Fundamentals of Plasma Physics

258 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

Γx =

1

2

Re

[

−

n 0 qe

Te

(

1 −i(ω∗−ω)τ‖

)iky|φ 1 |^2

B

]

= −

1

2

Re

[

n 0 qe

Te

(ω∗−ω)τ‖

ky|φ 1 |^2

B

]

=

(

k^2 yρ^2 s

1 +k^2 yρ^2 s

)

ky^2 |φ 1 |^2

2 LB^2

νein 0 me

kz^2 Te

= −Dnl

dn 0

dx

.

(8.138)

Thus, it is seen that collisional drift waves cause an outward diffusion of plasma with a
nonlinear wave-induced diffusion coefficient

Dnl=

(

ky^2 ρ^2 s

1 +k^2 yρ^2 s

)

k^2 y|φ 1 |^2

2 B^2

νeime

kz^2 Te

. (8.139)

This diffusion coefficient has a proportionality similar toωi.The drift wave thus has the
following properties: (i) the pressure gradient provides free energy andalso an equilib-
rium where drift waves are a normal mode, (ii) collisions allow the drift wave to feed on
the available free energy and grow, (iii) nonlinear diffusionflattens the pressure gradient
thereby depleting the free energy and also undermining the confining effectof the mag-
netic field. A nonlinear saturated amplitude can result if some external source continuously
replenishes the pressure gradient.
It was shown earlier that the most unstable drift waves are those havingkyρsof order
unity. This property can be adapted to the more physically realistic situation of a cylindrical
plasma by realizing that in a cylindrical plasmakyis replaced bym/rwheremis the
azimuthal mode number. Periodicity inθforcesmto be an integer. If the most unstable
mis a small integer, then the drift waves tend to be coherent, but ifmis a large integer,
many azimuthal wavelengths fit into the circumference of the cylinder. In this case, the
periodicity condition is only a weak constraint and the waves typically become turbulent.
Largemcorresponds to smallρswhich occurs when the magnetic field is strong. Thus,
plasmas with strong magnetic fields tend to have turbulent, short perpendicular wavelength
drift waves, whereas plasmas with weak magnetic fields have coherent, long perpendicular
wavelength drift waves.
Drift wave turbulence has been found to be the main reason why the radial diffusion in
tokamak magnetic plasma confinement devices is much worse than can be explained by
simple random walks due to particle collisions.

8.6.3 Vlasov theory of drift waves: collisionless drift waves

Drift waves also exist when the plasma is so hot that the collision frequency becomes
insignificant and the plasma can be considered collisionless. The previous section showed
that a collision-induced phase lag between the density and potentialfluctuations produces
a destabilizing imaginary term in the dispersion relation. The analysis of collisionless
plasma waves showed that Landau damping also causes a phase shift between thedensity