Drift Motion in a Constant Electric Field

Suppose now that there is a constant electric field E present in the plasma in addition to the constant magnetic field B. The relevant force in (2.19) is in this case

FIGURE 2.2 Larmor orbits plus the perpendicular Ex В drift in a presence of an electric field.

The field component £ц along the magnetic field will accelerate the charged particle along B, while the effect of the electric field component perpendicular to the magnetic field E± will cause an electric drift velocity of Larmor circles determined by (2.26) and (2.27)

Because the electric drift velocity (2.28) does not depend on the charge or mass of the particle, in the presence of a constant electric field, both electrons and ions drift together in the same direction and with the same drift velocities. No net current is produced due to the drift in a constant electric field. Figure 2.2 shows how the super-position of the gyro-motion and drift due to electric field (2.28) looks like in the plane perpendicular to B, with the magnetic field vector directed towards us. It may be noted that (2.28) is only valid for

which is satisfied in most cases to be considered in magnetic fusion.

Drift Motion in Inhomogeneous Static Magnetic Field

A precise description of charged particle orbit in inhomogeneous magnetic field usually requires a significant computational effort. Here, we consider two most important cases, both could be solved in a rather simple manner.

First, consider a static magnetic field that is no longer spatially uniform in its magnitude. When the magnetic field does not vary significantly over the Larmor radius, that is, when (2.18) is valid, the particle gyro-centre drifts due to V£. Let us split the Larmor motion in the inhomogeneous magnetic field into two parts, one of which has a smaller radius p = p_: in the stronger field area, while the other one has a larger p = p₂ in the weaker field area. If half of the cyclotron rotation is done with p_u and the other half with p₂, then the averaged position of the Larmor circle shifts by Ar = 2(p₂ - Pi) = 2Дp in one cyclotron period 2л / co_B thus making drift velocity |v_l_S| ~ (OgAp/n. Taking into account that the Larmor radius varies not in a single step but gradually, the shift of the Larmor circle can be estimated more accurately as

where the variation of the Larmor radius over its length is

The VB drift velocity takes the form

or, by substituting (0_B = v_±/p_s,

Equation (2.33) determines the gradient drift velocity in absolute value but not in directivity. For the component of the magnetic field gradient in the direction perpendicular to B, we represent V_XZ? = VB x b = -b x V£, where b = BIB is the unit vector along the magnetic field. By substituting this expression and (2.13) in (2.33), one obtains the gradient drift velocity in the vector form:

We note that v_1B depends on the particle charge in contrast to v_1ExB given by (2.28). In particular, v_ifi is in opposite directions for ions and electrons and gives rise to a current in plasma.

Here, (2.34) could be obtained from (2.16) and (2.26) via the force

Figure 2.3 shows how the particle motion combining its gyro-motion and drift due to V# looks in the plane perpendicular to В, with the magnetic field vector directed towards us.

Drift Motion in Static Magnetic Field with Curvature

Here, we consider the second simple case when the direction of a static magnetic field varies. In this case, the centre of Larmor circle moves along the curved magnetic field line with some curvature radius R. so a centrifugal force arises:

which is directed along the curvature radius.

By substituting (2.36) in the general expression (2.26), we obtain the expression for the curvature-й drift velocity:

Here, v_±(. depends on the particle charge, as in the case of V# drift, but opposite to the case of ExB drift. Hence, ions and electrons in curved magnetic field move in opposite directions and induce drift currents causing charge separation.

Note that (2.37) depends on the parallel velocity of the particles, in contrast to (2.34) depending

V|,

on the perpendicular velocity. For charged particles injected along the magnetic field, — » 1. the

v±

FIGURE 2.3 Particle motion consisting of Larmor orbits plus the perpendicular VB drift in a static inhomogeneous magnetic field.

value of VS drift could be small, while the value of the curvature drift (2.37) is large. This is often essential for describing charged fast ions produced from a neutral beam injection parallel to the magnetic field.

Drift Motion in Time-Dependent Electric Field

We saw in Section 2.2.1 that a constant electric field perpendicular to the magnetic field causes both electrons and ions to drift at the same drift velocity (2.28) so that no net electric current is generated in the plasma. However, a time-dependent electric field causes a drift perpendicular to В that depends on the mass and charge of charged particles and, consequently, produces a current in the plasma consisting of ions and electrons. We start from equation

and represent the perpendicular velocity as a sum of the velocity vlof cyclotron rotation with gyro- frequency (2.9), as well as the perpendicular drift velocity of the guiding centre. The drift velocity can be represented as

where v_1ExB is given by Eq. (2.28), and the additional drift t>v_xis to be found. If the characteristic time of the electric field variation is much longer than the inverse gyro-frequency, and |5v_x / Vj._£;|

The drift determined by (2.40) is called the “polarisation drift.” It gives drift velocities in opposite directions for charges of opposite signs and, consequently, provides polarisation current density.