AbstractThis study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.
Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise