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
dr-= B-[r(ℓ)]-. (1 ) dℓ |B [r(ℓ)]|
(It can be seen that this definition yields |dr∕d ℓ| = 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 Φ. The tube’s net flux is found by integrating ∫ B ⋅ da over any surface pierced by the entire tube.

As with any equation of its form, Equation (1View Equation) 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, 1973Jump To The Next Citation Point). 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 (1View Equation). 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 × ˆz + Bz (x,y)ˆz, (2 )
and the flux function A(x,y ) is the ˆz component of the magnetic vector potential. In the case where ϕ is ignorable (azimuthal symmetry) the field takes the form
B (r,z) = ∇f × ∇ ϕ + B (r,z )ϕˆ = r−1∇f × ϕˆ+ B (r,z)ϕˆ, (3 ) ϕ ϕ
where the flux function f is related to the vector potential as A ϕ(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 (1View Equation) 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, 1950Jump To The Next Citation PointDungey, 1953Jump To The Next Citation PointStern, 1966Jump To The Next Citation PointSturrock 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 (5View Equation) at all. This very powerful method is therefore used far less frequently than are flux functions.
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