Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
the frequency domain correspond to the same number of angular frequencies
ω, ranging between 2π10 −1 rad/s and 2π10 3 rad/s. By taking advantage of
the fact that the results in the positive semi-axis of the frequency are complex
conjugates of those along the negative semi-axis, only half of the solution is
sought for.
The ‘indirect’ procedure is applied to calculate the source term to be
used in the wave-maker boundary condition (left boundary), being known
the elevation of the free surface at the points where the gauges are located.
The experimental results at the gauge 1 are used to estimate the source term,
while the results at gauges 2, 3, 4 and 5 are compared with the numerical
model results. The first step is to solve the field equation (3.29) together with
the boundary conditions described in section 3.1.2, where a unit source term
is imposed at the wave-maker boundary condition (3.38). Consequently the
unit solution N ′ (x, ω) is obtained in the computational domain; the value
of this variable at the point corresponding to the gauge 1 is referred to as
N ′ 1(ω). The records obtained at the gauge 1 are then used to calculate the
variable N1 (ω) as the Fourier Transform of the experimental time series of
the surface elevation. Since during the experiments, part of the generated
waves were reflected at the right boundary, the records obtained at the gauge
1 suffer of reflected and re-reflected (at the left boundary) waves. However it
was easy to estimate the time at which the reflected waves had affected the
records and to smoothly set the signal to zero, in order to avoid generation
of spurious waves in the numerical test. The source term S (ω) tobeusedin
the boundary condition is then calculated by an equation equivalent to the
(3.50). The results in the whole computational domain are given by
N (x, ω) =S (ω) N ′ (x, ω) . (4.1)
The water surface elevation η(x, t) is obtained using the Inverse Fast
Fourier Transform with respect to the time, for each computational point of
interest, such as those at the position of the wave gauges.
The results of one representative test are presented. The water depth
used in the experiment is of 0.23 m and the box released with the bottom
placed at the still water level. Figure 4.3 shows the surface elevation, η (t),
measured by the five gauges (dashed red lines) and those resulting from the
numerical simulations with the present model (solid black lines).
The waves typically present first a crest, then a trough; this is due to the
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