source: documentation/experimentation/smf.tex @ 2843

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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\usepackage{setspace}
12% \Add{} and \Del{} Corrections and \Mark{}
13%\usepackage[active,new,noold,marker]{xrcs}
14\usepackage{eurosym}
15\DeclareInputText{128}{\euro} % ANSI code for euro: € \usepackage{eurosym}
16\DeclareInputText{165}{\yen}  % ANSI code for yen:  ¥ \usepackage{amssymb}
17
18\usepackage{lscape} %landcape pages support
19%\input{definitions}
20\topmargin 0pt
21\oddsidemargin 10pt
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23\marginparwidth 0.5pt
24\textwidth \paperwidth 
25\advance\textwidth -2.5in
26\setstretch{1.5}
27
28%\title{Application of SMF surface elevation function in inundation modelling}
29\date{}
30
31\begin{document}
32
33%\maketitle
34
35\noindent May 2006
36
37\noindent Dr Phil Watts
38
39\noindent Applied Fluids Engineering
40
41\noindent Long Beach California
42
43\noindent USA
44
45\noindent phil.watts@appliedfluids.com
46
47\noindent Dear ,
48
49{\bf Ref: Application of sediment mass failure surface elevation function
50in inundation modelling}
51
52Geoscience Australia (GA) is a federal government agency playing a
53critical role in enabling government and the community to make
54information decisions about exploration of resources, the management
55of the environment, the safety of critical infrastructure and the
56resultant wellbeing of all Australians. GA does this by producing
57first-class geoscientific information and knowledge.
58
59The Risk Research Group (RRG) within GA is researching natural and
60human-caused hazards to enhance Australia's risk mitigation
61capabilities through policy and decision-maker support. The group is
62working with other agencies to develop and collect information on
63natural disasters, and develop risk models for forecasting the
64impact of future hazard events.
65
66The risks posed by tsunamis is one of the natural hazards areas which
67the RRG is investigating. GA can model the propogation of an event
68generated through a submarine earthquake
69through to inundation ashore. Currently, we are
70employing the Method of Splitting Tsunami (MOST) [1] for the event
71and subsequent propogation in deep water, and then use ANUGA to
72propagate the resultant waves in shallow water and onshore.
73
74ANUGA has been developed by GA and ANU to solve the nonlinear shallow water
75wave equation using the finite volume technique (described in [2]).
76An advantage of this technique is that the cell resolution can be changed
77according to areas of interest. ANUGA is under constant development and
78validation investigations.
79
80A recent tsunami inundation study called for the tsunami source to
81be a slump and as such, we implemented the surface elevation
82function as described in equation 14 of Watts et al 2005, [3].
83Which brings us to the reason for contacting you as we have two questions.
84
85{\bf Question 1:}   Is there a physical explanation to why the volume
86of the surface elevation function should not be zero?
87
88Investigating the long term behaviour of the
89system, we found that water was being lost from the system when
90the slump was added to the system. Further investigation showed that
91the depressed volume was greater than the volume displaced above the
92water surface with approximately 2-3 \% loss. Figure 2 of [3] shows
93a series of the surface elevation functions for various parameters
94which indicate that volume is not conserved.
95
96Setting the integral of the elevation function to zero will
97ensure that volume is conserved. As a result,
98
99$$\kappa' = [
100{\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) / 
101{\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 }) 
102]_{x_{\rm min}}^{x_{\rm max}} \ .$$
103
104\noindent Figure \ref{fig:vol_cons} shows the relationship between
105$\kappa$ and $\Delta x$. It must be noted, that whilst
106$\kappa'$ is technically less than 1 for $\Delta x < 5.93$ it is
107effectively equal to 1 for $0 \le \Delta x \approx 5.93$. Therefore
108it is not possible for $\kappa' = 0.83$; a parameter chosen in [1].
109
110Figure 2 in [3]
111could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to
112ensure volume conservation within the system. Using the above
113formulation, the values of interest shown in Figure 2 of [3] would
114be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
115Figure \ref{fig:eta_vary}.
116
117
118\begin{figure}[hbt]
119
120  \centerline{ \includegraphics[width=100mm, height=75mm]{volume_conservation.png}}
121
122  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
123  \label{fig:vol_cons}
124\end{figure}
125
126\begin{figure}[hbt]
127
128  \centerline{ \includegraphics[width=100mm, height=75mm]{redo_figure.png}}
129
130  \caption{Surface elevation functions for
131($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.}
132  \label{fig:eta_vary}
133\end{figure}
134
135The next question is then how this alteration affects the impact onshore?
136It is of course expected to increase the inundation depth
137due to the increased volume of water which can
138be propagated ashore. In one investigation, we saw little
139change to the inundation extent, but some significant increases in
140maximum inundation depth in some locations.
141
142{\bf Question:}   Is the substitution of $x_g$ into the elevation function
143a realistic one?
144
145Watts et al [3] provide additional information on the value of
146$\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated
147as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge
148located above the SMF initial submergence location in [4]). Here $d$
149represents the depth at where the SMF is situated, $T$ the thickness
150and $\theta$ the slope of the bed. As a result, $\kappa'$ can be
151recast as
152
153$$\kappa\approx {\rm erf} ( \frac{x - x_0}{\sqrt\lambda_0} ) / 
154{\rm erf} ( \frac{x - 2 x_0
155- x_g}{\sqrt \lambda_0 } )$$
156
157\noindent thereby eliminating $\Delta x$ from the surface elevation
158function, $\eta(x,y)$.
159
160We look forward to your response on these questions.
161
162Yours sincerely,
163
164Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu.
165
166Risk Research Group, Geoscience Australia.
167
168\noindent {\bf References}
169
170\noindent [1]
171Titov, V.V., and F.I. Gonzalez (1997), Implementation and testing of
172the Method of Splitting Tsunami (MOST) model, NOAA Technical Memorandum
173ERL PMEL-112.
174
175\noindent 
176[2] Nielsen, O., S. Robers, D. Gray, A. McPherson, and A. Hitchman (2005)
177Hydrodynamic modelling of coastal inundation, MODSIM 2005 International
178Congress on Modelling and Simulation. Modelling and Simulation Society
179of Australian and New Zealand, 518-523, URL:
180http://www.msanz.org.au/modsim05/papers/nielsen.pdf
181
182\noindent
183[3] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005,
184Tsunami generation by submarine mass failure Part II: Predictive
185equations and case studies, Journal of Waterway, Port, Coastal, and
186Ocean Engineering, 131, 298 - 310.
187
188\noindent
189[4] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
190submarine mass failure Part I: Modeling, experimental validation,
191and sensitivity analyses, Journal of Waterway, Port, Coastal, and
192Ocean Engineering, 131, 283 - 297.
193
194
195
196\end{document}
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