source: documentation/experimentation/smf.tex @ 2801

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updates to smf doc

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