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.Comment: 45 pages, 11 figure