A New Vortex-in-cell Method for Solving the Shallow Water Equations on the Sphere

We present a new vortex-in-cell method for computational modeling of the quasi-geostrophic shallow water equations on the sphere. We assume that, ψ, the stream function and, h_s, the deviation of the water free surface from its resting position are related by the local linear balance, g h_s = f ψ, following the arguments by Kuo (1959) and Charney and Stern (1962), where g is gravity and f is the Coriolis parameter. For small h_s compared to the water depth d(φ,θ), the Global quasi-geostrophic shallow water equation is expressed as ∇²ψ - ψ/L_d² ≈ Q d - f (eq 1), where the Rossby radius is L_d = (gd(φ,θ))^1/2/f(φ,θ), and Q(φ,θ,t) is the potential vorticity, which is conserved along particles trajectories. Spatio-temporal variations in the water depth h(φ,θ,t), contribute to an irrotational component of the velocity, with the potential, Φ, governed by Poisson's equation ∇²Φ=-h^-1 Dh/Dt, where D is the Lagrangian derivative and h(φ,θ,t)=h_s(φ,θ,t)+ d(φ,θ). In the proposed hybrid method, Lagrangian particles carrying the potential vorticity are advected using a second order Runge Kutta time integration scheme. The velocity at the particles locations is interpolated from the velocity field computed on the grid using Grid-based Poisson's solvers for ψ and Φ. Assessment of the proposed method is carried out by exploring its accuracy as a function of the grid size and time step and by comparing with existing Vortex-in-cell methods.