250 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
direction but are constrained to make Larmor orbits in the perpendicular direction. Since
the concept of pressure gradient has no meaning for an individual particle, consideration of
the effect of pressure gradients requires afluid or Vlasov point of view. This will be done
first using a two-fluid analysis, then using a Vlasov analysis.
From the two-fluid point of view, the radial pressure gradient implies an equilibrium
force balance
0 =qσuσ×B−n−σ^1 ∇(nσκTσ). (8.97)Solving Eq.(8.97) foruσshows that each species has a steady-state perpendicular motion
at thediamagnetic drift velocity
udσ=−∇(nσκTσ)×B
qσnσB^2(8.98)
which is in the azimuthal direction. The corresponding diamagnetic drift current is
Jd =∑
σnσqσudσ= −
1
B^2
∑
σ∇(nσκTσ)×B= −
1
B^2
∇P×B (8.99)
which is just the azimuthal current associated with the MHD equilibrium equationJ×B=
∇PwhereP=Pi+Pe.
The electron and ion diamagnetic drift velocities thus provide the current needed to
establish the magnetic force which balances the MHD pressure gradient. It turns out that
magnetized plasmas with pressure gradients are unstable to a class of electrostatic modes
called ‘drift’ waves. These modes exist in the same frequency regime as MHD but do not
appear in standard MHD models, since standard MHD models do not provide enough detail
on the difference between electron and ion dynamics.
From the particle point of view, the diamagnetic velocity is entirelyfictitious because
no particle actually moves at such a velocity. The diamagnetic drift velocity is nevertheless
quite genuine from thefluid point of view and so afluid-based wave analysis must linearize
about an equilibrium that includes this equilibrium drift. As will be seen later, the Vlasov
point of view confirms and extends the conclusions of thefluid theory, providing care is
taken to start with an equilibrium velocity distribution function which is both valid and
consistent with existence of a pressure gradient.