Force-free Fluxons

Force-Free Fields
    As magnetic flux tubes rise up through the convection zone of the sun and reach the surface, they quickly expand into the low density of the solar corona. When this occurs, two prominent regions are created: one with positive magnetic flux through the photosphere, and one with negative flux. The corona is composed of a low-beta plasma, in which the magnetic pressure is much greater than the gas pressure. Therefore, the magnetic stress is balanced only by the repulsion it experiences from the gradient of other neighboring fields and the magnetic images created below the photosphere. These conditions are generally referred to as magnetohydrostatics (MHS), and this type of force-free magnetic field and is governed the Lorentz equation:

    Although the equations of magnetohydrodynamics reduce to this simplified form, the solutions are still quite complicated. In 1990, Low and Lou proposed a limited set of nonlinear solutions to this rather complicated equation. The use of their solution set effectively generates the magnitude and direction of the magnetic field at any point in the space z>0.



The Fluxon Model
    A "fluxon" is simply an area of constant magnetic flux. In this model, each fluxon is represented simply by a line that runs through the center of the flux bundle. To accomplish this, a line-of-sight magnetogram (as generated by the Low and Lou equatios) is divided up into small squares of constant positive flux and a dot is placed in each square, which represents the starting points for each fluxon.
    The above Lorentz equation can be used, in combination with some vector identities, to derive the following form of the force-free field equation that was our main focus:

This equation can then be separated into a more explicit expression of the magnetic field direction and magnitude as follows:

    In this form, it is more obvious that the first term is the amount of curvature of a circle locally tangent to a segment of a specific magnetic field line. The second term is, of course, the perpendicular gradient of the spacing of flux lines. The combination of these two interpretations shows that the Lorentz Force is always directed perpendicular to the field lines. Furthermore, the first term can be said to represent the tension of the field lines, while the second term represents the outward pressure gradient of the field. In order to further understand this concept, consider the more simplified model of a fluxon as a short, relatively straight flux tube that is constantly seeking to minimize its energy. Since we are primarily concerned with magnetic interactions and properties, the total energy is given by the fluxon's magnetic energy.

where the magnetic field strength (B) can be represented as
    From the above energy equation, it is obvious that the lowest energy results from lower values for the field strength and for the volume of the flux tube. The cross sectional area of this flux tube, however, is a contradictory term. A larger area creates a smaller field strength, but also creates a larger volume. Even so, the field strength is squared in the energy equation, and thus the area parameter is more strongly manifested here. This tendency for the area to expand is responsible for the "pressure gradient" term of the model. Consequently, the easiest way for the flux tube to reduce its volume is to reduce it's length. This tendency is responsible for the "tension" term of the model.
    For obvious reasons, this flux tube can only expand to a certain size, and can only shrink to a certain length. As multiple flux tubes undergo this energy minimization, they begin to interact with one another (magnetic repulsive) and find the lowest energy of the system as a whole, which satisfies the force-free criteria.
    In order to find the Lorentz force at a particular point, each of the above terms must be calculated. The tension term can be calculated by by the following equation

where j is a point along a specific field line (i), and the magnetic field directions are calculated by the following method.

Next, the magnetic pressure gradient must be calculated. To do this, it is best to rewrite this term as

and the magnetic field can be expressed by
where i represents the fluxon number, j represents the points on each fluxon, and delta s represents the distance between two consecutive points on a fluxon. This equation is very convinient because the amount of flux in each fluxon is constant, and therefore Phi only affects the magnitude of the whole field within the fluxon. Furthermore, since this amount of flux is never explicitly defined, the amount of flux can essentially be ignored (or at least assumed to be a unit quantity). The "kernel function" (G) is employed to weight the affects of each point accordingly. This kernel function approximates the Dirac Delta, and therefore has an integral value over three-dimensional space of 1. The exact function used is a three-dimensional parabola spliced onto a power law function, which makes it somewhat smoother and more realistic than the delta itself in that it encorporates more of the surrounding points into the calculation. Specifically, this kernel function is given by
    The x vector represents a point in three dimensional space upon which the kernel function is centered, and gamma is the width coefficient of the kernel. For this example, a gamma value of 3.5 was used.
    Once all of these components have been calculated, the net force at all points on all lines can be found. While the force-free eqations state that this force should be 0, it usually will not be (especially after the perturbation algorithm has been executed. See next secion). The model can then be advanced one "time step" forward in the direction of the net force on each point, which will make the model one step closer to equilibrium.