Deriving consistent approximate models of the global atmosphere using Hamilton's principle

Abstract

A quartet of dynamically consistent approximate models of the global atmosphere in non‐spherical coordinates has recently been developed, according to whether approximations of shallow and/or quasi‐hydrostatic nature are made. A model is considered to be dynamically consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity. The development of these approximate models involved detailed examination of conservation budgets to determine which terms to omit, modify, or retain in the equations for the three components of momentum, according to the type of approximation made.An alternative, complementary, derivation of this quartet of approximate models is developed herein using Hamilton's principle. With this approach, a single term in the Lagrangian is identified, which can then be optionally modified to apply the quasi‐hydrostatic approximation via an on–off switch. As in the previous work, application of the shallow‐atmosphere approximation is achieved by choosing an orthogonal curvilinear coordinate system having an appropriate shallow metric. Application of Hamilton's principle and Noether's theorem then intrinsically preserves the conservation principles for axial angular momentum, energy and potential vorticity. This obviates the need to examine conservation budgets to determine which terms to omit, modify, or retain in the equations for the three components of momentum; everything just falls into place via the application of variational calculus. The resulting unified equation set is then identical to that previously obtained by trial and error examination of conservation budgets.