An Alfven wave is a type of plasma wave that occurs in magnetized plasmas. It was first described by the Swedish physicist Hannes Alfvén, who won the Nobel Prize in Physics in 1970 for his work on magnetohydrodynamics.

Alfven waves are characterized by the interaction between magnetic fields and plasma particles. They propagate along the magnetic field lines, and their motion involves the oscillation of the magnetic field and the plasma particles in a coordinated manner. These waves are considered to be magnetohydrodynamic (MHD) waves, meaning they involve the collective behaviour of the plasma as a fluid under the influence of magnetic and fluid forces.

Here are some key characteristics of Alfven waves:

Propagation along the magnetic field: Alfven waves propagate parallel to the magnetic field lines. The wave energy is transmitted along the magnetic field with minimal dispersion or attenuation.
Compressional and transverse motion: Alfven waves have both compressional (changes in density and pressure) and transverse (changes in magnetic field strength) components. The plasma particles move in spiral or helical paths around the magnetic field lines as the wave propagates.
Dispersion relation: The dispersion relation of an Alfven wave describes the relationship between its frequency, wavelength, and the properties of the plasma and magnetic field. In a uniform plasma with a uniform magnetic field, the Alfven wave has a characteristic frequency determined by the Alfven speed, which is the speed at which the wave propagates.
Resonant absorption: Alfven waves can exhibit a phenomenon known as resonant absorption, where they can transfer energy to plasma particles that have a similar frequency to the wave. This energy transfer can lead to heating and acceleration of particles in the plasma.

Hydromagnetic wave concerns with low frequency ion oscillations in the presence of magnetic field.

Let us consider hydromagnetic wave in plane geometry so that k is along Bo

The wave equation is

The current density is contributed by both electrons and ions so at equilibrium, the current density is given by

From eq 1 and 2

Thermal motions are not taken in this wave, so we may consider Ti=0. In this case the wave equation for ion becomes

The resulting quantities are assumed to be varying sinusoidally as in the form e^i(kx -wt) then

The corresponding solution to the electron equation of motion is found by putting M =m, e = -e, Ω_c = - wc and then taking the limit wc²>>w² such that

In this limit, the Larmer gyration of electrons are neglected and electrons have simply as E x B drift in y direction putting these solution ( in x component of)

This is the dispersion relation for Alfven waves.

For low frequency plasma, the relative dielectric constant is given as

Er = E / Eo = 1 + 𝜌 𝜇oc²/ Bo² .......12

Then equation 11 becomes

w²/k² = c²/Er

=> w/k = V𝜙 = c / (Er)^1/2 .....13

Equation 13 gives the phase velocity of electromagnetic wave in a dielectric medium.

For E>>1,

𝜌 𝜇oc²/ Bo² >> 1

Then equation 11 gives

These hydromagnetic waves travel along Bo at constant velocity VA called the Alfven velocity.

To understand physically an Alfven wave, the electromagnetic wave a fluctuating magnetic field B given by

The perturbed 'By' gives a sinusoidal ripple as shown in figure 2. The electric field Ex gives the plasma an E x Bo drift in a negative y direction. Since we have taken the limit w²<< Ωc², both ions and electrons will have same drift vy. Thus the fluid moves up and down in y direction as shown in figure. The magnitude of this velocity is l Ex/Bo l . Since the ripple in the field is moving at the phase velocity w/k, the line of force is also moving downward. The downward velocity line of force is (w/k) l By/Bo l which is equal to fluid velocity l Ex/ Bo l and thus fluid and field lines oscillated together as if particles were stuck to the lines.

The lines of force behave as though they were mass-landed, tensioned springs, and the Alfven wave can be viewed as a spreading disturbance that happens when a spring is pulled.

The polarization drift Vp in the direction of E allows a current to flow in the x direction as the electron E fluctuates due to the inertia of the ions, which lag behind the electron.

Alfven waves have various applications and play significant roles in plasma physics and astrophysics. They are observed in laboratory plasma experiments, the Earth's magnetosphere, the Sun's corona, and other astrophysical environments. Understanding Alfven waves is crucial for studying plasma dynamics, magnetic confinement, energy transport, and phenomena such as magnetic reconnection and solar flares.

This note is a part of the Physics Repository.

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