The decay of Saffman and batchelor turbulence subject to rotation, stratification or an imposed magnetic field

Abstract

We consider unforced, statistically-axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification, or a uniform magnetic field. We focus on two canonical cases: Saffman turbulence, in which E(κ → 0) ∼ κ2, and Batchelor turbulence, in which E(κ → 0) ∼ κ4. It has recently been shown that, provided the large scales evolve in a self-similar manner, then u ⊥2ℓ⊥2ℓ // = constant in Saffman turbulence and u⊥2ℓ⊥4ℓ// = constant in Batchelor turbulence (Davidson, 2009, 2010). Here the subscripts ⊥ and // indicate directions perpendicular and parallel to the axis of symmetry, and ℓ⊥, ℓ//, and u⊥ are suitably defined integral scales. These constraints on the integral scales allow us to make simple, testable predictions for the temporal evolution of ℓ⊥, ℓ//, and u⊥ in rotating, stratified and MHD turbulence.