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 $ ablatheta$ grows without bound. This growth is due to the stretching term $( ablathetacdot abla)u$ 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 $ ablau$ 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 $alphage2$, for which the growth of $ ablatheta$ 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 $ ablau$, thereby enhancing the production of $ ablatheta$. 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 $ ablau$ 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.