source: documentation/experimentation/smf.tex @ 2702

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2\documentclass[reqno]{article}
3\usepackage{ae} % or {zefonts}
4\usepackage[T1]{fontenc}
5\usepackage[ansinew]{inputenc}
6\usepackage{amsmath}
7\usepackage{amssymb}
8\usepackage{graphicx}
9\usepackage{color}
10\usepackage[colorlinks]{hyperref}
11% \Add{} and \Del{} Corrections and \Mark{}
12%\usepackage[active,new,noold,marker]{xrcs}
13\usepackage{eurosym}
14\DeclareInputText{128}{\euro} % ANSI code for euro: € \usepackage{eurosym}
15\DeclareInputText{165}{\yen}  % ANSI code for yen:  ¥ \usepackage{amssymb}
16
17\usepackage{lscape} %landcape pages support
18%\input{definitions}
19
20\title{Application of SMF surface elevation function in inundation modelling}
21\date{}
22
23\begin{document}
24
25\maketitle
26
27
28Geoscience Australia (GA) is a federal government agency playing a
29critical role in enabling government and the community to make
30information decisions about exploration of resources, the management
31of the environment, the safety of critical infrastructure and the
32resultant wellbeing of all Australians. GA does this by producing
33first-class geoscientific information and knowledge.
34
35The Risk Research Group (RRG) within GA is researching natural and
36human-caused hazards to enhance Australia's risk mitigation
37capabilities through policy and decision-maker support. The group is
38working with other agencies to develop and collect information on
39natural disasters, and develop risk models for forecasting the
40impact of future hazard events.
41
42In a recent inundation study, we implemented the surface elevation
43function as described in equation 14 of Watts et al 2005, [1], for a
44slump tsunami scenario. Investigating the long term behaviour of the
45system, it was found that water was being lost from the system when
46the slump was added to the system. Further investigation showed that
47the depressed volume was greater than the volume displaced above the
48water surface with approximately 2-3 \% loss. Figure 2 of [1] shows
49a series of the surface elevation functions for various parameters
50which indicate that volume is not conserved.
51
52{\bf Question:}   Is there a physical explanation to why the volume
53of the surface elevation function should not be zero?
54
55Integrating equation 14 and solving to zero for $\kappa'$ ensures
56the system volume is conserved. As a result,
57
58$$\kappa' = [{\rm erf} ( \frac{(x - x_0)}{\sqrt \lambda_0 }) /
59{\rm erf} ( \frac{(x - \Delta x - x_0)}{\sqrt \lambda_0 })]_{x_{\rm
60min}}^{x_{\rm max}} \ .$$
61
62\noindent The relationship between $\kappa$ and $\Delta_x$ can be
63seen in Figure \ref{fig:vol_cons} where $\kappa$ approaches $\inf$
64quickly.Additionally, it is not possible for $\kappa' = 0.83$ as
65shown in Figure 2 of [1] as {\rm erf(x)} = 1 for ${\rm abs} x >
665.93$. For the example described in Figures 2 and 3 of [1], whilst
67$\kappa'$ is technically less than 1 for $\Delta x < 5$ it is
68effectively equal to 1 for $0 \le \Delta x \approx 5$.
69
70
71Figure 2 in [1]
72could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to
73ensure conservation of mass within the system. Using the above
74formulation, the values of interest shown in Figure 2 of [1] would
75be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
76Figure \ref{fig:eta_vary}.
77
78
79
80\begin{figure}[hbt]
81
82  %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.eps}}
83
84  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
85  \label{fig:vol_cons}
86\end{figure}
87
88\begin{figure}[hbt]
89
90  %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.eps}}
91
92  \caption{Surface elevation functions for
93($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.}
94  \label{fig:eta_vary}
95\end{figure}
96
97
98 {\bf TO DO:} Need a discussion in here on whether the
99modified slump surface elevation function makes a difference to the
100final impact onshore.
101
102Watts et al [1] also provide additional information on the value of
103$\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated
104as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge
105located above the SMF initial submergence location in [2]). Here $d$
106represents the depth at where the SMF is situated, $T$ the thickness
107and $\theta$ the slope of the bed. As a result, $\kappa'$ can be
108recast as
109
110$$\kappa\approx {\rm erf} ( \frac{(x - x_0)}{\sqrt\lambda_0} ) / 
111{\rm erf} ( \frac{(x - 2 x_0
112- x_g)}{\sqrt \lambda_0 )}$$
113
114\noindent thereby eliminating $\Delta x$ from the surface elevation
115function, $\eta(x,y)$. Implementing this formulation for values in
116[2] (T = 0.052m, d = 0.259m) provides the following figure
117describing the relationship between $x_0$ and $\kappa'$.
118
119%{\caption Utilising $x_g$ in determining $\kappa'$ to ensure volume
120%conservation}
121
122{\bf Question:}   Is this a realistic substitution?
123
124{\bf TO DO:} Need a discussion in here on "characteristic distance
125of motion".
126
127\section{References}
128
129[1] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005,
130Tsunami generation by submarine mass failure Part II: Predictive
131equations and case studies, Journal of Waterway, Port, Coastal, and
132Ocean Engineering, 131, 298 - 310.
133
134[2] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
135submarine mass failure Part I: Modeling, experimental validation,
136and sensitivity analyses, Journal of Waterway, Port, Coastal, and
137Ocean Engineering, 131, 283 - 297.
138
139
140\end{document}
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