Chaos in thermal convection and the wavelet analysis of geophysical fieldsLudk VecseyScope 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: ConclusionsDept. Of Geophysics, Charles University, PragueGeophysical Institute, Academy of Sciences of the Czech Republic, Prague

Time - frequency analysisLinear representations- windowed Fourier transform- wavelet transform

Quadratic representations- Wigner distribution

Nonlinear, nonquadratic...

(a) signal, (b) Morlet and (c) Gaussian wavelet transform, (d)-(g) Gabor windowed FT

Continuous wavelet transformWavelet 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-1/2 yo((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

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

Mexican-hat wavelet:yo(t) = 2/31/2 p-1/4 (1-t2) e-1/2 t2

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 scienceEmax and kmax 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 structurek=1/a|Wf (k,b)|2k-maxE-max

Thermal convectionBoussinesq approximation- nondimensional equations, infinite Prandtl number, without internal heat sources- behavior of the system depends only on Rayleigh number Ra . v = 0- P + 2v + Ra Q er = 0dQ/dt = 2Q - v . Q - v . To

Axisymmetrical shell geometryv = (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)

Nonlinear systems, chaos sensitive dependence on the initial conditionsRoutes 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

Convection of different Rayleigh numbersRa=1.7x104Ra=106Ra=108Ra=1010Two-cell convection, steadyTwo-cell convection, secondary instabilities inside the cellsPlume convection, whole-mantle plumesTurbulent convection, layered

Low-Ra convection: Ra=1.7x104SYMASYMSteady regime

Symmetrical initial temperature - 4-cell symmetrical attractor (unstable)Asymmetrical initial temperature- 2-cell symmetrical attractor (stable)

Low-Ra convection: Ra=105SYMASYM

Intermediate-Ra convection: Ra=105 - 106 RaKin.en. dev. Nudev1.7x1044.7x104 0 % 3.60 % 1053.5x10514 % 5.16 % 1064.2x10631 % 11.69 % 1071.8x10733 % 23.97 % 1084.1x10829 % 54.56 % 10101.9x109 3 %213.01 %

High-Ra convection: Ra=107 - 109Ra=106Ra=107Ra=108Whitehead instabilitiesRa=109

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

Wavelet featured geoid Mercator projected non-hydrostatic geoid with 4 degree latitude and longitude resolution (from Rapp and Paulis, 1990, converted and truncated by adek) 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)

Wavelet featured geoidRelief of the Earth surface, ETOPO5 (1988)Small-scale wavelet spectrum of the 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

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 mediumGlobal wavelet spectra (like the Fourier spectrum)

Thermal convection