Van Kampen and Case Formalism Applied to Linear and Weakly Nonlinear Initial Value Problems in Unmagnetized Plasma

Lambert, A.J.D.; Best, R.W.B.; Sluijter, F. W.

doi:10.1002/ctpp.19820220202

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…e.g. Case 1959;Bernstein et al 1964;Roos 1969;Lambert et al 1981). Hence our i]f k<T (q) is indeed the Fourier transform of the van Kampen-Case eigenmode XkA v ) with respect to velocity.…”

Section: A B)mentioning

confidence: 61%

“…Prom (4.23) it can be seen by inspection that the oscillations corresponding to the Best pole are of this latter form, thus giving this pole the nature of a second-order singularity (see e.g. Lambert et al 1981).…”

Section: -"]«' H(t-t*)mentioning

confidence: 98%

“…Since f(it, t*) depends upon k x and k 2 only through their ratio, we can envisage two 'geometrical' or 'kinematic' situations (cf. Best 1974;Lambert et al 1981).…”

Section: N (J) P L) 1 W P Jj 2 (O P I) 3 Lcmentioning

confidence: 99%

“…The full potential of this approach is, however, far from being completely exploited. In this paper we intend to use it for the analysis of some nonlinear plasma oscillation phenomena related to the coupling of plasma waves and free-streaming perturbations like the echo phenomenon and the second-order waves found by Best (Best 1973(Best , 1974aLambert, Best & Sluijter 1981). We use second-order perturbation theory and start from the linear solutions of the Fourier-transformed set of equations that correspond to the van Kampen-Case continuum eigenmodes.…”

Section: Introductionmentioning

confidence: 99%

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Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Sedláček

Nocera

1992

J. Plasma Phys.

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We study the Vlasov–Poisson system of equations in Fourier-transformed velocity space. First we reformulate some results of the linear theory: in the new representation the van Kampen–Case eigenmodes are found to be ordinary functions with convenient continuity properties. We give a transparent derivation of the free-streaming temporal echo in terms of the kinematics of wave packets in Fourier-transformed velocity space. We further extend this analysis to include Coulomb interactions, which allows us to establish a connection between the echo theory, the second-order oscillations of Best and the phenomenon of linear side bands. The calculation of the time evolution of the global second-order electric field is performed in detail in the case of a Maxwellian equilibrium distribution function. We conclude that the phenomenon of linear side bands may be properly explained in terms of the intrinsic features of the equilibrium distribution function.

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“…e.g. Case 1959;Bernstein et al 1964;Roos 1969;Lambert et al 1981). Hence our i]f k<T (q) is indeed the Fourier transform of the van Kampen-Case eigenmode XkA v ) with respect to velocity.…”

Section: A B)mentioning

confidence: 61%

Section: -"]«' H(t-t*)mentioning

confidence: 98%

“…Since f(it, t*) depends upon k x and k 2 only through their ratio, we can envisage two 'geometrical' or 'kinematic' situations (cf. Best 1974;Lambert et al 1981).…”

Section: N (J) P L) 1 W P Jj 2 (O P I) 3 Lcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Sedláček

Nocera

1992

J. Plasma Phys.

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“…Moreover, for unmagnetized equilibrium or parallel propagation, the three electromagnetic polarization states are decoupled and can be analysed independently. Thus, most of the normal-mode-based research carried out so far has been limited to these unmagnetized equilibrium or parallel propagation situations (Van Kampen 1955; Pradhan 1957; Case 1959; Felderhof 1963 a , b ; Van Kampen & Felderhof 1967; Lambert, Best & Sluijter 1982; Ignatov 2017). The works of McCune (1966) and Watanabe (1968) did tackle the far more difficult problem of wave propagation oblique to a non-zero equilibrium magnetic field.…”

Section: Introductionmentioning

confidence: 99%

Normal-mode-based theory of collisionless plasma waves

Ramos

2019

J. Plasma Phys.

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The Van Kampen normal-mode method is applied in a comprehensive study of the linear wave perturbations of a homogeneous, magnetized and finite-temperature plasma, described by the collisionless Vlasov–Maxwell system in its non-relativistic version. The analysis considers a stable, Maxwellian background, but is otherwise completely general in that it allows for arbitrary wave propagation direction relative to the equilibrium magnetic field, multiple plasma species and general polarization states of the perturbed electromagnetic fields. A convenient formulation is introduced whereby the generator of the time advance is a Hermitian operator, analogous to the Hamiltonian in the Schrödinger equation of quantum mechanics. This guarantees a real frequency spectrum and complete bases of normal modes. Expansions in these normal-mode bases yield immediately the solutions of initial-value problems for general initial conditions. With standard initial conditions and propagation direction parallel to the equilibrium magnetic field, all the familiar results obtained following Landau’s Laplace transform approach are recovered. Considering such parallel propagation, the present work shows also explicitly and provides an example of how to construct special initial conditions that result in different, damped but otherwise arbitrarily prescribed time variations of the macroscopic variables. The known dispersion relations for perpendicular propagation are also recovered.

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The energy of an electrostatic plasma wave packet

Best

1984

Physica B+C

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Van Kampen and Case Formalism Applied to Linear and Weakly Nonlinear Initial Value Problems in Unmagnetized Plasma

Cited by 6 publications

References 18 publications

Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Normal-mode-based theory of collisionless plasma waves

The energy of an electrostatic plasma wave packet

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