Braginskii Equations

Let now consider the problem of closure in plasma fluid equations. There are, in fact, two possible small parameters in plasmas upon which we could base an asymptotic closure scheme. The first is the ratio of the mean-free-path, $l$, to the macroscopic lengthscale, $L$. This is only appropriate to collisional plasmas. The second is the ratio of the Larmor radius, $\rho$, to the macroscopic lengthscale, $L$. This is only appropriate to magnetized plasmas. There is, of course, no small parameter upon which to base an asymptotic closure scheme in a collisionless, unmagnetized plasma. However, such systems occur predominately in accelerator physics contexts, and are not really plasmas at all, because they exhibit virtually no collective effects. Let us investigate Chapman-Enskog-like closure schemes in a collisional, quasi-neutral plasma consisting of equal numbers of electrons and ions. We shall treat the unmagnetized and magnetized cases separately.

The first step in our closure scheme is to approximate the actual collision operator for Coulomb interactions by an operator that is strictly bilinear in its arguments. (See Section 3.10.) Once this has been achieved, the closure problem is formally of the type that can be solved using the Chapman-Enskog method.

The electron-ion and ion-ion collision times are written

$$\tau_e = \frac{6\!\sqrt{2}\pi^{3/2}\,\epsilon_0^{2}\,\sqrt{m_e}\,\,T_e^{3/2}} {\ln{\mit\Lambda}_c\, e^4\, n},$$ (4.70)

and

$$\tau_i = \frac{ 12\pi^{3/2}\,\epsilon_0^{2}\,\sqrt{m_i}\,\,T_i^{3/2}} {\ln{\mit\Lambda}_c\, e^4\, n},$$ (4.71)

respectively. (See Section 3.14.) Here, $n=n_e=n_i$ is the number density of particles, and $\ln{\mit\Lambda}_c$ is the Coulomb logarithm, whose origin is the slight modification to the collision operator mentioned previously. (See Section 3.10.)

The basic forms of Equations (4.70) and (4.71) are not hard to understand. From Equation (4.58), we expect

$$\tau \sim \frac{l}{v_t} \sim \frac{1}{n\,\sigma^2\,v_t},$$ (4.72)

where $\sigma^2$ is the typical “cross-section” of the electrons or ions for Coulomb “collisions” (i.e., large angle scattering events). Of course, this cross-section is simply the square of the distance of closest approach, $r_c$, defined in Equation (1.17). Thus,

$$\tau \sim \frac{1}{n\,r_c^{2}\,v_t} \sim \frac{\epsilon_0^{2}\sqrt{m}\,\,T^{3/2}} {e^4\,n}.$$ (4.73)

The most significant feature of Equations (4.70) and (4.71) is the strong variation of the collision times with temperature. As the plasma gets hotter, the distance of closest approach gets smaller, so that both electrons and ions offer much smaller cross-sections for Coulomb collisions. The net result is that such collisions become far less frequent, and the collision times (i.e., the mean times between $90^\circ$ degree scattering events) get much longer. It follows that as plasmas are heated they become less collisional very rapidly.

The electron and ion fluid equations in a collisional plasma take the form [see Equations (4.47)–(4.49)]:

$$\frac{d_e n}{dt} + n\,\nabla\cdot{\bf V}_e =0,$$ (4.74)

$$m_e \,n\,\frac{d_e {\bf V}_e}{dt} + \nabla p_e+ \nabla\cdot \mbox{\boldmath$\pi$} _e + e\, n\, ({\bf E} + {\bf V}_e\times {\bf B}) = {\bf F},$$ (4.75)

$$\frac{3}{2}\frac{d_e p_e}{dt} + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e + \mbox{\boldmath$\pi$} _e:\nabla{\bf V}_e+ \nabla\cdot{\bf q}_e = w_e,$$ (4.76)

and

$$\frac{d_i n}{dt} + n\,\nabla\cdot{\bf V}_i =0,$$ (4.77)

$$m_i \,n\,\frac{d_i {\bf V}_i}{dt} + \nabla p_i + \nabla\cdot \mbox{\boldmath$\pi$} _i - e\, n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- {\bf F},$$ (4.78)

$$\frac{3}{2}\frac{d_i p_i}{dt} + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i + \mbox{\boldmath$\pi$} _i:\nabla{\bf V}_i+ \nabla\cdot{\bf q}_i = w_i,$$ (4.79)

respectively. Here, use has been made of the momentum conservation law, Equation (4.25). Equations (4.74)–(4.76) and (4.77)–(4.79) are called the Braginskii equations, because they were first obtained in a celebrated article by S.I. Braginskii (Braginskii 1965).