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    Hamiltonian description of the ideal fluid

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    Morrison_1998.pdf (807.3Kb)
    Date
    1998-04
    Author
    Morrison, Phillip J.
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    Abstract
    The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.
    Department
    Physics
    Subject
    wave-wave interactions
    nonlinear instability
    clebsch variables
    stability
    magnetohydrodynamics
    energy
    system
    equation
    hydrodynamics
    formulation
    URI
    http://hdl.handle.net/2152/61087
    xmlui.dri2xhtml.METS-1.0.item-citation
    Morrison, Philip J. "Hamiltonian description of the ideal fluid." Reviews of modern physics 70, no. 2 (1998): 467.
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