Fundamentals of Plasma Physics

300 Chapter 10. Stability of static MHD equilibria

adequate to support the inverted water. From a mathematical point of view, the cardboard
places a constraint on the system by imposing a boundary condition which prevents ripple
formation.
When the cardboard is removed so that there is no longer a constraint against ripple
formation, the ripples grow from noise to large amplitude, and the water falls out. This is
an example of an unstable equilibrium.

light fluid light fluid

heavy fluid heavy fluid

equilibrium perturbation

(a) (b)

wavenumber

of perturbation

y
(^0) k
y
h
Figure 10.2: (a)Top-heavyfluid equilibrium, (b) rippling instability.
The geometry shown in Fig.10.2 is now used to analyze the stability of a heavyfluid
supported by a lightfluid in the situation where no constraint exists at the interface. Here
y corresponds to the vertical direction so that gravity is in the negativeydirection. To
simplify the analysis, it is assumed thatρl<<ρhin which case the mass of the lightfluid
can be ignored. The water and air are assumed to be incompressible, i.e.,ρ=const.,so
that the continuity equation
∂ρ
∂t
+v·∇ρ+ρ∇·v=0 (10.4)
reduces to
∇·v=0. (10.5)
The linearized continuity equation in the water therefore reduces to
∂ρ 1
∂t
+v 1 ·∇ρ 0 =0 (10.6)
and the linearized equation of motion in the water is
ρ 0
∂v 1
∂t
=−∇P 1 −ρ 1 gˆy. (10.7)
The locationy=0is defined to be at the unperturbed air-water interface and the top of the
glass is aty=h. The water fills the glass to the top and thus is constrained from moving
at the top, giving the top boundary condition
vy=0aty=h. (10.8)