Changeset 2801


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Timestamp:
May 4, 2006, 5:39:51 PM (18 years ago)
Author:
sexton
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updates to smf doc

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  • documentation/experimentation/smf.tex

    r2748 r2801  
    1717\usepackage{lscape} %landcape pages support
    1818%\input{definitions}
     19\topmargin 0pt
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    1925
    20 \title{Application of SMF surface elevation function in inundation modelling}
     26%\title{Application of SMF surface elevation function in inundation modelling}
    2127\date{}
    2228
    2329\begin{document}
    2430
    25 \maketitle
     31%\maketitle
    2632
     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}
    2749
    2850Geoscience Australia (GA) is a federal government agency playing a
     
    4062impact of future hazard events.
    4163
    42 In a recent inundation study, we implemented the surface elevation
    43 function as described in equation 14 of Watts et al 2005, [1], for a
    44 slump tsunami scenario. Investigating the long term behaviour of the
    45 system, it was found that water was being lost from the system when
     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
    4688the slump was added to the system. Further investigation showed that
    4789the depressed volume was greater than the volume displaced above the
    48 water surface with approximately 2-3 \% loss. Figure 2 of [1] shows
     90water surface with approximately 2-3 \% loss. Figure 2 of [3] shows
    4991a series of the surface elevation functions for various parameters
    5092which indicate that volume is not conserved.
    5193
    52 {\bf Question:}   Is there a physical explanation to why the volume
    53 of the surface elevation function should not be zero?
     94Setting the integral of the elevation function to zero will
     95ensure that volume is conserved. As a result,
    5496
    55 Integrating equation 14 and solving to zero for $\kappa'$ ensures
    56 the system volume is conserved. As a result,
     97$$\kappa' = [
     98{\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) /
     99{\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 })
     100]_{x_{\rm min}}^{x_{\rm max}} \ .$$
    57101
    58 $$\kappa' = [{\rm erf} ( \frac{(x - x_0)}{\sqrt \lambda_0 }) /
    59 {\rm erf} ( \frac{(x - \Delta x - x_0)}{\sqrt \lambda_0 })]_{x_{\rm
    60 min}}^{x_{\rm max}} \ .$$
     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].
    61107
    62 \noindent The relationship between $\kappa$ and $\Delta_x$ can be
    63 seen in Figure \ref{fig:vol_cons} where $\kappa$ approaches $\inf$
    64 quickly.Additionally, it is not possible for $\kappa' = 0.83$ as
    65 shown in Figure 2 of [1] as {\rm erf(x)} = 1 for ${\rm abs} x >
    66 5.93$. 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
    68 effectively equal to 1 for $0 \le \Delta x \approx 5$.
    69 
    70 
    71 Figure 2 in [1]
     108Figure 2 in [3]
    72109could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to
    73 ensure conservation of mass within the system. Using the above
    74 formulation, the values of interest shown in Figure 2 of [1] would
     110ensure volume conservation within the system. Using the above
     111formulation, the values of interest shown in Figure 2 of [3] would
    75112be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
    76113Figure \ref{fig:eta_vary}.
    77114
    78115
    79 
    80116\begin{figure}[hbt]
    81117
    82   %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.eps}}
     118  \centerline{ \includegraphics[width=100mm, height=75mm]{volume_conservation.eps}}
    83119
    84120  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
     
    88124\begin{figure}[hbt]
    89125
    90   %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.eps}}
     126  \centerline{ \includegraphics[width=100mm, height=75mm]{redo_figure.eps}}
    91127
    92128  \caption{Surface elevation functions for
     
    95131\end{figure}
    96132
    97 The impact onshore is altered if the surface elevation function is altered
    98 as described. This is of course expected as there is an increased volume
    99 of water which can propagate ashore. In one investigation, we saw little
     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
    100137change to the inundation extent, but some significant increases in
    101138maximum inundation depth in some locations.
    102139
    103 Watts et al [1] also provide additional information on the value of
     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
    104144$\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated
    105145as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge
    106 located above the SMF initial submergence location in [2]). Here $d$
     146located above the SMF initial submergence location in [4]). Here $d$
    107147represents the depth at where the SMF is situated, $T$ the thickness
    108148and $\theta$ the slope of the bed. As a result, $\kappa'$ can be
    109149recast as
    110150
    111 $$\kappa'  \approx {\rm erf} ( \frac{(x - x_0)}{\sqrt\lambda_0} ) /
    112 {\rm erf} ( \frac{(x - 2 x_0
    113 - x_g)}{\sqrt \lambda_0 )}$$
     151$$\kappa'  \approx {\rm erf} ( \frac{x - x_0}{\sqrt\lambda_0} ) /
     152{\rm erf} ( \frac{x - 2 x_0
     153- x_g}{\sqrt \lambda_0 } )$$
    114154
    115155\noindent thereby eliminating $\Delta x$ from the surface elevation
    116 function, $\eta(x,y)$. Implementing this formulation for values in
    117 [2] (T = 0.052m, d = 0.259m) provides the following figure
    118 describing the relationship between $x_0$ and $\kappa'$.
     156function, $\eta(x,y)$.
    119157
    120 %{\caption Utilising $x_g$ in determining $\kappa'$ to ensure volume
    121 %conservation}
     158We look forward to your response on these questions.
    122159
    123 {\bf Question:}   Is this a realistic substitution?
     160Yours sincerely,
    124161
    125 {\bf TO DO:} Need a discussion in here on "characteristic distance
    126 of motion".
     162Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu.
    127163
    128 \section{References}
     164Risk Research Group, Geoscience Australia.
    129165
    130 [1] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005,
     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,
    131182Tsunami generation by submarine mass failure Part II: Predictive
    132183equations and case studies, Journal of Waterway, Port, Coastal, and
    133184Ocean Engineering, 131, 298 - 310.
    134185
    135 [2] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
     186\noindent
     187[4] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
    136188submarine mass failure Part I: Modeling, experimental validation,
    137189and sensitivity analyses, Journal of Waterway, Port, Coastal, and
     
    139191
    140192
     193
    141194\end{document}
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