Reconnection: Theory and Computation Programs and Plans C. C. Hegna Presented for E. Zweibel University of Wisconsin CMSO Meeting Madison, WI August 4, 2004

Outline Brief review of the intermediate term (1-3 year) theoretical areas of studies Three brief discussions of ongoing research efforts in these areas: –L. Malyshkin –A. Lazarian –C. C. Hegna Brief review of intermediate term computational tasks Open discussion

Theoretical Topics Asymptotic scaling of reconnection in systems with disparate scales between system size and reconnection layer –L. Malyshkin et al, "Studies of Two-dimensional Anomalous Magnetic Reconnection" Reconnection in chaotic magnetic fields –A. Lazarian "3D Reconnection in the presence of turbulence" Magnetic topology, rational surfaces –C. C. Hegna et al Reconnection in line-tied systems

Reconnection in Line-Tied Systems Toroidal geometry vs. line-tied systems –The vast majority of toroidal confinement devices are doubly periodic - (stellarators, tokamaks, RFPs, …) –Systems where equilibrium magnetic fields permeate a conducting surface have additional boundary conditions - e. g., for cylindrical columns with conducting end plates - B n = E t = 0 Applications include - solar physics, Rotating Wall Experiment, accretion discs The role of reconnection in open systems? How does one describe reconnection processes in these differing geometries?

Magnetic Field Topology in Toroidal Systems are characterized by sets of magnetic surfaces and a winding number q Equilibrium magnetic fields –Toroidal magnetic field –Poloidal magnetic field For symmetric systems, magnetic field lines lie on topological concentric magnetic surface Straight-field line coordinates can be constructed, q( )

Reconnection in toroidal systems occur at rational surfaces Singular nature of marginal ideal MHD equations occurs at rational surfaces in toroidal confinement devices. –For a doubly-periodic system –Singular differential equation at rational surfaces (q = m/n) - general solution allows for current sheets to form at rational surfaces –Formulation allows for descriptions of field-error penetration, resistive wall mode physics, …

Linear mode structure is not described by Fourier harmonics in line tied systems Ideal MHD stability properties of line-tied systems are understood. For example, for the ideal kink properties of the line-tied equilibrium, the linear eigenmode equation satisfies an ODE of the form After some manipulations, the eigenvalue equation reduces to –Line tying provides the boundary condition (z=0) = z=L) = 0

Line tied boundary conditions lead to quantized eigenvalues Linear eigenmode equation –Line tying provides the boundary condition (z=0) = z=L) = 0 –Nonsingular at rational surfaces

With the absence of a rational surface, the singular nature of the line tied equilibrium is not clear Simulations from Lionello et al, indicate the presence of a singular current in the nonlinear phase of an ideal kink mode –Localized to the center of the column - analytic understanding? Does the nonlinear phase of an ideal MHD instability always end up with a current singularity? –In toroidal geometry, the nonlinear phase of the ideal internal kink mode saturates with a current sheet, Rosenbluth, et al 73 Forced reconnection? What are the proper boundary conditions? Analog of the Taylor problem in sheared-slab 2-D problems? Relation to the Parker model of the Solar corona?

Intermediate term computational tasks Simulation efforts to complement theory and experimental programs. Specific implementations include: –Implement 2-fluid capability in NIMROD - Hall-MHD physics, energetic particles, electron viscous stress-tensor, … Vigorous research area amongst the NIMROD team –Implement implicit timestepping in FLASH –Implement AMR in FLASH –Collaborations with codes written outside the Center –Accretion disc flare simulations To be discussed in the momentum transfer section