models in geophysical fluid dynamics in nambu form€¦ · models in geophysical fluid dynamics in...
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![Page 1: Models in Geophysical Fluid Dynamics in Nambu Form€¦ · Models in Geophysical Fluid Dynamics in Nambu Form Richard Blender Meteorological Institute, University of Hamburg Thanks](https://reader030.vdocument.in/reader030/viewer/2022040603/5e9b486d8913eb150f7641bb/html5/thumbnails/1.jpg)
Models in Geophysical Fluid Dynamics in Nambu Form
Richard Blender
Meteorological Institute, University of Hamburg
Thanks to: Peter Névir (Berlin), Gualtiero Badin and Valerio Lucarini
Hamburg, May, 2014
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Overview • Nambu‘s (1973) extension of Hamiltonian mechanics
• Hydrodynamics in Nambu form (Névir and Blender 1993)
• Conservative codes (Salmon 2005)
• Review on applications of GFD
• Perspectives and Summary
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Nambu mechanics (Nambu 1973) two stream-functions C and H Yōichirō Nambu
Nobel prize 2008
Nambu and Hamiltonian mechanics Hamiltonian dynamics Basic ingredient: Liouville‘s theorem
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Low dimensional Nambu systems Nambu (1973)
Euler equations rigid top Névir and Blender (1994)
Lorenz-1963 conservative parts Chatterjee (1996)
Isotropic oscillator, Kepler problem, Vortices Phase space geometry and chaotic attractors
Axenides and Florates (2010), Roupas (2012)
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Conservation laws
The Lorenz Equations in Nambu form
Nambu
Névir and Blender (1994)
Conservative dynamics
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Minos Axenides and Emmanuel Floratos (2010) Strange attractors in dissipative Nambu mechanics
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Point vortices
Conservation
Nambu form for three point vortices (Müller and Névir 2014)
Kirchhoff 1876
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2D Vorticity equation: Nambu Representation
(Névir and Blender, 1993)
Nambu: Enstrophy and energy as Hamiltonians
cyclic
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3D Euler equations: Nambu Representation
(Névir and Blender, 1993)
Helicity and energy as Hamiltonians
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Hamiltonian Systems Physical system is Hamiltonian if a Poisson bracket exists
Casimir functions of the Poisson bracket (degenerate, noncanonical)
for all f
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Relationship between Noncanonical Hamiltonian Theory and Nambu Mechanics
Contraction/Evaluation
Poisson and Nambu brackets
• Noncanonical Hamiltonian theory is embedded in a Nambu hierarchy
• Casimirs are a second ‘Hamiltonian’
• Nambu bracket is nondegenerate (no Casimir)
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Nambu-Bracket (anti-symmetric due to the anti-symmetry of J)
Numerical Conservation of H and Z (Arakawa, 1966) • Jacobian J( , ) anti-symmetric • Discrete approximation of the integrals arbitrary • Energy and enstrophy arbitrary accuracy • Time stepping less relevant
Salmon (2005): 2D Euler equations in Nambu form (Z enstrophy)
Rewrite bracket
Conservative numerical codes using Nambu brackets
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Advantages of conservative algorithms • Improve nonlinear interaction terms
• Improve energy flow across spatial scales, cascades, and avoid spurious accumulation of energy
• Impact on nonequilibrium flows
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Geophysical Fluid Dynamics Models
• Quasigeostrophy (Névir and Sommer, 2009)
• Rayleigh-Bénard convection (Bihlo 2008, Salazar and
Kurgansky 2010)
• Shallow water equations (Salmon 2005, 2007, Névir and
Sommer 2009)
• Primitive equations (Salmon 2005, Nevir 2005, 2009, Herzog
and Gassmann 2008)
• Baroclinic atmosphere (Nevir and Sommer 2009)
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Quasigeostrophic Model (Névir and Sommer, 2009) QG Potential Vorticity
Conservation
Energy and Potential enstrophy conserved
Bracket
Nambu
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Surface Quasi-Geostrophy and Generalizations
Fractional Poisson equation for active scalar and stream-function
Conservation laws
2. Total Energy (Euler) 1. Arbitrary function G
α = 1: SQG, α = 2; Euler
Kinetic Energy (SQG)
Blumen (1978), Held et al. (1995), Constantin et al. (1994)
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Nambu representation for SQG and Gen-Euler
Nambu bracket for functions F(q)
Functional derivatives
Blender and Badin (2014)
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2D Boussinesq Approximation
Vorticity
Conserved: Energy and helicity analogue
buoyancy
Bihlo (2008), Salazar and Kurgansky (2010)
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3D Boussinesq approximation
absolute Vorticity
Energy (conserved)
buoyancy
Helicity (constitutive)
Buoyancy (constitutive)
Salazar and Kurgansky (2010) Salmon (2005)
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Rayleigh–Bénard convection with viscous heating (2D) Lucarini and Fraedrich (2009)
Nambu and metric bracket
Blender and Lucarini (2011)
Eckert number
Properties and impacts of viscous heating • Conservation of total energy (Hamiltonian) • Satisfies the LiouvilleTheorem
• Modifies available potential energy, source in Lorenz energy cycle
• Alters convective processes, implications for complex models
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Shallow water equations
Vorticity and divergence
Energy
Enstrophy
Salmon (2005, 2007) Sommer and Névir (2009)
Nambu
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Derivatives
Shallow water equations Brackets
Flow
2D vorticity
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Global Shallow Water Model using Nambu Brackets Sommer and Névir (2009)
ICON grid structure
ν vorticity, vertices i mass. triangle centers l wind, edges
Variables
Grid: functional derivatives, operators (div and curl), Jacobian and Nambu brackets Time step: leap-frog with Robert-Asselin filter
ICON model (Isosahedric grid, Non-hydrostatic, German Weather Service and Max Planck Institute for Meteorology, Hamburg).
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Sommer and Névir (2009)
Computational instability
Energy and potential enstrophy
ICOSWP
Nambu
Energy
Enstrophy
potential enstrophy conserving
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Nambu ICOSWP
Energy spectra and height snapshots (decaying)
Sommer and Névir (2009)
Kinetic Energy
Height
k -3
k -5/3
Initial (dotted)
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Névir (1998), Névir and Sommer (2009), Herzog and Gassmann (2008)
Energy
Helicity
Baroclinic Atmosphere
Mass, entropy
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Baroclinic Atmosphere as a Modular Nambu-system
3D incompr barotropic baroclinic
Névir (1998), Névir and Sommer (2009)
Casimir
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Gassmann and Herzog (2008)
Towards a consistent numerical code using Nambu brackets
• Compressible non-hydrostatic equations
• Turbulence-averaged
• Dry air and water in three phases
• Poisson bracket for full-physics equation
Gassmann (2012)
Non-hydrostatic core (ICON) with energetic consistency
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Standard approach
Equations
Identify conservation laws
Dynamic equations from conservation laws
A perspective for modelling and parameterizations
A Nambu approach
Conservation laws and operators
Equations
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Summary
• Application of Nambu Mechanics in Geophysical Fluid Dynamics
Second ‘Hamiltonian’ due to particle relabeling symmetry
Includes noncanonical Hamiltonian fluid dynamics (Casimirs)
• Flexible approach (constitutive integrals)
Modular design and approximations
• Construction of conservative numerical algorithms
Improves nonlinear terms, energy and vorticity cascades
Thank you
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References
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Appendix
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Nambu brackets (Takhtajan 1994) Skew symmetry (permutations σ with parity ε)
Leibniz rule
Fundamental (Jacobi) Identity
Hierarchy of Nambu brackets
Example: Poisson bracket with Casimir C from Nambu bracket
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Jacobi Identity and contraction
Salmon (2005)
Consider Nambu bracket
This contraction does not obey the Jacobi Identity for Poisson brackets
Contraction with H (evaluate)
Applications require only that the Nambu brackets are anti-symmetric Consequence: Not insist on the generalized (Nambu) Jacobi Identity
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Types of Nambu Representations
Second kind NB II: also constitutive quantities not necessarily constants of motion
Advantages: • Flexibility in the construction of a Nambu representation • Modular decomposition and approximation
Nambu brackets first kind NBI: different triplets, same conserved quantities
Roberto Salazar and Michael Kurgansky (2010), see also Nambu (1973)
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Nonconservative forces: Metriplectic systems Kaufman (1984), Morrison (1986), Bihlo (2008) Metriplectic systems conserve energy but are irreversible Brackets: symplectic and metric Entropy S: Casimir of the symplectic bracket Entropy increases