Are inverse-cascade vorticity isolines exactly conformally invariant? · revision history
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- Title
- Are inverse-cascade vorticity isolines exactly conformally invariant?
- Plain
- In flat turbulence, the curves where the local spin of the fluid changes sign look statistically identical to the boundaries of random clusters in a classic model of percolation, a symmetry known from equilibrium critical points. Why a driven, dissipative flow should have this symmetry is unexplained.
- Precise
- Forced 2D incompressible Navier-Stokes turbulence with large-scale friction, inverse-cascade range $\ell_f \ll r \ll L$. Zero-vorticity isolines are statistically consistent with Schramm-Loewner evolution $\mathrm{SLE}_{\kappa}$ with $\kappa = 6$ (Bernard, Boffetta, Celani, Falkovich 2006), the class of critical percolation cluster boundaries. Is this conformal invariance exact as $L/\ell_f \to \infty$, and which property of the dynamics implies it? Answer: a derivation, or a measured deviation of $\kappa$ beyond error bars.
- Settled by
- An analytic derivation from the $2\mathrm{D}$ Navier-Stokes dynamics, or a high-precision numerical test of $\kappa$ and of the conformal covariance of multipoint observables.
- Kind
- well-posed
- Answer type
- yes-no
- References
- (empty)
- Literature status
- open
- Status note
- Evidence is numerical and experimental; a 2025 theoretical proposal models the cascade as constant-flux domains whose boundaries carry a Liouville conformal field theory (Eling, arXiv:2505.09657), without derivation from the Navier-Stokes equations.
- Posed since
- 2006