Generalized quasilinear approximation of the interaction of convection and mean flows in a thermal annulus

Tobias, Steven M.; Oishi, Jeffrey S.; Marston, J. B.

doi:10.1098/rspa.2018.0422

Cited by 8 publications

(17 citation statements)

References 26 publications

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“…While its development has been motivated by the study of turbulent flows in astrophysics and geophysics, Dedalus is capable of solving a much broader range of PDEs. To date, it has been used for applications and publications in applied mathematics [13][14][15][16], astrophysics [17][18][19][20][21][22][23][24][25][26][27][28][29][30], atmospheric science [31][32][33][34][35], biology [36][37][38], fluid dynamics [39][40][41][42][43][44][45][46][47], glaciology [48], numerical analysis [49][50][51][52], oceanography [53][54][55], planetary science [56,57], and plasma physics [58][59][60].…”

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confidence: 99%

Dedalus: A flexible framework for numerical simulations with spectral methods

Burns

Vasil

Oishi

et al. 2020

Phys. Rev. Research

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Numerical solutions of partial differential equations enable a broad range of scientific research. The Dedalus Project is a flexible, open-source, parallelized computational framework for solving general partial differential equations using spectral methods. Dedalus translates plain-text strings describing partial differential equations into efficient solvers. This paper details the numerical method that enables this translation, describes the design and implementation of the codebase, and illustrates its capabilities with a variety of example problems. The numerical method is a first-order generalized tau formulation that discretizes equations into banded matrices. This method is implemented with an object-oriented design. Classes for spectral bases and domains manage the discretization and automatic parallel distribution of variables. Discretized fields and mathematical operators are symbolically manipulated with a basic computer algebra system. Initial value, boundary value, and eigenvalue problems are efficiently solved using high-performance linear algebra, transform, and parallel communication libraries. Custom analysis outputs can also be specified in plain text and stored in self-describing portable formats. The performance of the code is evaluated with a parallel scaling benchmark and a comparison to a finite-volume code. The features and flexibility of the codebase are illustrated by solving several examples: the nonlinear Schrodinger equation on a graph, a supersonic magnetohydrodynamic vortex, quasigeostrophic flow, Stokes flow in a cylindrical annulus, normal modes of a radiative atmosphere, and diamagnetic levitation. The Dedalus code and the example problems are available online at http://dedalus-project.org/. CONTENTS

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confidence: 99%

Dedalus: A flexible framework for numerical simulations with spectral methods

Burns

Vasil

Oishi

et al. 2020

Phys. Rev. Research

Self Cite

354

231

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“…For additional information on the geometry, see Tobias et al. (2018, particularly figure 1). The system is non-dimensionalised with the width of the annulus in the -direction () such that , the viscous time scale and the temperature difference between the inner and outer walls .…”

Section: Model Equations Formulation Parity and Numerical Methodsmentioning

confidence: 99%

“…Thus, the Busse annulus system presents an important challenge for CE2, not least because it is known to host multiple solutions at modest Rayleigh numbers (Brummell & Hart 1993). Moreover, it has been shown (Tobias, Oishi & Marston 2018) that the simplest quasilinear dynamical theory can perform poorly in describing this system and only works well when the definition of mean fields is generalised to large-scale modes via the so-called generalised quasilinear (GQL) approximation. How does a quasilinear statistical theory fare when faced with multiple potential solution basins?…”

Section: Introductionmentioning

confidence: 99%

Direct statistical simulation of the Busse annulus

et al. 2022

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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 quasilinear 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.

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“…The utility of the GQL approximation compared with that of QL has been tested on a number of paradigm turbulent fluid problems and MHD. These include the stochastic driving of jets on a spherical surface and β-plane (Marston et al 2016), three dimensional plane Poiseuille and rotating Couette flow (Kellam 2019, Hernández et al 2022a,b, Tobias & Marston 2017, convectively driven zonal flows in a rotating annulus (Tobias et al 2018) and the helical magnetorotational instability that is crucial to angular momentum transport in disks (Child et al 2016). Depending on the nature of the problem -in particular the degree of non-normality of the linear operator (see below) -the QL and GQL approximations may perform well or poorly in describing the statistics of the full system.…”

Section: Omittedmentioning

confidence: 99%

Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence

Marston¹,

Tobias²

2022

Preprint

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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.

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Generalized quasilinear approximation of the interaction of convection and mean flows in a thermal annulus

Cited by 8 publications

References 26 publications

Dedalus: A flexible framework for numerical simulations with spectral methods

Dedalus: A flexible framework for numerical simulations with spectral methods

Direct statistical simulation of the Busse annulus

Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence

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