Differential Geometry of the Semi-

Geostrophic and Euler Equations

Ian RoulstoneIan Roulstone

University of SurreyUniversity of Surrey

OutlineOutline

• Semi-geostrophic theory – Legendre Semi-geostrophic theory – Legendre duality and Hamiltonian structureduality and Hamiltonian structure

• Higher-order balanced modelsHigher-order balanced models

• Complex structuresComplex structures

• KKähler geometryähler geometry

• Complex manifolds and the Complex manifolds and the incompressible Navier-Stokes incompressible Navier-Stokes equationsequations

Semi-geostrophic theorySemi-geostrophic theory• Jets and fronts – two length scalesJets and fronts – two length scales

Semi-geostrophic equations: shallow Semi-geostrophic equations: shallow waterwater

Geostrophic Geostrophic windwind

Conservation lawsConservation laws

• The SG equations conserve energy The SG equations conserve energy and potential vorticity, and potential vorticity, qq

Geostrophic momentum Geostrophic momentum coordinatescoordinates

Equations of motion becomeEquations of motion become

Potential vorticityPotential vorticity

Legendre transformationLegendre transformation

DefineDefine

thenthen

PV and a PV and a Monge-AmpMonge-Ampère ère equationequation

andand Singularities/FrontsSingularities/Fronts

GeometricGeometric model (Cullen et al, 1984)model (Cullen et al, 1984)

Hamiltonian structureHamiltonian structure

DefineDefine

ThenThen

Higher-order balanced modelsHigher-order balanced models

There exists a family (Salmon 1985) of balanced There exists a family (Salmon 1985) of balanced models that conserve a PV of the formmodels that conserve a PV of the form

McIntyre and Roulstone (1996)McIntyre and Roulstone (1996)

Complex structureComplex structure

Introduce a symplectic structureIntroduce a symplectic structure

and a two-formand a two-form

On the On the graph of of φφ

Monge-Ampère Monge-Ampère eqneqn

Define the PfaffianDefine the Pfaffian

then then ωω (M-A eqn) is (M-A eqn) is ellipticelliptic, and, and

is an is an almost-complex structure Iω 2 = -Id

is an almost-Kähler manifold(Delahaies & Roulstone, Proc. R.

Soc. Lond. 2009)

Legendrian StructureLegendrian Structure..

(Delahaies & R. (2009), R. & Sewell (2012))(Delahaies & R. (2009), R. & Sewell (2012))

A canonical A canonical exampleexample

Cubic Cubic

y(x)y(x) = = x x 33/3/3

Incompressible Navier-Incompressible Navier-StokesStokes

Apply div v = Apply div v = 00

2d2d: Stream : Stream functionfunction

Complex structureComplex structure

Poisson eqn

Components

Complex structure

Vorticity and Rate of StrainVorticity and Rate of Strain(Weiss Criterion)(Weiss Criterion)

QQ>0 >0 implies almost-complex implies almost-complex structure (ellipticity)structure (ellipticity)

3d Incompressible Flows3d Incompressible Flows

• J.D. Gibbon (Physica D 2008 – J.D. Gibbon (Physica D 2008 – Euler, 250 years onEuler, 250 years on): ): “The elliptic equation for the pressure is by no means “The elliptic equation for the pressure is by no means fully understood and fully understood and locallylocally holds the key to the holds the key to the formation of vortical structures through the sign of formation of vortical structures through the sign of the Laplacian of pressure. In this relation, which is the Laplacian of pressure. In this relation, which is often thought of as a constraint, may lie a deeper often thought of as a constraint, may lie a deeper knowledge of the geometry of both the Euler and knowledge of the geometry of both the Euler and Navier-Stokes equations…The fact that vortex Navier-Stokes equations…The fact that vortex structures are dynamically favoured may be structures are dynamically favoured may be explained by inherent geometrical properties of the explained by inherent geometrical properties of the Euler equations but little is known about these Euler equations but little is known about these features.”features.”

Geometry of 3-forms Geometry of 3-forms (Hitchin)(Hitchin)

Lychagin-Rubtsov (LR) metricLychagin-Rubtsov (LR) metric

Metric and PfaffianMetric and Pfaffian

Construct a linear Construct a linear operator, operator, KKωω, using , using LR metric and LR metric and symplectic structuresymplectic structure

The The “pfaffian”“pfaffian”

Complex structureComplex structure

SummarySummary

• Vorticity-dominated incompressible Vorticity-dominated incompressible Euler flows in 2D are associated with Euler flows in 2D are associated with almost-Kalmost-Kähler structure – a geometric ähler structure – a geometric version of the “Weiss criterion”, much version of the “Weiss criterion”, much studied in turbulencestudied in turbulence

• Using the geometry of 3-forms in six Using the geometry of 3-forms in six dimensions, we are able to generalize dimensions, we are able to generalize this criterion to 3D incompressible this criterion to 3D incompressible flowsflows

• These ideas originate in models are These ideas originate in models are large-scale atmospheric flows, in which large-scale atmospheric flows, in which rotation dominates and an elliptic pde rotation dominates and an elliptic pde relates the flow velocity to the pressure relates the flow velocity to the pressure fieldfield

• Roubtsov and R (1997, 2001), Delahaies Roubtsov and R (1997, 2001), Delahaies and R (2009) showed how hyper-Kand R (2009) showed how hyper-Kähler ähler structures provide a geometric structures provide a geometric foundation for understanding Legendre foundation for understanding Legendre duality (singularity theory), Hamiltonian duality (singularity theory), Hamiltonian structure and Monge-Ampère structure and Monge-Ampère equations, in semi-geostrophic theoryequations, in semi-geostrophic theory