The mathematical problem governing the circulation of a barotropic ocean with bottom friction is re-examined by a procedure which allows us to systematically span the entire region of the parameter space R, ε << 1, where R and ε are the Rossby and 'friction' numbers. The procedure consists in first identifying three dynamically distinct sectors in parameter space. Next, single variable asymptotic expansions are carried out along the boundaries of these sectors. In addition to affording us the opportunity to study the regimes in both of the adjoining sectors, this procedure avoids the possibility of the series becoming disordered. The sector R < ε2 corresponds to the dynamical regime first studied by Stommel: the interior circulation satisfies the Sverdrup balance and is closed by a viscous western boundary layer. The regime in the sector ε2 < R < ε is characterized by a large, free Fofonoff mode, which exhibits boundary layer features. In particular, the western boundary layer has the inertial balance first proposed by Charney. The actual structure of this mode is found and the role played by an integral constraint, which differs from the familiar Prandtl-Batchelor one, is discussed. Finally, in R > ε, the Fofonoff mode, which is still present, reaches its maximal strength; however, it no longer exhibits any boundary layer structure. The details of this mode are given by the solution of a highly nonlinear Poisson equation. This novel mathematical problem generalizes one previously solved by Zimmerman (1993).
Barcilon, Victor. 1998. "On the barotropic ocean with bottom friction." Journal of Marine Research 56, (4). https://elischolar.library.yale.edu/journal_of_marine_research/2289