commit 381b8ae3559e9dfa5e009da020fc2051043c0e67
parent 4f4ab27f2fff4eeb493417d60e4cd49fce763e07
Author: miksa <milutin@popovic.xyz>
Date: Tue, 7 Jun 2022 11:55:27 +0200
done, only presentation|
Diffstat:
7 files changed, 196 insertions(+), 24 deletions(-)
diff --git a/app_pde/appendix.tex b/app_pde/appendix.tex
@@ -23,15 +23,12 @@ Which in integral representation reads
- f(t, a(t)) \frac{\partial a(t)}{\partial t}
\end{align}
-\subsection{Gaussian Integration Law}
-\label{appendix:gauss integration}
-This should explain the Gaussian integration law
-
\subsection{Identity for Vorticity}
+\label{appendix:diff identity}
We start off with the standard material derivative
\begin{align}
\frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
- (\mathbf{u}\nabla)\mathbf{u}.
+ +(\mathbf{u}\nabla)\mathbf{u}.
\end{align}
We will use Einstein's Summation Convention, where we sum over indices that
both appear at as the bottom as the top index, to rewrite the second part of
@@ -58,6 +55,7 @@ which is
\left(\mathbf{u}\times (\nabla \times \mathbf{u})\right)
\end{align}
\subsection{Middle Curvature of an Implicit Function}
+\label{appendix:curvature}
In our case the implicit function for fixed time reads
\begin{align}
z-h\left(x_1,x_2\right) = 0.
diff --git a/app_pde/build/main.pdf b/app_pde/build/main.pdf
Binary files differ.
diff --git a/app_pde/chap1.tex b/app_pde/chap1.tex
@@ -74,23 +74,36 @@ The figure bellow \ref{fig:volume}, expresses the above described picture.
\end{figure}
Since we want to figure out the fluid's dynamics, we can consider the rate
-of change in the completely arbitrary $V$. The rate of change of mass needs to
-disappear, i.e. it is equal to zero since we cannot lose mass. Matter (mass) is
-neither created nor destroyed anywhere in the fluid, leading us to
+of change in the completely arbitrary $V$
\begin{align}
- \frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = 0.
+ \frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = \int_V
+ \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \ dV
\end{align}
-\textbf{NOT SURE HERE YET!!!!!!!!!!!, CHECK LEIBINZ FORMULA}
-To get more information we simply ''differentiate under the integral
-sign``, also known as the Leibniz Rule of Integration, see appendix
-\ref{appendix:leibniz}, the integral equation representing the rate of change
-of mass reads
+On the other hand we have that density is mass over volume or
+\begin{align}
+ dm = \rho \cdot dV.
+\end{align}
+The infinitesimal volume has the base area $dS$ with hight $h$, which is the
+distance in the direction perpendicular to to the base area, leaving us with
+$dV = h dS$. By definition $\mathbf{n}$ is perpendicular to $dS$, we have
+that $h = l \mathbf{n}$. Where $l$ is the unit of length, or velocity times
+time $l = \mathbf{u} dt$, since mass is flowing out of the surface we
+change the sign of the flow leading us to
+\begin{align}
+ dm = -\rho dV = -\rho \mathbf{u} \cdot \mathbf{n} dt dS,
+\end{align}
+All together we have
+\begin{align}
+ \frac{dm}{dt} = -\int_{\partial V} \rho(\mathbf{x},t) dS
+ \mathbf{u}\cdot\mathbf{n}.
+\end{align}
+Putting both equations on one side leaves us with a variation of the mass
+conservation equation
\begin{align}\label{eq:mass balance}
- \frac{dm}{dt} = \int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
+ \int_V \frac{\partial \rho(\mathbf{x}, t)}{\partial t}\ dV
+\int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u}\cdot\mathbf{n}\ dS
= 0.
\end{align}
-\textbf{----------------------}
The above equation in \ref{eq:mass balance} is an underlying equation, describing that the rate of
change of mass in V is brought about, only by the rate of mass flowing into
V across S, and thus the mass does not change.
diff --git a/app_pde/chap2.tex b/app_pde/chap2.tex
@@ -8,7 +8,7 @@ scaling each quantity appropriately. The appropriate length scales are that
of the typical water depth $h_0$ and the typical wavelength $\lambda$ of a
surface wave.
-\subsection{Nondimensionalisation}
+\subsection{Nondimensionalisation\label{sec:nondim}}
In summary we use these adaptations
\begin{itemize}
\item $h_0$ for the typical water depth
diff --git a/app_pde/chap3.tex b/app_pde/chap3.tex
@@ -235,7 +235,7 @@ The solution is a $\text{sech}$ function
\begin{align}
f = 2c\ \text{sech}^2\left( \sqrt{\frac{3}{2K}}(\xi-ct)\right)
\end{align}
-\subsection{KdV Equation}
+\subsection{KdV Equation\label{sec:kdv}}
In this section we will go over the more general prerequisites and therefore
a more convincing expedition for the Korteweg-de Vries equation. We still
want to derive the wave profile of a wave in shallow water, small amplitude
@@ -271,18 +271,18 @@ and bottom boundary condition
\begin{align}
w = 0 \quad \text{on}\;\; z=b =0.
\end{align}
-We note here thatthe soluition for such a wave is a solitary wave as in
-described in the previous section. In principel we expect to wind such waves
-reather rarely in nature, since $\delta^2 = O(\varepsilon)$ is a very special
+We note here that the solution for such a wave is a solitary wave as in
+described in the previous section. In principle we expect to wind such waves
+rather rarely in nature, since $\delta^2 = O(\varepsilon)$ is a very special
case. Never the less this is not the case. We demonstrate that $\forall\
\delta$ as $\varepsilon$ goes to $0$ there exists a region in the position
space $(x, t)$ where the KdV balance in terms of linearity and dispersion
is observed. Indeed we can ''generate`` KdV solitary waves, provided a small
-enough amplitude in the sence of $\varepsilon$ goes to $0$. First of all we
+enough amplitude in the sense of $\varepsilon$ goes to $0$. First of all we
introduce a rescaling of the variables adjusted to our problem definition
-\begin{align}
+\begin{align}\label{eq:epsdelta}
x \rightarrow \frac{\delta}{\sqrt{\varepsilon} }\tilde{x}, \quad t
- \rightarrow \frac{\delta}{\sqrt{\varepsilon} }\tilde{t}\\
+ \rightarrow \frac{\delta}{\sqrt{\varepsilon} }\tilde{t}\quad
w \rightarrow \frac{\sqrt{\varepsilon} }{\delta}\tilde{w}.
\end{align}
Then the material derivative is transformed to be
diff --git a/app_pde/chap4.tex b/app_pde/chap4.tex
@@ -0,0 +1,160 @@
+\section{Modeling the 2004 Tsunami}
+\subsection{Description}
+On the 26. December 2004, time 7:58 a powerful earthquake generated a tsunami
+killing more than 275000 people and leaving millions homeless. The
+hypocenter f the earthquake was 30 km under the floor of the Indian Ocean,
+100 km away from Sumatra, an island in Indonesia. The earthquake displaced
+an enormous amount of water, sending tsunami waves westwards across the
+Indian Ocean to Sri Lanka and India and eastwards across the Andaman Basin.
+to Thailand and Indonesia. The Earthquake occurred over 10 minutes along a
+1000km long roughly straight line. Thereby we can model the corresponding
+fluid dynamics as 2 dimensional where in Cartesian coordinates the
+propagation of the tsunami wave is in $x$ direction and the $z$ direction
+pointing upwards perpendicular to the flat ocean surface. However the
+modeling assumption for two dimensions for the region outside the Bay of
+Bengal is not valid, since the diffraction around islands and reflection from
+steep shores pays a major role in the influence of the wave dynamics. Coming
+back to the $2004$ tsunami, which raised the ocean floor a few meters to the
+west and lowering it a few meters to the east, displacing the tectonic pates.
+The tsunami waves featured westwards a wave of elevation, meaning a wave of
+high amplitude, followed by a wave of depression, a wave of long wavelength
+hitting the coastal areas of Sri Lanka and India in roughly three hours,
+propagating a distance of approximately $1600\ \text{km}$. On the other hand
+eastwards featuring a first a wave of depression following a wave of
+elevation, propagating $700\ \text{km}$ in roughly two hours with a maximal
+amplitude of $10\ \text{m}$. Observations tell us that as the tsunami waves
+reached the shore the shape of the initial disturbance was not altered, which
+is supported by measurements by a radar altimeter two hours after the
+earthquake showing first an wave of elevation and then a wave of depression
+westwards and respectively vice versa eastwards. A conclusion is made that t
+he shape of the tsunami remained approximately constant. Additionally it
+should be mentioned that the tsunami waves reach very high amplitudes due to
+the diminishing depth effect as they approach the shore, yet at open sea are
+barely noticeable. A boat on open sea positioned at high depth in the region
+of the tsunami during the which the tsunami waves passed, captured the raise
+from $\pm 0.8\ \text{m}$ of the boat over a period of $10\ \text{min}$.
+This means that the wavelength of the tsunami wave was about $100\
+\text{km}$.
+\subsection{Long Wave, Shallow Water}
+At weakly nonlinear levels dispersion balances linearity/nonlinearity in
+certain regimes, such balance is found in the KdV equation. Among the KdV
+equation, the Carissa-Holm equation (CH) also features such balance since it
+arises as a high order approximation to KdV. Further there is also the
+regularized long wave equation usually called Benjamin–Bona–Mahony equation, or simply
+BBM. It should be noted that KdV is a solitary wave while BBM is not.
+Localized disturbances of flat water surfaces propagating without change of
+form need to be two dimension waves of elevation symmetric about the crest.
+The linear theory does not provide any approximation to solitary waves, only
+nonlinear or weakly linear approximations. These are KdV, BBM. The KdV
+equation is orbitaly stable, meaning the shape and form of the profile is
+stable under small perturbations (also CH \& BBM). Each solitary wave retains
+is local identity, where large waves are faster than small ones. Further the
+solution of the KdV equation evolves into a set of solitary waves, with
+tallest in from front followed by an oscillatory tail. The KdV is the
+proper equation, for our modeling purposes of tsunami waves. The main
+question arises if KdV enters the regime of validity, in our case for the
+$2004$ tsunami in the Andaman Basin. Or in other words are the involved
+geophysical scales leading to time and space scales that are compatible with
+the KdV weak nonlinearity balance.
+\subsection{Governing equations}
+The last two section show the derivation of the modeling equation for fluid
+dynamics, following with nondimensionalisation and rescaling. The scaling
+takes the same form as in section \ref{sec:nondim}, we only introduce the
+parameter $\alpha = h_0\sqrt{gh_0} $ and get the following equations
+\begin{align}
+ \begin{drcases}
+ u_x + w_z = 0\\
+ u_t + \varepsilon(uu_x + wu_z) = -p_x\\
+ \delta^2\left(w_t + \varepsilon(w w_x + w w_z)\right) = -p_y\\
+ u_z - \delta^2 w_x = 0
+ \end{drcases}
+\end{align}
+on $(x, z) \in \mathbb{R}\times [0, 1+\varepsilon \eta(x, t)]$, with boundary
+conditions
+\begin{align}
+ \begin{drcases}
+ p = \eta \\
+ w = h_t + \varepsilon u h_x
+ \end{drcases}
+ \text{on}\;\; z = 1+\varepsilon \eta(x, t)\\
+ w = 0 \quad \text{on}\;\; z = 0.
+\end{align}
+The KdV validity arises in the region $\varepsilon = O(\delta^2)$, we are
+going to thereby rescale $\delta$ in favour of $\varepsilon = \beta
+\delta^2$, where $\beta = O(1)$ as in equation \ref{eq:epsdelta}:
+\begin{align}\label{eq:epsdelta}
+ x \rightarrow \frac{\delta}{\sqrt{\varepsilon} }x, \quad t
+ \rightarrow \frac{\delta}{\sqrt{\varepsilon} }t\quad,
+ w \rightarrow \frac{\sqrt{\varepsilon} }{\delta}w.
+\end{align}
+This opens up the possibility to prove that provided a suitable length- \&
+time-scales for some $\delta$ the KdV will arise as a valid approximation for
+the evolution of the free surface waves. Given some $\varepsilon>0$ there
+exists a time in the such that the KdV balance holds, where we introduce the
+variables $\xi = x- t$ and $\tau = \varepsilon t$ with equation for the wave profile
+\begin{align}
+ \eta_\tau - \frac{3}{2} \eta_{\xi\xi\xi} + \frac{1}{6} \eta \eta_\xi = 0,
+\end{align}
+for $\xi \in \mathbb{R}$ and $\tau > 0 $ and the boundary condition $\eta(\xi
+,0)$ of the initial profile at $\tau = t = 0$. On the basis of satellite
+measurements for the Bay of Bengal we have
+\begin{align}
+ a = 1\ \text{km}, \quad \lambda = 100\ \text{km},\quad h_0 = 4\
+ \text{km}.
+\end{align}
+giving us
+\begin{align}
+ \varepsilon = \frac{a}{h_0} = 25 \cdot 10^{-5}\\
+ \delta = \frac{h_0}{\lambda} \simeq 4*10^{-2}
+\end{align}
+giving us a $\beta \simeq 6,4 = O(1)$ for $\varepsilon = \beta \delta^2$.
+The main issue is if the KdV balance can occur within the geophysical scales
+.Thee conditions $x-t = O(1)$ and $\tau = O(1)$ give
+\begin{align}
+ \frac{x - t \sqrt{gh_0} }{\lambda} = 0(1),\quad \frac{\varepsilon t
+ \sqrt{gh_0} }{\lambda} = 0(1).
+\end{align}
+Combining the above equations, we have
+\begin{align}
+ & \frac{x}{\lambda} = O(\varepsilon^{-1})\\
+ &x = O(\varepsilon^{-1}\lambda)
+\end{align}
+For the tsunami wave in the Bay of Bengal westwards towards India and Sri
+Lanka we have
+\begin{align}
+ \lambda = 10\ \text{km}, \quad \varepsilon = 25 \cdot 10^{-5},
+\end{align}
+therefore a propagation distance of $x \simeq 4 \cdot 10^{5}$ is needed for
+the KdV to enter the range of validity. This is however not the case since
+the tsunami waves propagated $1600\ \text{km}$ westwards.
+
+For the wave in the Andaman Basin towards Indonesia and Thailand we have
+\begin{align}
+ h_0 = 1\ \text{km}, \quad a = 1\ \text{m}, \quad \lambda = 100\
+ \text{km}.
+\end{align}
+Giving us the parameters
+\begin{align}
+ \varepsilon = 10^{-3}, \quad \delta = 10^{-2}.
+\end{align}
+this satisfys the range of validity $\varepsilon = O(\delta^2)$, with the
+requiring length scale $x = 10^{5}$.
+
+Setting $h_0 = 4\ \text{km}$ for the Bay of Bengal and $h_0 = 1\ \text{km}$
+for the Andaman Basin we have that the westwards tsunami propagated at speed
+$\sqrt{gh_0} = 712 \frac{\text{km}}{\text{h}}$ hitting the shore in about
+$2h\ 10min$, while the tsunami waves eastwards propagated at speed $356
+\frac{\text{km}}{\text{h}}$ hitting the coast in $1h\ 57min$. These
+predictions align with the observations. As the waves approach the shore, the
+tsunami dynamics enter the region of long waves over variable depth. In this
+case dispersion and their front steepness play an important role, where
+faster wave fronts catch up to slower ones and result in large amplitudes
+with devastating effects.
+
+
+
+
+
+
+
+
diff --git a/app_pde/main.tex b/app_pde/main.tex
@@ -11,6 +11,7 @@
\include{./chap1.tex}
\include{./chap2.tex}
\include{./chap3.tex}
+\include{./chap4.tex}
\newpage
