Atmospheric flows are governed by the equations of fluid dynamics. These equations are nonlinear, and 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 nontrivial 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. Here we show how truncations at second order can be used to model and elucidate the dynamics of turbulent 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. However, it does not capture important turbulent transport terms in the turbulence kinetic energy budget. Second, we study the evolution of two-dimensional large-scale waves, which are representative of waves seen in Earthʼs 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 strongly nonlinear eddyeddy interactions that generate small-scale filaments in surf zones around critical layers 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 angular momentum fluxes in the upper troposphere.

The extratropical eddy momentum flux (EMF) is controlled by generation, propagation, and dissipation of large-scale eddies and is concentrated in Earth's upper troposphere. An idealized GCM is used to investigate how this EMF structure arises. In simulations in which the poles are heated more strongly than the equator, EMF is concentrated near the surface, demonstrating that surface drag generally is not responsible for the upper-tropospheric EMF concentration. Although Earth's upper troposphere favors linear wave propagation, quasi-linear simulations in which nonlinear eddy-eddy interactions are suppressed demonstrate that this is likewise not primarily responsible for the upper-tropospheric EMF concentration. The quasi-linear simulations reveal the essential role of nonlinear eddy-eddy interactions in the surf zone in the upper troposphere, where wave activity absorption away from the baroclinic generation regions occurs through the nonlinear generation of small scales. In Earth-like atmospheres, wave activity that is generated in the lower troposphere propagates upward, then turns meridionally, eventually to be absorbed nonlinearly in the upper troposphere. The level at which the wave activity begins to propagate meridionally appears to be set by the typical height reached by baroclinic eddies. This can coincide with the tropopause height but also can lie below it if convection controls the tropopause height. In the latter case, EMF is maximal well below the tropopause. The simulations suggest that EMF is concentrated in Earth's upper troposphere because typical baroclinic eddies reach the tropopause.

The midwinter suppression of eddy activity in the North Pacific storm track is a phenomenon that has resisted reproduction in idealized models that are initialized independently of the observed atmosphere. Attempts at explaining it have often focused on local mechanisms that depend on zonal asymmetries, such as effects of topography on the mean flow and eddies. Here an idealized aquaplanet GCM is used to demonstrate that a midwinter suppression can also occur in the activity of a statistically zonally symmetric storm track. For a midwinter suppression to occur, it is necessary that parameters, such as the thermal inertia of the upper ocean and the strength of tropical ocean energy transport, are chosen suitably to produce a pronounced seasonal cycle of the subtropical jet characteristics. If the subtropical jet is sufficiently strong and located close to the midlatitude storm track during midwinter, it dominates the upper-level flow and guides eddies equatorward, away from the low-level area of eddy generation. This inhibits the baroclinic interaction between upper and lower levels within the storm track and weakens eddy activity. However, as the subtropical jet continues to move poleward during late winter in the idealized GCM (and unlike what is observed), eddy activity picks up again, showing that the properties of the subtropical jet that give rise to the midwinter suppression are subtle. The idealized GCM simulations provide a framework within which possible mechanisms giving rise to a midwinter suppression of storm tracks can be investigated systematically.

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic NavierStokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion κ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent λ with two times τ and τ . The time 1/λ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while τ measures it in the recent past before t, and τ in the early part of the trajectory. We show that the chemical speed scales like κ 1/2 and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.