From what I've read so far, there are the scalar variables and vector variables that make up finite difference equations that would all normally be placed on a coordinate point (i,j) on a collocated grid. On a staggered grid however, the scalar variables are placed on a standard grid point at a position (i,j) and the vector variables are placed such that the horizontal velocity component at (i,j) is displaced to $(i + .5,j)$ and the vertical velocity component at (i,j) is displaced to $(i, j + .5)$. This grid is illustrated below for Navier-Stokes equation:
I want to use this grid to solve the following Poisson equation (I'll worry about the boundary conditions later):
$$\nabla^2 \psi = \zeta$$
Where $\zeta = \textbf{k} \cdot \nabla \times \textbf{V} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$ i.e. a scalar field, and we know that $\nabla^2 = \nabla \cdot \nabla$ i.e. the divergence of the gradient and therefore a scalar field. So, our scalar fields would be $\psi$ and I'm guessing $\zeta$. Or would we not use $\zeta$ at all in the grid and instead use the vectors $u \text{ and } v$ which make it up as our vector fields? I think it's the latter, but I don't know enough about these grids to be certain of that.
Is it also possible to have no vector fields and only scalar fields on a staggered grid?