Numerical modeling of waves for a tsunami early warning system

Chapter 2
Tsunami wave modelling
In fluid mechanics the fluid is considered as a continuum; this allows all fluid
properties to be described by mathematical functions that are continuous and
differentiable. Thus modelling the fluids kinematics and dynamics, means
applying the laws of conservation of mass and of momentum to a finite
volume of fluid. Depending on the assumptions made in order to simplify
the problem, different mathematical problem are obtained. The Navier-
Stokes equations, within the continuity equation, represent a complete set
of equations for fluid flows. Assuming that the fluid viscosity neglectable,
i.e. away from the boundary layers and turbulence, it means that the fluid
motion is irrotational (zero vorticity); in this case the Navier-Stokes equations
reduce to the Euler equations. In this chapter the main irrotational water
wave theories are outlined, and a literature review of the tsunami modelling
is given.
2.1 Irrotational water waves theories
In general for the description of the water wave motion a system of reference
as the one sketched in figure 2.1 is used, where x and y are the horizontal
coordinates and z is the vertical one pointing upward. The plane x, y for z =
0 represents the still water free surface and the distance between this plane
and the see bottom is defined by the water depth function h (x, y, t). The fluid
velocity and the fluid pressure inside the domain are defined respectively by
the vector v = {u, v, w} and the scalar p (x, y, z, t), while η (x, y, t) defines the
free water surface vertical displacement. Assuming an irrotational fluid, the
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