The parameters of plasmas, including their spatial and temporal extent, vary by many orders of magnitude. Nevertheless, there are significant similarities in the behaviors of apparently disparate plasmas. It is not only of theoretical interest to understand the scaling of plasma behavior, it also allows the results of laboratory experiments to be applied to larger natural or artificial plasmas of interest. The situation is similar to testing aircraft or studying natural turbulent flow in wind tunnels.
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 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 oscillations, it is essential to use the proper value.
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 Alfvén and Carl-Gunne Fälthammar to plasmas. They can be summarised as follows:
Property | Scale Factor | |
---|---|---|
length, time, inductance, capacitance | x^{1} | x |
particle energy, velocity, potential, current, resistance | x^{0}=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/x^{2} |
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:
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}.
As an example, take an auroral sheet with a thickness of 1 km. A laboratory simulation might have a thickness of 10 cm, a factor of 10^{4} smaller. To satisfy the condition of this similarity transformation, the gaseous density would have to be increased by a factor of 10^{4} from 10^{4} m^{-3} to 10^{8} m^{-3} (10^{10} cm^{-3} to 10^{14} 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 10^{4} times faster so that a pulse that takes 100 s in nature would take only 10 ms in the laboratory.
Region | Characteristic dimension (cm) | Density (particles/cm^{3}) | Magnetic field (gauss) | Characteristic time | |||||
Actual | Scaled | Scale Factor | Actual | Scaled | Actual | Scaled | Actual | Scaled | |
Ionosphere | 10^{6} - 10^{7} | 10 | 10^{-5} - 10^{-6} | 10^{10} | 10^{15} - 10^{16} | 0.5 | 5x10^{4} - 5x10^{5} | Period of Giant pulsation | |
100 s | 0.1 - 1 ms | ||||||||
Exosphere | 10^{9} | 10 | 10^{-8} | 10^{5} - 10 | 10^{13} - 10^{9} | 0.5 - 5x10^{-4} | 5x10^{7} - 5x10^{4} | One Day | |
10^{5 }s | 1 ms | ||||||||
Interplanetary space | 10^{13} | 10 | 10^{-12} | 1 - 10 | 10^{12} - 10^{13} | 10^{-4} | 10^{8} | One Solar Rotation | |
2x10^{6 }s | 2 μs | ||||||||
Interstellar space | 3x10^{22} | 10 | 3x10^{-22} | 1 | 3x10^{21} | 10^{-6} - 10^{-5} | 3x10^{15} - 3x10^{16} | Period of galactic rotation | |
1x10^{16 }s | 3 μs | ||||||||
Intergalactic space | >3x10^{27} | 10 | <3x10^{-27} | 10^{-4}? | >3x10^{22} | 10^{-7}? | >3x10^{19} | Age of the Universe | |
4x10^{17}s | 1x10^{-9}s | ||||||||
Solar chromosphere | 10^{8} | 10 | 10^{-7} | 10^{11} - 10^{14} | 10^{18} - 10^{21} | 10^{3} - 1 | 10^{10} - 10^{7} | Life of Solar Flare | |
10^{3 }s | 100 μs | ||||||||
Life of Solar Prominence | |||||||||
10^{5 }s | 10 ms | ||||||||
Solar corona | 10^{10} - 10^{11} | 10 | 10^{-9} - 10^{-10} | 10^{8} - 10^{6} | 10^{17} - 10^{16} | 10^{2} - 10^{-1} | 10^{11} - 10^{9} | Life of Coronal Arc | |
10^{3 }s | 10^{-1} to 1 µs | ||||||||
Solar Cycle | |||||||||
22 years | 70 to 700 ms |
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 10^{16} cm^{-3} for the ionosphere, compared to the atmospheric density of about 10^{19} particles per cm^{3}. 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.
One of the central questions in fusion power 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:
The scaling of the magnetic field with the minus 3/4 power of the size implies that a 1:3 scale model of a power-producing tokamak with a magnetic field of 10 T at the coils would require a field of 30 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:
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. 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 where atomic physics is important, is small enough not to determine the overall energy confinement.