Local and Nonlocal Dispersive Turbulence

TL;DR

The paper studies a family of 2D dispersive active-scalar models with locality parameter $α$ and dispersion strength $ε$, focusing on how advection and dispersion shape inverse energy and forward enstrophy cascades. Using forced-dissipative pseudospectral simulations, it finds that small $α$ yields coherent eddies while large $α$ produces filamentary structures, and that for $ε≈1$ inverse-energy slopes align with non-dispersive estimates whereas forward enstrophy slopes are steeper than dispersionless predictions. The inverse cascade becomes more sensitive to dispersion as $ε$ decreases, shortening the inertial range and lowering the slope, while the forward cascade remains comparatively universal across $ε$ values. The analysis links resonant and near-resonant interactions to zonal-flow formation for $α>0.5$ and discusses the special mathematical status of $α=0.5$, outlining avenues for further exploration of wave–turbulence interplay in geophysical contexts.

Abstract

We consider the evolution of a family of 2D dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, the locality of the streamfunction-scalar relation is denoted by $α$, with smaller $α$ implying increased locality. The dispersive nature arises via a linear term whose strength is characterized by a parameter $ε$. Setting $0 < ε\le 1$, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward (inverse) transfer of enstrophy (energy) under large-scale (small-scale) random forcing. Straightforward arguments suggest that for small $α$ the scalar field should consist of progressively larger eddies, while for large $α$ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly non-local to local conditions. For $ε\sim 1$, there is quantitative agreement between non-dispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding non-dispersive estimate. Additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term : specifically, as $ε$ decreases, the inertial-range shortens and we also observe that the slope of the power-law decreases. On the other hand, for the same range of $ε$ values, the forward regime scaling is fairly universal.

Figures (7)

• Figure 1: Non-dispersive evolution of $\theta$ rings. From left to right, $\alpha=0.5,1$ and 1.5 respectively. Quite clearly, for smaller $\alpha$ the deformation is more local in character.
• Figure 2: The first panel is the spatially un-correlated initial condition (smoothened via a diffusive stencil). The second and third panels show the emergent scalar field for $\alpha=0.5$ and $2$ respectively. Quite clearly, for $\alpha=0.5$ we have a field composed of coherent $\theta$ eddies while for $\alpha=2$ we obtain a filamentary geometry reminiscent of a passive field when subjected to large-scale advection.
• Figure 3: The dispersion relation (\ref{['2']}) with $\epsilon =1$ for, from left to right, $\alpha=0.25,0.5$ and 1 respectively. Note that for $\alpha=0.5$ the frequencies are bounded while on either side we obtain $|\omega| \rightarrow \infty$ in particular limits.
• Figure 4: Inversion of a single "saw-tooth" $\theta$ profile for varying $\alpha$. As is evident, in addition to the east-west asymmetry, smaller $\alpha$ (more local) gives stronger and narrower jets.
• Figure 5: $\theta, u$ fields for $\alpha = 0.25,1$ and 1.25 in the upper and lower panels respectively. In all cases $\epsilon=0.1$. Note the finer scale of the flow as compared to the scalar field when $\alpha=0.25$. Further for increasing $\alpha$, we obtain coherent zonal flows.