1. Lorentz Centre 2 October, 2006

2. The Energy Spectrum of the Atmosphere Peter Lynch University College Dublin Geometric & Multi-scale Methods for Geophysical Fluid Dynamics Lorentz Centre, University of Leiden

3. Lorentz Centre 2 October, 2006 Background “Big whirls have little whirls … ”

4. Lorentz Centre 2 October, 2006

6. Figure from Davidson: Turbulence

7. Lorentz Centre 2 October, 2006 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.

8. Lorentz Centre 2 October, 2006

9. Observational evidence of the -5/3 spectrum

10. Lorentz Centre 2 October, 2006 Quasi-Geostrophic Turbulence The characteristic aspect ratio of the atmosphere is 100:1 L/H ~ 100

11. Lorentz Centre 2 October, 2006 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?

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

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

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

15. Lorentz Centre 2 October, 2006 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.

16. Lorentz Centre 2 October, 2006 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.

17. Lorentz Centre 2 October, 2006 Quasi-Geostrophic Potential Vorticity The equation of Conservation of Potential Vorticity is:  relative vorticity –f - planetary vorticity –h - fluid height

18. Lorentz Centre 2 October, 2006 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.

19. Lorentz Centre 2 October, 2006 QGPV Equation The Barotropic Quasi-Geostrophic Potential Vorticity Equation is: where.

20. Lorentz Centre 2 October, 2006 Digression on Resonant Triads (and the swinging spring … maybe … )

21. Lorentz Centre 2 October, 2006 2D versus QG 2D Case: QG Case:

22. Lorentz Centre 2 October, 2006 QG Turbulence: 2D or 3D? 2D Turbulence –Energy & Enstrophy conserved –No vortex stretching

23. Lorentz Centre 2 October, 2006 QG Turbulence: 2D or 3D? 2D Turbulence –Energy & Enstrophy conserved –No vortex stretching 3D Turbulence –Enstrophy not conserved –Vortex stretching present

24. Lorentz Centre 2 October, 2006 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)

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

26. Lorentz Centre 2 October, 2006 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.

27. Lorentz Centre 2 October, 2006

28. Inverse cascade to largest scales

29. Lorentz Centre 2 October, 2006 Inverse cascade to largest scales Inverse cascade to isolated vortices

30. Lorentz Centre 2 October, 2006 Inverse Energy Cascade matlab examples (Demo-01: QG01, QG24)

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

32. Lorentz Centre 2 October, 2006 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.

33. Lorentz Centre 2 October, 2006 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).

34. Lorentz Centre 2 October, 2006 2D Turbulence Standard 2D turbulence theory predicts:

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

36. Lorentz Centre 2 October, 2006 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)

37. Lorentz Centre 2 October, 2006 Decaying turbulence Some results for a 1024x1024 grid

38. Lorentz Centre 2 October, 2006

39. E/E(1) S/S(1)

40. Lorentz Centre 2 October, 2006 -3

41. Lorentz Centre 2 October, 2006

42. 2D Turbulence Inverse Energy Cascade Forward Enstrophy Cascade

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

44. Lorentz Centre 2 October, 2006

45. Two Mexican physicists, José Luis Aragón and Gerardo Naumis, have examined the patterns in van Gogh’s Starry Night

46. Lorentz Centre 2 October, 2006 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.

47. Lorentz Centre 2 October, 2006

48. 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.

49. Lorentz Centre 2 October, 2006 The Nastrom & Gage Spectrum

50. Lorentz Centre 2 October, 2006 Observational Evidence No evidence of a broad mesoscale “energy gap”.

51. Lorentz Centre 2 October, 2006 Observational Evidence No evidence of a broad mesoscale “energy gap”. Velocity and Temperature spectra have nearly the same shape.

52. Lorentz Centre 2 October, 2006 Observational Evidence No evidence of a broad mesoscale “energy gap”. Velocity and Temperature spectra have nearly the same shape. Little seasonal or latitudinal variation.

53. Lorentz Centre 2 October, 2006 Observed Power-Law Behaviour Two power laws were evident:

54. Lorentz Centre 2 October, 2006 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.

55. Lorentz Centre 2 October, 2006 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.

56. Lorentz Centre 2 October, 2006 The Nastrom & Gage Spectrum (again)

57. Lorentz Centre 2 October, 2006 The Spectral “Kink” The observational evidence outlined above showed a kink at around 600 km –Surely too large for isotropic 3D effects?

58. Lorentz Centre 2 October, 2006 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)

59. Lorentz Centre 2 October, 2006 Larsen’s Suggested Spectrum

60. Lorentz Centre 2 October, 2006 The Spectral “Kink” (cont.) Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.

61. Lorentz Centre 2 October, 2006 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.

62. Lorentz Centre 2 October, 2006 The Spectral Kink Tung and Orlando reproduced the N&G spectrum using QG dynamics alone. (They employed sub-grid diffusion.)

63. Lorentz Centre 2 October, 2006 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)

64. Lorentz Centre 2 October, 2006 No Physics With Physics Where is the small scale energy in the observed spectrum coming from? Atlantic case, NMM-B, 15 km, 32 Levels (Thanks to Zavisa Janjic for this slide)

65. Lorentz Centre 2 October, 2006 An Additive Spectrum? Charney (1973) noted the possibility of an additive spectrum: Tung & Gkioulekas (2005) proposed a similar form:

66. Lorentz Centre 2 October, 2006 Current View of Spectrum Energy is injected at scales associated with baroclinic instability.

67. Lorentz Centre 2 October, 2006 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)

68. Lorentz Centre 2 October, 2006 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.

69. Lorentz Centre 2 October, 2006 Current Picture (cont.) It is likely that a small portion of the injected energy cascades to smaller scales.

70. Lorentz Centre 2 October, 2006 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.

71. Lorentz Centre 2 October, 2006 Current Picture (cont.) Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.

72. Lorentz Centre 2 October, 2006 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.

73. Lorentz Centre 2 October, 2006 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.

74. Lorentz Centre 2 October, 2006 Inverse Enstrophy Cascade? It is possible that a small portion of the enstrophy inversely cascades from synoptic to planetary scales.

75. Lorentz Centre 2 October, 2006 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 Earth’s circumference dictates the size of the largest scale.

76. Lorentz Centre 2 October, 2006 ECMWF Model Output The “kink” at mesoscales is not evident in the ECMWF model output.

77. Lorentz Centre 2 October, 2006 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)

78. Lorentz Centre 2 October, 2006 Energy spectrum in T799 run E(n) n = spherical harmonic order missing energy

79. Lorentz Centre 2 October, 2006 ECMWF Model Output Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping.

80. Lorentz Centre 2 October, 2006 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.

81. Lorentz Centre 2 October, 2006 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.

82. Lorentz Centre 2 October, 2006 Energy spectrum in T799 run E(n) n = spherical harmonic order missing energy

83. Lorentz Centre 2 October, 2006 Energy spectrum in ECMWF model with backscatter T799 E(n)

84. Lorentz Centre 2 October, 2006 Some Outstanding Issues Flux Variability –Direction of (-5/3) short-wave energy cascade –Dependence on convective activity

85. Lorentz Centre 2 October, 2006 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

86. Lorentz Centre 2 October, 2006 Some Outstanding Issues Is it not possible for both Energy and Enstrophy to flow in both directions?

87. Lorentz Centre 2 October, 2006 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.

88. Lorentz Centre 2 October, 2006 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.

89. Lorentz Centre 2 October, 2006 Some Outstanding Issues The validity of an additive spectrum needs to be justified.

90. Lorentz Centre 2 October, 2006 The Okubo-Weiss Criterion ----------------------------

91. Lorentz Centre 2 October, 2006 Eample of interacting vortices (matlab program Vortex01)

92. Lorentz Centre 2 October, 2006 Thank You