curve.tex

\section{Horizontal curvilinear coordinates}
\label{Curve}
The requirement for a boundary-following coordinate system and for a
laterally variable grid resolution can both be met (for suitably
smooth domains) by introducing an appropriate orthogonal coordinate
transformation in the horizontal.  Let the new coordinates be
$\xi(x,y)$ and $\eta(x,y)$ where the relationship of horizontal arc
length to the differential distance is given by:
\begin{equation}
   (ds)_{\xi} = \left( {1 \over m} \right) d \xi
\end{equation}
\begin{equation}
   (ds)_{\eta} = \left( {1 \over n} \right) d \eta
\end{equation}
Here, $m(\xi,\eta)$ and $n(\xi,\eta)$ are the scale factors which
relate the differential distances $(\Delta \xi,\Delta \eta)$ to the
actual (physical) arc lengths.

It is helpful to write the equations in vector notation and to use
the formulas for div, grad, and curl in curvilinear coordinates
\citep[see][Appendix 2]{Batchelor}:
\begin{equation}
   \nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +
                 \hat{\eta} n {\partial \phi \over \partial \eta}
\end{equation}
\begin{equation}
   \nabla \cdot \vec{a} = mn \left[
   {\partial \over \partial \xi} \!\! \left( {a \over n} \right) +
   {\partial \over \partial \eta} \!\! \left( {b \over m} \right)
   \right]
\end{equation}
\begin{equation}
   \nabla \times \vec{a} = mn \left| \begin{array}{ccc}
   \vspace{1 mm}
   {\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \\
   \vspace{1 mm}
   {\partial \over \partial \xi} &
   {\partial \over \partial \eta} &
   {\partial \over \partial z} \\
   {a \over m} & {b \over n} & c
   \end{array} \right|
\end{equation}
\begin{equation}
   \nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[ 
   {\partial \over \partial \xi} \!\! \left( {m \over n} 
   {\partial \phi \over \partial \xi} \right) +
   {\partial \over \partial \eta} \!\! \left( {n \over m} 
   {\partial \phi \over \partial \eta} \right) \right]
\end{equation}
where $\phi$ is a scalar and $\vec{a}$ is a vector with components
$a$, $b$, and $c$.