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Chaos in thermal convection and the wavelet analysis of geophysical fields

Chaos in thermal convection and the wavelet analysis of geophysical fields. Lud ě k Vecsey. Dept. Of Geophysics, Charles University, Prague Geophysical Institute, Academy of Sciences of the Czech Republic, Prague. Scope of the thesis: Chapter 1: Introduction

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Chaos in thermal convection and the wavelet analysis of geophysical fields

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  1. Chaos in thermal convection and the wavelet analysis of geophysical fields Luděk Vecsey Dept. Of Geophysics, Charles University, Prague Geophysical Institute, Academy of Sciences of the Czech Republic, Prague • Scope of the thesis: • Chapter 1: Introduction • Chapter 2: Continuous wavelet transform • - definitions, kinds of wavelets, Morlet and Gaussian wavelet, scalograms, Emax and kmax • Chapter 3: Thermal convection & chaos • - model, chaos theory, low, intermediate, high and ultra-high Ra convection • Chapter 4: Results of wavelet analysis • - 2-D wavelet analysis of geoid, mixing medium and convection fields • Chapter 5: Conclusions

  2. Time - frequency analysis (a) signal, (b) Morlet and (c) Gaussian wavelet transform, (d)-(g) Gabor windowed FT Linear representations - windowed Fourier transform - wavelet transform Quadratic representations - Wigner distribution Nonlinear, nonquadratic ...

  3. Continuous wavelet transform Wavelet transform: Wf (a,b) =f(t)y*((t-b)/a) dt Fourier spectrum: FWf (a,B) = a Ff (B) Fy*(aB) - FFT for computation of the CWT Mother wavelet: y((t-b)/a) = a-1/2yo((t-b)/a) Conditions on a wavelet: - well localized in both physical and Fourier space - satisfies admissibility condition, what implies: yo(t)dt = 0 - unit L2 norm: |yo(t)|2dt = 1

  4. What kind of wavelet? • complex or real • width • shape • even or odd • vanishing moments: • tm yo(t) dt = 0 Morlet wavelet: yo(t) = p-1/4eiwote-1/2 t2 Mexican-hat wavelet: yo(t) = 2/31/2p-1/4 (1-t2) e-1/2 t2

  5. Scalograms • wavelet analysis of 1-D signal results in 2-D field (scale and shift) • wavelet analysis of 2-D field results (generally) in 4-D field (shift vector, scale and rotation) • - isotropic 2-D wavelet analysis results in 3-D field (shift vector and scale) • Problems with graphical visualization: • - slides for some fixed scales (e.g., small-, medium- and large-scale behavior) • - profile in physical space • - movie • - 3-D graphical science |Wf (k,b)|2 Emax and kmax E-max • reduction of the wavelet spectrum into two proxy quantities, Emax and kmax • 3-D wavelet spectrum will result in the two 2-D fields • detection of small-scale structure k-max k=1/a

  6. Thermal convection Boussinesq approximation - nondimensional equations, infinite Prandtl number, without internal heat sources - behavior of the system depends only on Rayleigh number Ra . v = 0 - P + 2v + RaQer = 0 dQ/dt = 2Q - v .Q - v .To Axisymmetrical shell geometry v = (vr(r,q), vq(r,q), 0) Computational aspects - code developed by Moser (1994) - finite-difference scheme - computed mostly in Minnesota Supercomputing Institute - Ra varies from 1.7x104 (grid size 50 x 100) to 1011 (grid size 1100 x 5100)

  7. Nonlinear systems, chaos • sensitive dependence on the initial conditions Routes to chaos (for finite Prandtl number) • phase space, attractors • bifurkations • strange attractor • dimension of the attractor - fractal dimension, information dimension, Lyapunov exponents and Lyapunov dimension - correlation dimension, reconstruction of the phase space

  8. Convection of different Rayleigh numbers Two-cell convection, steady Ra=1.7x104 Two-cell convection, secondary instabilities inside the cells Ra=106 Plume convection, whole-mantle plumes Ra=108 Turbulent convection, layered Ra=1010

  9. Low-Ra convection: Ra=1.7x104 SYM ASYM Steady regime Symmetrical initial temperature - 4-cell symmetrical attractor (unstable) Asymmetrical initial temperature - 2-cell symmetrical attractor (stable)

  10. Low-Ra convection: Ra=105 SYM ASYM

  11. Intermediate-Ra convection: Ra=105 - 106 Ra Kin.en. dev. Nu dev 1.7x104 4.7x104 0 % 3.6 0 % 105 3.5x105 14 % 5.1 6 % 106 4.2x106 31 % 11.6 9 % 107 1.8x107 33 % 23.9 7 % 108 4.1x108 29 % 54.5 6 % 1010 1.9x109 3 % 213.0 1 %

  12. High-Ra convection: Ra=107 - 109 Ra=106 Ra=107 Ra=108 Whitehead instabilities Ra=109

  13. Ultra-high Ra convection: Ra=1010 (1011) • qualitative change of convection, from whole-mantle plumes to layered • kinetic energy does not satisfy the power law

  14. Wavelet featured geoid • Mercator projected non-hydrostatic geoid with 4 degree latitude and longitude resolution (from Rapp and Paulis, 1990, converted and truncated by Čadek) • long wavelength anomalies have source mainly in the lower mantle (Chase, 1979) • short wavelength anomalies have a lithospheric source (Hager, 1983; Le Stunff and Ricard, 1995)

  15. Wavelet featured geoid (1) Peru-Chile Trench (2) Aleutian Trench (3) Kuril Trench (4) Japan Trench (5) Ariana Trench (6) Philippine Trench (7) New Hebrides Trench (8) Tonga & Karmadec Trench (9) Java Trench (10) South Sandwich Trench (11) Andes (12) Himalayas (13) Zagros Mts. (Alpine-Tethys Trench) (14) Congo Basin (15) Atlas Mts. (16) Mid-Atlantic Ridge (17) South-West Indian Ridge (18) Hawaii (19) Cape Verde (20) Yellowstone hotspot Small-scale wavelet spectrum of the geoid Relief of the Earth surface, ETOPO5 (1988)

  16. Mixing • Newtonian thermal convection (Ten et al., 1996), a flow is covered by a scalar field • isotropic wavelets in a strongly anisotropic medium • strong dependence of the wavelet spectra on the shape of unmixed parts in a medium • time- and scale similarity of the wavelet spectra in a well-mixed medium Global wavelet spectra (like the Fourier spectrum)

  17. Thermal convection

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