Modelling of Charles Darwin earthquake reports as catastrophic wave phenomena

Abstract

As it is the 200th anniversary of Darwin’s birth, 2009 has also been marked as 170 years since the publication of his book Journal of Researches, commonly referred to as The Voyage of the Beagle. During the voyage Darwin landed at Valdivia and Concepcion, Chile, just before, during and after a great earthquake which demolished hundreds of buildings, killing and injuring many people. It was a giant natural catastrophe. He saw the land rise before his eyes. Land was waved, lifted and cracked, volcanoes awoke and giant ocean waves attacked the coast. There are two main goals of this book. The first is emphasising the priority of Darwin in the description and the analysis of the results of the severe earthquakes (Chapter I). Extracts from Darwin’s Diary and Narrative 2 , ‘Journal of Researches’ and ‘The autobiography of Charles Darwin’ are presented. In the extracts Darwin described a few days of his work. Perhaps, those days were among the most important days of his life. We group the material of the extracts so that a reader can trace the evolution of Darwin’s thoughts. The key observations and ideas of Darwin, presented in the material, are shortly formulated. Then these ideas are analysed and compared with modern experimental and theoretical data. In particular, Darwin wrote ‘… the whole body of the sea retires from the coast, and then returns in great waves of overwhelming force...’, ‘ …Santa Maria was upheaved nine feet…’ (Santa Maria is the island), ‘… the displacement at first appears to be owing to a vorticose movement beneath each point thus affected;…’. Taking into account Darwin’ key ideas we construct the mathematical models of natural catastrophic phenomena. Chapter II is devoted to catastrophic ocean waves. The Lagrangian description is used. Highly nonlinear wave equations are derived, which describe the evolution of the waves propagating over a variable depth. An attention is focused on the transresonant evolution of periodic ocean waves, catastrophic waves and tsunami. An appearance of extreme waves is explained by resonant effects. The theory of uplift, loosening and rupture of weakly cohesive geomaterials, gassy soils and magma under sharp decompression within tension seismic waves is developed in Chapter III. In particular, our attention was attracted by Darwin’ words ‘… a severe earthquake, may, I think, be attributed to the disturbance of mud containing organic matter…’. Because of the global warming the mathematical description of properties of gassy liquids and soils becomes more and more timely. The last Chapter is devoted to Nonlinear Science problems, in particular, to the evolution of initially smooth wave motion into vortex motion, and turbulence. This evolution can take place in many layered systems: ground, ocean, air and plasma. The generation of elastica (mushroom)-like waves, surface drops and jets, vortices and turbulence is simulated by the same highly nonlinear wave equation. The transresonant evolution of highly nonlinear waves is studied. The transition to turbulence of these waves is modelled. It is emphasised that respectively simple nonlinear wave equations can describe wide spectre of catastrophic wave phenomena.