Kinetic model

GuidingCenter

Guiding-center equation for a single species in a static background magnetic field.

Overview

GuidingCenter is the reduced kinetic model used when fast gyromotion is

averaged out and only guiding-center motion needs to be resolved. It evolves particles in a fixed magnetic background without any self-consistent field solve.

Use cases

This model is appropriate for:

test-particle guiding-center orbit calculations
strongly magnetized plasmas with scale separation from gyromotion
verification of 5D guiding-center pushers and diagnostics

Governing equations

PDEs solved by model:

Guiding-center Vlasov 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

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

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

Normalization

The reference speed is Alfvénic and the key small parameter is the

normalized cyclotron time:

v^= v^_{A}, ε = 1/ (Ω^_{c} t^) .

Discretization

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

Diagnostics

The following scalars are tracked during simulation:

Parallel kinetic energy: en_fv
Magnetic-moment energy contribution: en_fB
Total particle energy: en_tot
Lost markers: n_lost_particles

Examples

Guiding-center orbits in a tokamak

Guiding centers with different pitch angles follow the field of a circular tokamak: particles with large parallel velocity circle the magnetic axis; others are reflected on the high-field side and bounce in banana-shaped orbits. At the logical launch point $\boldsymbol{\eta}_0=(0.5,0,0)$ , the markers have $v_{\parallel,0}=3\xi,\qquad \mu=\frac{9(1-\xi^2)}{2B_{\mathrm{start}}},$ with pitch $\xi\in\{\pm0.25,\pm0.5,\pm0.75,\pm0.9\}$ and $B_{\mathrm{start}}=|\mathbf{B}(\boldsymbol{\eta}_0)|$ .

Guiding-center orbits in a GVEC stellarator

GVEC creates a five-field-period stellarator equilibrium, then Struphy follows passing and mirror-trapped guiding centers in its three-dimensional magnetic field. At logical position $\boldsymbol{\eta}_0=(0.55,0,0)$ , each marker starts with $v_{\parallel,0}=1.5\xi,\qquad \mu=\frac{1.5^2(1-\xi^2)}{2B_{\mathrm{start}}},$ using pitches $\xi\in\{-0.85,-0.5,-0.2,0.1,0.2,0.5\}$ and the local field strength $B_{\mathrm{start}}$ .