Physics of Fluids, 12, 1377-1381.
Adam H. Sobel
Department of Applied Physics and Applied Mathematics and Department of Earth and Environmental Sciences,
Columbia University, New York, NY.
Glenn R. Flierl
Program in Atmospheres, Oceans, and Climate, Massachusetts
Institute of Technology, Cambridge, MA.
Abstract
The cross-channel tracer flux due to the combined effects of advection and diffusion is considered for two-dimensional incompressible flow in a channel, where the flow is that due to a single traveling wave and the boundary conditions at the walls are fixed tracer mixing ratio. The tracer flux is computed numerically over a wide range of the parameters $\epsilon=\frac{U}{c}$ and $\delta=\frac{K}{cL}$, with $U$ the maximum fluid velocity, $c$ the wave phase speed, $K$ the tracer diffusivity, and $L$ the channel width. Prior work has used analytical methods to obtain solutions for $\delta$ either infinite (stationary overturning cells) or small. In addition to the full numerical solutions, solutions obtained using mean field theory are presented, as well as a new asymptotic solution for small $\epsilon$, and one for small $\delta$ due to Flierl and Dewar. The various approximations are compared with each other and with the numerical solutions, and the domain of validity of each is shown. Mean field theory is fairly accurate compared to the full numerical solutions for small $\delta$, but tends to underpredict the tracer flux by 30-50\% for larger $\delta$. The asymptotic solution derived by Flierl and Dewar for small $\delta$ is found to break down when $\delta \sim \epsilon ^{-2}$ rather than when $\delta \sim 1$ as suggested by the original derivation, and a scaling argument is presented which explains this.