A new Hamiltonian formulation for fluids and plasmas. Part 1. The perfect fluid

Abstract

A new formulation of the Hamiltonian structure underlying the perfect fluid equations is presented. Besides time, a parameter c is also used. Correspondingly, there are two interdependent systems of equations expressing time evolution and e evolution respectively. The accessibility equations define the e dynamics and give the variation in the usual Eulerian fluid variables as determined by the generating functions. The time evolutions of both the Eulerian fluid variables and the generating functions are obtained from an action principle. The consistency of the e and the time dynamics is crucial for this formulation, i.e. the accessibility equations must be propagated in time by the Euler–Lagrange equations. The reason for introducing this new formulation is its power in certain applications where the existing Hamiltonian alternatives seem less convenient to use. In particular, it is a promising tool for Hamiltonian perturbation theory. We consider the small-amplitude expansion, and find, very simply and naturally, the Hermitian structure of the linearized ideal fluid equations as well as coupling coefficients for resonant three-wave interaction exhibiting the Manley–Rowe relations.

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

“…For reasons explained in the Introduction, we feel that it is essential not to restrict our discussion to flat spaces. Otherwise we would have used a 'half-way house' approach where all the differential geometry is expressed in the standard language of vector operations, as done, for example in the paper of Larsson (1996).…”

Convective derivatives and Reynolds transport in curvilinear time-dependent coordinates

“…Drobot and Rybarski (1958), Serrin (1959), Eckart (1960, Seliger and Whitham (1968), Finlayson (1972), Mobbs (1982), Salmon (1988), Morrison (1998)), mostly due to the appearance of velocity potentials whose physical significance is somehow obscure. To overcome these problems Bretherton (1970) first proposed a hybrid Hamilton's principle for perfect fluids based on constrained variations (see, e.g., Larsson (1996Larsson ( , 2003, Wilhelm (1979) for recent developments and applications). The key idea is to compute a field variation induced by a small perturbation of particle paths in convected coordinates and transform it back to fixed coordinates 1 .…”

Practical use of a Lie algebra structure in non-dissipative fluid and plasma models

“…The adjoint structure of the linear term in (1.2) saves us from this work, and, furthermore, the corresponding properties for the second-order term in (1.2) lead to a general proof of the Manley-Rowe relations (see Sec. 6.3 below and also Larsson 1996a). Applications of this general Hamiltonian theory for concrete calculations are given for example in Larsson (1998b) for drift waves, Larsson and Wiklund (1999) for Rossby waves, Wiklund (1998) for shallow-water equations and Axelsson (1999) for the ideal MHD modes.…”