Stellarator coils and field lines

How do three-dimensional coils twist a field, and what do repeated crossings of one plane reveal? Three-dimensional coils supply both toroidal and poloidal field components, so a field line winds around the tube as it follows the large ring. Repeated crossings of one plane reveal its finite computed pattern. Estimating rotational transform also requires a continuous angle about the moving magnetic axis and checks of step size and trace length.

Compare two Wendelstein 7-X current references, follow computed field lines, and inspect fixed-plane crossings and finite winding about the moving axis.

Choose currents and a starting point

Numerical current programme

0

Start outward from axis

0.05\,\mathrm{m}

Start above axis

0\,\mathrm{m}

Requested full toroidal turns

64

Non-planar

1.62\,\mathrm{MA}

Planar

0\,\mathrm{MA}

Effective currents include winding turns. New currents need a newly computed axis and trace.

Numerical window

Numerical window half-height

1.4\,\mathrm{m}

This window is an integration limit, not a material wall. The bounded current programme joins two source references; it is a numerical input family.

Model assumptions

A traceable W7-X CAD single-filament model supplies the prescribed-current vacuum field; it is not an as-built coil survey. Precomputed references retain actual crossings and numerical checks. New currents require a new axis and trace. Finite winding does not prove a unique surface or predict particle transport, wall heat or fusion power. Displayed wire thickness does not enter the field calculation, and integration limits are not material walls.

Return plotGlobal plane

Continuous winding about the computed moving axis

How to read the numerical record

Successive visits to one global half-plane are a full toroidal circuit apart. When the seed is off that plane, the first hit is an entry, not a completed turn from the seed. The field-period overlay uses five physical planes. Winding is normalized by actual signed toroidal angle.

The finite winding is about the computed axis at each position. Point clouds and short trajectories do not automatically identify islands, chaos or an invariant magnetic surface. Imported claims of qualification are checked against the complete known reference payload.

Finite numerical checks

Refinement differences and independent-solver comparisons describe finite numerical checks. They are not Gaussian uncertainties or a proof of infinite-time regularity.

Source geometry and display scales

All 70 main coils use the traceable single-filament CAD source. Cyan denotes non-planar coils, copper planar coils; planar coils remain visible at zero current. Conductor thickness and gold controls are illustrative. Right-handed SI coordinates have Z upward; hardware uses one uniform display scale.

The gold moving point replays accepted drawing states, with no physical flight-time meaning. White dots show actual computed-axis section positions. The transparent plane is mathematical. No vessel, divertor or particle-transport geometry is supplied.

Field arrows use a separate drawing scale: $1\,\mathrm{T}\longmapsto 0.12\,\mathrm{m}$

IPP · Stellarator Simsopt · W7-X source geometry Smiet et al. · Field-line sections

Retained crossings and stopping reason

Event	$R\;/\;\mathrm{m}$	$Z\;/\;\mathrm{m}$	$\phi\;/\;\mathrm{rad}$	Arc length	Base physical plane

Save and reopen numerical records

Saved record

Import complete numerical history

The archive retains actual source settings, axis solve, crossing events, continuous angles and stopping reasons. A new import replaces this session only after complete validation and derivation.

Open an empty session

Read how three-dimensional coils create a twisting field

Focus question: How do current ratios, starting position and tracing length change an actual section record?
One-sentence intuition: The line and crossings use the same coil field. Continuous signed angles about the moving computed axis preserve the winding direction.

Core mathematical model

Add the source-coil fields

\mathbf B(\mathbf x)=\frac{\mu_0}{4\pi}\sum_j I_j\oint\frac{d\boldsymbol\ell_j\times(\mathbf x-\mathbf r_j)}{|\mathbf x-\mathbf r_j|^3}

The source supplies each filament curve and effective current, already including winding turns. Illustrative conductor thickness never enters the field calculation.

Integrate in that same field

\frac{dR}{d\phi}=R\frac{B_R}{B_\phi},\qquad\frac{dZ}{d\phi}=R\frac{B_Z}{B_\phi}

R is distance from the central vertical axis, Z is height and φ advances around the ring. Events are located on actual global half-planes. A small toroidal component or numerical limit stops the calculation while retaining completed events.

Retain signed continuous winding

\iota_N=\frac{\Delta\theta_{\mathrm{unwrapped}}}{\Delta\phi},\qquad N=\frac{\Delta\phi}{2\pi}

θ is measured about the moving computed axis, from outward radial towards upward. Positive φ is counterclockwise viewed from above. N is signed full toroidal turns. A finite winding estimate alone does not establish a unique invariant magnetic surface.

Common difficulties

Is the moving dot a particle?

Typical misconceptionThe marker follows an electron orbit and therefore measures particle loss.

Better mental modelIt replays numerical drawing points in order. Real particles also gyrate, drift and collide; the field-line calculation provides no loss fraction.

Must the field line close?

Typical misconceptionA closed section curve requires a line to return to its starting point after a few turns.

Better mental modelThe curve can be the slice of a magnetic surface. One line may wind over that surface without closing exactly on itself.

Run the experiment

01
Compare traceable current references
Choose Standard reference and Author-code reference. Inspect the effective non-planar and planar currents.
What to observe: Both use the same 70 source coils. The default loads a numerical reference with recorded checks; computing again creates a separate single-run record.
02
Let new currents produce a new line
Drag the gold current handle on a coil, then compute the trace.
What to observe: The quick arrows are three actual field probes at the new currents. A new axis and crossings require integration; old paths are never reshaped by the slider.
03
Compare the two outer starting points
Compare Outer starting point and Author outer start in the section view.
What to observe: Every point is a retained crossing event. When the toroidal field becomes too small, the toroidal-angle method stops and preserves earlier crossings. This is a numerical stop, not particle loss.
04
Compare section conventions
Switch between one global plane and the five repeating sections. Inspect event counts and angle labels.
What to observe: The default reference completes 64 full turns: 320 crossings of five physical half-planes spaced a fifth of a turn apart are overlaid by rotational symmetry, with 64 returns to the same base plane. If the seed is off the chosen plane, the first hit is an entry. Winding uses actual signed toroidal angle, with no further factor of five.
05
Inspect short and terminated records
Choose One full turn and Restricted numerical window.
What to observe: Completed crossings and stopping reasons remain available. The window limits integration; leaving it is not a particle hitting a wall.

Stellarator coils and field lines

Follow a magnetic line from its source coils

Core mathematical model

Add the source-coil fields

Integrate in that same field

Retain signed continuous winding

Common difficulties

Is the moving dot a particle?

Must the field line close?

Run the experiment

Compare traceable current references

Let new currents produce a new line

Compare the two outer starting points

Compare section conventions

Inspect short and terminated records

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