Normalization of Braginskii Equations

Let us consider a magnetized plasma. It is convenient to split the friction force into a component corresponding to resistivity, and a component corresponding to the thermal force. Thus,

(4.138) |

where

(4.139) | ||

(4.140) |

Likewise, the electron collisional energy gain term is split into a component corresponding to the energy lost to the ions (in the ion rest frame), a component corresponding to work done by the friction force , and a component corresponding to work done by the thermal force . Thus,

(4.141) |

where

(4.142) | ||

(4.143) |

Finally, it is helpful to split the electron heat flux density into a diffusive component and a convective component . Thus,

(4.144) |

where

(4.145) | ||

(4.146) |

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

(4.147) | ||

(4.148) | ||

(4.149) |

Let , , , , and , 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 , and the typical variation lengthscale is . Let

(4.150) | ||

(4.151) | ||

(4.152) |

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

We define the following normalized quantities:

The normalization procedure is designed to make all hatted quantities . The normalization of the electric field is chosen such that the velocity is of similar magnitude to the electron fluid velocity. Note that the parallel viscosity makes an contribution to , whereas the gyroviscosity makes an contribution, and the perpendicular viscosity only makes an contribution. Likewise, the parallel thermal conductivity makes an contribution to , whereas the cross conductivity makes an contribution, and the perpendicular conductivity only makes an contribution. Similarly, the parallel components of and are , whereas the perpendicular components are .

The normalized electron fluid equations take the form:

(4.153) | ||

(4.154) | ||

(4.155) |

The only large or small (compared to unity) quantities in these equations are the parameters , , , and . Here, . It is assumed that .

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

(4.156) | ||

(4.157) | ||

(4.158) |

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 , , , , and , 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 , and the typical variation lengthscale is . Let

(4.159) | ||

(4.160) | ||

(4.161) |

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

We define the following normalized quantities:

As before, the normalization procedure is designed to make all hatted quantities . The normalization of the electric field is chosen such that the velocity is of similar magnitude to the ion fluid velocity. Note that the parallel viscosity makes an contribution to , whereas the gyroviscosity makes an contribution, and the perpendicular viscosity only makes an contribution. Likewise, the parallel thermal conductivity makes an contribution to , whereas the cross conductivity makes an contribution, and the perpendicular conductivity only makes an contribution. Similarly, the parallel component of is , whereas the perpendicular component is .

The normalized ion fluid equations take the form:

(4.162) | ||

(4.163) | ||

(4.164) |

The only large or small (compared to unity) quantities in these equations are the parameters , , , and . Here, .

Let us adopt the ordering

(4.165) |

which is appropriate to a collisional, highly magnetized, plasma. In the first stage of our ordering procedure, we shall treat and as small parameters, and , , and as . In the second stage, we shall take note of the smallness of , , and . Note that the parameters and are ``free ranging." In other words, they can be either large, small, or . In the initial stage of the ordering procedure, the ion and electron normalization schemes we have adopted become essentially identical [because ], and it is convenient to write

(4.166) | ||

(4.167) | ||

(4.168) | ||

(4.169) | ||

(4.170) |

There are three fundamental orderings in plasma fluid theory. These are analogous to the three orderings in neutral gas fluid theory discussed in Section 4.9.

The first ordering is

This corresponds to

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

Here, is conventionally termed the

and

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 (4.174)-(4.175) and (4.176)-(4.177) are called the *cold-plasma equations*, because
they can be obtained from the Braginskii equations by formally taking the
limit
. Likewise, the ordering (4.171) 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.,
),
which they impart to
plasma fluid elements, so there is usually
no difficulty satisfying the inequality (4.172).

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 ordering is

which corresponds to

(4.179) |

In other words, the fluid velocities are of similar magnitudes to the respective thermal velocities. Keeping only the largest terms in Equations (4.153)-(4.155) and (4.162)-(4.164), the Braginskii equations reduce to (in unnormalized form):

and

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 (4.180)-(4.182) and (4.183)-(4.184) are called the *magnetohydrodynamical equations*,
or *MHD equations*, for short. Likewise, the ordering (4.178) 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 final ordering is

which corresponds to

(4.187) |

where 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):

(4.188) | ||

(4.189) | ||

(4.190) |

and

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 (4.188)-(4.190) and (4.191)-(4.193) are called the *drift equations*. Likewise,
the ordering (4.186) 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.