energy & enstrophy cascades in the atmosphere prof. peter lynch michael clark university college...

Energy & Enstrophy Cascades in the Atmosphere

Prof. Peter LynchProf. Peter Lynch

Michael ClarkMichael Clark

University College DublinUniversity College Dublin

Met & Climate CentreMet & Climate Centre

Introduction

A full theoretical understanding of the A full theoretical understanding of the atmospheric energy spectrum remains atmospheric energy spectrum remains elusive.elusive.

At synoptic and sub-synoptic scales, the At synoptic and sub-synoptic scales, the energy spectrum exhibits k^(-3) power law energy spectrum exhibits k^(-3) power law behaviour, consistent with an enstrophy behaviour, consistent with an enstrophy cascade.cascade.

Introduction (cont.)

A k^(-5/3) law is evident at the mesoscales A k^(-5/3) law is evident at the mesoscales (below 600 km).(below 600 km).

Attempts using 2D, 3D and Quasi-Attempts using 2D, 3D and Quasi-geostrophic turbulence theory to explain the geostrophic turbulence theory to explain the “spectral kink” at around 600 km have not “spectral kink” at around 600 km have not been wholly satisfactory.been wholly satisfactory.

Introduction (cont.)

In this presentation, we will examine In this presentation, we will examine observational evidence and review attempts observational evidence and review attempts to explain the spectrum theoretically.to explain the spectrum theoretically.

We will also consider the reasons why the We will also consider the reasons why the spectral kink is not found in the ECMWF spectral kink is not found in the ECMWF model.model.

Quasi-Geostophic Turbulence

The typical aspect ratio of the atmosphere is The typical aspect ratio of the atmosphere is 100:1 (assuming 1000 km in the horizontal 100:1 (assuming 1000 km in the horizontal and 10 km in the vertical).and 10 km in the vertical).

Quasi-Geostophic Turbulence

The typical aspect ratio of the atmosphere is The typical aspect ratio of the atmosphere is 100:1 (assuming 1000 km in the horizontal 100:1 (assuming 1000 km in the horizontal and 10 km in the vertical).and 10 km in the vertical).

Is quasi-geostrophic turbulence more like Is quasi-geostrophic turbulence more like 2D or 3D turbulence?2D or 3D turbulence?

QG Turbulence: 2D or 3D?

2D Turbulence2D Turbulence Energy and Enstrophy conservedEnergy and Enstrophy conserved No vortex stretchingNo vortex stretching


2D Turbulence2D Turbulence Energy and Enstrophy conservedEnergy and Enstrophy conserved No vortex stretchingNo vortex stretching

3D Turbulence3D Turbulence Enstrophy not conservedEnstrophy not conserved Vortex stretching presentVortex stretching present


Quasi-Geostrophic TurbulenceQuasi-Geostrophic Turbulence

Energy & Enstrophy conserved (like 2D)Energy & Enstrophy conserved (like 2D)

Vortex stretching present (like 3D)Vortex stretching present (like 3D)


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


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

The mathematical similarity of 2D and QG The mathematical similarity of 2D and QG flows prompted Charney (1971) to conclude flows prompted Charney (1971) to conclude that an energy cascade to small-scales is that an energy cascade to small-scales is impossible in QG turbulence.impossible in QG turbulence.

Some Early Results

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

Some Early Results


Charney (1971) used Charney (1971) used FjFjørtoftørtoft’s proofs to derive ’s proofs to derive

the conservation laws for QG turbulence.the conservation laws for QG turbulence.

Some Early Results


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

The proof used is really just a convergence The proof used is really just a convergence requirement for a spectral representation of requirement for a spectral representation of enstrophy.enstrophy. (Tung & Orlando, 2003) (Tung & Orlando, 2003)

2D Turbulence

Standard 2D turbulence theory predicts:Standard 2D turbulence theory predicts:

2D Turbulence


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

2D Turbulence


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

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

2D Turbulence

Inverse Energy Inverse Energy CascadeCascade

Forward Enstrophy Forward Enstrophy CascadeCascade

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)(−

∝ kkE ε

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E(k)∝η2

3k−3

The Nastrom & Gage Spectrum

Observational Evidence

The primary source of observational evidence of The primary source of observational evidence of the atmospheric spectrum remains (over 20 years the atmospheric spectrum remains (over 20 years later!) the study undertaken by Nastrom and Gage later!) the study undertaken by Nastrom and Gage (1985)(1985)

They examined data collated by nearly 7,000 They examined data collated by nearly 7,000 commercial flights between 1975 and 1979.commercial flights between 1975 and 1979.

80% of the data was taken between 3080% of the data was taken between 30º and 55ºN.º and 55ºN.


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



Velocity and Temperature spectra have the Velocity and Temperature spectra have the same nearly universal shape.same nearly universal shape.



Velocity and Temperature spectra have the Velocity and Temperature spectra have the same nearly universal shape.same nearly universal shape.

Little seasonal or latitudinal variation.Little seasonal or latitudinal variation.

Observed Power-Law Behaviour

Two robust power laws were evident:Two robust power laws were evident:



The spectrum has slope close to –(5/3) for The spectrum has slope close to –(5/3) for the range of scales up to 600 km.the range of scales up to 600 km.



The spectrum has slope close to –(5/3) for The spectrum has slope close to –(5/3) for the range of scales up to 600 km.the range of scales up to 600 km.

At larger scales, the spectrum steepens At larger scales, the spectrum steepens considerably to a slope close to –3.considerably to a slope close to –3.

The N & G Spectrum (again)

The Spectral “Kink”

The observational evidence outlined above The observational evidence outlined above showed a kink at around 600 kmshowed a kink at around 600 km


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


The observational evidence outlined above The observational evidence outlined above showed a kink at around 600 kmshowed a kink at around 600 km

Surely too large for isotropic 3D effects?Surely too large for isotropic 3D effects?

Nastrom & Gage (1986) suggested the Nastrom & Gage (1986) suggested the shortwave –5/3 slope could be explained by shortwave –5/3 slope could be explained by another inverse energy cascade from storm another inverse energy cascade from storm scales. (after Larsen, 1982)scales. (after Larsen, 1982)

Larsen’s Suggested Spectrum

The Spectral “Kink” (cont.)

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

The Spectral “Kink” (cont.)

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

Tung and Orlando (2002) suggested that the Tung and Orlando (2002) suggested that the shortwave k^(-5/3) behaviour was due to a shortwave k^(-5/3) behaviour was due to a small downscale energy cascade from the small downscale energy cascade from the synoptic scales.synoptic scales.

The Spectral Kink

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

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

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

Tung & Gkioulekas (2005) proposed a similar form.Tung & Gkioulekas (2005) proposed a similar form.

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E(k) ≈ Ak−3 + Bk−5

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Current View of Spectrum

Energy is injected at scales associated with Energy is injected at scales associated with baroclinic instability.baroclinic instability.



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



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

Large-scale energy may be dissipated by Large-scale energy may be dissipated by Ekman damping.Ekman damping.

Current Picture (cont.)

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


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

At synoptic scales, the downscale energy At synoptic scales, the downscale energy cascade is spectrally dominated by the cascade is spectrally dominated by the k^(-3) enstrophy cascade.k^(-3) enstrophy cascade.


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



The k^(-5/3) slope is evident at scales smaller than The k^(-5/3) slope is evident at scales smaller than this.this.



The k^(-5/3) slope is evident at scales smaller than The k^(-5/3) slope is evident at scales smaller than this.this.

The k^(-5/3) slope is probably augmented by an The k^(-5/3) slope is probably augmented by an inverse energy cascade from storm scales.inverse energy cascade from storm scales.

Inverse Enstrophy Cascade?

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



We are unlikely, however, to find evidence We are unlikely, however, to find evidence of large-scale k^(-3) behaviour.of large-scale k^(-3) behaviour.



We are unlikely, however, to find evidence We are unlikely, however, to find evidence of large-scale k^(-3) behaviour.of large-scale k^(-3) behaviour. The Earth’s circumference dictates the The Earth’s circumference dictates the

size of the largest scale.size of the largest scale.

ECMWF Model Output

The k^(-5/3) “kink” at mesoscales is not The k^(-5/3) “kink” at mesoscales is not evident in the ECMWF model output.evident in the ECMWF model output.

ECMWF Model Output

The k^(-5/3) “kink” at mesoscales is not The k^(-5/3) “kink” at mesoscales is not evident in the ECMWF model output.evident in the ECMWF model output.

Excessive damping of energy is likely to be Excessive damping of energy is likely to be the cause.the cause.

(Thanks to Tim Palmer of ECMWF for the following figures)(Thanks to Tim Palmer of ECMWF for the following figures)

Energy spectrum in T799 run

E(n)

)(log10 n

3/5−k

3−k

n = spherical harmonic order

missing energy

3/5−k

3−k

ECMWF Model Output

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

ECMWF Model Output


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

ECMWF Model Output


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

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

Energy spectrum in T799 run

E(n)

)(log10 n

3/5−k

3−k

n = spherical harmonic order

missing energy

3/5−k

3−k

Energy spectrum in ECMWF forecast model with backscatter

3−k

T799

E(n)

)(log10 n

3−k

3/5−k

Outstanding Issues and Conclusions

IntermittencyIntermittency Direction of (-5/3) short-wave energy cascade?Direction of (-5/3) short-wave energy cascade? Dependent on convective activityDependent on convective activity


IntermittencyIntermittency Direction of (-5/3) short-wave energy cascade?Direction of (-5/3) short-wave energy cascade? Dependent on convective activityDependent on convective activity

Geographic VariabilityGeographic Variability Strong convective activityStrong convective activity Little data collated in tropical areasLittle data collated in tropical areas


We believe that both Energy and Enstrophy We believe that both Energy and Enstrophy flow in both directions.flow in both directions.



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



In an unbounded system, a “W-spectrum” In an unbounded system, a “W-spectrum” may arise.may arise. Enstrophy and Energy cascades exerting Enstrophy and Energy cascades exerting

spectral dominance alternately.spectral dominance alternately.


The validity of an additive spectrumThe validity of an additive spectrum

needs to be justified.needs to be justified.

€

E(k) ≈ Ak−3 + Bk−5

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energy & enstrophy cascades in the atmosphere prof. peter lynch michael clark university college...

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