Vertical S-coordinate
ROMS has a generalized vertical, terrain-following, coordinate system. Currently, two vertical transformation
equations,
, are available which can support numerous vertical stretching 1D-functions when several
constraints are satisfied.
Transformation Equations
The following vertical coordinate transformations are available:
![{\displaystyle {\begin{aligned}z(x,y,\sigma ,t)&=S(x,y,\sigma )+\zeta (x,y,t)\left[1+{\frac {S(x,y,\sigma )}{h(x,y)}}\right],\\S(x,y,\sigma )&=h_{c}\,\sigma +\left[h(x,y)-h_{c}\right]\,C(\sigma )\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/6466f7caf947aba25a5591634397968857c92f81)
or
![{\displaystyle {\begin{aligned}z(x,y,\sigma ,t)&=\zeta (x,y,t)+\left[\zeta (x,y,t)+h(x,y)\right]\,S(x,y,\sigma ),\\S(x,y,\sigma )&={\frac {h_{c}\,\sigma +h(x,y)\,C(\sigma )}{h_{c}+h(x,y)}}\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/babbb934465e3692ec8d85fe094a8cd8d664f00f)
where
is a nonlinear vertical transformation functional,
is the time-varying free-surface,
is the unperturbed water column thickness and
corresponds to the ocean bottom,
is a fractional
vertical stretching coordinate ranging from
,
is a nondimensional, monotonic, vertical
stretching function ranging from
, and
is a positive thickness controlling the stretching.
In sediment applications,
is changed at every time-step since it is affected by erosion and deposition processes.
Warning: We are now very strict about the value of
and other input vertical transformation parameters. Since the beginning of ROMS,
where
and
is the stretching parameter specified in ocean.in. Notice that it is not possible to build the vertical stretching coordinates when
and using transformation (1) because it yields
. This has been a legacy issue in ROMS for years, and you are either consistent with the previous set-up of your application or change the value of
in all input NetCDF files and ocean.in. Therefore, in new applications you need to be sure that
when using the transformation (1). Notice that this restriction is removed in transformation (2) and
can be any positive value. It works for both
or
. Again, we are now checking internally for all stretching parameters for consistency in all input NetCDF files that have such variables.
We find it convenient to define:
![{\displaystyle H_{z}\equiv {\partial z \over \partial \sigma }}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/3c5501b1ef13d87fdecdf0af7cb2ac02429f1d85)
where
are the vertical grid thicknesses. In ROMS,
is computed discretely as
since this leads to the vertical sum of
being exactly the total water column thickness
.
Transformation (1) has been available in ROMS since 1999. It is activated by setting Vtransform = 1 in ocean.in. Notice that,
![{\displaystyle S(x,y,\sigma )={\begin{cases}0,&{\text{if}}\;\;\sigma =\;\;\;\,0,\;\;&C(\sigma )=\;\;\;\,0,&{\hbox{at the free-surface;}}\\-h(x,y),&{\text{if}}\;\;\sigma =-1,\;\;&C(\sigma )=-1,&{\hbox{at the ocean bottom.}}\end{cases}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/8ce45241c7abe3379dff9779d0d71f339dc06dd8)
In an undisturbed ocean state, corresponding to zero free-surface,
. Shchepetkin and
McWilliams (2005) denotes this transformation as an unperturbed coordinate system since all the depths are not affected by the
displacements of the free-surface. This ensures that the vertical mass fluxes generated by a purely barotropic motion will vanish at every interface
Transformation (2) has been available in UCLA-ROMS since 2005. It is activated by setting Vtransform = 2 in ocean.in. Notice that,
![{\displaystyle S(x,y,\sigma )={\begin{cases}\;\;0,&{\text{if}}\;\;\sigma =\;\;\;\,0,\;\;&C(\sigma )=\;\;\;\,0,&{\hbox{at the free-surface;}}\\-1,&{\text{if}}\;\;\sigma =-1,\;\;&C(\sigma )=-1,&{\hbox{at the ocean bottom.}}\end{cases}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2b509a3ab78ad775560c520c4e17f0a46933cae1)
which is different to the behavior of the original functional in (1). Shchepetkin (personal communication) points out that (2) offers several advantages:
- Regardless of the design of
, it behaves like equally-spaced sigma-coordinates in shallow regions, where
. This is advantageous because it avoids excessive resolution and associated CFL limitation is such areas.
- Near-surface refinement behaves more or less like geopotential coordinates in deep regions (level thicknesses,
, do not depend or weakly depend on bathymetry), while near-bottom like sigma coordinates (
is roughly proportional to depth). This reduces the extreme r-factors near the bottom and reduces pressure gradient errors.
- The true sigma-coordinate system is recovered as
. This is useful when configuring applications with flat bathymetry and uniform level thickness. Practically, you can achieve this by setting Tcline to 1.0d+16 in ocean.in. This will set
. Although not necessary, we also recommend that you set
and
.
In an undisturbed ocean state,
, transformation (2) yields the following unperturbed depths,
,
![{\displaystyle {\hat {z}}(x,y,\sigma )\equiv h(x,y)\,S(x,y,\sigma )=h(x,y)\left[{\frac {h_{c}\,\sigma +h(x,y)\,C(\sigma )}{h_{c}+h(x,y)}}\right]}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/cbfd5c93e7736a3e52b52bd973dbf18e6da9197e)
and
![{\displaystyle d{\hat {z}}=d\sigma \;h(x,y)\;\left[{\frac {h_{c}}{h_{c}+h(x,y)}}\right]}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/73a5c5b5889cee40d3766a46dbe0dcfa0949e8a2)
As a consequence, the uppermost grid box retains very little dependency from bathymetry in deep areas, where
.
For example, if
and
changes from
to
, the uppermost grid box changes only by a factor
of 1.08 (less than
).
Vertical Stretching Functions
The above generic vertical transformation design facilitates numerous vertical stretching functions,
. This function
is defined in terms of several parameters which are specified in the standard input file, ocean.in.
Stretching Function Properties:
is a dimensionless, nonlinear, monotonic function;
is a continuous differentiable function, or a differentiable piecewise function with smooth transition;
must be discretized in terms of the fractional stretched vertical coordinate
,
![{\displaystyle \sigma (k)={\begin{cases}{\frac {k-N}{N}},&{\text{at vertical }}W{\hbox{-points,}}&k=0,\dots ,N{\hbox{,}}\\\\{\frac {k-N-0.5}{N}},&{\text{at vertical }}\rho {\hbox{-points,}}&k=1,\dots ,N\end{cases}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/58d0710554e7e7994a25d570da2ec84a405735fd)
must be constrained by
, that is,
![{\displaystyle C(\sigma )={\begin{cases}\;\;0,&{\text{if}}\;\;\sigma =\;\;\;\,0,\;\;&C(\sigma )=\;\;\;\,0,&{\hbox{at the free-surface;}}\\-1,&{\text{if}}\;\;\sigma =-1,\;\;&C(\sigma )=-1,&{\hbox{at the ocean bottom.}}\end{cases}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/c2e34e839818ed14efa275775d4e01ba95e482f8)
Available Stretching Functions:
- Song and Haidvogel (1994) function available in ROMS since its beginning, Vstretching = 1.
is defined as:
![{\displaystyle C(\sigma )=(1-\theta _{B}){\sinh(\theta _{S}\,\sigma ) \over \sinh \theta _{S}}+\theta _{B}\left[{\tanh[\theta _{S}(\sigma +{1 \over 2})] \over 2\tanh({1 \over 2}\theta _{S})}-{\frac {1}{2}}\right]}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/c14f2fb44197aed4dcf6ec3624e6c467df762995)
where
and
are the surface and bottom control parameters. Their ranges are
and
, respectively. This function has the following features:
- It is infinitely differentiable in
.
- The larger the value of
, the more resolution is kept above
.
- For
, the resolution all goes to the surface as
is increased.
- For
, the resolution goes to both the surface and the bottom equally as
is increased.
- For
, there is a subtle mismatch in the discretization of the model equations, for instance in the horizontal viscosity term. We recommend that you stick with reasonable values of
, say
.
- Some applications turn out to be sensitive to the value of
used.
- A. Shchepetkin (2005) UCLA-ROMS deprecated function, Vstretching = 2.
is defined as a piecewise function:
![{\displaystyle {\begin{aligned}C(\sigma )&=\mu \,C_{{sur}}(\sigma )+(1-\mu )\,C_{{bot}}(\sigma ),\\\\C_{{sur}}(\sigma )&={\frac {1-\cosh(\theta _{S}\,\sigma )}{\cosh(\theta _{S})-1}},\qquad {\hbox{for }}\theta _{S}>0,\qquad C_{{bot}}(\sigma )={\frac {\sinh[\theta _{B}\,(\sigma +1)]}{\sinh(\theta _{B})}}-1,\qquad {\hbox{for }}\theta _{B}>0,\\\\\mu &=(\sigma +1)^{{\alpha }}\;\left[1+{\frac {\alpha }{\beta }}\left(1-(\sigma +1)^{{\beta }}\right)\right]\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/f4ed87fed088d3bf7bf480ece2965d96d49f8bbf)
This function is similar in meaning to the Song and Haidvogel (1994), but note that hyperbolic function in
is
instead of
and
![{\displaystyle {\frac {\partial C}{\partial \sigma }}\rightarrow 0\qquad {\hbox{as}}\;\;\sigma \rightarrow 0.}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/67c53ba97afefe986fa53013a4f00e522be65f85)
- R. Geyer function for high bottom boundary layer resolution in relatively shallow applications, Vstretching = 3.
is defined as a piecewise function:
![{\displaystyle {\begin{aligned}C(\sigma )&=\mu \,C_{{bot}}(\sigma )+(1-\mu )\,C_{{sur}}(\sigma ),\\\\C_{{sur}}(\sigma )&=-{\frac {\log[\cosh(\gamma \,{\hbox{abs}}(\sigma )^{{\theta _{S}}}]}{\log[\cosh(\gamma )]}},\qquad C_{{bot}}(\sigma )={\frac {\log[\cosh(\gamma \,(\sigma +1)^{{\theta _{B}}}]}{\log[\cosh(\gamma )]}}-1,\\\\\mu &={\frac {1}{2}}\left[1-\tanh \left(\gamma \left(\sigma +{\frac {1}{2}}\right)\right)\right]\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/945a7dd7ae17190355eaecac33c1726de1178c01)
where the power exponents
and
are the surface and bottom control parameters specified in standard input file ocean.in, as before. Here,
is a scale factor for all hyperbolic functions. Currently,
. Typical values for the control parameters are:
|
minimal increase of surface resolution
|
|
significant surface amplification
|
|
no bottom amplification
|
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significant bottom amplification
|
|
good bottom resolution for gravity flows
|
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super-high bottom resolution
|
- A. Shchepetkin (2010) UCLA-ROMS current function, Vstretching = 4.
is defined as a continuous, double stretching function:
Surface refinement function:
![{\displaystyle C(\sigma )={\frac {1-\cosh(\theta _{S}\,\sigma )}{\cosh(\theta _{S})-1}},\qquad {\hbox{for }}\theta _{S}>0,\qquad C(\sigma )=-{\sigma }^{{2}},\qquad {\hbox{for }}\theta _{S}\leq 0}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/8c733e4061f7d9c034d7da35d35a024bd57979ce)
Bottom refinement function:
![{\displaystyle C(\sigma )={\frac {\exp(\theta _{B}\,C(\sigma ))-1}{1-\exp(-\theta _{B})}},\qquad {\hbox{for }}\theta _{B}>0}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/3f8550f5ee377d591e0b6a4c7786b0ba5d2c21ab)
Notice that the bottom function (9) is the second stretching of an already stretched transform (8). The resulting stretching function is continuous with respect to
and
as their values approach zero. The range of meaningful values for
and
are:
![{\displaystyle 0\leq \theta _{S}\leq 10\qquad {\hbox{and}}\qquad 0\leq \theta _{B}\leq 4}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/90824e57412eeda697700a1f6792d7e1b6dc066b)
However, users need to pay attention to extreme r-factor (rx1) values near the bottom.
Due to its functionality and properties Vtransform = 2 and Vstretching = 4 are now the default values for ROMS.
Metadata Considerations
The above vertical transformations have been registered with the CF Conventions Committee and the NetCDF-Java group. We expect that it will take some time for the CF Committee to approve them. However, the NetCDF-Java is expected to be coded right away. Two new standard_name variable attributes: ocean_s_coordinate_g1 and ocean_s_coordinate_g2 have been assigned for transformations (1) and (2), respectively. The terms in the definition are associated with file variables by the formula_terms attribute as follows:
Transformation (1):
double s_rho(s_rho) ;
s_rho:long_name = "S-coordinate at RHO-points" ;
s_rho:valid_min = -1. ;
s_rho:valid_max = 0. ;
s_rho:positive = "up" ;
s_rho:standard_name = "ocean_s_coordinate_g1" ;
s_rho:formula_terms = "s: s_rho C: Cs_r eta: zeta depth: h depth_c: hc" ;
double s_w(s_w) ;
s_w:long_name = "S-coordinate at W-points" ;
s_w:valid_min = -1. ;
s_w:valid_max = 0. ;
s_w:positive = "up" ;
s_w:standard_name = "ocean_s_coordinate_g1" ;
s_w:formula_terms = "s: s_w C: Cs_w eta: zeta depth: h depth_c: hc" ;
Transformation (2):
double s_rho(s_rho) ;
s_rho:long_name = "S-coordinate at RHO-points" ;
s_rho:valid_min = -1. ;
s_rho:valid_max = 0. ;
s_rho:positive = "up" ;
s_rho:standard_name = "ocean_s_coordinate_g2" ;
s_rho:formula_terms = "s: s_rho C: Cs_r eta: zeta depth: h depth_c: hc" ;
double s_w(s_w) ;
s_w:long_name = "S-coordinate at W-points" ;
s_w:valid_min = -1. ;
s_w:valid_max = 0. ;
s_w:positive = "up" ;
s_w:standard_name = "ocean_s_coordinate_g2" ;
s_w:formula_terms = "s: s_w C: Cs_w eta: zeta depth: h depth_c: hc" ;
Notice that the formula_terms are identical for both transformations and are independent of the vertical stretching function used. This metadata conventions allows to process/visualize any NetCDF file generated by ROMS easily and correctly compute the associated vertical grid depths.
Vertical Grid Examples
The following figures shows the
-surfaces for both vertical transformations and available vertical stretching functions. These plots were generated in Matlab using the script scoord.m.
Vtransform=1, Vstretching=1, , , ![N=30](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1cae4bd0e7179a47f24bda656d5f5969c9d92507)
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Vtransform=2, Vstretching=2, , , ![N=30](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1cae4bd0e7179a47f24bda656d5f5969c9d92507)
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Vtransform=2, Vstretching=4, , , ![N=30](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1cae4bd0e7179a47f24bda656d5f5969c9d92507)
|
Vtransform=2, Vstretching=4, , , ![N=30](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1cae4bd0e7179a47f24bda656d5f5969c9d92507)
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Vtransform=1, Vstretching=1, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=2, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=4, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=4, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=1, Vstretching=1, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=2, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=3, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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Vtransform=2, Vstretching=4, , , ![N=20](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a5c78ec997d223a010921f350195268f242c37a)
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