Cumulant expansions for atmospheric flows

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 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.

Is spontaneous generation of coherent baroclinic flows possible?

“…Analysis of the SSD and the resulting instability is only possible through a closure assumption for the dynamics, as a straightforward calculation leads to an infinite hierarchy of equations for the moments and is therefore intractable (Hopf 1952;Kraichnan 1964;Frisch 1995). There is now a large number of studies of barotropic turbulence (Farrell & Ioannou 2007;Marston 2010;Srinivasan & Young 2012), shallow-water turbulence (Farrell & Ioannou 2009a), baroclinic turbulence (DelSole 1996;Farrell & Ioannou 2008Marston et al 2016), turbulence in pipe flows (Constantinou et al 2014b;Farrell et al 2016, turbulence in a convectively unstable flows (Herring 1963;Fitzgerald & Farrell 2014;Ait-Chaalal et al 2016) and turbulence in plasma and astrophysical flows (Farrell & Ioannou 2009b;Tobias et al 2011;Parker & Krommes 2013) providing evidence that whenever there is a coherent flow coexisting with the turbulent field, the SSD can be accurately captured by a second-order closure. Such closures of the SSD are either termed Stochastic Structural Stability Theory (S3T) (Farrell & Ioannou 2003) or second-order cumulant expansion (CE2) (Marston et al 2008).…”

Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation

“…If one projects Eq. (3.7) on the double-physical coordinate space (x, x ) using multiplication by t, x | and | t, x , then one obtains the so-called CE2 equations (Farrell & Ioannou 2003Marston et al 2008;Srinivasan & Young 2012;Ait-Chaalal et al 2016). Likewise, if one projects Eq.…”

“…Notably, Eq. (2.1) with L D → ∞ also describes Rossby turbulence in planetary atmospheres (Farrell & Ioannou 2003Marston et al 2008;Srinivasan & Young 2012;Ait-Chaalal et al 2016). For isolated systems, where Q = 0, Eq.…”

Direct statistical simulation of jets and vortices in 2D flows

“…Here we define the cumulants in terms of zonal averages over the x-direction (see Refs. 6,7,[18][19][20] as opposed to ensemble averages [21][22][23] . Thus…”