Kinetic model

LinearVlasovAmpereOneSpecies

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

Overview

LinearVlasovAmpereOneSpecies is the linearized delta-f counterpart of

VlasovAmpereOneSpecies. It is intended for weakly perturbed kinetic problems around a prescribed Maxwellian equilibrium and is especially useful for linear instability or damping studies.

Use cases

This model is appropriate for:

linear delta-f kinetic instabilities
weakly perturbed Landau-like damping studies with electrostatic fields
verification of linearized field-particle coupling

Governing equations

PDEs solved by model:

Linearized Ampère's law:

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

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

Normalization

The light speed defines the velocity scale:

v^= c, E^= B^v^, ϕ^= E^x^.

The species parameters are $α = Ω^_{p} / Ω^_{c}$ and $ε = 1/ (Ω^_{c} t^)$ .

Discretization

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

struphy.propagators.push_eta.PushEta
struphy.propagators.push_vin_efield.PushVinEfield (if with_E0 is True)
struphy.propagators.efield_weights_coupling.EfieldWeightsCoupling
struphy.propagators.push_vxb.PushVxB (if with_B0 is True)

Diagnostics

The following scalars are tracked during simulation:

Electric field energy: en_E
Perturbation particle energy: en_w
Total energy: en_tot

Example

Create and initialize the linear Vlasov-Ampère model:

from struphy.models import LinearVlasovAmpereOneSpecies

model = LinearVlasovAmpereOneSpecies()
model.em_fields.e_field
model.em_fields.phi

model.kinetic_ions.var