Project Description
The devastating effects of the 2004 tsunami have motivated increased investigation into the model- ing of water waves. To this end, the shallow water (SW) equations, which can describe the behavior of tsunami waves in the ocean, were investigated.
In the shallow water model of waves, the horizontal length scale must be much greater than the
vertical length scale. This is applicable to tsunami waves, which have a large wavelength relative
to the depth. The derivation of the SW equations results from Euler’s equations for an inviscid,
incompressible fluid. In particular, the linear SW equations were considered in two coordinate
systems: in (x, t) for Cartesian coordinates, and in (r, t) for polar coordinates.
The Cartesian form of the linear SW equations is :
†where η(x, t) is the amplitude of the wave, u(x, t) is the velocity of the wave, g is the acceleration
due to gravity, and h(x) is the depth of the water, as depicted in Figure 1. It was assumed at first
that h was constant.
The axisymmetric SW equations are :
where η(r, t), ur (r, t), g, and h(r) are defined similarly to the Cartesian case. Observe that the
primary difference between the two sets of equations is the factor of 1/r found in the ∂/∂r term.
In each coordinate system, the linear SW equations are a coupled set of partial differential
equations, which were solved for the amplitude of the wave η, and the velocity u. The method
of characteristics, which reduces partial differential equations into ordinary differential equations, was used in order to find numerical solutions to the SW equations. These solutions model the propagation of a wave in shallow water, given certain initial data and boundary conditions. Figures 2 and 3 show sample solutions to the linear SW equations in Cartesian coordinates, In this instance, u = 0 was enforced on the boundaries, and the initial amplitude and velocity of the wave at t = 0 were constant, and nonzero only in some small localized region.
The flow has exact invariants, which were used to check the accuracy of the computations. That is, the partial differential equations that make up the SW equations were integrated over x (or r), and it was detereined that the resulting invariants were exactly conserved in the Cartesian numerical solutions, and to within 10−14 in the polar solutions. This difference is a result of finite difference approximations that had to be made in the polar solution.
Although the linear SW equations for a one-dimensional surface are one of the simplest possible
models for tsunami waves, further work will expand and improve upon this model. For instance,
the work above assumes a constant ocean depth, but it is simple to generalize to variable depth
instead. Additionally, a comparison between the linear and nonlinear SW equations is expected to be useful, as is the implementation of an additional dimension. Eventually, the goal is to use the solutions to the SW equations in order to examine the behavior of tsunami waves near shore given different ocean floor topographies, as well as initial and boundary conditions.
