The large scale structure of decaying stratified Saffman turbulence

We consider freely decaying turbulence of incompressible fluid evolving in the presence of imposed stratification in which the energy spectrum E(k) ~ k^2 at small wavenumber k. Turbulence with this kind of spectrum is called here Saffman turbulence. The turbulence field can be represented in terms of three eigenmodes of a linear operator including the buoyancy effect, say zeta_1 (voritical mode), zeta_2 and zeta_3 (wave modes). Within the linearized approximation ignoring the nonlinear coupling between the modes, the velocity correlation spectrum <|u_i(p)|^2> (no summation over i) in general oscillates in time due to the presence of buoyancy force, while the spectra Z_a(p)=<|zeta_a(p)|^2> (a=1,2,3) do not oscillate, where p is the wave vector. A simple analysis under certain assumptions shows that there are an infinite number of invariants associated with each of the spectra Z_1, Z_2 and Z_3 at small |p|. The invariants may depend on the direction of the wave vector p. Theoretical conjectures based on the analysis are examined by comparison with direct numerical simulation data.