Martin F. Heyn

Evaluation of 1/ν neoclassical transport in stellarators

Using an analytic solution of the kinetic equation in the 1/ν regime, a new formula for the neoclassical transport coefficients is obtained which takes into account all classes of trapped particles. This formula holds in any coordinate system and no simplifying assumptions about the magnetic field are needed. Therefore it is also applicable to complex magnetic fields given in real space coordinates. The method and the results can be used to optimize magnetic field configurations with respect to the 1/ν regime. The method is bench-marked against Monte Carlo calculation both for the l=3 classical stellarator model and also for the original Helias configuration [J. Nuhrenberg and R. Zille, Phys. Lett. A 114, 129 (1986)] with a more complex magnetic field structure. Some features of transport for Helias are clarified by analyzing the bounce-averaged drift velocity.

Rapid reconnection in compressible plasma

A study of set‐up, propagation, and interaction of non‐linear and linear magnetohydrodynamic waves driven by magnetic reconnection is presented. The source term of the waves generated by magnetic reconnection is obtained explicitly in terms of the initial background conditions and the local reconnection electric field. The non‐linear solution of the problem found earlier, serves as a basis for formulation and extensive investigation of the corresponding linear initial‐boundary value problem of compressible magnetohydrodynamics. In plane geometry, the Green’s function of the problem is obtained and its properties are discussed. For the numerical evaluation it turns out that a specific choice of the integration contour in the complex plane of phase velocities is much more effective than the convolution with the real Green’s function. Many complex effects like intrinsic wave coupling, anisotropic propagation characteristics, generation of surface and side wave modes in a finite beta plasma are retained in th...

A comparison and review of steady-state and time-varying reconnection

Abstract Reconnection is a ubiquitous energy conversion process operating in current sheets. It appears in a variety of applications and is, for example, the dominant coupling process at the Earths magnetopause, the current sheet which separates the solar wind and the terrestrial magnetosphere. Reconnection at the magnetopause is investigated theoretically and experimentally, and in both areas a division exists between so-called steady-state and time-dependent reconnection. In theoretical research the former is associated with the time-invariant analysis of Petschek and coworkers, and this is often put into contrast with an intrinsically time-varying process such as tearing. In experimental research, manifestations of reconnection have been classified either as large scale and (quasi) steady-state, or time dependent, with the former corresponding to accelerated plasma flows along the magnetopause, and the latter to the flux transfer event signature. This division is a source of confusion, in particular since it is unlikely that a true steady-state is ever achieved in nature. To clarify the relationship between steady-state and time-dependent reconnection we discuss here an extension of Petscheks analysis to include time variations in the reconnection rate. In this generalized analysis the reconnection electric field is imposed as an initial-boundary condition which can be specified as an arbitrary function of space and time. Different types of reconnection behaviour can therefore be investigated and we take advantage of this to compare steady-state and time-varying reconnection. We show that the former is just a special case of the latter and that there are no jumps in conceptual understanding required from one to the other. Furthermore, the time-dependent analysis is easily understood and gives a framework which unifies the interpretation of reconnection phenomena observed at the magnetopause. In particular, the theoretical results indicate that the same reconnection rate can give rise to both accelerated plasma flows and the flux transfer event signature; thus there is no physical reason to make a distinction in the underlying process giving rise to different reconnection phenomena.

Time-dependent localized reconnection of skewed magnetic fields

We describe and analyze a model for time-varying, localized reconnection in a current sheet with skewed magnetic field orientations on opposite sides. As in Petscheks description, disruption is initiated in a localized part of the current sheet known as the diffusion region, and the disturbances are subsequently propagated into the system at large through magnetohydrodynamic (MHD) waves. The MHD waves therefore play the dominant role in energy conversion, and collectively they form an outflow for plasma streaming toward the current sheet and a field reversal region joining magnetic field lines from opposite sides. We restrict the analysis to an incompressible plasma, in which case the Alfven wave and the slow shock merge to form shocks bounding the field reversal or outflow region, and to the case of weak reconnection, which implies that the reconnection electric field is much smaller than the product of the characteristic values of the external field strength and Alfven speed. It is then possible to perform a perturbation analysis of the MHD equations which govern the plasma and field behavior. The analysis can be formulated as a mixture of three well-known problems. The problem of determining the appropriate combination of MHD waves corresponds to the Riemann problem, which also specifies the tangential field and flow components in the field reversal region. These results, it is important to note, are not sensitive to variations in the reconnection rate. Reconnection also acts as a source of surface waves, and their analysis determines the behavior of the perpendicular field and flow components and the shape of the shocks. Lastly, the field reversal region can be considered as a thin boundary layer in our treatment, and the external disturbances can therefore be solved in a way similar to the flow around a thin aerofoil. The model presented here can be applied to the Earths magnetopause, where reconnection is considered to be the dominant process coupling the solar wind and the magnetosphere. In particular, the results can be used to interpret different manifestations of reconnection such as accelerated plasma flows along the magnetopause and flux transfer events.

Magnetic reconnection with space and time varying reconnection rates in a compressible plasma

Fast magnetic reconnection of Petschek-type including moving shock waves and discontinuities in a compressible plasma is studied. Magnetic flux tubes of finite size are reconnected by a localized dissipative electric field pulse. This process generates nonlinear perturbations propagating along the initial current surface. The linear wave problem in the outer regions is solved analytically in terms of the reconnection induced sources which move in different directions and with different speeds along the surface. The time-coordinate representation of the solution is given in form of convolution integrals over the reconnection initializing electric field. As an example, reconnection of flux tubes in a sheared magnetic field geometry is analyzed.

Time-dependent reconnection in a current sheet with velocity shear

Abstract Magnetic field reconnection is a macroscopic energy-conversion and transport process which operates in current sheets. Here we investigate analytically a physico-mathematical model of reconnection in a 2-D configuration consisting of a current sheet separating two different plasma regions with antiparallel magnetic field orientations. Reconnection is initiated in a localized region of the current sheet known as the diffusion region. The disruption of the current sheet generates MHD waves, which propagate the local disturbances into the system at large. We apply the MHD equations to describe and analyse the macroscopic response of the plasma and magnetic field to an imposed reconnection rate, which is generally a function of time. The dissipative process leading to disruption is not specified, and instead an arbitrary reconnection rate is used as an initial-boundary condition for solving the MHD equations. Analysis is restricted to an incompressible plasma, and to the case of weak reconnection, which implies that the magnetic field component perpendicular to the current sheet remains small relative to the field strength specified in the initial configuration. Time dependency and the inclusion of different plasma parameter values and field strengths on opposite sides of the current sheet lead to new features which are not evident in Petscheks analysis of steady-state reconnection in a symmetric current sheet configuration. These features include an asymmetric evolution of the outflow region on opposite sides of the current sheet, and a shift of the reconnection line into the region of higher field strength. The outflow region is partially bounded by a tangential discontinuity as a result of the different propagation speeds of the shocks and the surface waves. Once the reconnection rate drops to zero, i.e. reconnection is no longer active, the outflow region splits at the reconnection line and propagates like two solitary waves moving in opposite directions along the current sheet. The previous shift of the reconnection line now corresponds to a displacement of the current sheet downstream of the outflow region. Although no more flux is reconnected at this stage, the outflow region continues to grow in size and gathers up an increasingly large volume of reconnected plasma. The growth is eventually confined to a stretching of the outflow region along the current sheet. With velocity shear there is an additional asymmetry in the evolution on opposite sides of the reconnection line, and the conditions for the appearance and outward propagation of MHD waves are shown to be related to the criterion for Kelvin-Helmholtz stability of the current sheet.

Evaluation of the toroidal torque driven by external non-resonant non-axisymmetric magnetic field perturbations in a tokamak

The toroidal torque driven by external non-resonant magnetic perturbations (neoclassical toroidal viscosity) is an important momentum source affecting the toroidal plasma rotation in tokamaks. The well-known force-flux relation directly links this torque to the non-ambipolar neoclassical particle fluxes arising due to the violation of the toroidal symmetry of the magnetic field. Here, a quasilinear approach for the numerical computation of these fluxes is described, which reduces the dimension of a standard neoclassical transport problem by one without model simplifications of the linearized drift kinetic equation. The only limiting condition is that the non-axisymmetric perturbation field is small enough such that the effect of the perturbation field on particle motion within the flux surface is negligible. Therefore, in addition to most of the transport regimes described by the banana (bounce averaged) kinetic equation also such regimes as, e.g., ripple-plateau and resonant diffusion regimes are naturally included in this approach. Based on this approach, a quasilinear version of the code NEO-2 [W. Kernbichler et al., Plasma Fusion Res. 3, S1061 (2008).] has been developed and benchmarked against a few analytical and numerical models. Results from NEO-2 stay in good agreement with results from these models in their pertinent range of validity.

Analysis of time-dependent reconnection in compressible plasmas

Time-dependent Petschek-type reconnection including moving shock waves and discontinuities in a compressible plasma is studied. Reconnection is initiated by a dissipative electric field pulse along a reconnection line and generates nonlinear perturbations propagating along an initial current sheet. The induced linear wave perturbations in the adjacent half-spaces are obtained analytically in the form of a convolution integral. The results are evaluated for typical dayside magnetopause parameters and presented for comparison with observations of dayside reconnection events.

Kinetic modelling of the interaction of rotating magnetic fields with a radially inhomogeneous plasma

The interaction of rotating magnetic fields (RMFs) with a plasma is modelled in the linear approximation. A kinetic Hamiltonian model for the rf plasma conductivity is used. A radially inhomogeneous periodic cylindrical plasma with a rotational transform of the magnetic field is studied with parameters relevant to the dynamic ergodic divertor (DED) of TEXTOR. For the case of a finite electron diamagnetic velocity it is shown that the torque resulting from the RMF tends to bring the electron fluid approximately to the rest frame of this field. This result is in qualitative agreement with long mean-free path drift MHD theory. In contrast to that theory where a resonant behaviour is found at electron and ion diamagnetic frequencies, in the present kinetic analysis, the RMF frequency where the torque passes through zero is smaller than the electron diamagnetic frequency if there is an electron temperature gradient present. The relation of these results with recent experimental measurements of the DED-induced plasma rotation in TEXTOR is discussed.

Time‐varying reconnection: Implications for magnetopause observations

We discuss the implications of results arising from an analysis of a Petschek-type reconnection model for the interpretation of data obtained at the terrestrial magnetopause. In this model, reconnection is initiated through the introduction of a reconnection electric field in the diffusion region. The magnitude of the electric field is considered to be small compared to the product of characteristic values of the magnetic field strength and Alfven speed in the system; that is, we study the case of weak reconnection only. Outside the diffusion region, the behavior of the plasma is governed by the ideal MHD equations. Petscheks original analysis is generalized through the introduction of a spatially and temporally varying reconnection rate, that is, the reconnection line has a finite length and the reconnection electric field along it varies in time. Additionally, the magnetic fields on either side of the current sheet (although uniform initially) may have arbitrary strength and are skewed relative to each other. New features are that (1) the plasma velocity may have a shear across the current layer, and (2) the densities on either side of the current sheet may be different in general. The reconnection electric field initiates a localized disruption of the current sheet, and the associated disturbances are propagated into the system by MHD waves. With this model we are able to explain and interpret various features observed at the terrestrial magnetopause, such as accelerated plasma flows and flux transfer events. We describe magnetic field signatures predicted by our model. We also show that reconnection is capable of generating surface waves. A property of our model is that it predicts a displacement of the magnetopause when time-dependent reconnection is occurring.