Abstract
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 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 strongly nonlinear eddy–eddy 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.
Highlights
Atmospheric flows shape Earths climate and are governed by the equations of fluid dynamics, the Navier– Stokes equations augmented by the Coriolis force and thermodynamic equations (e.g., Ooyama 2001, Vallis 2006, Pauluis 2008), and equations for the microphysical processes describing, for example, the formation and re-evaporation of cloud droplets (Pruppacher et al 1998)
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
Both objectives require the development of closure models for the hierarchy of statistical moment or cumulant equations associated with the equations of fluid dynamics
Summary
Atmospheric flows shape Earths climate and are governed by the equations of fluid dynamics, the Navier– Stokes equations augmented by the Coriolis force and thermodynamic equations (e.g., Ooyama 2001, Vallis 2006, Pauluis 2008), and equations for the microphysical processes describing, for example, the formation and re-evaporation of cloud droplets (Pruppacher et al 1998). It may lead to insight into how climate is maintained and how it varies on timescales of seasons to millennia Both objectives require the development of closure models for the hierarchy of statistical moment or cumulant equations associated with the equations of fluid dynamics. Because the nonlinearity of turbulent interactions in many atmospheric flows may be limited, truncating the hierarchy of moment or cumulant equations at a low order has potential to be successful. QL approximations capture sheared stably stratified flows when the dynamics involve the linear excitation and absorption of internal gravity waves (Orr 1907, Lindzen 1988, Bakas and Ioannou 2007) They reproduce aspects of thermal convection, such as the dependence of the heat flux on the Rayleigh number (e.g., Malkus 1954, Herring 1963, Toomre et al 1977, Busse 1978, Niemela et al 2000). The results will demonstrate the potential and limitations of CE2 approaches
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