 / 3
missing energy
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log10 (n)
2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy
backscattering approach to compensate for
the overdamping.
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2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy
backscattering approach to compensate for
the overdamping.
• His modifications allow for a substantially
higher amount of energy at smaller scales.
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2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy
backscattering approach to compensate for
the overdamping.
• His modifications allow for a substantially
higher amount of energy at smaller scales.
• The backscatter approach does produce the
spectral kink at the mesoscales.
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2 October, 2006
Energy spectrum in T799
run
k 3
E(n)
k
3
k 5 / 3
n = spherical
harmonic order
k 5 / 3
missing energy
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log10 (n)
2 October, 2006
Energy spectrum in ECMWF model with backscatter
T799
k k3
3
E(n)
k
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log10 (n)
5 / 3
2 October, 2006
Some Outstanding Issues
• Flux Variability
– Direction of (-5/3) short-wave energy cascade
– Dependence on convective activity
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2 October, 2006
Some Outstanding Issues
• Flux Variability
– Direction of (-5/3) short-wave energy cascade
– Dependence on convective activity
• Geographic Variability
– Strong convective activity
– Little data collated in tropical areas
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2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and
Enstrophy to flow in both directions?
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2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and
Enstrophy to flow in both directions?
• In an unbounded system, a “W-shaped
spectrum” may arise.
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2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and
Enstrophy to flow in both directions?
• In an unbounded system, a “W-shaped
spectrum” may arise.
• For an additive spectrum, dominance
will alternate between -5/3 and -3 terms.
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2 October, 2006
Some Outstanding Issues
• The validity of an additive spectrum
3
5
E(k)  Ak  Bk
3
needs to be justified.
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2 October, 2006
Thank You
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