Hierarchies of balance conditions for the f-plane shallow-water equations

For the f-plane shallow-water primitive equations (PEs), hierarchies of balance conditions relating the gravity manifold (divergence delta and ageostrophic vorticity gamma = f zeta - g del (2)h) to the Rossby manifold (linearized potential vorticity q(l) = zeta - fh/H) are introduced. These hierarchies are partial derivative (N)gamma/partial derivativet(N) = partial derivative (N+1)gamma/partial derivativet(N+1) = 0 (delta balance), partial derivative (N)delta/partial derivativet(N) = partial derivative (N)gamma/partial derivativet(N) = 0 (delta-gamma balance), and partial derivative (N)gamma/partial derivativet(N) = partial derivative (N+1)gamma/partial derivativet(N+1) = 0 (gamma 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 Ro(max) (=) over dot 0.73 and Fr-max (=) over dot 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 gamma balance at N = 2. The most accurate balance conditions (e.g., gamma and delta balances at N 5 2) all exhibit slightly different energy spectra for the imbalance at medium to largest scales. Further, the greatest improvement shown by these accurate balance conditions over the less accurate conditions like quasigeostrophy occurs at large scales.