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The Energy Spectrum of the Atmosphere

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## The Energy Spectrum of the Atmosphere

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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 Lorentz Centre2 October, 2006**Background**“Big whirls have little whirls … ” Lorentz Centre2 October, 2006**Figure from Davidson: Turbulence**Lorentz Centre2 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. Lorentz Centre2 October, 2006**Quasi-Geostrophic Turbulence**• The characteristic aspect ratio of the atmosphere is 100:1 L/H ~ 100 Lorentz Centre2 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? Lorentz Centre2 October, 2006**2D Vorticity Equation**• In 2D flows, the vorticity is a scalar: • For non-divergent, non-rotating flow: Lorentz Centre2 October, 2006**2D Vorticity Equation**• If we introduce a stream function , we can write the vorticity equation as • The velocity is Lorentz Centre2 October, 2006**Quasi-Geostrophic Potential Vorticity**• In the QG formulation we seek to augment the 2D picture in two ways: Lorentz Centre2 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. Lorentz Centre2 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. Lorentz Centre2 October, 2006**Quasi-Geostrophic Potential Vorticity**• The equation of Conservation of Potential Vorticity is: - relative vorticity • f - planetary vorticity • h - fluid height Lorentz Centre2 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. Lorentz Centre2 October, 2006**QGPV Equation**• The Barotropic Quasi-Geostrophic Potential Vorticity Equation is: where . Lorentz Centre2 October, 2006**Digression on Resonant Triads(and the swinging spring …**maybe … ) Lorentz Centre2 October, 2006**2D versus QG**• 2D Case: • QG Case: Lorentz Centre2 October, 2006**QG Turbulence: 2D or 3D?**• 2D Turbulence • Energy & Enstrophy conserved • No vortex stretching Lorentz Centre2 October, 2006**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**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**QG Turbulence: 2D or 3D?**• The prevailing view has been that QG turbulence is more like 2D turbulence. Lorentz Centre2 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. Lorentz Centre2 October, 2006**Inverse cascade to largest scales**Lorentz Centre2 October, 2006**Inverse cascade to largest scales**Inverse cascade to isolated vortices Lorentz Centre2 October, 2006**Inverse Energy Cascadematlab examples(Demo-01: QG01, QG24)**Lorentz Centre2 October, 2006**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**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**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**2D Turbulence**• Standard 2D turbulence theory predicts: Lorentz Centre2 October, 2006**2D Turbulence**• Standard 2D turbulence theory predicts: • Upscale energy cascade from the point of energy injection (spectral slope –5/3) Lorentz Centre2 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) Lorentz Centre2 October, 2006**Decaying turbulenceSome results for a1024x1024 grid**Lorentz Centre2 October, 2006**E/E(1)**S/S(1) Lorentz Centre2 October, 2006**-3**Lorentz Centre2 October, 2006**2D Turbulence**• Inverse Energy Cascade • Forward Enstrophy Cascade Lorentz Centre2 October, 2006**2D Turbulence**• Inverse Energy Cascade • Forward Enstrophy Cascade What observational evidence do we have? Lorentz Centre2 October, 2006**Two Mexican physicists,**José Luis Aragón and Gerardo Naumis, have examined the patterns in van Gogh’s Starry Night Lorentz Centre2 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. Lorentz Centre2 October, 2006**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**The Nastrom & Gage Spectrum**Lorentz Centre2 October, 2006**Observational Evidence**• No evidence of a broad mesoscale “energy gap”. Lorentz Centre2 October, 2006**Observational Evidence**• No evidence of a broad mesoscale “energy gap”. • Velocity and Temperature spectra have nearly the same shape. Lorentz Centre2 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. Lorentz Centre2 October, 2006**Observed Power-Law Behaviour**• Two power laws were evident: Lorentz Centre2 October, 2006