Magnetic Field-Line Tracing

Following field lines through a magnetic configuration reveals confinement structure, flux surfaces, and where confinement breaks down.

A specialized streamline

Tracing a magnetic field line is integrating dx/ds = B(x)/|B|, following the field direction through space. It is a streamline problem, but the physics gives it special meaning: in a confined plasma, field lines lie on nested flux surfaces, and their behavior classifies the whole configuration.

What the traces show

• Lines that stay on a torus map out a flux surface.
• Lines that fill a volume ergodically reveal a stochastic (chaotic) region.
• A Poincare section, marking where lines pierce a plane, turns 3D structure into a readable 2D map of islands and surfaces.

Poincare plots

Rather than draw tangled 3D curves, one records each crossing of a field line through a chosen poloidal plane over many transits. Points that fall on a closed curve indicate an intact flux surface; scattered points indicate island chains or stochasticity. This is the standard, information-dense view of magnetic topology.

Numerical care

Field lines are sensitive to integration error; a small step and an accurate scheme are needed to distinguish a true surface from numerical diffusion. The field itself comes from an equilibrium or a coil model, so the trace is only as good as the field.

python

def field_line_step(x, B, ds):
    b = B(x); b = b/np.linalg.norm(b)
    return x + ds*b  # use RK4 in practice

Kronos use

Field-line tracing and Poincare sections visualize the simulated confinement geometry of the breeder tokamak and the burner mirror, showing where surfaces are intact.