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%\title{Application of SMF surface elevation function in inundation modelling}
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May 2006

Dr Phil Watts

Applied Fluids Engineering

Long Beach California

USA

phil.watts@appliedfluids.com

Dear Phil,
\parindent 15pt

{\bf Ref: Application of sediment mass failure surface elevation function
in inundation modelling}

Geoscience Australia (GA) is a federal government agency playing a
critical role in enabling government and the community to make
information decisions about exploration of resources, the management
of the environment, the safety of critical infrastructure and the
resultant wellbeing of all Australians. GA does this by producing
first-class geoscientific information and knowledge.

The Risk Research Group (RRG) within GA is researching natural and
human-caused hazards to enhance Australia's risk mitigation
capabilities through policy and decision-maker support. The group is
working with other agencies to develop and collect information on
natural disasters, and develop risk models for forecasting the
impact of future hazard events.

The risks posed by tsunamis is one of the natural hazards areas which
the RRG is investigating. GA can model the propagation of an event 
generated through a submarine earthquake
through to inundation ashore. Currently, we are
employing the Method of Splitting Tsunami (MOST) [1] for the event
and subsequent propagation in deep water, and then use ANUGA to 
propagate the resultant waves in shallow water and onshore.

ANUGA has been developed by GA and ANU to solve the nonlinear shallow water 
wave equation using the finite volume technique (described in [2]). 
An advantage of this technique is that the cell resolution can be changed 
according to areas of interest. ANUGA is under constant development and 
validation investigations.

A recent tsunami inundation study called for the tsunami source to
be a slump and as such, we implemented the surface elevation
function as described in equation 14 of Watts et al 2005, [3]. The reason
then for our contact is that we have two questions.

{\bf Question 1:}   Is there a physical explanation to why the volume
of the surface elevation function should not be zero?

Investigating the long term behaviour of the
system, we found that water was being lost from the system when
the slump was added to the system. Further investigation showed that
the depressed volume was greater than the volume displaced above the
water surface with approximately 2-3 \% loss. Figure 2 of [3] shows
a series of the surface elevation functions for various parameters
which indicate that volume is not conserved.

Setting the integral of the elevation function to zero will 
ensure that volume is conserved. As a result,

$$\kappa' = [
{\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) / 
{\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 }) 
]_{x_{\rm min}}^{x_{\rm max}} \ .$$

\noindent Figure \ref{fig:vol_cons} shows the relationship between
$\kappa$ and $\Delta x$. It must be noted, that whilst 
$\kappa'$ is technically less than 1 for $\Delta x < 5.93$ it is
effectively equal to 1 for $0 \le \Delta x \approx 5.93$. Therefore
it is not possible for $\kappa' = 0.83$; a parameter chosen in [1].

Figure 2 in [3]
could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to
ensure volume conservation within the system. Using the above
formulation, the values of interest shown in Figure 2 of [3] would
be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
Figure \ref{fig:eta_vary}.


\begin{figure}

  \centerline{ \includegraphics[width=100mm, height=75mm]{volume_conservation.png}}

  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
  \label{fig:vol_cons}
\end{figure}

\begin{figure}[hbt]

  \centerline{ \includegraphics[width=100mm, height=75mm]{redo_figure.png}}

  \caption{Surface elevation functions for
($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.}
  \label{fig:eta_vary}
\end{figure}

The next question is then how this alteration affects the impact onshore?
It is of course expected to increase the inundation depth
due to the increased volume of water which can
be propagated ashore. In one investigation, we saw little 
change to the inundation extent, but some significant increases in
maximum inundation depth in some locations.

{\bf Question:}   Is the substitution of $x_g$ into the elevation function
a realistic one?

Watts et al [3] provide additional information on the value of
$\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated
as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge
located above the SMF initial submergence location in [4]). Here $d$
represents the depth at where the SMF is situated, $T$ the thickness
and $\theta$ the slope of the bed. As a result, $\kappa'$ can be
recast as

$$\kappa'  \approx {\rm erf} ( \frac{x - x_0}{\sqrt\lambda_0} ) / 
{\rm erf} ( \frac{x - 2 x_0
- x_g}{\sqrt \lambda_0 } )$$

\noindent thereby eliminating $\Delta x$ from the surface elevation
function, $\eta(x,y)$. 

\parindent 0pt

We look forward to your response on these questions.

Yours sincerely,

Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu.

Risk Research Group, Geoscience Australia.

{\bf References}

[1] 
Titov, V.V., and F.I. Gonzalez (1997), Implementation and testing of
the Method of Splitting Tsunami (MOST) model, NOAA Technical Memorandum
ERL PMEL-112.

[2] Nielsen, O., S. Robers, D. Gray, A. McPherson, and A. Hitchman (2005)
Hydrodynamic modelling of coastal inundation, MODSIM 2005 International
Congress on Modelling and Simulation. Modelling and Simulation Society 
of Australian and New Zealand, 518-523, \newline URL: 
http://www.msanz.org.au/modsim05/papers/nielsen.pdf

[3] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005,
Tsunami generation by submarine mass failure Part II: Predictive
equations and case studies, Journal of Waterway, Port, Coastal, and
Ocean Engineering, 131, 298 - 310.

[4] Grilli, S.T. and Watts, P., 2005, Tsunami generation by
submarine mass failure Part I: Modeling, experimental validation,
and sensitivity analyses, Journal of Waterway, Port, Coastal, and
Ocean Engineering, 131, 283 - 297.

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