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Energy & Enstrophy Cascades in the Atmosphere

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.

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Energy & Enstrophy Cascades in the Atmosphere

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

  2. Introduction • 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.

  3. Introduction (cont.) • 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.

  4. Introduction (cont.) • 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.

  5. 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).

  6. 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). • Is quasi-geostrophic turbulence more like 2D or 3D turbulence?

  7. QG Turbulence: 2D or 3D? • 2D Turbulence • Energy and Enstrophy conserved • No vortex stretching

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

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

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

  11. QG Turbulence: 2D or 3D? • 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.

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

  13. 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.

  14. 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)

  15. 2D Turbulence • Standard 2D turbulence theory predicts:

  16. 2D Turbulence • Standard 2D turbulence theory predicts: • Inverse energy cascade from the point of energy injection (spectral slope –5/3)

  17. 2D Turbulence • 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)

  18. 2D Turbulence • Inverse Energy Cascade • Forward Enstrophy Cascade

  19. The Nastrom & Gage Spectrum

  20. 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) • They examined data collated by nearly 7,000 commercial flights between 1975 and 1979. • 80% of the data was taken between 30º and 55ºN.

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

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

  23. Observational Evidence • No evidence of a broad mesoscale “energy gap”. • Velocity and Temperature spectra have the same nearly universal shape. • Little seasonal or latitudinal variation.

  24. Observed Power-Law Behaviour • Two robust power laws were evident:

  25. Observed Power-Law Behaviour • Two robust power laws were evident: • The spectrum has slope close to –(5/3) for the range of scales up to 600 km.

  26. Observed Power-Law Behaviour • Two robust 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.

  27. The N & G Spectrum (again)

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

  29. The Spectral “Kink” • The observational evidence outlined above showed a kink at around 600 km • Surely too large for isotropic 3D effects?

  30. 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 storm scales. (after Larsen, 1982)

  31. Larsen’s Suggested Spectrum

  32. The Spectral “Kink” (cont.) • Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.

  33. 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.

  34. 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)

  35. An Additive Spectrum? • Charney (1973) noted the possibility of an additive spectrum. • Tung & Gkioulekas (2005) proposed a similar form.

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

  37. 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)

  38. 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 may be dissipated by Ekman damping.

  39. Current Picture (cont.) • It is likely that a small portion of the injected energy cascades to smaller scales.

  40. 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.

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

  42. Current Picture (cont.) • 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.

  43. Current Picture (cont.) • 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 probably augmented by an inverse energy cascade from storm scales.

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

  45. Inverse Enstrophy Cascade? • 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.

  46. Inverse Enstrophy Cascade? • 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. • The Earth’s circumference dictates the size of the largest scale.

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

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

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

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

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