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Tsunami modeling
Philip L-F. Liu Class of 1912 Professor
School of Civil and Environmental Engineering Cornell University
Ithaca, NY USA
PASI 2013: Tsunamis and storm surges Valparaiso, Chile
January 2-‐13, 2013
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Analytical solutions of shallow water equations
for wave runup
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Analytical solutions Two-dimensional shallow water equations
o Linear model o Nonlinear model
Simply geometry o A uniform beach o A sloping beach connected to a constant depth
Non-breaking waves o Theoretical criterion:
Frictionless sea bottom
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Analytical solution by the transform technique
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Carrier & Greenspan (1958)
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Linear waves running up a simple beach
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Shallow water equations Non-breaking solitary waves Basic idea: Keller & Keller + Carrier & Greenspan Analytical solutions on the sloping beach Run-up laws: 1/17/13 7
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Synolakis (1987): maximum runup height of solitary waves on a 1:19.85 beach
Theory
Exp (non-‐breaking)
Exp (breaking)
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Analytical solutions Bathymetry
o The 1+1 beach (e.g. Synolakis) is more useful than the uniform sloping beach (Carrier & Greenspan)
For the 1+1 beach o Linear wave model: no shoreline solutions o Nonlinear wave model: restricted to solitary waves
Questions… o Is solitary wave a good model wave for tsunamis? o Extension of the analytical approach to a more
general (non-breaking) waveform? 1/17/13 9
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Solitary waves vs. leading tsunami waves
Solitary wave o SoluJon of the KdV equaJon o Permanent form
o Very aQracJve to analyJcal and laboratory studies Solitary waves and leading tsunami waves are generally not comparable o Field observaJons o AnalyJcal arguments
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2011 Japan Tsunami: GPS wave gauge data
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Evolutions of solitary waves?
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Leading tsunami waves evolve into solitary waves?......NO!
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A better model wave for tsunamis? Solitary wave paradigm
o Extensive studies have been based on the solitary wave theory o They are still important o They are still useful: use with care
o How good (or bad) of these past findings? o Improvement on the analytical approach? o Suggestion on the laboratory experiments?
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Effect of incident waveform on the runup height
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Effect of the incident waveform on the runup height
Runup of three different sech2(·∙)-‐shape waves with the same wave height but different wavelengths. (a) Incident wave forms; (b) EvoluJons of the shoreline Jps.
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Same acceleraJng phase; different deceleraJng phase
(a) Incident wave forms; (b) EvoluJons of the shoreline. 1/17/13 17
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Analytical estimation of the runup height
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Extended analytical (asymptotic) solutions
1+1 beach model (beach slope s) Extend the approach by Synolakis (JFM 1987) Shoreline solutions in the transformed domain
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New runup solutions for cnoidal waves (Chan & Liu 2012)
Alternative expressions for the cnoidal waves
Runup solutions
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Runup of cnoidal waves
Symbols are the experimental data of Ohyama (1987). Solid lines plot the new analyJcal soluJons. Water depth is fixed at 0.2 m. Wave frequencies are different in cases (a) to (c).
Maximum runup height on a 1:1 slope.
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Extension of the analytical approach
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Runup of leading tsunami waves
Analytical solution o A quick estimation on the runup height
Use the 2011 Japan Tsunami as an example o Record at a GPS-based station as the input
Assuming a 1+1 beach with a 1-on-10 slope o Analytical approach works only for a simple beach o Estimation on the runup height: the final slope (the
closest to the shoreline) has the dominant effect on the runup (Kanoglu & Synolakis 1998)
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Estimation of the runup height
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