2.2 Physical significance of magnetic field lines

Many of the statements collected under the heading magnetic topology are simply mathematical consequences of the first-order ordinary differential equation (1

). These same properties apply to integral curves found in other areas of physics, such as phase-space trajectories, flow stream lines or vortex lines. Unlike these more abstract curves, however, magnetic field lines are often closely related to physical structures, so that their topological properties can have direct physical significance.

It goes without saying that the magnetic field is physically significant in many situations. It is not so clear, however, how magnetic field lines themselves, the integral curves satisfying Equation (1 ) at a single instant, have physical significance. Indeed, elementary physics texts often contain the warning that lines of force (i.e. field lines) are not physically meaningful. There are certain circumstances in space physics, however, in which this warning may be disregarded and field lines are related to physical properties of the plasma. The list below mentions a few mechanisms by which field lines are rendered physically meaningful. We will have occasion to revisit these circumstances in order to decide which topological properties are truly relevant.

Single particle motion

A charged particle subject to no other forces will remain close to a single field line. Drifts will displace the particle’s guiding center by several gyro-radii after it has traversed a length comparable to the field’s curvature radius or gradient scale. Most space plasmas are characterized by global scales much, much greater than the gyro-radii of their particles, especially their electrons. Field lines are therefore excellent approximations to the electron orbits, at least between scattering events.

The heliosphere, for example, includes a population of high-energy electrons (halo electrons) for which collisions are so rare that each one remains effectively confined to a single field line from the Sun to beyond $1 AU$ (Feldman et al., 1975 ). The properties of the electrons at a given point may therefore be attributed to events occurring elsewhere on that field line. Electrons flowing in both directions along the field lines imply that both ends of the field line are connected to the Sun (Gosling et al., 1987 ; McComas et al., 1995 ).

Solar flares produce an accelerated population of electrons which follow the field line on which they are produced until they impact the dense chromosphere. This impact produces signatures such as $Ha$ ribbons and hard X-ray footpoint emission, which betray the magnetic field configuration above. When footpoints or ribbons appear in pairs they are assumed to be conjugate footpoints of a single magnetic field line.

As a beam of flare-accelerated electrons passes through the ambient plasma, they excite radio waves at the local plasma frequency. The radio frequency changes as the beam propagates into regions of higher or lower ambient density, producing a characteristic emission pattern knows as a type-III radio burst (Bastian et al., 1998 ). Frequencies decreasing below $~ 10 MHz$ are taken as a evidence that the electrons are propagating on open field lines, while those which remain at higher frequencies are assumed to be trapped on closed field lines.

Thermal conductivity and coronal loops

In a diffuse, high temperature plasma thermal energy is conducted principally by electrons. When electrons are strongly magnetized ( $_O_e » nei$ ) their orbits will follow field lines between collisions making thermal conductivity highly anisotropic (Braginskii, 1965 ). Heat is conducted parallel to the magnetic field far more readily than perpendicular to the field.

Due to this anisotropic conductivity, heat deposited somewhere in a plasma is rapidly and efficiently conducted to all points on the same field line. Plasma flows are also mechanically confined by the field, so a bundle of field lines will behave as one-dimensional autonomous atmosphere (Rosner et al., 1978 ). High resolution images of the corona made in soft X-ray (SXR) or extreme ultraviolet (EUV) are characterized by thin coronal loops which are each assumed to be a single bundle of field lines for the reasons just mentioned.

Figure 2 shows an example of an image made at the EUV wavelength $171 ºA$ by the TRACE spacecraft (Handy et al., 1999 ). The majority of emission at this wavelength is believed to originate in coronal plasma around $T -~ 106 K$ . The numerous dark, thin curves are coronal loops, which are believed to follow bundles of coronal field lines. The coronal loops in this figure appear to connect polarity regions from three different active regions located at the center, left and lower-left of the field of view.

Coronal loop images in SXR or EUV provide one of the best observational indicators of magnetic topology in the corona. In the end, though, a coronal loop is not a magnetic field line, but is a column of plasma characterized by its excess emission. The neighboring corona is filled with other field lines which, as far as we know, are magnetically identical but which do not appear in these images. Partly this is due to the temperature response of the particular instrument, especially for narrow-band EUV images such as Figure 2 . Here we are seeing only that plasma which happens to be within a narrow range of temperatures. Isolated loops also appear in broad-band instruments, such as Yohkoh SXT, which are sensitive to a much broader range of temperatures. Indeed, these same loops are sometimes observed in narrow-band EUV images at a slightly later time (Winebarger and Warren, 2005 ), so temperature response alone cannot explain why so much of the coronal magnetic field is free of loops at a given time.

One interpretation of various multi-temperature observational studies is that the corona has a tendency to form density enhancements along selected bundles of field lines, which then appear in imaging instruments as loops. No complete explanation has yet emerged as to why some bundles are selected while the majority are not (see Litwin and Rosner, 1993 , for one proposed explanation). We henceforth assume only that coronal images reveal a sampling (perhaps rather sparse) of field-line bundles from the coronal magnetic field. It is also difficult to associate the time evolution of a loop with the motion of a given field line, since it is always possible that a pattern of sequential brightenings on neighboring loops has produced an apparent motion in a stationary magnetic field.

Figure 2:

An EUV image of the coronal plasma made by TRACE at $171 ºA$ on October 26, 1999. The intensity of $171 ºA$ emission from a small portion of the solar disk is indicated by a reverse color table (darker indicates higher emission). An inset shows the location of the irregular field of view on the solar disk.

Alfvén wave propagation: Low-frequency waves in a magnetized plasma comprise three branches: slow magnetosonic, fast magnetosonic and shear Alfvén waves. The group velocity of the shear Alfvén wave is exactly parallel to the local magnetic field. Within the WKB limit any small localized disturbance will therefore propagate along a path following a magnetic field line. This means that a given field line will “learn” of perturbations anywhere along its arc at the Alfvén speed.
When equilibrium is established in a magnetic field, the distribution of current and pressure is dictated by equations whose characteristics are the field lines. For example, in a force-free equilibrium, the current is proportional to the field: $\~/ × B = aB$ , where $B . \~/ a = 0$ . This means that the field line twist parameter $a(x)$ must be constant along each field line. In this way the field lines are the mathematical characteristics for the equilibrium equation (see Parker, 1979 , for a discussion of characteristics in magnetostatic equations).

Frozen field lines

Space plasmas are often approximated as perfect conductors, meaning that the electric field vanishes in a frame moving with the plasma’s local center of mass, $' E = 0$ . Expressing this fact in terms of the corresponding laboratory electric field $E = - (v × B)/c$ , and using it in Faraday’s law yields the ideal induction equation,

$@B ----- \~/ × (v× B) = 0, (6) @t$

governing the evolution of the magnetic field $B(x, t)$ under the influence of a plasma flow $v(x,t)$ ⁵ . Equation (6

) provides a recipe for updating the magnetic vector field at all points in space under the influence of a perfectly conducting fluid with velocity field $v(x, t)$ . It is one of the basic equations of magnetohydrodynamics (MHD); the others being mass continuity equation, momentum equation and some form of energy equation. Considerations of magnetic field topology and its evolution concern only the consequences of Equations (1

, 6

), independent of the other equations. Among the direct consequences of these two equations are the fact that field lines move with the plasma, and that flux is frozen into the plasma. First put forward by Alfvén (1943 ), developed further in following years (Sweet, 1950 ; Dungey, 1953 ), and rigorously formulated by Newcomb (1958

), versions of these relationships are derived in most plasma physics texts (Moffatt, 1978 ; Parker, 1979 ; Priest, 1982

; Sturrock, 1994 ), in many review papers (Stern, 1966 ; Axford, 1984 ; Greene, 1993

), and among the preliminaries of topologically-oriented investigations (Vasyliunas, 1975

; Hesse and Schindler, 1988

; Hornig and Schindler, 1996

). We present still another derivation below since these concepts are central to all that follows.

The relation between plasma motion and magnetic field lines follows from the manner by which Equation (6 ) relates the magnetic field and the velocity field. This inter-relation leads to an inter-relation between field lines, defined by Equation (1 ), and fluid trajectories. A particle moving at the plasma’s flow velocity, i.e. a fluid element, follows a trajectory $r(t)$ satisfying

Parameterizing field lines by mass-column $integral m = (r/ |B |)dl$ , rather than arc length $l$ , transforms the field line equation (1

) into

The reality of field lines follows from the fact that the field line equation (8 ) may be integrated either before or after trajectories of the fluid elements forming that field line are followed. Consider following a field line for $dm$ from a point $r0$ to a point $rm$ . Next follow the trajectory from $rm$ for an interval $dt$ to a point $rmt$ . This point is displaced by $d2rmt$ from $r0$ as shown in Figure 3 .

The two operations may be performed in the opposite order by first following, for $dt$ , the trajectory of $r0$ , and denoting by $rtm$ the point $dm$ along the later field line. The difference in separations resulting from these two processes can be shown to be

$( d dr d dr) d2rmt- d2rtm = dtdm ----- - ------ [(dtdm dm d)t ( ) ] @ B B = dtdm ---+ v . \~/ -- - -- . \~/ v (9) @t r r$

to leading order in $dm$ and $dt$ . The term in square brackets can be written, after using mass continuity, as the left hand side of Equation (6

) divided by $r$ . This means that for a perfectly conducting plasma, i.e. $' E = 0$ , the displacement difference vanishes to second order, proving that the processes of field line tracing and trajectory following commute with one another.

The calculation above can be integrated to finite $Dt$ and $Dm$ to show that two fluid elements on the same field line at one time, will also be on the same field line at all later times. Following this reasoning, we see that a field line evolves in time exactly like the curve of fluid elements initially lying along it. So long as the fluid velocity $v(x, t)$ remains bounded and continuous then it will transform the field line continuously, without breaking it.

The frozen field line theorem reviewed above is related to, but not the same as, the frozen-flux theorem. The latter is the magnetic analogue of Kelvin’s circulation theorem for inviscid fluid flow. According to the frozen-flux theorem, the magnetic flux, $integral P = B .da$ , enclosed by a closed loop of fluid elements (not necessarily a field line) will not change as the loop moves. This is a straightforward consequence of the fact that the electromotive force, $gf E'.dl$ , must vanish in any perfect conductor, since $' E = 0$ .

Figure 3:

An illustration of how the ideal induction equation implies that the operations of tracing field lines and following trajectories commute with one another. Field lines (green lines) are tangent to the magnetic field vectors (green arrows), while trajectories (red lines) are tangent to velocity vectors (red arrows). These are followed beginning with the point $r0$ in both orders to the points $rmt$ and $rtm$ . The net displacements $d2rmt$ and $d2rtm$ are shown by blue arrows (they are not infinitesimal in the figure). In the case that $B$ and $v$ are related through the ideal induction equation, shown on the right, $rmt = rtm$ , so the order of operations does not matter.

It is important to note that the final item, concerning frozen field lines, provides the only sense in which field lines are persistent. To be sure, field lines may be found for the magnetic field $B(x)$ at a given instant and these will be physically significant for any of the other reasons presented above. In order to consider the evolution of a given field line, however, one must be able to track field lines in time. This tracking is unambiguous only when the field evolves according to the ideal induction equation (6

). There are a slightly broader set of circumstances where it is possible to formally define field evolution (Hesse and Schindler, 1988

; Hornig and Schindler, 1996

), however, this definition will not be physically meaningful unless it coincides with the motion of something physical such as the plasma or the electron fluid. That is to say, while it is mathematically possible to define rules for tracking field lines, these rules are not observed by objects such as electrons or Alfvén waves.