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The equations governing atmospheric flows are nonlinear. Consequently, thehierarchy of cumulant equations is not closed. But because atmospheric flowsare inhomogeneous and anisotropic, the nonlinearity may manifest itself onlyweakly through interactions of mean fields with disturbances such as thermalsor eddies. In such situations, truncations of the hierarchy of cumulantequations hold promise as a closure strategy.  We review how truncations at second order can be used to model and elucidatethe dynamics of atmospheric flows. Two examples are considered. First, we studythe 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, inwhich interactions of disturbances among each other are neglected butinteractions with mean fields are taken into account, can capture the growth ofthe convective boundary layer even if it does not capture important turbulenttransport terms. Second, we study the evolution of two-dimensional large-scalewaves representing waves in Earth's upper atmosphere. We demonstrate that acumulant expansion truncated at second order (CE2) can capture the evolution ofsuch waves and their nonlinear interaction with the mean flow in somecircumstances, for example, when the wave amplitude is small enough or theplanetary rotation rate is large enough. However, CE2 fails to capture the flowevolution when nonlinear eddy--eddy interactions in surf zones becomeimportant. Higher-order closures can capture these missing interactions.  The results point to new ways in which the dynamics of turbulent boundarylayers may be represented in climate models, and they illustrate differentclasses of nonlinear processes that can control wave dissipation and momentumfluxes in the troposphere.

Cumulant expansions for atmospheric flows