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curve.tex
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curve.tex
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\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 $c$.