The Navier-Stokes equations in $\mathbb{R}^n$ read $$u_t+u\cdot \nabla u -\mu \Delta u +\nabla p =f$$ $$\nabla \cdot u =0.$$ I had a look at a paper by Maryam Samavaki and Jukka Tuomela titled "Navier-Stokes equations on Riemannian manifolds" and they propose that on a Riemannian manifold $(M,g)$ the Navier-Stokes equations become $$u_t+\nabla_u u-\mu Lu +\operatorname{grad}(p)=0$$ $$\operatorname{div} (u)=0$$ where $Lu=\Delta_B u+\operatorname{grad}(\operatorname{div}(u)) +\operatorname{Ric}(u)$ and $\Delta_B u=\operatorname{div}(g^{ij} u^k_{;i}).$
But I was wondering how can we write the above equations on a rotating sphere, I know that in theoretical meteorology they use the Navier-Stokes equations on rotating sphere extrinsically but I would like to know what the intrinsic approach would be.
So how can we do this?