General vertical coordinates

As a preparation of the discretisation, the physical space is vertically divided into $N$ layers. This is done by introducing internal surfaces $z_k$, $k=1,\dots,N-1$ which do not intersect, each depending on the horizontal position $(x,y)$ and time $t$. Let

$$-H(x,y)=z_0(x,y)<z_1(x,y,t)<\dots<z_{N-1}(x,y,t)<z_N(x,y,t)=\zeta(x,y,t)$$ (12)

define the local layer depths $h_k$ with

$$h_k=z_k-z_{k-1}.$$ (13)

for $1\leq k\leq N$. For simplicity, the argument $(x,y,t)$ is omitted in most of the cases.

The most simple layer distribution is given by the so-called $\sigma$ transformation (see Phillips (1957) for a first application in meteorology and Freeman et al. (1972) for a first application in hydrodynamics) with

$$\sigma_k=\frac{k}{N}-1$$ (14)

and

$$z_k=D\sigma_k$$ (15)

for $0\leq k\leq N$.

The $\sigma$-coordinates can also be refined towards the surface and the bed:

$$\beta_k = \frac{\mbox{tanh}\left( (d_l+d_u)(1+\sigma_k)-d_l\right... ...mbox{tanh}(d_l)}{\mbox{tanh}(d_l)+\mbox{tanh}(d_u)}-1, \qquad k=0,\dots,N\qquad$$ (16)

such that $z$-levels are obtained as follows:

$$z_k=D\beta_k$$ (17)

for $0\leq k\leq N$.

The grid is refined towards the surface for $d_u>0$ and refined towards the bottom for $d_l>0$. When both, $d_u$ and $d_l$ are larger than zero, then refinement towards surface and bed is obtained. For $d_u=d_l=0$ the $\sigma$-transformation (14) with $\beta_k=\sigma_k$ is retained. Figure 1 shows four examples for vertical layer distributions obtained with the $\sigma$-transformation.

Due to the fact that all layer thicknesses are proportional to the water depth, the equidistant and also the non-equidistant $\sigma$-transformations, (14) and (16), have however one striking disadvantage. In order to sufficiently resolve the mixed layer also in deep water, many layers have to be located near the surface. The same holds for the bottom boundary layer. This problem of $\sigma$-coordinates has been discussed by several authors (see e.g. Deleersnijder and Ruddick (1992), de Kok (1992), Gerdes (1993), Song and Haidvogel (1994), Burchard and Petersen (1997)) who suggested methods for generalised vertical coordinates not resulting in layer thicknesses not proportional to the water depth.

The generalised vertical coordinate introduced here is a generalisation of the so-called mixed-layer transformation suggested by Burchard and Petersen (1997). It is a hybrid coordinate which interpolates between the equidistant and the non-equidistant $\sigma$-transformations given by (14) and (16). The weight for the interpolation depends on the ratio of a critical water depth $D_{\gamma }$ (below which equidistant $\sigma$-coordinates are used) and the actual water depth:

$$z_k = D\left(\alpha_{\gamma} \sigma_k + (1-\alpha_{\gamma}) \beta_k\right)$$ (18)

with

$$\alpha_{\gamma} = \min\left(\frac{ (\beta_k-\beta_{k-1})-\frac{D_... ...igma_k-\sigma_{k-1})} {(\beta_k-\beta_{k-1})-(\sigma_k-\sigma_{k-1})},1\right).$$ (19)

and $\sigma_k$ from (14) and $\beta_k$ from (16).

For inserting $k=N$ in (19) and $d_l=0$ and $d_u>0$ in (16), the mixed layer transformation of Burchard and Petersen (1997) is retained, see the upper two panels in figure 2. Depending on the values for $D_{\gamma }$ and $d_u$, some near-surface layer thicknesses will be constant in time and space, allowing for a good vertical resolution in the surface mixed layer.

The same is obtained for the bottom with the following settings: $k=1$, $d_l>0$ and $d_u=0$, see the lower two panels in figure 2. This is recommended for reproducing sedimentation dynamics and other benthic processes. For $d_l=d_u>0$ and $k=1$ or $k=N$ a number of layers near the surface and near the bottom can be fixed to constant thickness. Intermediate states are obtained by intermediate settings, see figure 3. Some pathological settings are also possible, such as $k=1$, $d_l=1.5$ and $d_u=5$, see figure 4.

Figure 1: $\sigma$-transformation with four different zooming options. The plots show the vertical layer distribution for a cross section through the North Sea from Scarborough in England to Esbjerg in Denmark. The shallow area at about $x=100$ nm is the Doggerbank.

Figure 2: Boundary layer transformation (or $\gamma$ transformation) with concentration of layers in the surface mixed layer (upper two panels) and with concentration of layers in the bottom mixed layer (lower two panels). The critical depth $D_{\gamma }$ is here set to 20 m, such that at all shallower depths the equidistant $\sigma$-transformation is used. The same underlying bathymetry as in figure 1 has been used.

Figure 3: Boundary layer transformation (or $\gamma$ transformation) with concentration of layers in both, the surface mixed layer and the bottom mixed layer. Four different realisations are shown. The critical depth $D_{\gamma }$ is here set to 20 m, such that at all shallower depths the equidistant $\sigma$-transformation is used. The same underlying bathymetry as in figure 1 has been used.

Figure 4: Two pathological examples for the boundary layer transformation. The critical depth $D_{\gamma }$ is here set to 20 m, such that at all shallower depths the equidistant $\sigma$-transformation is used. The same underlying bathymetry as in figure 1 has been used.

The strong potential of the general vertical coordinates concept is the extendibility towards vertically adaptive grids. Since the layers may be redistributed after every baroclinic time step, one could adapt the coordinate distribution to the internal dynamics of the flow. One could for example concentrate more layers at vertical locations of high stratification and shear, or force certain layer interfaces towards certain isopycnals, or approximate Lagrangian vertical coordinates by minimising the vertical advection through layer interfaces. The advantages of this concept have recently been demonstrated for one-dimensional water columns by Burchard and Beckers (2004). The three-dimensional generalisation of this concept of adaptive grids for GETM is currently under development.