Abstract

We consider freely-decaying, two-dimensional, isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k->0)=Ik^{3 , where I is the two-dimensional version of Loitsyanskys integral. However, a second possibility is E(k->0)=Lk , where the pre-factor, L, is the two-dimensional analogue of Saffmans integral. We show that, as in three dimensions, L is an invariant and that E_{Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k->0)=Jk}}-1, where J, like L, is an invariant. However, while EJk^{-1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E_Jk}-1 spectra. The constraint imposed by the conservation of energy is particularly important,ruling out EJk^-1 spectra for certain classes of initial conditions.

We illustrate these ideas with some direct numerical simulations.