Lorentz Centre 2 October, 2006. The Energy Spectrum of the Atmosphere Peter Lynch University College...

Lorentz Centre 2 October, 2006


The Energy Spectrumof the

Atmosphere

Peter LynchUniversity College Dublin

Geometric & Multi-scale Methods for Geophysical Fluid Dynamics

Lorentz Centre, University of Leiden


Background

“Big whirls have little whirls … ”


Figure from Davidson: Turbulence


The Problem

• A complete understanding of the atmospheric energy spectrum remains elusive.

• Attempts using 2D and 3D and Quasi-Geostrophic turbulence theory to explain the spectrum have not been wholly satisfactory.


Quasi-Geostrophic Turbulence

• The characteristic aspect ratio of the atmosphere is 100:1

L/H ~ 100


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?


2D Vorticity Equation

• In 2D flows, the vorticity is a scalar:

• For non-divergent, non-rotating flow:

vx

uy

t

ux

vy

ddt

0


2D Vorticity Equation

• If we introduce a stream function , we can write the vorticity equation as

• The velocity is

t

2 J ,2 0

(u,v) y

,x


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 include the effect of the Earth’s rotation.

– We allow for horizontal divergence.



• The equation of Conservation of Potential Vorticity is:

relative vorticity– f - planetary vorticity– h - fluid height

d

dt

fh

0



• To derive a single equation for a single variable, we assume geostrophic balance:

• This allows us to relate the mass and wind fields.

V g

fkh k


QGPV Equation• The Barotropic Quasi-Geostrophic Potential Vorticity

Equation is:

where .

t

2 F J ,2 x

0

F f0

2

gH


Digression on Resonant Triads

(and the swinging spring … maybe … )


2D versus QG

• 2D Case:

• QG Case:

t

2 F J ,2 x

0

t

2 J ,2 0


QG Turbulence: 2D or 3D?

• 2D Turbulence– Energy & Enstrophy conserved– No vortex stretching




• 3D Turbulence– Enstrophy not conserved– Vortex stretching present




• 3D Turbulence– Enstrophy not conserved– Vortex stretching present

• QG Turbulence– Energy & Enstrophy conserved (like 2D)– Vortex stretching present (like 3D)



• The prevailing view has been that QG turbulence is more like 2D turbulence.



• 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.


Inverse cascade to largest scales


Inverse cascade to largest scales

Inverse cascade to isolated vortices


Inverse Energy Cascade

matlab examples

(Demo-01: QG01, QG24)


Some Early Results

• Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales.


Some Early Results


• Charney (1971) used Fjørtoft’s proofs to derive the conservation laws for QG turbulence.


Some Early Results


• 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).


2D Turbulence

• Standard 2D turbulence theory predicts:


2D Turbulence


– Upscale energy cascade from the point of energy injection (spectral slope –5/3)


2D Turbulence


– Upscale energy cascade from the point of energy injection (spectral slope –5/3)

– Downscale enstrophy cascade to smaller scales (spectral slope –3)


Decaying turbulence

Some results for a

1024x1024 grid


E/E(1)

S/S(1)


-3


2D Turbulence

• Inverse Energy Cascade

• Forward Enstrophy Cascade

35

32

)(

kkE

E(k) 2

3k 3


2D Turbulence

• Inverse Energy Cascade

• Forward Enstrophy Cascade

35

32

)(

kkE

E(k) 2

3k 3

What observational evidence do we have?


Two Mexican physicists,José Luis Aragón andGerardo Naumis, haveexamined the patterns in van Gogh’s

Starry Night


Two Mexican physicists,José Luis Aragón andGerardo Naumis, haveexamined 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.


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.


The Nastrom & Gage Spectrum



• No evidence of a broad mesoscale “energy gap”.




• Velocity and Temperature spectra have nearly the same shape.




• Velocity and Temperature spectra have nearly the same shape.

• Little seasonal or latitudinal variation.


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.




• 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.


The Nastrom & Gage Spectrum (again)


The Spectral “Kink”

• The observational evidence outlined above showed a kink at around 600 km– Surely too large for isotropic 3D effects?


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)


Larsen’s Suggested Spectrum


The Spectral “Kink” (cont.)

• Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.


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.


The Spectral Kink

• Tung and Orlando reproduced the N&G spectrum using QG dynamics alone. (They employed sub-grid diffusion.)

20


The Spectral Kink

• Tung and Orlando reproduced the N&G spectrum using QG dynamics alone. (They employed sub-grid diffusion.)

• The NMM model also reproduces the spectral kink at the mesoscales when physics is included (Janjic, EGU 2006)

20


No Physics With Physics

Where is the small scale energy in the observed spectrum coming from?

Atlantic case, NMM-B, 15 km, 32 Levels

(Thanks to Zavisa Janjic for this slide)


An Additive Spectrum?• Charney (1973) noted the possibility of an additive

spectrum:

• Tung & Gkioulekas (2005) proposed a similar form:

E(k) Ak 3 Bk 5

3 Ck 2

E(k) Ak 3 Bk 5

3


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)




• Most injected energy inversely cascades to larger scales (-5/3 spectral slope)

• Large-scale energy is lost through radiative dissipation & Ekman damping.


Current Picture (cont.)

• It is likely that a small portion of the injected energy cascades to smaller scales.



• 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.



• Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.




• The slope is evident at scales smaller than this.

k 5

3




• The slope is evident at scales smaller than this.

• The slope is probably augmented by an inverse energy cascade from convective scales.

k 5

3

k 5

3


Inverse Enstrophy Cascade?

• It is possible that a small portion of the enstrophy inversely cascades from synoptic to planetary scales.


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 evidence of large-scale behaviour:– The Earth’s circumference dictates the

size of the largest scale.

k 3


ECMWF Model Output

• The “kink” at mesoscales is not evident in the ECMWF model output.

k 5

3


ECMWF Model Output

• The “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)

k 5

3


Energy spectrum in T799 run

E(n)

)(log10 n

3/5k

3k

n = spherical harmonic order

missing energy

3/5k

3k


ECMWF Model Output

• Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping.


ECMWF Model Output


• His modifications allow for a substantially higher amount of energy at smaller scales.


ECMWF Model Output


• His modifications allow for a substantially higher amount of energy at smaller scales.

• The backscatter approach does produce the spectral kink at the mesoscales.


Energy spectrum in T799 run

E(n)

)(log10 n

3/5k

3k

n = spherical harmonic order

missing energy

3/5k

3k


Energy spectrum in ECMWF model with backscatter

3k

T799

E(n)

)(log10 n

3k

3/5k


Some Outstanding Issues

• Flux Variability– Direction of (-5/3) short-wave energy cascade– Dependence on convective activity



• 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



• Is it not possible for both Energy and Enstrophy to flow in both directions?




• In an unbounded system, a “W-shaped spectrum” may arise.




• In an unbounded system, a “W-shaped spectrum” may arise.

• For an additive spectrum, dominance

will alternate between -5/3 and -3 terms.



• The validity of an additive spectrum

needs to be justified.

E(k) Ak 3 Bk 5

3


Thank You

Lorentz Centre 2 October, 2006. The Energy Spectrum of the Atmosphere Peter Lynch University College...

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