V. S. Titov

Ideal kink instability of a magnetic loop equilibrium

The force-free coronal loop model by Titov & Demoulin (1999) is found to be unstable with respect to the ideal kink mode, which suggests this instability as a mechanism for the initiation of flares. The long-wavelength ( m= 1) mode grows for average twists � > 3.5� (at a loop aspect ratio of≈ 5). The threshold of instability increases with increasing major loop radius, primarily because the aspect ratio then also increa ses. Numerically obtained equilibria at subcritical twist are very close to the approximate analytical equilibrium; they do not show indications of sigmoidal shape. The growth of kink perturbations is eventually slowed down by the surrounding potential field , which varies only slowly with radius in the model. With this field a global eruption is not obtained in the ideal MHD limit. Kink perturbations with a rising loop apex lead to the formation of a vertical current sheet below the apex, which does not occur in the cylindrical approximation.

Formation of current sheets and sigmoidal structure by the kink instability of a magnetic loop

We study dynamical consequences of the kink instability of a twisted coronal flux rope, using the force-free coronal loop model by Titov & Demoulin (1999) as the initial condition in ideal-MHD simulations. When a critical value of the twist is exceeded, the long-wavelength (m=1) kink mode develops. Analogous to the well-known cylindrical approximation, a helical current sheet is then formed at the interface with the surrounding medium. In contrast to the cylindrical case, upward-kinking loops form a second, vertical current sheet below the loop apex at the position of the hyperbolic flux tube (generalized X line) in the model. The current density is steepened in both sheets and eventually exceeds the current density in the loop (although the kink perturbation starts to saturate in our simulations without leading to a global eruption). The projection of the field lines that pass through the vertical current sheet shows an S shape whose sense agrees with the typical sense of transient sigmoidal ( forward or reverse S-shaped) structures that brighten in soft X rays prior to coronal eruptions. The upward-kinked loop has the opposite S shape, leading to the conclusion that such sigmoids do not generally show the erupting loops themselves but indicate the formation of the vertical current sheet below them that is the central element of the standard flare model.

Steady linear X-point magnetic reconnection

In the main part of this paper a model for linear reconnection is developed with a current spike around the X-point and vortex current sheets along the separatrices, which are resolved by the effects of viscosity and magnetic diffusivity. The model contains three regions. In the external ideal region, diffusion effects are negligible, and the flow is purely radial but becomes singular both along the separatrices and at the X-point. Near the separatrix there is a self-similar boundary layer with strong electric current and vorticity, where resistivity and viscosity resolve the singularity and allow the flow to cross the separatrix. A composite solution is set up that matches the external and separatrix solutions. Near the origin diffusion also resolves the singularity and is described approximately by a biharmonic solution. A classification of steady two-dimensional reconnection regimes is proposed into viscous reconnection (Me > Re), extra slow (linear) reconnection (Me < Rme−1), slow reconnection (Rme−1 < Me Rme−½), and fast reconnection (Rme−½ < Me), where Me is the dimensionless reconnection rate, Rme the magnetic Reynolds number, and Re the Reynolds number, all based on the Alfven speed far from the reconnection point. Also, an antireconnection theorem is proved, which has profound effects on the nature of linear reconnection. It states that steady two-dimensional MHD reconnection with plasma flow across the separatrices is impossible in a plasma which is inviscid, highly sub-Alfvenic, and has uniform magnetic diffusivity.

Exact solutions for reconnective magnetic annihilation

A family of exact solutions of the steady resistive nonlinear magnetohydrodynamic equations in two dimensions (x, y) is presented for reconnective annihilation, in which the magnetic field is advected across one pair of separatrices and diffuses across the other pair. They represent a two-fold generalization of the previous Craig-Henton solution, since a dimensionless free parameter (γ) in the new solutions equals unity in the previous solutions and the components (vxe, vye) and (Bxe, Bye) of plasma velocity and magnetic field at a fixed external point (x, y) = (1, 0), say, may all be imposed, whereas only three of these four components are free in the previous solutions. The solutions have the exact forms A= A 0 (x)+ A 1 (x)y,ψ= ψ 0 (x)+ ψ 1 (x)y for the magnetic flux function (A) and stream function (ψ), so that the electric current is no longer purely a function of x as it was previously. The origin (0,0) represents both a stagnation point and a magnetic null point, where the plasma velocity (v = ∇ × ψ) and magnetic field (B = ∇ × ψ) both vanish. A current sheet extends along the y-axis. The nonlinear fourth-order equations for A1 and ψ1 are solved in the limit of small dimensionless resistivity (large magnetic Reynolds number) using the method of matched asymptotic expansions. Although the solution has a weak boundary layer near x = 0, we show that a composite asymptotic representation on 0 ⩽ x⩽ 1 is given by the leading-order outer solution, which has a simple closed-form structure. This enables the equations for A0 and ψ0 to be solved explicitly, from which their representation for small resistivity is obtained. The effect of the five parameters (vxe, vye, Bxe, Bye, γ) on the solutions is determined, including their influence on the width of the diffusion region and the inclinations of the streamlines and magnetic field lines at the origin. Several possibilities for generalizing these solutions for asymmetric reconnective annihilation in two and three dimensions are also presented.

Magnetohydrodynamic flows sustaining stationary magnetic nulls

Exact solutions of the resistive magnetohydrodynamic equations are derived which describe a stationary incompressible flow near a generic null point of a three-dimensional magnetic field. The properties of the solutions depend on the topological skeleton of the corresponding magnetic field. This skeleton is formed by one-dimensional and two-dimensional invariant manifolds (so-called spine line and fan plane) of the magnetic field. It is shown that configurations of generic null points may always be sustained by stationary field-aligned flows of the stagnation type, where the null points of the magnetic and velocity fields have the same location. However, if the absolute value |j∥| of the current density component parallel to the spine line exceeds a critical value jc, the solution is not unique—there is a second nontrivial solution describing spiral flows with the stagnation point at the magnetic null. The characteristic feature of these new flows is that they cross magnetic field lines but they do not cr...

Exact solutions for steady reconnective annihilation revisited

This work complements the previous studies on steady reconnective magnetic annihilation in three different geometries: the two-dimensional Cartesian and polar ones and the three-dimensional (3D) cylindrical one. A special class of diffusive solutions is found analytically in explicit form for all of the three geometries. In the 3D case it is extended to a much wider class of exact solutions describing reconnective magnetic annihilation at the separatrix spine line of a magnetic null point. One of the obtained solutions provides an explicit expression for the Craig-Fabling solution. It is also identified which of the steady flow regimes found are dynamically accessible.

Exact solutions for magnetic annihilation in curvilinear geometry

New exact solutions of the steady and incompressible 2D MHD equations in polar coordinates are presented. The solutions describe the process of reconnective magnetic annihilation in a curved current layer. They are particularly interesting for modeling magnetic reconnection in solar flares caused by the interaction of three photospheric magnetic sources.

Exact solutions for reconnective annihilation in magnetic configurations with three sources

Exact solutions of the steady resistive three-dimensional magnetohydrodynamics equations in cylindrical coordinates for an incompressible plasma are presented. The solutions are translationally invariant along one direction and in general they describe a process of reconnective annihilation in a curved current layer with nonvanishing magnetic field. In the derivation of the solutions the ideal case with vanishing resistivity and electric field is considered first and then generalized to include the effects of finite electric field and resistivity. Particular attention is devoted to the analysis of how the latter ones influence the presence of singularities in the solutions. In this respect comparisons with the purely two-dimensional case are made and the resulting important differences are highlighted. Finally, applications of the solutions for modeling an important class of solar flares are discussed.

New classes of exact solutions for magnetic reconnective annihilation

Analytical solutions for reconnective annihilation in curvilinear geometry are presented. These solutions are characterized by current density distributions varying both along the radial and azimuthal coordinates. They represent an extension of previously proposed models, based on purely radially dependent current densities. Possible applications of these solutions to the modeling of solar flares are also discussed.

Analytical solutions for reconnective magnetic annihilation

Exact solutions of the resistive magnetohydrodynamics (MHD) equations for a stationary and incompressible plasma are presented. These solutions describe a particular kind of magnetic reconnection known as magnetic reconnective annihilation. The two dimensional (2D) case in polar coordinates is described first. Subsequently two ansatzs for 3D solutions in cylindrical and cartesian coordinates are presented. The former consists of a generalization of the previously described form for 2D solutions in curvilinear coordinates. The latter is its analogous counterpart in cartesian coordinates and it allows to derive new 3D solutions for fan reconnection.