Cumulant expansions for atmospheric flows
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.
© 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. For paper published after November 2012, authors, their institutions and third parties all have the same rights to reuse articles published in New Journal of Physics in accordance with the Creative Commons Attribution 3.0 Unported (CC-BY) license. Published 19 February 2016. Collection: Focus on Stochastic Flows and Climate Statistics. We thank Kyle Pressel for helping in the development of the QL LES and Joe Skitka for sharing results about the oceanic boundary layer. We are grateful for helpful discussions with Freddy Bouchet, Greg Chini, Baylor Fox-Kemper, Cesare Nardini, and Steve Tobias. The comments of two anonymous reviewers improved the manuscript significantly. This work was supported by the US National Science Foundation under grants CCF-1048575 (FAC and TS) and CCF-1048701 (FAC and JBM).
Published - njp_18_2_025019.pdf
Submitted - 1505.07643v3.pdf