INVITED: Slow manifold reduction for plasma science

Burby, J. W.; Klotz, Taylor

doi:10.1016/j.cnsns.2020.105289

Cited by 19 publications

(19 citation statements)

References 78 publications

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“…A goal of future work will be to apply our results to infinite dimensional systems, especially systems with slow manifolds such as ideal magnetohydrodynamics, Burby (2017), kinetic magnetohydrodynamics, Burby & Sengupta (2018) and Lorentz loop dynamics, Burby (2020). See Burby & Klotz (2020) for an in-depth discussion of the role of slow manifolds in plasma physics. Just as Cotter & Reich (2004) shows that the long-time persistence of quasigeostrophic balance in non-dissipative geophysical fluid flows may be explained by finding an appropriate adiabatic invariant, adiabatic invariants in these plasma-dynamical systems may explain subtle notions such as the persistence time scale for gyrotropy in strongly magnetized plasmas.…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

Burby

Squire

2020

J. Plasma Phys.

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While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. A less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

Burby

Squire

2020

J. Plasma Phys.

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“…The method has been used in particular in fluid dynamics for model-order reduction in different problems, see, e.g. [27,82,109,151,159,180,278], but also in unsteady magnetic dynamos [151] and plasma physics [21]. For conservative or near-conservative systems, a straightforward application of centre manifold is, however, more difficult due to the small (or vanishing) decay rates.…”

Section: Invariant Manifolds For Dynamical Systemsmentioning

confidence: 99%

“…This will be detailed next since it gives the first significant terms in the developments, that can be used for direct comparisons with other methods. In subsequent developments, They also propose to solve (21) numerically. This will be reviewed in Sect.…”

Section: Two-dimensional Invariant Manifoldmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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“…Mathematically, the formal invariant manifold underlying Vlasov-Poisson, Vlasov-Darwin and higher-order corrections thereof is an example of a slow manifold. The general theory of slow manifolds is reviewed in MacKay ( 2004) for a mathematical audience and in Burby (2020b) for an audience of plasma physicists. Since a slow manifold in a Hamiltonian system necessarily inherits a Hamiltonian structure, it follows immediately that the dark slow manifold in the Vlasov-Maxwell system has a natural Hamiltonian structure.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold

Miloshevich

Burby

2021

J. Plasma Phys.

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We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

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INVITED: Slow manifold reduction for plasma science

Cited by 19 publications

References 78 publications

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold

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