The basic algorithm employed for stepping forward the momentum equations is based on retaining non-divergence of the flow at all times. This is most naturally done if the components of flow are staggered in space in the form of an Arakawa C grid.

(1) Using the provided shallow-water model (or your previously implemented code), simulate the two-dimensional linearised rotating shallow-water equations on a square domain using an Arakawa C-grid. Initialize a cell-centered geopotential field with a localized positive perturbation at the domain center (zero elsewhere), define staggered zonal and meridional velocities on the C-grid, and advance the coupled system forward in time using an explicit scheme with second-order centered finite differences, including Coriolis coupling and appropriate velocity averaging. Report the gravity-wave Courant number for your chosen grid spacing and time step, and visualize the evolving geopotential and velocity fields over several time steps.

(2) Modify your implementation from part (1) so that it instead uses an Arakawa A-grid (i.e., all variables defined at the same grid points), while keeping the same numerical parameters and initial condition. Compare the results obtained with the C-grid and A-grid formulations, and discuss the differences in wave behavior and numerical properties.