AIP Digital Archive
A formulation of inviscid fluid dynamics based on the density F(x,v,t) in a single-particle phase space [x=(x1,x2,x3), v=(v1,v2,v3)] is presented. This density evolves in time according to a Poisson bracket of F with H(x,v,t)—a Hamiltonian in the same single-particle phase space. Compressible flows of barotropic fluid and homogeneous, incompressible flows are disscussed. The main advantage of the phase space density formulation over either Euler or Lagrange formulations is the algebraic and conceptual ease in making fully Hamiltonian approximations to the flow by altering H(x,v,t) and the Poisson brackets appropriately. The example of a shallow layer of rapidly rotating fluid where a Rossby number expansion is desired will be discussed in some detail. Changes of phase space coordinates that give an approximate H (expanded in Rossby number) and exact Poisson brackets will be exhibited. The resulting quasigeostrophic equations for F are two-dimensional partial differential equations to every order in Rossby number. The extension to multiple layers will be presented.
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