Kinetic model

LinearVlasovMaxwellOneSpecies

Linearized Vlasov-Maxwell equations for one kinetic species around a Maxwellian background.

Overview

LinearVlasovMaxwellOneSpecies is the electromagnetic extension of the

linear delta-f Vlasov model. It is intended for weakly perturbed kinetic electromagnetic waves and instabilities around a prescribed equilibrium.

This model is appropriate for:

Linearized Ampère's law:

\frac{\partial E ~}{\partial t} = \nabla \times B ~ - \frac{α ^{2}}{ε} \int_{R^{3}} v f ~ d^{3} v

Linearized Faraday's law:

\frac{\partial B ~}{\partial t} = - \nabla \times E ~

Linearized Vlasov equation:

\frac{\partial f ~}{\partial t} + v \cdot \nabla f ~ + \frac{1}{ε} (E_{0} + v \times B_{0}) \cdot \frac{\partial f ~}{\partial v} = \frac{1}{v _{th}^{2} ε} E ~ \cdot v f_{0}

where $Z = - 1$ and $A = 1/1836$ for electrons. The background distribution function $f_{0}$ is a uniform Maxwellian

f_{0} = \frac{n _{0} ( x )}{( 2 π v _{th} ) ^{3}} exp (- \frac{∣ v ∣ ^{2}}{2 v _{th}^{2}})

and the background electric field has to verify the following compatibility condition between with background density

\nabla_{x} ln (n_{0} (x)) = \frac{1}{v _{th}^{2} ε} E_{0}

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss' law,

\int_{Ω} \nabla ψ^{⊤} \cdot \nabla ϕ d x E ~ (t = 0) = \frac{α ^{2}}{ε} \int_{Ω} \int_{R^{3}} ψ f ~ d^{3} v d x \forall ψ \in H^{1} = - \nabla ϕ (t = 0)

Moreover, it is assumed that

\int_{R^{3}} v f_{0} d^{3} v = 0

The normalization matches the linear Vlasov-Ampère model:

v^= c, E^= B^v^, ϕ^= E^x^.

The model retains the same $α$ and $ε$ parameters while adding magnetic perturbation dynamics.

Time integration is performed by the following propagators (in sequence):

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