9.6.2 Conservation of angular momentum about the north pole

In general, for unforced motion on a manifold which contains a translational symmetry, momentum in the direction of this symmetry is conserved. Likewise, if the manifold contains an axis of symmetry, the angular momentum about this axis is conserved. In either case, the form of the equation describing the evolution of the density of the conserved quantity will take the form of a conservation law. To be more specific, for unforced motion on a rotating sphere, it is angular momentum about the axis of rotation which is conserved. In contrast, zonal and meridional momentum are not conserved on the sphere, even in the absence of external forces, since the manifold is not flat. As such, the time tendency of the zonal and meridional momentum will not appear in a conservative form. This is the physical/geometric explanation for why the momentum equations on the sphere contain both ``advective metric terms'' and ``frictional metric terms.'' The following discussion provides an elaboration of this point. Consider the case of spherical coordinates for describing motion on a rotating sphere. In this subsection, it is sufficient to employ the more familiar vector notation introduced in Section 4.6. In this special case, the friction components (9.182) and (9.183) can be written in the compact form

$$(\cos\phi)^{-1} \, \nabla_{h} \cdot {\bf P}$$ F^u = (9.148)

$$(\cos\phi)^{-1} \, \nabla_{h} \cdot {\bf Q},$$ F^v = (9.149)

where ${\bf P}$ and ${\bf Q}$ are horizontal vectors given by

$${\bf P} A \, \cos\phi \, ( \hat{\lambda} \, D_{T} + \hat{\phi} \, D_{S})$$ = (9.150)

$${\bf Q} A \, \cos\phi \, ( \hat{\lambda} \, D_{S} - \hat{\phi} \, D_{T}).$$ = (9.151)

As a result, the zonal momentum equation takes the form

 

$$\rho \, dV \, [u_{,t} + {\bf u} \cdot \nabla u - (u \, v/a) \, \tan\phi - fv] = dV ( -p_{, x} + (\cos\phi)^{-1} \, \nabla_{h} \cdot {\bf P} ),$$ (9.152)

where again $\rho \, dV$ is the conserved mass of a parcel. Notably, this equation cannot be written in the form of a conservation equation, as expected since the zonal momentum $\rho \, dV \, u$ is not conserved on a sphere. However, the angular momentum about the north pole can be written such, as now shown. For the purpose, multiply equation (9.152) by $\cos\phi$ and use the conservation of mass to find

$$(\partial_{t} + {\bf u} \cdot \nabla ) \, [ \rho \, dV \, a \, \c... ...a \, a \, \cos\phi)] = a \, dV \, (- p_{, \lambda} + \nabla_{h} \cdot {\bf P} )$$ (9.153)

where $\Omega$ is the Earth's rotation rate. This equation states that the angular momentum per unit volume

$$\rho \mathcal{M} = \rho \, a \, \cos\phi \, ( u + \Omega \, a \, \cos\phi)$$ (9.154)

satisfies the conservation equation

$$\partial_{t} (\rho \, \mathcal{M}) + \nabla \cdot (\rho \, \mathcal{M} \, {\bf u}) = - p_{, \lambda} + \nabla_{h} \cdot {\bf P}.$$ (9.155)

That is, for motion on a smooth sphere, $\int dV \, \rho \mathcal{M}$ is a constant in time.