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\markboth{\hfil On the tidal motion around the earth \hfil EJDE--2000/35}
{EJDE--2000/35\hfil Ranis N. Ibragimov \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~35, pp.~1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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On the tidal motion around the earth complicated by the circular
geometry of the ocean's shape
\thanks{ {\em Mathematics Subject Classifications:} 35Q35, 76C99.
\hfil\break\indent
{\em Key words:} Cauchy-Poisson free boundary problem, shallow water theory,
\hfil\break\indent conformal mapping
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted January 5, 2000. Published May 16, 2000. \hfil\break\indent
Dedicated to Professor L. V. Ovsyannikov on his 80-th birthday.} }
\date{}
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\author{ Ranis N. Ibragimov }
\maketitle
\begin{abstract}
We study the Cauchy-Poisson free boundary problem on the stationary motion
of a perfect incompressible fluid circulating around the Earth.
The main goal is to find the inverse conformal mapping of the unknown
free boundary in the hodograph plane onto some fixed boundary
in the physical domain. The approximate solution to the problem is
obtained as an application of this method.
We also study the behaviour of tidal waves around the Earth. It is shown
that on a positively curved bottom the problem admits two different high
order systems of shallow water equations, while the classical problem for
the flat bottom admits only one system.
\end{abstract}
\newtheorem{theorem}{Theorem}
\section{Introduction}
We study the Cauchy-Poisson problem on the stationary motion of a perfect
fluid which
has a free boundary and has a solid bottom represented by a circle with a
sufficiently large radius. We have shown in [4]
that such problem can be associated with a two dimensional model to an
oceanic motion around the Earth since we consider strictly longitudinal
flow. Since the problem is a free boundary problem, the analysis is rather
difficult.
The permanent water waves have been considered in a great number of papers.
However, most researchers are concerned with fluid motion which is infinitely
deep and extends infinitely both rightward and leftward. See Crapper [1],
Stoker [9] or Stokes [10] for the history. Such problems are usually called
Stoke problem (if the surface tension is neglected) and Wilton's problem (if the
surface tension is taken into account).
We consider water waves for which ratio of depth of fluid above the circular
bottom to the radius of the circle is small (shallow water). In these the
disturbance to the water does not penetrate unchanged to the bottom and the
effective inertia of the water is therefore reduced.
Our primary concern is to find the conformal mapping (for the Stoke problem) of
the unknown free boundary onto fixed one. The resulting Dirichlet problem can
be solved numerically using Okamoto method [3]. The more detailed structure of
the bifurcation of solutions for the related problem was numerically computed by
Fujita at al. [3]. The existence of nontrivial solutions for the analogous
reduced Dirichlet problem can be found in Okamoto [7], Ibragimov [4] as well as
in classical literature (see e.g., [8] or [9]).
The higher order shallow water equations in the non-stationary case are derived
in this paper. It is shown that the present problem admits two different
systems of shallow water equations while the classical problem for the flat
bottom admits only one system (see [2]).
We note that papers [3], [6] and [7] are concerned with fluid whose surface
tension is taken into account. In fact, the surface tension plays the role
of a ''regulator'' of the problem which simplifies the analysis
substantially. Furthermore the nature of the problem requires the surface
tension to be neglected. Thus, the present paper represents more systematic
approach to the problem.
The present paper aims to investigate the problem by using a conformal
mapping which distinguishes our paper from [3], [4], [6] and [7].
\section{Basic Equations}
The analysis of this problem is performed in the following notation: $R$ is
the radius of the circle, $r$ is a distance from the origin, $\theta $ is a
polar angle, $h_0$ is the undisturbed level of the liquid above the circle
and $h=h(\theta )$ is the level of the disturbance of the free
boundary above the circle. For the sake of simplicity we assume that the
pressure is constant on the free boundary. The stream function $\psi =\psi
(r,\theta )$ defines the velocity vector, i.e.,
\[
v^r=-\frac{1}{r}\psi_\theta ,\quad v^\theta =\psi_r\,.
\]
Hence irrotational motion of an ideal incompressible fluid of the constant
pressure in the homogeneous gravity field $g=$const is described by the
stream function $\psi $ in the domain
\[
\Omega_h=\left\{ (r,\theta ): 0\leq \theta \leq 2\pi ,
R\leq r\leq R+h_0+h(\theta )\right\}
\]
which is bounded by the bottom $\Gamma _{R}=\left\{ (r,\theta )
:r=R, \theta \in [0,2\pi ]\right\} $ and the free
boundary with equation $\Gamma_h=\left\{ (r,\theta )
:r=R+h_0+h(\theta ), \theta \in [0,2\pi ]
\right\} $. Note that $\psi $ is a harmonic function in $\Omega_h$, since
we assumed that the flow is irrotational. More specifically, we assume that
the fluid is incompressible and inviscid and that the flow is stationary.
Then the problem is to find the function $h(\theta )$ and the
stationary, irrotational flow beneath the free boundary
$r=R+h_0+h(\theta )$ given by the stream function $\psi $ which satisfy the
following differential equations
\begin{eqnarray}
&\Delta \psi =0(\mbox{in }\Omega_h),\qquad \psi =0(
\mbox{on }\Gamma _{R}),\qquad \psi =a\mbox{ (on $\Gamma_h$),}& \\
&\left| \nabla \psi \right| ^{2}+2gh=\mbox{constant}\quad (\mbox{ on }
\Gamma_h), & \\
&\frac{1}{2}\int_0^{2\pi } (R+h_0+h(\theta))^{2}\, d\theta =\pi
(R+h_0)^{2},&
\end{eqnarray}
where the constant $a$ denotes the flow rate.
Equations (1)-(3) represent the free boundary Cauchy-Poisson problem in
which
the boundary $\Gamma_h$ is unknown as well as the stream function.
\section{The Inverse Transforms Principle}
\subsection*{Constant flow}
The exact solution
\begin{equation}
h\equiv 0 \quad \mbox{and}\quad \psi =\psi _0=a\log r
\end{equation}
of (1)-(3) corresponds to the constant flow with undisturbed free
boundary. The trivial solution (4) represents a flow whose streamlines are
concentric circles with the common center at the origin.
The following non-dimensional quantities are introduced:
\[
r=R+h_0r',\quad h=h_0h',\quad \psi
=a\psi ',\quad \epsilon =\frac{h_0}{R},\quad %
\mathcal{F}=\frac{h_0\sqrt{gh_0}}{a},
\]
where $\mathcal{F}$ is a Froude number and $R$ is used as a vertical scale.
We consider $\epsilon $ as the small parameter of the problem. After
dropping the prime, Equations (1)-(3) are written by $\psi ',h'$, and
$(r',\theta )$ as follows
\begin{eqnarray}
&\Delta _{(\epsilon )}\psi =0\quad \mbox{(in $\Omega_h$)}, &\\
&\psi =0 \mbox{(on $\Gamma _{R}$)}, &\\
&\psi =1\mbox{(on $\Gamma_h$)}, &\\
&\left| \nabla _{(\epsilon )}\psi \right| ^{2}+2\mathcal{F}
^{-2}h=\mbox{ constant } \quad \mbox{(on $\Gamma_h$)}, & \\
&\frac{1}{2} \int_0^{2\pi} (1+\epsilon+\epsilon h(\theta ))^{2}d\theta
=\pi (1+\epsilon )^{2}\quad \mbox{(on $\Gamma_h$)}. &
\end{eqnarray}
Here the Laplace and gradient operators are given by
\[
\Delta _{(\epsilon )}=(\epsilon \partial
_{\theta })^{2}+[(1+\epsilon r)\partial_r]^{2},
\quad \nabla _{(\epsilon )}
=\Big(\frac{\epsilon \partial_\theta }{(1+\epsilon r)},\partial_r\Big),
\]
where the subscripts imply the differentiation.
We further consider the complex potential $\omega (\zeta )
=\phi +i\psi $ where $\zeta =(1+\epsilon r)e^{i\theta }$ is the
independent complex variable and $\phi (\zeta )$ is the
velocity potential which is characterized by the analyticity of $\phi
+i\psi $, i.e.,
\[
\phi_r=\frac{\epsilon \psi_\theta }{(1+\epsilon r)},
\quad \frac{\epsilon \phi_\theta }{(1+\epsilon r)}=-\psi_r.
\]
We note that the complex velocity $d\omega /d\zeta $ is a single-valued
analytic function of $\zeta $, although $\omega $ is not single-valued. In
fact, when we turn around the bottom $r=1$ once, $\phi $ increases by
\[
-\int_0^{2\pi} \psi_r(1,\theta )\,d\theta
\]
which has a positive sign by the maximum principle (Hopf's lemma). Hence,
if
we remove the width of annulus region $\theta =0$,
$r\in [1,1+\epsilon ]$, then at every point $(r,\theta )$,
$\omega (\zeta )$ is single-valued analytic function which maps
the rectangular (in the $\omega (\alpha )$-hodograph plane) domain with
$\phi$ in $[0,-2\pi/\log (1+\epsilon)]$ and $\psi$ in $[0,1]$ as
coordinates
onto the annulus
\[
\Gamma_h^{0}=\left\{ (r,\theta ):1