# REU Summer 2013

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For summer 2013, the Research Experience for Undergraduates (REU) program in Fairbanks, Alaska will build on the success of the summer 2012 REU program and will focus on the mathematical modeling of tsunami waves. You can read about tsunamis here: http://ngm.nationalgeographic.com/2012/02/tsunami/folger-text/1

You can also watch some rare footage of the recent Japan tsunami here: http://video.nationalgeographic.com/video/news/environment-news/japan-tsunami-2011-vin/

Tsunamis are typically generated by either the motion of tectonic plates or landslides. Tectonic tsunamis are the most common and can cause devastating destruction. (e.g., the 2011 Japan and 2004 Sumatra events). Landslides can also generate noticeable tsunamis such as the one in Lituya Bay, Alaska in 1958 that produced a wave that surged 1,700 feet up a hillside.

## Tsunami-type wave generated from Child's glacier calving, REU Summer 2012 |

While it is nearly impossible to be an eyewitness to a real tsunami - you literally have to be in the wrong place at the wrong time — the REU program participants will, in fact, have a rare opportunity to watch this unique phenomenon in action. A field trip to observe a tidal bore wave, which is produced by the gravitational pull of the Moon or tsunami waves produced by glacier calving. While the magnitude of these phenomena is not as large as that of catastrophic events, it is a fascinating event and its occurrence is quite predictable.

Tsunami waves are modeled by nonlinear partial differential equations and the most important issues are their formation, propagation, and inundation. We are going to concentrate on the last one, also called the tsunami wave run-up problem. The tsunami run-up/run-down process can be effectively modeled within the theory of shallow water waves by a nonlinear hyperbolic system. In the influential paper *G.F. Carrier, H.P. Greenspan (1958) Water waves of finite amplitude on a sloping beach, J. Fluid Mech. 4, 97-109*, this system was explicitly solved for the case of a sloping (inclined) plane beach through a hodograph type transformation. More recently, in *I. Didenkulova, E. Pelinovsky (2011) Runup of Tsunami Waves in U-Shaped Bays. Pure Appl. Geophys. 168, 1239-1249*, the Carrier-Greenspan approach was generalized to sloping U-shaped bays and in *A. Rybkin, E. Pelinovsky, and I. Didenkulova (2013) Nonlinear wave runup in bays of arbitrary cross-section: generalization of the Carrier-Greenspan approach (preprint)*, to a wide range of bathymetries of arbitrary shape.

Besides a plane sloping beach, explicit solutions have however been obtained and analyzed for parabolic bays only. For any other shapes numerical computations become necessary. The main result of last summer's REU program was the development of a fast and extremely stable numerical algorithm to model a trapezoidal bay (see final presentation) that allows producing real time tsunami run-up/run-down simulations. This can directly be used to estimate run-up of the leading tsunami wave and the subsequent inundation without time consuming numerical computations. The latter is very important for tsunami broadcast in the coastal zone of Alaska where the use of expensive full-scale numerical models is not feasible due to the low population density.

## Child's glacier calving, REU Summer 2012 |

The summer 2013 REU program will build on this progress. Namely, we will be modeling tsunami inundation in more complicated bathymetries. Our ambitious goal is to test our approach on the 2011 Japan tsunami.

From the mathematical point of view, the problem essentially comes down to solving the Klein-Gordon equation which spatial component is the Schrödinger equation — a famous equation in quantum mechanics. Thus our approach connects a tsunami wave and quantum mechanics and students will be able to learn about and use a variety of mathematics from different areas. Among the variety of interesting problems to tackle is the study of resonance phenomena, i.e., what kinds of bays tend to amplify tsunami waves propagating through them.

This program is supported by a National Science Foundation grant. I intend to offer up to three (3) eight-week-long summer scholarships. The stipend amount is competitive. Lodging and travel expenses will be covered for out-of-state participants. Travel expenses within Alaska will also be paid for.

Prerequisites:

- Active interest in the topic of the program.
- Active status of an undergraduate student during the program.
- Minimum qualification: calculus through differential equations.
- Desired qualifications: previous exposure to partial differential equations, numerical analysis, and computer programming. Active knowledge of MATLAB is particularly desirable. Any other skills related to the project (e.g. background in hydrodynamics) can also be valuable.