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 $\theta ={\left(-\Delta \right)}^{\alpha /2}\psi$ is advected by the incompressible flow $\mathit{u}=\left(-{\psi }_y,{\psi }_x\right)$. Here $\psi$ is the stream function, $\Delta$ is the Laplace operator, and $\alpha$ is a positive number. The dynamics of this family are characterized by the material conservation of $\theta$, whose variance $⟨{\theta }^2⟩$ is preferentially transferred to high wave numbers (direct transfer). As this transfer proceeds to ever-smaller scales, the gradient $\nabla \theta$ grows without bound. This growth is due to the stretching term $\left(\nabla \theta \cdot \nabla \right)\mathit{u}$ whose “effective degree of nonlinearity” differs from one member of the family to another. This degree depends on the relation between the advecting flow $\mathit{u}$ and the active scalar $\theta$ (i.e., on $\alpha$) and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $\nabla \mathit{u}$ 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 $\nabla \theta$ is approximately exponential in time and nonaccelerated. For $\alpha <2$, superlinear dynamics are realized as the direct transfer of $⟨{\theta }^2⟩$ entails a growth in $\nabla \mathit{u}$, 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 $\left(\alpha =2\right)$ is the border line, where $\nabla \mathit{u}$ 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.