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2-3-gr-current.tex\\
{\bf Reference:} \\
C. C. Mei, (1966), {\it J. Math. \& Phys.} pp.
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\section {A gravity current}
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For the highly nonlinear equation, a relatively simple solution is that of a stationary (or permanent) wave which is profile advancing at a constant speed without changing its shape. Mathematically the profile is describable as
\be h(x,t) = h(x-Ct)=h(\sigma), ~~~\sigma=x-Ct\ee
By the chain rule of differentiation,
\[ \f{\p h(x-Ct)}{\p t}= \f{dh}{d\sigma}\f{\p \sigma}{\p t}=-C\f{dh}{d\sigma}, ~~~~\f{\p h(x-Ct)}{\p x}=\f{dh}{d\sigma}\f{\p \sigma}{\p x}=\f{dh}{d\sigma}\]
Hence (2.3.30) reduces to an ordinary differential equation,
\be -C\f{dh}{d\sigma}+\f{\rho g\cos \theta}{3\mu}\f{d}{d\sigma}\lb h^3\lp\tan\theta-\f{dh}{d\sigma}\rp\rb=0\ee
Integrating once we get
\[
-Ch+\f{\rho g\cos \theta}{3\mu}\lb h^3\lp\tan\theta-\f{dh}{d\sigma}\rp\rb=\mbox{constant}\]
Let the gravity current advance along a dry bed, then $h=0$ is a part of the solution. The constant of integration must be set to zero. Introducing the dimensionless variables
\be h=H_ch', ~~\sigma=L_c\sigma', ~~~\mbox{with}~~L_c=H_c/\tan\theta, \ee
where $H_c$ is the maximum depth far upstream, we get
\be -\f{3C\mu}{\rho g H_c^2\sin\theta} h'+h'^3\lp 1-\f{dh'}{d\sigma'}\rp=0, \ee
Let the gravity current be uniform far upstream, then
\be h'\to 1, ~~\f{dh'}{d\sigma'}\to 0, ~~\mbox{as}~~\sigma'\to -\infty.\ee
It follows that
\[ \f{3C\mu}{\rho g H_c^2\sin\theta} =1\]
or, \be \fbox{$\displaystyle C=\f{\rho g H_c^2\sin\theta}{3\mu}$}
\label{wavespeed}\ee
and \be h'\lb -1+h'^2\lp 1-\f{dh'}{d\sigma'}\rp\rb =0, \ee
One of the solution is $h'=0$, representing the dry bed. For the nontrivial solution, we
rewrite
\be d\sigma' = -\f{h^2 dh}{1 -h^2} = dh \lb 1-\f{1}{2}\lp\f{1}{1-h} +\f{1}{1+h}\rp \rb \label{ode}\ee
which can be integrated to give
\be \fbox{$\displaystyle h'+\f{1}{2} \log \lp\f{1-h'}{1+h'}\rp = \sigma'-\sigma'_o $}\ee
This is an implicit relation between $h'$ and $\sigma'$, and represents a smooth surface decreasing monotonically from $h = 1$ at $\sigma'\sim -\infty$ to $h'=0$ at the front $\sigma'=\sigma'_o$, as plotted in Figure \ref{gravity- current}. Note from (\ref{ode})
that $d\sigma'/dh'=0$ when $h'=0$, implying infinite slope at the tip of the gravity current. This infinity violates the original approximation that $dh'/d\sigma'=O(1)$. Fortunately it is highly localized and does not affect the validity of the theory elsewhere (see Liu \& Mei, 1989, JFM).
\begin{figure}[h]\begin{center}
\includegraphics[scale=1]{gracur.eps}\end{center}
\caption{Gravity current down an inclined plane}
\label{gravity- current}
\end{figure}
Eq. (\ref{wavespeed}) tells us that the speed of the front is higher for a thicker layer, steeper slope or smaller viscosity. This relation can be confirmed by a quicker argument. In the fixed frame of reference, the total flux must be equal to $CH$. therefore $C$ must be equal to the depth-averaged velocity $\overline u$ which is given by (2.3.19) with $\p h/\p x=0$.
A similar analysis has been applied to a fluid-mud which is non-Newtionian characterized by the yield stress. Laboratory simulations have been reported by Liu \& Mei ({\it J. Fluid Mech.} 207, 505-529.) who used a kaolinite/water mixture. Figure \ref{fig:exptsetup}
shows the setup of the inclined flume and Figure \ref{fig:experiment-mud} shows the recorded profiles of the gravity current along with the theory . The agreement is very good, despite the steep front where the approximation is locally invalid.
\begin{figure}[h]
\begin{center}\includegraphics[scale=1]{mud-exp-1.eps}\end{center}
\caption{Experiment setup for a mud current down an inclined plane. From Liu \& Mei 1989.}
\label{fig:exptsetup}
\end{figure}
\begin{figure}[h]\begin{center}\includegraphics[scale=1]{mud-exp-2.eps}\end{center}
\label{fig:experiment-mud}
\end{figure}
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\begin{figure}[h]
\vspace{3.5in}\hskip 0.5in
\special{bmp:mud-exp-1.bmp x=4.5in y=3in}
\caption{Experiment setup for a mud current down an inclined plane. From Liu \& Mei 1989.}
\label{fig:exptsetup}
\end{figure}
\begin{figure}[h]
\vspace{4.5in}\hskip 0.5in
\special{bmp:mud-exp-2.bmp x=6in y=3.5in}
\caption{Profiles of a mud current down an inclined plane. From Liu \& Mei 1989.}
\label{fig:experiment-mud}
\end{figure}