Abstract
Asymmetric current sheets are likely to be prevalent in both astrophysical and laboratory plasmas with complex three dimensional (3D) magnetic topologies. This work presents kinematic analytical models for spine and fan reconnection at a radially symmetric 3D null (i.e., a null where the eigenvalues associated with the fan plane are equal) with asymmetric current sheets. Asymmetric fan reconnection is characterized by an asymmetric reconnection of flux past each spine line and a bulk flow of plasma across the null point. In contrast, asymmetric spine reconnection is characterized by the reconnection of an equal quantity of flux across the fan plane in both directions. The higher modes of spine reconnection also include localized wedges of vortical flux transport in each half of the fan. In this situation, two definitions for reconnection rate become appropriate: a local reconnection rate quantifying how much flux is genuinely reconnected across the fan plane and a global rate associated with the net flux driven across each semiplane. Through a scaling analysis, it is shown that when the ohmic dissipation in the layer is assumed to be constant, the increase in the local rate bleeds from the global rate as the sheet deformation is increased. Both models suggest that asymmetry in the current sheet dimensions will have a profound effect on the reconnection rate and manner of flux transport in reconnection involving 3D nulls.
We wish to acknowledge the financial support of EPSRC.
I. INTRODUCTION
II. GENERAL METHOD
III. ASYMMETRIC FAN RECONNECTION
A. The symmetric case
B. The asymmetric case
IV. ASYMMETRIC SPINE RECONNECTION
A. The symmetric case
B. A simple asymmetric case
V. RECONNECTION RATE: THE SIMPLE ASYMMETRIC CASE
VI. ASYMMETRIC SPINE RECONNECTION: GENERAL EXAMPLES
A. Reconnection rate vs ohmic dissipation
VII. CONCLUDING REMARKS
Key Topics
 Magnetic reconnection
 33.0
 Plasma flows
 29.0
 Rotating flows
 17.0
 Magnetic fields
 13.0
 Electric fields
 11.0
Figures
(a) and (b) isosurface of at 25% of the maximum and the current flow in the y = 0 plane for the symmetric model. (c) and (d) the equivalent figures for the asymmetric model with p = 0.5 and m = 0.5 (see Eq. (20) ). Both have the parameter set .
(a) and (b) isosurface of at 25% of the maximum and the current flow in the y = 0 plane for the symmetric model. (c) and (d) the equivalent figures for the asymmetric model with p = 0.5 and m = 0.5 (see Eq. (20) ). Both have the parameter set .
(a) and (b) in the x = 0 and z = 4 planes, respectively, for the symmetric model. (c) and (d) in the same planes for the asymmetric case with m = 0.5. The contours and arrows denote and , respectively. The spine is shown in blue as a line in the x = 0 plane and a square in the z = 4 plane. The fan plane is shown in red. The parameters are as in Figure 1 .
(a) and (b) in the x = 0 and z = 4 planes, respectively, for the symmetric model. (c) and (d) in the same planes for the asymmetric case with m = 0.5. The contours and arrows denote and , respectively. The spine is shown in blue as a line in the x = 0 plane and a square in the z = 4 plane. The fan plane is shown in red. The parameters are as in Figure 1 .
Evolution of flux in the symmetric (a)–(c) and asymmetric (d)–(f) fan cases. For the parameter set given in Figure 1 .
Evolution of flux in the symmetric (a)–(c) and asymmetric (d)–(f) fan cases. For the parameter set given in Figure 1 .
(a) and (b) isosurface of at 25% of the maximum and the current flow in the z = 0 plane for the symmetric model. (c) and (d) are the equivalent figures for the asymmetric model with m = 0.5. Both have the parameter set .
(a) and (b) isosurface of at 25% of the maximum and the current flow in the z = 0 plane for the symmetric model. (c) and (d) are the equivalent figures for the asymmetric model with m = 0.5. Both have the parameter set .
(a) in the x = 0 plane with contours showing the strength of . The spine is in blue and the fan plane red. (b) evaluated on the fan plane (Z = 0) with the dotted circle showing the cut taken in Figure 6 . (c) and (d) are the corresponding figures for the simple asymmetric case. For the parameters given in Figure 4 .
(a) in the x = 0 plane with contours showing the strength of . The spine is in blue and the fan plane red. (b) evaluated on the fan plane (Z = 0) with the dotted circle showing the cut taken in Figure 6 . (c) and (d) are the corresponding figures for the simple asymmetric case. For the parameters given in Figure 4 .
with (a) n = 0, (b) n = 1, and (c) n = 3. To be compared against Figures 5(b) , 5(d) , and 10(b) , respectively. For the parameter set .
(a) Integral loops constructed along paths either or to the magnetic field. Such paths enable potential drops ( ) along field lines crossing to be compared with flux movement in the ideal region. (b) The induced flux transport in the ideal region threading with the edge of depicted by blue lines.
(a) Integral loops constructed along paths either or to the magnetic field. Such paths enable potential drops ( ) along field lines crossing to be compared with flux movement in the ideal region. (b) The induced flux transport in the ideal region threading with the edge of depicted by blue lines.
Reconnection rate diagram. The edge of a general asymmetric nonideal region is shown in red on the fan plane. The points A and B lie between the positive and negative regions of flux transport across this plane. These points can be connected by a path through the ideal region around the edge of the large side of the nonideal region (C 3), around a path circuiting the small side (C 2) or though the nonideal region and the null (C 1).
Reconnection rate diagram. The edge of a general asymmetric nonideal region is shown in red on the fan plane. The points A and B lie between the positive and negative regions of flux transport across this plane. These points can be connected by a path through the ideal region around the edge of the large side of the nonideal region (C 3), around a path circuiting the small side (C 2) or though the nonideal region and the null (C 1).
The shape of the nonideal region shown in red on the fan plane when . The distance L indicates the length of the nonideal region along the line AB.
The shape of the nonideal region shown in red on the fan plane when . The distance L indicates the length of the nonideal region along the line AB.
Asymmetric spine reconnection with n = 3. (a) Current flow in the z = 0 plane. (b) in the fan plane (Z = 0) to be compared against Figure 6(c) . (c) (red) overlayed with (green). The overlaid dashed grid highlights the relationship between the two quantities.
Asymmetric spine reconnection with n = 3. (a) Current flow in the z = 0 plane. (b) in the fan plane (Z = 0) to be compared against Figure 6(c) . (c) (red) overlayed with (green). The overlaid dashed grid highlights the relationship between the two quantities.
Loglog plots of , and vs n. (a) when all other parameters are held fixed (given in Figure 6 ). (b) when a stall is introduced heuristically into (see Eq. (52) ).
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