Fundamentals of Plasma Physics

10.2 MHD Rayleigh-Taylor instability 305

Finitek·B 0 opposes the effect of the destabilizing positive density gradient, reducing
the growth rate toγ^2 ∼g∆yρ− 01 ∂ρ 0 /∂y−(k·B 0 )^2 /μ 0. A sufficiently strong field
will stabilize the system. However, thermally excited noise excites modes with all possible
values ofkand those modes aligned such thatk·B 0 =0will not be stabilized. A shear in
the magnetic field has the effect of making it possible to havek·B 0 (y)=0at only a single
value ofy. The lack of any spatial dependence ofkresults from the translational invariance
of the plasma with respect to bothxandzimplying thatk=kxxˆ+kzzˆis independent of
position. Thus, magnetic shear constrains the instability to a narrowystratum.
The stabilizing effect of finitek·B 0 can be understood by considering Eq.(10.32)
which shows that the amount ofB 1 yassociated with a givenv 1 yis proportional tok·B 0.
Since the original equilibrium field had noycomponent, introducing a finiteB 1 ycorre-
sponds to ‘plucking’ the equilibrium field. The plucking varies sinusoidally alongthe equi-
librium field and stretches the equilibrium field like a plucked violin string.The plucked
field has a restoring force which pulls the field back to its equilibrium position. If the energy
associated with plucking the magnetic field exceeds the energy liberated by the interchange
of heavy upper magnetofluid with light lower magnetofluid, the mode is stable.

‘good ’curvature ‘bad ’curvature

last

flux

surface

last

flux

surface plasma

vacuum vacuum

plasma

Figure 10.3: Good and bad curvature of the plasma-vacuum interface.

As discussed earlier, the gravitational force in the magnetofluid model represents the
centrifugal force resulting from guiding center motion along curved field lines. This leads
to the concept of ‘good’ and ‘bad’ curvature illustrated in Fig.10.3. A plasma has bad
curvature if the field lines at the plasma-vacuum boundary have a convex shape as seen
by an observer outside the plasma since in this case the centrifugal force is outwards from
the plasma. Bad curvature gives a centrifugal force that can drive interchange instabilities
whereas good curvature corresponds to a concave shape so that the centrifugal force is
always inwards. Mirror magnetic fields as sketched in Fig.10.4 havegood curvature in
the vicinity of the mirrors and bad curvature in the vicinity of the mirror minimum so that
a detailed analysis of interchange instabilities requires averaging the‘goodness/badness’
along the portion of theflux tube experienced by the particle. Cusp magnetic fields (cf.
Fig.10.4) have good curvature everywhere, but have singular behavior at the cusps. Plasmas
with internal currents such as tokamaks have significant shear everywhere since∂Bx/∂y∼