Critical Rossby number for the inverse cascade in rotating turbulence
In plain words
Rapid rotation makes turbulence nearly two-dimensional and can reverse the direction of energy flow, sending energy to large scales. The rotation rate at which this switch happens, and whether it is sharp like a phase transition, are unsettled.
Precise statement
In forced rotating turbulence with rotation rate $\Omega$, forcing scale $\ell_f$ and Rossby number $\mathrm{Ro} = u_f/(2\Omega \ell_f)$, an upscale energy flux appears below a critical $\mathrm{Ro}_c$. Determine $\mathrm{Ro}_{c}$ in the limits $\mathrm{Re} \to \infty$ and $\text{domain size} \to \infty$, the scaling of the upscale flux fraction near $\mathrm{Ro}_c$, and whether the transition is continuous with critical exponents or discontinuous with hysteresis.
What would settle it
DNS at increasing Re and domain size showing convergence of $\mathrm{Ro}_{c}$ and of the exponent of the upscale flux near $\mathrm{Ro}_{c}$.
Status in the literature
Simulations find a transition with hysteresis and metastable flux-loop states (Clark di Leoni, Alexakis, Biferale, Buzzicotti 2020); the large-Re, large-domain limit is not settled.
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