Fluid Closure Schemes

The fluid equations for the electron and ion species take the respective forms:

$$\frac{d_en_e}{dt} + n_e\,\nabla\cdot{\bf V}_e =0,$$ (2.26)

$$m_e \,n_e\,\frac{d_e {\bf V}_e}{dt} + e\, n_e\, ({\bf E} + {\bf V}_e\times {\bf B})+ \nabla p_e+ \nabla\cdot \mbox{\boldmath$\pi$} _e = {\bf F}_e,$$ (2.27)

$$\frac{3}{2}\frac{d_e p_e}{dt} + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e + \mbox{\boldmath$\pi$} _e:\nabla{\bf V}_e+ \nabla\cdot{\bf q}_e = W_e,$$ (2.28)

and

$$\frac{d_i n_e}{dt} + n_e\,\nabla\cdot{\bf V}_i =0,$$ (2.29)

$$m_i \,n_e\,\frac{d_i {\bf V}_i}{dt} - e\, n_e\, ({\bf E} + {\bf V}_i\times {\bf B})+ \nabla p_i + \nabla\cdot \mbox{\boldmath$\pi$} _i =- {\bf F}_e,$$ (2.30)

$$\frac{3}{2}\frac{d_i p_i}{dt} + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i + \mbox{\boldmath$\pi$} _i:\nabla{\bf V}_i+ \nabla\cdot{\bf q}_i = W_i.$$ (2.31)

Here, use has been made of Equations (2.11)–(2.13) and (2.16), as well as the fact that ${\bf F}_i=-{\bf F}_e$ (because collisions conserve momentum) [18]. In their present forms, the electron and ion fluid equations relate interesting fluid quantities, such as the electron number density, $n_e$, the mean flow velocities, ${\bf V}_e$ and ${\bf V}_i$, and the scalar pressures, $p_e$ and $p_i$, to unknown quantities, such as the viscosity tensors, $\pi$$_e$ and $\pi$$_i$, the heat fluxes, ${\bf q}_e$ and ${\bf q}_i$, and the moments of the collision operator, ${\bf F}_e$, $W_e$, and $W_i$. In order to complete our set of equations, we need to employ some additional information to express the latter quantities in terms of the former; this process is known as closure.

There are two basic types of fluid closure schemes. In truncation schemes, high-order velocity-space moments of the distribution function are assumed to vanish, or are prescribed in terms of low-order moments [24,36]. Truncation schemes are relatively straightforward to implement, but the error associated with the closure cannot easily be determined. Asymptotic schemes, on the other hand, depend on a rigorous expansion of the kinetic equation in terms of some dimensionless parameter that is small compared to unity [10]. Asymptotic closure schemes have the advantage of providing some estimate of the error involved in the closure. However, the asymptotic approach to closure is mathematically demanding, because it involves working closely with the kinetic equation. In this book, we shall rely on a mixture of truncation and asymptotic closure schemes.