9.6.3 The advective and frictional metric terms

It is useful to provide a more mathematical statement concerning the origin of the metric terms. For this purpose, consider the momemtum equations for fluid motion on an arbitrary manifold

 

$$\rho \frac{D u^{m}}{D t} T^{mn}_{; n} + \rho \, f^{m}.$$ = (9.156)

The acceleration of a fluid parcel takes the form

$$\rho \, \frac{D u^{m}}{D t} \rho (u^{m}_{,t} + u^{n} \, u^{m}_{; n})$$ =

$$(\rho \, u^{m})_{,t} + (\rho \, u^{m} \, u^{n})_{; n}.$$ = (9.157)

To reach this result, the conservation of mass on a curved manifold has been used

$$\rho_{,t} + (\rho \, u^{n})_{; n} = 0.$$ (9.158)

As such, the time tendency for the momentum density is given by

$$(\rho \, u^{m})_{,t} (T^{mn} - \rho \, u^{m} \, u^{n})_{; n} + \rho \, f^{m}.$$ = (9.159)

This equation is written in the same form as the Cartesian equivalent (9.143), except that now the derivatives are covariant and so contain information about the generally curved manifold. Expanding the covariant divergence using equation (9.110) yields

$$(\rho \, u^{m})_{,t} (\sqrt{\mathcal{G}})^{-1} \, [\sqrt{\mathcal{G}} \, (T^{mn} - \rh... ...)]_{,n} + (T^{ab} - \rho \, u^{a} \, u^{b}) \, \Gamma^{m}_{ab} + \rho \, f^{m}.$$ = (9.160)

The Christoffel symbol $\Gamma^{m}_{np}$ accounts for the spatial dependence of the basis vectors. The $\Gamma^{m}_{ab} \, \tau^{ab}$ term is the ``frictional metric term'' and the $\rho \, u^{a} \, u^{b} \, \Gamma^{m}_{ab}$ term is the ``advective metric term.''

Integration of a quantity over the volume of a finite grid cell in arbitrary coordinates means performing an integral of the form

$$\int dV \psi \int \sqrt{\mathcal{G}} \, d\xi^{1} \, d\xi^{2} \, d\xi^{3} \, \psi,$$ = (9.161)

where

$$dV = \sqrt{\mathcal{G}} \, d\xi^{1} \, d\xi^{2} \, d\xi^{3}$$ (9.162)

is the invariant volume element. For example, in spherical coordinates

$$dV = a^{2} \cos\phi \, d\lambda \, d\phi \, dz.$$ (9.163)

Hence, the metric terms cannot be written as the difference of fluxes across grid cells faces, whereas the remaining terms can. For flat space using Cartesian coordinates, the metric terms drop out, thus recovering the results discussed in Section 9.6.1.