Decay exponent of vortex number in freely decaying 2D turbulence
In plain words
In freely decaying flat turbulence, small vortices merge into fewer, larger ones. The rate at which the number of vortices falls with time has been measured, and a scaling theory exists, but whether the exponent is universal is not settled.
Precise statement
In freely decaying 2D Navier-Stokes turbulence at large Re, the number density of coherent vortices decays as $n(t) \sim t^{-\xi}$. Early simulations gave $\xi \sim 0.7\text{ to }0.75$, approximately (Carnevale et al. 1991); a scaling theory for inviscid decay predicts a vortex number density per unit vortex area $n(A,t) \sim t^{-2/3}/A$ (Dritschel et al. 2008). Determine $\xi$ in the limit $\mathrm{Re} \to \infty$, and whether it is universal or depends on initial conditions and Re.
What would settle it
A derivation of $\xi$ verified by ensembles of high-resolution decaying DNS with varied initial spectra and Re.
Status in the literature
The 2008 scaling theory (Dritschel, Scott, Macaskill, Gottwald, Tran, PRL 2008) is supported by high-resolution inviscid simulations; its agreement with finite-Re values near 0.75 and its universality across initial conditions are not established.