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Title: Decay exponent of vortex number in freely decaying 2D turbulence
Plain: 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: 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.
Settled by: A derivation of $\xi$ verified by ensembles of high-resolution decaying DNS with varied initial spectra and Re.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: contested
Status note: 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.
Posed since: 1991