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 θ=(−Δ)α/2ψ is advected by the incompressible flow u=(−ψy,ψx). Here ψ is the stream function, Δ is the Laplace operator, and α is a positive number. The dynamics of this family are characterized by the material conservation of θ, whose variance ⟨θ2⟩ is preferentially transferred to high wave numbers (direct transfer). As this transfer proceeds to ever-smaller scales, the gradient ∇θ grows without bound. This growth is due to the stretching term (∇θ⋅∇)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 θ (i.e., on α) and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when ∇u is a quantity of no smaller scales than θ, so that it is insensitive to the direct transfer of the variance of θ, which is nearly passively advected. This case corresponds to α≥2, for which the growth of ∇θ is approximately exponential in time and nonaccelerated. For α<2, superlinear dynamics are realized as the direct transfer of ⟨θ2⟩ entails a growth in ∇u, thereby, enhancing the production of ∇θ. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at α=1 and surpasses that for α<1. The usual vorticity equation (α=2) is the border line, where ∇u and θ 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.