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Monge-Ampere Geometry and Vortices

Lewis Napper, Ian Roulstone, Vladimir Rubtsov, Martin Wolf

TL;DR

This work reframes the Poisson equation for pressure in incompressible flows within a higher Monge–Ampère geometric framework, linking vorticity–strain dynamics to Monge–Ampère structures on cotangent bundles. In 2D, it builds a pull-back metric on a Lagrangian submanifold and uses Gauss–Bonnet to connect curvature with vortex topology, while classifying elliptic vs hyperbolic regions via the sign of a diagnostic $f$ built from vorticity and strain. Extending to 3D, the authors employ higher (k-plectic) geometry and higher Monge–Ampère structures, enabling higher-symplectic reductions that yield 2D stream-function descriptions for complex flows such as Arnol’d–Beltrami–Childress and Hicks–Moffatt vortices, with helicity arising naturally from the geometric data. The framework provides curvature-based diagnostics for vortical structures, clarifies topological features of vortex tubes, and suggests geometric pathways to study Euler and Navier–Stokes regularity via a geometric flow of Monge–Ampère data.

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.

Monge-Ampere Geometry and Vortices

TL;DR

This work reframes the Poisson equation for pressure in incompressible flows within a higher Monge–Ampère geometric framework, linking vorticity–strain dynamics to Monge–Ampère structures on cotangent bundles. In 2D, it builds a pull-back metric on a Lagrangian submanifold and uses Gauss–Bonnet to connect curvature with vortex topology, while classifying elliptic vs hyperbolic regions via the sign of a diagnostic built from vorticity and strain. Extending to 3D, the authors employ higher (k-plectic) geometry and higher Monge–Ampère structures, enabling higher-symplectic reductions that yield 2D stream-function descriptions for complex flows such as Arnol’d–Beltrami–Childress and Hicks–Moffatt vortices, with helicity arising naturally from the geometric data. The framework provides curvature-based diagnostics for vortical structures, clarifies topological features of vortex tubes, and suggests geometric pathways to study Euler and Navier–Stokes regularity via a geometric flow of Monge–Ampère data.

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.
Paper Structure (54 sections, 6 theorems, 223 equations, 12 figures)

This paper contains 54 sections, 6 theorems, 223 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Fluid flows and Monge--Ampère geometry
  4. Incompressible fluid flows
  5. Navier--Stokes equations.
  6. Pressure constraint.
  7. Geometric properties of fluids in two dimensions
  8. Monge--Ampère structures.
  9. Monge--Ampère geometry of two-dimensional fluid flows.
  10. Almost (para-)Hermitian structure.
  11. Pull-back metric.
  12. Local Gauß--Bonnet theorem.
  13. Christoffel symbols and curvatures.
  14. Examples in two dimensions
  15. Preliminaries.
  16. ...and 39 more sections

Key Result

theorem 1

Let $\Sigma$ be a two-dimensional, compact, oriented Riemannian manifold with metric $g$. Suppose that $\Sigma$ has a boundary composed of disjoint, simple, closed, piecewise regular, piecewise arc-length parametrised curves $\gamma_\alpha$, that is, $\partial\Sigma=\bigcup_\alpha\gamma_\alpha$. Let

Figures (12)

  • Figure 1: Plot of the streamlines for stream function \ref{['eq:streamFunctionMoffat']} with $t=-1$. The streamlines around the elliptic point $(0,-1)$ form closed contours, whilst those near the hyperbolic point $(0,1)$ diverge.
  • Figure 2: A selection of plots of the Legendre-dual stream function \ref{['eq:moffattDualStream']}, at time $t=-1$. The multivalued behaviour of $\psi'$ is associated with the corresponding Lagrangian submanifold $\iota:L\hookrightarrow T^*\IR^2$ being a generalised solution to \ref{['eq:MongeLegendre']}.
  • Figure 3: Plots of the iso-lines of the stream function and half the Laplacian of pressure for the Taylor--Green vortex with parameters $a=b=1$ and $F(t)\equiv 1$, which shall be used for the remainder of the plots for this example. Streamlines corresponding to values of sufficiently large magnitude are closed contours contained in regions of positive $f$, where vorticity dominates. The vorticity is proportional to the stream function, $\zeta=-(a^2+b^2)\psi$.
  • Figure 4: Plots of the eigenvalues \ref{['eq:eigenvaluesTGV']} of the pull-back metric \ref{['eq:pullbackMetricTGV']} for the Taylor--Green vortex with parameters $a=b=1$ and $F(t)\equiv1$.
  • Figure 5: Contour plots of the curvatures \ref{['eq:curvatureLR_TGV']} and \ref{['eq:curvaturePullbackTGV']} respectively, for the Taylor Green vortex with parameters $a=b=1$ and $F(t)\equiv 1$.
  • ...and 7 more figures

Theorems & Definitions (16)

  • remark 1
  • theorem 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • theorem 2
  • theorem 3
  • remark 7
  • ...and 6 more