Chapter 4 - Electromagnetism
November 4, 2011

4.1 The Electromagnetic Field Equation

Scientists attempt to explain physical systems in terms of mathematical models which describe and predict the behavior of the system.

For example, Kepler explained the movement of the planets with his three laws. In the same way, plasma behavior is governed by the electromagnetic field equations, which describe the motions of charged particles and their interaction with electric and magnetic fields.

There are two components of the electromagnetic field equations:

• Maxwell's Equations

• Lorentz Force Law

The two components act in tandem as a feedback loop:

Maxwell's Equations determine the electric and magnetic fields based on the position and motion of charged particles. They also determine the interaction of the electric and magnetic fields if either is changing.

The Lorentz Force Law determines the electric and magnetic forces on a charged particle moving within the fields. This force will cause each particle to move (accelerate) in accordance with Newton's Laws. The changes in the positions and motions of the charged particles in turn cause changes in the electric and magnetic fields.

Computer programs have been constructed to follow these interacting phenomena in plasmas.

They typically involve a series of steps, each representing a very short span of time. First, given the state of magnetic and electric fields present and the mass, charge, speed and direction of each particle, using the Lorentz Force Law, the forces applied on each particle by the field values at its position (x, y, z coordinates) are calculated.

The vector sum of the contributing forces is calculated, and the resulting acceleration of the particle moves it a small distance in some direction in the interval of the tiny time step (Newton's Laws of Motion). This is accomplished for the entire set of particles.

Then, considering the new coordinates and kinematic conditions of each particle, Maxwell's equations are used to determine the values of the electric and magnetic fields. After this, the program loops back to the first step, where the electric and magnetic forces acting on each particle are calculated once again using Lorentz Law.

The loop is controlled by the program's directing it to stop when a defined condition is reached, such as a certain number of repetitions, or if a certain value in the variables is reached, changed, or exceeded, or an error of some kind is encountered, and so on.

Once a set of starting conditions has been defined (number of particles, their charges, masses, initial velocities, and a description of the intensities of the assumed electric and magnetic fields throughout a defined volume of space), the loop process above might be outlined as follows:

• Calculate all the forces acting on each particle via Lorentz Law

• Calculate new locations and velocities for a very short increment of time using Newton's Laws of Motion

• Calculate E and B at each charged particle's new location after this time increment

• If an End-Loop condition is not satisfied yet, go back to 1. and continue calculating

Other aspects can be added in for greater accuracy or a better approximation to "reality", such as collisions of particles, viscous and gravity forces, etc. for more complete modeling.

This is a complex undertaking, and large models with many particles may take months of supercomputer time to run. This feedback loop can rapidly result in highly complex behavior, which is extremely difficult to model mathematically.

Simplifications are often introduced.

However, simplifying assumptions often lead to the omission of precisely those sorts of behavior which distinguish plasma behavior from that of a gas or other fluid.
 

A bubble chamber within a magnetic field creates visible tracks of charged particles,

allowing evaluation of particle energies, interactions and collision by-products,

when installed in line with a particle accelerator.

Image credit: Bubble chamber tutorial provided by CERN (link below)

Bubble chamber tutorial by CERN

A full description of the electromagnetic field equations can be found in Appendix II. What follows is a summary of the key points.
 

4.2 Maxwell's Equations

The implications of Maxwell's Equations and the underlying research are:

• A static electric field can exist in the absence of a magnetic field; e.g., a capacitor or a dust particle with a static charge Q has an electric field without a magnetic field.
   

• A constant magnetic field can exist without an electric field; e.g., a conductor with a constant current I has a magnetic field without an electric field.
   

• Where electric fields are time-variable, a nonzero magnetic field must exist.
   

• Where magnetic fields are time-variable, a nonzero electric field must exist.
   

• Magnetic fields can only be generated in two ways other than by permanent magnets: by an electric current, or by a changing electric field.
   

• Magnetic monopoles cannot exist; all lines of magnetic flux are closed loops.

4.3 The Lorentz Force Law

The Lorentz Force Law expresses the total force on a charged particle exposed to both electric and magnetic fields.

The resultant force dictates the motion of the charged particle by Newtonian mechanics. As the Lorentz equation is fundamental to all plasma behavior, it is worth spending a little time understanding what it means.

The equation is:

F = Q(E + U × B)
(Vectors are given in bold text and are explained below)

...where,

• F is the Lorentz force on the particle

• Q is the charge on the particle

• E is the electric field intensity

• U is the velocity of the particle

• B is the magnetic flux density

• "×" is the vector cross product symbol, not merely a multiplication sign.

Read it as "U cross B".

In order to understand what the equation actually means, we need to know a little about vectors.

A vector is a quantity which has both magnitude and direction. Examples include velocity and force. It is like an arrow: it has a length and it points in a direction. By contrast, a scalar quantity only has magnitude. Examples include speed and temperature. Vector algebra is the mathematics which deals with vectors.

For those wanting to know, further details of vector algebra are given in Appendix III. The Hyperphysics explanation is also a good introduction. The essentials for understanding the Lorentz equation will be explained here.

First, multiplying a vector by a scalar quantity is like putting a number of similar arrows together end to end. The vector is the first arrow; the scalar quantity is the number of similar arrows.

The result is a bigger arrow in the same direction as the original vector.

A simplified example is increasing the speed of a car to three times its initial speed as it moves in a straight line. Imagine that the car's velocity vector is simply an arrow pointing straight ahead down the roadway, with its base or starting point always at the center of the car.

Picture this arrow as being 20 cm long to represent a starting speed of 20 km/hour. Then you push down on the accelerator pedal to make the wheels of the car turn faster and push (accelerate) the car to a higher speed. As the car speeds up, the length of the arrow increases so that it always matches the car's speed.

At 60 km/hour the arrow is 60 cm long, and its direction is still parallel to the roadway. If you press the brake pedal, the car accelerates in the opposite direction, slowing down, and the arrow becomes shorter and shorter.

As the car stops, its speed drops to zero, and the velocity arrow or vector becomes zero in length.

"That is easy to understand", you say. "What happens if I turn the steering wheel to, say, the left?"

That kind of an action introduces an additional force on the car, in a different direction from that pointing parallel to the centerline of the car.

It does not increase or decrease its speed (neglecting friction!) but something changes because the car is turning! The velocity vector from the wheels making it go 60 km/hour has not changed length, but an additional force in a different direction has been applied, so now the velocity vector becomes the result of two different forces (two arrows acting on the center of the car).

As long as you hold the steering wheel at the same angle, the same force is being applied that wants to turn the car, and it moves around on a circle at a constant speed.

You can see that there are two kinds of acceleration: changes in the speed of motion, either faster or slower - just a plain numerical value change in the distance per unit time ratio without reference to any direction - and changes to the direction of motion - just an angular change of the direction in space that something is heading, without reference to how fast along its path or trajectory it is moving.

Both types of change are the result of a force being applied to an object.

Multiplying two vectors together is a little more complicated. Think of a very large screw in a wood board where the slot in the head represents the first vector and the second vector is drawn on the board.

As the screw is twisted clockwise until the slot aligns with the second vector, the screw will move into the board at right angles to both the slot and the second vector. The amount of movement depends on the dimensions of the screw and the amount it is turned.

The vector cross product is a bit like this.

Multiplying two vectors together using the cross product results in another vector at right angles to both the previous vectors - that is, perpendicular to the plane containing the two previous vectors.

The direction of the new vector is given by the direction of movement of our imaginary screw. The magnitude (length) of the new vector depends both on the angle turned and on the size of the original vectors.

As in the case of our screw, if the vectors are aligned (parallel) in the first place, then no movement of the screw takes place. The cross product of aligned vectors is zero.

More formally, in Cartesian coordinates, if a vector in the x direction is crossed with a vector in the y direction, then the result is a vector in the z direction. The magnitude of the resultant vector is the triple product of the lengths of the two original vectors and the sine of the smaller angle between them. If they are parallel, the angle between them is zero. Since sine(0°) is zero, in that case there is no resultant force in the z direction.

The effect is very similar to the gyroscopic effect in rotating solids: a force in one direction results in motion in a direction at right angles. This is known as precession.

Going back to the Lorentz Force Law, we see that the total force is made up of two parts.

The first part is QE, which is the product of the scalar value of the charge on the particle and the electric field strength vector. The magnitude of the force due to the electric field is the product of the charge on the particle and the strength of the electric field.

Note that the force due to the electric field is constant and in the direction of E, so it will cause constant acceleration of the particle in the direction of E according to Newton's Laws of Motion, one direction for a positive charge, and the opposite direction for a negative charge.

The second part of the equation, Q(U × B) is more interesting.

Here we have two vectors multiplied together using the cross product and then multiplied by the charge on the particle. Assuming that the particle was not moving in alignment with the field in the first place, when the force would be zero, then the result will be a force which is at right angles to both the direction of motion of the particle and the magnetic field.

This explanation of the Right Hand Rule will explain the "steering" force that a magnetic field, in a specified direction, exerts on a charged particle entering the field.

A force at right angles to the motion is a centripetal force (definition: "toward the center"). The magnetic field will therefore cause the charged particle to move in a circle in a plane perpendicular to the direction of the magnetic field.

As the particle is moving round the circle its velocity at any point will still have a component at right angles to the magnetic field, and so it will still experience a centripetal force which keeps it moving in the circle. Its direction is constantly changing, but its scalar speed (m/s) is unchanged, under this condition.

A simple case is to consider what happens when a moving charged particle enters a (fixed) magnetic field.

For simplicity, we will ignore any effects that the particle might have upon the magnetic field. If it enters the field parallel to the direction of the field, it experiences no force and nothing about its velocity (scalar speed or direction) changes. If it enters the field at a right angle to the direction of the field, its path will simply curve into a circle which closes upon itself.

Without an electric field, the Lorentz law reads (centripetal force) F = Q(U × B). The force applied to the charged particle is directly proportional to Q, the particle's charge, to U, the velocity vector, and to B, the magnetic field vector.

The meaning of U × B is U times B times the sine of the smaller angle between the two vectors, which means that UB is multiplied by the sine of an angle, so its effect ranges from zero to 1. In the comparative illustration below, the particle's charge and the magnetic field are held constant and the velocity of the particle as it enters the field increases from left to right.

The faster the particle is moving, the larger the radius of the resultant circular motion, because the radius r is a measure of the particle's linear momentum mU where m is the particle mass: r = mU ÷ (|Q|B).

The same result would apply if the charge were to increase while the other two variables were held constant.

If the charged particle enters the magnetic field at an oblique angle, with a component of its motion vector in the direction of the field, i.e., at an angle between zero and 90 degrees to the field direction, it will "drift" in the direction parallel with the field, while the field forces the particle into a circular motion. This "drifting" circular path traces out a helix or spiral.

The "guiding center" of the circle follows a field line of the magnetic field.

The radius r is known as the Larmor radius or cyclotron radius. In the three illustrations below, the angle of entry by the particle and the strength of the magnetic field, B, remain the same, with a small drift motion toward the right. The initial entry velocity is increased step-wise from left to right, to show that the faster a charged particle enters a magnetic field, the larger its radius of curvature.

In the series of images below, the green entry vector touching the magnetic and electric field lines shows which way a positively charged particle (by convention) is moving as it "enters" the field(s).

The particle could be going in either direction along this vector line at entry, so there are two trajectories coming out of the tip of the green vector, as you will see.

If the particle were charged negatively, it would accelerate in the opposite direction, and if it were heavier or moving faster, it would have a larger diameter circle than depicted. Similarly, if the magnetic or electric fields were changed, holding other factors constant, that would similarly change the particle's behavior.

The narrow orange "tubes" represent the particle's trajectory resulting from the entry conditions.
 

As a charged particle enters a uniform magnetic field B,

its path is bent into a circle whose radius r is proportional to its linear momentum, mass times velocity (mU).

The particle's speed does not change, so its kinetic energy is unchanged,

and the field does no work on the particle.

This is analogous to gravity's exerting a continuous centripetal force on an orbiting satellite in space.

The magnetic field direction is shown by a blue axial line; particle entry angle by a green radial line.
 

As the particle's entry angle into the B-field changes

from perpendicular to parallel, its trajectory will change to a spiral.

The spiral will decrease in radius as the angle decreases from 90 degrees

to the magnetic field direction and approaches zero or parallel to the field.

Note the changing angle of the green entry vector, left to right,

and the helical stretching.

Images above created with Mathematica Demonstrations
 

The total force will be the vector resultant of the electric and magnetic forces and depends on the angle between the two fields.

If the electric and magnetic fields are parallel (as in the field-aligned current situation we will consider later), then a charged particle approaching radially to the axial direction of the fields will be constrained to move in a helical path aligned with the direction of the fields.

That is to say, the particle will accelerate (constantly change its direction to spiral around the axial direction of the magnetic field) as a result of the Lorentz force, and will simultaneously accelerate (change its scalar speed) in the direction of the electric field.

This makes successive revolutions farther and farther apart as the particle's velocity component in the E-field direction increases over time.
 

In this field-aligned situation (E and B fields parallel) a particle trajectory

has the centripetal circularizing magnetic force applied at

the same time that the E-field vector (red) forces it to accelerate axially.

Over time the particle is moving nearly parallel to the fields.
 

If the charged particle enters the combined, aligned field axially (parallel to the magnetic field), it experiences no magnetic field, so force to revolve around a guiding center is not exerted.

The electric field, however, will still accelerate the particle along the field lines. Depending on its charge, if the particle enters in the direction of the accelerating force, its velocity increases. If it enters counter to this force, it decelerates and may stop and accelerate back in the opposite direction.

Recall that the "direction" of an electric field is defined as the direction that its force is applied to a positively-charged particle.

If the fields are not aligned, various trajectory combinations can occur depending on the particulars of the charge, field strengths, entry direction and angular misalignment of the magnetic and electric fields.
 

With a constant electric field present, its general tendency will be

to accelerate particles ever more closely aligned with its field lines,

and to increasing velocities.

Images above created with Mathematica Demonstrations
 

Although these trajectories may look complex, they involve only a single charged particle at a time, with constant electric and magnetic fields, with the same entry velocity. In practice many charged particles of different polarities and velocity vectors may occupy a volume of space at once, and their electric and magnetic interactions will affect the field values in which they move.

There may also be neutral particles present, as well as dust and grains and large bodies, all of which may exert other forces (gravity, viscous, collisions) on the plasma interactions, too.

We note in passing that secondary effects of relativistic electrons spiraling around magnetic field lines in space are often detected in the form of synchrotron radiation.

From consideration of the Lorentz Force Law, we know that there must therefore be an electric field aligned with the magnetic field and that the axial movement of the spiraling electrons with a velocity component parallel to the magnetic field constitutes a field-aligned current.

These currents are Birkeland currents; they occur at many cosmic scales.

4.4 Other Effects of the Field Equations

It is worth remembering some basic results arising from the application of the electromagnetic field equations.

1. Electric fields cause a force on all charged particles.
    

2. The electric force will be in opposite directions for oppositely charged particles; therefore, an electric field will produce opposite velocities of ions and electrons and so tend to separate them. Charge separation in space is important in plasma physics.
    

3. Magnetic fields only act on moving charged particles having a component of motion perpendicular to the magnetic field. Because the force depends on the cross-product of the velocity and field vectors, the effect will be different in different directions. This results in a direction-dependent electrical resistance. Think of trying to swim straight across a river rather than with the water's current.
    

4. The direction of the magnetic force is momentum and charge-dependent; ions and electrons will therefore circle in opposite directions with different radii and periods of rotation.
    

5. Bulk plasma moving across the direction of a magnetic field will cause a local electric field to develop which itself will cause new forces on the charged particles.
    

6. Changes in the distribution of charged particles cause a change in the electric field between them; a changing electric field generates a change in the magnetic field.
    

7. The Maxwell Equations and the Lorentz Force Law act together as a feedback loop modifying the motions of the charged particles and the fields in complex ways.

4.5 Replacing Currents With Magnetic Fields

The question arises as to whether electric currents can be replaced by magnetic fields using Maxwell's Equations, which would make the solutions much easier.

The answer is, technically, yes they can in certain simple situations, and this is often done in magneto-hydrodynamic theories and models because it is more convenient for studying certain plasma phenomena. However, there are many aspects of plasma behavior where it is necessary and crucial to consider the movement of the charged particles because simply considering the field behavior cannot model the observed complexity of plasma behavior.

The situation is analogous to the wave-particle duality in particle physics: there are some situations where it is necessary to use the particle description.

Examples of plasma behavior requiring use of the particle or current description include cellularization and filamentation, energy transport, and instabilities. Consideration of electric currents and circuits also necessitates the use of a particle-based description.

Simply considering only the field effects in these situations will miss the true complexity of plasma behavior.

We shall look at some of these more complex behaviors next.
 

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Chapter 5 - Plasma Sheaths, Cells, and Current-Free Double Layers
December 3, 2011

The Saturn aurora's reddish color is characteristic of ionized hydrogen plasma.

Powered by the Saturnian equivalent of (filamentary) Birkeland currents,

streams of charged particles from the interplanetary medium and solar wind

interact with the planet's magnetic field and funnel down to the polar regions.

Double layers are associated with filamentary currents and current sheets,

and their electric fields accelerate ions and electrons.

Image credits: Wiki Commons; J.Trauger (JPL), NASA, Hubble Space Telescope
 

5.1 Plasma Temperature and Potential

We have seen that temperature is a measure of the thermal energy of the particles in matter. More specifically, temperature is a measure of the kinetic energy of the particles' random thermal motion.

An electron has only 1/1840th the mass (approximately) of a proton, so electrons will have much higher velocities than ions at the same temperature.

This is because kinetic energy is proportional to the mass of the particle and the square of its velocity, K.E. = 1/2 mv². Therefore, at the same temperature, the ratio of velocities will be inversely proportional to the square root of the particle masses.

For example, average electron velocity will be around 43 (i.e., √1840) times higher than the velocity of a single proton. If the positive ions in the plasma are heavier than a single proton then the difference will increase accordingly.

What is more, because of the Principle of Conservation of Momentum, an electron will tend to undergo a larger change in its velocity than an ion does in a collision between the two particles.

The electrons' higher velocity results in more rapid interactions, which means that the electrons reach thermodynamic equilibrium ("the same temperature") amongst themselves much faster than the ions do. Any increase in velocity, whether from collisions or external energy inputs, is therefore 'shared out' amongst the electrons very rapidly.

For these reasons, it is common for the electron temperature in a plasma to be different from the ion temperature.

Often the electron temperature will be higher than the ion or the ambient temperatures. This is especially common in weakly ionized plasmas, where the ions are often near the ambient temperature while the faster-moving electrons have high temperatures. Wikipedia reference on plasma temperature here.

In a plasma the temperature is often expressed as a thermal potential which is equal to the potential drop (change in voltage) through which the particles would have to fall in order to gain the same amount of energy. The kinetic energy can then be expressed in electron-volts or eV.

The hotter the plasma, the faster the electrons and ions are moving in random thermal motion and the higher their potential. A potential of 1 eV is equivalent to a temperature of 11,604.5 K.

Particles with potentials many orders of magnitude higher are common in space.

Note:

one must be cautious about the conversion between electron volts and thermal temperatures in plasma.

 

Plasmas can become ordered so that charged particles follow paths that are aligned with the local direction of the accompanying magnetic field. Such current flows are termed field-aligned currents.

 

Under this condition, charged particles are moving approximately parallel to one another, and, partly due to the low density of particles, collisions of the thermal variety can become very rare.

The high temperature alleged for the solar corona is based on spectroscopic observations of the light (electromagnetic radiation including frequencies outside visible light) which indicate how much ionization of atoms has occurred.

The ionization energy in eV is inferred from the wavelengths of light emitted, and converted by the formula above to equivalent temperature.

The thermal aspect of temperature which is caused by large numbers of random collisions is not necessarily present, however, even if there has been sufficient energy input to strip electrons away from their nuclei. The electrons can be fast (energetic) while their (thermal) collision rates are low.

The high velocity of the electrons is especially important in understanding many aspects of plasma behavior, including radio galaxies, galactic and stellar jets, and production of synchrotron radiation and cosmic rays.

5.2 Development of Surface Sheaths

If plasma is contained within a laboratory tube or other vessel, the electrons and ions in the plasma will impact the walls of the vessel with a frequency proportional to their velocity. On impact, particles are absorbed by the walls.

As the electrons have much higher velocities than the ions, the rate of electron impact will be many times that of the ion impacts. As a result, the walls of the vessel will acquire a negative charge.

As the negative charge on a surface develops, arriving electrons will tend to be repelled from the surface. Only those electrons with sufficient velocity to overcome the repulsion will still be able to impact the surface. The negative charge on the surface will continue to increase until the number of electrons hitting the surface equals the number of positive ions arriving.

The plasma and the surface will have achieved a balance, or steady state.

In the steady state, only the fastest electrons will still be able to get through the adverse potential gradient from the negative surface. Most electrons will be prevented from approaching the surface. This results in a layer of plasma adjacent to the surface in which the ions outnumber the electrons. This positive layer is known as a Debye Sheath.

Similar effects are found if the surface is charged negatively or positively by connecting a source of potential such as a battery. The charge on the surface repels like charges in the plasma, leaving behind an oppositely-charged sheath.

5.3 Extent of a Sheath

A surface sheath does not have a definite physical boundary but may be considered to end where the potential resulting from the negative surface and the positive sheath acting together balances the potential of the plasma itself.

In other words, the sheath boundary is where the potential is just sufficient to repel electrons with energy equal to the plasma potential.

For example, if the plasma potential is +1V then the nominal boundary will have a potential of -1V.

The explanation is as follows:

The boundary has a negative potential because the sheath must repel approaching electrons. The electrons in the plasma have a kinetic energy of 1eV. Therefore, the sheath needs -1V potential to stop the approaching electrons from reaching the surface.

This is analogous to rolling a ball up a hill. If the ball has enough kinetic energy then it will reach the top.

If not, it will get a part of the way up before coming to a stop and then rolling down again. The sheath potential is analogous to the height of the hill.

It can be seen that the sheath does not have a 'hard' edge and in fact the potential field arising from the negative surface continues past the sheath 'boundary'. Nevertheless, the boundary may be taken as the point at which the negative surface is effectively 'neutralized' by the sheath because electrons with the plasma potential are 'reflected' back into the plasma at that point.

American chemist and Nobel laureate Irving Langmuir developed measurement methods and observations of plasma actions.

An interesting and useful PDF lecture, 'Plasma, Sheaths and Surfaces - The Discharge Science of Irving Langmuir,' can be found here.

5.4 Charged Bodies in a Plasma

Similar sheaths will form around any charged body in a plasma where the body has a different potential from the plasma itself.

The plasma effectively isolates the foreign body by forming a sheath round it. The sheath will tend to screen out the electrostatic field from the alien charge in the same way that a sheath tends to isolate a negatively charged surface.

The body eventually may be neutralized by opposite charges that it absorbs.

If the charged body can artificially be given a positive or negative charge by connecting it to an external source such as a battery, ions or electrons, depending on the charge, will be attracted to the body and so a current will flow. By careful measurement of the current for a range of voltages, it is possible to measure the potential of the plasma itself.

One such device is named a Langmuir Probe after Irving Langmuir, 1881-1957.
 

5.5 Cellularization in Plasma

Similar effects also occur between two adjacent regions of plasma with different characteristics.

For example, the two regions may have different temperatures, densities, or degrees of ionization. In this situation, the different velocity distributions in the two regions will set up a double sheath at the boundary whereby each region effectively insulates itself from the other.

The double sheath will consist of adjacent thin layers of positive and negative charge, separated by a relatively small distance. It is one type of Double Layer.

Because no externally driven currents are involved, sheaths between different plasma regions are known as Current-Free Double Layers (CFDL). More on double layers in plasma here.

Note especially the external links, linked reference papers and publications at the bottom of this article.

Double layers and sheaths are well-known phenomena in plasma dynamics, described in textbooks and best described in Wiki's discussion of the Vlasov-Poisson equation:

"In general the plasma distributions near a double layer are necessarily strongly non-Maxwellian, ¹ and therefore inaccessible to fluid models.

 

In order to analyze double layers in full generality, the plasma must be described using the particle distribution function, which describes the number of particles of species α having approximately the velocity v near the place x and time t"

1 - From Wikipedia, Physical Applications of Maxwell-Boltzman Distributions: The Maxwell–Boltzmann distribution applies to ideal gases close to thermodynamic equilibrium with negligible quantum effects and at non-relativistic speeds. It forms the basis of the kinetic theory of gases, which explains many fundamental gas properties, including pressure and diffusion.
 

Importance of the reference above: This is the reason that conventional hydrodynamic and magnetohydrodynamic equations of fluid flow are inadequate to a full and reasonably accurate mathematical description of plasma dynamics.

Consequently the computational method called particle-in-cell (PIC) simulation was developed for plasma modeling in massively parallel computer systems in the 1980s.

Here is a Wikipedia article on PIC, and here is a more technical paper on the subject.

5.6 Formation of a Current-Free Double Layer (CFDL)

We have seen that CFDLs form between regions of plasma with different characteristics. As an example, let us consider the effect of a temperature difference (in electron volts, ref. 5.1 above).

This causes an electric field to build up, which will accelerate electrons back to the hotter region.

A net flow of electrons to the cold region will continue to build up the electric field until a balance is achieved between the numbers of hotter electrons moving to the cool region and the number of electrons being accelerated back to the hot region by the electric field.

The thin regions near the boundary containing an excess of ions or electrons constitute a Double Layer at the boundary which has an electric field and associated potential drop across it.

The formation of sheaths at boundaries between different plasma regions creates cells of plasma. This cellularization is a defining characteristic of plasma behavior.

Gases do not behave in this fashion, which is one reason why it is not possible to apply gas laws to plasmas.
 

5.7 Similarity to Fluid Mechanics

At first sight, a Double Layer (DL) appears to be something like a shock wave in fluid dynamics.

Indeed, a DL does share some characteristics of a shock wave in that it separates regions of differing characteristics and acts to accelerate the medium.

In the case of DLs, however, the acceleration occurs as a result of the strong electric field set up between the oppositely charged layers. As the force from the electric field depends on the charge on the particle, ions and electrons are accelerated in opposite directions. Neutral particles are not accelerated at all by the electric field, but may be entrained through viscous or other effects.

Note that the formation of Double Layers cannot be effectively modeled by fluid analyses such as magneto-hydrodynamics (MHD) because it is caused by and is dependent on motions of different individual particles, not on the bulk motion of the plasma.

A good introduction to Plasma Physics from Wikipedia's perspective can be found here, including properties, phenomena and mathematical models.

While Wikipedia often has well-written articles, like anything else it can sometimes be unreliable or incomplete, or prone to biased editing, so always use care when evaluating articles from Wiki, as well as other sources.

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Chapter 6 - Currents, Filaments and Pinches
December 6, 2011

Planetary nebulas often exhibit characteristic

bi-polar symmetries with a centered plasma pinch,

polar jets, and an equatorial torus.

Image courtesy NASA, ESA and the Hubble Heritage Team
 

6.1 Thermal Motion and Current

It is important to distinguish between random thermal motion and uniform linear motion in a plasma.

The latter is an electric current which flows due to the presence of an electric field.

The random thermal motion is measured by the temperature of the plasma, or by the temperatures of the ions and electrons separately if their temperatures are different. This motion, being a motion of charged particles, is also a form of current, but one which oscillates about an average position, as opposed to moving in one direction only.

Strictly speaking, temperature can only be an accurate measure of the energy if the distribution of velocities of individual particles is Maxwellian, that is, if the distribution is equivalent to that which would result from elastic collisions between the particles.

Uniform linear motion results from an electric field and represents a drift current. All particles with the same sign charge (positive or negative) move in the same direction under the influence of the electric field.

That is, in a plasma, where there are roughly equal numbers of positive and negative particles ("net neutrality"), we find the positively charged particles moving together in a direction opposite that of the electrons' motion.

The particles all have kinetic energy, which may be high, but they do not have a temperature as a result of this linear motion.

That's because temperature is only used to measure the energy of particles with random velocities, undergoing collisions. As both types tend to move along more or less aligned or parallel trajectories, and plasma densities are relatively low, collisions are less common and Maxwellian collision conditions are not obtained.

Both types of motion exist simultaneously wherever a current is flowing. The current motion or drift of the particles is superimposed on the random motions.

Another way of looking at this is to think of the mean position of the random motion as moving with the drift velocity in the direction of the current.
 

6.2 Electron and Ion Currents

We have seen that the electrons acquire much higher velocities than ions due to their smaller mass.

However, an electron carries the same magnitude of (negative) charge as a positively charged proton, the lightest form of ion. Therefore, the higher velocity of the electrons means that they are more effective than the ions at carrying current in a plasma.

The ratio of the electron current to the ion current in a non-relativistic plasma current is proportional to the square root of the inverse ratio of the masses.

For the lightest positive ion, a proton, this means that the electron current is around 43 times greater than the ion current. [ If the mass of an electron is taken as 1, the mass of a proton would be about 1836 times greater): √(1836 ÷ 1) = 42.85 ]

In many situations, it is the motion of the electrons which determines the plasma behavior.
 

6.3 Current in Laboratory Discharge Tubes

Plasma has been studied in laboratory experiments for over a hundred years, and a vast amount of experimental data and analysis is now available.

One of the fundamental experiments involves a Glow Discharge Tube in which a current is passed through a low-pressure gas such as mercury vapor. This causes ionization of the gas and creation of a plasma within the tube.
 

Evacuated (low pressure gas) tube

with anode and cathode and high-voltage power source.

Image credit: Wiki Creative Commons
 

6.4 Glow Discharge Tubes

Many descriptions of discharge tubes are available and will not be repeated here in detail.

The salient points for the present purposes are as follows:

• Within the tube, there are visible bands along the axis wherein the plasma is seen to glow, interspersed with 'dark' bands where there is no such glow. The different bands represent two of the three possible modes of operation of plasma when carrying a current.

   

• The dark bands represent, unsurprisingly, the Dark Current Mode. In these regions the electron velocity is below that necessary to cause visible excitation of the atoms of neutral gas, although ionization will start to occur at higher currents. However, radiation will be given off at wavelengths outside the visible even in the Dark Current Mode and so may be detected by non-optical means.
   

• The glowing bands represent the Normal Glow Mode. Here, the velocity of the electrons causes ionization to occur. The glow is caused by radiation from the electrons of neutral atoms after they have been excited by collisions with fast free electrons.
   

• The third possible mode of plasma operation is the Arc Mode, familiar in painfully bright welding applications or lightning, for example.
   

• Returning to the glow discharge tube, one might expect that the potential difference between the electrodes would cause a uniform electric field along the length of the tube. However, the plasma behaves differently.
   

• It is found that a Double Layer (DL) develops in the tube which modifies the externally applied electric field between the anode and cathode. The DL forms in such a way that the majority of the potential drop occurs across the DL. Away from the DL region, much of the remainder of the plasma is a glow discharge region known as the positive column. This can extend for a significant part of the length of the discharge tube.

   

• Within the positive column there are approximately equal numbers of electrons and ions. The plasma here is therefore quasi-neutral. Because most of the potential drop occurs across the DL, only a small but constant voltage gradient, or electric field, exists within the positive column.

   

• There appear to be analogies between the positive column in a discharge tube and the plasma within the Sun's heliosphere. Another result of the discharge tube experiments is also relevant to our discussion of plasma behavior and will be discussed in the next section.

6.5 The Voltage-Current Density Curve

voltage-current curve
 

If the voltage V is plotted against the current density J in a discharge tube (current density is the current divided by the area of the discharge tube), then it is found that the three different plasma glow modes correspond to three different sections of a discontinuous graph, known as the voltage vs current or V-J curve.

In the dark discharge mode, the V-J curve rises with increasing voltage, although not regularly. Once the voltage reaches a high enough value, ionization begins and the current starts to rise very rapidly for very little increase in voltage.

The discharge will then change rapidly into the glow discharge mode. This is accompanied by a dramatic step change in the voltage. The voltage drops right down because, when large numbers of electrons have been produced by ionization, only a small voltage is needed in order to generate a large current.

A very significant effect often occurs in the lower current density part of the glow discharge region.

The voltage actually decreases with increasing current density. In other words, the plasma finds it more efficient to transmit the current at a higher current density because the voltage drop is less.

At still higher current densities, the voltage increases again, meaning that the glow discharge section of the V-J curve has a minimum at a particular value of current density. This minimum represents the point of lowest resistance for transmission of the total current.

In cosmic plasmas, this effect may be significant in causing the formation of current filaments by confining the current within a particular cross-sectional area.

Similarly, in the extremely bright arc discharge mode, the voltage once again decreases with increasing current density. If plasma is forced into the arc mode, it will again tend to filament in order to reduce the voltage drop.
 

6.6 Current Filamentation

Filamentation is observed to be a normal behavior mode for currents in a plasma, as evidenced by the J-V curve and by physical structures in space itself.

A paper by Dr. Anthony Peratt regarding filamentation can be found here.

In particular, current sheets (which we will consider later) tend to break up into individual filaments due to the development of vortices. These vortices are somewhat similar to those found in fluid flows with adjacent layers of different flow velocities (Kelvin-Helmholtz instabilities).

Clearly, the conditions inside a current filament are going to be different to those in the rest of the plasma.

This causes a current-free double layer (CFDL) to form at the boundary of the filament in the normal way, such that the faster electrons are confined to the filament by the electric field within the DL.

We can now see that filaments are current-carrying elongated plasma cells with CFDLs at their boundaries.

Evidence of filaments and electric currents in space is widespread. Filamentary structure is acknowledged by most astronomers to exist at all levels, from the solar system to galactic and intergalactic scales.

The only area of disagreement between the Electric Model and the Gravity Model is whether these filaments are current-carrying structures, naturally following the laws of plasma electrodynamics, or somehow fluid 'jets' thousands of light-years long, gravitationally driven in accordance with computer simulations of the hypothesized gravity forces due to cold dark matter (CDM).

In a fluid, jets tend to dissipate rapidly into low-velocity plumes.

An aircraft's turbines expel jets of gas, seen here as contrails of ice crystals precipitating some distance aft of the engines, which quickly expand and decelerate to a stop in the upper atmosphere

However, some jets in space, for example the 4,000 light-years-long jet from the elliptical galaxy M87, appear to remain in the jet state for enormous distances before dissipating into a plume.

This might indicate that the jets are not fluid jets but electrical filaments.
 

The jet from galaxy M87.

Galaxy is the bright knot, upper left, in visible light (reddish);

the jet extends down and to the right, seen here in UV light (white and blue).

Image credit: NASA/ Hubble
 

A significant paper titled, "Measurement of the Current in a Kpc-Scaled Jet" was published in 2011 in arXiv by Kronberg, Lovelace, et al, based on their investigations of a jet emanating from radio galaxy 3C303.

If we assume they are electric filaments, then we need to know what theory and experiment might tell us about how electric filaments keep their shape over astronomical distances.

This is discussed next.

6.7 Current Pinches

Any current I flowing in a conductor or filament will cause a magnetic field B around it. The lines of equal magnetic force will be in the form of rings around the axis of the current.

The magnetic force will decrease with radial distance from the axis.

From consideration of the Lorentz Force, it can be shown that the interaction of the current I with its own magnetic field B will cause a pressure radially inward on the current filament, written as I × B (that is, "I cross B" in vector terminology).

This is called a 'pinch' or 'z-pinch' (when defining the current flow as parallel with the 'z' co-ordinate's direction).

In a metal conductor, the I × B pressure is resisted by the atomic ion lattice. In a plasma current, the pressure can be balanced by the pressure of the plasma inside the filament. This results in a steady state where the current can flow axially across its own azimuthal or circling magnetic field. The balancing equation is known as the Bennett Pinch equation.

Lab demonstrations can use the pinch effect to crush aluminum cans by applying a strong magnetic field very quickly. The can is crushed before the pressure in the can is able to build up sufficiently to resist the pinch force.

Magnetic field forces in lightning can create an inward pinch that will crush a solid copper grounding rod.
 

Left: The field generated by a fast 2 kj discharge through 3-turn heavy wire crushed this can.

Right: Nature's lightning z-pinch deformed this metal rod.

Images credit: Wiki Creative Commons

6.8 Field-Aligned Currents

In space, the neutral gas pressure is usually negligible, and so the balance between the I × B force and the pressure force cannot occur.

The only way the situation can be resolved is for the I × B force to disappear. This implies that I and B (current direction and magnetic field direction) are parallel and, by vector algebra, the cross product is zero.

If other magnetic fields are present, as they are known to be through much of cosmic space, then the I × B force must be calculated using the total magnetic field, that is, by adding the current's own B to the general B, added using vector algebra.

Thus in a space plasma, the current I and the total magnetic field B realign so as to be parallel. In other words, the current follows the magnetic field: it is a 'field-aligned' current.

Even if there is no external magnetic field, any small elements of current flowing in a plasma will tend to accumulate naturally into larger currents which generate their own magnetic fields and so preserve the filament of current.

What happens is that electrons nearer the centre of the filament flow in almost straight lines and generate an azimuthal magnetic field around them. Electrons further from the centre are influenced by this azimuthal component of the magnetic field and move in a more helical path aligned with the main current direction.

This helical motion creates the straighter magnetic field lines near the axis, as shown in the following diagram. The nearer the centre of the filament, the straighter are the magnetic field lines and the paths of the electrons.
 

Electron flows in a magnetic-field-aligned current

varying with distance from center of filament

Wiki Commons

Any individual electron in the current is thus flowing along the magnetic field direction in its own vicinity, but collectively the filament is preserved even without an external magnetic field. This means that very large currents can be assembled out of small current elements and transmitted over huge distances.

Another way of looking at this is to consider the electrical resistance of the plasma.

Current flowing across the magnetic field direction will experience more resistance than current flowing along the magnetic field direction because of the U × B term in the Lorentz Force Law.

Effectively, the parallel resistance is less than the perpendicular resistance, so the current tends to flow in alignment with the magnetic field.

6.9 Self-Constriction of Currents

Detailed mathematical analysis shows that I and B interact in such a way that both I and B tend to spiral parallel to each other around an axis aligned with the external B.

The net effect is that I and B both follow a helical path aligned with the direction of the external B field.

It is also found that the interaction of the axial and azimuthal (ring) components of the helical I and B cause both I and B to be largely confined to a cylinder of definite radius centered on the axis.

To summarize, the absence of significant pressure in space plasmas causes currents to flow in cylindrical filaments aligned with the general magnetic field direction. Within the cylindrical filament, both the current and the magnetic field will spiral around the axis of the cylinder whilst remaining parallel to each other.

Note that if for any reason the parallel alignment between I and the total B is disturbed, then an I × B force will arise and cause either radial compression or radial expansion, depending on which of the two components is more axial.

Thus pinching of a filament could occur because, for example, of changes in the fields through which the current filament was flowing.

6.10 Stability of Current Filaments

Another significant factor emerges from the mathematical analysis.

The force-free or field-aligned arrangement is a minimum energy state for the current to flow in. This means that the field-aligned arrangement is inherently stable. Unless disturbed by external factors, currents will tend to remain aligned with the magnetic field.

We can now see how field-aligned currents can persist over vast distances.

Field-aligned currents are therefore a much more likely explanation of the collimated (parallel-flow) 'jets' seen to be extending for hundreds to thousands of light-years than is the Gravity Model explanation based on conventional fluid flows.

The confinement of field-aligned filamentary currents to definite cylinders of current by electromagnetic forces is also consistent with the falling characteristic of the J-V curve seen in laboratory experiments in discharge tubes.

If the plasma is in Glow Mode, which in space plasmas may mean a glow in wavelengths outside the visible range, then the radius of the current cylinder will be determined by a combination of the effects of the electric and magnetic fields and the shape of the current density-Voltage curve.

Read more about the filamentation process in dense cosmic z-pinches in this paper by Russian physicists A.B Kukushkin and V.A. Rantsev-Kartinov of the Kurchatov Institute, Moscow.

6.11 Condensation of Matter
 

A further effect related to the I × B force can also be determined by analysis.

Suppose that the current I is caused by an electric field E. Now consider the force arising from the interaction of E and B. Remember that I tends to become aligned with the total B due to the forces on the current itself.

Then the E causing the current will not be entirely aligned with the total B, which is the vector sum of the external magnetic field through which the current flows and the azimuthal magnetic field generated by the current itself.
 

As with the I × B force, there is also an E × B force, whenever E is not parallel to B.

This E × B force acts on charged particles in the current cylinder and causes both ions and electrons to move towards the centre of a filament. Plasmas often contain a high proportion of charged dust grains, which will also be drawn into the filament. Viscous drag between the charged particles and neutral atoms will tend to draw the neutral atoms towards the filament as well.

Therefore, current filaments in space will tend accumulate matter in them as a result of the misalignment of the electric field causing the current and the total magnetic field.

Remembering that pinches can occur if any misalignment of I and B occurs, any matter that has been drawn into the filament will also be compressed if a misalignment of I and B occurs. If the pinch force is large enough, it can fragment the filament into discrete spherical or toroidal plasmoids along the axis of the current. Any matter in the pinch zone would then become compressed into the same form.

Because the electromechanical forces are vastly stronger than gravity, this mechanism offers a means by which diffuse matter can be accumulated and compressed in a much more efficient way than gravitational compression of diffuse clouds of fine dust particles.

Of course, once the matter has been sufficiently compressed and if it is neutralized by recombination of ions and electrons, then the electromagnetic forces may be reduced to the point that gravity becomes significant and continues the compression started by the electromagnetic forces.

6.12 Marklund Convection

In the case of a cylindrical current, the E × B force is radially inwards and results in the self-constriction of a current filament, as we have seen. This results in an increase in the particle density near the axis of the current.

Two things can then happen.

• The first is that radiative cooling from the regions of increased density can result in a temperature decrease nearer the center, contrary to the increase one might intuitively expect from increasing the density.

   

• The second is that recombination of ions and electrons starts to occur.

Every chemical element has a particular energy level, known as its ionization energy, at which it will either ionize or recombine. This is analogous to the boiling point of a liquid such as water: at a particular temperature, the phase or state of the matter will change from one state to another.

If the kinetic energy of motion is equated with the ionization energy, then a characteristic velocity, known as the Critical Ionization Velocity (CIV), can be derived for each element.

Because temperature is a measure of thermal energy, CIV can be related to temperature.

The CIV values of elements commonly found in space are not distributed randomly but are grouped into four distinct bands around certain velocity values. Within each band, all the elements in that band have similar CIVs to each other.

In the vicinity of a field-aligned current, the E × B force causes a radial drift of ions and electrons towards the cooler central axis. Because of their differing CIVs, different ions will recombine at different radii as they move towards the centre and enter progressively cooler regions.

This process is known as Marklund Convection after the Swedish physicist who discovered it, Göran Marklund.
 

Marklund convection and sorting

in a magnetically pinched current.

Image courtesy of Wal Thornhill, www.holoscience.com
 

The net result is that Marklund Convection sorts any elements present in the locality into different groups according to their ionization potentials. The groups of elements are arranged in cylindrical shells at different radii within a cylindrical field-aligned current.

As hydrogen has a high CIV compared to the other elements, it will recombine first, in a cylindrical shell of larger radius than the shells of the other elements.

This type of electrical sorting may be responsible for some of the non-random distribution of elements that we observe in the cosmos. In particular, it may explain the preponderance of neutral hydrogen in thread-like structures throughout the galaxy that have been detected by radio telescopes.
 

Could this Eagle Nebula image by the Hubble Space Telescope

be an illustration of a cosmic magnetic pinch and resultant dusty plasma

surrounded by a hydrogen-helium environment?

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Chapter 7 - Birkeland Currents, Magnetic Ropes and Current-Carrying Double Layers
January 4, 2012

7.1 Birkeland Currents
 

There is another cause of filamentation of currents in plasma.

This is due to the fact that there is a force of attraction between any two parallel currents. Each current generates a magnetic field which circles the first current

and attracts the other current according to the normal laws of electro-magnetism.

Therefore the two currents are drawn together.

This effect will apply to individual electron streams as well as to wires carrying currents. Therefore, in a plasma, a diffuse current will tend to become concentrated into a filament, as we have seen.

Similarly, a sheet of current will also tend to coalesce into individual filaments, rather like a sheet of falling water breaks up into individual streams.
 

Braided current sheets glow softly in visible and infrared light

along the Cygnus Loop of the Veil Nebula.

Image credit: W. P. Blair, R. Sankrit

Johns Hopkins University / NASA
 

If two parallel filaments occur in the same vicinity, or form out of a sheet of current due to the filamentation processes, then they will attract each other and initially move toward one another under under the magnetic attraction described by the Biot-Savart Law.

Therefore there is a tendency for current cylinders to occur in pairs.

The inverse-distance dependency of the Biot-Savart force law between current-conducting filaments leads - curiously - to pairing of filaments.

This shows 3 current filaments in a particle-in-cell (PIC) computer simulation, where only two will interact strongly while the third remains quiescent. This leads directly to "twoness" or "doubleness" when many filaments are present in a plasma with a significant magnetic field. Credit: adapted from Fig. 3.21, Physics of the Plasma Universe, Peratt, Springer Verlag,1992

A point of balance is reached when the long-range attraction force is balanced by a shorter-range repulsion between the two counter-parallel spiraling currents' azimuthal components.

Analysis shows that there is an offset in the centers of the attractive forces that results in a couple, or force of rotation, acting on each current.

The twin currents will therefore tend to spiral around a common axis in a helical motion. As before, the axis of the helix will tend to be aligned with the general magnetic field.

This arrangement of current pairs is known as a Birkeland Current after the Norwegian physicist Kristian Birkeland, who first studied them in the early part of the 20th Century.

7.2 Magnetic Ropes

The spiraling effect of currents around each other gives the appearance of twisted ropes.

Because the currents are aligned with the magnetic field, Birkeland Currents are often called 'magnetic ropes' or 'flux tubes'. Although not inaccurate, this description tends to disguise the current-carrying nature of the filaments and imply that the effect is due to magnetic forces alone.

As we have seen, this is not correct.

Birkeland currents can also attract matter from the surrounding region.

This is because the azimuthal magnetic fields created by each axial current form a pressure gradient radially inwards with a minimum between the two currents, while the magnetic fields extend beyond the current rope itself. This causes charged matter and ionized species external to the current rope to be attracted toward the centre of the current rope, the process known as Marklund Convection (see 6.12).

Although the effect is similar to the I × B force of a single current cylinder, the magnetic pressure minimum between the twin currents can be a more efficient mechanism for concentration of matter.

The plasma density outside the Birkeland Current is reduced while the density inside the rope is increased. Birkeland Currents are therefore often associated with density variations in the plasma.

7.3 Visible Effects of Currents in Space

 

Twisted current filaments

in the Double Helix Nebula near the center of the Milky Way,

in infrared light.

Image credit: NASA/JPL - CalTech/UCLA
 

Filamentary structures of the type just described are common in space: examples include,

• Auroral filaments

• flux ropes from Venus

• Solar prominences and coronal streamers

• cometary tails

• interstellar nebulae where webs of filaments are often seen

Filamentary neutral hydrogen structures have already been mentioned (see Marklund Convection in 6.12 above).

Filamentary structure has also been observed in the arrangement of clusters of galaxies.
 

7.4 Current-Carrying Double Layers

We have already seen that Double Layers can form in glow discharge tubes in the laboratory.

Obviously these DLs permit the transmission of current through them, as well as having the property of accelerating ions and electrons in the strong electric field within the DL.

To distinguish them from CFDLs they are known as Current-Carrying Double Layers (CCDL).

A CCDL forms in a different way from a CFDL. It is usually triggered by some form of instability or change in the current flow.

As an example of a change which causes a CCDL to form, consider what happens when a current passes into a region where the plasma density is lower. As the current is mainly carried by the lighter electrons we can consider the situation relative to the ions in the first instance.

If the electron current did not change then the lower density region would rapidly acquire an excess of electrons due to the 'stream' of incoming (electron) current. This would result in a potential difference in the lower density region which would repel further electrons and disrupt the current flow.

Remembering that current is proportional to the product of electron density and velocity, the only way for the electron density to be reduced to the appropriate level whilst the total current is maintained is for the electron velocity to be increased.

The way this is achieved is by formation of a CCDL at the boundary of the lower density region which accelerates the electrons into the region. The strength of the DL will increase until it is just sufficient to provide the electron velocity necessary to reduce their density to match the lower ion density and maintain charge neutrality.

Of course the ions are also affected by the DL but the overall effect is similar to that just described.

Also, the faster electrons can cause additional ionization which modifies the requirement for additional velocity but a DL will still be necessary to provide the necessary acceleration.

7.5 Flow Instabilities and CCDLs

CCDLs can also be formed as a result of flow instabilities in the counter-streaming electrons and ions comprising the current.

Various types of instability can occur.

One example is the Buneman or two-stream instability which occurs when the streaming velocity of the electrons (basically the current density divided by the electron density) exceeds the electron thermal velocity of the plasma. In other words, the drift velocity due to the current is higher than the random thermal velocity.

The actual mechanism of the Buneman instability is complicated. However, in essence, the density of ions and electrons in a plasma will always vary locally from absolute neutrality. The plasma then self-adjusts to correct any imbalance.

These density variations occur at a frequency dependent on the plasma temperature and the current passing through it. If the current density is high enough then the frequency of the density variations is too fast for the plasma to adjust itself. The situation has become unstable.

This type of instability has been found to lead to the formation of a CCDL. The variations in ion and electron densities cause local electric fields to develop. These fields exchange energy with the ions, which begin to oscillate with large amplitude and so amplify the density variations. Areas of differing charge density set up electric fields between them.

As the electric field increases due to these density variations, the electron flow in the current is disrupted and some electrons become 'trapped', or start to flow backwards in local vortices. The result is formation of a CCDL with populations of accelerated electrons and ions, and trapped electrons and ions downstream of the DL.

This process is similar in some respects to fluid flow instabilities.

The CCDL is in some ways like a hydraulic jump, where the fluid velocities are different on either side of the jump; the jump contains vortices of trapped fluid; and the jump itself is 'fixed' in position.

However this is not to say that fluid analyses are complex enough to model the electro-dynamical motions of charged particles in the fields they themselves create. A principle difference is that the DL accelerates particles, in opposite directions depending on their charge, while a hydraulic jump reduces the fluid flow velocity by introducing turbulence.

A CCDL will always concentrate a part of the current-producing potential drop within the DL region and so reduce the potential gradient in the remainder of the flow.

As CCDLs occur when changes in the flow characteristics occur, pinches in the current, where the area of the flow is constricted, may also cause DLs to form at the point where the flow area changes.

7.6 Energy Dissipation in DLs

Electrons accelerated across the potential drop of a CCDL will tend to lose their energy in collisions with neutral atoms beyond the DL.

These excited atoms will in turn lose energy by radiation as they return to the ground state. Formation of a DL therefore acts as a means whereby the plasma can dissipate excess energy in a manner analogous to a resistor in an electrical circuit.

This mechanism contributes to the stability of plasma circuits by 'safely' dissipating the energy which might otherwise result in more turbulent instabilities developing.

7.7 Classification of DLs

As already discussed, there is a principal difference between current-carrying double layers (CCDL) and current-free double layers (CFDL), which are formed by different mechanisms and distinguished by whether or not the DL allows a significant electric current to pass across it.

Another classification is based on the strength of the DL. Depending on the potential drop across it, a DL may be classified as weak, strong or relativistic. Each class will have different effects on charged particles in the surrounding plasma.

If the potential drop across the DL is larger than the plasma potential, then the DL is classified as a strong DL. A strong DL will reflect particles that approach the DL with energies less than the plasma potential. Only those particles with energies above the plasma potential will enter the DL and be accelerated.

A weak DL will decelerate particles with the plasma potential that approach from the 'wrong' side, but then re-accelerate them as they pass through the DL.

If the potential drop across the DL is sufficient to cause particles to acquire energy larger than the rest mass energy of the electron then it is known as a relativistic DL.

A relativistic DL will therefore accelerate electrons to near the speed of light as they pass through the potential drop.

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