The existence of an inverse cascade is demonstrated for three-dimensional incompressible flow displaying the Anisotropic Kinetic Alpha (AKA) instability (Frisch, She & Sulem). By means of full three-dimensional simulations of the Navier–Stokes equations, it is shown that flow stirred at small scales by an anisotropic force lacking parity-invariance (i.e. lacking any centre of symmetry) can generate strongly helical structures on larger scales. When there is a substantial range of linearly unstable modes, the most unstable ones emerge at first, but are eventually dominated by modes with the smallest wavenumbers.
The key observation for the theory of this inverse cascade is that, in the presence of forcing, the small-scale Reynolds stresses will become dependent on the large-scale flow. Elimination of the small scales produces the nonlinear AKA equations for the large-scale flow. The latter have non-trivial one-dimensional solutions also displaying an inverse cascade, qualitatively similar to the one reported above. This cascade has been numerically simulated over a range of more than two decades. For a simple choice of the forcing, a steady state is eventually reached; it can be described analytically and presents interesting geometric features in the limit of very extended systems. The corresponding energy spectrum has a k−4 range. A number of other scaling relations are also derived.
The multi-dimensional extension of the theory is briefly considered. The resulting large-scale structures are conjectured to correspond to solutions of the incompressible Euler equation.