Changeset 2801
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- May 4, 2006, 5:39:51 PM (19 years ago)
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documentation/experimentation/smf.tex (modified) (5 diffs)
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documentation/experimentation/smf.tex
r2748 r2801 17 17 \usepackage{lscape} %landcape pages support 18 18 %\input{definitions} 19 \topmargin 0pt 20 \oddsidemargin 10pt 21 \evensidemargin 10pt 22 \marginparwidth 0.5pt 23 \textwidth \paperwidth 24 \advance\textwidth -2.5in 19 25 20 \title{Application of SMF surface elevation function in inundation modelling}26 %\title{Application of SMF surface elevation function in inundation modelling} 21 27 \date{} 22 28 23 29 \begin{document} 24 30 25 \maketitle31 %\maketitle 26 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 48 in inundation modelling} 27 49 28 50 Geoscience Australia (GA) is a federal government agency playing a … … 40 62 impact of future hazard events. 41 63 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 64 The risks posed by tsunamis is one of the natural hazards areas which 65 the RRG is investigating. GA can model the propogation of an event 66 generated through a submarine earthquake 67 through to inundation ashore. Currently, we are 68 employing the Method of Splitting Tsunami (MOST) [1] for the event 69 and subsequent propogation in deep water, and then use ANUGA to 70 propagate the resultant waves in shallow water and onshore. 71 72 ANUGA has been developed by GA and ANU to solve the nonlinear shallow water 73 wave equation using the finite volume technique (described in [2]). 74 An advantage of this technique is that the cell resolution can be changed 75 according to areas of interest. ANUGA is under constant development and 76 validation investigations. 77 78 A recent tsunami inundation study called for the tsunami source to 79 be a slump and as such, we implemented the surface elevation 80 function as described in equation 14 of Watts et al 2005, [3]. 81 Which 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 84 of the surface elevation function should not be zero? 85 86 Investigating the long term behaviour of the 87 system, we found that water was being lost from the system when 46 88 the slump was added to the system. Further investigation showed that 47 89 the depressed volume was greater than the volume displaced above the 48 water surface with approximately 2-3 \% loss. Figure 2 of [ 1] shows90 water surface with approximately 2-3 \% loss. Figure 2 of [3] shows 49 91 a series of the surface elevation functions for various parameters 50 92 which indicate that volume is not conserved. 51 93 52 {\bf Question:} Is there a physical explanation to why the volume 53 of the surface elevation function should not be zero? 94 Setting the integral of the elevation function to zero will 95 ensure that volume is conserved. As a result, 54 96 55 Integrating equation 14 and solving to zero for $\kappa'$ ensures 56 the system volume is conserved. As a result, 97 $$\kappa' = [ 98 {\rm erf} ( \frac{x - x_0 } {\sqrt \lambda_0 } ) / 99 {\rm erf} ( \frac{x - \Delta x - x_0}{\sqrt \lambda_0 }) 100 ]_{x_{\rm min}}^{x_{\rm max}} \ .$$ 57 101 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 105 effectively equal to 1 for $0 \le \Delta x \approx 5.93$. Therefore 106 it is not possible for $\kappa' = 0.83$; a parameter chosen in [1]. 61 107 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] 108 Figure 2 in [3] 72 109 could then be reproduced for appropriate values of $\kappa'$ and $\Delta x$ to 73 ensure conservation of masswithin the system. Using the above74 formulation, the values of interest shown in Figure 2 of [ 1] would110 ensure volume conservation within the system. Using the above 111 formulation, the values of interest shown in Figure 2 of [3] would 75 112 be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in 76 113 Figure \ref{fig:eta_vary}. 77 114 78 115 79 80 116 \begin{figure}[hbt] 81 117 82 %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.eps}}118 \centerline{ \includegraphics[width=100mm, height=75mm]{volume_conservation.eps}} 83 119 84 120 \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.} … … 88 124 \begin{figure}[hbt] 89 125 90 %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.eps}}126 \centerline{ \includegraphics[width=100mm, height=75mm]{redo_figure.eps}} 91 127 92 128 \caption{Surface elevation functions for … … 95 131 \end{figure} 96 132 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 133 The next question is then how this alteration affects the impact onshore? 134 It is of course expected to increase the inundation depth 135 due to the increased volume of water which can 136 be propagated ashore. In one investigation, we saw little 100 137 change to the inundation extent, but some significant increases in 101 138 maximum inundation depth in some locations. 102 139 103 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 141 a realistic one? 142 143 Watts et al [3] provide additional information on the value of 104 144 $\Delta x$; $x_0 - \Delta x \approx x_g$, where $x_g$ is formulated 105 145 as $x_g = d/\tan \theta + T/ \sin \theta$ (described as a gauge 106 located above the SMF initial submergence location in [ 2]). Here $d$146 located above the SMF initial submergence location in [4]). Here $d$ 107 147 represents the depth at where the SMF is situated, $T$ the thickness 108 148 and $\theta$ the slope of the bed. As a result, $\kappa'$ can be 109 149 recast as 110 150 111 $$\kappa' \approx {\rm erf} ( \frac{ (x - x_0)}{\sqrt\lambda_0} ) /112 {\rm erf} ( \frac{ (x - 2 x_0113 - x_g )}{\sqrt \lambda_0 )}$$151 $$\kappa' \approx {\rm erf} ( \frac{x - x_0}{\sqrt\lambda_0} ) / 152 {\rm erf} ( \frac{x - 2 x_0 153 - x_g}{\sqrt \lambda_0 } )$$ 114 154 115 155 \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 $\kappa'$. 156 function, $\eta(x,y)$. 119 157 120 %{\caption Utilising $x_g$ in determining $\kappa'$ to ensure volume 121 %conservation} 158 We look forward to your response on these questions. 122 159 123 {\bf Question:} Is this a realistic substitution? 160 Yours sincerely, 124 161 125 {\bf TO DO:} Need a discussion in here on "characteristic distance 126 of motion". 162 Jane Sexton, Ole Nielsen, Adrian Hitchman and Trevor Dhu. 127 163 128 \section{References} 164 Risk Research Group, Geoscience Australia. 129 165 130 [1] Watts, P., Grilli, S.T., Tappin, D.R. and Fryer, G.J., 2005, 166 \noindent {\bf References} 167 168 \noindent [1] 169 Titov, V.V., and F.I. Gonzalez (1997), Implementation and testing of 170 the Method of Splitting Tsunami (MOST) model, NOAA Technical Memorandum 171 ERL PMEL-112. 172 173 \noindent 174 [2] Nielsen, O., S. Robers, D. Gray, A. McPherson, and A. Hitchman (2005) 175 Hydrodynamic modelling of coastal inundation, MODSIM 2005 International 176 Congress on Modelling and Simulation. Modelling and Simulation Society 177 of Australian and New Zealand, 518-523, URL: 178 http://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, 131 182 Tsunami generation by submarine mass failure Part II: Predictive 132 183 equations and case studies, Journal of Waterway, Port, Coastal, and 133 184 Ocean Engineering, 131, 298 - 310. 134 185 135 [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 136 188 submarine mass failure Part I: Modeling, experimental validation, 137 189 and sensitivity analyses, Journal of Waterway, Port, Coastal, and … … 139 191 140 192 193 141 194 \end{document}
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