Toy model

ToyDrift

Electrostatic drift toy model for a single ion species in a given background magnetic field.

Overview

ToyDrift is a stripped-down guiding-center model used to isolate the

electrostatic drift part of the dynamics. It is intended for algorithm prototyping and reduced verification problems rather than production drift-kinetic studies.

Use cases

This model is appropriate for:

reduced electrostatic guiding-center benchmarks
testing the field solve plus $E \times B$ pusher
algorithm prototyping before moving to the full drift-kinetic model

Governing equations

PDEs solved by model:

Drift equation:

\frac{\partial f}{\partial t} + \frac{E \times b _{0}}{B _{∥}^{*}} \cdot \frac{\partial f}{\partial X} = 0

Poisson equation:

- \nabla \cdot \nabla ϕ = \int f B_{∥}^{*} d v_{∥} d μ

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

E = - \nabla ϕ, 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 \nabla ϕ = \int (f - f_{0}) B_{∥}^{*} d v_{∥} d μ

Normalization

The reference speed is the ion thermal velocity:

v^= v^_{i}, E^= v^_{i} B^, ϕ^= E^x^.

Discretization

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

Diagnostics

The following scalars are tracked during simulation:

Electrostatic field energy: en_phi
Particle kinetic energy: en_particles
Total energy: en_tot

Examples

Diocotron instability

A sheared E×B ring of charge is unstable: a tiny azimuthal perturbation grows, rippling the ring's edges — the onset of a rotating pattern of vortices. The charge ring starts from $n(r,\theta,0)=\mathbf{1}_{[4,5)}(r)\,[1+10^{-6}\cos(4\theta)],$ inside conducting cylinders at $r=1$ and $r=10$ ; the indicator is one inside the ring and zero outside.

Vortex merger

Two blobs of charge of the same sign circle each other under E×B drift and merge into one, the guiding-centre analogue of the merger of two like-signed vortices. The initial density is $n(x,y,0)=3\sum_{s=\pm1}\exp\!\left[-\frac{(x-x_s)^2+(y-y_s)^2}{2(0.55)^2}\right],$ with blob centres $(x_s,y_s)=(\sqrt{3.5^2-1},s)$ , separated by two length units.