Rapid Rotation and Cubic Damping in Navier-Stokes Equations: A Fourier-Galerkin Framework and Reproducible Numerical Audit

Abstract: We investigate the dynamics of the three-dimensional Navier-Stokes equations under rapid rotation and strong cubic absorption damping using a high-precision Fourier-Galerkin spectral framework. We establish rigorous analytical bounds on the enstrophy and energy dissipation rates, demonstrating how rotational constraints and nonlinear damping regularize potential finite-time singularities. Reproducible Framework & Contents: This record contains the complete computational and analytical package: Rapid_Rotation_Cubic_Damping_Galerkin_Framework.pdf : The formal preprint manuscript detailing the mathematical proofs, spectral expansions, and stability theorems. RRNS_Fourier_Galerkin_Audit.ipynb : The master Jupyter Notebook executing the full Fourier-Galerkin pseudospectral simulation and energy spectrum verification. RRNS_Fourier_Galerkin_Audit.py & .tex / .md / .html : Standalone Python script, LaTeX source, Markdown notes, and interactive HTML execution reports for full multi-platform reproducibility. RRNS.zip : Comprehensive archive containing all numerical outputs, diagnostic logs, and configuration files. All results, spectral coefficients, and energy audits are fully reproducible via the accompanying code and computational scripts.