Monitoreo de la calidad del aire en Montevideo

An alternative set of macroscopic equations is obtained by intro- ducing truncation at the energy-conservation equation. Thermal motions are accounted for but it is assumed that the kinetic pressure tensor is diagonal, with equal diagonal terms, so that∇ ·  = ∇ p. Physically, this means that viscous forces are neglected. We then have ∂ N ∂t + ∇ · [Nu] = 0 (4.33a) mN _∂u ∂t + (u · ∇)u = −∇ p + q N(E + u × B) + Si j. (4.33b)

The system (4.33) still does not form a closed set, since the

scalar pressure is now a third variable. In principle, the energy-

conservation equation (4.27) is needed to determine p, but it

contains a fourth unknown, q. However, it is often possible to truncate the system of equations at this point by adopting simpli- fying assumptions which either make the energy-transport equation

4.6 Summary 103

unnecessary or reduce it to simpler forms expressed in terms of p.5 The simplest method of truncation is to assume a thermodynamic equation of state in order to relate p to the number density N. The actual form of the equation of state varies from case to case. The two most common equations of state are the so-called isothermal and adiabatic ones. The isothermal equation of state is

p= Nk_BT or ∇ p = kBT∇ N, (4.34)

and holds for relatively slow time variations, allowing temperatures to reach equilibrium. In this case, the plasma fluid can exchange energy with its surroundings, and the simpler version of the energy- conservation equation,(4.28), is also required. Alternatively, we can use the adiabatic equation of state given by

p N−γ = C or ∇ p

p = γ

∇ N

N , (4.35)

where C is a constant andγ is the ratio of specific heat at constant pressure to that at constant volume. Typically,γ = 1 + 2/nd, where

n_d is the number of degrees of freedom. The adiabatic relation holds for fast time variations, as in the case of plasma waves, when the plasma fluid does not exchange energy with its surroundings; thus a separate energy-conservation equation is not needed, since it leads to (4.34). The use of the adiabatic gas law to close the system of equations is equivalent to assuming that there is no energy interchange due to collisional interactions and that there is no heat flow. In cases where the explicit use of the energy conservation equation is required, the heat-flow (or heat-flux) vector q would be the outstanding unknown, and we would need to adopt a physical assumption in order to close the system of moment equations. For electrons, the approximation most commonly used is one which is derived from thermodynamics:

q= −K ∇T,

where K is the thermal conductivity.

4.6 Summary

In this chapter we set out to avoid solving the Boltzmann equa- tion for the velocity distribution function since this is often not straightforward and since the averages obtained by integrating over

5 _{For simplifications particularly suited to ionospheric and magnetospheric plasmas, see}

the distribution function are more useful in practical applications. Instead, we set out to solve for the moments of the distribution, which are averages of different powers of particle velocity. The moments of the distribution function represent average quantities such as particle density (zeroth-order moment) and mean veloc- ity (first-order moment) and kinetic energy (third-order moment). These average bulk quantities are the primary variables in modeling a plasma as a fluid, which we will discuss in the next chapters. Taking any moment of the Boltzmann equation yields an expression that contains the next-highest moment. In principle, the velocity distribution function is known once we know all of its moments. However, finding a large number of successively higher moments is not practical.

The power of the moment approach lies in truncating a finite set of moment equations using an approximation for the highest moment appearing in the system. Such a truncation yields a closed set of equations that can be solved, and are known as a plasma model. We presented two of the most commonly used plasma models: the cold- and warm-plasma approximations. In the cold-plasma model, thermal motions are assumed to be negligible, which means that the pressure term in the moment equations is equal to zero. In the warm-plasma model, thermal motions are taken into account using either an isothermal or an adiabatic approximation, and the heat-flow term is set to be zero. In the warm-plasma model the pressure term is a scalar value like that used to describe non-ionized gases. Although not covered in this text, it is possible to develop more complicated plasma models where truncation is done at higher-level moments.

4.7 Problems

4-1. Using the ionospheric parameters given in Example 4-1, use a numerical technique to find how much time after sunrise it will take the electron density to change from its nighttime ambient value to its daytime value. Make a plot of the electron density versus time. You may ignore any plasma flows.

4-2. In a laboratory plasma experiment a plasma density of 1019m−3 is created by a rapid burst of ultraviolet radiation. The plasma density is observed to decay to half of its original value in 10 ms. Find the value of the recombination coefficient να, assuming that attachment is negligible.

4-3. The strength of the Earth’s magnetic field at the surface is approximately 30 μT. Show that this field has a negligible

References 105

effect on the physics of the fluorescent lamp discussed in Example 4.2.

4-4. For the fluorescent lamp discussed in Example 4.2, calculate the Debye length for the electrons in the positive column region and compare it to the dimensions of the lamp.

4-5. An engineer proposes to double the length of the 120 cm fluor- escent lamp in Example 4-2. If making the tube longer causes the electric field in the positive column to drop by a factor of two, calculate the power the lamp will draw if the electron and gas density, temperature, and voltage all remain the same. 4-6. Consider the one-dimensional plasma configuration shown

below, which is clearly not in equilibrium. Calculate an electric field that could be used to maintain the density profile, assum- ing mobile electrons and stationary ions.

1018 _m–3

0 1 cm

Distance

Plasma density

99 cm 100 cm

4-7. Consider a 200 V m−1 electric field applied to a partially ionized plasma with an electron density of 1015 m−3. The effective collision frequency for the electrons is 3.5 GHz. Using the cold-plasma model and ignoring ion motion, (a) find the steady-state electron fluid velocity; (b) use the energy trans- port equation to find the energy dissipated, ignoring convective terms and interactions with other species.

References

[1] G. G. Lister, Low pressure gas discharge modelling. J. Phys. D: Appl.

Phys., 25 (1992), 1649–80.

[2] G. G. Lister, J. E. Lawler, W. P. Lapatovich, and V. A. Godyak, The physics of discharge lamps. Rev. Mod. Phys., 76 (2004), 541–98. [3] P. Baille, J.-S. Chang, A. Claude, et al., Effective collision frequency of

electrons in noble gases. J. Phys. B: At., Mol. Opt. Phys., 14 (1981), 1485–95.

[4] T. E. Cravens, Physics of Solar System Plasmas (Cambridge: Cambridge University Press, 1997).

5

Multiple-fluid theory of plasmas