Large-scale structure of velocity and passive scalar fields in freely decaying homogeneous anisotropic turbulence

We consider freely decaying homogeneous anisotropic turbulence whose energy spectrum $E\left(k\right)$ at $k\to 0$ is given by $E\left(k\right)=C{k}^2+o\left(k^2\right)$ at an initial instant, where $k$ is the wave number and $C$ is a $k$-independent positive number. An argument is given to show that there are an infinite number of invariants characterizing the large-scale structure of the turbulence. This is a generalization of Saffman's argument, which shows the existence of a finite number of invariants [P. G. Saffman, J. Fluid Mech. 27, 581 (1967)]. By applying a similar argument to homogeneous anisotropic passive scalar turbulence without any scalar source, we show that there are an infinite number of invariants characterizing the large-scale structure of passive scalar fields. Theoretical analysis based on the invariance and a self-similarity assumption for the large-scale evolution shows that the anisotropy of the velocity and passive scalar fields is persistent at large scales. The decay laws of the velocity and passive scalar fields are derived by a simple dimensional analysis.