Finite predictability horizon in multiscale chaotic flows
In plain words
In 1969 Lorenz argued that in flows with motion on many scales, shrinking the initial error buys almost no extra forecast time, because errors pass quickly from small to large scales. Whether this hard limit exists in real turbulence, and what sets its value, is still studied.
Precise statement
For flows with energy spectrum $E(k) \sim k^{-\beta}$ over an inertial range, Lorenz predicted that for $\beta < 3$ the predictability time $T_p$ stays bounded as the initial error amplitude $\delta_0 \to 0$ at fixed error scale, since error moves upscale in eddy turnover times $\tau(k) \sim k^{((\beta-3)/2)}$. Determine whether this holds for 3D Navier-Stokes turbulence ($\beta = 5/3$) as the Reynolds number grows, and how $T_p$, in units of the large-eddy turnover time, depends on the dissipation range and on the scale of the initial error. Answer: yes or no, with $T_p$ and its dependences.
What would settle it
Identical-twin direct numerical simulations over many decades of $\delta_0$ and a range of Reynolds numbers, showing saturation of $T_p$ and its value.
Status in the literature
Identical-twin simulations of 3D turbulence and of atmospheric models support a finite horizon; its dependence on the dissipation range is debated.
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