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| As a trade-off for this geometric | | As a trade-off for this geometric |
| simplification, the [[Equations of motion|dynamic equations]] become somewhat more | | simplification, the [[Equations of Motion|dynamic equations]] become somewhat more |
| complicated. The resulting dynamic equations are, after dropping the | | complicated. The resulting dynamic equations are, after dropping the |
| carats: | | carats: |
Revision as of 01:52, 7 November 2006
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 \cite{Phil} and Freeman et al.\ \cite{FHD}).
To proceed, we make the coordinate transformation:
![{\hat {x}}=x](https://www.myroms.org/myroms.org/v1/media/math/render/svg/cbae757cddcfd0bd5ee3e969a2b592487132ebc5)
![{\hat {y}}=y](https://www.myroms.org/myroms.org/v1/media/math/render/svg/24ecc9d7dad36560c8a6d8030cd6371a518bc13f)
![s=s(x,y,z)\!\,](https://www.myroms.org/myroms.org/v1/media/math/render/svg/aa89c71fc29b2a7d35cb78d26c9be623a04b680e)
![z=z(x,y,s)\!\,](https://www.myroms.org/myroms.org/v1/media/math/render/svg/b651612f7225f8bdffa46864a359cf1a916f4573)
![{\hat {t}}=t](https://www.myroms.org/myroms.org/v1/media/math/render/svg/11294106c763b810c8bc8d0753d25e8ecad94127)
See S-coordinate for the form of
used here.
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:
![\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}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/8da36be44b1f3a886655e344463d9fe7f8c1bb8a)
![\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}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/77be582251fcd2b00031fbe4e6ddc552be2261ff)
![{\partial \over \partial z}=\left({\partial s \over \partial z}\right){\partial \over \partial s}={1 \over H_{z}}{\partial \over \partial s}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/fd3eb8f795424b57662fea06aaa83b44b2b6b794)
where
![H_{z}\equiv {\partial z \over \partial s}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/824eb80d604356a1ab65602e62921dba7ee45772)
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}+{F}_{u}+{D}_{u}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/fbc1a5eb5421cb9a4e651d631d3d84aa4ba8eabc)
![{\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}+{F}_{v}+{D}_{v}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/88bb8239d1fc7ba3d79be469c0be6fa3163e974d)
![{\frac {\partial T}{\partial t}}+{\vec {v}}\cdot \nabla T={F}_{{T}}+{D}_{{T}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/d4e0a99fa49a9c19ec5c1a160f6944bd7a9fca89)
![{\frac {\partial S}{\partial t}}+{\vec {v}}\cdot \nabla S={F}_{{S}}+{D}_{{S}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/60062cdf5a8df44f6d97d3f0771db8d9eff556f4)
![\rho =\rho (T,S,P)\!\,](https://www.myroms.org/myroms.org/v1/media/math/render/svg/75ffa0a0cb497a1fb24634d3a97532fce457e718)
![{\frac {\partial \phi }{\partial s}}=\left({\frac {-gH_{z}\rho }{\rho _{o}}}\right)](https://www.myroms.org/myroms.org/v1/media/math/render/svg/50508d6b980ecef7de7812df3e765dfa6e2e9250)
![{\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 s}=0](https://www.myroms.org/myroms.org/v1/media/math/render/svg/53fda5a587ff7f5ad4a03e186399dae081b826ad)
where
![{\vec {v}}=(u,v,\Omega )](https://www.myroms.org/myroms.org/v1/media/math/render/svg/b68da24340a5b6f78df7ca8aa3277e22dfd48167)
![{\vec {v}}\cdot \nabla =u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}+\Omega {\frac {\partial }{\partial s}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/054b94b382aa8df1b78ab3bc67c4c6f26cc0b16a)
The vertical velocity in
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]](https://www.myroms.org/myroms.org/v1/media/math/render/svg/351b2674c8914d5ba96e99e20c97cb7e7cb8e861)
and
![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 (
):
![\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial u}{\partial s}}=\tau _{s}^{x}(x,y,t)](https://www.myroms.org/myroms.org/v1/media/math/render/svg/7be36294d08d85517df543093672435364416792)
![\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial v}{\partial s}}=\tau _{s}^{y}(x,y,t)](https://www.myroms.org/myroms.org/v1/media/math/render/svg/ea4ac84b4514c8efaa80e24862d3a6e265d47226)
![\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}}}})](https://www.myroms.org/myroms.org/v1/media/math/render/svg/235ac5031be3b27df41ca98f4addbb78801b0881)
![\left({\frac {K_{S}}{H_{z}}}\right){\frac {\partial S}{\partial s}}={(E-P)S \over \rho _{o}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/a092f0fb3cb4c5a3a6b77e0ff3a317ef366e91c7)
![\Omega =0\!\,](https://www.myroms.org/myroms.org/v1/media/math/render/svg/f8443301cc55a978969273f786ce8fc0f84641c5)
and bottom (
):
![\left({\frac {K_{m}}{H_{z}}}\right){\frac {\partial u}{\partial s}}=\tau _{b}^{x}(x,y,t)](https://www.myroms.org/myroms.org/v1/media/math/render/svg/9e1e074aa24db7137e5536420e6ab94c49752a8f)
& \left(\frac{K_m}{H_z}\right) \frac{\partial v}{\partial s} = \tau^y_b (x,y,t)</math>
& \left(\frac{K_T}{H_z}\right) \frac{\partial T}{\partial s} = 0</math>
![\left({\frac {K_{S}}{H_{z}}}\right){\frac {\partial S}{\partial s}}=0](https://www.myroms.org/myroms.org/v1/media/math/render/svg/222706fd34aec1b1550575097e6f09b3d491e1be)
![\Omega =0\!\,](https://www.myroms.org/myroms.org/v1/media/math/render/svg/f8443301cc55a978969273f786ce8fc0f84641c5)
Note the simplification of the boundary conditions on vertical
velocity that arises from the
coordinate transformation.