Hierarchies of Balance Conditions for thef-Plane Shallow-Water Equations

Abstract

Abstract For the f-plane shallow-water primitive equations (PEs), hierarchies of balance conditions relating the gravity manifold (divergence δ and ageostrophic vorticity γ = fζ − g∇2h) to the Rossby manifold (linearized potential vorticity ql = ζ − fh/H) are introduced. These hierarchies are ∂Nδ/∂tN = ∂N+1δ/∂tN+1 = 0 (δ balance), ∂Nδ/∂tN = ∂Nγ/∂tN = 0 (δ–γ balance), and ∂Nγ/∂tN = ∂N+1γ/∂tN+1 = 0 (γ balance), for N = 0, 1, … . How well these balance conditions represent the balance accessible to a given PE flow is explored. Detailed numerical experiments are carried out on an idealized potential vorticity distribution for which the domain maximum Rossby and Froude numbers are Romax ≐ 0.73 and Frmax ≐ 0.28. The numerical results reveal that all these hierarchies are asymptotic: as N increases, imbalance first decreases and then increases, as measured for instance by a linearized available energy. The minimum imbalance, over all the balance conditions considered, is attained by γ balance at N = 2. The most ...