Closure in Collisionless Magnetized Plasmas

In the limit
, the cold-plasma equations reduce to

Here, we have neglected the resistivity term, since it is . Note that none of the remaining terms in these equations depend explicitly on collisions. Nevertheless, the absence of collisions poses a serious problem. Whereas the magnetic field effectively confines charged particles in directions perpendicular to magnetic field-lines, by forcing them to execute tight Larmor orbits, we have now lost all confinement along field-lines. But, does this matter?

The typical frequency associated with fluid motion
is the transit frequency, . However, according to Eq. (345),
the cold-plasma ordering implies that the transit frequency is of order a
typical gyrofrequency:

(412) |

Let us now consider the MHD limit. In this case, the typical
transit frequency is

(414) |

In fact, in collisionless plasmas, MHD theory is replaced by a theory
known as *kinetic-MHD*.^{} The latter theory is a combination of a
one-dimensional kinetic theory, describing particle motion along magnetic
field-lines, and a two-dimensional fluid theory, describing perpendicular
motion. As can well be imagined, the equations of kinetic-MHD are
considerably more complicated that the conventional MHD equations.
Is there any situation in which we can salvage the simpler MHD equations in
a collisionless plasma? Fortunately, there
is one case in which this is possible.

It turns out that in both varieties of MHD the motion of the plasma
parallel to magnetic field-lines is associated with the dynamics of
*sound waves*, whereas the motion perpendicular to field-lines
is associated with the dynamics of a new type of wave called an *Alfvén
wave*. As we shall see, later on, Alfvén
waves involve the ``twanging'' motion of magnetic field-lines--a bit
like the twanging of guitar strings. It is only the sound wave dynamics
which are significantly modified when we move from a collisional to a
collisionless plasma. It follows, therefore, that the MHD equations
remain a reasonable approximation in a collisionless plasma in situations where
the dynamics of sound waves, parallel to the magnetic field, are unimportant
compared to the dynamics of Alfvén waves, perpendicular to the field.
This situation arises whenever the parameter

(415) |

(416) |

(417) |

We conclude, therefore, that in a low-, collisionless, magnetized
plasma the MHD equations,

(418) | |||

(419) | |||

(420) | |||

(421) |

fairly well describe plasma dynamics which satisfy the basic MHD ordering (413).

Let us, finally, consider the drift limit. In this case, the typical
transit frequency is

(422) |

(423) |

Now, in the drift limit, the perpendicular drift velocity of
charged particles, which is a combination of
drift,
grad- drift, and curvature drift (see Sect. 2), is of order

(424) |

(425) |

In fact, in collisionless plasmas, Braginskii-type transport theory--conventionally
known as *classical transport theory*--is replaced by a new theory--known as
*neoclassical transport theory*^{}--which is a combination of a two-dimensional
kinetic theory, describing particle motion on *drift surfaces*, and
a one-dimensional fluid theory, describing motion perpendicular to the drift
surfaces. Here, a drift surface is a closed surface formed by the locus of a
charged particle's drift orbit (including
drifts parallel and perpendicular to the magnetic field).
Of course, the orbits only form closed surfaces if the plasma is *confined*,
but there is little point in examining transport in an unconfined plasma.
Unlike classical
transport theory, which is strictly *local* in nature,
neoclassical transport theory is *nonlocal*, in the sense that the
transport coefficients depend on the *average* values of
plasma properties taken over
drift surfaces. Needless to say, neoclassical transport theory
is horribly complicated!