a geometric description of large-scale atmospheric flows roulstone.pdf · large-scale atmospheric...
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A Geometric Description of Large-Scale Atmospheric
Flows (and Navier-Stokes)
Ian Roulstone
University of Surrey
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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
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Semi-geostrophic theory
• Jets and fronts – two length scales
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Semi-geostrophic equations: shallow water
Geostrophic wind
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Conservation laws
• The SG equations conserve energy and potential vorticity, q
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Geostrophic momentum coordinates
Equations of motion become
Potential vorticity
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Legendre transformation
Define
then
PV and Monge-Ampère equation
and Singularities/Fronts
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Geometric model (Cullen et al, 1984)
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Hamiltonian structure
Define
Then
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Higher-order balanced models
There exists a family (Salmon 1985) of balanced models that conserve a PV of the form
McIntyre and Roulstone (1996)
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Complex structure
Introduce a symplectic structure
and a two-form
On the graph of φ
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Define the Pfaffian
then ω is elliptic, and
is an almost-complex structure Iω 2 = -Id
is an almost-Kähler manifold
(Delahaies & Roulstone, Proc. R. Soc. Lond. 2010)
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Incompressible Navier-Stokes
Apply div v = 0
2d: Stream function
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Complex structure
Poisson eqn
Components
Complex structure
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Vorticity and Rate of Strain (Weiss Criterion)
Q>0 implies almost-complex structure (ellipticity)
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3d Incompressible Flows
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• J.D. Gibbon (Physica D 2008 – Euler, 250 years on): “The elliptic equation for the pressure is by no means fully understood and locally holds 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.”
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Geometry of 3-forms (Hitchin)
Lychagin-Roubtsov (LR) metric
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Metric and Pfaffian
Construct a linear operator, Kω, using LR metric and symplectic structure
The “pfaffian”
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Complex structure
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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
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• 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 Roulstone (1997, 2001) 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