Vorticity and symplecticity in Lagrangian fluid dynamics

Abstract

The relationship between potential vorticity (PV) and the symplectic form is explored, for the shallow-water equations governing Lagrangian particle paths. Starting with the symplectic form, the PV is found by the pullback operation to the reference space. At first sight, the encoding of PV in the symplectic form appears to be independent of the particle relabelling symmetry. The analysis is carried a step further in two ways. Using the 'conservation of symplecticity' as a starting point, the fluxes of symplecticity arise as differential forms, and a complete pull back of the flux forms leads to a geometric description of PV conservation. Secondly, symmetry methods are used to give a rigorous connection between particle relabelling, symplecticity and PV conservation. Generalizations of these issues to semi-geostrophic flow and three-dimensional Lagrangian fluid flows, and the connection with Ertel's theorem are also discussed.

Ion acoustic traveling waves

“…In the latter case, the Hamiltonian evolution operator is the advective Lagrangian time derivative operator d/dτ = u x d/dx. These dual variational principles are analogous to the dual or multi-symplectic variational principles obtained by Bridges (1992) in studies of travelling water waves (see also Bridges (1997a,b); Bridges et al (2005)). McKenzie et al (2006) cast the spatial evolution equations for solitary travelling waves in a Hall current plasma in Hamiltonian form, in which the energy flux integral ε is the Hamiltonian and the longitudinal momentum flux inte-gral P x = const.…”

Multi-symplectic formulation of near-local Hamiltonian balanced models

“…Following Bridges et al (2005), we present the shallow-water theory in a Lagrangian formulation. The position and velocity of fluid particles are denoted by x = (x 1 , x 2 ) and u = (u 1 , u 2 ), respectively, and we label each fluid particle by its position at t = 0, which is denoted by m = (m 1 , m 2 ); then, all variables are treated as functions of m and t. The total derivatives with respect to t and m a are denoted by subscript 't' or 'a' after a comma, for example,…”

“…We now review some relevant details of the MS version of the shallowwater model, which was derived in Bridges et al (2005). The shallow-water Lagrangian (2.8) is not affine linear as it stands because it contains a multiple of The MS structure emerges when we reorganize this system of first-order partial differential equations as and W is the 12 × 12 skew-symmetric matrix…”

“…The structural conservation law and the 1-form quasi-conservation law for each of these models is obtained by substituting the components into the general forms (3.1)-(3.3) (see Bridges et al (2005), Hydon (2005) and Cotter et al (2007) for details).…”

“…Recently, Bridges et al (2005) showed that the shallow-water model and the semigeostrophic model each have a multi-symplectic (MS) formulation; moreover, these two formulations are very similar. MS systems have a common geometric structure: a vector of closed 2-forms is conserved by the flow.…”