Instanton calculus in shell models of turbulence

Area of Science:

• * Physics
• * Fluid Dynamics
• * Statistical Mechanics

Background:

• * Turbulence intermittency in the Gledzer-Ohkitani-Yamada (GOY) shell model is linked to singular structures interacting with background fluctuations.
• * Existing models require a statistical theory to explain the dynamics of these singular objects within the inertial range.

Purpose of the Study:

• * To develop a statistical theory for singular structures in turbulent systems.
• * To model the incoherent background as a Gaussian white-noise forcing.
• * To compute the Cramer function for singularity exponent distributions.

Main Methods:

• * Developed a general scheme for constructing instantons in spatially discrete dynamical systems.
• * Modeled background fluctuations using a Gaussian white-noise forcing of small strength (Gamma).
• * Employed a semiclassical expansion to compute the Cramer function up to first order in Gamma.

Main Results:

• * Computed the Cramer function governing the probability distribution of effective singularity exponents (z).
• * The theory provides predictions for the statistics of coherent structures.
• * Results are presented up to first order in the semiclassical expansion parameter Gamma.

Conclusions:

• * The proposed statistical theory successfully models singular structures in turbulent systems.
• * Predictions derived from the theory align with statistical data from high-Reynolds number GOY model simulations.
• * This work offers a framework for understanding intermittency through the dynamics of singular structures.