Terrain-Following Coordinate Transformation

Vertical Terrain-Following Coordinates

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From the point of view of the computational model, it is highly convenient to introduce a stretched vertical coordinate system which essentially "flattens out" the variable bottom at $z = -h(x,y)$. Such "$s$" coordinate systems have long been used, with slight appropriate modification, in both meteorology and oceanography [e.g., Phillips (1957) and Freeman et al. (1972)]. To proceed, we make the coordinate transformation:

$$\hat{x} = x$$ $$\hat{y} = y $$ $$s = s(x,y,z)$$ $$z = z(x,y,s)$$ $$\hat{t} = t$$

See S-coordinate for the form of $s$ used here. In the stretched system, the vertical coordinate $s$ spans the range $-1 \leq s \leq 0$; we are therefore left with level upper ($s = 0$) and lower ($s = -1$) bounding surfaces. The chain rules for this transformation are:

$$\left( { \partial \over \partial x } \right)_z =\left( { \partial \over \partial x } \right)_s - \left( { 1 \over H_z } \right) \left( { \partial z \over \partial x } \right)_s { \partial \over \partial s}$$

$$\left( { \partial \over \partial y } \right)_z = \left( { \partial \over \partial y } \right)_s - \left( { 1 \over H_z } \right) \left( { \partial z \over \partial y } \right)_s { \partial \over \partial s}$$

$${ \partial \over \partial z } = \left( { \partial s \over \partial z } \right) { \partial \over \partial s} = { 1 \over H_z } { \partial \over \partial s }$$

where

$$H_z \equiv { \partial z \over \partial s }$$

As a trade-off for this geometric simplification, the dynamic equations become somewhat more complicated. The resulting dynamic equations are, after dropping the carats:

$${\partial u \over \partial t} - fv + \vec{v} \cdot \nabla u = - {\partial \phi \over \partial x} - \left( \frac{g\rho} {\rho_o} \right) \frac{\partial z}{\partial x} - g {\partial \zeta \over \partial x} + {\cal F}_u + {\cal D}_u$$

$$\frac{\partial v}{\partial t} + fu + \vec{v} \cdot \nabla v = - \frac{\partial \phi}{\partial y} - \left( \frac{g\rho} {\rho_o} \right) \frac{\partial z}{\partial y} - g {\partial \zeta \over \partial y} + {\cal F}_v + {\cal D}_v$$

$$\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T = {\cal F}_{T} + {\cal D}_{T}$$

$$\frac{\partial S}{\partial t} + \vec{v} \cdot \nabla S = {\cal F}_{S} + {\cal D}_{S}$$

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

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

$${\partial H_z \over \partial t} + {\partial (H_zu) \over \partial x} + {\partial (H_zv) \over \partial y} + {\partial (H_z \Omega) \over \partial s} = 0$$ where

$$\vec{v} = (u,v,\Omega)$$

$$\vec{v} \cdot \nabla = u \frac{\partial}{\partial x} + v

 \frac{\partial}{\partial y} + \Omega \frac{\partial}{\partial s}$$

The vertical velocity in $s$ coordinates is

$$\Omega (x,y,s,t) = {1 \over H_z} \left[ w - (1+s) {\partial \zeta \over \partial t} - u {\partial z \over \partial x} - v {\partial z \over \partial y} \right]$$

and

$$w = {\partial z \over \partial t} + u {\partial z \over \partial x}

 + v {\partial z \over \partial y} + \Omega H_z$$

Vertical Boundary Conditions

In the stretched coordinate system, the vertical boundary conditions become:

top ($s = 0$):

$\left(\frac{K_m}{H_z}\right) \frac{\partial u}{\partial s} = \tau^x_s (x,y,t)$

$\left(\frac{K_m}{H_z}\right) \frac{\partial v}{\partial s} = \tau^y_s(x,y,t)$

$\left(\frac{K_T}{H_z}\right) \frac{\partial T}{\partial s} = {Q_T \over \rho_o c_P} + {1 \over \rho_o c_P} {dQ \over dT} (T - T_{\rm ref})$

$\left(\frac{K_S}{H_z}\right) \frac{\partial S}{\partial s} = {(E - P) S \over \rho_o}$

$\Omega = 0$

and bottom ($s = -1$):

$\left(\frac{K_m}{H_z}\right) \frac{\partial u}{\partial s} = \tau^x_b (x,y,t)$

$\left(\frac{K_m}{H_z}\right) \frac{\partial v}{\partial s} = \tau^y_b (x,y,t)$

$\left(\frac{K_T}{H_z}\right) \frac{\partial T}{\partial s} = 0<$

$\left(\frac{K_S}{H_z}\right) \frac{\partial S}{\partial s} = 0$

$\Omega = 0$

Note the simplification of the boundary conditions on vertical velocity that arises from the $s$ coordinate transformation.