Manifestly invariant Lagrangians for geophysical fluids

Abstract

Three manifestly invariant Lagrangians are presented from which the covariant equations of motion for inviscid classical fluids are derived using the least action principle. Invariance and covariance are here defined with respect to synchronous, but otherwise arbitrary, coordinate transformations, i.e. supposing that time intervals are absolute as required by Newtonian mechanics. In the first Lagrangian, the flow is formulated in terms of fluid particles, but conservation of mass and entropy is assumed a priori. In the second Lagrangian, the flow is described by fields, and conservation of mass and entropy is obtained with Lagrange multipliers. The third Lagrangian is also based on a field formulation, but has no Lagrange multipliers and produces all the desired equations of motion, including conservation of mass and entropy. The differences and similarities between these formulations are discussed. Hydrostatic equations are rederived from an asymptotic expansion of the action using the covariant field formulation.

On the triviality of potential vorticity conservation in geophysical fluid dynamics

“…The coordinate x 0 =t is the time, and the x i ʼs are any curvilinear spatial coordinates. An invariant spatial volume element is written º dx dx dx g d x g (the parentheses around the symbol β indicate that it is not a space-time index-see Zadra and Charron (2015) where these equations are derived from a least action principle). The symbol ρ represents the fluid density, º m m u dx dtthe 4-velocity field with u 0 =1, º -mn mn m n h g g g g of time intervals in Newtonian mechanics imposes a constraint on the space-time metric tensor: the contravariant component g 00 must be a non-zero constant (taken here as unity).…”

“…This assumption is unnecessary when using Clebsch potentials as dynamical fields, see e.g Zadra and Charron (2015)…”

Dynamical consistency and covariance: reply to Staniforth and White

“…This point of view is justified by the fact that any approximation (geometric and/or dynamical) of the governing equation that remains covariant will automatically be consistent with the fundamental assumptions underlying Newtonian mechanics. Equivalently, any approximation that preserves the scalar property of the action functional for inviscid fluids, will always lead to approximated equations of motion compatible with Newton's fundamental assumptions (Charron et al , give definitions and details; Charron and Zadra, ; , ; Zadra and Charron, ).…”

“…Dirac, ). To generate covariant equations of motion, a Lagrangian must be a scalar under admissible coordinate transformations (Zadra and Charron, ). Therefore, we conclude that this criterion of White et al () is incomplete.…”

“…will always lead to approximated equations of motion compatible with Newton's fundamental assumptions Zadra, 2014, 2015;Zadra and Charron, 2015, give definitions and details).…”