  The equations governing atmospheric flows are nonlinear. Consequently, the
hierarchy of cumulant equations is not closed. But because atmospheric flows
are inhomogeneous and anisotropic, the nonlinearity may manifest itself only
weakly through interactions of mean fields with disturbances such as thermals
or eddies. In such situations, truncations of the hierarchy of cumulant
equations hold promise as a closure strategy.
  We review how truncations at second order can be used to model and elucidate
the dynamics of atmospheric flows. Two examples are considered. First, we study
the growth of a dry convective boundary layer, which is heated from below,
leading to turbulent upward energy transport and growth of the boundary layer.
We demonstrate that a quasilinear truncation of the equations of motion, in
which interactions of disturbances among each other are neglected but
interactions with mean fields are taken into account, can capture the growth of
the convective boundary layer even if it does not capture important turbulent
transport terms. Second, we study the evolution of two-dimensional large-scale
waves representing waves in Earth's upper atmosphere. We demonstrate that a
cumulant expansion truncated at second order (CE2) can capture the evolution of
such waves and their nonlinear interaction with the mean flow in some
circumstances, for example, when the wave amplitude is small enough or the
planetary rotation rate is large enough. However, CE2 fails to capture the flow
evolution when nonlinear eddy--eddy interactions in surf zones become
important. Higher-order closures can capture these missing interactions.
  The results point to new ways in which the dynamics of turbulent boundary
layers may be represented in climate models, and they illustrate different
classes of nonlinear processes that can control wave dissipation and momentum
fluxes in the troposphere.
