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Differential Geometry of the Semi-Geostrophic and Euler Equations

Differential Geometry of the Semi-Geostrophic and Euler Equations. Ian Roulstone University of Surrey. Outline. Semi-geostrophic theory – Legendre duality and Hamiltonian structure Higher-order balanced models Complex structures K ähler geometry

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Differential Geometry of the Semi-Geostrophic and Euler Equations

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  1. Differential Geometry of the Semi-Geostrophic and Euler Equations Ian Roulstone University of Surrey

  2. Outline • Semi-geostrophic theory – Legendre duality and Hamiltonian structure • Higher-order balanced models • Complex structures • Kähler geometry • Complex manifolds and the incompressible Navier-Stokes equations

  3. Semi-geostrophic theory • Jets and fronts – two length scales

  4. Semi-geostrophic equations: shallow water Geostrophic wind

  5. Conservation laws • The SG equations conserve energy and potential vorticity, q

  6. Geostrophic momentum coordinates Equations of motion become Potential vorticity

  7. Legendre transformation Define then Singularities/Fronts and PV and a Monge-Ampère equation

  8. Geometricmodel (Cullen et al, 1984)

  9. Hamiltonian structure Define Then

  10. Higher-order balanced models There exists a family (Salmon 1985) of balanced models that conserve a PV of the form McIntyre and Roulstone (1996)

  11. Complex structure Introduce a symplectic structure and a two-form On the graph of φ Monge-Ampère eqn

  12. then ω (M-A eqn) is elliptic, and Define the Pfaffian is an almost-complex structure Iω 2 = -Id is an almost-Kähler manifold (Delahaies & Roulstone, Proc. R. Soc. Lond. 2009)

  13. Legendrian Structure.(Delahaies & R. (2009), R. & Sewell (2012))

  14. A canonical example Cubic y(x) = x 3/3

  15. Incompressible Navier-Stokes Apply div v = 02d: Stream function

  16. Complex structure Poisson eqnComponentsComplex structure

  17. Vorticity and Rate of Strain(Weiss Criterion) Q>0 implies almost-complex structure (ellipticity)

  18. 3d Incompressible Flows

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

  20. Geometry of 3-forms (Hitchin) Lychagin-Rubtsov (LR) metric

  21. Metric and Pfaffian Construct a linear operator, Kω, using LR metric and symplectic structure The “pfaffian”

  22. Complex structure

  23. Summary • Vorticity-dominated incompressible Euler flows in 2D are associated with almost-Kähler structure – a geometric version of the “Weiss criterion”, much studied in turbulence • Using the geometry of 3-forms in six dimensions, we are able to generalize this criterion to 3D incompressible flows

  24. These ideas originate in models are large-scale atmospheric flows, in which rotation dominates and an elliptic pde relates the flow velocity to the pressure field • Roubtsov and R (1997, 2001), Delahaies and R (2009) showed how hyper-Kähler structures provide a geometric foundation for understanding Legendre duality (singularity theory), Hamiltonian structure and Monge-Ampère equations, in semi-geostrophic theory

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