Terrain-Following Coordinate Transformation
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
.
Such "
" 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:
![{\displaystyle {\begin{aligned}{\hat {x}}&=x\\{\hat {y}}&=y\\\sigma &=\sigma (x,y,z)\\z&=z(x,y,\sigma )\\{\hat {t}}&=t\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/3ceec31829b2a562fd6fe3d5c523d00765b2bd37)
See Vertical S-coordinate for the form of
used here. Also, see Shchepetkin and McWilliams, 2005 for a discussion about the nature of this form of
and how it
differs from that used in SCRUM.
In the stretched system, the vertical coordinate
spans the range
; we are therefore left with level upper (
) and lower (
) bounding surfaces. The chain rules for this transformation are:
![{\displaystyle {\begin{aligned}\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 }\\\\\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 }\\\\{\partial \over \partial z}&=\left({\partial \sigma \over \partial z}\right){\partial \over \partial \sigma }={1 \over H_{z}}{\partial \over \partial \sigma }\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/048172c185a28510859813704156b7a4d5c07934)
where
![{\displaystyle H_{z}\equiv {\partial z \over \partial \sigma }}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/3c5501b1ef13d87fdecdf0af7cb2ac02429f1d85)
As a trade-off for this geometric simplification, the dynamic equations become somewhat more complicated. The resulting dynamic equations, after dropping the
carats, are:
![{\displaystyle {\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}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/487b4e188a579e162dabc07d33966bd1c9b47f0a)
![{\displaystyle {\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}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/8dc2f5338bce5024907b81f40dad0c97c2e1eb0e)
![{\displaystyle {\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}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/af6892023a11694d2a145679e95ac2f95f360ebf)
![{\displaystyle \rho =\rho (T,S,P)}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/b639c0ff8aeb716adc0b3277afea3d105d28eff1)
![{\displaystyle {\frac {\partial \phi }{\partial \sigma }}=\left({\frac {-gH_{z}\rho }{\rho _{o}}}\right)}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/a141769b9c1b88268b6910ac58465bc336071180)
![{\displaystyle {\partial H_{z} \over \partial t}+{\partial (H_{z}u) \over \partial x}+{\partial (H_{z}v) \over \partial y}+{\partial (H_{z}\Omega ) \over \partial \sigma }=0}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/d14a9ca9a620ce5f2a5f8f9837db6515645490c3)
where
![{\displaystyle {\vec {v}}=(u,v,\Omega )}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/b68da24340a5b6f78df7ca8aa3277e22dfd48167)
![{\displaystyle {\vec {v}}\cdot \nabla =u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}+\Omega {\frac {\partial }{\partial \sigma }}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/0921d5afccdf761c817e9f03d26f8a70a8c0ebc6)
The vertical velocity in
coordinates is
![{\displaystyle \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]}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/e4770e99d7e01eec1bb1529d1f8e9c766f4a8b7d)
and
![{\displaystyle w={\partial z \over \partial t}+u{\partial z \over \partial x}+v{\partial z \over \partial y}+\Omega H_{z}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1979105e3053fd5a5bb78e2e79f80dae93829b6f)
Vertical Boundary Conditions
In the stretched coordinate system, the vertical boundary conditions
become:
top (
):
![{\displaystyle {\begin{aligned}\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial u}{\partial \sigma }}&=\tau _{s}^{x}(x,y,t)\\\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial v}{\partial \sigma }}&=\tau _{s}^{y}(x,y,t)\\\left({\frac {K_{C}}{H_{z}}}\right){\frac {\partial C}{\partial \sigma }}&={Q_{C} \over \rho _{o}c_{P}}\\\Omega &=0\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/0b09ae092845e86bef5dd8a076ef432b8347bcfb)
and bottom (
):
![{\displaystyle {\begin{aligned}\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial u}{\partial \sigma }}&=\tau _{b}^{x}(x,y,t)\\\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial v}{\partial \sigma }}&=\tau _{b}^{y}(x,y,t)\\\left({\frac {K_{C}}{H_{z}}}\right){\frac {\partial C}{\partial \sigma }}&=0\\\Omega &=0\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2221de977d1b9db28704585ffce3cff4dd340b00)
Note the simplification of the boundary conditions on vertical
velocity that arises from the
coordinate transformation.