Long-wave Runup

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The goal of this verification test to asses the model performance in simulating nonlinear runup/rundown over a plane slopt. Carrier et al. (2003) presented an analytical solution to the nonlinear shallow water equations over a plane slope for several initial wave forms. Here the analytical solution for a the initial wave form given by leading-depression N-wave is used.

Initial Condition

The bed has a constant slope of 1/10 with the initial shoreline located at x=0. Figure 1 shows the initial water level (is given by a leading depression N-wave (characteristic of the waves caused by submarine landslides). The initial current velocity is equal to zero everywhere.

Figure 1. Initial water level

Model Setup

The computational grid has a 3 m resolution for x<300 and increases to 10 m with an aspect ratio of 1.05. The general model parameters used in the simulation are shown in Table 1.

Table 1. Model Parameters

Parameter Value
Time step 0.1 s
Ramp period 0.0 s
Drying depth 0.01 m
Wall friction Off
Mixing terms Off
Manning's coefficient 0.0


Figure 2 shows a comparison of computed and analytical water surface elevations near the shoreline at 4 different time steps. The goodness of fit statistics are shown in Table 2.

Figure 2. Comparison of calculated and analytical water levels for different time steps.

Table 2. Goodness of Fit Statistics

Time, s RRMSE, % RMAE, % R^2 Bias, m
160 3.7 3.8 0.999 -0.012
175 6.5 5.9 0.997 -0.113
220 4.6 5.4 1.000 -0.066

A comparison of the calculated and analytical shoreline position are shown in Figure 3.

Figure 3. Time series comparison of calculated and analytical shoreline position.


  • Carrier, G., Wu, T.T., and Yeh, H. (2003). "Tsunami runup and draw-down on a plane beach", Journal of Fluid Mechanics, 475, 79-99.

Test Cases

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