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28 | %\title{Application of SMF surface elevation function in inundation modelling} |
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29 | \date{} |
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30 | |
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31 | \begin{document} |
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32 | |
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33 | %\maketitle |
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34 | |
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35 | \noindent May 2006 |
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36 | |
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37 | \noindent Dr Phil Watts |
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38 | |
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39 | \noindent Applied Fluids Engineering |
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40 | |
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41 | \noindent Long Beach California |
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42 | |
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43 | \noindent USA |
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44 | |
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45 | \noindent phil.watts@appliedfluids.com |
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46 | |
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47 | \noindent Dear , |
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48 | |
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49 | {\bf Ref: Application of sediment mass failure surface elevation function |
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50 | in inundation modelling} |
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51 | |
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52 | Geoscience Australia (GA) is a federal government agency playing a |
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53 | critical role in enabling government and the community to make |
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54 | information decisions about exploration of resources, the management |
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55 | of the environment, the safety of critical infrastructure and the |
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56 | resultant wellbeing of all Australians. GA does this by producing |
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57 | first-class geoscientific information and knowledge. |
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58 | |
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59 | The Risk Research Group (RRG) within GA is researching natural and |
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60 | human-caused hazards to enhance Australia's risk mitigation |
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61 | capabilities through policy and decision-maker support. The group is |
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62 | working with other agencies to develop and collect information on |
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63 | natural disasters, and develop risk models for forecasting the |
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64 | impact of future hazard events. |
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65 | |
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66 | The risks posed by tsunamis is one of the natural hazards areas which |
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67 | the RRG is investigating. GA can model the propogation of an event |
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68 | generated through a submarine earthquake |
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69 | through to inundation ashore. Currently, we are |
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70 | employing the Method of Splitting Tsunami (MOST) [1] for the event |
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71 | and subsequent propogation in deep water, and then use ANUGA to |
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72 | propagate the resultant waves in shallow water and onshore. |
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73 | |
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74 | ANUGA has been developed by GA and ANU to solve the nonlinear shallow water |
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75 | wave equation using the finite volume technique (described in [2]). |
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76 | An advantage of this technique is that the cell resolution can be changed |
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77 | according to areas of interest. ANUGA is under constant development and |
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78 | validation investigations. |
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79 | |
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80 | A recent tsunami inundation study called for the tsunami source to |
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81 | be a slump and as such, we implemented the surface elevation |
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82 | function as described in equation 14 of Watts et al 2005, [3]. |
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83 | Which brings us to the reason for contacting you as we have two questions. |
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84 | |
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85 | {\bf Question 1:} Is there a physical explanation to why the volume |
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86 | of the surface elevation function should not be zero? |
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87 | |
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88 | Investigating the long term behaviour of the |
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89 | system, we found that water was being lost from the system when |
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90 | the slump was added to the system. Further investigation showed that |
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91 | the depressed volume was greater than the volume displaced above the |
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92 | water surface with approximately 2-3 \% loss. Figure 2 of [3] shows |
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93 | a series of the surface elevation functions for various parameters |
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94 | which indicate that volume is not conserved. |
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95 | |
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96 | Setting the integral of the elevation function to zero will |
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97 | ensure that volume is conserved. As a result, |
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98 | |
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99 | $$\kappa' = [ |
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100 | {\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) / |
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101 | {\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 }) |
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102 | ]_{x_{\rm min}}^{x_{\rm max}} \ .$$ |
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103 | |
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104 | \noindent Figure \ref{fig:vol_cons} shows the relationship between |
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105 | $\kappa$ and $\Delta x$. It must be noted, that whilst |
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106 | $\kappa'$ is technically less than 1 for $\Delta x < 5.93$ it is |
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107 | effectively equal to 1 for $0 \le \Delta x \approx 5.93$. Therefore |
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108 | it is not possible for $\kappa' = 0.83$; a parameter chosen in [1]. |
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109 | |
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110 | Figure 2 in [3] |
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111 | could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to |
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112 | ensure volume conservation within the system. Using the above |
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113 | formulation, the values of interest shown in Figure 2 of [3] would |
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114 | be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in |
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115 | Figure \ref{fig:eta_vary}. |
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116 | |
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117 | |
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118 | \begin{figure}[hbt] |
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119 | |
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120 | \centerline{ \includegraphics[width=100mm, height=75mm]{volume_conservation.png}} |
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121 | |
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122 | \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.} |
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123 | \label{fig:vol_cons} |
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124 | \end{figure} |
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125 | |
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126 | \begin{figure}[hbt] |
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127 | |
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128 | \centerline{ \includegraphics[width=100mm, height=75mm]{redo_figure.png}} |
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129 | |
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130 | \caption{Surface elevation functions for |
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131 | ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.} |
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132 | \label{fig:eta_vary} |
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133 | \end{figure} |
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134 | |
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135 | The next question is then how this alteration affects the impact onshore? |
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136 | It is of course expected to increase the inundation depth |
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137 | due to the increased volume of water which can |
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138 | be propagated ashore. In one investigation, we saw little |
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139 | change to the inundation extent, but some significant increases in |
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140 | maximum inundation depth in some locations. |
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141 | |
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142 | {\bf Question:} Is the substitution of $x_g$ into the elevation function |
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143 | a realistic one? |
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144 | |
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145 | Watts et al [3] provide additional information on the value of |
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146 | $\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated |
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147 | as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge |
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148 | located above the SMF initial submergence location in [4]). Here $d$ |
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149 | represents the depth at where the SMF is situated, $T$ the thickness |
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150 | and $\theta$ the slope of the bed. As a result, $\kappa'$ can be |
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151 | recast as |
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152 | |
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153 | $$\kappa' \approx {\rm erf} ( \frac{x - x_0}{\sqrt\lambda_0} ) / |
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154 | {\rm erf} ( \frac{x - 2 x_0 |
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155 | - x_g}{\sqrt \lambda_0 } )$$ |
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156 | |
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157 | \noindent thereby eliminating $\Delta x$ from the surface elevation |
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158 | function, $\eta(x,y)$. |
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159 | |
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160 | We look forward to your response on these questions. |
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161 | |
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162 | Yours sincerely, |
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163 | |
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164 | Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu. |
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165 | |
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166 | Risk Research Group, Geoscience Australia. |
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167 | |
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168 | \noindent {\bf References} |
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169 | |
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170 | \noindent [1] |
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171 | Titov, V.V., and F.I. Gonzalez (1997), Implementation and testing of |
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172 | the Method of Splitting Tsunami (MOST) model, NOAA Technical Memorandum |
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173 | ERL PMEL-112. |
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174 | |
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175 | \noindent |
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176 | [2] Nielsen, O., S. Robers, D. Gray, A. McPherson, and A. Hitchman (2005) |
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177 | Hydrodynamic modelling of coastal inundation, MODSIM 2005 International |
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178 | Congress on Modelling and Simulation. Modelling and Simulation Society |
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179 | of Australian and New Zealand, 518-523, URL: |
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180 | http://www.msanz.org.au/modsim05/papers/nielsen.pdf |
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181 | |
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182 | \noindent |
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183 | [3] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005, |
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184 | Tsunami generation by submarine mass failure Part II: Predictive |
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185 | equations and case studies, Journal of Waterway, Port, Coastal, and |
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186 | Ocean Engineering, 131, 298 - 310. |
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187 | |
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188 | \noindent |
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189 | [4] Grilli, S.T. and Watts, P., 2005, Tsunami generation by |
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190 | submarine mass failure Part I: Modeling, experimental validation, |
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191 | and sensitivity analyses, Journal of Waterway, Port, Coastal, and |
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192 | Ocean Engineering, 131, 283 - 297. |
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193 | |
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194 | |
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195 | |
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196 | \end{document} |
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