Analytic and Numerical Solutions of a Nonlinear Boundary‐Layer Problem

Abstract

Solutions of a viscoinertial western‐boundary‐layer problem arising in physical oceanography are discussed for both no‐slip and stress‐free boundary conditions. A fairly complete characterization of the behavior of the solutions in a number of regimes is presented using the techniques of regular perturbation expansion, matched asymptotic expansion, and rescaling of both regular and singular solutions. A comparison with the actual solutions found by extensive numerical computations is given. Parametrization of solutions in terms of the unknown first (or second) derivative at the origin, for which a useful analytic approximation is developed, provides a convenient characterization of the entire family.

Wind-driven barotropic gyre I: Circulation control by eddy vorticity fluxes to an enhanced removal region

“…The ratio of inertial terms to friction terms can be considered as a comparison of the inertial boundary layer width (Charney, 1955) to the frictional boundary layer width (Munk, 1950;Stommel, 1948). When the inertial boundary current is signi cantly wider than the frictional, Il'in and Kamenkovich (1964) and Ierley and Ruehr (1986) demonstrate that a steady western boundary layer structure is no longer possible in regions of out ow from the boundary layer. If viscosity is decreased until the frictional boundary layer becomes thinner than the inertial boundary layer width, the closed innermost streamlines of the time-mean ow do not enter the frictional boundary layer and cannot release the vorticity input within them (as pointed out by Niiler, 1966).…”

Eddies and friction : removal of vorticity from the wind-driven gyre

“…However, as pointed out by Ierley and Ruehr (1986) these steady solutions soon disappear. For a small range of viscosities beyond this point, unsteady westernintensified equilibria are found.…”

“…(1987) and Ierley and Young (1991a)) at the same Reynolds number as the viscoinertial solutions of Ierley and Ruehr (1986) break down, a recirculation gyre forms in the exit region from the boundary current. Since a recirculation is a two-dimensional phenomenon, it need not obey the one-dimensional boundary-layer approximations used by Ierley and Ruehr (1986) Also, Ierley and Sheremet (1995) found that, because of the nonlinearity of the inertial terms, more than one solution is possible for the inertial Munk problem with the same forcing.…”

“…In order to compare the solutions generated here with traditional boundary layer theories (such as Stommel (1948), Munk (1950), and Charney (1955) As shown by Il'in and Kamenkovich (1964) (and in English in Kamenkovich (1966)) and Ierley and Ruehr (1986), there is no steady-state boundary layer solution here.…”

“…Before going on to two-dimensional calculations, it is the appropriate point to mention the results of Ierley and Ruehr (1986). They solved a one-dimensional approximation to the inertial Munk model in the boundary layer numerically, and showed that the diffcultly in connecting to the interior flow in the exit region that troubled Charney's 28 model persists even when friction is present, once the Reynolds number is greater than one.…”