On the decay of Saffman turbulence subject to rotation, stratification or an imposed magnetic field

We consider freely decaying, statistically axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification or a uniform magnetic field. We focus on the case of Saffman turbulence in which E(k -> 0) similar to k(2) and show that, if the large scales evolve in a self-similar manner, then u(perpendicular to)(2)l(perpendicular to)(2)l(parallel to) = constant in rotating, stratified and magnetohydrodynamic turbulence. This may be contrasted with E(k -> 0) similar to k(4) turbulence, in which u(perpendicular to)(2)l(perpendicular to)(4)l(parallel to) approximate to constant. (Here the subscripts perpendicular to and parallel to indicate directions perpendicular and parallel to the axis of symmetry, and l(perpendicular to), l(parallel to) and u(perpendicular to) are suitably defined integral scales.) This constraint on the integral scales allows us to make simple, testable predictions for the temporal evolution of l(perpendicular to), l(parallel to) and u(perpendicular to) in all three systems. There are only limited data sets against which to compare these predictions, but they are consistent with those data which are currently available.