Dynamical consistency and covariance: reply to Staniforth and White

Abstract

It is argued that the concept of dynamical consistency of approximated equations of motion should be based on their compatibility with the fundamental assumptions underlying Newtonian mechanics. This is achieved by preserving covariance of the approximated equations under synchronous coordinate transformations. A discussion is presented on how this definition of dynamical consistency relates to and differs from White et al.'s.

Manifestly invariant Lagrangians for geophysical fluids

“…This, among other things, ensures that existing conservation laws are preserved under arbitrary coordinate transformations, and that the resulting approximated equations obey the principle of Newtonian relativity and are therefore consistent with Newton's laws of mechanics (Charron et al . ). In section 5, a field formulation is used to demonstrate how to consistently derive approximated governing equations.…”

“…the metric tensor, its determinant and its properties; the definition of covariant and contravariant tensors and derivatives; the implicit summation over repeated indices in all formulae–has been presented in Charron et al . () and Charron and Zadra (, ). The reader is therefore invited to consult those articles if needed.…”

On the dissociation between potential vorticity conservation and symmetries

“…To render covariance explicit, governing equations of classical fluid mechanics may be written in terms of tensor components suitable for any curvilinear, non-inertial coordinate system in which time intervals are absolute (Charron et al, 2014). This ensures that the underlying theory is relativistic (Charron et al, 2015), obviously not in the traditional sense of describing fluids with velocities comparable to that of light but in the literal sense of "obeying a principle of relativity"-in this case, the principle of Newtonian relativity.…”