Geostrophic Turbulence Atmospheric Energy

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The Energy Spectrumof theAtmosphere

• 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 Quasi-Geostrophic
  turbulence theory to explain the spectrum have
  not been wholly satisfactory.

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Observational evidence of the -5/3 spectrum
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Quasi-Geostrophic Turbulence

• The characteristic aspect ratio of the atmosphere
  is 1001
• L/H 100

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Quasi-Geostrophic Turbulence

• The characteristic aspect ratio of the atmosphere
  is 1001
• 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
• For non-divergent, non-rotating flow

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2D Vorticity Equation

• If we introduce a stream function ?, we can write
  the vorticity equation as
• The velocity is

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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 Earths 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 Earths rotation.
• We allow for horizontal divergence.

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Quasi-Geostrophic Potential Vorticity

• The equation of Conservation of Potential
  Vorticity is
• - ????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
• This allows us to relate the mass and wind
  fields.

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QGPV Equation

• The Barotropic Quasi-Geostrophic Potential
  Vorticity Equation is
• where .

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Digression on Resonant Triads(and the swinging
spring maybe )
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2D versus QG

• 2D Case
• QG Case

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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 Cascadematlab 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 Fjørtofts 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 Fjørtofts 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 turbulenceSome results for
a1024x1024 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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2D Turbulence

• Inverse Energy Cascade
• Forward Enstrophy Cascade

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 Goghs Starry Night
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Two Mexican physicists, José Luis Aragón
and Gerardo Naumis, have examined the patterns
in van Goghs 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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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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Larsens 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 NG spectrum
  using QG dynamics alone. (They employed
  sub-grid diffusion.)

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The Spectral Kink

• Tung and Orlando reproduced the NG 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)

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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
With Physics
(Thanks to Zavisa Janjic for this slide)
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An Additive Spectrum?

• Charney (1973) noted the possibility of an
  additive spectrum
• Tung Gkioulekas (2005) proposed a similar form

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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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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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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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Current Picture (cont.)

• Below about 600 km, the downscale energy cascade
  begins to dominate the energy spectrum.
• The 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.
• The slope is evident at scales smaller
  than this.
• The slope is probably augmented by an
  inverse energy cascade from convective scales.

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Inverse Enstrophy Cascade?

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

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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 Earths circumference dictates the size of
  the largest scale.

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ECMWF Model Output

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

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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)

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Energy spectrum in T799 run
E(n)
n spherical harmonic order
missing energy
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ECMWF Model Output

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

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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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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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Energy spectrum in T799 run
E(n)
n spherical harmonic order
missing energy
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Energy spectrum in ECMWF model with backscatter
T799
E(n)
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Some Outstanding Issues

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

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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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Some Outstanding Issues

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

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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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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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Some Outstanding Issues

• The validity of an additive spectrum
• needs to be justified.

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The Okubo-Weiss Criterion
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Eample of interacting vortices(matlab program
Vortex01)
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Thank You