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Earthquakes

Earthquakes. A real world application of chaos Earth’s crust made up of tectonic plates Over time crust deforms - forces build up between plates Energy released by sudden movement of one plate relative to another - earthquake. Questions. Would like to know when will earthquake occur ?

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Earthquakes

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  1. Earthquakes • A real world application of chaos • Earth’s crust made up of tectonic plates • Over time crust deforms - forces build up between plates • Energy released by sudden movement of one plate relative to another - earthquake

  2. Questions • Would like to know • when will earthquake occur ? • how big will it be ? • Economic and human costs • Very complex problem • use a simple model • See that earthquakes are intrinsically chaotic phenomenon

  3. Gutenberg-Richter law • Magnitude=log(E released) • Very many more small earthquakes than large ones • f(M)=Aexp(-bM) • b approx 1.0 • Would like a model that reproduces this feature.

  4. Gutenberg Richter law

  5. Model • Imagine two plates move at speed v relative to each other • Material between plates modeled by blocks of mass m • These blocks sit on the bottom plate and are attached to the top via springs • Coupled to each other through further springs • Frictional force between blocks and bottom

  6. Model

  7. Comments • Springs are simple model for elastic forces in rocks and between plates and crust material • Want simplest model which can reproduce Gutenberg-Richter • Need to find out what essential physics is needed to produce interesting phenomena

  8. Frictional Force • Force opposes motion of blocks • Maximum when stationary - decreases as speed increases F v

  9. Forces on blocks • Force on block i due to neighbors F=-k(xi-xi+1)-k(xi-xi-1) • Force due to leaf similar F=-c(xi-vt) • Add friction F - when blocks not moving takes on whatever value it needs to keep stationary up to some maximum.

  10. Newton’s laws • F=ma: mdvi/dt=k(xi+1+xi-1-2xi)+ c(vt-xi)+F dxi/dt=vi • Discretize time - time step Dt • Get nonlinear maps for each block position • N blocks

  11. What happens ... • Initially all blocks are at rest • As top plate moves leaf springs stretch and frictional force builds • At some point friction cannot hold blocks - several slip at the same time - an earthquake ! • Eventually come to a halt because leaf springs become unstretched and friction slows down blocks

  12. Note • Detailed set of earthquakes depends sensitively on the initial positions (chaotic ?) • Can make a histogram of the number of earthquakes as a function of their magnitude • See GR law ! • (Need N=100 or so)

  13. Distribution of Magnitudes

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