Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence

We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $\theta =(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}\psi$ is advected by the incompressible flow $\breve{=}(-{\psi }_y,{\psi }_x)$. The dynamics of this family are characterized by the material conservation of $\theta$, whose variance $<{\theta }^2>$ is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $\mathrm{\nabla }\theta$ grows without bound. This growth is due to the stretching term (\nabla\theta\cdot\nabla)\u whose ``effective degree of nonlinearity’’ differs from one member of the family to another. This degree depends on the relation between the advecting flow \u and the active scalar $\theta$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when \nabla\u is a quantity of no smaller scales than $\theta$, so that it is insensitive to the direct transfer of the variance of $\theta$, which is nearly passively advected. This case corresponds to $\alpha \ge 2$, for which the growth of $\mathrm{\nabla }\theta$ is approximately exponential in time and non-accelerated. For $\alpha <2$, superlinear dynamics are realized as the direct transfer of $<{\theta }^2>$ entails a growth in \nabla\u, thereby enhancing the production of $\mathrm{\nabla }\theta$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $\alpha =1$ and surpasses that for $\alpha <1$. The usual vorticity equation ($\alpha =2$) is the border line, where \nabla\u and $\theta$ are of the same scale, separating the linear and nonlinear regimes of the small-scale dynamics. We discuss these regimes in detail, with an emphasis on the locality of the direct transfer.