### 5.4 Stability of force-free fields

In principle, the MHD stability criteria can also be applied to force-free equilibria. Typical approaches
(see the book by Priest, 1982) to investigate the stability of ideal MHD equilibria (which correspond to the
assumption of infinite electrical conductivity) are normal mode analysis and an energy criterion. The basic
question is how a small disturbance to the equilibrium evolves. Analytic methods typically linearize the
problem around an equilibrium state, which leads to the so-called linear stability analysis. One has to
keep in mind, however, that a linearly-stable configuration might well be nonlinearly unstable.
The nonlinear stability of a system is usually investigated numerically with the help of time
dependent simulations, e.g., with an MHD code (see also Section 5.5 for an application to NLFFF
equilibria). In the following, we concentrate on linear stability investigations by using an energy
criterion.
For a force-free configuration the energy is given by

where the subscript corresponds to the equilibrium state. This equilibrium becomes disturbed
by an displacement in the form with . This
form of the magnetic field displacement has its origin from the linearized induction equation
, where the velocity field has been replaced by the displacement . The MHD
energy principle (Bernstein et al., 1958) reduces for force-free fields to (Molodensky, 1974):
A configuration is stable if , unstable for , and marginally stable for . For
force-free fields and using the perturbed vector potential , Equation (46) can be written as:
From Equation (47) it is obvious that the potential field with is stable. If we approximate
with a typical length scale of the system, the first term may remain larger than the
second term (i.e., stability) in Equation (47) if
This means that the scale of twist in the system, , should be larger than the system size
for it to be stable. This criterion is known as Shafranov’s limit in plasma physics. More
precise criteria for stability can be obtained for specific geometries. For example the case of
cylindrical linear force-free field (Lundquist’s field) was studied by Goedbloed and Hagebeuk
(1972).