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19 | |
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20 | \title{Application of SMF surface elevation function in inundation modelling} |
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21 | \date{} |
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22 | |
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23 | \begin{document} |
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24 | |
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25 | \maketitle |
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26 | |
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27 | |
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28 | Geoscience Australia (GA) is a federal government agency playing a |
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29 | critical role in enabling government and the community to make |
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30 | information decisions about exploration of resources, the management |
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31 | of the environment, the safety of critical infrastructure and the |
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32 | resultant wellbeing of all Australians. GA does this by producing |
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33 | first-class geoscientific information and knowledge. |
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34 | |
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35 | The Risk Research Group (RRG) within GA is researching natural and |
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36 | human-caused hazards to enhance Australia's risk mitigation |
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37 | capabilities through policy and decision-maker support. The group is |
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38 | working with other agencies to develop and collect information on |
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39 | natural disasters, and develop risk models for forecasting the |
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40 | impact of future hazard events. |
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41 | |
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42 | In a recent inundation study, we implemented the surface elevation |
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43 | function as described in equation 14 of Watts et al 2005, [1], for a |
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44 | slump tsunami scenario. Investigating the long term behaviour of the |
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45 | system, it was found that water was being lost from the system when |
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46 | the slump was added to the system. Further investigation showed that |
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47 | the depressed volume was greater than the volume displaced above the |
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48 | water surface with approximately 2-3 \% loss. Figure 2 of [1] shows |
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49 | a series of the surface elevation functions for various parameters |
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50 | which indicate that volume is not conserved. |
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51 | |
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52 | {\bf Question:} Is there a physical explanation to why the volume |
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53 | of the surface elevation function should not be zero? |
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54 | |
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55 | Integrating equation 14 and solving to zero for $\kappa'$ ensures |
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56 | the system volume is conserved. As a result, |
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57 | |
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58 | $$\kappa' = [{\rm erf} ( (x - x_0)/ \sqrt \lambda_0 ) / |
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59 | {\rm erf} ( (x - \Delta x - x_0)/ \sqrt \lambda_0 )]_{x_{\rm |
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60 | min}}^{x_{\rm max}} \ .$$ |
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61 | |
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62 | \noindent The relationship between $\kappa$ and $\Delta_x$ can be |
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63 | seen in Figure \ref{fig:vol_cons} where $\kappa$ approaches $\inf$ |
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64 | quickly.Additionally, it is not possible for $\kappa' = 0.83$ as |
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65 | shown in Figure 2 of [1] as {\rm erf(x)} = 1 for ${\rm abs} x > |
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66 | 5.93$. For the example described in Figures 2 and 3 of [1], whilst |
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67 | $\kappa'$ is technically less than 1 for $\Delta x < 5$ it is |
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68 | effectively equal to 1 for $0 \le \Delta x \approx 5$. |
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69 | |
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70 | |
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71 | Figure 2 in [1] |
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72 | could then be reproduced for appropriate values of $\kappa'$ and $\Delta_x$ to |
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73 | ensure conservation of mass within the system. Using the above |
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74 | formulation, the values of interest shown in Figure 2 of [1] would |
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75 | be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in |
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76 | Figure \ref{fig:eta_vary}. |
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77 | |
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78 | |
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79 | |
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80 | \begin{figure}[hbt] |
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81 | |
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82 | %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.ps}} |
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83 | |
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84 | \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.} |
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85 | \label{fig:vol_cons} |
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86 | \end{figure} |
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87 | |
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88 | \begin{figure}[hbt] |
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89 | |
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90 | %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.ps}} |
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91 | |
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92 | \caption{Surface elevation functions for |
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93 | ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.} |
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94 | \label{fig:eta_vary} |
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95 | \end{figure} |
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96 | |
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97 | |
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98 | {\bf TO DO:} Need a discussion in here on whether the |
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99 | modified slump surface elevation function makes a difference to the |
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100 | final impact onshore. |
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101 | |
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102 | Watts et al [1] also provide additional information on the value of |
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103 | $\delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated |
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104 | as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge |
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105 | located above the SMF initial submergence location in [2]). Here $d$ |
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106 | represents the depth at where the SMF is situated, $T$ the thickness |
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107 | and $\theta$ the slope of the bed. As a result, $\kappa'$ can be |
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108 | recast as |
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109 | |
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110 | $$\kappa' \approx {\rm erf} ( (x - x_0)/ \sqrt\lambda_0 ) / {\rm erf} ( (x - 2 x_0 |
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111 | - x_g)/ \sqrt \lambda_0 )$$ |
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112 | |
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113 | \noindent thereby eliminating $\Delta x$ from the surface elevation |
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114 | function, $\eta(x,y)$. Implementing this formulation for values in |
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115 | [2] (T = 0.052m, d = 0.259m) provides the following figure |
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116 | describing the relationship between $x_0$ and $\kappa'$. |
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117 | |
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118 | %{\caption Utilising $x_g$ in determining $\kappa'$ to ensure volume |
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119 | %conservation} |
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120 | |
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121 | {\bf Question:} Is this a realistic substitution? |
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122 | |
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123 | {\bf TO DO:} Need a discussion in here on "characteristic distance |
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124 | of motion". |
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125 | |
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126 | \section{References} |
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127 | |
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128 | [1] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005, |
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129 | Tsunami generation by submarine mass failure Part II: Predictive |
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130 | equations and case studies, Journal of Waterway, Port, Coastal, and |
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131 | Ocean Engineering, 131, 298 - 310. |
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132 | |
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133 | [2] Grilli, S.T. and Watts, P., 2005, Tsunami generation by |
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134 | submarine mass failure Part I: Modeling, experimental validation, |
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135 | and sensitivity analyses, Journal of Waterway, Port, Coastal, and |
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136 | Ocean Engineering, 131, 283 - 297. |
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137 | |
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138 | |
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139 | \end{document} |
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