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The Energy Spectrum of the Atmosphere

The Energy Spectrum of the Atmosphere. Peter Lynch University 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.

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The Energy Spectrum of the Atmosphere

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  1. Lorentz Centre2 October, 2006

  2. The Energy Spectrumof theAtmosphere Peter Lynch University College Dublin Geometric & Multi-scale Methods for Geophysical Fluid Dynamics Lorentz Centre, University of Leiden Lorentz Centre2 October, 2006

  3. Background “Big whirls have little whirls … ” Lorentz Centre2 October, 2006

  4. Lorentz Centre2 October, 2006

  5. Lorentz Centre2 October, 2006

  6. Figure from Davidson: Turbulence Lorentz Centre2 October, 2006

  7. 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. Lorentz Centre2 October, 2006

  8. Quasi-Geostrophic Turbulence • The characteristic aspect ratio of the atmosphere is 100:1 L/H ~ 100 Lorentz Centre2 October, 2006

  9. 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? Lorentz Centre2 October, 2006

  10. 2D Vorticity Equation • In 2D flows, the vorticity is a scalar: • For non-divergent, non-rotating flow: Lorentz Centre2 October, 2006

  11. 2D Vorticity Equation • If we introduce a stream function , we can write the vorticity equation as • The velocity is Lorentz Centre2 October, 2006

  12. Quasi-Geostrophic Potential Vorticity • In the QG formulation we seek to augment the 2D picture in two ways: Lorentz Centre2 October, 2006

  13. 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. Lorentz Centre2 October, 2006

  14. 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 allow for horizontal divergence. Lorentz Centre2 October, 2006

  15. Quasi-Geostrophic Potential Vorticity • The equation of Conservation of Potential Vorticity is: - relative vorticity • f - planetary vorticity • h - fluid height Lorentz Centre2 October, 2006

  16. 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. Lorentz Centre2 October, 2006

  17. QGPV Equation • The Barotropic Quasi-Geostrophic Potential Vorticity Equation is: where . Lorentz Centre2 October, 2006

  18. Digression on Resonant Triads(and the swinging spring … maybe … ) Lorentz Centre2 October, 2006

  19. 2D versus QG • 2D Case: • QG Case: Lorentz Centre2 October, 2006

  20. QG Turbulence: 2D or 3D? • 2D Turbulence • Energy & Enstrophy conserved • No vortex stretching Lorentz Centre2 October, 2006

  21. QG Turbulence: 2D or 3D? • 2D Turbulence • Energy & Enstrophy conserved • No vortex stretching • 3D Turbulence • Enstrophy not conserved • Vortex stretching present Lorentz Centre2 October, 2006

  22. 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) Lorentz Centre2 October, 2006

  23. QG Turbulence: 2D or 3D? • The prevailing view has been that QG turbulence is more like 2D turbulence. Lorentz Centre2 October, 2006

  24. 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. Lorentz Centre2 October, 2006

  25. Inverse cascade to largest scales Lorentz Centre2 October, 2006

  26. Inverse cascade to largest scales Inverse cascade to isolated vortices Lorentz Centre2 October, 2006

  27. Inverse Energy Cascadematlab examples(Demo-01: QG01, QG24) Lorentz Centre2 October, 2006

  28. Some Early Results • Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales. Lorentz Centre2 October, 2006

  29. 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. Lorentz Centre2 October, 2006

  30. 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). Lorentz Centre2 October, 2006

  31. 2D Turbulence • Standard 2D turbulence theory predicts: Lorentz Centre2 October, 2006

  32. 2D Turbulence • Standard 2D turbulence theory predicts: • Upscale energy cascade from the point of energy injection (spectral slope –5/3) Lorentz Centre2 October, 2006

  33. 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) Lorentz Centre2 October, 2006

  34. Decaying turbulenceSome results for a1024x1024 grid Lorentz Centre2 October, 2006

  35. Lorentz Centre2 October, 2006

  36. E/E(1) S/S(1) Lorentz Centre2 October, 2006

  37. -3 Lorentz Centre2 October, 2006

  38. Lorentz Centre2 October, 2006

  39. 2D Turbulence • Inverse Energy Cascade • Forward Enstrophy Cascade Lorentz Centre2 October, 2006

  40. 2D Turbulence • Inverse Energy Cascade • Forward Enstrophy Cascade What observational evidence do we have? Lorentz Centre2 October, 2006

  41. Lorentz Centre2 October, 2006

  42. Two Mexican physicists, José Luis Aragón and Gerardo Naumis, have examined the patterns in van Gogh’s Starry Night Lorentz Centre2 October, 2006

  43. Two Mexican physicists, José Luis Aragón and Gerardo Naumis, have examined 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. Lorentz Centre2 October, 2006

  44. Lorentz Centre2 October, 2006

  45. 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. Lorentz Centre2 October, 2006

  46. The Nastrom & Gage Spectrum Lorentz Centre2 October, 2006

  47. Observational Evidence • No evidence of a broad mesoscale “energy gap”. Lorentz Centre2 October, 2006

  48. Observational Evidence • No evidence of a broad mesoscale “energy gap”. • Velocity and Temperature spectra have nearly the same shape. Lorentz Centre2 October, 2006

  49. Observational Evidence • No evidence of a broad mesoscale “energy gap”. • Velocity and Temperature spectra have nearly the same shape. • Little seasonal or latitudinal variation. Lorentz Centre2 October, 2006

  50. Observed Power-Law Behaviour • Two power laws were evident: Lorentz Centre2 October, 2006

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