Recent calculations from high resolution aircraft data have shown lower stratospheric tracer power spectra that appear to decay approximately as k^-2, where k is horizontal wavenumber (Strahan and Mahlman JGR 1994, Bacmeister et al. JGR 1996). Simple theories of tracer spectra in flows dominated by large-scale advection, which is widely believed to be the regime relevant to the stratosphere, predict, in contrast, decay as k^-1 (the `Batchelor spectrum'). Ngan and Shepherd (JGR 1997) have suggested that the observed spectrum may be explained if the tracer field is dominated by sharp jumps associated with the edge of the polar vortex. If such jumps are well-separated in space then the `Saffman spectrum' decaying as k^-2 is expected. However the distribution of tracer in the lower stratosphere seems to be most naturally considered as the solution of a forced-dissipative problem in which there is a forcing of tracer variation on the large scale and mixing at small scales. It is not at all clear why the situation described by Ngan and Shepherd should arise in this case.
We consider a forced-dissipative problem for the tracer, but extend the problem originally considered by Batchelor and others to include new ingredients that seem particularly relevant to the atmosphere. The flow is considered to be horizontal, but varying in the vertical. This implies a reduction in vertical as well as horizontal scales. For realistic parameter regimes the vertical scales are much smaller than horizontal and diffusive mixing is preferentially in the vertical rather than the horizontal. Both stochastic models of the time variation of the flow (with a range of correlation times) and flows taken from atmospheric observations are considered. The mathematical model used to predict the tracer spectrum exploits a WKB-like approach rather than solving the full 3-D PDEs for the tracer field following Antonsen et al (Phys. Fluids 1996).
The Batchelor spectrum is expected only if there is sufficient separation between the forcing scale and the dissipation scale. Our results show that including the effect of vertical shear and also realistic time dependence of the flow significantly increases the required separation over that deduced in simpler models. With realistic values of diffusivity it appears that there may simply not be a large enough separation for the Batchelor spectrum to be apparent. Instead the spectrum decays more steeply with wavenumber and our results show that, with realistic parameters, a calculated spectral slope would be close to that of -2 suggested by the observations.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics