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)). 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 del theta grows without bound. This growth is due to the stretching term (del theta.del)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 (i.e., on alpha) and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when del 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 >= 2, for which the growth of del 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 del u, thereby, enhancing the production of del 0. 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 del 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.