The parameters of

plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...

s, including their spatial and temporal extent, vary by many

orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

. Nevertheless, there are significant similarities in the behaviors of apparently disparate plasmas. Understanding the

scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...

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

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or studying natural

turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...

wind tunnel Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...

s with smaller-scale models. Similarity transformations (also called similarity laws) help us work out how plasma properties change 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 oscillation In plasma physics, 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. Othe ...

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),.von Engel, A., and Steenbeck, M., 1934: ''ElektrischeGasentladungen'', Springer-Verlag, Berlin. See also von Engel, 1955: ''Ionized Gases'', Clarendon Press, Oxford They can be summarised as follows: 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 induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...

and Ampère's law. * Assuming that the speed of the

Alfvén wave In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines. Definition An Alfvén wave is a ...

is important and must remain invariant, the ''ion density'' (and with it the electron density) must scale with ''B''², 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

beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...

, remains constant. Furthermore, in regions where quasineutrality is violated, this scaling is required by

Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...

. * 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.

References

{{reflist Plasma physics Space plasmas