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 airplanes or studying natural turbulent flow in wind tunnels. The word plasma has a Greek root which means to be formed or molded (the word plastic shares this root). ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... The term scaling can have several manings: Scaling can be defined as the determination of the interdependency of variables in a physical system. ... Fixed-wing aircraft is a term used to refer to what are more commonly known as aeroplanes in Commonwealth English (excluding Canada) or airplanes in North American English. ... Turbulent flow around an obstacle; the flow further away is laminar Laminar and turbulent water flow over the hull of a submarine Turbulence creating a vortex on an airplane wing In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by low-momentum diffusion, high momentum convection, and... A wind tunnel is a research tool developed to assist with studying the effects of air moving over or around solid objects. ...

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. Several equivalence relations in mathematics are called similarity. ... Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

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. A lower hybrid oscillation is a longitudinal oscillation of ions and electrons in a magnetized plasma. ...

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 Alfvén and Carl-Gunne Fälthammar to plasmas. They can be summarised as follows: Hannes Olof Gösta Alfvén (May 30, 1908; Norrköping, Sweden - April 2, 1995; Djursholm, Sweden) was a Swedish electrical power engineer. ...

The table shows, for example, that if the length of a plasma region decreases by x, (eg. from 1km to 1m, then x=1000) then:

• the time taken for the phenomenon also decreases by x.
• the particle energy, velocity, potential, current, resistance of the plasma are all unchanged.
• the electric and magnetic fields would need to be increased by x (ie. it is inversely proportional to the length), in order for the plasma to behave in the same way.
• the current density and space charge density inceases by x²

Explanation

Maxwell's equations require that time and length are proportional to each other. The interactions between the particles in the plasma (electrons, positive ions), require that the energies and hence the electrostatics potentials, remain the same. For example if the linear dimensions of a plasma (l) are decreased by x: Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

• the electric field (E) must be increased by x since the voltage is proportional to l.E.
• From Maxwell's equations, the electric displacement field, magnetic field strength and magnetic flux density must change in the same way as the electric field.
• the current density (i), which is equivalent to the displayment current, must be increased by x² since the total current, I = i.l² is unchanged.
• the conductivity (i/E) changes inversely as the length.
• the inductance (dI/dt) and the capacitance is proportional to the length.

Limitations

Alfvén and Fälthammar note that while the similarity transformations are useful for some basic properties of plasma, they do not necessarily work for other. For example, hydrodynamic waves do not obey the transformation. See magnetohydrodynamics. MHD Simulation of Solar Wind Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics), is the academic discipline which studies the dynamics of electrically conducting fluids. ...

Cosmic Application

As an example, if we were to decrease the size of a 1 km auroral sheet down in scale to a 1 m laboratory simulation (i.e. a 1000-fold decease in magnitude), then we would find that it would appear a 1000 times faster, and would require a magnetic field 1000 times greater. Aurora borealis For other meanings, see Aurora In astronomy, an aurora is an optical phenomenon characterised by colourful displays of light in the night sky, caused by the interaction of charged particles from the solar wind with the upper atmosphere of a planet. ...

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 10cm, and (b) the characteristics of the plasma were to remain unchanged.

Of particular interest is that when a cosmic plasma is reduced in scale length, the plasma density needs to be increased to compensate. For example, the Solar Chromosphere if reduced in scale by about 10^-7, would be simulated by a laboratory plasma with a particle density of the order 10¹⁸ - 10²¹ particles per cm³; atmospheric pressure has about 10¹⁹ particles per cm³. In other words, the laboratory analogy of a low density space plasma is not a "vacuum chamber", but a high density laboratory plasma.

It also shows that cosmic plasmas with high magnetic fields can not be reproduced in the laboratory because technology is currently not able to create higher magnetic fields. The consequence of this is that charged particles moving in very highly magnetised space plasmas, are somewhat different to what is seen in the laboratory; charged particles in highly magnetised space plasmas move with a small radius of curvature. Additionally, charged particles (ions) that move across a magnetic field generate a strong electric field. For example, when an ion moves at 3 x 10⁵cm per second in a laboratory reduced magnetic field of 10⁶ gauss, it produces an electric field of 10esu = 3000V/cm. The same particle moving in the magentic field of a cosmic plasma of strength 10¹⁰ gauss, produces an electric field strength of 3 x 10⁷V/cm.

The table also shows that phenomenon such as solar flares are comparative short-lived, but that such transient phenomenon are still important.

Dimensionless parameters in tokamaks

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 n, T, B, and 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 Sun is a natural fusion reactor. ...

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 ^-3/4

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 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^*, the parameter considered least likely to harbor surprises, to vary and to extrapolate according to the dependence found. 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. 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.

References

• Alfvén, Hannes, and Fälthammar, Carl-Gunne, Cosmical Electrodynamics, 2nd Ed. 1963.
• Cobine, J. D ., 1941 : Gaseous Conductors, McGraw-Hill . New York.
• von Engel, A., and Steenbeck, M., 1934: ElektrischeGasentladungen, Springer-Verlag, Berlin. See also von Engel, 1955 : Ionized Gases, Clarendon Press, Oxford.