Unmagnetized Limit

In the unmagnetized limit, which actually corresponds to

$${\mit\Omega}_i\,\tau_i, \,\,\,{\mit\Omega}_e\,\tau_e \ll 1,$$ (4.80)

the standard two-Laguerre-polynomial Chapman-Enskog closure scheme yields

$${\bf F} = \frac{n\,e}{\sigma_\parallel}\,{\bf j} - 0.71\,n\,\nabla T_e,$$ (4.81)

$$w_i = \frac{3\,m_e}{m_i} \frac{n\,(T_e-T_i)}{\tau_e},$$ (4.82)

$$w_e = -w_i + \frac{ {\bf j}\cdot {\bf F} }{n \,e}= -w_i + \frac{j^{2}}{\sigma_\parallel} - 0.71\,\frac{{\bf j}\cdot \nabla T_e}{e}.$$ (4.83)

Here, ${\bf j} = - n\,e\,({\bf V}_e-{\bf V}_i)$ is the net plasma current density, and the electrical conductivity, $\sigma_\parallel$, is given by

$$\sigma_\parallel = 1.96\,\frac{n \,e^2\,\tau_e}{m_e}.$$ (4.84)

Moreover, use has been made of the conservation law, Equation (4.32).

Let us examine each of the previous collisional terms, one by one. The first term on the right-hand side of Equation (4.81) is a friction force (per unit volume) caused by the relative motion of electrons and ions, and obviously controls the electrical conductivity of the plasma. The form of this term is fairly easy to understand. The electrons lose their ordered velocity with respect to the ions, ${\bf U} = {\bf V}_e - {\bf V}_i$, in an electron-ion collision time, $\tau_e$, and consequently lose momentum $m_e\,{\bf U}$ per electron (which is given to the ions) in this time. This means that a frictional force $(m_e\,n/\tau_e)\,{\bf U} \sim n\,e\,{\bf j}/(n\,e^2\,\tau_e/m_e)$ is exerted on the electrons. An equal and opposite force is exerted on the ions. Because the Coulomb cross-section diminishes with increasing electron energy (i.e., $\tau_e\sim T_e^{3/2}$), the conductivity of the fast electrons in the distribution function is higher than that of the slow electrons (because $\sigma_\parallel \sim \tau_e$). Hence, electrical current in plasmas is carried predominately by the fast electrons. This effect has some important and interesting consequences.

One immediate consequence is the second term on the right-hand side of Equation (4.81), which is called the thermal force. To understand the origin of a frictional force (per unit volume) proportional to minus the gradient of the electron temperature, let us assume that the electron and ion fluids are at rest (i.e., $V_e=V_i =0$). It follows that the number of electrons moving from left to right (along the $x$-axis, say) and from right to left per unit time is exactly the same at a given point (coordinate $x_0$, say) in the plasma. As a result of electron-ion collisions, these fluxes experience frictional forces, ${\bf F}_-$ and ${\bf F}_+$, respectively, of approximate magnitude $m_e\,n\,v_e/\tau_e$, where $v_e$ is the electron thermal velocity. In a completely homogeneous plasma, these forces balance exactly, and so there is zero net frictional force. Suppose, however, that the electrons coming from the right are, on average, hotter than those coming from the left. It follows that the frictional force ${\bf F}_+$ acting on the fast electrons coming from the right is less than the force ${\bf F}_-$ acting on the slow electrons coming from the left, because $\tau_e$ increases with electron temperature. As a result, there is a net frictional force acting to the left: that is, in the direction of $-\nabla T_e$.

Let us estimate the magnitude of the frictional force. At point $x_0$, collisions are experienced by electrons that have traversed distances of similar magnitude to a mean-free-path, $l_e\sim v_e\,\tau_e$. Thus, the electrons coming from the right originate from regions in which the temperature is approximately $l_e\,\partial T_e/\partial x$ greater than the regions from which the electrons coming from the left originate. Because the friction force is proportional to $T_e^{-1}$, the net force ${\bf F}_+ - {\bf F}_-$ is approximately

$${\bf F}_T \sim- \frac{l_e}{T_e} \frac{\partial T_e}{\partial x} \... ...n\,\frac{\partial T_e} {\partial x} \sim - n\,\frac{\partial T_e} {\partial x}.$$ (4.85)

It must be emphasized that the thermal force is a direct consequence of collisions, despite the fact that the expression for the thermal force does not contain $\tau_e$ explicitly.

The term $w_i$, specified in Equation (4.82), represents the rate (per unit volume) at which energy is acquired by the ions due to collisions with the electrons. The most striking aspect of this term is its smallness (note that it is proportional to an inverse mass ratio, $m_e/m_i$). The smallness of $w_i$ is a direct consequence of the fact that electrons are considerably lighter than ions. Consider the limit in which the ion mass is infinite, and the ions are at rest on average: that is, $V_i=0$. In this case, collisions of electrons with ions take place without any exchange of energy. The electron velocities are randomized by the collisions, so that the energy associated with their ordered velocity, ${\bf U} = {\bf V}_e - {\bf V}_i$, is converted into heat energy in the electron fluid [this is represented by the second term on the extreme right-hand side of Equation (4.83)]. However, the ion energy remains unchanged. Let us now assume that the ratio $m_i/m_e$ is large, but finite, and that $U=0$. If $T_e=T_i$ then the ions and electrons are in thermal equilibrium, so no heat is exchanged between them. However, if $T_e>T_i$ then heat is transferred from the electrons to the ions. As is well known, when a light particle collides with a heavy particle, the order of magnitude of the transferred energy is given by the mass ratio $m_1/m_2$, where $m_1$ is the mass of the lighter particle. For example, the mean fractional energy transferred in isotropic scattering is $2\,m_1/m_2$. Thus, we would expect the energy per unit time transferred from the electrons to the ions to be roughly

$$w_i \sim \frac{n}{\tau_e} \,\frac{2\,m_e}{m_i}\,\frac{3}{2} \,(T_e-T_i).$$ (4.86)

In fact, $\tau_e$ is defined so as to make the previous estimate exact.

The term $w_e$, specified in Equation (4.83), represents the rate (per unit volume) at which energy is acquired by the electrons because of collisions with the ions, and consists of three terms. Not surprisingly, the first term is simply minus the rate at which energy is acquired by the ions due to collisions with the electrons. The second term represents the conversion of the ordered motion of the electrons, relative to the ions, into random motion (i.e., heat) via collisions with the ions. This term is positive definite, indicating that the randomization of the electron ordered motion gives rise to irreversible heat generation. Incidentally, this term is usually called the ohmic heating term. Finally, the third term represents the work done against the thermal force. This term can be either positive or negative, depending on the direction of the current flow relative to the electron temperature gradient, which indicates that work done against the thermal force gives rise to reversible heat generation. There is an analogous effect in metals called the Thomson effect (Doolittle 1959).

The electron and ion heat flux densities are given by

$${\bf q}_e = -\kappa_\parallel^e\,\nabla T_e - 0.71\,\frac{T_e}{e}\,{\bf j},$$ (4.87)

$${\bf q}_i = -\kappa_\parallel^i\,\nabla T_i,$$ (4.88)

respectively. The electron and ion thermal conductivities are written

$$\kappa_\parallel^e = 3.2\,\,\frac{n\,\tau_e\,T_e}{m_e},$$ (4.89)

$$\kappa_\parallel^i = 3.9\,\,\frac{n\,\tau_i\,T_i}{m_i},$$ (4.90)

respectively.

It follows, by comparison with Equations (4.63)–(4.68), that the first term on the right-hand side of Equation (4.87), as well as the expression on the right-hand side of Equation (4.88), represent straightforward random-walk heat diffusion, with frequency $\nu$, and step-length $l$. Recall, that $\nu=\tau^{-1}$ is the collision frequency, and $l=\tau\,v_t$ is the mean-free-path. The electron heat diffusivity is generally much greater than that of the ions, because $\kappa_\parallel^e/\kappa_\parallel^i\sim \sqrt{m_i/m_e}$, assuming that $T_e\sim T_i$.

The second term on the right-hand side of Equation (4.87) describes a convective heat flux due to the motion of the electrons relative to the ions. To understand the origin of this flux, we need to recall that electric current in plasmas is carried predominately by the fast electrons in the distribution function. Suppose that $U$ is non-zero. In the coordinate system in which $V_e$ is zero, more fast electrons move in the direction of ${\bf U}$, and more slow electrons move in the opposite direction. Although the electron fluxes are balanced in this frame of reference, the energy fluxes are not (because a fast electron possesses more energy than a slow electron), and heat flows in the direction of ${\bf U}$: that is, in the opposite direction to the electric current. The net heat flux density is of approximate magnitude $n\,T_e\,U$, because there is no near cancellation of the fluxes due to the fast and slow electrons. Like the thermal force, this effect depends on collisions, despite the fact that the expression for the convective heat flux does not contain $\tau_e$ explicitly.

Finally, the electron and ion viscosity tensors take the form

$$\pi_{e\,\alpha\beta} = - \eta_0^e\, \left( \frac{\partial V_\alpha}{\partial r_\beta} ... ...rtial r_\alpha} - \frac{2}{3}\,\nabla\cdot{\bf V}\,\delta_{\alpha\beta}\right),$$ (4.91)

$$\pi_{i\,\alpha\beta} = - \eta_0^i\, \left( \frac{\partial V_\alpha}{\partial r_\beta} ... ...tial r_\alpha} - \frac{2}{3}\,\nabla\cdot{\bf V}\,\delta_{\alpha\beta} \right),$$ (4.92)

respectively. Obviously, $V_\alpha$ refers to a Cartesian component of the electron fluid velocity in Equation (4.91) and the ion fluid velocity in Equation (4.92). Here, the electron and ion viscosities are given by

$$\eta_0^e = 0.73\,n\,\tau_e\,T_e,$$ (4.93)

$$\eta_0^i = 0.96\,n\,\tau_i\,T_i,$$ (4.94)

respectively. It follows, by comparison with Equations (4.62)–(4.68), that the previous expressions correspond to straightforward random-walk diffusion of momentum, with frequency $\nu$, and step-length $l$. Again, the electron diffusivity exceeds the ion diffusivity by the square root of a mass ratio (assuming $T_e\sim T_i$). However, the ion viscosity exceeds the electron viscosity by the same factor (recall that $\eta\sim n\,m\,\chi_v$): that is, $\eta_0^i/\eta_0^e\sim\sqrt{m_i/m_e}$. For this reason, the viscosity of a plasma is determined essentially by the ions. This is not surprising, because viscosity is the diffusion of momentum, and the ions possess nearly all of the momentum in a plasma by virtue of their large masses.