chap4.tex (8690B)
1 \section{Modeling the 2004 Tsunami} 2 \subsection{Description} 3 On the 26. December 2004, time 7:58 a powerful earthquake generated a tsunami 4 killing more than 275000 people and leaving millions homeless. The 5 hypocenter of the earthquake was 30 km under the floor of the Indian Ocean, 6 100 km away from Sumatra, an island in Indonesia. The earthquake displaced 7 an enormous amount of water, sending tsunami waves westwards across the 8 Indian Ocean to Sri Lanka and India and eastwards across the Andaman Basin. 9 to Thailand and Indonesia. The Earthquake occurred over 10 minutes along a 10 1000km long roughly straight line. Thereby we can model the corresponding 11 fluid mechanics as 2 dimensional where in Cartesian coordinates the 12 propagation of the tsunami wave is in $x$ direction and the $z$ direction 13 pointing upwards perpendicular to the flat ocean surface. However the 14 modeling assumption for two dimensions for the region outside the Bay of 15 Bengal is not valid, since the diffraction around islands and reflection from 16 steep shores pays a major role in the influence of the wave mechanics. Coming 17 back to the $2004$ tsunami, which raised the ocean floor a few meters to the 18 west and lowering it a few meters to the east, displacing the tectonic pates. 19 The tsunami waves featured westwards a wave of elevation, meaning a wave of 20 high amplitude, followed by a wave of depression, a wave of long wavelength 21 hitting the coastal areas of Sri Lanka and India in roughly three hours, 22 propagating a distance of approximately $1600\ \text{km}$. On the other hand 23 eastwards featuring a first a wave of depression following a wave of 24 elevation, propagating $700\ \text{km}$ in roughly two hours with a maximal 25 amplitude of $10\ \text{m}$. Observations tell us that as the tsunami waves 26 reached the shore the shape of the initial disturbance was not altered, which 27 is supported by measurements by a radar altimeter two hours after the 28 earthquake showing first an wave of elevation and then a wave of depression 29 westwards and respectively vice versa eastwards. The conclusion is made that 30 the shape of the tsunami remained approximately constant. Additionally it 31 should be mentioned that the tsunami waves reach very high amplitudes due to 32 the diminishing depth effect as they approach the shore, yet at open sea are 33 barely noticeable. A boat on open sea positioned at high depth in the region 34 of the tsunami during which the tsunami waves passed, captured the raise 35 from $\pm 0.8\ \text{m}$ of the boat over a period of $10\ \text{min}$. 36 This means that the wavelength of the tsunami wave was about $100\ 37 \text{km}$. 38 \subsection{Long Wave, Shallow Water} 39 At weakly nonlinear levels dispersion balances linearity/nonlinearity in 40 certain regimes, such balance is found in the KdV equation. Among the KdV 41 equation, the Carissa-Holm equation (CH) also features such balance since it 42 arises as a high order approximation to KdV. Further there is also the 43 regularized long wave equation usually called Benjamin–Bona–Mahony equation, or simply 44 BBM. It should be noted that KdV is a solitary wave while BBM is not. 45 Localized disturbances of flat water surfaces propagating without change of 46 form need to be two dimension waves of elevation symmetric about the crest. 47 The linear theory does not provide any approximation to solitary waves, only 48 nonlinear or weakly linear approximations. These are KdV, BBM. The KdV 49 equation is orbitaly stable, meaning the shape and form of the profile is 50 stable under small perturbations (also CH \& BBM). Each solitary wave retains 51 is local identity, where large waves are faster than small ones. Further the 52 solution of the KdV equation evolves into a set of solitary waves, with 53 tallest in from front followed by an oscillatory tail. The KdV is the 54 proper equation, for our modeling purposes of tsunami waves. The main 55 question arises if KdV enters the regime of validity, in our case for the 56 $2004$ tsunami in the Andaman Basin. Or in other words are the involved 57 geophysical scales leading to time and space scales, compatible with 58 the KdV weak nonlinearity balance. 59 \subsection{Governing equations} 60 The last two section show the derivation of the modeling equation for fluid 61 mechanics, following with nondimensionalisation and rescaling. The scaling 62 takes the same form as in section \ref{sec:nondim}, we only introduce the 63 parameter $\alpha = h_0\sqrt{gh_0} $ and get the following equations 64 \begin{align} 65 \begin{drcases} 66 u_x + w_z = 0\\ 67 u_t + \varepsilon(uu_x + wu_z) = -p_x\\ 68 \delta^2\left(w_t + \varepsilon(w w_x + w w_z)\right) = -p_y\\ 69 u_z - \delta^2 w_x = 0 70 \end{drcases} 71 \end{align} 72 on $(x, z) \in \mathbb{R}\times [0, 1+\varepsilon \eta(x, t)]$, with boundary 73 conditions 74 \begin{align} 75 \begin{drcases} 76 p = \eta \\ 77 w = h_t + \varepsilon u h_x 78 \end{drcases} 79 \text{on}\;\; z = 1+\varepsilon \eta(x, t)\\ 80 w = 0 \quad \text{on}\;\; z = 0. 81 \end{align} 82 The KdV validity arises in the region $\varepsilon = O(\delta^2)$, we are 83 going to thereby rescale $\delta$ in favour of $\varepsilon = \beta 84 \delta^2$, where $\beta = O(1)$ as in equation \ref{eq:epsdelta}: 85 \begin{align}\label{eq:epsdelta} 86 x \rightarrow \frac{\delta}{\sqrt{\varepsilon} }x, \quad t 87 \rightarrow \frac{\delta}{\sqrt{\varepsilon} }t\quad, 88 w \rightarrow \frac{\sqrt{\varepsilon} }{\delta}w. 89 \end{align} 90 This opens up the possibility to prove that provided a suitable length- \& 91 time-scales for some $\delta$ the KdV will arise as a valid approximation for 92 the evolution of the free surface waves. Given some $\varepsilon>0$ there 93 exists a time in the such that the KdV balance holds, where we introduce the 94 variables $\xi = x- t$ and $\tau = \varepsilon t$ with equation for the wave profile 95 \begin{align} 96 \eta_\tau - \frac{3}{2} \eta_{\xi\xi\xi} + \frac{1}{6} \eta \eta_\xi = 0, 97 \end{align} 98 for $\xi \in \mathbb{R}$ and $\tau > 0 $ and the boundary condition $\eta(\xi 99 ,0)$ of the initial profile at $\tau = t = 0$. On the basis of satellite 100 measurements for the Bay of Bengal we have 101 \begin{align} 102 a = 1\ \text{km}, \quad \lambda = 100\ \text{km},\quad h_0 = 4\ 103 \text{km}. 104 \end{align} 105 giving us 106 \begin{align} 107 \varepsilon = \frac{a}{h_0} = 25 \cdot 10^{-5}\\ 108 \delta = \frac{h_0}{\lambda} \simeq 4*10^{-2} 109 \end{align} 110 giving us a $\beta \simeq 6,4 = O(1)$ for $\varepsilon = \beta \delta^2$. 111 The main issue is if the KdV balance can occur within the geophysical scales 112 .The conditions $x-t = O(1)$ and $\tau = O(1)$ give 113 \begin{align} 114 \frac{x - t \sqrt{gh_0} }{\lambda} = O(1),\quad \frac{\varepsilon t 115 \sqrt{gh_0} }{\lambda} = O(1). 116 \end{align} 117 Combining the above equations, we have 118 \begin{align} 119 & \frac{x}{\lambda} = O(\varepsilon^{-1})\\ 120 &x = O(\varepsilon^{-1}\lambda) 121 \end{align} 122 For the tsunami wave in the Bay of Bengal westwards towards India and Sri 123 Lanka we have 124 \begin{align} 125 \lambda = 10\ \text{km}, \quad \varepsilon = 25 \cdot 10^{-5}, 126 \end{align} 127 therefore a propagation distance of $x \simeq 4 \cdot 10^{5}$ is needed for 128 the KdV to enter the range of validity. This is however not the case since 129 the tsunami waves propagated $1600\ \text{km}$ westwards. 130 131 For the wave in the Andaman Basin towards Indonesia and Thailand we have 132 \begin{align} 133 h_0 = 1\ \text{km}, \quad a = 1\ \text{m}, \quad \lambda = 100\ 134 \text{km}. 135 \end{align} 136 Giving us the parameters 137 \begin{align} 138 \varepsilon = 10^{-3}, \quad \delta = 10^{-2}. 139 \end{align} 140 this also does not satisfys the range of validity $\varepsilon = 141 O(\delta^2)$, with the requiring length scale $x = 10^{5}$. 142 143 Setting $h_0 = 4\ \text{km}$ for the Bay of Bengal and $h_0 = 1\ \text{km}$ 144 for the Andaman Basin we have that the westwards tsunami propagated at speed 145 $\sqrt{gh_0} = 712 \frac{\text{km}}{\text{h}}$ hitting the shore in about 146 $2h\ 10min$, while the tsunami waves eastwards propagated at speed $356 147 \frac{\text{km}}{\text{h}}$ hitting the coast in $1h\ 57min$. These 148 predictions align with the observations. As the waves approach the shore, the 149 tsunami mechanics enter the region of long waves over variable depth. In this 150 case dispersion and their front steepness play an important role, where 151 faster wave fronts catch up to slower ones and result in large amplitudes 152 with devastating effects. 153 154 For the tsunami in 2004 the propagation distance across the Indian Ocean in 155 both directions to Sri Lanka, India and Thailand was too short for nonlinear 156 KdV dynamics take place. The linear wave equation model discribes the tsunami 157 clear across the Indian Ocean untill it reaches costal regions. So for short 158 times the Tsunami waves are governed by the linear wave equation $\eta_{x x } 159 = \eta_{t t}$. 160 161 162 163 164 165 166 167