Plasma physics · Viv posts

Particle motion

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Adam Zettel@adamSep 25, 2026

A gyro-orbit and the E×B drift by the Boris push

θ = 2 arctan (\frac{ω _{c} Δ t}{2}), v ˉ = \frac{E \times B}{B ^{2}}

The Boris push moves a charged particle through electric and magnetic fields in almost every particle-in-cell code. In a magnetic field alone it keeps the speed to rounding and puts every position on one circle; the gyroperiod comes out 6.28449 instead of $2 π$ , longer by $(ω_{c} Δ t)^{2} /12$ . With an electric field added, a particle started at $E \times B / B^{2}$ moves at exactly that velocity forever, and any other start averages to it.

Boris positions

Gyroperiod error of the Boris push

measured

Boris trajectory

✓ 7 tests passchecked when the site was built#plasma #particle-pushers #boris #gyration #drift #notebook

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Instabilities

demoAIPlasma physicswritten with an agent#d764b8f8e6

Adam Zettel@adamSep 25, 2026

The two-stream instability in a particle-in-cell code

γ_{m a x} = \frac{ω _{p}}{2 2} at k v_{0} = 3/8 ω_{p}

Two cold electron beams passing through each other are unstable. A particle-in-cell code with 2000 particles measures the growth rate of the fastest mode as 0.35366 $ω_{p}$ ; cold-beam linear theory gives $1/ (22) = 0.35355$ . The phase-space film shows the beams rolling up into vortices of trapped electrons, which is where the growth stops.

Phase space of the two beams (press play)

21/21

32 × 48 grid · 21 frames

Amplitude of the seeded mode

PIC, 2000 particles

✓ 9 tests passchecked when the site was built#plasma #particle-in-cell #instability #kinetic #animation #notebook

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ladderAIPlasma physicswritten with an agent#b7c5ad40e0

Adam Zettel@adamSep 25, 2026

The Hasegawa-Wakatani drift-wave model

ω = \frac{κ k _{y}}{1 + k _{⊥}^{2}}, \frac{d}{d t} (ζ n) = (- C / k^{2} - C / k^{2} + iκ k_{y} / k^{2} - C - C) (ζ n)

The Hasegawa-Wakatani equations are the simplest model of drift-wave turbulence at the edge of a tokamak that contains the instability driving it. Five rungs build their linear theory: the drift-wave frequency, the growth rate as the eigenvalue of a 2×2 matrix (0.098684 at $C = 1$ , as numpy's eigensolver gives), the return to the drift wave as $C \to \infty$ at the predicted rates $1/ C$ and $1/ C^{2}$ , a zonal flow that keeps $k^{2} / (1 + k^{2})$ of its start as it relaxes, and the one-line change in the modified model that leaves it untouched.

classical Hasegawa-Wakatani

The same band in the two models

classicalmodified

1The drift wave+1 function · +3 tests
2The growth rate is an eigenvalue+3 functions · +3 tests
3Stiff coupling gives back the drift wave+2 tests
4A zonal flow relaxes to k^2 / (1 + k^2) of its start+1 function · +2 tests
5The modified model leaves the band alone+1 function · +2 tests

✓ 12 tests passon 5 rungs, each rerunning every earlier test · checked when the site was built#plasma #drift-waves #turbulence #zonal-flows #eigenvalues #ladder

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Kinetic theory

demoAIPlasma physicswritten with an agent#827ef11264

Adam Zettel@adamSep 25, 2026

Landau damping from the root of the dispersion relation

1 + \frac{1 + ζ Z ( ζ )}{k ^{2} λ _{D}^{2}} = 0, ζ = \frac{ω}{2 k v _{t h}}

A Langmuir wave in a collisionless plasma still decays: electrons moving near its phase velocity take energy from it. Newton's method on the kinetic dispersion relation, with the plasma dispersion function from lib/zfunc, puts the least-damped root at $k λ_{D} = 0.5$ at $ω = 1.415662 - 0.153359 i$ (in units of $ω_{p e}$ ), matching a 30-digit mpmath root to $1 0^{- 12}$ . The usual small- $k$ damping formula is 1.3% off there and 59% off at $k λ_{D} = 0.3$ .

48 × 64 grid

Damping rate of the least-damped Langmuir wave

exact rootsmall-k formula

Frequency of the least-damped Langmuir wave

exact root

The field of a Langmuir wave at k lambda_D = 0.5, from the linearized Vlasov equation

Re E1Im E1decay at the root's rate

✓ 9 tests passchecked when the site was built#plasma #kinetic-theory #landau-damping #dispersion-relation #complex-roots #newton #notebook

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demoAIPlasma physicswritten with an agent#91420950b3

Adam Zettel@adamSep 25, 2026

The bounce period of an electron trapped in a plasma wave

T_{B} = \frac{4}{ω _{B}} K (sin^{2} \frac{θ _{0}}{2}), ω_{B} = k \frac{e ϕ _{0}}{m}

In the frame of a large-amplitude plasma wave, an electron near the wave's phase velocity moves like a pendulum. Released 0.1 radian from the bottom of a trough, it takes $6.287115$ time units to bounce back, which is $4 K (sin^{2} 0.05)$ from the complete elliptic integral, and $2 π (1 + θ_{0}^{2} /16)$ to six digits. Electrons with speed below $2 ω_{B} / k$ at the trough stay in it; faster ones pass over the crests.

velocity Verlet, dt = 0.002

orbits from the trough

✓ 9 tests passchecked when the site was built#plasma #nonlinear-waves #trapping #landau-damping #pendulum #elliptic-integral

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Equilibrium

demoAIPlasma physicswritten with an agent#ff0590fe04

Adam Zettel@adamSep 25, 2026

Grad-Shafranov by finite differences, against Solov'ev's closed form

Δ^{*} ψ \equiv R \frac{\partial}{\partial R} (\frac{1}{R} \frac{\partial ψ}{\partial R}) + \frac{\partial ^{2} ψ}{\partial Z ^{2}} = - μ_{0} R^{2} p^{'} (ψ) - F F^{'} (ψ)

A tokamak equilibrium solved on a grid and graded against an exact one. Solov'ev's $ψ = (R^{2} - 1)^{2} /8 + R^{2} Z^{2} / (2 κ^{2})$ solves the Grad-Shafranov equation with constant pressure gradient; the five-point solver, relaxed by SOR until the residual is below $1 0^{- 11}$ , misses it by $1.68 \times 1 0^{- 4}$ , $4.21 \times 1 0^{- 5}$ and $1.06 \times 1 0^{- 5}$ on grids of spacing 0.1, 0.05 and 0.025, an observed order of 1.995 and then 1.994. Those errors are scipy's direct solve of the same equations, to $1 0^{- 10}$ .

Grad-Shafranov error against the Solov'ev solution

five-point difference equations, solved by SORslope 2

57 × 33 grid

✓ 7 tests passchecked when the site was built#plasma #mhd #tokamak #equilibrium #grad-shafranov #finite-differences #sor

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