Nondispersive evolution of rings. From left to right, , 1, and 1.5, respectively. Quite clearly, for smaller the deformation is more local in character.
The first panel is the spatially uncorrelated initial condition (smoothened via a diffusive stencil). The second and third panels show the emergent scalar field for and 2, respectively. Quite clearly, for we have a field composed of coherent eddies while for we obtain a filamentary geometry reminiscent of a passive field when subjected to large-scale advection.
The dispersion relation (2) with for, from left to right, , 0.5, and 1, respectively. Note that for the frequencies are bounded while on either side we obtain in particular limits.
Inversion of a single sawtooth profile for varying . As is evident, in addition to the east-west asymmetry, smaller (more local) gives stronger and narrower jets.
and fields for , 1, and 1.25 in the upper and lower panels, respectively. In all cases . Note the finer scale of the flow as compared to the scalar field when . Further for increasing , we obtain coherent zonal flows.
The first panel shows energy spectra in the weakly local to local inverse energy transfer regime with . We notice well-developed power laws, and the slopes are in reasonable agreement with nondispersive estimates. Note that these are slopes from individual realizations, not temporal or ensemble averages. The second panel shows the inverse transfer for , with , 0.1, 0.5, and 1. The sensitivity of the scaling with respect to is evident; also note that the range up to which the scaling extends decreases with .
Enstrophy spectra in the forward weakly local to local enstrophy transfer regime for . We notice well-developed power laws, but the slopes are significantly steeper than nondispersive estimates. The second panel shows the same experiment for but with varying (, 0.1, 0.5, and 1 from top to bottom). Quite clearly, the slopes are fairly insensitive to . Once again, we note that our estimates of the slopes are based on individual realizations as opposed to a temporal or ensemble average that might be more suitable at higher resolutions for accurate quantitative slopes and error estimates.
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