On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence

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A powerful feature-tracking tool is applied to several high-resolution, very long-duration, regularized contour dynamics (contour surgery) simulations of unforced nearly-inviscid two-dimensional turbulence (2DT) on the surface of a sphere. Particularly low density gases of vortices (i.e. on average, very widely separated vortices) are examined to ascertain the nature of their interactions. The simplest (minimal) model system is studied, namely a set of vortex patches of just two vorticity values, ±ω0, whose total circulation is zero. The areas of the patches are selected initially from a pre-assigned, stable (nearly time invariant) power-law distribution. When the vorticity occupation fraction f≳0.01, often more than three vortices are found relatively close together at the onset of a strong interaction. But, when f≲0.01, all such interactions involve only three nearby vortex patches, not all having the same sign of vorticity. This is related to the well-known collapse of three singular (point) vortices. Thus, under these conditions, isolated two-vortex interactions, which have figured in recent ad hoc theories and models for decaying 2DT, cannot occur. Taking into account these results, we propose an asymptotically-motivated and computationally-efficient ‘‘reduced’’ model.