## Fluid Dynamics and the Navier-Stokes Equations

### Plasma Confinement and Pinch Effect

The confinement of a plasma by self-magnetic fields is very important in thermonuclear reactor and other applications. However, long-lived pinched plasmas are extremely difficult to maintain. The plasma column is observed to break up rapidly. The reason for the disintegration of the column is the growth of instabilities. The column is unstable against various departures from cylindrical geometry. Small distortions are amplified rapidly and destroy the column in a very short time. The mechanisms of instability in plasma physics are nearly unlimited. Some instabilities are comparable to examples borrowed from fluid mechanics, as the Rayleigh Taylor's instability, which consists of superposing two fluids with the heaviest on top. Imagine for example a vessel in which you pour water and then carefully add oil over the top. The system is then in a state of meta-stable equilibrium. The slightest nudge will provoke a change with the heavier fluid dropping to the bottom, which corresponds to a stable equilibrium. Another type of instability are kink instabilities, which occur when a current parallel to the magnetic field cause twisting of the field lines, recalling the effect obtained if we twist a rope too much: the rope twists out and kinks. The sausage or neck instability causes a greater inwards pressure at the neck of a constriction. This serves to enhance the existing distortion.

#### Figure 26 Confinement of Plasma [view large image]

The following provides a little bit more detail on the basic mechanism from a simplified configuration, to the pinch effect and finally the instabilities.

The principle of plasma confinement by self-magnetic fields can be illustrated with an infinite cylinder of conducting fluid with an axial current Jz = J(r) and a resulting azimuthal magnetic induction B = B(r). In steady-state dv/dt = 0, and negligible gravitational interaction, the equation of motion in Magnetohydrodynamics (MHD) is simplified to (Figure 27,a):

#### Figure 27 Plasma Confinement, Diagram [view large image]

See Ampere's Law in both forms.

The static equilibrium considered above does not happen in the real world. The whole thing is dynamic in a process of time evolution. So let's re-draw the plasma column as in Figure 28,a where the plasma column with cross section R(t) is contained inside a cylinder of radius R0 with an electric field E = V/L driving the current I along at the surface. Initially the pressure p is much too small to resist the magnetic pressure outside (see case b above), the plasma column pinches inward with a radial velocity |dR/dt| ~ v0 = (c2E2/4)1/2, where is the initial mass density.
For small scale lab experiment involving R0 ~ 10 cm, E ~ 103 volts/cm, ~ 10-8 gm/cm3 of deuterons, v0 ~ 107 cm/sec; while the current I ~ c2R0E/v0 would be in the order 105 amp. Eventually in a time interval of less than = R0/v0 ~ 10-6 sec the inward flow bounces back and the cycles repeat as shown in Figure 28,b. It is conjectured that the movement will approach to a steady state at some radius less than R0 .

#### Figure 28 Pinch Effect [view large image]

The pinching action would be very desirable as the effect pulls the plasma column away from the wall and thus prevents burning it up.

However, the previous consideration is also incomplete as instabilities develop in a time scale comparable to . The plasma column is observed to break up rapidly. There are many types of instabilities. In the kink instability, the B magnetic lines are bunched together above, and parted
below causing the column to bend downwards as shown in Figure 29,a. The sausage or neck instability develops more compression at one point in the column causing an increased pressure inside to oppose the magnetic pressure (Figure 29,b). Instability is the result of distortion of the trapped axial field lines Bz. Thus, the disturbance can be stabilized by bumping up the axial field such that (Bz)2 > (B)2/2 (Figure 29,d). Another stabilizing technique for long-wavelength kink is provided by the container wall to straighten up the distorted field lines (Figure 29,c).

#### Figure 29 Instabilities [view large image]

See "Plasma Stability" for further detail.

.