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

∫ 2 B-0- W0 = 2μ0dV, (45 )
where the subscript 0 corresponds to the equilibrium state. This equilibrium becomes disturbed by an displacement ξ(r0,t) in the form B = B0 + B1 with B1 = ∇0 × (ξ × B0 ). This form of the magnetic field displacement has its origin from the linearized induction equation ∂B1-= ∇ × (v1 × B0 ) ∂t, where the velocity field has been replaced by the displacement ξ. The MHD energy principle (Bernstein et al., 1958) reduces for force-free fields to (Molodensky, 1974):
∫ -1-- [ 2 ] W = 2μ (∇ × (ξ × B )) − (∇ × (ξ × B )) ⋅ (ξ × (∇ × B )) dV. (46 ) 0 V
A configuration is stable if W > 0, unstable for W < 0, and marginally stable for W = 0. For force-free fields and using the perturbed vector potential A1 = ξ × B, Equation (46View Equation) can be written as:
∫ -1-- [ 2 ] W = 2μ0 (∇ × A1 ) − αA1 ⋅ ∇ × A1 dV. (47 ) V
From Equation (47View Equation) it is obvious that the potential field with α = 0 is stable. If we approximate |∇ × A1| ∼ |A1|∕ℓ with a typical length scale ℓ of the system, the first term may remain larger than the second term (i.e., stability) in Equation (47View Equation) if
|α| ≲ 1∕ℓ. (48 )
This means that the scale of twist in the system, 1∕α, should be larger than the system size ℓ 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).
View Image

Figure 9: Numerical simulations starting from an unstable branch of the Titov–Démoulin equilibrium in comparison with TRACE observations of an eruption. Image reproduced by permission from Figure 1 of Török and Kliem (2005Jump To The Next Citation Point), copyright by AAS.

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