Hermitian structure for linearized ideal MHD equations with equilibrium flows

Brizard, Alain J.

doi:10.1016/0375-9601(92)90518-q

Cited by 12 publications

(17 citation statements)

References 25 publications

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“…Note also that (29) agrees with Eq. (22) of Brizard,9 when that expression is reduced to one spatial dimension, and zero magnetic field. Now introduce the following variational principle (the overall factor of 1/2 will ensure that the canonical momentum is equal to the physical momentum density.…”

Section: A Acoustic Wavesmentioning

confidence: 99%

“…9. We use a cold ideal fluid model in one spatial dimension, x, which has as dependent variables the density, qðx; tÞ, the velocity, v(x, t), and the pressure p(x, t).…”

Section: One-dimensional Hydrodynamicsmentioning

confidence: 99%

“…This important fact is a general characteristic of Brizard's theory. 9 To isolate the tunneling process as much as possible, we need to bring the symbol matrix (5) into normal form. Then, the 2 Â 2 operator associated with that symbol will provide us with the simplest local wave equation, valid in the immediate vicinity of the event horizon.…”

Section: Heuristic Treatmentmentioning

confidence: 99%

“…This will allow us to use standard methods to derive conservation laws and will lead us to a Hamiltonian formulation of the linearized wave equation. We follow Brizard 9 for this part of the paper, and note that the symplectic inner product that plays an important role in our work is also invoked in Ref. 4, but they do not use a ray phase space approach.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the variational methodology used here was originally introduced by Brizard 9 to study linearized MHD waves in the presence of flows. But we postpone consideration of those matters because of the greater complexity of the problem if we include magnetic fields.…”

Section: Introductionmentioning

confidence: 99%

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Hawking radiation and classical tunneling: A ray phase space approach

Tracy

Zhigunov

2016

Physics of Plasmas

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Acoustic waves in fluids undergoing the transition from sub-to supersonic flow satisfy governing equations similar to those for light waves in the immediate vicinity of a black hole event horizon.This acoustic analogy has been used by Unruh and others as a conceptual model for 'Hawking radiation.' Here we use variational methods, originally introduced by Brizard for the study of linearized MHD, and ray phase space methods, to analyze linearized acoustics in the presence of background flows. The variational formulation endows the evolution equations with natural Hermitian and symplectic structures that prove useful for later analysis. We derive a 2 × 2 normal form governing the wave evolution in the vicinity of the 'event horizon.' This shows that the acoustic model can be reduced locally (in ray phase space) to a standard (scalar) tunneling process weakly coupled to a unidirectional non-dispersive wave (the 'incoming wave'). Given the normal form, the Hawking 'thermal spectrum' can be derived by invoking standard tunneling theory, but only by ignoring the coupling to the incoming wave. Deriving the normal form requires a novel extension of the modular ray-based theory used previously to study tunneling and mode conversion in plasmas. We also discuss how ray phase space methods can be used to change representation, which brings the problem into a form where the wave functions are less singular than in the usual formulation, a fact that might prove useful in numerical studies.

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Section: A Acoustic Wavesmentioning

confidence: 99%

“…9. We use a cold ideal fluid model in one spatial dimension, x, which has as dependent variables the density, qðx; tÞ, the velocity, v(x, t), and the pressure p(x, t).…”

Section: One-dimensional Hydrodynamicsmentioning

confidence: 99%

Section: Heuristic Treatmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hawking radiation and classical tunneling: A ray phase space approach

Tracy

Zhigunov

2016

Physics of Plasmas

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On the relation between pseudo-Hermiticity and dissipation

Brizard

1994

Physics Letters A

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An action principle for the Vlasov equation and associated Lie perturbation equations. Part 2. The Vlasov–Maxwell system

Larsson

1993

J. Plasma Phys.

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An action principle for the Vlasov–Maxwell system in Eulerian field variables is presented. Thus the (extended) particle distribution function appears as one of the fields to be freely varied in the action. The Hamiltonian structures of the Vlasov–Maxwell equations and of the reduced systems associated with small-ampliltude perturbation calculations are easily obtained. Previous results for the linearized Vlasov–Maxwell system are generalized. We find the Hermitian structure also when the background is time-dependent, and furthermore we may now also include the case of non-Hamiltonian perturbations within the Hamiltonian-Hermitian context. The action principle for the Vlasov–Maxwell system appears to be suitable for the derivation of reduced dynamical equations by expanding the action in various small parameters.

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Hermitian structure for linearized ideal MHD equations with equilibrium flows

Cited by 12 publications

References 25 publications

Hawking radiation and classical tunneling: A ray phase space approach

Hawking radiation and classical tunneling: A ray phase space approach

On the relation between pseudo-Hermiticity and dissipation

An action principle for the Vlasov equation and associated Lie perturbation equations. Part 2. The Vlasov–Maxwell system

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