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Resonance Cones

Source lecture(s): PC368 Lec 13

Intuition

Poke a small antenna into a magnetised plasma at the right frequency and its field does not spread out smoothly. It concentrates on the surface of a cone whose opening angle you can calculate — and measure. The reason is startling: in part of parameter space, Laplace's equation stops being elliptic and turns hyperbolic, and a static-looking field problem acquires characteristics like a supersonic flow.

The equation changes type

Near a resonance the wave becomes quasi-electrostatic, \(\mathbf{E}\approx-\nabla\phi\), and the potential obeys

\[\nabla\cdot\left(\overleftrightarrow{K}\cdot\nabla\phi\right) = 0 \qquad\Longrightarrow\qquad S\left(\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}\right) + P\frac{\partial^2\phi}{\partial z^2} = 0\]

with \(S\) and \(P\) the Stix parameters. Everything hinges on the sign of \(SP\):

  • \(SP > 0\) — elliptic. Rescale \(\xi = x/\sqrt{|S|}\), \(\eta = z/\sqrt{|P|}\) and it becomes Laplace's equation. A point source gives a smooth potential with ellipsoidal equipotentials: ordinary anisotropic screening, nothing exotic.
  • \(SP < 0\) — hyperbolic. The same rescaling produces a wave equation in which \(z\) plays the role of time. It has real characteristics,
\[z = \pm\sqrt{\left|\frac{P}{S}\right|}\;x \qquad\Longrightarrow\qquad \tan\theta_{\rm cone} = \sqrt{\left|\frac{S}{P}\right|}\]

and the potential of an oscillating point source diverges on that cone.

What this means physically

A small antenna radiating into the hyperbolic region deposits its energy on two nested cone surfaces rather than spreading it isotropically. Energy propagates along the cone; the group velocity is directed along the resonance-cone surface while the phase velocity is perpendicular to it — one of the cleanest demonstrations anywhere that \(\mathbf{v}_g\) and \(\mathbf{v}_p\) need not be parallel.

The divergence on the cone is an artefact of the cold model. Warm-plasma corrections — finite Larmor radius and thermal dispersion — smear the singularity into a narrow but finite-amplitude structure with interference fringes. The field stays large; it just stops being infinite.

Fisher and Gould measured these cones in the laboratory in 1969, and the agreement with the cold-plasma angle is excellent — a satisfying case of an abstract change in PDE type being directly visible on an oscilloscope.

Where it matters

  • Lower-hybrid current drive. LH waves launched from a tokamak antenna propagate on resonance cones; predicting where they deposit their momentum is precisely a ray-tracing problem along these characteristics.
  • Whistler-mode antennas in the magnetosphere and in laboratory devices show the same structure.
  • Spacecraft interference. An antenna on a satellite in the ionosphere radiates anisotropically for exactly this reason, which matters when the same craft is trying to make quiet field measurements.

Common mistakes

  • Assuming a static-looking equation must be elliptic. The type of a PDE depends on the signs of its coefficients, and in an anisotropic dielectric those signs are frequency- dependent. Change \(\omega\) and you change the character of the mathematics.
  • Taking the divergence literally. It signals the breakdown of the cold-plasma approximation, not infinite energy density.
  • Confusing \(\theta_{\rm cone}\) with the resonance angle. They are complementary: \(\theta_{\rm cone}\) is measured from the field direction to the cone surface.

Knowledge graph position

Prerequisites: Dielectric tensor, magnetised waves. Leads to: lower-hybrid heating and current drive, ray tracing in inhomogeneous plasma.

Quiz

Q1 (conceptual). What changes about the mathematics when \(SP\) turns negative?

Answer

The quasi-electrostatic equation changes from elliptic to hyperbolic. It acquires real characteristics, so influence propagates along preferred directions instead of diffusing smoothly in all directions — and the potential of a point source concentrates on those characteristic surfaces.

Q2 (computational). At some frequency \(S = 2\) and \(P = -8\). What is the cone angle?

Answer

\(SP = -16 < 0\), so cones exist. \(\tan\theta_{\rm cone} = \sqrt{|S/P|} = \sqrt{2/8} = 0.5\), giving \(\theta_{\rm cone} \approx 26.6°\) from the magnetic field.

Q3 (MCQ). On a resonance cone the group velocity is:

  • (a) parallel to the phase velocity
  • (b) along the cone surface, while the phase velocity is perpendicular to it
  • (c) zero
  • (d) always along \(\mathbf{B}_0\)
Answer

(b). Energy flows along the cone; the wavefronts move across it. Anisotropic media routinely separate \(\mathbf{v}_g\) from \(\mathbf{v}_p\), and this is the most dramatic plasma example.