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In recent years, the Generalised Quasilinear (GQL) approximation has been developed and its efficacy tested against purely quasilinear (QL) approximations. GQL systematically interpolates between QL and fully non-linear dynamics by employing a generalised Reynolds decomposition. Here, we examine an exact statistical closure for the GQL equations on the doubly periodic $\beta$-plane. Closure is achieved at second order using a generalised cumulant approach which we term GCE2. GCE2 is shown to yield improved performance over statistical representations of purely QL dynamics (CE2) and thus enables Direct Statistical Simulation (DSS) of complex mean flows that do not entirely fall within the remit of pure QL theory. Despite the existence of an exact closure, GCE2 like CE2 admits the possibility of a rank instability that leads to differences with statistics obtained from GQL. Recognition of this instability is a necessary step before further progress can be made with the GCE2 statistical closure., Comment: 19 pages, 10 figures. Version to appear in JFM

Understanding inhomogeneous and anisotropic fluid flows require mathematical and computational tools that are tailored to such flows and distinct from methods used to understand the canonical problem of homogeneous and isotropic turbulence. We review some recent developments in the theory of inhomogeneous and anisotropic turbulence, placing special emphasis on several kinds of quasilinear approximations and their corresponding statistical formulations. Aspects of quasilinear theory that have received insufficient attention in the literature are discussed, and open questions are framed., Comment: Some added material and references and corrected typos. To appear in Annual Reviews of Fluid Mechanics. 29 pages, 7 figures. When citing this paper, please use the following: J. B. Marston and S. M. Tobias. 2022. "Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence," Ann. Rev. Fluid Mech. Vol. 55. Accepted. http://doi.org/10.1146/annurev-fluid-120720-031006

It is widely believed that statistical closure theories for dynamical systems provide statistics equivalent to those of the governing dynamical equations from which the former are derived. Here, we demonstrate counterexamples in the context of the widely used mean-field quasilinear (QL) approximation applied to 2D fluid dynamical systems. We compare statistics of QL numerical simulations with those obtained by direct statistical simulation via a cumulant expansion closed at second order (CE2). We observe that, though CE2 is an exact statistical closure for QL dynamics, its predictions disagree with the statistics of the QL solution for identical parameter values. These disagreements are attributed to instabilities, which we term rank instabilities, of the second cumulant dynamics within CE2 that are unavailable in the QL equations., Comment: 12 pages, 6 figures

Direct Statistical Simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by Direct Numerical Simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low order statistics typically have higher dimension than the underlying fields. Here we show that it is possible to go much further by using Proper Orthogonal Decomposition (POD) to address the "curse of dimensionality." We apply POD directly to DSS in the form of expansions in the equal-time cumulants to second order (CE2). We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models on a rotating sphere (a jet that relaxes deterministically towards an unstable profile, and a stochastically-driven flow that spontaneously organizes into jets). Order-of-magnitude savings in computational cost are obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS., Comment: 17 pages and 11 figures. Corrections, improved figures, and new section on "Continuation in Parameter Space."

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. In this paper we present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through non-local spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta-plane and show that it is accurate even for a small number of large-scale modes. As this approximation is formally linear in the small zonal scales it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems., Comment: 5 pages and 5 figures

We review progress that has been made in utilizing one form of Direct Statistical Simulation (DSS) to describe geophysical and astrophysical flows that are anisotropic and inhomogeneous. We first explain the approach, which is based upon a systematic and conservative expansion of the equations of motion for low-order equal-time cumulants. We place the method into context with other statistical procedures. Truncation at second order in the hierarchy of cumulants is equivalent to retaining the interaction between zonal mean flows and eddies. Eddy-eddy interactions appear at higher orders, but care must be taken to keep the higher-order expansions realizable with non-negative probability distribution functions. The strengths and weaknesses of different levels of approximation are assessed with numerical experiments on the fiducial problem of a stochastically forced jet on a spherical surface. The results give an insight into the mechanisms that may control jet spacing and strength, and indicate interesting avenues for future research., Comment: 22 pages and 9 figures. Updated references, typos corrected, Figure 1 clearer