| description abstract | The combined effects of advection and diffusion on the equilibrium spatial structure of a tracer whose spatial variation is maintained by a large-scale forcing are considered. Motivated by the lower stratosphere, the flow is taken to be large-scale, time-dependent, and purely horizontal but varying in the vertical, with the vertical shear much larger than horizontal velocity gradients. As a result, the ratio α between horizontal and vertical tracer scales is large. (For the lower stratospheric surf zone α has been shown to be about 250.) The diffusion parameterizes the mixing effects of small-scale processes. The three space dimensions and the large range between the forcing scale and the diffusive scale mean that direct numerical simulation would be prohibitively expensive for this problem. Instead, an ensemble approach is used that takes advantage of the separation between the large scale of the flow and the small scale of the tracer distribution. This approach, which has previously been used in theoretical investigations of two-dimensional flows, provides an efficient technique to derive statistical properties of the tracer distributions such as horizontal-wavenumber spectrum. First, the authors consider random-strain models in which the velocity gradient experienced by a fluid parcel is modeled by a random process. The results show the expected k?1 Batchelor spectrum at large scales, with a deviation from this form at a scale that is larger by a factor α than the diffusive scale found in the absence of vertical shear. This effect may be crudely captured by replacing the diffusivity ? by an &ldquo=uivalent diffusivity? α2?, but the diffusive dissipation is then substantially overestimated, and the spectrum at large k is too steep. This may be attributed to the failure of the equivalent diffusivity to capture the variability of the vertical shear. The technique is then applied to lower-stratospheric velocity fields. For realistic values of the diffusivity ?, the spectrum is found to be affected by diffusion at surprisingly large scales. For the value ? ? 10?2 m2 s?1 suggested in several recent papers, the diffusion is sufficiently strong that there is no clear k?1 regime, consistent with observations. The spectrum is then relatively well approximated by the observed k?2 power law in the range 20?200 km, but significantly steeper at smaller scales. For the molecular value ? = 10?4 m2 s?1, in contrast, an unrealistic k?1 regime appears. | |
