The ionosphere or the ionospheric plasma

 

The ionosphere is the ionized section of the Earth's upper atmosphere between 80 and 600 Km altitude where solar radiation ionizes atoms and molecules or, in other terms, extracts electrons electron from those, thereby creating the corresponding positively charged molecules or atoms i.e. ions and a large content of electrons. Solar radiation consists of extreme UV and x-ray photons which are highly energetic and can therefore dislodge electrons upon collision with molecules and atoms found at this area, such as molecular and atomic oxygen, molecular nitrogen, hydrogen and helium. The generated ions and electrons of the ionosphere constitute the ionospheric plasma. The total electron content (TEC) is one of the most significant ionospheric parameters, as electromagnetic signals transmitted in the ionosphere, e.g. from GPS satellites, may be scattered by the electrons and be deflected causing signal distortion and delay.

 

 

 

 

The lines of the Incoherent Scatter Radar spectrum

 

A typical incoherent scatter spectrum shown in Figure 1 (from Bhatt A. 2010) consists of a double humped ion line (I) in the middle, the gyroline (GL) and the plasma line (PL). The gyroline and the plasma line are much weaker (by approximately three orders of magnitude). The following sections explain to what these lines correspond.

 

 

Figure 1:  ISR spectrum showing the double humped ion line (I), the plasma line (PL) and the gyroline (GL). (From Bhatt A. 2010).

 

 

 

Oscillation frequency and cyclotron frequency for electrons and ions - Linkage to electrostatic plasma waves

 

Introduction

 

Electrons and ions are characterized by an oscillation frequency and a cyclotron frequency, which represent plasma parameters. These frequencies are linked to plasma waves and specifically electrostatic plasma waves. Plasma waves are distinguished in electrostatic and electromagnetic. When the magnetic field is not oscillatory, the plasma wave is considered to be electrostatic. Electrostatic plasma waves are distinguished in electron and ion corresponding waves. Electron waves are distinguished in electron plasma waves or Langmuir waves and in upper hybrid frequency waves. Ion waves are distinguished in ion acoustic waves, ion cyclotron waves and lower hybrid frequency waves. Table 1 which is an abridged version of the table found in Wikipedia article "Waves in plasma", presents the waves in plasma.

 

 

EM character

oscillating species

name

Electrostatic

electrons

plasma oscillation (or Langmuir wave)

upper hybrid oscillation

ions

ion acoustic wave

electrostatic ion cyclotron wave

lower hybrid oscillation

Electromagnetic

electrons

light wave

O wave

X wave

R wave (whistler mode)

L wave

ions

 

Alfvén wave

magnetosonic wave

 

 

 

 

(Electron) Plasma oscillations: Electrons are oscillating at the electron plasma frequency

(Electron) Plasma oscillations are linked to electron plasma waves or Langmuir waves

Representation by the plasma line 


 

Electrons in the ionospheric plasma are oscillating or, in simple terms, they are moving back and forth as if they were attached to a spring. In formal terms, they are performing a simple harmonic motion with a specific angular frequency ω, which corresponds to a frequency f termed (electron) plasma frequency. In order to describe this motion, we consider the following analysis.

 

If an electron in the ionosphere plasma is displaced from its equilibrium position, the associated positive ion will exert an electrostatic attractive force on the electron. Upon its effect, the electron will move as if it was attached to the end of a spring. It will come back to its initial position and then due to its inertia, it will surpass this position and move away, until a restoring force brings it back to the initial position. This trajectory will be repeated, resulting in an oscillation of the electron in the plasma which is due to the electrostatic Coulomb force by the ions. This oscillation is so fast that the massive ions don't have the time to respond to it and are considered fixed for the electron case analysis. Because the interaction between electrons is strong, they all oscillate together at a characteristic frequency. This is termed (electron) plasma oscillation. As a result, an electron pressure is created in the plasma. 

 

The frequency of the electron plasma oscillation ω depends only on the density of the plasma n:

 

ω(ep)/2π = f(ep) = 9 * SQRT(n)

 

For instance, in a plasma of density n = 10^-12 m^-3, we have f(ep)= 9*10^6= 9 MHz.

 

Radiation at f(ep) normally lies in the microwave range.

 

The oscillation leads to electron bunching (the oscillations bunch the electrons). In other words, compressions are rarefactions of electrons are created. The resulting charge bunching causes a spatially periodic E field which tends to restore the electrons to their initial position. The E field is behaving sinusoidally. We consider that the fringing electric field causes a coupling of the disturbance to adjacent layers leading to the propagation of the oscillation. The thermal motion of the electron can also lead to propagation and in this case the thermal velocities of the electrons provide information about what is occurring in the oscillating region (similarly to the modulation of a carrier wave). Due to the above, the (electron) plasma oscillation can be called an electron plasma wave or Langmuir wave.

 

It is possible to excite plasma waves by applying an oscillating potential to a grid in a plasma. Alternatively, an electron beam can be used: if the electrons in the beam are bunched so that they pass by any fixed point at a frequency f(ep), they would generate an electric field at that frequency and excite plasma oscillations.

 
 

 

 

 

Electrons are rotating at the cyclotron resonance frequency (Larmor gyration) while oscillating at the (electron) plasma frequency; the frequency of the composite motion is the upper hybrid frequency

Upper hybrid oscillations are linked to corresponding waves, also termed whistler waves - Representation by the gyroline

 

In the absence of a magnetic field, electrons are performing plasma oscillations under the influence of the restoring Coulomb force. In the presence of a perpendicular magnetic field B, a perpendicular Lorenz force will be exerted on the electrons and will act as a centripetal. On its own, it would lead to a circular motion by the cyclotron resonance frequency; when combined with the horizontal Coulomb electrostatic force, the trajectrories of the electrons will be elliptical. 

 

The frequency of motion will include a combination of the plasma frequency and the cyclotron resonance frequency, in a frequency termed upper hybrid frequency. The frequency will be higher than the plasma oscillation frequency as an additional restoring force, the Lorenz force is added to the original one, the Coulomb force. It is given by the following equation:

 

ω(h)2=ω(p)2+ω(c)2

 

"The existence of the upper hybrid frequency has been verified experimentally by microwave transmission across a magnetic field. As the plasma density is varied, the transmission through the plasma takes a dip at the density that makes ω(h) equal to the applied frequency. This is because the upper hybrid oscillations are excited, and energy is absorbed from the beam."*

 

"As the magnetic field goes to zero, ω(c) goes to zero in the above equation, and one recovers a plasma oscillation. As the plasma density goes to zero, ω(p) goes to zero, and one has a simple Larmor gyration, since the electrostatic forces vanish with density."*

 

Upper hybrid oscillations are linked to the corresponding waves which belong to the whistler mode branch and are sometimes termed "whistler waves". The gyroline is known as "resonance line" and "whistler line".

 

Note that the gyroline depends on the scattering geometry i.e. the angle with the magnetic field as mentioned at the study of Bjørna et al. (1990).

 

*Reference: https://link.springer.com/book/10.1007/978-3-319-22309-4 (p. 109-112)

 

 

Figure 1: Motion of electrons in an upper hybrid oscillation. From https://link.springer.com/book/10.1007/978-3-319-22309-4 Figure 4.19 p.105.

 

 

Figure 1: Cyclotron motion of electrons and ions.

 

 

 

Ion plasma oscillations: ions are oscillating at the ion plasma frequency

Ion plasma oscillations are linked to ion acoustic waves (compressions and rarefactions)

Representation by the ion line

 

Similarly to the electrons, the ions are oscillating with a frequency f which is called ion plasma frequency.

 

The oscillation leads to ion bunching or in other words to compressions and rarefactions (Figure 2). Ion bunches which are positively charged would be expected to disperse but this only occurs partially due to the following reason: ions pull with them the electrons which are canceling significantly the dispersing effect. However, dispersion also occurs due to the thermal motion of the ions. 

 

The oscillatory cycles lead to cycles of compression and rarefaction of ions. The phenomenon of compression and rarefaction of molecules and specifically air molecules is encountered in the transmission of sound waves or acoustic waves. This has led to naming the above waves "ion acoustic waves". It is noted that acoustic waves are longitudinal waves.

 

 

Figure 2: Ions create compressions and rarefactions.

 

Reference: https://link.springer.com/book/10.1007/978-3-319-22309-4 (p. 94-99)

 

 

Ions are rotating at the ion cyclotron resonance frequency while oscillating at the ion plasma frequency

The composite motion leads to either ion cyclotron waves or lower hybrid waves (lower hybrid frequency)


 

In the presence of a perpendicular π/2 or near perpendicular ~π/2 (π/2-θ) magnetic field B, a corresponding Lorenz force will be exerted on the ions and will act as a centripetal during their trajectory. On its own, it would lead to a circular motion by the ion cyclotron resonance frequency; when combined with the horizontal Coulomb electrostatic force generating compressions and rarefactions, the final trajectory will be a composite one. The angle θ of the magnetic field is very important: at θ=0 or π/2, while the electrons are doing circular trajectories, their small Larmor radii don't allow them to move significantly on the x axis. As a result they won't be providing shielding to the ion dispersion. In this case we will have lower hybrid oscillations. Near π/2 (when θ is not zero but small), electrons will be moving in the x axis, they will be providing shielding to ions and as a result we will have ion cyclotron waves.

 

 

 

References

 

1. Francis F. Chen, "Introduction to Plasma Physics and Controlled Fusion", Springer 2016 (Chapter 4 p. 82-113) (reference link)

2. Wikipedia article "Waves in Plasma" (reference link)