302 Chapter 10. Stability of static MHD equilibria
Substitution of Eq.(10.17) into Eq.(10.19) gives the dispersion relationγ^2 =kgtanh(k⊥h) (10.20)which shows that the configuration is always unstable sinceγ^2 > 0 .Equation (10.20)
furthermore shows that short wavelengths are most unstable, but a more detailed analysis
taking into account surface tension (which is stronger for shorter wavelengths) would show
thatγ^2 has a maximum at some wavelength. Above this most unstable wavelength, surface
tension would decrease the growth rate.
10.2 MHD Rayleigh-Taylor instability
We define a ‘magnetofluid’ as afluid which satisfies the MHD equations. A given plasma
may or may not behave as a magnetofluid, depending on the validity of the MHD approx-
imation for the circumstances of the given plasma. The magnetofluid concept provides
a legalism which allows consideration of the implications of MHD without necessarily
accepting that these implications are relevant to a specific actual plasma. In effect, the
magnetofluid concept can be considered as a tentative model of plasma.
Let us now replace the water in the Rayleigh-Taylor instability by magnetofluid. We
further suppose that instead of atmospheric pressure supporting the magnetofluid, a vertical
magnetic field gradient balances the downwards gravitational force, i.e., at eachythe up-
ward force of−∇B^2 / 2 μ 0 supports the downward force of the weight of the plasma above.
Although gravity is normally unimportant in actual plasmas, the gravitational model is nev-
ertheless quite useful for characterizing situations of practical interest, because gravity can
be considered as a proxy for the actual forces which typically have a more complex struc-
ture. An important example is the centrifugal force associated with thermal particle motion
along curved magnetic field lines acting like a gravitational force in the direction of the ra-
dius of curvature of these field lines. The curved field with associated centrifugal force due
to parallel thermal motion is replaced by a Cartesian geometry model having straight field
lines and, perpendicular to the field lines, a gravitational force is invokedto represent the
effect of the centrifugal force. Theydirection corresponds to the direction of the radius of
curvature.
In order for−∇B^2 to point upwards in theydirection, the magnetic field must depend
onysuch that its magnitude decreases with increasingy.Furthermore it is required that
By=0so that∇B^2 is perpendicular to the magnetic field and the field can be considered
as locally straight (field line curvature has already been taken into account by introducing
the fictitious gravity). Thus, the equilibrium magnetic field is assumed to be of the general
form
B 0 =Bx 0 (y)ˆx+Bz 0 (y)ˆz. (10.21)
The unit vector associated with the equilibrium field is
Bˆ 0 =√Bx^0 (y)ˆx+Bz^0 (y)ˆz
Bx 0 (y)^2 +Bz 0 (y)^2. (10.22)
For the special case whereBx 0 (y)andBz 0 (y)are proportional to each other, the field line
direction is independent ofy,but in the more general case where this is not so,Bˆ 0 depends