# Ice Spiral Patterns on the Ocean Surface

## TL;DR

This paper introduces a novel hydrodynamic model to explain large-scale ice swirl patterns on the ocean surface, analyzing their formation, stability, and geometric properties through multiple mathematical approaches.

## Contribution

The study develops a new two-dimensional compressible Navier-Stokes model for ice swirls, deriving analytical solutions and stability analysis that align with experimental observations.

## Key findings

- Logarithmic spiral solutions are derived from the nonlinear equations.
- Multiple spiral modes, including pure radial and azimuthal, are obtained.
- The stability analysis reveals geometric phase transitions in the patterns.

## Abstract

We investigate a new two-dimensional compressible Navier-Stokes hydrodynamic model design to explain and study large scale ice swirls formation at the surface of the ocean. The linearized model generates a basis of Bessel solutions from where various types of spiral patterns can be generated and their evolution and stability in time analyzed. By restricting the nonlinear system of equations to its quadratic terms we obtain swirl solutions emphasizing logarithmic spiral geometry. The resulting solutions are analyzed and validated using three mathematical approaches: one predicting the formation of patterns as Townes solitary modes, another approach mapping the nonlinear system into a sine-Gordon equation, and a third approach uses a series expansion. Pure radial, azimuthal and spiral modes are obtained from the fully nonlinear equations. Combinations of multiple-spiral solutions are also obtained, matching the experimental observations. The nonlinear stability of the spiral patterns is analyzed by Arnold's convexity method, and the Hamiltonian of the solutions is plotted versus some order parameters showing the existence of geometric phase transitions.