Monge-Ampere Geometry and Vortices

Abstract

We introduce a new approach to Monge-Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampère 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 the (higher) Lagrangian submanifold it defines in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampère 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-Ampère 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.

Godbillon-Vey invariants of Non-Lorentzian spacetimes and Aristotelian hydrodynamics

“…Then the streamlines are unbounded and M 2 is foliated by rectangular quarterhyperbolas for r ̸ = 0. In this example, the torsion-free regions coincide with the singular regions observed in [40] where the Monge-Ampère metric is Kleinian. In this sense, torsion is a desirable feature of an Aristotelian fluid.…”

“…which is also discussed by [40] in connection with the occurence of scalar curvature singularities of the Monge-Ampère geometry associated to topological bifurcations in the fluid flow. The fluid domain is M 2 = R 2 , and here we restrict to flows in a time parameter t > 0, so that ψ has no critical points.…”

“…which is discussed by [40] The leaves of the one-parameter family of foliations F are given by the streamlines…”

“…The torsion-free parabolas change with time, but they always contains points with y = 1 3 at which the vorticity vanishes and where the scalar curvature of the Monge-Ampère metric is singular [40]. On the other hand, there are torsion-free points with non-zero vorticity, as well as points in R 2 with vanishing vorticity but non-zero torsion.…”