Abstract
The nonlinear behavior of reconnecting modes in three spatial dimensions (3D) is investigated, on the basis of a collisionless fluid model in slab geometry, assuming a strong constant guide field in one direction. Unstable modes in the socalled large regime are considered. Single helicity modes, i.e., modes with the same orientation with respect to the guide field, depending on all three spatial coordinates correspond to “oblique” modes with, in general, mixed parity around the corresponding resonant magnetic surface, giving rise to a nonlinear drift of the magnetic island point. The nonlinear coupling of initial perturbations with different helicities introduces additional helicities that evolve in time in agreement with quasilinear estimates, as long as their amplitudes remain relatively small. Magnetic field lines become stochastic when islands with different helicities are present. Basic questions such as the proper definition of the reconnection rate in 3D are addressed.
Fruitful discussions with R. J. Hastie, F. Malara, M. Ottaviani, L. Rondoni, and P. L. Veltri are gratefully acknowledged. Special thanks go to C. Cavazzoni for his guidance in the code parallelization.
This work was supported by the INFM Parallel Computing Initiative and by the Italian Ministry for University and Scientific Research (Grant No. PRIN2002).
I. INTRODUCTION
II. MODEL EQUATIONS
III. SINGLEHELICITY INITIAL PERTURBATIONS
IV. DOUBLEHELICITY INITIAL PERTURBATIONS
A. Case I
B. Case II
V. MAGNETIC FIELD STOCHASTICITY
VI. DEFINITION OF RECONNECTED FLUX IN 3D
VII. CONCLUSIONS
Key Topics
 Magnetic fields
 33.0
 Magnetic reconnection
 32.0
 Magnetic islands
 28.0
 Current density
 27.0
 Surface magnetism
 18.0
Figures
Comparison between the position of the point as a function of time (crosses) and the displacement calculated by integrating in time the velocity . Note the very good agreement, especially in the nonlinear phase.
Comparison between the position of the point as a function of time (crosses) and the displacement calculated by integrating in time the velocity . Note the very good agreement, especially in the nonlinear phase.
Perturbed magnetic flux and current density profiles at and for two different simulation times. Note the asymmetry of these functions with respect to the rational surfaces located at , in the linear phase, and the nonlinear drift of the current density peaks.
Perturbed magnetic flux and current density profiles at and for two different simulation times. Note the asymmetry of these functions with respect to the rational surfaces located at , in the linear phase, and the nonlinear drift of the current density peaks.
Left frame, for different modes. Right frame, normalized to the growth rate of the (1, 1) mode. Note that the time interval has been selected in such a way that all the represented modes have a significant amplitude.
Left frame, for different modes. Right frame, normalized to the growth rate of the (1, 1) mode. Note that the time interval has been selected in such a way that all the represented modes have a significant amplitude.
Contour plots of the current density and vorticity on the plane, at different simulation times. Superimposed is the magnetic island separatrix (black solid line). Note that the plots have been shifted by in the direction so that the cross structure appears in the middle of the box.
Contour plots of the current density and vorticity on the plane, at different simulation times. Superimposed is the magnetic island separatrix (black solid line). Note that the plots have been shifted by in the direction so that the cross structure appears in the middle of the box.
Contour plots of on the plane, at different simulation times. Superimposed is the magnetic island separatrix (black solid line).
Contour plots of on the plane, at different simulation times. Superimposed is the magnetic island separatrix (black solid line).
Left frame, for different modes. Right frame, normalized to the growth rate of the (1, 1) mode. Note that the time interval has been selected in such a way that all the represented modes have a significant amplitude.
Left frame, for different modes. Right frame, normalized to the growth rate of the (1, 1) mode. Note that the time interval has been selected in such a way that all the represented modes have a significant amplitude.
Perturbed magnetic flux (top panel) and current density (bottom panel) profiles as function of , on the plane , at different simulation times: .
Perturbed magnetic flux (top panel) and current density (bottom panel) profiles as function of , on the plane , at different simulation times: .
Isosurfaces and contour plots of the current density field at two different simulation times. The values of the isosurfaces have been chosen close to the value of the maximum of the current density at the corresponding time.
Isosurfaces and contour plots of the current density field at two different simulation times. The values of the isosurfaces have been chosen close to the value of the maximum of the current density at the corresponding time.
Difference between each term of the energy, as defined in Eq. (6), and the corresponding value at the initial time, divided by the total initial energy plotted time.
Difference between each term of the energy, as defined in Eq. (6), and the corresponding value at the initial time, divided by the total initial energy plotted time.
Conservation properties of a singlehelicity case and of a doublehelicity case. The data of the singlehelicity case refer to the case presented in Sec. III, while the data of the doublehelicity case refer to the case I presented in this section. denotes the difference between the final and the initial values. Note that both integrals are Casimir invariants for a single helicity case in which case they are numerically conserved within the same percentage range. On the contrary, for a doublehelicity case is not invariant and its variation grows in time as the nonlinear interactions become important.
Conservation properties of a singlehelicity case and of a doublehelicity case. The data of the singlehelicity case refer to the case presented in Sec. III, while the data of the doublehelicity case refer to the case I presented in this section. denotes the difference between the final and the initial values. Note that both integrals are Casimir invariants for a single helicity case in which case they are numerically conserved within the same percentage range. On the contrary, for a doublehelicity case is not invariant and its variation grows in time as the nonlinear interactions become important.
In the first column are the contour plots of the current density in the plane. In the second column are the sections of for on the same plane. Note how the presence of a smaller perturbation on the (1, 1) mode significantly alters the structure of the (1, 0) mode when the merging between the two corresponding peaks has occurred.
In the first column are the contour plots of the current density in the plane. In the second column are the sections of for on the same plane. Note how the presence of a smaller perturbation on the (1, 1) mode significantly alters the structure of the (1, 0) mode when the merging between the two corresponding peaks has occurred.
Left frame: current density isosurface corresponding to a value close to the maximum of the current density. Right frame: isosurface at a lower value.
Left frame: current density isosurface corresponding to a value close to the maximum of the current density. Right frame: isosurface at a lower value.
(Color). Poincarè maps for case I at different evolution times. The maps have been computed by integrating the field line equations vs the variable using the Hamiltonian obtained from the MHD code. Stochasticity starts to develop near the separatrices of the magnetic islands, corresponding to the two initially excited modes. Subsequently, it grows to such a level that in the Poincarè map corresponding to the last Hamiltonian output of the MHD simulation (last frame), only a few KAM magnetic surfaces are preserved. The curves drawn in red represent the KAM surfaces utilized in Sec. VI to define the reconnected flux.
(Color). Poincarè maps for case I at different evolution times. The maps have been computed by integrating the field line equations vs the variable using the Hamiltonian obtained from the MHD code. Stochasticity starts to develop near the separatrices of the magnetic islands, corresponding to the two initially excited modes. Subsequently, it grows to such a level that in the Poincarè map corresponding to the last Hamiltonian output of the MHD simulation (last frame), only a few KAM magnetic surfaces are preserved. The curves drawn in red represent the KAM surfaces utilized in Sec. VI to define the reconnected flux.
Plot of the average Lyapunov exponent vs the number of periods along , , for the Hamiltonians corresponding to different times. The average has been performed over field lines with initial conditions taken in the stochastic region.
Plot of the average Lyapunov exponent vs the number of periods along , , for the Hamiltonians corresponding to different times. The average has been performed over field lines with initial conditions taken in the stochastic region.
Plot of the firstorder, , and secondorder moments, , vs the number of periods along , , at . Each moment oscillates around its mean value represented by the superimposed horizontal line. After a few periods the value of , averaged over different initial conditions, becomes of the order of the maximum width of the stochastic region.
Plot of the firstorder, , and secondorder moments, , vs the number of periods along , , at . Each moment oscillates around its mean value represented by the superimposed horizontal line. After a few periods the value of , averaged over different initial conditions, becomes of the order of the maximum width of the stochastic region.
(Color). Contour plots and profiles of the current density on the plane, corresponding to the last two frames of Poincarè maps of Fig. 13.
(Color). Contour plots and profiles of the current density on the plane, corresponding to the last two frames of Poincarè maps of Fig. 13.
(Color). Poincarè maps for case II at different evolution times. The curves drawn in red represent the KAM surfaces utilized in Sec. VI to define the reconnected flux.
(Color). Poincarè maps for case II at different evolution times. The curves drawn in red represent the KAM surfaces utilized in Sec. VI to define the reconnected flux.
Logarithmic plot of the toroidal magnetic flux for case I and case II. The time has been normalized to the linear growth rate for , . The two curves have been shifted to highlight their similar behavior when the transition to the global stochasticity is encountered.
Logarithmic plot of the toroidal magnetic flux for case I and case II. The time has been normalized to the linear growth rate for , . The two curves have been shifted to highlight their similar behavior when the transition to the global stochasticity is encountered.
Logarithmic plot of the magnetic energy variation, , dashed line, and of the reconnected magnetic flux , solid line, vs the normalized time . The data refer to case I.
Logarithmic plot of the magnetic energy variation, , dashed line, and of the reconnected magnetic flux , solid line, vs the normalized time . The data refer to case I.
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