notes

basics_fluids.tex (42231B)

---

 \include{./preamble.tex}
 
 \usepackage{amsmath}
 \numberwithin{equation}{section}
 
 \begin{document}
 
 \maketitle
 \tableofcontents
 
 \section{Governing Equations of Fluid Dynamics}
 We first start of with a fluid with a density
 \begin{align}
     \rho(\mathbf{x}, t),
 \end{align}
 in three dimensional Cartesian coordinates $\mathbf{x} = (x, y, z)$ at time
 $t$. For water-wave applications, we should note that we take
 $\rho=\text{constant}$, but we will go into this fact later. The fluid moves
 in time and space with a velocity field
 \begin{align}
     \mathbf{u}(\mathbf{x}, t) = (u, v, w).
 \end{align}
 Additionally it is also described by its pressure
 \begin{align}
     P(\mathbf{x}, t),
 \end{align}
 generally depending on time and position. When thinking of e.g. water the
 pressure increases the deeper we go, that is with decreasing or increasing $z$
 direction (depending how we set up our system $z$ pointing up or down
 respectively).
 
 The general assumption in fluid dynamics is the \textbf{Continuum
 Hypothesis}, which assumes continuity of $\textbf{u}, \rho$ and $P$ in
 $\mathbf{x}$ and $t$. In other words, we premise that the velocity field,
 density and pressure are ''nice enough`` functions of position and time, such
 that we can do all the differential operations we desire in the framework of
 differential analysis.
 \subsection{Mass Conservation}
 Our aim is to derive a model of the fluid and its dynamics, with respect to
 time and position, in the most general way. This is usually done thinking
 of the density of a given fluid, which is a unit mass per unit volume,
 intrinsically  an integral representation to derive these equations suggests
 by itself.
 
 Let us now thing of an arbitrary fluid. Within this fluid we define a fixed
 volume $V$ relative to a chosen inertial frame and bound it by a surface $S$
 within the fluid, such that the fluid motion $\mathbf{u}(\mathbf{x}, t)$ may
 cross the surface $S$. The fluid density is given by $\rho(\mathbf{x}, t)$,
 thereby the mass of the fluid in the defined Volume $V$ is an integral
 expression
 \begin{align}
     m = \int_V \rho(\mathbf{x}, t) dV.
 \end{align}
 The figure bellow \ref{fig:volume}, expresses the above described picture.
 \begin{figure}[H]
     \centering
   \begin{tikzpicture}[>=latex,scale=1, xscale=1, opacity=.8]
 % second sphere
     \begin{scope}[rotate=10, xscale=3, yscale=2, shift={(2.3,-0.2)}]
       \coordinate (O) at (0,0);
       \shade[ball color=gray!10!] (0,0) coordinate(Hp) circle (1) ;
 
       \draw[thick] (O) circle (1);
       \draw[rotate=5] (O) ellipse (1cm and 0.66cm);
       \draw[rotate=90] (O) ellipse (1cm and 0.33cm);
 \node[circle, fill=black, inner sep=1pt] at (0.15, 0.25) {} ; \draw[-latex, thick] (0.15, 0.25) -- (1, 1) ;
       \node[right] at (1, 1) {$\mathbf{u}(\mathbf{x}, t)$};
 
       \node[] at (O) {$V$};
       \node[] at (0.55, -0.25) {$\rho(\mathbf{x}, t)$};
 
       \draw[-] (0.76, -0.66) -- (1.2, -0.7);
       \node[right] at (1.2, -0.7) {$S$};
 
       \draw[-latex, thick] (-0.25, -0.65) -- (-1, -1);
       \node[left] at (-1, -1) {$\mathbf{n}$};
 
     \end{scope}
 
 % axis
   \end{tikzpicture}
   \caption{Volume bounded by a surface in a fluid with density and momentum,
   with a surface normal vector $\mathbf{n}$ \label{fig:volume}}
 \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
 \begin{align}
     \frac{d}{dt}\left( \int_V \rho(\mathbf{x}, t)\ dV \right) = 0.
 \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
 \begin{align}\label{eq:mass balance}
     \frac{dm}{dt} = \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.
 
 For the second integral in \ref{eq:mass balance} we utilize the Gaussian
 integration law to acquire an integral over the volume
 \begin{align}
     \int_{\partial V} \rho(\mathbf{x}, t) \mathbf{u} \cdot \mathbf{n} \ dS =
     \int_V \nabla (\rho \mathbf{u})\ dV.
 \end{align}
 Thereby we can put everything inside the volume integral
 \begin{align}
     \frac{d m}{dt} = \int_V \left(\partial_t \rho + \nabla(\rho \mathbf{u}) \right) \ dV = 0.
 \end{align}
 Everything under the integral sign needs to be zero, thus we obtain
 the \textbf{Equation of Mass Conservation} or in the general sense also
 called the \textbf{Continuity Equation}
 \begin{align}\label{eq:continuity}
     \partial_t \rho + \nabla(\rho \mathbf{u}) = 0
 \end{align}
 
 In light of the results of the equation of mass conservation
 in \ref{eq:continuity}, an product rule gives
 \begin{align}
     \partial_t \rho + (\nabla \rho)\mathbf{u} + \rho(\nabla \mathbf{u}),
 \end{align}
 for notational purposes, we define the \textbf{material/convective derivative}
 as follows
 \begin{align}
     \frac{D}{Dt} = \partial_t + \mathbf{u}\nabla.
 \end{align}
 With the material derivative the equation of mass conservation reads
 \begin{align}
     \frac{D\rho}{Dt} + \rho \nabla\mathbf{u} = 0
 \end{align}
 We may undertake the first case separation, initiating $\rho = \text{cosnt.}$
 called \textbf{incompressible flow} causes the material derivative of $\rho$ to
 be zero, and thereby
 \begin{align}
     \frac{D\rho}{Dt} = 0 \quad \Rightarrow \quad \nabla \mathbf{u} = 0,
 \end{align}
 following that the divergence of the velocity field is zero, in this case
 $\mathbf{u}$ is called \textbf{solenoidal}.
 \subsection{Euler's Equation of Motion}
 Additional consideration we undertake is the assumption of an
 \textbf{inviscid} fluid, that is we set viscosity to zero. Otherwise we would
 get a viscous contribution under the integral which results in the
 Navier-Stokes equation. In this regard we apply Newton's second law to our
 fluid in terms of infinitesimal pieces $\delta V$ of the fluid. The
 acceleration divides into two terms, a \textbf{body force} given by gravity
 of earth in the $z$ coordinate $\mathbf{F} = (0, 0, -g)$ and a
 \textbf{local/short-rage force} described by the stress tensor in the fluid.
 In the inviscid case we the local force retains the pressure $P$, producing a
 normal force, with respect to the surface, acting onto any infinitesimal
 element in the fluid. The integral formulation of the force would be
 \begin{align}
     \int_V \rho \mathbf{F}\ dV - \int_S P\mathbf{n}\ dV.
 \end{align}
 Now applying the Gaussian rule of integration on the second integral over the
 surface, the resulting force in per unit volume is
 \begin{align}
     \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
 \end{align}
 The acceleration of the fluid particles is given by $\frac{D\mathbf{u}}{Dt}$,
 and thus the total force per unit volume on the other hand is
 \begin{align}
     \int_V \rho \frac{D\mathbf{u}}{Dt}\ dV =
     \int_V \left(\rho \mathbf{F} - \nabla P\right)\ dV.
 \end{align}
 Newton's Second Law for a fluid in an Volume is essentially saying that the
 rate of change of momentum of the fluid in the fixed volume $V$, which is the particle
 acceleration is the resulting force acting on V together with the rate of
 flow of momentum across the surface $S$ into the volume $V$. Hence we arrive
 at the \textbf{Euler's Equation(s) of Motion}
 \begin{align}
     \frac{D\mathbf{u}}{Dt} = \left(\frac{\partial \mathbf{u}}{\partial t}
     (\mathbf{u}\nabla)\mathbf{u}\right)  =
     -\frac{1}{\rho}\nabla P + \mathbf{F}.
 \end{align}
 As a side note we have mentioned that there is another contribution if the
 fluid is viscid. Indeed there is a tangential force due to the velocity
 gradient, which into introduces the additional term
 \begin{align}
     \mu \nabla^2 \mathbf{u}, \qquad
     \mu = \text{viscosity of the Fluid}.
 \end{align}
 Thereby the equations become
 \begin{align}
     \rho\frac{D\mathbf{u}}{Dt}
     =  -\nabla P + \rho \mathbf{F} + \mu \nabla^2 \mathbf{u}.
 \end{align}
 
 For now we have separated two simplifications, that define an
 \textbf{idealized/perfect fluid}
 \begin{enumerate}
     \item \textbf{incompressible} $\qquad \mu=0$
     \item  \textbf{inviscid} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
         0$
 \end{enumerate}
 \subsection{Vorticity and irrotational Flow}
 The curl of the velocity field $\mathbf{\omega} = \nabla \times \mathbf{u}$
 of a fluid (i.e. the vorticity), describes a spinning motion of the fluid
 near a position $\mathbf{x}$ at time $t$. The vorticity is an important
 property of a fluid, flows or regions of flows where $\mathbf{\omega}=0$ are
 \textbf{irrotational}, and thus can be modeled and analyzed following well
 known routine methods. Even though real flows are rarely irrotational
 anywhere (!), in water wave theory wave problems, from the classical aspect
 of vorticity have a minor contribution. Hence we can assume irrotational flow
 modeling water waves. To arrive at the vorticity in the equations of motions
 derived in the last section we resort to a differential identity derived in appendix
 \ref{appendix:diff identity}, which gives for the material derivative
 \begin{align}
     \frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t}
     \nabla(\frac{1}{2}\mathbf{u}\mathbf{u)}
     - \left( \mathbf{u}\times (\nabla \times  \mathbf{u} \right).
 \end{align}
 Thus the equations of motion become
 \begin{align}
     \frac{\partial \mathbf{u}}{\partial t} + \nabla\left(
     \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega \right)
     = \mathbf{u} \times  \mathbf{\omega},
 \end{align}
 where $\Omega$ is the force potential per
 unite mass given by $\mathbf{F} = -\nabla \Omega$.
 
 At this point we may differentiate between \textbf{stead and unsteady flow}.
 For \textbf{Steady Flow} we assume that $\mathbf{u}, P$ and $\Omega$ are time
 independent, thus we get
 \begin{align}
       \nabla\left( \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega
       \right)  = \mathbf{u} \times  \mathbf{\omega}.
 \end{align}
 It is general knowledge that the gradient of a function $\nabla f$ is
 perpendicular the level sets of $f(\mathbf{x})$, where $f(\mathbf{x}) =
 \text{const.}$. Thus $\mathbf{u} \times  \mathbf{\omega}$ is orthogonal to
 the surfaces  where
 \begin{align} \label{eq:bernoulli}
     \frac{1}{2}\mathbf{u}\mathbf{u} + \frac{P}{\rho} + \Omega =
     \text{const.},
 \end{align}
 The above equation is called \textbf{Bernoulli's Equation}.
 
 Secondly \textbf{Unsteady Flow} but irrotational (+ incompressible), first of
 all gives us the condition for the existence of a velocity potential $\phi$
 in the sense
 \begin{align}
     \mathbf{\omega} = \nabla \times  \mathbf{u} = 0  \quad \Rightarrow \quad
     \mathbf{u} = \nabla \phi,
 \end{align}
 where $\phi$ needs to satisfy the Laplace equation
 \begin{align}
     \Delta \phi = 0.
 \end{align}
 According to the Theorem of Schwartz we may exchange $\frac{\partial
 }{\partial t}$ and $\nabla$, giving us an expression for the material
 derivative
 \begin{align}
     \nabla\left( \frac{\partial \phi}{\partial t} +\frac{1}{2}
     \mathbf{u}\mathbf{u} + \frac{P}{\rho}  + \Omega \right) = 0
 \end{align}
 Thus the expression differentiated by the $\nabla$ operator is an arbitrary
 function $f(\mathbf{x}, t)$, writing
 \begin{align}
      \frac{\partial \phi}{\partial t} +\frac{1}{2}
     \mathbf{u}\mathbf{u} + \frac{P}{\rho}  + \Omega = f(\mathbf{x}, t).
 \end{align}
 The function $f(\mathbf{x}, t)$ can be removed by gauge transformation of
 $\phi \rightarrow \phi + \int f(\mathbf{x}, t)\ dt$, never the less this is
 not further discussed and left to the reader in the reference.
 \subsection{Boundary Conditions for water waves}
 The boundary conditions for water-wave problems vary, generally on the
 simplification we undertake. At the surface, called the free surface as in
 free from the velocity conditions, we have the atmospheric stress on the
 fluid. The stress component would again have a viscid component, this however
 is only relevant when modeling surface wind, in this review we model the
 fluid as unaffectedly and within reason as inviscid. The atmosphere employs
 only a pressure on the surface, this pressure is taken to be the atmospheric
 pressure, dependent on time and point in space. Thereby  any surface tension
 effects can also include a scenario at a curved surface (e.g. wave), giving
 rise to the pressure difference across the surface. A more precise
 description would use Thermodynamics to derive boundary conditions coupling
 water surface and the air above it, yet the density component of air
 compared to that of water makes our ansatz viable. The described conditions
 are called the \textbf{dynamic conditions}
 
 An additional condition revolves around the fluid particles on the moving
 surface, called the \textbf{kinematic condition}. This condition bounds
 the vertical velocity component on the surface.
 
 The logical step now is to define boundary conditions on the bod of the
 fluid, i.e. the bottom. If the viscid case bottom is impermeable, we a no
 slip condition to all fluid particles $\mathbf{u}_\text{bottom}= 0$. If we
 assume that the fluid is inviscid then the bottom becomes a surface of the
 fluid in the sense that the fluid particles in contact with the bed move in
 the surface, we more or less mirror the kinematic condition of the surface.
 For many problems the condition is going to vary, in most cases the bottom
 will be rigid and fixed not necessarily horizontal. This condition is simply
 called the \textbf{bottom condition}.
 \subsubsection{Kinematic Condition}
 Obtaining the free surface is the primary objective in the theory of modeling
 water waves, represented by
 \begin{align}
     z = h(\mathbf{x}_\perp, t),
 \end{align}
 where $\mathbf{x}_\perp = (x, y)$ in Cartesian, or $\mathbf{x}_\perp = (r,
 \theta)$ in cylindrical coordinates. A surfaces that moves with the fluid,
 always contains the same fluid particles, described as
 \begin{align}
     \frac{D}{Dt}\left(z - h(\mathbf{x}_\perp, t  \right) = 0.
 \end{align}
 Upon expanding the derivative we get
 \begin{align}
     \frac{Dz}{Dt} - \frac{Dh}{Dt}
     &= \frac{\partial z}{\partial t}+
     (\mathbf{u}\nabla)z - \frac{\partial h}{\partial t} -(\mathbf{u}\nabla)\\
     &= w - \left(h_t - (\mathbf{u}_\perp \nabla_\perp) h\right) = 0,
 \end{align}
 where the subscript $\perp$ describes the components with regard to
 $\mathbf{x}_\perp$. The \textbf{kinematic condition} reads
 \begin{align}
     w = h_t - (\mathbf{u}_\perp \nabla_\perp) h \qquad \text{on}\;\;
     z=h(\mathbf{u}_\perp, t).
 \end{align}
 
 \subsubsection{Dynamic Condition}
 As described in the prescript of this section, the case of an inviscid fluid,
 requires that only the pressure $P$ needs to be described on the free surface
 $z = h(\mathbf{x}_\perp, t)$. Assuming incompressible, irrotational,
 unsteady flow and setting $P=P_a$ for atmospheric pressure and $\Omega =
 g\cdot z$ for the force per unit mass potential the equations of motion are
 \begin{align}
     \frac{\partial \phi}{\partial t} +\frac{1}{2}\mathbf{u}\mathbf{u}
     + P_\frac{a}{\rho}+gh = f(t) \qquad \text{on}\;\; on z=h.
 \end{align}
 Somewhere $\|\mathbf{x}_\perp\| \rightarrow \infty$ the fluid reaches
 equilibrium and is thereby stationary, thereby has no motion and the pressure
 is $P=P_a$ and the surface is a constant $h = h_0$ $f(t)$ is
 \begin{align}
     f(t) = \frac{P_a}{\rho}+gh_0.
 \end{align}
 The simplest description for the \textbf{dynamic condition} may be written as
 \begin{align}
     \frac{\partial \phi}{\partial t}
     +\frac{1}{2}\mathbf{u}\mathbf{u}+g(h-h_0) = 0 \qquad \text{on}\;\; z=h.
 \end{align}
 
 Regarding the pressure difference on a curved surface, we may expand the
 dynamic condition by introducing the pressure difference known as the
 \textbf{Young-Laplace Equation}
 \begin{align}
     \Delta P = \frac{\Gamma}{R},
 \end{align}
 where $\Gamma>0$ is the coefficient of surface tension and $\frac{1}{R}$ is
 the curvature representing an implicit function, in our case the implicit
 function is $z - h(\mathbf{x}_\perp, t)$ for fixed time. The curvature in
 Cartesian coordinates takes the form
 \begin{align}
     \frac{1}{R} = \frac{(1+h_y^2)h_{x x}+(1+h_y^2)h_{yy} -
     2h_xh_yh_{xy}}{\left( h_x^2+h_y^2+1 \right)^{\frac{3}{2}} },
 \end{align}
 the derivation is precisely described in \ref{appendix:curvature}
 
 
 
 \subsubsection{The Bottom Condition}
 The representation for the bottom is
 \begin{align}
    z = b(\mathbf{x}_\perp, t),
 \end{align}
 where the fluid surface needs to satisfy
 \begin{align}
     \frac{D}{Dt} \left(z - b(\mathbf{x}_\perp) \right)  = 0.
 \end{align}
 Hence we arrive at the bottom boundary conditions
 \begin{align}
     w = b_t + (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b ,
 \end{align}
 where $b(\mathbf{x}_\perp, t)$ is already known for most water wave
 problems. If we consider a stationary bottom then the time derivative
 vanishes, leaving us with the following condition
 \begin{align}
     w = (\mathbf{u}_\perp \nabla_\perp)b \qquad \text{on}\;\; z=b
 \end{align}
 
 
 \subsubsection{Integrated Mass Condition}
 In this section we want to combine the kinematics of both the free and the
 bottom surface with the mass conservation equation on the perpendicular
 components
 \begin{align}
     \nabla \mathbf{u} = \nabla_\perp \mathbf{u}_\perp + w_z = 0 .
 \end{align}
 Integrating the above expression from bottom to surface, i.e. from
 $z=b(\mathbf{x}_\perp,t)$ to $z = h (\mathbf{x},t)$ gives
 \begin{align}
     \int_b^h \nabla_\perp \mathbf{u}_\perp\ dz + w\bigg|_{z=b}^{z=h} = 0,
 \end{align}
 where we insert the conditions on the free surface and on the bottom surface
 \begin{align}
     w &= h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h \quad
     \text{on}\;\; z = h\\
     w &= b_t + (\mathbf{u}_{\perp \text{b}} \nabla_\perp) h \quad
     \text{on}\;\; z =b,
 \end{align}
 with the subscript $s$ and $b$ indicating the evaluation of a quantity
 on the free surface and the bottom surface respectively. Inserting the
 boundary conditions we get
 \begin{align}
     \int_b^h \nabla_\perp \mathbf{u}_\perp
     + h_t + (\mathbf{u}_{\perp \text{s}} \nabla_\perp) h
     - b_t - (\mathbf{u}_{\perp \text{b}} \nabla_\perp) b= 0.
 \end{align}
 To simplify the equation we resort again to the Leibniz Rule of Integration
 \begin{align}
      \int_b^h \nabla_\perp\mathbf{u}_\perp =
     \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz - (\mathbf{u}_{\perp \text{s}}
     \nabla_\perp)h - (\mathbf{u}_{\perp \text{b}})b.
 \end{align}
 As a consequence the \textbf{Integrated Mass Condition} is given by
 \begin{align}
     \nabla_\perp \int_b^h \mathbf{u}_\perp\ dz  + \underbrace{h_t -
     b_t}_{=d_t} = 0.
 \end{align}
 \subsection{Energy Equation}
 To derive the energy equation we start off with Euler's Equation of Motion
 \begin{align}
   \mathbf{u} _t + \nabla
   (\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega) = \mathbf{u}\times
   \mathbf{w},
 \end{align}
 multiplying the equation with $\mathbf{u}$ we get
 \begin{align}
   &\mathbf{u}\mathbf{u} _t \label{eq:energy1} \\
   &+(\mathbf{u}\nabla)(\frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}+\Omega)\label{eq:energy2}\\
   &= \mathbf{u}(\mathbf{u}\times
   \mathbf{w})\label{eq:energy3}.
 \end{align}
 The first equation given in \ref{eq:energy1} can we rewritten using inverse
 product rule of differentiation
 \begin{align}
     \mathbf{u}\frac{\partial \mathbf{u}}{\partial t}
     &= \frac{\partial
     }{\partial t} (\mathbf{u}\mathbf{u}) - \frac{\partial \mathbf{u}}{\partial t}
     \mathbf{u} \\
     &= \frac{\partial
     }{\partial t} (\mathbf{u}\mathbf{u}) - \mathbf{u}\frac{\partial
     \mathbf{u}}{\partial t}\\
     \Rightarrow\quad & \mathbf{u} \frac{\partial \mathbf{u}}{\partial t}  =
     \frac{1}{2}\frac{\partial }{\partial t} (\mathbf{u}\mathbf{u}).
 \end{align}
 Then we may add
 \begin{align}
     \left(\frac{1}{2} \mathbf{u}\mathbf{u}+\frac{P}{\rho} +\Omega  \right)
     \underbrace{(\nabla u)}_{=0} = 0,
 \end{align}
 to above not changing anything. Thereby getting
 \begin{align}
     \frac{\partial }{\partial t} (\frac{1}{2}\mathbf{u}\mathbf{u})
     +(\mathbf{u}\nabla \mathbf{u})\left(
     \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} \right)
     +\left( \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho} + \Omega \right)
     (\nabla \mathbf{u}) = 0.
 \end{align}
 Applying the product rule we can simplify
 \begin{align}
     \frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}\right)
     +\nabla \left(\mathbf{u}\left(\mathbf{u}(
     \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right) \right) = 0,
 \end{align}
 additionally adding $\frac{\partial \Omega}{\partial t}  =0$ leads us to
 \begin{align}
     \underbrace{\frac{\partial }{\partial t} \left(\frac{1}{2}\mathbf{u}\mathbf{u}
     +\Omega\right)}_{\text{change of total energy density}}
     +\underbrace{\nabla \left(\mathbf{u}\left(\mathbf{u}(
     \frac{1}{2}\mathbf{u}\mathbf{u}+\frac{P}{\rho}\right)
 \right)}_{\text{energy flow of the velocity field}} = 0.\label{eq:energy}
 \end{align}
 This is called the \textbf{energy equation} and is a general result for a
 inviscid and incompressible fluids, which we can apply to study water waves.
 We start off with replacing $\nabla = \nabla_\perp + \frac{\partial }{\partial
 z} $ and $\Omega = g z$ and multiplying by $\rho$, then our energy equation
 in \ref{eq:energy} becomes
 \begin{align}
     \frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
     g z\right)  + \nabla_\perp\left( \mathbf{u}_\perp\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right)  \right)
     \frac{\partial}{\partial z} \left( w\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right)  \right)  = 0.
 \end{align}
 Integrating from bottom to top, i.e. from bed to free surface gets us to
 \begin{align}
     &\int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
     g z\right)\ dz  \label{eq:e-int1}\\
     &+ \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right)  \right)\
     dz\label{eq:e-int2}\\
     &+ \left(\frac{\partial}{\partial z} \left( w\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho g z \right)
 \right)\right)\Bigg|_b^h \label{eq:e-int3}
     = 0.
 \end{align}
 For equation \ref{eq:e-int1} we use Leibniz Rule of Integration, leaving us
 with
 \begin{align}
     \int_b^h\frac{\partial }{\partial t}\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho
     g z\right)\ dz
     &= \frac{\partial }{\partial t} \int_b^h
     \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho gz \ dz\\
     &+ \left( \frac{1}{2}\rho \mathbf{u}_s \mathbf{u}_s  + \rho g h \right)
     h_t\\
     &- \left( \frac{1}{2}\rho \mathbf{u}_b \mathbf{u}_b  + \rho g b \right)
     b_t
 \end{align}
 For equation \ref{eq:e-int2} we again take note of the Leibniz Rule of
 Integration, getting
 \begin{align}
     \int_b^h \nabla_\perp\left( \mathbf{u}_\perp\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u}+P+\rho gz \right)  \right)\
     dz
     &= \nabla_\perp \int_b^h \mathbf{u}_\perp\left(
     \frac{1}{2}\rho\mathbf{u}\mathbf{u} + P + \rho g z \right) \ dz\\
     &- \left( \frac{1}{2}\rho \mathbf{u}_s\mathbf{u}_s + P + \rho g h \right)
     \left( \mathbf{u}_{\perp s} \nabla_\perp \right) h\\
     &+\left( \frac{1}{2}\rho \mathbf{u}_b\mathbf{u}_b + P + \rho g b \right)
     \left( \mathbf{u}_{\perp b} \nabla_\perp \right) b
 \end{align}
 Thereby transforming our equation into
 \begin{align}
     \frac{\partial }{\partial t} \underbrace{\int_b^h \frac{1}{2}\rho
     \mathbf{u}\mathbf{u}+\rho g z\ dz}_{=:\mathcal{E}}
     + \nabla_\perp&\underbrace{\int_b^h
     \mathbf{u}_\perp\left( \frac{1}{2}\rho\mathbf{u}\mathbf{u} + \rho g z
 \right)\ dz}_{:=\mathcal{F}}
 + \underbrace{P_s h_t - P_b b_t}_{:=\mathcal{P}} = 0\\
 \nonumber\\
     &\frac{\partial \mathcal{E}}{\partial t}
     + \nabla_\perp \mathcal{F} + \mathcal{P} = 0,
 \end{align}
 where $\mathcal{E}$ represents the energy in the flow per unit horizontal
 area, since we are integrating from bed to free surface. Where $\mathcal{F}$
 is the horizontal energy flux vector and lastly $\mathcal{P} = P_s h_t -
 P_b b_t$ is the net energy input due to the pressure forces doing work on the
 upper and lower boundaries, i.e. bottom and free surface of the fluid.
 Assuming stationary rigid bottom condition and constant surface pressure, we
 can set $P_s=0$, such that $\mathcal{P} =0$ leaving us with the equation
 \begin{align}
     \frac{\partial \mathcal{E}}{\partial t}
     + \nabla_\perp \mathcal{F} = 0.
 \end{align}
 We note that the assumption $P_s=0$ is only possible if the coefficient of
 surface tension is set to 0, which usually is not the case.
 \section{Dimensional Analysis}
 Our derived model of fluid dynamics yields formal connections between
 physical quantities. These quantities bear units, e.g. the velocity of fluid
 particles $\mathbf{u}$ has the ``SI'' unites of $\frac{m}{s}$, meters per
 second. The idea is the make use of these scales and formulate a model, where
 the quantities are nondimensionalized, i.e. to get rid of physical units by
 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}
 In summary we use these adaptations
 \begin{itemize}
     \item $h_0$ for the typical water depth
     \item $\lambda$ for the typical wavelength
     \item $\frac{\lambda}{\sqrt{g h_0}}$ time scale of wave propagation
     \item $\sqrt{g h_0}$ velocity scale of waves in $(x, y)$
     \item $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in the $z$
         direction.
 \end{itemize}
 $(x, z, t)$, then
 \begin{align}
     u = \psi _z, \qquad w = - \psi_x;
 \end{align}
 and the scale of $\psi$ must be $h_0\sqrt{g h_0}$. Additionally we write the
 boundary condition on the free surface as follows
 \begin{align}
     h  = h_0 + a \eta (\mathbf{x}_\perp, t) = z,
 \end{align}
 where $a$ is the typical amplitude and $\eta$ nondimensional function. All in
 all we have the following scaling for the physical quantities of our context
 \begin{align}
     &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, \\
       &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad
       t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\\
       &z \rightarrow\ h_0 z, \quad w \rightarrow
     \frac{h_0\sqrt{gh_0}}{\lambda} w.
 \end{align}
 with
 \begin{align}
     h = h_0 + a \eta, \qquad  b \rightarrow h_0 b.
 \end{align}
 The pressure is also rewritten into
 \begin{align}
     P = P_a + \rho g(h_0 -z) + \rho g h_0 p,
 \end{align}
 where $P_a$ is the atmospheric pressure, the term $h_0-z$ represent the
 hydrostatic pressure distribution, i.e. pressure at depth and the term with the pressure
 variable $p$  measures the deviation from the hydrostatic pressure
 distribution. Indeed $p\neq 0 $ for wave propagation. Now we can perform a
 rescaling of the Euler's Equation of Motion, we introduce the notation
 \begin{align}
     &t = \frac{\lambda}{\sqrt{gh_0}}\tau,\quad x = \lambda \xi,\quad u =
     \sqrt{gh_0} \tilde{u}\\
     &y = \lambda \chi,\quad v = \sqrt{gh_0} \tilde{v}\\
     &z = h_0 \zeta, \quad w = \frac{h_0\sqrt{gh_0} }{\lambda}\tilde{w}.
 \end{align}
 We start off with the $x$ coordinate, substitute and apply the chain rule
 leading us to
 \begin{align}
     \frac{Du}{Dt}
     &= \frac{\partial u}{\partial t} +u \frac{\partial
     u}{\partial x} \\
     &= \sqrt{gh_{0}}\frac{\partial \tilde{u}}{\partial \tau} \frac{\partial
     \tau}{\partial t} +gh_0 \tilde{u} \frac{\partial \tilde{u}}{\partial \xi}
     \frac{\partial \xi}{\partial x} \\
     &= \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
     \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right),
 \end{align}
 on the other hand
 \begin{align}
     \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau}
     \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right)
     &=-\frac{1}{\rho}\frac{1}{\lambda}\frac{\partial P}{\partial x} \\
     &=-\frac{ g h_0 }{\lambda}\rho \frac{\partial p}{\partial \xi}.
 \end{align}
 Thereby the rescaling evolves to
 \begin{align}
     \frac{D \tilde{u}}{D\tau} = -\frac{\partial p}{\partial \xi}.
 \end{align}
 Because of the same scaling in $y$ we get the same result as in $x$, that is
 \begin{align}
     \frac{D \tilde{v}}{D\tau} = -\frac{\partial p}{\partial \chi}.
 \end{align}
 In the $z$ coordinate we have
 \begin{align}
     \frac{Dw}{Dt}
     &= \frac{\partial w}{\partial t} +w \frac{\partial
     w}{\partial \zeta} \\
     &= \frac{h_0\sqrt{gh_0}}{\lambda} \frac{\sqrt{gh_0}}{\lambda}
     \frac{\partial \tilde{w}}{\partial \tau}  + \frac{1}{h_0}
     \frac{h_0\sqrt{gh_0} }{\lambda} \frac{h_0\sqrt{gh_0}}{\lambda}
     \tilde{w}\frac{\partial \tilde{v}}{\partial \zeta}\\
     &= \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
     + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) .
 \end{align}
 On the other side we have
 \begin{align}
     \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau}
     + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right)
     &=
     -\frac{1}{h_0\rho} \frac{\partial P}{\partial z} +g \\
     &=-\frac{1}{h_0\rho}(-\rho gh_0 \frac{\partial \zeta}{\partial \zeta}
     \rho gh_0
     \frac{\partial p}{\partial \zeta} ) + g  \\
     &= -g \frac{\partial p}{\partial z}.
 \end{align}
 In total for the $z$ direction we get
 \begin{align}
    \underbrace{\left( \frac{h_0}{\lambda} \right)^2}_{=: \delta^2}
     \frac{Dw}{Dt} = -\frac{\partial p}{\partial z},
 \end{align}
 where $\delta$ is the \textbf{long wavelength} or \textbf{shallowness}
 parameter, a very important constant for developing model hierarchies. For
 clarity we resubstitute for $x, y, z, t, u, v$ and $w$, and for completeness
 the we display the equations again, which are
 \begin{align}\label{eq:nondim-motion}
     \frac{Du}{Dt} = - \frac{\partial p}{\partial x}&, \quad
     \frac{Dv}{Dt} = - \frac{\partial p}{\partial y}, \quad
     \delta^2\frac{Dw}{Dt} = - \frac{\partial p}{\partial z}, \\
     &\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
     +\frac{\partial w}{\partial z}  = 0.
 \end{align}
 We can now turn our attention to the boundary conditions, on both free
 surface $z=h$ and the bottom $z=b$ we have $z \Rightarrow h_0 z$ and thereby
 \begin{align}
     z = 1+
     \underbrace{\frac{a}{h_0}}_{:=\varepsilon}\eta(\mathbf{x}_\perp,t) \quad
     \text{and}\quad z= b,
 \end{align}
 where we arrive at our second very important parameter $\varepsilon$ called
 the \textbf{amplitude} parameter. As for the kinematic condition, we
 substitute the free surface $z=h = 1+\varepsilon \eta$ and get
 \begin{align}
     \frac{Dz}{Dt} = \varepsilon\left(\eta_t + (\mathbf{u}_\perp
         \nabla_\perp)\eta\right) \qquad \text{on}\;\; z= 1+\varepsilon \eta.
 \end{align}
 Respectively the bottom condition is not changed
 \begin{align}
     w = b_t + (\mathbf{u}_\perp \nabla_\perp) b \quad \text{on}\;\; z= b.
 \end{align}
 The general dynamic condition for $h = h(x, y, t)$ yields a rescaling of the
 curvature in terms of
 \begin{align}
    \frac{1}{R}
    &= \frac{(1+h_y^2)h_{x x} + (1+h_x^2)h_yy - 2h_xh_yh_{xy}
    }{\left(h_x^2+h_y^2 +1  \right)^{\frac{3}{2}} } \\
    &= -\frac{\varepsilon h_0}{\lambda^2} \frac{(
    1+\varepsilon^2\delta^2\eta_y^2 )\eta_{x x}+
     (1+\varepsilon^2\delta^2\eta_x^2)\eta_{yy} -
     2\varepsilon^2\delta^2\eta_x\eta_y\eta_{xy}}{\left(
     1+\varepsilon^2\delta^2\eta_x^2+\varepsilon^2\delta^2\eta_y^2
     \right)^{\frac{3}{2}} },
 \end{align}
 together with the pressure difference
 \begin{align}
     \Delta P = \rho g h_0(p - \varepsilon \eta) = \frac{\Gamma}{R},
 \end{align}
 leaving us ultimately with the dynamic condition
 \begin{align}
     p-\varepsilon\eta= \varepsilon\left( \frac{\Gamma}{\rho g\lambda^2}
     \right) \left(\frac{\lambda^2}{\varepsilon h_0}\frac{1}{R}\right),
 \end{align}
 where $W_e = \frac{\Gamma}{\rho g h_0^2}$ is the \textbf{Weber number}. This
 dimensionless parameter can be considered as a measure of the fluid's inertia
 compered to its surface tension, which satisfies the relation
 \begin{align}
     \delta^2 W_e = \frac{\Gamma}{\rho g \lambda^2}.
 \end{align}
 \subsection{Scaling of Variables}
 Admits a simple observation of the governing equations in the last chapter we
 notice that $w$ and $p$ on the free surface $z = 1 + \varepsilon\eta$ are
 directly proportional to $\varepsilon$. Hence we want to ''scale this way``
 by introducing the following transformation
 \begin{align}
     p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad
     \mathbf{u}_\perp \rightarrow \varepsilon \mathbf{u}_\perp.
 \end{align}
 Because of this scaling our material derivative changes slightly to
 \begin{align}\label{eq:mod-material}
     \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon\left(u
     \frac{\partial }{\partial x}  + v \frac{\partial }{\partial y}  + w
     \frac{\partial }{\partial z} \right)
 \end{align}
 A simple recalculation yields the rescaled, nondimensionalized Euler's
 Equation of motion are the same as in equations \ref{eq:nondim-motion} with
 the modified material derivative from \ref{eq:mod-material}, and the boundary
 conditions are
 \begin{align}
     p &= \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\
     w &= \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta
     \quad \text{on}\;\; z = 1+\varepsilon\eta\\
     w &=\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad
     \text{on}\;\; z=b
 \end{align}
 \subsection{Model Hierarchies}
 As we have derived a model of fluid dynamics, with small parameters
 $\varepsilon$ and $\delta$, we can conduct a series of classifications and
 perform asymptotic analysis on them. The main hierarchies important in this
 review are derived from the following problem classifications
 \begin{itemize}
     \item $\varepsilon\rightarrow 0$: linearized problem, small amplitude
     \item $\delta\rightarrow 0$: shallow Water, long-wave
     \item$\delta \rightarrow 0;\; \varepsilon~1$: shallow Water, large
         amplitude
     \item $\delta\ll 1;\; \varepsilon~\delta$: shallow water, medium
         amplitude
     \item $\delta\ll 1;\; \varepsilon~\delta^2$: shallow water, small
         amplitude
     \item $\delta \gg 1;\; \varepsilon\delta\ll 1$: deep water, small
         steepness.
 \end{itemize}
 
 \section{The Solitary Wave and The KdV Equation}
 The solitary wave is a wave of translation, it is stable and can travel long
 distances additionally the speed depends on the size of the wave. An
 interesting feature is that two solitary waves do not merge together to form
 one solitary wave, rather the small wave is overtaken by a larger one. If a
 solitary wave is too big for the depth it splits into two, a big and a small
 one. Solitary waves arise in the region $\varepsilon=O(\delta^2)$.
 
 
 \subsection{Solitary Wave}
 To describe
 a solitary wave we begin with Euler's Equation of Motion, where we assume
 there is no surface tension we set $W_e = 0$ and additionally assume
 irrotational flow $\mathbf{\omega}=\nabla \times  \mathbf{u} = 0$. This means
 that there exists a velocity potential $\phi(\mathbf{x},t)$ given
 by$\mathbf{u} = \nabla \phi$ satisfying the Laplace equation. In regard of a
 solitary wave being a plane wave, we rotate our coordinate system such that
 the propagation is in the $x$-direction and a stationary \& fixed bottom
 $b=0$. Ultimately leaving us with the following model
 \begin{align}\label{eq:soliton}
 \begin{drcases}
    & \phi_{zz} + \delta \phi_{x x }  = 0,\\
    &\text{with the boundary conditions}\\
    &\begin{drcases}
     &\phi_z = \delta^2 (\eta_t + \varepsilon \phi_x \eta_x) \\
     &\phi_t + \eta +  \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
     + \phi_x^2\right)  =0
   \end{drcases}\quad \text{on}\;\; z = 1+\varepsilon\eta,\\
    &\text{and}\\
    & \phi_z =0 \quad \text{on}\;\; z = b = 0.
 \end{drcases}
 \end{align}
 Since the model arises $\varepsilon = O(\delta^2)$, for convince we set
 $\varepsilon=1$. The fact of the matter is we are seeking a traveling wave
 solution, thereby we can go into the coordinate system of the traveling wave,
 one in the variable $\xi = x - ct$ for a from left to right traveling wave,
 where $c$ is the nondimensional speed of the wave. Our goal is to find the
 solution for the velocity potential $\phi(\xi, z)$ and the wave profile
 $\eta(\xi)$. The chain rule gives us
 \begin{align}
     \frac{\partial }{\partial x} &= \frac{\partial \xi}{\partial x}
     \frac{\partial }{\partial \xi}  = \frac{\partial }{\partial \xi}, \\
     \frac{\partial }{\partial t} &= \frac{\partial \xi}{\partial t}
     \frac{\partial }{\partial \xi}  = -c\frac{\partial }{\partial \xi}.
 \end{align}
 Together with the equations in \ref{eq:soliton} we obtain
 \begin{align}\label{eq:soliton-xi}
     \begin{drcases}
    & \phi_{zz} + \delta \phi_{\xi\xi}  = 0,\\
    &\text{with the boundary conditions}\\
    &\begin{drcases}
     &\phi_z = \delta^2 (\phi_\xi -c)\eta_\xi \\
     &-c\phi_\xi + \eta +  \frac{1}{2}\varepsilon\left( \frac{1}{\delta^2}\phi^2_z
     + \phi_\xi^2\right)  =0
   \end{drcases}\quad \text{on}\;\; z = 1+\eta,\\
    &\text{and}\\
    & \phi_z =0 \quad \text{on}\;\; z = b = 0.
     \end{drcases}
 \end{align}
 \subsubsection{Exponential Decay}
 We would like to analyze if the equation in \ref{eq:soliton-xi} gives viable a
 solution that decays exponentially, we make the ansatz
 \begin{align}
     \eta \simeq a e^{-\alpha |\psi|},\quad \phi \simeq \psi(z)e^{-\alpha
     |\xi|}, \qquad  \mid \xi \mid \rightarrow \infty,
 \end{align}
 where $\alpha>0$ is the exponent. The equations in \ref{eq:soliton-xi}
 transforms to
 \begin{align}
     \psi'' + \alpha^2 \delta^2\psi = 0.
 \end{align}
 The above equation is a standard well known ordinary differential equation
 reading
 \begin{align}
     \psi = A \cos(\alpha\delta z),
 \end{align}
 where $A$ is the integration constant. On the free surface $z\simeq 1$ gives
 \begin{align}
     &-cA\alpha\sin(\alpha\delta) = ca\alpha,\label{eq:sol1}\\
     &cA\alpha \cos(\alpha\delta) = -a \label{eq:sol2}.
 \end{align}
 Dividing equation \ref{eq:sol1} with equation \ref{eq:sol2} gives
 \begin{align} \label{eq:soliton-dispersion}
     c^2 = \frac{\tan\left(\alpha\delta  \right) }{\alpha\delta}.
 \end{align}
 We conclude that the solution for such a wave exists provided that the
 dispersion relation on the wave propagation speed holds, thereby all solitary
 waves exhibit exponential decay in their tail and satisfy the dispersion
 relation in equation \ref{eq:soliton-dispersion}.
 \subsubsection{Asymptotic Analysis}
 The underlining equations in \ref{eq:soliton} extend from $-\infty$ to
 $\infty$, so the length scale is much greater than any finite depth of
 water. Therefore the classification $\delta \rightarrow 0$ is appropriate for
 a solitary wave, this however goes with the assumption
 $\varepsilon\rightarrow 0$ otherwise we cannot make an appropriate expansion.
 Let us look at the main equation
 \begin{align}\label{eq:sol-laplace}
     \phi_{zz} + \delta \phi_{x x} = 0.
 \end{align}
 For small $\delta$ we conduct the $\delta^2 = O(\varepsilon)$ standard ansatz
 in asymptotic analysis
 \begin{align}
     \phi_{\delta}(x, t, z) \simeq \sum_{n=0}^{\infty} \delta^{2n}\phi_n(x, t,
     z).
 \end{align}
 Substituting $\phi_\delta$ into equation \ref{eq:sol-laplace} we get
 \begin{align}
     \delta^{2\cdot 0}\left( \phi_{0zz} \right)  + \delta^{2\cdot 1}\left(
     \phi_{1zz}+\phi_{0 x x} \right)  + \delta^{2\cdot 2}\left( \phi_{2zz}+
     \phi_{1 x x} \right)  + O(\delta^{2\cdot 3}) = 0.
 \end{align}
 We start off with $O(\delta^{2\cdot0}) $, which gives us an arbitrary function
 $\phi_{0} = \theta(x, t)$. Next we may generalize the results for all
 $O(\delta^{2\cdot n})$ in the means of
 \begin{align}
     \phi_{n+1zz}  = -\phi_{nx x}\qquad \forall n\in \mathbb{N} .
 \end{align}
 Therefore leaving us for $\phi_1$ and $\phi_2$ with
 \begin{align}
     &\phi_1 = -\frac{1}{2} z^2 \theta_0(x,t) + \theta(x, t),\\
     \Rightarrow& \phi_2 =
     \frac{1}{24}z^4\theta_0(x,t)-\frac{1}{2}z^2\theta_1(x,t) + \theta_2(x,t).
 \end{align}
 The boundary condition on the bottom comes around to be
 \begin{align}
     \phi_{nz} =0 \quad \text{on}\;\; z=0.
 \end{align}
 The free surface boundary condition $z= 1+\varepsilon\eta$ n evolves more calculation, we consider
 only terms up the order of $\delta^2$, initializing with
 \begin{align}
     &\phi_z = \delta^2(\eta_t + \varepsilon\phi_x \eta_x)\\
     \Leftrightarrow &\frac{1}{\delta}\phi_z = \eta_t + \varepsilon\phi_x
     \eta_x,
 \end{align}
 substituting $\phi_\delta$ into the above proceeds to be
 \begin{align}
     \frac{1}{\delta^2}\underbrace{\phi_{0z}}_{=0} + \phi_{1z}+ \delta^2\phi_{zz}
     O(\delta^{2\cdot 2})
     &= -z\theta_{x x} + \delta^2\left( \frac{1}{6}z^3\theta_{0 x x x x} - z
     \theta_{0x x} \right) + O(\delta^{2\cdot 2})\\
     &=-(1+\varepsilon\eta)\theta_{0 x x} + \delta^2\left(
     \frac{1}{6}(1+\varepsilon\eta)^3\theta_{0 x x} -
 (1+\varepsilon\eta)\theta_{0 x x} \right) +O(\delta^{2\cdot 2})\\
     &= \eta_t + \varepsilon\eta_x \left( \theta_{0x}
     \delta^2(\theta_{1x}-\frac{1}{2}( 1+ \varepsilon\eta)^2 \theta_{0x x x}
 \right).
 \end{align}
 The second condition is
 \begin{align}
     \phi_t + \eta + \frac{1}{2}\varepsilon \left( \frac{1}{\delta}\phi^2_z
     \phi_x^2\right)  = 0,
 \end{align}
 becomes after substitution
 \begin{align}
     \theta_{0t}+ \delta^2\left( -\frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x xt}
     + \theta_{1t}\right) + \eta + O(\delta^{2\cdot 2})
     &=-\frac{1}{2}\delta^2\varepsilon(-(1+\varepsilon\eta)\theta_{0 x x
     })^2\\
     &-\frac{1}{2}\left( \theta_{0 x} + \delta^2\left( \theta_{1x} -
     \frac{1}{2}(1+\varepsilon\eta)^2\theta_{0 x x x x}  \right)  \right) ^2
 \end{align}
 The simplest case is $\varepsilon,\delta \rightarrow 0$.
 
 
 
 
 
 
 
 
 
 
 
 
 \newpage
 \include{appendix.tex}
 
 
 
 \nocite{johnson_1997}
 \nocite{vallis_2017}
 \nocite{constantin_tsunami}
 \nocite{rupert_2009}
 \nocite{mathe-physik}
 
 \printbibliography
 
 \end{document}