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We study the small-scale behavior of generalized two-dimensional turbulencegoverned by a family of model equations, in which the active scalar$\theta=(-\Delta)^{\alpha/2}\psi$ is advected by the incompressible flow$\u=(-\psi_y,\psi_x)$. The dynamics of this family are characterized by thematerial conservation of $\theta$, whose variance $<\theta^2>$ ispreferentially transferred to high wave numbers. As this transfer proceeds toever-smaller scales, the gradient $\nabla\theta$ grows without bound. Thisgrowth is due to the stretching term $(\nabla\theta\cdot\nabla)\u$ whose``effective degree of nonlinearity'' differs from one member of the family toanother. This degree depends on the relation between the advecting flow $\u$and the active scalar $\theta$ and is wide ranging, from approximately linearto highly superlinear. Linear dynamics are realized when $\nabla\u$ is aquantity of no smaller scales than $\theta$, so that it is insensitive to thedirect transfer of the variance of $\theta$, which is nearly passivelyadvected. This case corresponds to $\alpha\ge2$, for which the growth of$\nabla\theta$ 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 productionof $\nabla\theta$. This superlinearity reaches the familiar quadraticnonlinearity of three-dimensional turbulence at $\alpha=1$ and surpasses thatfor $\alpha<1$. The usual vorticity equation ($\alpha=2$) is the border line,where $\nabla\u$ and $\theta$ are of the same scale, separating the linear andnonlinear regimes of the small-scale dynamics. We discuss these regimes indetail, with an emphasis on the locality of the direct transfer.

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