Plasma scaling
Encyclopedia
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
s, including their spatial and temporal extent, vary by many orders of magnitude. Nevertheless, there are significant similarities in the behaviors of apparently disparate plasmas. Understanding the scaling
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
of plasma behavior is of more than theoretical value. It allows the results of laboratory experiments to be applied to larger natural or artificial plasmas of interest. The situation is similar to testing aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
or studying natural turbulent flow in wind tunnel
Wind tunnel
A wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.-Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...
s with smaller-scale models.
Similarity
Similitude (model)
Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity...
transformations (also called similarity laws) help us work out how plasma properties changes in order to retain the same characteristics. A necessary first step is to express the laws governing the system in a nondimensional
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...
form. The choice of nondimensional parameters is never unique, and it is usually only possible to achieve by choosing to ignore certain aspects of the system.
One dimensionless parameter characterizing a plasma is the ratio of ion to electron mass. Since this number is large, at least 1836, it is commonly taken to be infinite in theoretical analyses, that is, either the electrons are assumed to be massless or the ions are assumed to be infinitely massive. In numerical studies the opposite problem often appears. The computation time would be intractably large if a realistic mass ratio were used, so an artificially small but still rather large value, for example 100, is substituted. To analyze some phenomena, such as lower hybrid oscillation
Lower hybrid oscillation
A lower hybrid oscillation is a longitudinal oscillation of ions and electrons in a magnetized plasma. The direction of propagation must be very nearly perpendicular to the stationary magnetic field, within about √ radians...
s, it is essential to use the proper value.
A commonly used similarity transformation
One commonly used similarity transformation was derived for gas discharges by James Dillon Cobine (1941), Alfred Hans von Engel and Max Steenbeck (1934), and further applied by Hannes AlfvénHannes Alfvén
Hannes Olof Gösta Alfvén was a Swedish electrical engineer, plasma physicist and winner of the 1970 Nobel Prize in Physics for his work on magnetohydrodynamics . He described the class of MHD waves now known as Alfvén waves...
and Carl-Gunne Fälthammar
Carl-Gunne Fälthammar
Carl-Gunne Fälthammar is Professor Emeritus at the Royal Institute of Technology in Stockholm, Sweden, specialising in space and plasma physics in the School of Electrical Engineering...
to plasmas. They can be summarised as follows:
Similarity Transformations Applied to Gaseous Discharges and some Plasmas
| Property | Scale Factor | |
|---|---|---|
| length, time, inductance, capacitance | x1 | x |
| particle energy, velocity, potential, current, resistance | x0=1 | Unchanged |
| electric and magnetic fields, conductivity, neutral gas density, ionization fraction | x−1 | 1/x |
| current density, electron and ion densities | x−2 | 1/x2 |
This scaling applies best to plasmas with a relatively low degree of ionization. In such plasmas, the ionization energy of the neutral atoms is an important parameter and establishes an absolute energy scale, which explains many of the scalings in the table:
- Since the masses of electrons and ions cannot be varied, the velocities of the particles are also fixed, as is the speed of sound.
- If velocities are constant, then time scales must be directly proportional to distance scales.
- In order that charged particles falling through an electric potential gain the same energy, the potentials must be invariant, implying that the electric field scales inversely with the distance.
- Assuming that the magnitude of the E-cross-B drift is important and should be invariant, the magnetic field must scale like the electric field, namely inversely with the size. This is also the scaling required by Faraday's law of inductionFaraday's law of inductionFaraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...
and Ampère's lawAmpère's lawIn classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...
. - Assuming that the speed of the Alfvén waveAlfvén waveAn Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.-Definition:An Alfvén wave in a plasma is a low-frequency travelling oscillation of the ions and the magnetic field...
is important and must remain invariant, the ion density (and with it the electron density) must scale with B2, that is, inversely with the square of the size. Considering that the temperature is fixed, this also ensures that the ratio of thermal to magnetic energy, known as betaBeta (plasma physics)The beta of a plasma, symbolized by β, is the ratio of the plasma pressure to the magnetic pressure...
, remains constant. Furthermore, in regions where quasineutrality is violated, this scaling is required by Gauss's lawGauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
. - Ampère's law also requires that current density scales inversely with the square of the size, and therefore that current itself is invariant.
- The electrical conductivity is current density divided by electric field and thus scales inversely with the length.
- In a partially ionized plasma, the electrical conductivity is proportional to the electron density and inversely proportional to the neutral gas density, implying that the neutral density must scale inversely with the length, and ionization fraction scales inversely with the length.
Limitations
While these similarity transformations capture some basic properties of plasmas, not all plasma phenomena scale in this way. Consider, for example, the degree of ionization, which is dimensionless and thus would ideally remain unchanged when the system is scaled. The number of charged particles per unit volume is proportional to the current density, which scales as x -2, whereas the number of neutral particles per unit volume scales as x -1 in this transformation, so the degree of ionization does not remain unchanged but scales as x -1.Astrophysical application
As an example, take an auroral sheetAurora (astronomy)
An aurora is a natural light display in the sky particularly in the high latitude regions, caused by the collision of energetic charged particles with atoms in the high altitude atmosphere...
with a thickness of 1 km. A laboratory simulation might have a thickness of 10 cm, a factor of 104 smaller. To satisfy the condition of this similarity transformation, the gaseous density would have to be increased by a factor of 104 from 104 m−3 to 108 m−3 (1010 cm−3 to 1014 cm−3), and the magnetic field would have to be increased by the same factor from 50 microteslas to 500 milliteslas (0.5 gauss to 5 kilogauss). These values are large but within the range of technology. If the experiment captures the essential features of the aurora, the processes will be 104 times faster so that a pulse that takes 100 s in nature would take only 10 ms in the laboratory.
Similarity transformations applied to some astrophysical plasmas
Actual plasma properties compared to a laboratory plasma if the scale length is reduced to 10 cm.
| Region | Characteristic dimension (cm) | Density (particles/cm3) | Magnetic field (gauss) | Characteristic time | |||||
| Actual | Scaled | Scale Factor | Actual | Scaled | Actual | Scaled | Actual | Scaled | |
| Ionosphere Ionosphere The ionosphere is a part of the upper atmosphere, comprising portions of the mesosphere, thermosphere and exosphere, distinguished because it is ionized by solar radiation. It plays an important part in atmospheric electricity and forms the inner edge of the magnetosphere... | 106 - 107 | 10 | 10−5 - 10−6 | 1010 | 1015 - 1016 | 0.5 | 5x104 - 5x105 | Period of Giant pulsation | |
| 100 s | 0.1 - 1 ms | ||||||||
| Exosphere Exosphere The exosphere is the uppermost layer of Earth's atmosphere. In the exosphere, an upward travelling molecule moving fast enough to attain escape velocity can escape to space with a low chance of collisions; if it is moving below escape velocity it will be prevented from escaping from the celestial... | 109 | 10 | 10−8 | 105 - 10 | 1013 - 109 | 0.5 - 5x10−4 | 5x107 - 5x104 | One Day | |
| 105 s | 1 ms | ||||||||
| Interplanetary space | 1013 | 10 | 10−12 | 1 - 10 | 1012 - 1013 | 10−4 | 108 | One Solar Rotation | |
| 2x106 s | 2 μs | ||||||||
| Interstellar medium Interstellar medium In astronomy, the interstellar medium is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space... | 3x1022 | 10 | 3x10−22 | 1 | 3x1021 | 10−6 - 10−5 | 3x1015 - 3x1016 | Period of galactic rotation | |
| 1x1016 s | 3 μs | ||||||||
| Intergalactic space | >3x1027 | 10 | <3x10−27 | 10−4? | >3x1022 | 10−7? | >3x1019 | Age of the Universe | 4x1017s | 1x10−9s |
| Solar chromosphere Chromosphere The chromosphere is a thin layer of the Sun's atmosphere just above the photosphere, roughly 2,000 kilometers deep.... | 108 | 10 | 10−7 | 1011 - 1014 | 1018 - 1021 | 103 - 1 | 1010 - 107 | Life of Solar Flare | |
| 103 s | 100 μs | ||||||||
| Life of Solar Prominence | |||||||||
| 105 s | 10 ms | ||||||||
| Solar corona | 1010 - 1011 | 10 | 10−9 - 10−10 | 108 - 106 | 1017 - 1016 | 102 - 10−1 | 1011 - 109 | Life of Coronal Arc | |
| 103 s | 10−1 to 1 µs | ||||||||
| Solar Cycle | |||||||||
| 22 years | 70 to 700 ms | ||||||||
Particle density of the Earth's atmosphere at sea level is 1019 per cm3
Small bar magnet = 100 milliteslas. Big electromagnet = 2 teslas
109 cm = 10,000 km
The table shows the properties of some actual space plasma (see the columns labelled Actual). It also shows how other plasma properties would need to be changed, if (a) the characteristic length of a plasma were reduced to just 10 cm, and (b) the characteristics of the plasma were to remain unchanged.
The first thing to notice is that many cosmic phenomena cannot be reproduced in the laboratory because the necessary magnetic field strength is beyond the technological limits. Of the phenomena listed, only the ionosphere and the exosphere can be scaled to laboratory size. Another problem is the ionization fraction. When the size is varied over many orders of magnitude, the assumption of a partially ionized plasma may be violated in the simulation. A final observation is that the plasma densities needed in the laboratory are sizeable, up to 1016 cm−3 for the ionosphere, compared to the atmospheric density of about 1019 particles per cm3. In other words, the laboratory analogy of a low density space plasma is not a "vacuum chamber", but laboratory plasma with a pressure, when the higher temperature is taken into consideration, which can approach atmospheric pressure.
Dimensionless parameters in tokamaks
One of the central questions in fusion powerFusion power
Fusion power is the power generated by nuclear fusion processes. In fusion reactions two light atomic nuclei fuse together to form a heavier nucleus . In doing so they release a comparatively large amount of energy arising from the binding energy due to the strong nuclear force which is manifested...
research is to predict the energy confinement time in machines that are larger than any that have ever been built. A widely accepted approach to doing this is to express the scaling in terms of nondimensional parameters. Geometrical parameters, such as the ratio of the major to the minor radius, the shape of the plasma cross section, and the angle of the magnetic field, can be chosen in current experiments to equal the value desired for a full scale reactor. The remaining (dimensional) parameters can be taken to be the particle density n, the temperature T, the magnetic field B, and the size (major radius) R. These can be combined into the three dimensionless parameters β (the ratio of plasma pressure to magnetic pressure), ν* (the product of the collision frequency and the thermal transit time), and ρ* (the ratio of the Larmor radius to the torus radius). These have the following scalings:
- β ~ nTB -2
- ν* ~ nT -2R
- ρ* ~ T 1/2B -1R -1
The radius R can be varied while keeping these three parameters constant if n, T, and B are scaled in this way:
- n ~ R -2
- T ~ R -1/2
- B ~ R -5/4
Note that this similarity transformation is distinct from that considered above, which would yield n ~ R -1, T ~ R 0, and B ~ R -1. This is because the physical effects to be studied are different.
The scaling of the magnetic field with the minus 5/4 power of the size implies that a 1:3 scale model of a power-producing tokamak
Tokamak
A tokamak is a device using a magnetic field to confine a plasma in the shape of a torus . Achieving a stable plasma equilibrium requires magnetic field lines that move around the torus in a helical shape...
with a magnetic field of 10 T at the coils would require a field of about 40 T, which is technologically infeasible.
The next best alternative is to allow ρ* to vary and to extrapolate according to the dependence found. ρ* is the parameter considered least likely to harbor surprises, partly for theoretical considerations, but also simply because it is, in contrast to β and ν*, already much larger than unity. This can be done in a single machine (constant R) by varying the magnetic field and scaling density and temperature as:
- n ~ B 4/3
- T ~ B 2/3
It should be kept in mind that the assumption has been made that the important turbulent transport processes depend only on the parameters chosen. It is only physical reasoning, not mathematical necessity, that concludes that the ratio of the torus radius to the Larmor radius is important, and not, for example, the ratio to the Debye length
Debye length
In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...
. In the same way, it has been assumed that the absolute energy levels of atomic physics do not dictate an absolute temperature dependence, or equivalently, that the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...
where atomic physics is important, is small enough not to determine the overall energy confinement.