# Monge-Ampere Geometry and Vortices

## TL;DR

This paper develops a novel geometric framework using higher symplectic geometry to analyze the Monge-Ampere equation related to vortices in fluid flows, providing new insights into vortex formation and structure in turbulence.

## Contribution

It introduces a higher Monge-Ampere geometric approach to study vortex dynamics and links geometric invariants to physical properties in fluid flows.

## Key findings

- Derived topological information from Monge-Ampere geometry in 2D flows.
- Connected scalar curvature of metrics to vorticity and strain accumulation.
- Applied symplectic reduction to specific Navier-Stokes solutions like Hill's vortex.

## Abstract

We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampere structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampere geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.