Search

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

This Special Topic focuses on magnetohydrodynamic (MHD) processes in deep interiors of planets, in which their fluid dynamos are in operation. The dynamo-generated, global, magnetic fields provide a background for our solar-terrestrial environment. Probing the processes within the dynamos is a significant theoretical and computational challenge and any window into interior dynamics greatly increases our understanding. Such a window is provided by exploring rapid dynamics, particularly MHD waves about the dynamo-defined basic state. This field is the subject of current attention as geophysical observations and numerical modellings advance. We give here particular attention to torsional Alfv\'{e}n waves/oscillations and magnetic Rossby waves, which may be regarded as typical axisymmetric and nonaxisymmetric modes, respectively, amongst a wide variety of wave classes of the rapidly-rotating MHD fluids. The excitation of those waves is being evidenced for the geodynamo, whilst also being suggested for Jupiter. We shall overview their dynamics, summarise our current understanding, and give open questions for future perspectives., Comment: 26 pages, 12 figures, accepted for publication in Reviews of Modern Plasma Physics

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

We consider direct statistical simulation (DSS) of a paradigm system of convection interacting with mean flows. In the Busse Annulus model zonal jets are generated through the interaction of convectively driven turbulence and rotation; non-trivial dynamics including the emergence of multiple jets and bursting `predator-prey' type dynamics can be found. We formulate the DSS by expanding around the mean flow in terms of equal-time cumulants and arrive at a closed set of equations of motion for the cumulants. Here, we present results using an expansion terminated at the second cumulant (CE2); it is fundamentally a quasi-linear theory. We focus on particular cases including bursting and bistable multiple jets and demonstrate that CE2 can reproduce the results of direct numerical simulation if particular attention is given to symmetry considerations., Comment: 11 pages, 6 figures

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