Horizontal curvilinear coordinates

In this section, the layer-integrated equations from section 4 are transformed to horizontal orthogonal curvilinear coordinates. Similarly to general coordinates in the vertical, these allow for much more flexibility when optimising horizontal grids to coast-lines and bathymetry. Furthermore, this type of coordinates system includes spherical coordinates as a special case. The derivation of the transformed equations is carried out here according to Haidvogel and Beckmann (1999), see also Arakawa and Lamb (1977).

A rectangular domain with non-dimensional side lengths and with local Cartesian coordinates ${\cal X}$ and ${\cal Y}$ is mapped to a physical domain with four corners in such a way that the local coordinates of the physical space, $(\xi_x,\xi_y)$ are orthogonal to each others everywhere:

$${\cal X} \rightarrow \xi_x,\quad {\cal Y} \rightarrow \xi_y.$$ (34)

The infinitesimal increments in the physical space, $d\,\xi_x$ and $d\,\xi_y$ are related to the infinitesimal increments in the transformed space, $d\,{\cal X}$ and $d\,{\cal Y}$ by so-called metric coefficients $m(x,y)$ and $n(x,y)$:

$$d\,\xi_x = \left(\frac{1}{m} \right) d\,{\cal X}, \quad d\,\xi_y = \left(\frac{1}{n} \right) d\,{\cal Y}.$$ (35)

These metric coefficients have the physical unit of [m$^{-1}$]. With $m=n=$const, Cartesian coordinates are retained, and with

$$m=\frac{1}{r_E\cos\phi},\quad n=\frac{1}{r_E},$$ (36)

spherical coordinates with ${\cal X}=\lambda$and ${\cal Y}=\phi$ are retained (with the Earth's radius $r_E$, longitude $\lambda$ and latitude $\phi$).

With these notations, the layer-integrated equations (25), (26), and (27) given in section 4 can be formulated as follows:

Continuity equation:

$$\partial_t \left(\frac{h_k}{mn}\right) + \partial_{\cal X} \left(... ...al_{\cal Y} \left(\frac{q_k}{m} \right) + \frac{\bar w_k - \bar w_{k-1}}{mn}=0.$$ (37)

Momentum in $\xi_x$ direction:

$$\begin{array}{l} \displaystyle \partial_t \left(\frac{p_k}{mn... ...ht)\Bigg\} = -g\frac{h_k}{n}\partial_{\cal X}\zeta. \end{array}$$ (38)

Momentum in $\xi_y$ direction:

$$\begin{array}{l} \displaystyle \partial_t \left(\frac{q_k}{mn... ...ht)\Bigg\} = -g\frac{h_k}{m}\partial_{\cal Y}\zeta. \end{array}$$ (39)

In (38) and (39), the velocity and momentum components $u_k$ and $p_k$ are now pointing into the $\xi_x$-direction and $v_k$ and $q_k$ are pointing into the $\xi_y$-direction. The stresses $\tau^{\cal X}_k$ and $\tau^{\cal Y}_k$ are related to these directions as well. In order to account for this rotation of the velocity and momentum vectors, the rotational terms due to the Coriolis rotation are extended by terms related to the gradients of the metric coefficients. This rotation is here not considered for the horizontal diffusion terms in order not to unnecessarily complicate the equations. Instead we use the simplified formulation by Kantha and Clayson (2000b), who argue that it does not make sense to use complex formulations for minor processes with highly empirical parameterisations.

Finally, the tracer equation is of the following form after the transformation to curvilinear coordinates:

$$\begin{array}{l} \displaystyle \partial_t \left(\frac{h_k c^i... ...} n\partial_{\cal Y}c^i_k\right) =\frac{Q^i_k}{mn}. \end{array}$$ (40)