Equations of Atmospheric Motion in Non-Eulerian Vertical Coordinates: Vector-Invariant Form and Quasi-Hamiltonian Formulation

Abstract

The curl form of equations of inviscid atmospheric motion in general non-Eulerian coordinates is obtained. Narrowing down to a general vertical coordinate, a quasi-Hamiltonian form is then obtained in a Lagrangian, isentropic, mass-based or z-based vertical coordinate. In non-Lagrangian vertical coordinates, the conservation of energy by the vertical transport terms results from the invariance of energy under the vertical relabeling of fluid parcels. A complete or partial separation between the horizontal and vertical dynamics is achieved, except in the Eulerian case. The horizontal–vertical separation is especially helpful for (quasi-)hydrostatic systems characterized by vanishing vertical momentum. Indeed for such systems vertical momentum balance reduces to a simple statement: total energy is stationary with respect to adiabatic vertical displacements of fluid parcels. From this point of view the purpose of (quasi-)hydrostatic balance is to determine the vertical positions of fluid parcels, for which no evolution equation is readily available. This physically appealing formulation significantly extends previous work. The general formalism is exemplified for the fully compressible Euler equations in a Lagrangian vertical coordinate and a Cartesian (x, z) slice geometry, and the deep-atmosphere quasi-hydrostatic equations in latitude–longitude horizontal coordinates. The latter case, in particular, illuminates how the apparent intricacy of the time-dependent metric terms and of the additional forces can be absorbed into a proper choice of prognostic variables. In both cases it is shown how the quasi-Hamiltonian form leads straightforwardly to the conservation of energy using only integration by parts. Relationships with previous work and implications for stability analysis and the derivation of approximate sets of equations and energy-conserving numerical schemes are discussed.

DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility

“…Equations (29)- (32) imitate at the discrete level the Hamiltonian formulations obtained in Dubos and Tort (2014). Discrete conservation of energy then appears as a consequence of the antisymmetry of the brackets {F, H } µ ,…”

“…Gassmann (2013) uses the Hamiltonian formulation of the fully compressible equations in Eulerian coordinates. This Hamiltonian theory has been recently extended for compressible hydrostatic flows and for non-Eulerian vertical coordinates Dubos and Tort, 2014) and serves as the basis to formulate the discretization of dynamics in DYNAMICO.…”

“…A Hamiltonian formulation of the hydrostatic primitive equations in a generalized vertical coordinate is used (Dubos and Tort, 2014). From this formulation the energy budget is obtained invoking only integration by parts, a structure easy to reproduce at the discrete level in order to conserve energy.…”

“…where f (n, η) = f d 2 n with the unit sphere and v = a 2 (u + n × ) is prognostic (Dubos and Tort, 2014). In Eq.…”

“…Assuming a Lagrangian vertical coordinate, the additional integral quantities conserved by the discrete equations of motion are, for each layer, the horizontally integrated mass and potential temperature i µ ik , i ik , which form a subset of the Casimir invariants of the continuous equations (Dubos and Tort, 2014). Stationary points of the pseudo-energy H = H − k k i µ ik − k π k i ik are such that ∂H /∂v ek = 0 (state of rest), ∂H /∂ ik = π ik = π k and ∂H /∂µ ik = i k = k .…”