9.4.2 Orthogonal coordinates

The metric tensor for orthogonal coordinates is diagonal

$$g_{ij} = \mbox{diag}(g_{11}, g_{22}, g_{33}),$$ (9.89)

where the components $g_{ij} = g_{ij}(t,\xi^{1},\xi^{2},\xi^{3})$ are generally functions of space-time. The infinitesimal arc-length measuring the distance between any two closely spaced points is therefore given by the diagonal quadratic-form

$$g_{11} (d\xi^{1})^{2} + g_{22} (d\xi^{2})^{2} + g_{33} (d\xi^{3})^{2}$$ ds² =

$$(h_{1} \, d\xi^{1})^{2} + (h_{2} \, d\xi^{2})^{2} + (h_{3} \, d\xi^{3})^{2},$$ = (9.90)

where the metric functions g_mm = (h_m)², with no sum, are often useful to introduce. Additionally, the relation between covariant and contravariant components of a tensor is given through a single multiplication. For example,

$$g_{mn} \, u^{n}$$ u_m =

$$g_{mm} \, u^{m},$$ = (9.91)

relates the covariant velocity components u_m to the contravariant components u^m. Importantly, there is no sum on the m label in the last expression.

For the purposes of large-scale ocean modeling, it is usually sufficient to assume the simpler form of the metric

$$g_{ij} = \mbox{diag}(g_{11}, g_{22}, 1),$$ (9.92)

where the nontrivial metric components are independent of time. This assumption follows from the quasi-hydrostatic approximation and will be made in the following.

In the following, the determinant of the metric tensor

$$\mathcal{G} = g_{11} \, g_{22} \, g_{33}$$ (9.93)

will appear quite frequently. With the quasi-hydrostatic approximation for which g₃₃ = 1, the determinant is given by

$$\mathcal{G} = g_{11} \, g_{22} = (h_{1} \, h_{2})^{2}.$$ (9.94)