Fluid theory

In the fluid description, information on the particle velocity distribution is replaced by values averaged over velocity space. The averaging is achieved by taking moments of Boltzmann's equation (3.11). We proceed by rewriting Boltzmann's equation in tensor form:

$$\frac{\partial f_{\alpha}}{\partial t} + v_{i} \frac{\partial... ...t( \frac{\partial f_{\alpha}} {\partial t} \right)_{\rm coll.}$$ (3.18)

Moment equations are obtained by multiplying () by an arbitrary function of velocity $\Theta ({\bf v})$ and integrating over velocity space. The moment of the first term in () is

$$\int {\rm d}^{3} {\bf v} \,\, \Theta({\bf v}) \frac{\partial ... ...ac{\partial}{\partial t} (n_{\alpha} \langle \Theta \rangle) .$$ (3.19)

Similarly the second term satisfies

$$\int {\rm d}^{3} {\bf v} \,\, \Theta({\bf v}) v_{i} \frac{\p... ...}{\partial x_{i}} (n_{\alpha} \langle v_{i} \Theta \rangle) .$$ (3.20)

After some algebra, the third term transforms to

$$\int {\rm d}^{3} {\bf v} \,\, \Theta({\bf v}) \frac{q_{\alpha... ...frac{\partial f_{\alpha}}{\partial v_{i}} \mbox{\hspace{1cm}}$$

$$\mbox{\hspace{2cm}} = - \frac{q_{\alpha}}{m_{\alpha}} E_{i} n... ...a} \langle \frac{\partial \Theta}{\partial v_{i}} \rangle \, .$$ (3.21)

The general moment equation has the form

$$\frac{\partial}{\partial t} (n_{\alpha} \langle \Theta \rangl... ...\alpha} \langle \frac{\partial \Theta}{\partial v_{i}} \rangle$$

$$-\frac{q_{\alpha}}{m_{\alpha} c} \varepsilon_{ijk} B_{k} \lan... ...rac{\partial f_{\alpha}} {\partial t} \right)_{\rm coll.} \, .$$ (3.22)

Taking the zeroth order moment, with $\Theta({\bf v})=1$, () gives

$$\frac{\partial n_{\alpha}}{\partial t} + \frac{\partial}{\partial x_{i}} (n_{\alpha} u_{i})=0 \, .$$ (3.23)

Note that the right-hand side is zero due to particle conservation for ions and electrons (for an ideal plasma, ignoring ionization, recombination and charge exchange effects); i.e.,

$$\left( \frac{\partial n_{p}} {\partial t} \right)_{\rm coll.}... ...rac{\partial n_{e}} {\partial t} \right)_{\rm coll.} = 0 \,\,.$$ (3.24)

Mass conservation equation: multiply () by $m_{\alpha}$ and sum over $\alpha$:

$$\frac{\partial \eta}{\partial t} +{\mbox{\boldmath$\nabla \cdot$ }} (\eta {\bf U}) = 0 \, .$$ (3.25)

Charge conservation equation: Multiply () by $q_{\alpha}$ and sum over $\alpha$:

$$\frac{\partial \rho}{\partial t} + {\mbox{\boldmath$\nabla \cdot$ }} (\eta {\bf J}) = 0 \, .$$ (3.26)

We take the first order moment ( $\Theta({\bf v})= {\bf v}$) by multiplying () by v_s and integrating over velocity space,

$$\frac{\partial}{\partial t} (m_{\alpha} n_{\alpha} u_{s}) + \... ...arepsilon_{sjk} u_{j} B_{k} -n_{\alpha} g_{s} = \pm P_{s} \, ,$$ (3.27)

where the sign of the momentum density P_s is opposite for electrons and ions, and P_sis defined by

$$P_{s} = m_{p} \int {\rm d}^{3} {\bf v}_{p} \, v_{p,s} \left( ... ...t( \frac{\partial f_{e}} {\partial t} \right)_{\rm coll.} \, ,$$ (3.28)

since collisions between electrons and ions within the plasma do not change the total momentum density of the system. Using the defined fluid quantities for u_i, U_i, w_i and p_ij, and using the relation:

$$\langle v_{i} v_{j} \rangle = \langle (U_{i} + w_{i})(U_{j} +... ...c{1}{n m} p_{ij} + U_{i} u_{j} + U_{j} u_{i} - U_{i} U_{j} \,,$$ (3.29)

equation () may be re-expressed in the form (omitting particle species $\alpha$),

$$\frac{\partial} {\partial t}(m n u_{s}) + \frac{\partial} {\p... ... m n (U_{i} u_{s} + U_{s} u_{i} - U_{i} U_{s})) \hspace{1.0cm}$$

$$\hspace{1.8cm} - n q E_{s} - n q \varepsilon_{sjk} u_{j} B_{k} -n g_{s} = \pm P_{s} \, .$$ (3.30)