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 $\theta=(-\Delta)^{\alpha/2}\psi$ is advected by the incompressible flow $\u=(-\psi_y,\psi_x)$. 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 $\nabla\theta$ 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 $\alpha\ge2$, 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 production of $\nabla\theta$. 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.