2.1 Magnetic field lines

Magnetic field lines are the fundamental element in all discussions of magnetic topology. A field line, sometimes called a line of force, is a space-curve $r(l)$ which is everywhere tangent to the local magnetic field vector $B(x)$ . It satisfies the differential equation

(It can be seen that this definition yields $|dr/dl |= 1$ , demonstrating that the parameter $l$ is the arc-length along the field line, measured forward¹ from the point $r(0)$ .) The field line is a curve, and therefore has volume zero. A flux tube may be constructed by bundling together a group of field lines with net flux $P$ . The tube’s net flux is found by integrating $integral B .da$ over any surface pierced by the entire tube. As with any equation of its form, Equation (1

) may be solved either forward or backward from any “initial” point $r(0)$ , except from so-called singular points where $B(x) = 0$ , since the vector’s direction would not be defined there (Arnold, 1973

). Any volume where $|B(x) |> 0$ may, in principle, be completely filled with field lines such that a unique field line passes through each point.

In general circumstances, the only way to find a field line is to integrate Equation (1 ). A useful shortcut is available, however, in cases with one symmetry dimension (i.e. in two dimensions). In these special circumstances a general magnetic field satisfying $\~/ .B = 0$ can be written in terms of a scalar function called the flux function, and an arbitrary component in the ignorable direction, both depending only on two coordinates² . When $z$ is the ignorable coordinate (planar symmetry), the expression is

$B(x, y) = \~/ A × ^z + Bz(x, y)^z, (2)$

and the flux function $A(x,y)$ is the $^z$ component of the magnetic vector potential. In the case where $f$ is ignorable (azimuthal symmetry) the field takes the form

$B(r, z) = \~/ f × \~/ f + B (r,z)f^ = r-1 \~/ f × f^+ B (r,z)f^, (3) f f$

where the flux function $f$ is related to the vector potential as $Af(r, z) = r-1f(r,z)$ . In either geometry the flux function has the useful property that it is constant along field lines, since its derivative

$dA - 1 --- = |B | B . \~/ A = 0, (4) dl$

or similarly for $df /dl$ . In two-and-a-half dimensional cases, i.e. $Bz /= 0$ , a field line equation like Equation (1

) must still be solved within the flux surface $A =$ constant. It is often the case, however, that the topology of the flux surface defines the topology of its field lines. Part of the appeal of working with two-dimensional models is the ability to easily draw a selection of field lines by contouring the flux function.

The analog of a flux function in three dimensions are the Euler potentials³ $u(x)$ and $v(x)$ , which generate the magnetic field⁴ (Sweet, 1950 ; Dungey, 1953 ; Stern, 1966 ; Sturrock and Woodbury, 1967 )

$B(x) = \~/ u × \~/ v = \~/ × (u \~/ v) = - \~/ × (v \~/ u). (5)$

The two expressions on the right show that the field is automatically divergence-free and that its vector potential can be written as $A = u \~/ v$ or through the gauge transformation $- \~/ (uv)$ as $A = - v \~/ u$ . It is easily verified that both potentials are constant along field lines, so a given field line may be identified by the pair of values $(u,v)$ . Finding potentials to generate a given magnetic field requires the solution non-linear differential equations. These can prove difficult even for simple cases such as a potential field, $\~/ .B = 0$ . Indeed, many useful magnetic fields cannot be even written in the form given by Equation (5

) at all. This very powerful method is therefore used far less frequently than are flux functions.