Chaos in thermal convection and the wavelet analysis of geophysical fields

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)

18. Thermal convection