Question 4.1.4: If the plasma thermal pressure is negligibly small compared ......
If the plasma thermal pressure is negligibly small compared to the pressure of the magnetic field (i.e., for β → 0), and the plasma is in magnetostatic equilibrium, then (\vec{∇} × \vec{B})× \vec{B} = 0; the magnetic force is absent.
Hence, such a field is called a force-free magnetic field. Derive the general form for a force-free field that does not vary along one of the cooordinates (say, z).
The most general form for an arbitrary magnetic field, \vec{B} (x, y), that satisfies identically a necessary condition \vec{∇} · \vec{B} = 0 is
\vec{B}(x, y) = [\vec{∇} Ψ (x, y) ×\vec{e}_{z}] + B_{z}(x, y)\vec{e}_{z}, (4.10)
where ψ(x, y) is called the poloidal flux function, and B_{z} the toroidal magnetic field. Now, B_{x} = \frac{∂Ψ}{∂y} , B_{y} = − \frac{∂Ψ}{∂x} , so that the equation Ψ(x, y) = const defines projections of magnetic field lines on the (x, y) plane. According to equation (4.10),
(\vec{∇} × \vec{B}) = [\vec{∇}B_{z}(x, y) × \vec{e}_{z}] − \vec{∇}^{2} Ψ(x, y)\vec{e}_{z}, (4.11)
and since, for a force-free field, vectors (4.10) and (4.11) must be parallel to each other, this should be also the case for vectors \vec{∇} Ψ(x, y) and \vec{∇}B_{z}(x, y).
This implies that B_{z}(x, y) = F(Ψ). Furthermore, the ratio of the poloidal components of (4.11) and (4.10), which is dF/dΨ , should be equal to the ratio of their toroidal components, which is −\vec{∇}^{2} Ψ/F. Hence, we have
\vec{∇}^{2} Ψ = −F \frac{dF}{dΨ} (4.12)
This is the sought after Grad-Shafranov equation for the force-free magnetic field (4.10). In general, this is a non-linear equation that is hardly tractable analytically. It becomes greatly simplified for the special case of the so-called linear force-free fields, for which F(Ψ) = αΨ , where α = const.
With this choice the equation (4.12) reduces to
\vec{∇}^{2} Ψ = −α^{2}Ψ (4.13)