Kinetic model

DriftKineticElectrostaticAdiabatic

Electrostatic drift-kinetic model for a single ion species with adiabatic electrons.

Overview

This model is an electrostatic drift-kinetic reduction for strongly

magnetized ions in a fixed magnetic equilibrium. Electrons are not evolved kinetically; instead they enter through the adiabatic response in the quasi-neutrality solve. The implementation supports control variates for the field solve.

Use cases

This model is appropriate for:

electrostatic drift-kinetic ion turbulence studies
strongly magnetized plasmas with adiabatic electrons
guiding-center PIC verification in realistic magnetic geometry
low-frequency regimes where full gyrophase resolution is unnecessary

Governing equations

PDEs solved by model:

Drift-kinetic equation:

\frac{\partial f}{\partial t} + [v_{∥} \frac{B ^{*}}{B _{∥}^{*}} + \frac{E ^{*} \times b _{0}}{B _{∥}^{*}}] \cdot \frac{\partial f}{\partial X} + [\frac{1}{ε} \frac{B ^{*}}{B _{∥}^{*}} \cdot E^{*}] \cdot \frac{\partial f}{\partial v _{∥}} = 0

Poisson equation:

- \nabla_{⊥} \cdot (\frac{n _{0}}{∣ B _{0} ∣ ^{2}} \nabla_{⊥} ϕ) + \frac{1}{ε} n_{0} (1 + \frac{1}{Z ε} \frac{1}{T _{0}} ϕ) = \frac{1}{ε} \int f B_{∥}^{*} d v_{∥} d μ

where $f (X, v_{∥}, μ, t)$ is the guiding center distribution and

E^{*} = - \nabla ϕ - ε μ \nabla∣ B_{0} ∣, B^{*} = B_{0} + ε v_{∥} \nabla \times b_{0}, B_{∥}^{*} = B^{*} \cdot b_{0}

The control variate method can be activated in the Poisson equation; if enabled, the following Poisson equation is solved:

- \nabla_{⊥} \cdot (\frac{n _{0}}{∣ B _{0} ∣ ^{2}} \nabla_{⊥} ϕ) + \frac{1}{Z ε ^{2}} \frac{n _{0}}{T _{0}} ϕ = \frac{1}{ε} \int (f - f_{0}) B_{∥}^{*} d v_{∥} d μ

Normalization

The reference speed is the ion thermal speed and the electrostatic

fields are scaled accordingly:

v^= v^_{i}, E^= v^_{i} B^, ϕ^= E^x^.

The small parameter is $ε = 1/ (Ω^_{c} t^)$ .

Discretization

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

struphy.propagators.implicit_diffusion.ImplicitDiffusion
struphy.propagators.push_guiding_center_bx_estar.PushGuidingCenterBxEstar
struphy.propagators.push_guiding_center_parallel.PushGuidingCenterParallel

Diagnostics

The following scalars are tracked during simulation:

Field energy: en_phi
Guiding-center particle energy: en_particles
Total energy: en_tot

Example

Create and initialize the drift-kinetic adiabatic-electron model:

from struphy.models import DriftKineticElectrostaticAdiabatic

model = DriftKineticElectrostaticAdiabatic()
model.em_fields.phi

model.kinetic_ions.var