A differential approximation model for passive scalar turbulence

Two-dimensional passive scalar turbulence is studied by means of a k-space diffusion model based on a third order differential approximation. This simple description of local nonlinear interactions in Fourier space is shown to provide a general expression, in line with previous seminal works, and appears to be suitable for various 2D turbulence problems. Steady state solutions for the spectral energy density of the flow are shown to recover the Kraichnan–Kolmogorov phenomenology of the dual cascade, while various passive scalar spectra, such as Batchelor or Obukhov–Corssin spectra are recovered as steady state solutions of the spectral energy density of the passive scalar. These analytical results are then corroborated by numerical solutions of the time evolving problem with energy and passive scalar injection and dissipation on a logarithmic wavenumber space grid over a large range of scales. The particular power law spectra that are obtained are found to depend mainly on the location of the kinetic and passive scalar energy injections. In particular, it is shown that by injecting the energy simultaneously at large and small scales, wave number spectra consistent with those of Nastrom–Gage can be obtained.