D-X5 · Magnetic tension carries a wave

Alfvén Waves

How can a magnetic field carry a sideways disturbance through a plasma? Bending the magnetic field creates a restoring force that moves the conducting fluid sideways. The disturbance travels along the field as neighbouring fluid responds. A stronger field makes the wave faster; a denser plasma makes it slower.

Pluck a magnetic field line and watch the bend separate into two travelling waves. Strengthen the field or load the plasma with more mass to change their speed.

Wave speed $71.4\,\mathrm{km}\,\mathrm s^{-1}$

Time to cross the domain $23.4\,\mathrm{min}$

Total wave energy per area $554\,\mathrm J\,\mathrm m^{-2}$

Kinetic share of wave energy $70.9\%$

Pluck the field

Let magnetic tension pull back

Magnetic field

80

B_0/\mathrm{\mu T}

Mass density

1

\rho/(10^{-12}\,\mathrm{kg}\,\mathrm m^{-3})

Pluck size

1.2

A/\mathrm{Mm}

Drag the bright field line, then let go. Use the presets to compare a pulse and a standing wave.

Model, scale and sources

The model solves linear, incompressible shear Alfvén waves parallel to a uniform background field in a periodic domain one hundred megametres long. The displacement is transverse; fluid elements do not travel along the field with the wave. Magnetic field perturbations follow the displacement gradient, and the model calculates kinetic and magnetic perturbation energy from that same solution. Packet profiles use twelve exact Fourier modes; periodic copies re-enter the opposite side. The displayed transverse displacement is enlarged fourteen times. Playback advances one hundred and eighty model seconds per real second. The density is uniform, the fluid is perfectly conducting, and the small physical slope stays within the linear approximation. Compressible waves, resistivity, viscosity, kinetic effects, reconnection, curved solar loops and observed solar data are omitted.

Alfvén · Original wave proposal, 1942 University of Wisconsin · Linear Alfvén waves

Simulated data

Where the wave stores its energy

Local kinetic and magnetic perturbation energies come from the same wave solution.

Kinetic energy

Magnetic perturbation energy

Start with the essentials

Focus question: Does a stronger field or a heavier plasma transmit the disturbance faster?
One-sentence intuition: Magnetic tension speeds the wave up; the inertia of the plasma slows it down.

Core mathematical model

The speed of a parallel shear Alfvén wave

v_A=\frac{B_0}{\sqrt{\mu_0\rho}}

The background magnetic field sets the restoring force and mass density sets the inertia. The displayed field is in microtesla and density is in trillionths of a kilogram per cubic metre.

One displacement determines both velocity and magnetic bending

v_\perp=\frac{\partial\xi}{\partial t},\qquad b_\perp=B_0\frac{\partial\xi}{\partial x}

Position along the background field is horizontal. Fluid displacement is transverse. A travelling bend need not carry a fluid element along with it.

Kinetic and magnetic perturbation energies

u_K=\frac12\rho v_\perp^2,\qquad u_B=\frac{b_\perp^2}{2\mu_0}

The plot below shows local wave-energy density, excluding the unperturbed background field. In the periodic ideal model, their integrated sum stays constant.

Common difficulties

The wave moves along the field; the fluid moves across it

Typical misconceptionA bright fluid element must travel down the field line with the pulse.

Better mental modelThe element oscillates sideways around its own horizontal position. The changing bend passes from one element to the next.

Run the experiment

01
Release a sideways bend
Choose One pluck, two waves, or drag the bright field line and release it.
What to observe: A displacement released from rest separates into two waves moving in opposite directions.
02
Change the restoring force
Keep the density fixed and double Magnetic field.
What to observe: The same disturbance crosses the domain in half the time.
03
Add inertia
Compare One pluck, two waves and A heavier plasma.
What to observe: Quadrupling density halves the speed. The brighter fluid markers still move across the field, not along it.
04
Watch energy exchange
Choose A standing wave and compare the energy-density curves below.
What to observe: The wave alternates between magnetic bending and fluid motion. Total perturbation energy is conserved.