Dimensional reduction of direct statistical simulation

Allawala, Altan; Tobias, Steven M.; Marston, J. B.

doi:10.1017/jfm.2020.382

Cited by 21 publications

(20 citation statements)

References 43 publications

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“…On the other hand, GQL appears to contain enough nonlinearity to represent the complicated nonlinear relaxation oscillation. This result is encouraging for a program of direct statistical simulation based on GQL or upon DSS formulations predicated on more sophisticated averaging procedures than simple zonal averaging (Bakas & Ioannou 2013, 2014; Allawala, Tobias & Marston 2017).…”

Section: Discussionmentioning

confidence: 76%

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

2019

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Global magnetohydrodynamic (MHD) instabilities are investigated in a computationally tractable two-dimensional model of the solar tachocline. The model's differential rotation yields stability in the absence of a magnetic field, but if a magnetic field is present, a joint instability is observed. We analyze the nonlinear development of the instability via fully nonlinear direct numerical simulation, the generalized quasilinear approximation (GQL), and direct statistical simulation (DSS) based upon low-order expansion in equaltime cumulants. As the magnetic diffusivity is decreased, the nonlinear development of the instability becomes more complicated until eventually a set of parameters are identified that produce a previously unidentified long-term cycle in which energy is transformed from kinetic energy to magnetic energy and back. We find that the periodic transitions, which mimic some aspects of solar variability-for example, the quasiperiodic seasonal exchange of energy between toroidal field and waves or eddies-are unable to be reproduced when eddy-scattering processes are excluded from the model.

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Section: Discussionmentioning

confidence: 76%

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

2019

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“…A study of rotating plane Couette flow [39] has provided an illustration of the ability of GQL to capture features that QL does not. Another type of mean flow for which the quasilinear approach can be used is the ensemble average [40]. Ensemble averaging, like temporal averaging, can also be combined with averaging over a homogeneous spatial direction as in Refs.…”

Section: Semilinear or Quasilinear Modelsmentioning

confidence: 99%

“…Other reduced-order models in which sets of modes or interactions are omitted have been proposed and implemented for many other hydrodynamic phenomena, notably in aeronautics and fluid mechanics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. and in geophysics and astrophysics [35][36][37][38][39][40]. Some of these models will be compared to RZIF and SCM in the next section.…”

Section: Introductionmentioning

confidence: 99%

Frequency prediction from exact or self-consistent mean flows

Bengana

Tuckerman

2021

Phys. Rev. Fluids

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A number of approximations have been proposed to estimate basic hydrodynamic quantities, in particular, the frequency of a limit cycle. One of these, real zero imaginary frequency (RZIF), calls for linearizing the governing equations about the mean flow and estimating the frequency as the imaginary part of the leading eigenvalue. A further reduction, the self-consistent model (SCM), approximates the mean flow as well, as resulting only from the nonlinear interaction of the leading eigenmode with itself. Both RZIF and SCM have proven dramatically successful for the archetypal case of the wake of a circular cylinder. Here, the SCM is applied to thermosolutal convection, for which a supercritical Hopf bifurcation gives rise to branches of standing waves and traveling waves. The SCM is solved by means of a full Newton method coupling the approximate mean flow and leading eigenmode. Although the RZIF property is verified for the traveling waves, the SCM reproduces the nonlinear frequency only very near the onset of the bifurcation and for another isolated parameter value. Thus, the nonlinear interaction arising from the leading mode is insufficient to reproduce the nonlinear mean field and frequency.

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“…On the other hand, the dynamics of the low-order statistics is usually slower than that of the original equations. In addition, it is possible to reduce the complexity of the problem by employing proper orthogonal decomposition (Allawala, Tobias & Marston 2017). This approach has not yet been applied to magnetohydrodynamics and the dynamo problem, but it has the potential of being a strong competitor in addressing the high Reynolds number dynamics of problems of astrophysical and geophysical relevance.…”

Section: Building Blocks Used In Modern Mean-field Theorymentioning

confidence: 99%

Advances in mean-field dynamo theory and applications to astrophysical turbulence

Brandenburg

2018

J. Plasma Phys.

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Recent advances in mean-field theory are reviewed and applications to the Sun, late-type stars, accretion disks, galaxies, and the early Universe are discussed. We focus particularly on aspects of spatio-temporal nonlocality, which provided some of the main new qualitative and quantitative insights that emerged from applying the test-field method to magnetic fields of different length and timescales. We also review the status of nonlinear quenching and the relation to magnetic helicity, which is an important observational diagnostic of modern solar dynamo theory. Both solar and some stellar dynamos seem to operate in an intermediate regime that has not yet been possible to model successfully. This regime is bracketed by antisolar-like differential rotation on one end and stellar activity cycles belonging to the superactive stars on the other. The difficulty in modeling this regime may be related to shortcomings in modelling solar/stellar convection. On galactic and extragalactic length scales, the observational constraints on dynamo theory are still less stringent and more uncertain, but recent advances both in theory and observations suggest that more conclusive comparisons may soon be possible also here. The possibility of inversely cascading magnetic helicity in the early Universe is particularly exciting in explaining the recently observed lower limits of magnetic fields on cosmological length scales. Such magnetic fields may be helical with the same sign of magnetic helicity throughout the entire Universe. This would be a manifestation of parity breaking. † Email address for correspondence: brandenb@nordita.org † In the following, we continue using the lowercase symbols u and b instead of U ′ and B ′ to denote fluctuations of U and B.

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Dimensional reduction of direct statistical simulation

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Cited by 21 publications

References 43 publications

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

Frequency prediction from exact or self-consistent mean flows

Advances in mean-field dynamo theory and applications to astrophysical turbulence

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