Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
This thesis consists of two major parts: the examination of the power of the continuous wavelet transform (CWT) followed by its application to geophysical fields and the modelling of convection in the Earth's mantle with main focus on the nonlinear behavior of the system.
The wavelet theory arose from the need of a good time-frequency representation of a signal, or a field. As there exist a number of different wavelets which could be, in principle, employed, we discuss properties of the CWT for a particular choice of the wavelet function. The Morlet wavelet used in the 1-D wavelet transform is found satisfactory for the time-frequency analysis of long signals, and the Mexican-hat wavelet used in the isotropic 2-D wavelet transform works well for finding structures of different scales in an analyzed field. The new method of E_max/k_max maps (Bergeron et al., 1999) is applied to increase the detection ability of the wavelets at small-scales.
The power of the 2-D wavelet transform is demonstrated by analysizing the non-hydrostatic geoid in the Mercator projection with a resolution of 4 degrees. The small-scale wavelet spectrum of the geoid reveals features created by lithospheric processes. The strongest signal originates from subduction zones (e.g., the "Ring of Fire"), and weaker signals are detected along several mountain belts (the Himalayas, the Andes, and the Atlas Mts.), oceanic rifts (the Mid-Atlantic Ridge), and hotspots (Hawaii). The wavelet analysis of the second field, the mixed medium, helps to monitor the mixing progress. We can take advantage of the sensitivity of the isotropic wavelet transform to unmixed anisotropic structures in the field to detect the degree of the mixing. The last case to which we applied the 2-D CWT is the temperature and vorticity fields from our thermal convection model. Here the wavelets prove their ability for both small-scale and large-scale detection. Small-scale wavelet analysis of the temperature reveals the complex structure of plumes in the ultra-high Ra convection as well as helps to find areas with numerical instabilities. Large-scale analysis of the vorticity determines the number of convection cells.
The second part of the thesis contains numerical studies of the nonlinear system of thermal convection in the classical Boussinesq approximation for infinite Prandtl number and axisymmetrical geometry. Complexity of the behavior of the nonlinear system results in different convection regimes dependending on the Rayleigh number (Ra), which was the only control parameter in the model. We classified the convection systems for Rayleigh numbers between Ra=1.7x10^4 - 10^11 into four basic regimes. The low Rayleigh number convection (for Ra varying from 1.7x10^4 to 10^5) reaches simple attractors (steady or periodic ones) of a two-cell structure. Intermediate Ra convection (Ra=10^5 - 10^6) still has a two-cell structure, but secondary plumes begin to appear. The correlation dimension for Ra=10^5 reveals the complex temporal evolution of the system. High Ra runs with Ra between 10^6 and 10^10 exhibit complex structure with waning large-scale motion and prevailing plume-plume interactions. Secondary (Whitehead) instabilities occur. Finally, ultra-high Ra convection (Ra>=10^10) has a very turbulent behavior with powerful mixing; no whole-mantle plumes exist. In the convection part of the thesis we depict the main characteristics of basic thermal convection regimes as they move from a calm, steady state to turbulent chaos with increasing Ra. These characteristics can help to substantially unify convection modelling with other fields of geodynamics. There is no doubt that without the excellent wavelet transform tool the analysis of convection systems (and many other geophysical fields) could not be satisfactorily completed.
The thesis is available in PDF (26 MB).
References
[1] Bergeron, S.Y., Vincent, A.P., Yuen, D.A., Tranchant, B.J.S.
and C. Tchong, Viewing seismic velocity anomalies with 3-D continuous
Gaussian wavelets, Geophys. Res. Lett., Vol. 26, No. 15, pp. 2,311-2,314,
1999.