Curvilinear Coordinates Transformation

Horizontal Curvilinear Coordinates

<wikitex>In many applications of interest (e.g., flow adjacent to a coastal boundary), the fluid may be confined horizontally within an irregular region. In such problems, a horizontal coordinate system which conforms to the irregular lateral boundaries is advantageous. It is often also true in many geophysical problems that the simulated flow fields have regions of enhanced structure (e.g., boundary currents or fronts) which occupy a relatively small fraction of the physical/computational domain. In these problems, added efficiency can be gained by placing more computational resolution in such regions.

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:

$$(ds)_\xi = \left( \frac{1}{m} \right) d \xi$$

$$(ds)_\eta = \left( \frac{1}{n} \right) d \eta$$

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. Curvilinear Coordinates contains the curvilinear version of several common vector quantities.

Denoting the velocity components in the new coordinate system by

$$\vec{v} \cdot \hat{\xi} = u$$

$$\vec{v} \cdot \hat{\eta} = v$$

the equations of motion can be re-written (see, e.g., Arakawa and Lamb, 1977) as:

$$\frac{\partial}{\partial t} \left( \frac{H_z u}{mn} \right) + \frac

  {\partial}{\partial \xi} \left( \frac{H_z u^2}{n} \right ) + \frac
  {\partial}{\partial \eta} \left( \frac{H_z uv}{m} \right) + \frac
  {\partial}{\partial s} \left( \frac{H_z u\Omega}{mn} \right)$$

$$ - \left\{\left(\frac{f}{mn} \right) + v \frac{\partial}{\partial \xi}

  \left( \frac{1}{n} \right) - u \frac{\partial}{\partial \eta} \left(
  \frac{1}{m} \right) \right\} H_z v =$$

$$ -\left( \frac{H_z }{n} \right )

  \left( \frac{\partial \phi}{\partial \xi} +
  {g \rho \over \rho_o} {\partial z \over \partial \xi} +
  g {\partial \zeta \over \partial \xi} \right) +
  { H_z \over mn}
  \left( {\cal F}_u + {\cal D}_u \right)$$

$$\frac{\partial}{\partial t} \left( \frac{H_z v}{mn} \right) + \frac

  {\partial}{\partial \xi} \left( \frac{H_z uv}{n} \right ) + \frac
  {\partial}{\partial \eta} \left( \frac{H_z v^2}{m} \right) + \frac
  {\partial}{\partial s} \left( \frac{H_z v\Omega}{mn} \right)$$

$$ + \left\{\left(\frac{f}{mn} \right) + v \frac{\partial}{\partial \xi}

  \left( \frac{1}{n} \right) - u \frac{\partial}{\partial \eta} \left(
  \frac{1}{m} \right) \right\} H_z u =$$

$$ -\left( \frac{H_z }{m} \right )

  \left( \frac{\partial \phi}{\partial \eta} +
  {g \rho \over \rho_o} {\partial z \over \partial \eta} +
  g {\partial \zeta \over \partial \eta} \right) +
  { H_z \over mn}
  \left( {\cal F}_v + {\cal D}_v \right)$$

$$\frac{\partial}{\partial t} \left( \frac{H_z T}{mn} \right) + \frac

  {\partial}{\partial \xi} \left( \frac{H_z uT}{n}
  \right ) + \frac
  {\partial}{\partial \eta} \left( \frac{H_z vT}{m}
  \right) +$$

$$\frac {\partial}{\partial s}

  \left( \frac{H_z \Omega T}{mn} \right) =
  { H_z \over mn}
  \left( {\cal F}_{T} + {\cal D}_{T} \right)$$

$$\frac{\partial}{\partial t} \left( \frac{H_z S}{mn} \right) + \frac

  {\partial}{\partial \xi} \left( \frac{H_z uS}{n}
  \right ) + \frac
  {\partial}{\partial \eta} \left( \frac{H_z vS}{m}
  \right) +$$

$$ \frac {\partial}{\partial s}

  \left( \frac{H_z \Omega S}{mn} \right) =
  { H_z \over mn}
  \left( {\cal F}_{S} + {\cal D}_{S} \right)$$

$$\rho = \rho(T,S,P)$$

$$\frac{\partial \phi}{\partial s} = -\left( \frac{gH_z \rho}

  {\rho_o} \right)
  \frac{\partial}{\partial t} \left( \frac{H_z}{mn} \right) +
  \frac{\partial}{\partial \xi} \left( \frac{H_z u}{n} \right) +
  \frac{\partial}{\partial \eta} \left( \frac{H_z v}{m} \right) +
  \frac{\partial}{\partial s}\left( \frac{H_z \Omega}{mn} \right)
  = 0.$$

Since $z$ is a linear function of $\zeta$, the continuity equation can be rewritten as:

$$\frac{\partial}{\partial t} \left( \frac{\zeta}{mn} \right) +

  \frac{\partial}{\partial \xi} \left( \frac{H_z u}{n} \right) +
  \frac{\partial}{\partial \eta} \left( \frac{H_z v}{m} \right) +
  \frac{\partial}{\partial s}\left( \frac{H_z \Omega}{mn} \right)
  = 0.$$

All boundary conditions remain unchanged. </wikitex>