Energy & Enstrophy Cascades in the Atmosphere Prof. Peter Lynch Michael Clark University College Dublin Met & Climate Centre

Introduction A full theoretical understanding of the atmospheric energy spectrum remains elusive. A full theoretical understanding of the atmospheric energy spectrum remains elusive. At synoptic and sub-synoptic scales, the energy spectrum exhibits k^(-3) power law behaviour, consistent with an enstrophy cascade. At synoptic and sub-synoptic scales, the energy spectrum exhibits k^(-3) power law behaviour, consistent with an enstrophy cascade.

Introduction (cont.) A k^(-5/3) law is evident at the mesoscales (below 600 km). A k^(-5/3) law is evident at the mesoscales (below 600 km). Attempts using 2D, 3D and Quasi- geostrophic turbulence theory to explain the “spectral kink” at around 600 km have not been wholly satisfactory. Attempts using 2D, 3D and Quasi- geostrophic turbulence theory to explain the “spectral kink” at around 600 km have not been wholly satisfactory.

Introduction (cont.) In this presentation, we will examine observational evidence and review attempts to explain the spectrum theoretically. In this presentation, we will examine observational evidence and review attempts to explain the spectrum theoretically. We will also consider the reasons why the spectral kink is not found in the ECMWF model. We will also consider the reasons why the spectral kink is not found in the ECMWF model.

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

Quasi-Geostophic Turbulence The typical aspect ratio of the atmosphere is 100:1 (assuming 1000 km in the horizontal and 10 km in the vertical). The typical aspect ratio of the atmosphere is 100:1 (assuming 1000 km in the horizontal and 10 km in the vertical). Is quasi-geostrophic turbulence more like 2D or 3D turbulence? Is quasi-geostrophic turbulence more like 2D or 3D turbulence?

QG Turbulence: 2D or 3D? 2D Turbulence 2D Turbulence  Energy and Enstrophy conserved  No vortex stretching

QG Turbulence: 2D or 3D? 2D Turbulence 2D Turbulence  Energy and Enstrophy conserved  No vortex stretching 3D Turbulence 3D Turbulence  Enstrophy not conserved  Vortex stretching present

QG Turbulence: 2D or 3D? Quasi-Geostrophic Turbulence Quasi-Geostrophic Turbulence  Energy & Enstrophy conserved (like 2D)  Vortex stretching present (like 3D)

QG Turbulence: 2D or 3D? The prevailing view had been that QG turbulence is more like 2D turbulence. The prevailing view had been that QG turbulence is more like 2D turbulence.

QG Turbulence: 2D or 3D? The prevailing view had been that QG turbulence is more like 2D turbulence. The prevailing view had 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. 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.

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

Some Early Results Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales. 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. Charney (1971) used Fjørtoft’s proofs to derive the conservation laws for QG turbulence.

Some Early Results Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales. 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. 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) 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: Standard 2D turbulence theory predicts:

2D Turbulence Standard 2D turbulence theory predicts: Standard 2D turbulence theory predicts:  Inverse energy cascade from the point of energy injection (spectral slope –5/3)

2D Turbulence Standard 2D turbulence theory predicts: Standard 2D turbulence theory predicts:  Inverse energy cascade from the point of energy injection (spectral slope –5/3)  Downscale enstrophy cascade to smaller scales (spectral slope –3)

2D Turbulence Inverse Energy Cascade Inverse Energy Cascade Forward Enstrophy Cascade Forward Enstrophy Cascade

The Nastrom & Gage Spectrum

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) The primary source of observational evidence of the atmospheric spectrum remains (over 20 years later!) the study undertaken by Nastrom and Gage (1985) They examined data collated by nearly 7,000 commercial flights between 1975 and They examined data collated by nearly 7,000 commercial flights between 1975 and % of the data was taken between 30º and 55ºN. 80% of the data was taken between 30º and 55ºN.

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

Observational Evidence No evidence of a broad mesoscale “energy gap”. No evidence of a broad mesoscale “energy gap”. Velocity and Temperature spectra have the same nearly universal shape. Velocity and Temperature spectra have the same nearly universal shape.

Observational Evidence No evidence of a broad mesoscale “energy gap”. No evidence of a broad mesoscale “energy gap”. Velocity and Temperature spectra have the same nearly universal shape. Velocity and Temperature spectra have the 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:

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

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

The N & G Spectrum (again)

The Spectral “Kink” The observational evidence outlined above showed a kink at around 600 km The observational evidence outlined above showed a kink at around 600 km

The Spectral “Kink” The observational evidence outlined above showed a kink at around 600 km 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 The observational evidence outlined above showed 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 shortwave –5/3 slope could be explained by another inverse energy cascade from storm scales. (after Larsen, 1982) Nastrom & Gage (1986) suggested the shortwave –5/3 slope could be explained by another inverse energy cascade from 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. 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. 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. 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.) 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) The NMM model also reproduces the spectral kink at the mesoscales when physics is included. (Janjic, EGU 2006)

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.

Current View of Spectrum Energy is injected at scales associated with baroclinic instability. Energy is injected at scales associated with baroclinic instability.

Current View of Spectrum Energy is injected at scales associated with baroclinic instability. 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)

Current View of Spectrum Energy is injected at scales associated with baroclinic instability. 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 may be dissipated by Ekman damping. Large-scale energy may be dissipated by 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.

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

Current Picture (cont.) Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum. Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.

Current Picture (cont.) Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum. Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum. The k^(-5/3) slope is evident at scales smaller than this. The k^(-5/3) slope is evident at scales smaller than this.

Current Picture (cont.) Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum. Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum. The k^(-5/3) slope is evident at scales smaller than this. The k^(-5/3) slope is evident at scales smaller than this. The k^(-5/3) slope is probably augmented by an inverse energy cascade from storm scales. The k^(-5/3) slope is probably augmented by an inverse energy cascade from storm scales.

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

Inverse Enstrophy Cascade? It is likely that a small portion of the enstrophy inversely cascades from synoptic to planetary scales. It is likely that a small portion of the enstrophy inversely cascades from synoptic to planetary scales. We are unlikely, however, to find evidence of large-scale k^(-3) behaviour. We are unlikely, however, to find evidence of large-scale k^(-3) behaviour.

Inverse Enstrophy Cascade? It is likely that a small portion of the enstrophy inversely cascades from synoptic to planetary scales. It is likely that a small portion of the enstrophy inversely cascades from synoptic to planetary scales. We are unlikely, however, to find evidence of large-scale k^(-3) behaviour. We are unlikely, however, to find evidence of large-scale k^(-3) behaviour.  The Earth’s circumference dictates the size of the largest scale.

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

ECMWF Model Output The k^(-5/3) “kink” at mesoscales is not evident in the ECMWF model output. The k^(-5/3) “kink” at mesoscales is not evident in the ECMWF model output. Excessive damping of energy is likely to be the cause. Excessive damping of energy is likely to be 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) n = spherical harmonic order missing energy

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

ECMWF Model Output Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping. 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. His modifications allow for a substantially higher amount of energy at smaller scales.

ECMWF Model Output Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping. 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. His modifications allow for a substantially higher amount of energy at smaller scales. The backscatter approach does produce the spectral kink at the mesoscales. The backscatter approach does produce the spectral kink at the mesoscales.

Energy spectrum in T799 run E(n) n = spherical harmonic order missing energy

Energy spectrum in ECMWF forecast model with backscatter T799 E(n)

Outstanding Issues and Conclusions Intermittency Intermittency  Direction of (-5/3) short-wave energy cascade?  Dependent on convective activity

Outstanding Issues and Conclusions Intermittency Intermittency  Direction of (-5/3) short-wave energy cascade?  Dependent on convective activity Geographic Variability Geographic Variability  Strong convective activity  Little data collated in tropical areas

Outstanding Issues and Conclusions We believe that both Energy and Enstrophy flow in both directions. We believe that both Energy and Enstrophy flow in both directions.

Outstanding Issues and Conclusions We believe that both Energy and Enstrophy flow in both directions. We believe that both Energy and Enstrophy flow in both directions. In an unbounded system, a “W-spectrum” may arise. In an unbounded system, a “W-spectrum” may arise.

Outstanding Issues and Conclusions We believe that both Energy and Enstrophy flow in both directions. We believe that both Energy and Enstrophy flow in both directions. In an unbounded system, a “W-spectrum” may arise. In an unbounded system, a “W-spectrum” may arise.  Enstrophy and Energy cascades exerting spectral dominance alternately.

Outstanding Issues and Conclusions The validity of an additive spectrum The validity of an additive spectrum needs to be justified.