[Paper Review] Conformal Invariance in Inverse Turbulent Cascades

Gregory Falkovich, S. Musacchio|arXiv (Cornell University)|Dec 17, 2010

Fluid Dynamics and Turbulent Flows1 references7 citations

TL;DR

This paper demonstrates that zero-isolines of scalar fields in inverse turbulent cascades of a class of 2D nonlinear models are conformally invariant, with statistical properties matching Schramm-Loewner Evolution (SLE) curves. The universality class $κ=4$ is found for $0 < m < 1$, indicating that conformal invariance depends on the velocity field's dynamics rather than the scalar field's properties, with a phase transition at $m=1$.

ABSTRACT

We study statistical properties of turbulent inverse cascades in a class of nonlinear models describing a scalar field transported by a two-dimensional incompressible flow. The class is characterized by a linear relation between the transported field and the velocity, and include several cases of physical interest, such as Navier-Stokes, surface quasi-geostrophic and Charney-Hasegawa-Mima equations. We find that some statistical properties of the inverse turbulent cascades in such systems are conformal invariant. In particular, the zero-isolines of the scalar field are statistically equivalent to conformal invariant curves within the resolution of our numerics. We show that the choice of the conformal class is determined by the properties of a transporting velocity rather than those of a transported field and discover a phase transition when the velocity turns from a large-scale field to a small-scale one.

Motivation & Objective

• To investigate whether conformal invariance observed in Navier-Stokes and SQG turbulence extends to a broader class of 2D turbulent models.
• To determine whether the conformal invariance of isolines is a property of the scalar field or driven by the velocity field's dynamics.
• To identify the universality class $\kappa$ of the isolines and relate it to the model parameter $m$ governing the relation between scalar field and velocity.
• To test whether the fractal dimension and winding statistics of isolines match predictions from SLE theory.

Proposed method

• Numerical simulations of a family of 2D incompressible flow models with scalar field $\theta$ related to stream function $\psi$ via $\theta(\mathbf{k}) = |\mathbf{k}|^m \psi(\mathbf{k})$.
• Forcing and dissipation are applied at a finite scale $\ell_f$, enabling study of inverse cascades for $m > 0$.
• Statistical analysis of zero-isolines includes measuring fractal dimension, winding angle variance, and comparison with SLE predictions.
• The Loewner driving function $\xi(t)$ is extracted from isolines mapped to the upper half-plane to test for Gaussian statistics with $\langle \xi^2(t) \rangle \sim \kappa t$.
• Random phase reshuffling in Fourier space is used to destroy dynamical correlations while preserving scaling, testing the necessity of turbulence for conformal invariance.
• Comparison of scaling exponents $h$ for $\theta$ and $\psi$ with theoretical predictions confirms the model's consistency with turbulent cascade theory.

Experimental results

Research questions

• RQ1 Are the isolines of the scalar field in inverse turbulent cascades conformally invariant across different values of $m$?
• RQ2 Does the universality class $\kappa$ of the isolines depend on the scalar field's properties or on the velocity field's dynamics?
• RQ3 Is the fractal dimension of isolines consistent with SLE predictions, and does it remain constant for $0 < m < 1$?
• RQ4 What is the relationship between the scaling exponent $h$ of the scalar field and the fractal dimension of its isolines?
• RQ5 Does conformal invariance persist when dynamical correlations are destroyed by randomizing Fourier phases?

Key findings

• For $0 < m < 1$, the zero-isolines of the scalar field exhibit conformal invariance with $\kappa = 4$, as confirmed by Gaussian statistics of the Loewner driving function and winding angle variance scaling as $\frac{2}{3}\log \ell$.
• The fractal dimension of isolines is $D = \frac{3}{2}$ for $0 < m < 1$, consistent with $\kappa = 4$ via $D = 1 + \frac{2}{\kappa}$, and independent of $m$ in this range.
• For $m > 1$, the universality class is $\kappa = \frac{12}{4 - m}$, which matches the predicted $\kappa$ from the fractal dimension and is consistent with numerical results.
• The phase transition at $m = 1$ separates regimes where the velocity field is large-scale ($m > 1$) from small-scale ($m < 1$), with distinct conformal behavior.
• Randomizing the Fourier phases of $\theta$ destroys conformal invariance, confirming that the symmetry arises from turbulent dynamics, not from the field's scaling properties alone.
• The scaling exponent $h = \frac{2 - 2m}{3}$ for $\theta$ and $h = \frac{2 + m}{3}$ for $\psi$ are confirmed numerically, supporting the theoretical cascade model.