Turbulence in noninteger dimensions by fractal Fourier decimation

Area of Science:

• Fluid Dynamics
• Statistical Mechanics
• Turbulence Theory

Background:

• Fractal decimation is a technique used to reduce the effective dimensionality of a fluid flow by selectively retaining Fourier modes.
• An equilibrium Gibbs state with a k(-5/3) spectrum is known to exist at the critical dimension D(c)=4/3, consistent with prior research.
• Understanding the behavior of turbulence at reduced dimensions is crucial for developing comprehensive fluid dynamics models.

Purpose of the Study:

• To investigate the persistence and characteristics of the inverse cascade in fractally decimated two-dimensional turbulence.
• To analyze the behavior of the Kolmogorov constant as the effective dimensionality is reduced below D=2.
• To explore the relationship between fractal decimation and established turbulence spectra.

Main Methods:

• Spectral simulations were employed to model fractally decimated two-dimensional turbulence.
• The study focused on analyzing the Fourier mode distribution, where the number of modes in a ball of radius k is proportional to k(D).
• The behavior of the inverse cascade and the Kolmogorov constant were examined across different effective dimensions D.

Main Results:

• The inverse cascade was observed to persist even when the effective dimensionality D was reduced below 2.
• A rapidly increasing Kolmogorov constant was detected as D approached the critical dimension of 4/3.
• The Kolmogorov constant is predicted to diverge as (D-4/3)(-2/3) near the critical dimension.

Conclusions:

• Fractal decimation allows for the study of turbulence at effective dimensions lower than the physical dimension.
• The findings suggest that the inverse cascade is a robust feature of turbulence, persisting under dimensional reduction.
• The divergence of the Kolmogorov constant near D(c)=4/3 highlights a critical transition in turbulent behavior.