The Vertical-Scale Cascade in Atmospheric Tracers due to Large-Scale Differential Advection

Abstract

Reduction in the vertical scale of atmospheric tracer fields occurs as a result of quasi-horizontal stirring by the large-scale flow, provided that the flow varies in the vertical. This scale reduction and the implications for dissipation and mixing are here analyzed using simple mathematical models. The first is based on a steady linear flow and gives simple insight into the interaction between horizontal strain, vertical shear, and diffusion. The second is a simple random-straining model, with random vertical shear being added to a horizontal random strain field. Analytical progress is possible in the limit where the correlation time for the flow is small or large compared to the inverse strain itself. Numerical integration allows investigation of the intermediate case. In all cases the vertical scale decreases exponentially fast, at the same rate as the horizontal scale, with the rate, say S, being controlled by the statistics of the horizontal strain field. As this exponential decrease occurs, the horizontal and vertical scales remain in a characteristic ratio, say α, which depends on the statistics of the vertical shear and on S. The third model is based on integration of the differential equation for the wavenumber vector associated with a tracer distribution along a parcel trajectory using velocity fields calculated from large-scale observations of the lower stratosphere. These calculations give S ? 0.3 days?1 and α ? 250. The large value of α implies that vertical contrasts in dynamical and tracer quantities are far more important than horizontal contrasts for dissipative processes such as radiative transfer or diffusion. The results suggest that the ?mix-down? time for a tracer field initially varying on horizontal scales of 1000 km to be distorted to have vertical scales on which molecular diffusion is important, that is, 5 meters or so, is about 20 days in the winter lower stratosphere.