Summary: We consider the formation of small-scale magnetic structures in solar coronal loops, with the aim of understanding the possible role of these structures in the process of coronal heating. A simplified model of a coronal loop is discussed. Neglecting loop curvature, we consider an initially uniform magnetic field embedded in a perfectly conducting plasma between two flat parallel plates \(z=0\) and \(z=L\), which represent the photosphere at the two ends of the loop. Slow, random motions at these boundary plates produce twists and braids in the magnetic field. We discuss the properties of such braided fields assuming the field evolves through a series of force-free equilibria.
Using a Lagrangean description of the field, the equilibrium problem is formulated as a boundary-value problem for the functions \(X(x_ 0,y_ 0,z,t)\) and \(Y(x_ 0,y_ 0,z,t)\) which describe the shape of field lines characterized by the initial coordinates \(x_ 0\) and \(y_ 0\). We argue that \(X(x_ 0,y_ 0,z,t)\) and \(Y(x_ 0,y_ 0,z,t)\) are continuous functions of \(x_ 0\) and \(y_ 0\) at time \(t=T\), provided X and Y are continuous in \(x_ 0\) and \(y_ 0\) at the boundary plates \((z=0\) and \(z=L)\) for all intermediate times \(0<t<T.\)
In particular, we show that isolated infinitesimally thin current sheets do not arise if the field between the plates is force free. This suggests that spatially continuous velocity fields at the boundary plates do not produce tangential discontinuities in the magnetic structure as first suggested by E. N. Parker [(*) Ap. J. 174, 499 ff. (1972)]. It also implies that ideal-MHD instabilities, if they occur in this model, do not lead to tangential discontinuities. We contrast our results with those obtained for more complicated field topologies containing multiple flux systems. Instead of the catastrophic “non-equilibrium” process of current-sheet formation proposed by Parker (*), we propose a more gradual process in which small-scale structures are produced by the random intermixing of magnetic footpoints in the solar photosphere.