Objective Observer-Relative Flow Visualization in Curved Spaces for Unsteady 2D Geophysical Flows.

Area of Science:

• Fluid dynamics
• Differential geometry
• Mathematical physics

Background:

• Objectivity in fluid flow analysis is crucial but limited to Euclidean domains.
• Standard objectivity definitions fail in curved spaces, hindering analysis of flows on spheres or other non-flat surfaces.
• Existing methods require 3D ambient space computations, which are inefficient and less direct for intrinsically 2D flows.

Purpose of the Study:

• To generalize the concept of objectivity for fluid flow feature detection to curved spaces.
• To develop a mathematical framework for computing objective observer fields in curved domains.
• To enable intrinsic 2D computation and visualization of geophysical flows on spherical surfaces.

Main Methods:

• Leveraging mathematical physics and Riemannian geometry to define objectivity based on symmetry groups.
• Developing a general framework for computing observer fields intrinsically in 2D.
• Applying the framework to unsteady 2D geophysical flows on spheres.

Main Results:

• A novel, general mathematical framework for objective observer field computation in curved spaces.
• The framework operates intrinsically in 2D, avoiding 3D ambient space computations.
• Demonstrated application to geophysical flows on spheres, enabling objective feature computation and visualization.
• Computed observer frames adapt to follow moving structures like vortices, making their time evolution appear steady.

Conclusions:

• The developed framework successfully generalizes objectivity to curved spaces for fluid flow analysis.
• Intrinsic 2D computation offers a significant advantage for analyzing flows on surfaces like Earth.
• Objective feature detection and visualization are enhanced, providing clearer insights into complex fluid dynamics.