I would write the Navier-Stokes equations on a Riemannian manifold $(\mathcal M,g)$ in a slightly different way. The unknown is still a time-dependent vector field $v$, to which you can associate a one-form $u$, defined in the charts by
$$
\langle u(x), T\rangle_{T_x^*(\mathcal M), T_x(\mathcal M)}=g(v(x),T), \quad \text{$u=gv$ for short.}
$$
Then the equation is
$$
\partial_t
u+\mathcal L_v(u)+\nu d^* du=dq,\quad \text{div} v=0,$$
where $\mathcal L_v$ is the Lie derivative with respect to $v$. Note that, if $\Omega_0$ is an orientation of $\mathcal M$, we define the divergence of $v$ by the formula
$
\mathcal L_v(\Omega_0)=(\text{div} v)\Omega_0.
$
We may define now the vorticity $\omega$ as $du$ and get the equations
$$
\partial_t
\omega+\mathcal L_v(\omega)+\nu dd^* \omega=0,\quad \text{div} v=0, \omega=d(gv).$$