Drift Equations

If we assume that then the dominant term in the electron energy conservation equation (4.202) yields

(4.206) |

The dominant term in the ion energy conservation equation (4.205) yields

(4.208) |

The dominant terms in the electron and ion momentum conservation equations, (4.201) and (4.204), yield

The sum of the preceding two equations gives

(4.212) |

(4.213) |

(4.214) |

(4.215) |

Equations (4.210)–(4.211) can be inverted to give

Here, is the velocity, whereas(4.218) |

(4.219) |

According to Equations (4.216)–(4.217), in the drift approximation, the velocity of the electron fluid perpendicular to the magnetic field is the sum of the velocity and the electron diamagnetic velocity. A similar statement can be made for the ion fluid. By contrast, in the MHD approximation the perpendicular velocities of the two fluids consist of the velocity alone, and are, therefore, identical to lowest order. The main difference between the two orderings lies in the assumed magnitude of the electric field. In the MHD limit

(4.220) |

(4.221) |

The diamagnetic velocities are so named because the *diamagnetic
current*,

(4.222) |

The electron diamagnetic velocity can be written

In order to account for this velocity, let us consider a simplified case in which the electron temperature is uniform, there is a uniform density gradient running along the -direction, and the magnetic field is parallel to the -axis. (See Figure 4.3.) The electrons gyrate in the - plane in circles of radius . At a given point, coordinate , say, on the -axis, the electrons that come from the right and the left have traversed distances of approximate magnitude . Thus, the electrons from the right originate from regions where the particle density is approximately greater than the regions from which the electrons from the left originate. It follows that the -directed particle flux is unbalanced, with slightly more particles moving in the -direction than in the -direction. Thus, there is a net particle flux in the -direction: that is, in the direction of . The magnitude of this flux is(4.224) |

The most curious aspect of diamagnetic flows is that they represent fluid flows for which there is no corresponding motion of the particle guiding centers. Nevertheless, the diamagnetic velocities are real fluid velocities, and the associated diamagnetic current is a real current. For instance, the diamagnetic current contributes to force balance inside the plasma, and also gives rise to ohmic heating.