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%\title{Application of SMF surface elevation function in inundation modelling}
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\begin{document}

%\maketitle

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}

We work at Geoscience Australia (GA) in the Risk Research Group 
researching risks posed by a range of natural hazards 
(http://www.ga.gov.au/urban/projects/risk/index.jsp). 
Due to recent
events and Australia's apparent vulnerabiliy to tsunami hazards, 
we are investigating the tsunami risk to Australia. To understand
impact ashore, we have developed in conjunction
with the Australian National University, a hydrodynamic model called
ANUGA which uses the finite volume technique, [1].

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 Watts et al 2005, [2]. We found this a very useful
way to incorporate another tsunami-genic event to our understanding
of tsunami risk. In trying
to implement this function however, we had some questions;

\begin{itemize}
\item
Is there a physical explanation to why the total volume
of the surface elevation function should not be zero?
\item
Should $\eta_{\rm min}$ used in the surface elevation function
be | ${\eta_{\rm min}}$ | instead?
\item
Is the substitution of $x_g$ into the elevation 
function realistic?
\end{itemize}

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. You can see from
Figure 2 of [2] that the
surface elevation function $\eta(x,y)$ indicates that
the total volume is not conserved.

However, we can alleviate this issue by finding the appropriate set of 
parameters which
will conserve volume. Setting the integral of the elevation function to zero 
and solving for $\kappa'$ yields the 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 The relationship between $\kappa'$ and $\Delta x$ is shown in
Figure \ref{fig:vol_cons}. It must be noted, that whilst 
$\kappa'$ is technically less than 1 for $\Delta x < 5.93$ it is
effectively equal to 1 for those values. From this calculation, it would
seem then that there is no appropriate $\Delta x$ for $\kappa'$ = 0.83 
(a parameter used in [2]) satisfying conservation of volume.

We've reproduced Figure 2 in [2]
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 in [2] would
be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
Figure \ref{fig:eta_vary}. Note, this has not been scaled by $\eta_{\rm min}$.


\begin{figure}

  \centerline{ \includegraphics[width=75mm, height=50mm]{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=75mm, height=50mm]{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}

For our particular test case, changing the surface elevation function 
in this way increases the inundation depth ashore by a factor greater than
the initial water loss of 2-3 \%. 

Turning to our question regarding the scaling of the surface elevation 
function formulation, we see that $\eta_{\rm min}$ is always negative
and hence
$- \eta_{O,3D} / \eta_{\rm min}$ would be always positive. This
would change the form of $\eta(x,y)$ and place the depressed volume behind
the submarine mass failure. Should then $\eta_{\rm min}$ be replaced
by |$\eta_{\rm min}$|?

Our final question is whether it is appropriate to substitute
the formulation for $x_g$ into the surface elevation function using
$x_0 - \Delta x \approx x_g$. 
($x_g$ is formulated
as $x_g = d/\tan \theta + T/ \sin \theta$ which is described as a gauge
located above the submarine mass failure
initial submergence location in [3].) In this
way, $\kappa'$ as described above would not 
be dependent on $\Delta x$, nor the subsequent surface elevation function.


We are continuing to seek out validation data sets to improve the
accuracy of our model. We recently had success in validating
the model against the Benchmark Problem $\#$2 Tsunami Run-up 
onto a complex 3-dimensional beach, as provided to the 3rd 
International Workshop on Long Wave Run-up in 2004, see [1].
We note in [4] your proposal for others to employ the benchmark
cases described there for experimental or numerical work.
Your model has been compared with the laboratory experiments in 2003 [5] and 
again in 2005 [3] with fairly good agreement. Given
the numerical model you implemented was the boundary element method, we would
be very interested in comparing our finite volume model using the
approximated surface elevation function with your 
experimental results. Would it therefore be possible for you to provide the 
experimental time series for comparison with ANUGA?

\parindent 0pt

Thanks for your time and we look forward to your response.

Yours sincerely,

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

Risk Research Group, Geoscience Australia.

\newpage
{\bf References}

[1] 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.mssanz.org.au/modsim05/papers/nielsen.pdf

[2] 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.

[3] 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.

[4] Watts, P., Imamura, F. and Grilli, S. (2000)
Comparing Model Simulations of Three Benchmark Tsunami Generation,
Science of Tsunami Hazards, 18, 2, 107-123.

[5] Enet, F., Grilli, S.T. and Watts, P. (2003), Laboratory Experiments for 
Tsunamis Generated by Underwater Landslides: 
Comparison with Numerical Modeling,
Proceedings of the Thirteenth (2003) International Offshore and
Polar Engineering Conference. The International Society of Offshore and
Polar Engineers.
\end{document}