Vertical S-coordinate

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Vertical S-coordinate

<wikitex> ROMS has a generalized vertical, terrain-following, coordinate system. Currently, two vertical transformation equations, $z=z(x,y,\sigma,t)$, are available which can support numerous vertical stretching 1D-functions when several constraints are satisfied. </wikitex>

Transformation Equations

<wikitex> The following vertical coordinate transformations are available: $$ \eqalign {

     z(x,y,\sigma,t) &= S(x,y,\sigma) + \zeta(x,y,t) \left[1 + \frac{S(x,y,\sigma)}{h(x,y)}\right], \cr
  \noalign{\medskip}
       S(x,y,\sigma) &= h_c \, \sigma + \left[h(x,y) - h_c\right] \, C(\sigma) } \eqno{(1)} $$

or $$ \eqalign {

     z(x,y,\sigma,t) &= \zeta(x,y,t) + \left[\zeta(x,y,t) + h(x,y)\right] \, S(x,y,\sigma), \cr
  \noalign{\medskip}
       S(x,y,\sigma) &= \frac{h_c \, \sigma + h(x,y)\, C(\sigma)}{h_c + h(x,y)} } \eqno{(2)} $$

where $S(x,y,\sigma)$ is a nonlinear vertical transformation functional, $\zeta(x,y,t)$ is the time-varying free-surface, $h(x,y)$ is the unperturbed water column thickness and $z=-h(x,y)$ corresponds to the ocean bottom, $\sigma$ is a fractional vertical stretching coordinate ranging from $-1 \le\sigma\le 0$, $C(\sigma)$ is a nondimensional, monotonic, vertical stretching function ranging from $-1 \le C(\sigma) \le 0$, and $h_c$ is a positive critical depth or thickness controlling the stretching. In sediment applications, $h=h(x,y,t)$ is changed at every time-step since it is affected by erosion and deposition processes.

We find it convenient to define:

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

where $H_z=H_z(x,y,\sigma,t)$ are the vertical grid thicknesses. In ROMS, $H_z$ is computed discretely as $\Delta z/ \Delta \sigma$ since this leads to the vertical sum of $H_z$ being exactly the total water column thickness $D$.

Note Transformation (1) has been available in ROMS since 1999. It is activated by setting Vtransform = 1 in ocean.in. Notice that,

$$ S(x,y,\sigma) = \cases{0,        &if $\;\;\sigma = \phantom{-}0,  \;\; C(\sigma) = \phantom{-}0, \qquad\hbox{at the free-surface;}$\cr
                          -h(x,y),  &if $\;\;\sigma = -1, \;\; C(\sigma) = -1$, \qquad\hbox{at the ocean bottom.}\cr}$$

In an undisturbed ocean state, corresponding to zero free-surface, $z=S(x,y,\sigma)$. 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

Note Transformation (2) has been available in UCLA-ROMS for awhile. It is activated by setting Vtransform = 2 in ocean.in. Notice that,

$$ S(x,y,\sigma) = \cases{0,        &if $\;\;\sigma = \phantom{-}0,  \;\; C(\sigma) = \phantom{-}0, \qquad\hbox{at the free-surface;}$\cr
                          -1,       &if $\;\;\sigma = -1, \;\; C(\sigma) = -1$, \qquad\hbox{at the ocean bottom;}\cr}$$

which is different to the behavior of the original functional in (1). Shchepetkin (personal communication) points out that (2) offers a couple of advantages:

  • Regardless of the design of $C(\sigma)$, it behaves like equally-spaced sigma-coordinates in shallow regions, where $h(x,y) \ll h_c$. 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, $H_z$, do not depend or weakly depend on bathymetry), while near-bottom like sigma coordinates ($H_z$ is roughly proportional to depth). This reduces the extreme r-factors near the bottom and reduces pressure gradient errors.

In an undisturbed ocean state, $\zeta\equiv 0$, transformation (2) yields the following unperturbed depths, $\hat{z}$,

$$ \hat{z}(x,y,\sigma) \equiv h(x,y) \, S(x,y,\sigma) = h(x,y) \left[\frac{h_c \, \sigma + h(x,y)\, C(\sigma)}{h_c + h(x,y)}\right] \eqno{(3)} $$

and

$$ d\hat{z} = d\sigma \; h(x,y) \; \left[\frac{h_c}{h_c + h(x,y)}\right] \eqno{(4)} $$

As a consequence, the uppermost grid box retains very little dependency from bathymetry in deep areas, where $h_c \ll h(x,y)$. For example, if $h_c = 250\,m$ and $h(x,y)$ changes from $2000$ to $6000\,m$, the uppermost grid box changes only by a factor of 1.08 (less than $10\%$). </wikitex>

Vertical Stretching Functions

<wikitex> The above generic vertical transformation design facilitates numerous vertical stretching functions, $C(\sigma)$. This function is defined in terms of several parameters which are specified in the standard input file, ocean.in.

Stretching Function Properties:

  • $C(\sigma)$ is a dimensionless, nonlinear, monotonic function;
  • $C(\sigma)$ is a continuous differentiable function, or a differentiable piecewise function with smooth transition;
  • $C(\sigma)$ must be discritized in terms of the fractional stretched vertical coordinate $\sigma$,
 $$ \sigma(k) = \cases{\frac{k-N}{N},        &at vertical $W\hbox{-points},    \;k=0,\dots,N$, \cr\cr
                       \frac{k-N-0.5}{N},    &at vertical $\rho\hbox{-points}, \;\;\;k=1,\dots,N$  \cr}$$
  • $C(\sigma)$ must be constrained by $-1 \le C(\sigma) \le 0$, that is,
$$ C(\sigma) = \cases{0,        &for $\;\;\sigma = \phantom{-}0,  \qquad\hbox{at the free-surface;}$\cr
                      -1,       &for $\;\;\sigma = -1, \qquad\hbox{at the ocean bottom.}$\cr}$$

Available Stretching Functions:

  1. Song and Haidvogel (1994) function available in ROMS since its beginning, Vstretching = 1. $C(\sigma)$ is defined as:

    $$C(\sigma) = (1 - \theta_B) {\sinh (\theta_S\,\sigma) \over \sinh \theta_S } +
                                     \theta_B \left[{ \tanh [\theta_S ( \sigma + {1\over 2})]  \over 
                                      2 \tanh ( {1\over 2} \theta_S) } - \frac{1}{2}\right] \eqno{(5)} $$ 
where ${\theta_S}$ and $\theta_B$ are the surface and bottom control parameters. Their ranges are $0 < \theta_S \leq 20$ and $0 \leq \theta_B \leq 1$, respectively. This function has the following features:
    • It is infinitely differentiable in $\sigma$.
    • The larger the value of ${\theta_S}$, the more resolution is kept above $h_c$.
    • For $\theta_B = 0$, the resolution all goes to the surface as $\theta_S$ is increased.
    • For $\theta_B = 1$, the resolution goes to both the surface and the bottom equally as $\theta_S$ is increased.
    • For $\theta_S \neq 0$, 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 $\theta_S$, say $\theta_S \leq 8$.
    • Some applications turn out to be sensitive to the value of $\theta_S$ used.
  1. A. Shchepetkin function available in UCLA-ROMS, Vstretching = 2. $C(\sigma)$ is defined as peicewise function:

    $$ C(\sigma) = \mu \, C_{sur}(\sigma) + (1 - \mu) \, C_{bot}(\sigma), $$ $$ C_{sur}(\sigma) = \frac{1 - \cosh(\theta_S \, \sigma)}{\cosh(\theta_S) - 1}, \qquad \hbox{for }\theta_S > 0, \qquad C_{bot}(\sigma) = \frac{\sinh[\theta_B \,(\sigma + 1)]}{\sinh(\theta_B)} - 1, \qquad \hbox{for }\theta_B > 0, \eqno{(6)} $$ $$ \mu = (\sigma + 1)^{\alpha} \; \left[1 + \frac{\alpha}{\beta} \left(1 - (\sigma + 1)^{\beta}\right)\right] $$ This function is similar in meaning to the Song and Haidvogel (1994), but note that hyperbolic function in $C_{sur}$ is $\cosh$ instead of $\sinh$ and
    $$ \frac {\partial C}{\partial\sigma} \rightarrow 0 \qquad \hbox{as} \;\;\sigma \rightarrow 0.$$
  2. R. Geyer function for high bottom boundary layer resolution in relatively shallow applications, Vstretching = 3. $C(\sigma)$ is defined as peicewise function:

    $$ C(\sigma) = \mu \, C_{bot}(\sigma) + (1 - \mu) \, C_{sur}(\sigma), $$ $$ C_{sur}(\sigma) = - \frac{\log[\cosh(\gamma\,\hbox{abs}(\sigma)^{\theta_S}] }{\log[\cosh(\gamma)]}, \qquad C_{bot}(\sigma) = \frac{\log[\cosh(\gamma\,(\sigma+1)^{\theta_B}]}{\log[\cosh(\gamma)]} - 1, \eqno{(7)} $$ $$ \mu = \frac{1}{2} \left[1 - \tanh\left(\gamma\left(\sigma + \frac{1}{2}\right)\right)\right] $$ where the power exponents ${\theta_S}$ and $\theta_B$ are the surface and bottom control parameters specified in standard input file ocean.in, as before. Here, $\gamma$ is a scale factor for all hyperbolic functions. Currently, $\gamma = 3$.

</wikitex>

Vertical Grid Examples

The following figure shows the $\sigma$-surfaces for several values of $\theta$ and $b$ for one of our domains. It was produced by a Matlab tool written by Hernan Arango which is available from our web site. vertical s-coordinate

Figure: The $\sigma$-surfaces for the North Atlantic with (a) $\theta = 0.0001$ and $b = 0$, (b) $\theta = 8$ and $b = 0$, (c) $\theta = 8$ and $b = 1$. (d) The actual values used in this domain were $\theta = 5$ and $b = 0.4$.