Modelling the Big One

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By now, most people living in the Pacific Northwest are aware of the danger posed by the Cascadia Subduction Zone that runs for more than 1000 km from Northern California to British Columbia. Every 500 years or so (on average, but with a large standard deviation!), it lets rip a Magnitude 9 earthquake comparable to the recent ones off the coast of Japan in 2011, or Sumatra in 2004, and also generates a comparable tsunami. A recent article in the New Yorker about the potential disastrous consequences helped to raise awareness (but you should also read the follow-up article that contains some clarifications as well as good advice on how to prepare in your own home). The last major quake occurred on January 26, 1700, as is known from geological evidence along the West Coast coupled with historical records of the devastating tsunami that reached Japan 10 hours later.

A tsunami propagating into the Strait of Juan de Fuca, one hour after a hypothetical Magnitude 9 earthquake offshore. Color scale is meters of surface displacement relative to sea level.

For the past several years, Professors Randy LeVeque and Loyce Adams have been working with students and other researchers across campus and beyond to apply numerical tsunami models to help the region better prepare for the next “Big One”. Randy recently appeared on King 5 News showing one of the simulations that was developed with Loyce and Frank Gonzalez (Affiliate Professor of Earth & Space Sciences), which was used in the design of the new Ocosta Elementary School. This school, on Gray’s Harbor in Westport, WA, is the first “vertical evacuation structure” to be constructed in the United States, as part of Project Safe Haven. A platform on the roof of the gymnasium can hold 1000 people and may be the only high ground that can be reached from this community before the tsunami arrives less than half an hour after the earthquake, as shown in this video. Randy, Loyce, and Frank have also done many simulations along the coast of the Olympic Peninsula and the Strait of Juan de Fuca that are being incorporated into the next generation of tsunami evacuation maps now being developed by the Washington State Department of Natural Resources. A number of related efforts have grown out of the NSF-funded M9 Project, a highly interdisciplinary study of all aspects of a Magnitude 9 event, including seismology, tsunamis, and landslides, and spanning basic geoscience, civil engineering, urban design, and public policy… and of course, applied mathematics! 

Ocosta Elementary School ground breaking ceremony. Left to Right: Brian Ho (TCF Architecture), Brian Fitzgerald (TCF Architecture), Cale Ash (Delenkolb Engineers), Randy LeVeque (UW), Loyce Adams (UW), Paula Akerlund (Superintendent, Ocosta School District), Tim Walsh (WA Division of Natural Resources), John Schelling (WA Emergency Management Division).

Where’s the math?

Tsunamis can be modeled with the shallow water equations, a two-dimensional system of nonlinear hyperbolic PDEs that are valid for modeling waves whose wavelength is much greater than the fluid depth. The ocean may seem deep with an average depth of 4 km, but is shallow compared to the wavelength of a major tsunami, which can be hundreds of kilometers. More surprisingly, the shallow water equations not only model propagation across the ocean, but can also do a very good job of modeling inundation and interaction with complex topographic features. Even though two-dimensional models of the fluid dynamics can be used, rather than full three-dimensional equations, it is also critical to use adaptive mesh refinement (AMR) since a community on the coast must generally be modeled with a very fine grid (10 meter resolution, for example), while propagation across the ocean must be modeled with much larger grid cells.

Randy has been working on software for wave propagation problems using AMR since the 1990s, with Marsha Berger at the Courant Institute and with a number of past and present students and other developers of the open source Clawpack software project (Conservation Laws Package). The application to tsunami modeling was initiated shortly before the 2004 Indian Ocean tsunami by David George (AMath PhD 2006) as his thesis project. This version of the code was originally called TsunamiClaw, but has since been renamed GeoClaw since it can also be applied to many other geophysical flow problems. In particular, Kyle Mandli (AMath PhD 2011) has concentrated on modeling hurricane storm surge in his thesis work, as a postdoc at UT-Austin, and now as an Assistant Professor at Columbia.

Dave George is now a staff scientist at the USGS Cascades Volcano Observatory, where he mostly works on modeling debris flows and landslides using a variant of GeoClaw that solves two-dimensional fluid dynamics equations for these more complex rheologies. This code has been used in particular to better understand the disastrous 2014 landslide in Oso, WA, as he described in this interview.

In addition to developing numerical models and software, the UW group has developed new techniques for probabilistic tsunami hazard assessment (PTHA), an approach to dealing with the multitude of uncertainties regarding potential future earthquakes. Working closely with seismologists, the idea is to develop a probability distribution of possible events, some of which would be much more devastating than others, and propagate this uncertainty through the tsunami model to obtain probabilistic hazard maps. There are a number of interesting mathematical challenges in representing and sampling this high-dimensional stochastic space, and combining this with other uncertainties such as the fact that the effect of a tsunami can be very different if it arrives at high tide or low tide. Some of this work is funded by FEMA for use in developing tsunami flood maps. Probabilistic hazard assessment, more broadly, is also a key component of the M9 Project.

Recently Randy has also started working on modeling seismic waves in order to better model the coupling of an earthquake with the tsunami it generates. NSF Postdoctoral Fellow Chris Vogl will be staying on another year in AMath with partial support from a Gordon and Betty Moore Foundation grant on Early Earthquake Warning to work on this project.

A number of current AMath PhD students are also involved in the tsunami modeling efforts, including Brisa Davis, Scott Moe, and Donsub Rim. In addition to Kyle Mandli and Dave George, several previous PhD students have remained active in Clawpack and GeoClaw development, including Donna Calhoun (AMath PhD 1999, now at Boise State University) and David Ketcheson (AMath PhD 2009, now at KAUST). The code is all developed openly on GitHub and many other researchers around the world have joined the community of developers. The most recent Clawpack Developers’ Workshop took place at UW in August 2016.

If you are interested in reading more about potential Cascadia Subduction Zone earthquakes, the science behind our current understanding, and many policy and preparedness issues, the recent book Full Rip 9.0 is recommended. For more details about GeoClaw modeling and the software, see