  We study 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. We extend two existing Galerkin approaches (A and B) and
develop a new Galerkin approximation (C). Approximation A, due to
\cite{flierl1978}, represents the streamfunction as a truncated Galerkin series
and defines the potential vorticity (PV) that satisfies the inversion problem
exactly. Approximation B, due to \cite{tulloch_smith2009b}, 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. We 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 have superior accuracy
than A at high wavenumbers. Because B lacks conservation of energy, we
recommend approximation C for constructing solutions to the surface-active QG
equations using Galerkin series with standard vertical modes.
