On freely decaying, anisotropic, axisymmetric Saffman turbulence

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which $E(k\ensuremath{\rightarrow} 0)\ensuremath{\sim} {k}^{2} $, where $E$ is the energy spectrum and $k$ the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor ${\Phi }_{ij} (\mathbi{k})$ at small $k$. These are ${M}_{\parallel } = {\Phi }_{\parallel } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$ and ${M}_{\perp } = 2{\Phi }_{\perp } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$, where the subscripts $\parallel $ and $\perp $ indicate quantities parallel and perpendicular to the axis of symmetry. Since ${M}_{\parallel } \ensuremath{\sim} { u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $ and ${M}_{\perp } \ensuremath{\sim} { u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $, $u$ and $\ell $ being integral scales, self-similarity of the large scales (when it applies) demands ${ u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $ and ${ u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $. This, in turn, requires that ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that ${M}_{\parallel } $ and ${M}_{\perp } $ are indeed invariants and that, in the fully developed state, ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} = \text{constant} $. Somewhat surprisingly, the same simulations also show that ${\ell }_{\parallel } / {\ell }_{\perp } $ is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ and ${\ell }_{\parallel } / {\ell }_{\perp } $ are both constant, the integral scales should evolve as ${ u}_{\perp }^{2} \ensuremath{\sim} { u}_{\parallel }^{2} \ensuremath{\sim} {t}^{\ensuremath{-} 6/ 5} $ and ${\ell }_{\perp } \ensuremath{\sim} {\ell }_{\parallel } \ensuremath{\sim} {t}^{2/ 5} $, irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.