Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

Area of Science:

• Fluid dynamics
• Computational physics
• Statistical mechanics

Background:

• Two-dimensional turbulence exhibits complex dynamics, including energy and enstrophy cascades.
• Existing shell models simplify these dynamics but often struggle with phenomena like energy-enstrophy equipartition.
• The need for versatile models that capture essential interactions without explicit conservation law enforcement is recognized.

Purpose of the Study:

• To derive a systematically discretized model for two-dimensional turbulence.
• To explore the flexibility in retaining local and disparate scale interactions.
• To investigate the dominance of energy and enstrophy equipartition over the dual cascade in numerical simulations.

Main Methods:

• Logarithmic discretization of wave-number space applied to fluid equations.
• Systematic derivation allowing for selective retention of interactions (local or disparate scale).
• Numerical integration of the derived model to observe dynamical behavior.

Main Results:

• A family of models is generated, including anisotropic and nonlocal interaction variants.
• The model recovers various limiting forms of existing shell models.
• Numerical simulations indicate that energy and enstrophy equipartition are favored over the dual cascade.

Conclusions:

• The derived model provides a flexible framework for studying two-dimensional turbulence.
• The model's ability to reproduce known shell models and its numerical behavior offer insights into turbulence dynamics.
• The observed dominance of equipartition highlights a key characteristic of the simulated two-dimensional turbulence.