Normalization of Braginskii Equations

As we have just seen, the Braginskii equations contain terms that describe a very wide range of different physical phenomena. For this reason, they are extremely complicated. Fortunately, however, it is not generally necessary to retain all of the terms in these equations when investigating a particular problem in plasma physics: for example, electromagnetic wave propagation through plasmas. In this section, we shall attempt to construct a systematic normalization scheme for the Braginskii equations that will, hopefully, enable us to determine which terms to keep, and which to discard, when investigating a particular aspect of plasma physics.

Let us consider a magnetized plasma. It is convenient to split the friction force ${\bf F}$ into a component ${\bf F}_U$ corresponding to resistivity, and a component ${\bf F}_T$ corresponding to the thermal force. Thus,

$${\bf F} = {\bf F}_U+{\bf F}_T,$$ (4.119)

where

$${\bf F}_U = n\,e\left(\frac{{\bf j}_\parallel}{\sigma_\parallel} +\frac{{\bf j}_\perp}{\sigma_\perp}\right),$$ (4.120)

$${\bf F}_T = - 0.71\,n\,\nabla_\parallel T_e -\frac{3\,n}{2\,\vert{\mit\Omega}_e\vert\,\tau_e}\,{\bf b}\times\nabla_\perp T_e.$$ (4.121)

Likewise, the electron collisional energy gain term $w_e$ is split into a component $-w_i$ corresponding to the energy lost to the ions (in the ion rest frame), a component $w_U$ corresponding to work done by the friction force ${\bf F}_U$, and a component $w_T$ corresponding to work done by the thermal force ${\bf F}_T$. Thus,

$$w_e = -w_i + w_U + w_T,$$ (4.122)

where

$$w_U = \frac{{\bf j}\cdot{\bf F}_U}{n\,e},$$ (4.123)

$$w_T = \frac{{\bf j}\cdot{\bf F}_T}{n\,e}.$$ (4.124)

Finally, it is helpful to split the electron heat flux density ${\bf q}_e$ into a diffusive component ${\bf q}_{Te}$ and a convective component ${\bf q}_{Ue}$. Thus,

$${\bf q}_e = {\bf q}_{Te} + {\bf q}_{Ue},$$ (4.125)

where

$${\bf q}_{Te} =-\kappa_\parallel^e\,\nabla_\parallel T_e -\kappa_\perp^e\, \nabla_\perp T_e -\kappa_\times^e\,{\bf b}\times\nabla_\perp T_e,$$ (4.126)

$${\bf q}_{Ue} = 0.71\,\frac{T_e}{e}\,{\bf j}_\parallel- \frac{3\,T_e}{2\,\vert{\mit\Omega}_e\vert\,\tau_e\,e}\,{\bf b}\times{\bf j}_\perp.$$ (4.127)

Let us, first of all, consider the electron fluid equations, which can be written:

$$\frac{d_e n}{dt} + n\,\nabla\cdot{\bf V}_e =0,$$ (4.128)

$$m_e \,n\,\frac{d_e {\bf V}_e}{dt} + \nabla p_e+ \nabla\cdot \mbox{\boldmath$\pi$} _e + e\, n\, ({\bf E} + {\bf V}_e\times {\bf B}) = {\bf F}_U +{\bf F}_T,$$ (4.129)

$$\frac{3}{2}\frac{d_e p_e}{dt} + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e + \mbox{\boldmath$\pi$} _e:\nabla{\bf V}_e+ \nabla\cdot{\bf q}_{Te} + \nabla\cdot{\bf q}_{Ue} = -w_i+w_U + w_T.$$ (4.130)

Let $\bar{n}$, $\bar{v}_e$, $\bar{l}_e$, $\bar{B}$, and $\bar{\rho}_e =\bar{ v}_e/(e\bar{B}/m_e)$, be typical values of the particle density, the electron thermal velocity, the electron mean-free-path, the magnetic field-strength, and the electron gyroradius, respectively. Suppose that the typical electron flow velocity is $\lambda_e\,\bar{v}_e$, and the typical variation lengthscale is $L$. Let

$$\delta_e =\frac{\bar{\rho}_e}{L},$$ (4.131)

$$\zeta_e = \frac{\bar{\rho}_e}{\bar{l}_e},$$ (4.132)

$$\mu = \sqrt{\frac{m_e}{m_i}}.$$ (4.133)

All three of these parameters are assumed to be small compared to unity.

We define the following normalized quantities:

$$\hat{n} = \frac{n}{\bar{n}}, \hat{v}_e = \frac{v_e}{\bar{v}_e}, \hat{\bf r} = \frac{{\bf r}}{ L},$$

$$\widehat{\nabla} = L\,\nabla, \hat{t} = \frac{\lambda_e\,\bar{v}_e\,t}{L}, \widehat{\bf V}_e = \frac{{\bf V}_e}{\lambda_e\,\bar{v}_e},$$

$$\widehat{\bf B} = \frac{{\bf B}}{\bar{B}}, \widehat{\bf E} = \frac{{\bf E}}{ \lambda_e\,\bar{v}_e\,\bar{B}}, \widehat{\bf U} =\frac{{\bf U} }{ (1+\lambda_e^{2})\,\delta_e\,\bar{v}_e},$$

$$\hat{p}_e = \frac{p_e}{m_e\,\bar{n}\,\bar{v}_e^{2}}, \widehat{\mbox{\boldmath$\pi$}}_e = \frac{\mbox{\boldmath$\pi$}_e}{\lambda_e\,\delta_e\,\zeta_e^{-1}\,m_e\,\bar{n}\,\bar{v}_e^{2}}, \widehat{\bf q}_{Te} = \frac{{\bf q}_{Te} }{ \delta_e\,\zeta_e^{-1}\,m_e\,\bar{n}\,\bar{v}_e^{3}},$$

$$\widehat{\bf q}_{Ue} = \frac{{\bf q}_{Ue} }{ (1+\lambda_e^{2})\,\delta_e\,m_e\,\bar{n}\,\bar{v}_e^{3}}, \widehat{\bf F}_U = \frac{{\bf F}_U}{(1+\lambda_e^{2}) \,\zeta_e\,m_e\,\bar{n}\,\bar{v}_e^{2}/L}, \widehat{\bf F}_T = \frac{{\bf F}_T}{m_e\,\bar{n}\,\bar{v}_e^{2}/L},$$

$$\widehat{w}_i = \frac{w_i}{\delta_e^{-1}\,\zeta_e\,\mu^2\, m_e\,\bar{n}\,\bar{v}_e^{3}/L}, \widehat{w}_U = \frac{w_U}{(1+\lambda_e^{2})^2\,\delta_e\,\zeta_e\, m_e\,\bar{n}\,\bar{v}_e^{3}/L}, \widehat{w}_T = \frac{w_T}{(1+\lambda_e^{2})\,\delta_e\, m_e\,\bar{n}\,\bar{v}_e^{3}/L}.$$

The normalization procedure is designed to make all hatted quantities ${\cal O}(1)$. The normalization of the electric field is chosen such that the ${\bf E}\times{\bf B}$ velocity is of similar magnitude to the electron fluid velocity. Note that the parallel viscosity makes an ${\cal O}(1)$ contribution to $\widehat{\mbox{\boldmath $\pi$}}_e$, whereas the gyroviscosity makes an ${\cal O}(\zeta_e)$ contribution, and the perpendicular viscosity only makes an ${\cal O}(\zeta_e^{2})$ contribution. Likewise, the parallel thermal conductivity makes an ${\cal O}(1)$ contribution to $\widehat{\bf q}_{Te}$, whereas the cross conductivity makes an ${\cal O}(\zeta_e)$ contribution, and the perpendicular conductivity only makes an ${\cal O}(\zeta_e^{2})$ contribution. Similarly, the parallel components of ${\bf F}_T$ and ${\bf q}_{Ue}$ are ${\cal O}(1)$, whereas the perpendicular components are ${\cal O}(\zeta_e)$.

The normalized electron fluid equations take the form:

$$\frac{d_e\hat{n}}{d\hat{t}} + \hat{n}\,\widehat{\nabla}\cdot\widehat{\bf V}_e =0,$$ (4.134)

$$\lambda_e^{2}\,\delta_e\,\hat{n}\,\frac{d_e\widehat{\bf V}_e}{d\h... ...\lambda_e\,\hat{n}\,(\widehat{\bf E} + \widehat{\bf V}_e \times\widehat{\bf B}) = (1+\lambda_e^{2})\,\delta_e\,\zeta_e\,\widehat{\bf F}_U + \delta_e \,\widehat{\bf F}_T,$$ (4.135)

$$\lambda_e\,\frac{3}{2}\frac{d_e\hat{p}_e}{d\hat{t}} + \lambda_e\,... ...-1}\,\widehat{\mbox{\boldmath$\pi$}}_e : \widehat{\nabla}\cdot\widehat{\bf V}_e$$

$$+\delta_e\,\zeta_e^{-1} \,\widehat{\nabla}\cdot\widehat{\bf q}_{Te} +(1+\lambda_e^{2})\,\delta_e\,\widehat{\nabla}\cdot\widehat{\bf q}_{Ue} = -\delta_e^{-1}\,\zeta_e\,\mu^2\,\widehat{w}_i$$ (4.136)

$$\phantom{=}+ (1+\lambda_e^{2})^2\,\delta_e\,\zeta_e\, \widehat{w}_U + (1+\lambda_e^{2})\,\delta_e\,\widehat{w}_T.$$

The only large or small (compared to unity) quantities in these equations are the parameters $\lambda_e$, $\delta_e$, $\zeta_e$, and $\mu$. Here, $d_e/d\hat{t}\equiv\partial /\partial\hat{t} + \widehat{\bf V}_e\cdot\widehat{\nabla}$. It is assumed that $T_e\sim T_i$.

Let us now consider the ion fluid equations, which can be written:

$$\frac{d_i n}{dt} + n\,\nabla\cdot{\bf V}_i =0,$$ (4.137)

$$m_i \,n\,\frac{d_i {\bf V}_i}{dt} + \nabla p_i + \nabla\cdot \mbox{\boldmath$\pi$} _i - e n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- {\bf F}_U -{\bf F}_T,$$ (4.138)

$$\frac{3}{2}\frac{d_i p_i}{dt} + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i + \mbox{\boldmath$\pi$} _i:\nabla{\bf V}_i+ \nabla\cdot{\bf q}_i =w_i.$$ (4.139)

It is convenient to adopt a normalization scheme for the ion equations which is similar to, but independent of, that employed to normalize the electron equations. Let $\bar{n}$, $\bar{v}_i$, $\bar{l}_i$, $\bar{B}$, and $\bar{\rho}_i =\bar{ v}_i/(e\bar{B}/m_i)$, be typical values of the particle density, the ion thermal velocity, the ion mean-free-path, the magnetic field-strength, and the ion gyroradius, respectively. Suppose that the typical ion flow velocity is $\lambda_i\,\bar{v}_i$, and the typical variation lengthscale is $L$. Let

$$\delta_i = \frac{\bar{\rho}_i}{L},$$ (4.140)

$$\zeta_i = \frac{\bar{\rho}_i}{\bar{l}_i},$$ (4.141)

$$\mu = \sqrt{\frac{m_e}{m_i}}.$$ (4.142)

All three of these parameters are assumed to be small compared to unity.

We define the following normalized quantities:

$$\hat{n} = \frac{n}{\bar{n}}, \hat{v}_i = \frac{v_i}{\bar{v}_i}, \hat{\bf r} = \frac{{\bf r}}{ L},$$

$$\widehat{\nabla} = L\,\nabla, \hat{t} = \frac{\lambda_i\,\bar{v}_i\,t}{L}, \widehat{\bf V}_i = \frac{{\bf V}_i}{\lambda_i\,\bar{v}_i},$$

$$\widehat{\bf B} = \frac{{\bf B}}{\bar{B}}, \widehat{\bf E} = \frac{{\bf E}}{ \lambda_i\,\bar{v}_i\,\bar{B}}, \widehat{\bf U} =\frac{{\bf U} }{ (1+\lambda_i^{2})\,\delta_i\,\bar{v}_i},$$

$$\hat{p}_i =\frac{ p_i}{m_i\,\bar{n}\,\bar{v}_i^{2}}, \widehat{\mbox{\boldmath$\pi$}}_i = \frac{\mbox{\boldmath$\pi$}_i}{\lambda_i\,\delta_i\,\zeta_i^{-1}\,m_i\,\bar{n}\,\bar{v}_i^{2}}, \widehat{\bf q}_{i} = \frac{{\bf q}_{i} }{ \delta_i\,\zeta_i^{-1}\,m_i\,\bar{n}\,\bar{v}_i^{3}},$$

$$\widehat{\bf F}_U = \frac{{\bf F}_U}{ (1+\lambda_i^{2}) \,\zeta_i\,\mu\,m_i\,\bar{n}\,\bar{v}_i^{2}/L}, \widehat{\bf F}_T =\frac{ {\bf F}_T}{m_i\,\bar{n}\,\bar{v}_i^{2}/L}, \widehat{w}_i = \frac{w_i}{\delta_i^{-1}\,\zeta_i\,\mu\, m_i\,\bar{n}\,\bar{v}_i^{3}/L}.$$

As before, the normalization procedure is designed to make all hatted quantities ${\cal O}(1)$. The normalization of the electric field is chosen such that the ${\bf E}\times{\bf B}$ velocity is of similar magnitude to the ion fluid velocity. Note that the parallel viscosity makes an ${\cal O}(1)$ contribution to $\widehat{\mbox{\boldmath $\pi$}}_i$, whereas the gyroviscosity makes an ${\cal O}(\zeta_i)$ contribution, and the perpendicular viscosity only makes an ${\cal O}(\zeta_i^{2})$ contribution. Likewise, the parallel thermal conductivity makes an ${\cal O}(1)$ contribution to $\widehat{\bf q}_{i}$, whereas the cross conductivity makes an ${\cal O}(\zeta_i)$ contribution, and the perpendicular conductivity only makes an ${\cal O}(\zeta_i^{2})$ contribution. Similarly, the parallel component of ${\bf F}_T$ is ${\cal O}(1)$, whereas the perpendicular component is ${\cal O}(\zeta_i\,\mu)$.

The normalized ion fluid equations take the form:

$$\frac{d_i \hat{n}}{d\hat{t}} + \hat{n}\,\widehat{\nabla}\cdot\widehat{\bf V}_i =0,$$ (4.143)

$$\lambda_i^{2}\,\delta_i\,\hat{n}\,\frac{d_i\widehat{\bf V}_i}{d\h... ...ta_i^{2}\,\zeta_i^{-1}\,\widehat{\nabla} \cdot\widehat{\mbox{\boldmath$\pi$}}_i$$ (4.144)

$$-\lambda_i\,\hat{n}\,(\widehat{\bf E} + \widehat{\bf V}_i \times\widehat{\bf B}) = -(1+\lambda_i^{2})\,\delta_i\,\zeta_i\,\mu\,\widehat{\bf F}_U - \delta_i \,\widehat{\bf F}_T,$$

$$\lambda_i\,\frac{3}{2}\frac{d_i\hat{p}_i}{d\hat{t}} + \lambda_i\,... ...hat{\bf V}_i +\delta_i\,\zeta_i^{-1} \,\widehat{\nabla}\cdot\widehat{\bf q}_{i} = \delta_i^{-1}\,\zeta_i\,\mu\,\widehat{w}_i.$$ (4.145)

The only large or small (compared to unity) quantities in these equations are the parameters $\lambda_i$, $\delta_i$, $\zeta_i$, and $\mu$. Here, $d_i/d\hat{t}\equiv\partial /\partial\hat{t} + \widehat{\bf V}_i\cdot\widehat{\nabla}$.

Let us adopt the ordering

$$\delta_e, \delta_i \ll \zeta_e, \zeta_i, \mu \ll 1,$$ (4.146)

which is appropriate to a collisional, highly magnetized, plasma. In the first stage of our ordering procedure, we shall treat $\delta_e$ and $\delta_i$ as small parameters, and $\zeta_e$, $\zeta_i$, and $\mu$ as ${\cal O}(1)$. In the second stage, we shall take note of the smallness of $\zeta_e$, $\zeta_i$, and $\mu$. Note that the parameters $\lambda_e$ and $\lambda_i$ are “free ranging." In other words, they can be either large, small, or ${\cal O}(1)$. In the initial stage of the ordering procedure, the ion and electron normalization schemes we have adopted become essentially identical [because $\mu\sim {\cal O}(1)$], and it is convenient to write

$$\lambda_e\sim \lambda_i \sim \lambda,$$ (4.147)

$$\delta_e \sim \delta_i \sim \delta,$$ (4.148)

$$V_e \sim V_i \sim V,$$ (4.149)

$$v_e \sim v_i \sim v_t,$$ (4.150)

$${\mit\Omega}_e \sim {\mit\Omega}_i \sim {\mit\Omega}.$$ (4.151)

There are three fundamental orderings in plasma fluid theory.

The first fundamental ordering is

$$\lambda \sim \delta^{-1}.$$ (4.152)

This corresponds to

$$V\gg v_t.$$ (4.153)

In other words, the fluid velocities are much greater than the respective thermal velocities. We also have

$$\frac{V}{L} \sim {\mit\Omega}.$$ (4.154)

Here, $V/L$ is conventionally termed the transit frequency, and is the frequency with which fluid elements traverse the system. It is clear that the transit frequencies are of approximately the same magnitudes as the gyrofrequencies in this ordering. Keeping only the largest terms in Equations (4.134)–(4.136) and (4.143)–(4.145), the Braginskii equations reduce to (in unnormalized form):

$$\frac{d_e n}{dt} + n\,\nabla\cdot{\bf V}_e =0,$$ (4.155)

$$m_e \,n\,\frac{d_e {\bf V}_e}{dt} + e\, n\, ({\bf E} + {\bf V}_e\times {\bf B}) = [\zeta]\,{\bf F}_U,$$ (4.156)

and

$$\frac{d_i n}{dt} + n\,\nabla\cdot{\bf V}_i =0,$$ (4.157)

$$m_i\, n\,\frac{d_i {\bf V}_i}{dt} - e\, n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- [\zeta]\,{\bf F}_U.$$ (4.158)

The factors in square brackets are just to remind us that the terms they precede are smaller than the other terms in the equations (by the corresponding factors inside the brackets).

Equations (4.155)–(4.156) and (4.157)–(4.158) are called the cold-plasma equations, because they can be obtained from the Braginskii equations by formally taking the limit $T_e, T_i\rightarrow 0$. Likewise, the ordering (4.152) is called the cold-plasma approximation. The cold-plasma approximation applies not only to cold plasmas, but also to very fast disturbances that propagate through conventional plasmas. In particular, the cold-plasma equations provide a good description of the propagation of electromagnetic waves through plasmas. After all, electromagnetic waves generally have very high velocities (i.e., $V\sim c$), which they impart to plasma fluid elements, so there is usually no difficulty satisfying the inequality (4.153).

The electron and ion pressures can be neglected in the cold-plasma limit, because the thermal velocities are much smaller than the fluid velocities. It follows that there is no need for an electron or ion energy evolution equation. Furthermore, the motion of the plasma is so fast, in this limit, that relatively slow “transport” effects, such as viscosity and thermal conductivity, play no role in the cold-plasma fluid equations. In fact, the only collisional effect that appears in these equations is resistivity.

The second fundamental ordering is

$$\lambda \sim 1,$$ (4.159)

which corresponds to

$$V \sim v_t.$$ (4.160)

In other words, the fluid velocities are of similar magnitudes to the respective thermal velocities. Keeping only the largest terms in Equations (4.134)–(4.136) and (4.143)–(4.145), the Braginskii equations reduce to (in unnormalized form):

$$\frac{d_e n}{dt} + n\,\nabla\cdot{\bf V}_e =0,$$ (4.161)

$$m_e \,n\,\frac{d_e {\bf V}_e}{dt} + \nabla p_e+ [\delta^{-1}]\,e\, n\, ({\bf E} + {\bf V}_e\times {\bf B}) = [\zeta]\,{\bf F}_U + {\bf F}_T,$$ (4.162)

$$\frac{3}{2}\frac{d_e p_e}{dt} + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e = -[\delta^{-1}\,\zeta\,\mu^2]\,w_i,$$ (4.163)

and

$$\frac{d_i n}{dt} + n\,\nabla\cdot{\bf V}_i =0,$$ (4.164)

$$m_i \,n\,\frac{d_i {\bf V}_i}{dt} + \nabla p_i - [\delta^{-1}]\, e \,n\, ({\bf E} + {\bf V}_i\times {\bf B}) =- [\zeta]\,{\bf F}_U -{\bf F}_T,$$ (4.165)

$$\frac{3}{2}\frac{d_i p_i}{dt} + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i =[\delta^{-1}\,\zeta\,\mu^2]\, w_i.$$ (4.166)

Again, the factors in square brackets remind us that the terms they precede are larger, or smaller, than the other terms in the equations.

Equations (4.161)–(4.163) and (4.164)–(4.165) are called the magnetohydrodynamical equations, or MHD equations, for short. Likewise, the ordering (4.159) is called the MHD approximation. The MHD equations are conventionally used to study macroscopic plasma instabilities possessing relatively fast growth-rates: for example, “sausage” modes and “kink” modes (Bateman 1978).

The electron and ion pressures cannot be neglected in the MHD limit, because the fluid velocities are similar in magnitude to the respective thermal velocities. Thus, electron and ion energy evolution equations are needed in this limit. However, MHD motion is sufficiently fast that “transport” effects, such as viscosity and thermal conductivity, are too slow to play a role in the MHD equations. In fact, the only collisional effects that appear in these equations are resistivity, the thermal force, and electron-ion collisional energy exchange.

The third fundamental ordering is

$$\lambda\sim \delta,$$ (4.167)

which corresponds to

$$V \sim \delta\,v_t \sim v_d,$$ (4.168)

where $v_d$ is a typical drift (e.g., a curvature or grad-B drift—see Chapter 2) velocity. In other words, the fluid velocities are of similar magnitude to the respective drift velocities. Keeping only the largest terms in Equations (3.113) and (3.116), the Braginskii equations reduce to (in unnormalized form):

$$\frac{d_e n}{dt} + n\,\nabla\cdot{\bf V}_e =0,$$ (4.169)

$$m_e \,n\,\frac{d_e {\bf V}_e}{dt} + [\delta^{-2}]\,\nabla p_e+ [\zeta^{-1}]\, \nabla\cdot \mbox{\boldmath$\pi$} _e$$

$$+ [\delta^{-2}]\,e \,n\, ({\bf E} + {\bf V}_e\times {\bf B}) =[\delta^{-2}\,\zeta]\, {\bf F}_U + [\delta^{-2}]\, {\bf F}_T,$$ (4.170)

$$\frac{3}{2}\frac{d_e p_e}{dt} + \frac{5}{2}\,p_e\,\nabla\cdot{\bf V}_e + [\zeta^{-1}]\,\nabla\cdot{\bf q}_{Te} + \nabla\cdot{\bf q}_{Ue} =-[\delta^{-2}\,\zeta\,\mu^2]\, w_i +[\zeta]\,w_U + w_T,$$ (4.171)

and

$$\frac{d_i n}{dt} + n\,\nabla\cdot{\bf V}_i =0,$$ (4.172)

$$m_i \,n\,\frac{d_i {\bf V}_i}{dt} + [\delta^{-2}]\,\nabla p_i + [\zeta^{-1}]\,\nabla\cdot \mbox{\boldmath$\pi$} _i$$

$$- [\delta^{-2}]\,e\, n\, ({\bf E} + {\bf V}_i\times {\bf B}) = -[\delta^{-2}\,\zeta]\, {\bf F}_U - [\delta^{-2}]\,{\bf F}_T,$$ (4.173)

$$\frac{3}{2}\frac{d_i p_i}{dt} + \frac{5}{2}\,p_i\,\nabla\cdot{\bf V}_i + [\zeta^{-1}]\,\nabla\cdot{\bf q}_i = [\delta^{-2}\,\zeta\,\mu^2]\,W_i.$$ (4.174)

As before, the factors in square brackets remind us that the terms they precede are larger, or smaller, than the other terms in the equations.

Equations (4.169)–(4.171) and (4.172)–(4.174) are called the drift equations. Likewise, the ordering (4.167) is called the drift approximation. The drift equations are conventionally used to study equilibrium evolution, and the slow growing “micro-instabilities” that are responsible for turbulent transport in tokamaks. It is clear that virtually all of the original terms in the Braginskii equations must be retained in this limit.

In the following sections, we investigate the cold-plasma equations, the MHD equations, and the drift equations, in more detail.