  Quasilinear theory is often utilized to approximate the dynamics of fluids
exhibiting significant interactions between mean flows and eddies. In this
paper we present a generalization of quasilinear theory to include dynamic mode
interactions on the large scales. This generalized quasilinear (GQL)
approximation is achieved by separating the state variables into large and
small zonal scales via a spectral filter rather than by a decomposition into a
formal mean and fluctuations. Nonlinear interactions involving only small zonal
scales are then removed. The approximation is conservative and allows for
scattering of energy between small-scale modes via the large scale (through
non-local spectral interactions). We evaluate GQL for the paradigmatic problems
of the driving of large-scale jets on a spherical surface and on the beta-plane
and show that it is accurate even for a small number of large-scale modes. As
this approximation is formally linear in the small zonal scales it allows for
the closure of the system and can be utilized in direct statistical simulation
schemes that have proved an attractive alternative to direct numerical
simulation for many geophysical and astrophysical problems.
