Terrain-Following Coordinate Transformation

Vertical Terrain-Following Coordinates

<wikitex>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 "$\sigma$" 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:

$$ \eqalign{ \hat{x} &= x \cr \hat{y} &= y \cr \sigma &= \sigma(x,y,z) \cr z &= z(x,y,\sigma) \cr \hat{t} &= t \cr } \eqno{(1)}$$

See S-coordinate for the form of $\sigma$ used here. Also, see Shchepetkin and McWilliams, 2005 for a discussion about the nature of this form of $\sigma$ and how it differs from that used in SCRUM.

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

$$ \eqalign{ \left( { \partial \over \partial x } \right)_z &=\left( { \partial \over \partial x } \right)_\sigma - \left( { 1 \over H_z } \right) \left( { \partial z \over \partial x } \right)_\sigma { \partial \over \partial \sigma}\cr \noalign{\smallskip} \left( { \partial \over \partial y } \right)_z &= \left( { \partial \over \partial y } \right)_\sigma - \left( { 1 \over H_z } \right) \left( { \partial z \over \partial y } \right)_\sigma { \partial \over \partial \sigma} \cr \noalign{\smallskip} { \partial \over \partial z } &= \left( { \partial \sigma \over \partial z } \right) { \partial \over \partial \sigma} = { 1 \over H_z } { \partial \over \partial \sigma } \cr} \eqno{(2)} $$

where

$$H_z \equiv { \partial z \over \partial \sigma } \eqno{(3)}$$

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

$${\partial u \over \partial t} - fv + \vec{v} \cdot \nabla u = - {\partial \phi \over \partial x} - \left( \frac{g\rho} {\rho_o} \right) \frac{\partial z}{\partial x} - g {\partial \zeta \over \partial x} + { 1 \over H_z } {\partial \over \partial \sigma} \left[ {(K_m+\nu) \over H_z} {\partial u \over \partial \sigma} \right] + {\cal F}_u + {\cal D}_u \eqno{(4)}$$

$$\frac{\partial v}{\partial t} + fu + \vec{v} \cdot \nabla v = - \frac{\partial \phi}{\partial y} - \left( \frac{g\rho} {\rho_o} \right) \frac{\partial z}{\partial y} - g {\partial \zeta \over \partial y} + { 1 \over H_z } {\partial \over \partial \sigma} \left[ {(K_m+\nu) \over H_z} {\partial v \over \partial \sigma} \right] + {\cal F}_v + {\cal D}_v \eqno{(5)}$$

$$\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = { 1 \over H_z } {\partial \over \partial \sigma} \left[ {(K_C+\nu) \over H_z} {\partial C \over \partial \sigma} \right] + {\cal F}_{C} + {\cal D}_{C} \eqno{(6)}$$

$$\rho = \rho(T,S,P) \eqno{(7)}$$

$$\frac{\partial \phi}{\partial \sigma} = \left( \frac{-gH_z\rho} {\rho_o} \right) \eqno{(8)}$$

$${\partial H_z \over \partial t} + {\partial (H_zu) \over \partial x} + {\partial (H_zv) \over \partial y} + {\partial (H_z \Omega) \over \partial \sigma} = 0 \eqno{(9)}$$ where

$$\vec{v} = (u,v,\Omega) \eqno{(10)}$$

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

 \frac{\partial}{\partial y} + \Omega \frac{\partial}{\partial \sigma} \eqno{(11)}$$

The vertical velocity in $\sigma$ coordinates is

$$\Omega (x,y,\sigma,t) = {1 \over H_z} \left[ w - {z+h \over \zeta + h} {\partial \zeta \over \partial t} - u {\partial z \over \partial x} - v {\partial z \over \partial y} \right] \eqno{(12)}$$

and

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

 + v {\partial z \over \partial y} + \Omega H_z \eqno{(13)}$$

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Vertical Boundary Conditions

<wikitex>In the stretched coordinate system, the vertical boundary conditions become:

top ($\sigma = 0$): $$ \eqalign{ \left( \frac{K_m}{H_z}\right)& \frac{\partial u}{\partial \sigma} = \tau^x_s (x,y,t) \cr \left( \frac{K_m}{H_z}\right)& \frac{\partial v}{\partial \sigma} = \tau^y_s(x,y,t)\cr \left( \frac{K_C}{H_z}\right)& \frac{\partial C}{\partial \sigma} = {Q_C \over \rho_o c_P}\cr &\Omega = 0 \cr} \eqno{(14)} $$

and bottom ($\sigma = -1$): $$ \eqalign{ \left( \frac{K_m}{H_z}\right)& \frac{\partial u}{\partial \sigma} = \tau^x_b (x,y,t) \cr \left( \frac{K_m}{H_z}\right)& \frac{\partial v}{\partial \sigma} = \tau^y_b (x,y,t) \cr \left( \frac{K_C}{H_z}\right)& \frac{\partial C}{\partial \sigma} = 0 \cr &\Omega = 0 \cr} \eqno{(15)}$$

Note the simplification of the boundary conditions on vertical velocity that arises from the $\sigma$ coordinate transformation. </wikitex>