10.1 The Rayleigh-Taylor instability of hydrodynamics 299The equation of motion for this system ismd^2 x
dt^2=±κx (10.1)whereκis assumed positive, the plus sign is chosen for the ball on hill case, andthe minus
sign is chosen for the ball in valley case. This equation has a solutionx∼exp(−iωt)
whereω=±
√
κ/mfor the valley case andω=±i√
κ/mfor the hill case. The hill
solutionω=−i
√
κ/m is unstable and corresponds to the ball accelerating down the hill
when perturbed from its initial equilibrium position.
If this configuration is extended to two dimensions, then stability would require an
absolute minimum in both directions. However, a saddle point potential would suffice for
instability, since the ball could always roll down from the saddle point. Thus, a multi-
dimensional system can only be stable if the equilibrium potential energy corresponds to
an absolute minimum with respect to all possible displacements.
The problem of determining MHD stability is analogous to having a ball on a multi-
dimensional hill. If the potential energy of the system increases for any allowed perturba-
tion of the system, then the system is stable. However, if there existseven a single allowed
perturbation that decreases the system’s potential energy, then the system is unstable.
10.1 The Rayleigh-Taylor instability of hydrodynamics
An important subset of MHD instabilities is formally similar to the Rayleigh-Taylor in-
stability of hydrodynamics;it is therefore useful to put aside MHD for the moment and
examine this classical problem. As any toddler learns from stacking building blocks, it is
possible to construct an equilibrium whereby a heavy object is supported by a light ob-
ject, but such an equilibrium is unstable. The corresponding hydrodynamic situationhas
a heavyfluid supported by a lightfluid as shown in Fig.10.2(a);this situation is unstable
with respect to the rippling shown in Fig.10.2(b). The ripples are unstable because they
effectively interchange volume elements of heavyfluid with equivalent volume elements
of lightfluid. Each volume element of interchanged heavyfluid originally had its center
of mass a distance∆above the interface while each volume element of interchanged light
fluid originally had its center of mass a distance∆below the interface. Since the potential
energy of a massmat heighthin a gravitational fieldgismghthe respective changes in
potential energy of the heavy and lightfluids are,
δWh=− 2 ρhV∆g, δWl=+2ρlV∆g (10.2)whereV is the volume of the interchangedfluid elements andρhandρlare the mass
densities of the heavy and lightfluids. The net change in the total system potential energy
is
δW=−2(ρh−ρl)V∆g (10.3)
which is negative so that the systemlowersits potential energy by forming ripples. This is
analogous to the ball falling off the top of the hill.
A well-known example of this instability is the situation of an inverted glass of wa-
ter. The heavyfluid in this case is the water and the lightfluid is the air. The system is
stable when a piece of cardboard is located at the interface betweenthe water and the air,
but when the cardboard is removed the system becomes unstable and the water falls out.
The function of the cardboard is to prevent ripple interchange from occurring. Thesys-
tem is in stable equilibrium when ripples are prevented because atmospheric pressure is