A variational formulation of geophysical fluid motion in non‐Eulerian coordinates

Abstract

Systematic methods to derive geophysical equations of motion possessing conservation laws for energy, momentum and potential vorticity have recently been developed. One approach is based on Hamilton's least action principle in Eulerian curvilinear coordinates and the other is based on the covariance upon time-dependent changes of spatial coordinates. The variational approach unifies, facilitates and generalizes the formulation of a wide range of approximations. The covariant approach makes it straightforward to rewrite approximate equations of motion derived in a particular coordinate system in another coordinate system.The variational approach encompasses models that, like the β-plane and non-traditional shallow-atmosphere approximations, lack a global inertial frame, so that the resulting motion cannot be interpreted as Newtonian motion observed in a non-inertial frame. It has been previously suggested that such models are not covariant.In this work a covariant variational formulation is developed. As a consequence, it is shown that all models previously obtained by the variational approach are covariant upon time-dependent changes of spatial coordinates.

Energy‐conserving finite‐difference schemes for quasi‐hydrostatic equations

“…Hence the NTEs system lacks an inertial frame. This peculiarity is discussed in more depth in recently submitted articles, and we refer the interested reader to them (Staniforth and White, 2015;T. Dubos, 2015;personal communication).…”

“…Hence the NTEs system lacks an inertial frame. This peculiarity is discussed in more depth in recently submitted articles, and we refer the interested reader to them (Staniforth and White, ; T. Dubos, ; personal communication). Here we shall only emphasize the fact that the absence of an inertial frame does not prevent the NTEs and their energy being formulated and discretized in a general vertical coordinate, in contrast to concerns recently raised by Charron and Zadra ().…”

Covariant Structure of Models of Geophysical Fluid Motion

“…Therefore, inertial frames do not exist. If geophysical fluid dynamics are not strictly equivalent to Newtonian hydrodynamics observed in a rotating frame [1], what kind of dynamics are they? In order to discuss similarities and differences between physical models, it is useful to characterize their formal structure.…”

“…In addition to their variational structure, Newtonian hydrodynamics possess a space-time covariant formulation closely parallel to that of relativistic hydrodynamics, based on the Milne-Cartan structure of space-time [4][5][6]. Geophysical models have been formulated from a least action principle [7][8][9] and, recently, as a space-time covariant mass-momentum budget involving a connection generalizing the Milne-Cartan structure [1]. However, this mass-momentum budget and space-time connection could not be related to a space-time invariant action, as in relativistic hydrodynamics [6].…”

“…the Euler-Lagrange equations ðd=dtÞðH μν u ν Þ¼ 1 2 u α u β ∂ μ H αβ yield the Newtonian free-fall equations in a Cartesian, inertial frame ðt; x; y; zÞ in the presence of a gravitational field Vðx; y; z; tÞ [1].…”