Long-wave Runup: Difference between revisions

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<font color=red>'''UNDER  CONSTRUCTION'''</font>
<font color=red>'''UNDER  CONSTRUCTION'''</font>
== Analytical Solution ==
== Overview ==
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.


where <math>\eta</math> is the water surface elevation, <math>\rho</math> is the water density, <math>\rho_a</math> is the air density, <math>g</math> is the gravitational acceleration, <math>W</math> is the wind speed, <math>h</math> is the water depth, and <math>C</math> is a constant of integration.
== 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.
 
[[Image:Long-wave_Runup_Initial_Water_Level.png|thumb|left|600px|  Figure 1. Initial water level]]
 
<br  style="clear:both" />


== Model Setup ==
== Model Setup ==
A computational grid with constant water depth of 5 m and irregular boundaries is used in order to test the model performance. The computational grid has 60 columns and 70 rows and a constant resolution of 500 m.  
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.  


== Initial Condition ==
Table  1. Model Parameters
[[Image:Long-wave_Runup_Initial_Water_Level.jpg|thumb|left|600px|   Figure 1. Computed water surface elevation for the irregular domain with constant water depth.]]
{|border="1"
!'''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
|}


== Results ==
== Results ==
‎[[Image:Long-wave_Runup_Water_Level.jpg|thumb|none|600px| Figure 2. Comparison of computed water surface elevation to the analytical solution for an irregular basin with constant depth.]]
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.
‎[[Image:Long-wave_Runup_Water_Level.png|thumb|none|600px| Figure 2. Comparison of calculated and analytical water levels for different time steps. ]]
 
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'''Table 2. Goodness of Fit Statistics'''
{| border="1"
!  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
|}
* For a definition of the goodness of fit statistics see [[Statistics |  Goodness of fit statistics]].
 
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A comparison of the calculated and analytical shoreline position are shown in Figure 3.
 
[[Image:Long-wave_Shoreline.png|thumb|none|600px| Figure 3. Time series comparison of calculated and analytical shoreline position.]]


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== References ==
== References ==
 
* 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.
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Latest revision as of 22:06, 11 May 2011

UNDER CONSTRUCTION

Overview

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

Results

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.


References

  • 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.

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