Diffusion in an Ekman Layer

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  • 1 University of Waterloo, Ontario, Canada
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Abstract

When a relatively large diffusing cloud is transported by the flow in an Ekman layer, the trajectory and the rate of spread of the cloud are affected by the velocity variations both along and across the geostrophic wind. In a first approximation, these effects may be calculated with the aid of the classical diffusion equation using a constant eddy diffusivity and the classical Ekman layer velocity profile, also derived on the basis of a constant eddy viscosity.

Analytical solutions (obtained by the “concentration-moment” method) show that the ground level trajectory of an instantaneously released cloud departs significantly from the surface wind track at distances from the source beyond a few kilometers, and that while this trajectory remains in the sector between surface and geostrophic winds, it does not follow asymptotically either wind direction. The spread of the cloud, as observed at ground level, is dominated, beyond the first few kilometers of travel by the “wind-shear effect” on diffusion, the mean-square dispersion increasing rather more rapidly than it would due to atmospheric turbulence alone. Asymptotically (and this means at very large distances, in practice on a global scale), the mean-square dispersion at ground level becomes proportional to an “effective” shear diffusivity several orders of magnitude greater than the eddy diffusivity.

Abstract

When a relatively large diffusing cloud is transported by the flow in an Ekman layer, the trajectory and the rate of spread of the cloud are affected by the velocity variations both along and across the geostrophic wind. In a first approximation, these effects may be calculated with the aid of the classical diffusion equation using a constant eddy diffusivity and the classical Ekman layer velocity profile, also derived on the basis of a constant eddy viscosity.

Analytical solutions (obtained by the “concentration-moment” method) show that the ground level trajectory of an instantaneously released cloud departs significantly from the surface wind track at distances from the source beyond a few kilometers, and that while this trajectory remains in the sector between surface and geostrophic winds, it does not follow asymptotically either wind direction. The spread of the cloud, as observed at ground level, is dominated, beyond the first few kilometers of travel by the “wind-shear effect” on diffusion, the mean-square dispersion increasing rather more rapidly than it would due to atmospheric turbulence alone. Asymptotically (and this means at very large distances, in practice on a global scale), the mean-square dispersion at ground level becomes proportional to an “effective” shear diffusivity several orders of magnitude greater than the eddy diffusivity.

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