Basic schematic of simulation domain and boundary conditions employed to mimic the MRX experiment. Computational grid covers the entire rectangular domain. Field boundary conditions are applied only on the outer boundary, while particle boundary conditions are applied at the surface of the flux cores and the outer wall. The current inside the flux cores is prescribed as a function of time. For all simulations presented in this article, , , , , and .
Comparison of the actual MRX poloidal field (PF) current waveform to the functional fit in Eq. (1) employed in this simulation study. For the case shown, the time scale of the current ramp-down .
Time evolution of the electron current sheet for a simulation with dimensionless parameters , (equivalent to ), , , and full particle absorption at the flux core surfaces. The initial density is corresponding to a flux core spacing of . Shown is out-of-plane component of the electron flow normalized to the initial electron thermal velocity ; black lines correspond to the magnetic flux surfaces. Movie showing the evolution is available in Ref. 38.
Typical structure of the reconnection layer for the same case as Fig. 3 at time . Top panel shows the electron density for the entire global domain while subsequent panels show various quantities of interest within the window indicated in the top panel. Black lines represent flux surfaces in all panels except the bottom, where the black lines indicate the streamlines for the electron flow. The density is normalized to , the ion and electron flow velocities are normalized to the initial thermal velocity for each species and is normalized to the initial reference magnetic field .
Normalized electric field at the center of the layer as a function of simulation time for various drive time times indicated (top). The parameters are , and . The bottom panel shows the scaling of the average electric field in the quasi-steady interval (indicated by vertical dashed lines in upper panel); error bars represent the standard deviation due to the time average. Simulations include a range of initial densities and mass ratios , but all cases have constant box geometry and . Some cases in the over-driven regime have a reflection coefficient parameter of 0.1 or 0.2 to prevent magnetic island formation (see Sec. III E). Circle sizes represent different initial densities; values given for assume hydrogen ions and are in units of .
Reconnection inflow rate as a function of simulation time for the various drive times indicated (top). The bottom panel shows the time-averaged rate over the quasi-steady interval as function of drive time . Parameters and notation are the same as Fig. 5.
Normalized inflow speed profile [as defined by Eq. (6)] at for a well-matched simulation compared with MRX measurements for a hydrogen discharge. Simulation parameters are , , , , and no reflection at the flux core surfaces. There is a mismatch on the inboard side of MRX due to toroidal geometry effects not included in the simulation.
Structure of the electron layer from experimental data [panels (a)–(c)] compared with the well-matched simulation from Fig. 7 [panels (d)–(f)]. Arrows in panels (b) and (e) represent in-plane electron flow; the color scale in the two figures is identical and represents , where is measured at ( is a half-layer width based on a Harris sheet fit to the profile) and is measured at the point in question. Note that although the electron layer dimensions in centimeters are the similar in the simulation and experiment, due to the artificial mass ratio used in the simulation, the dimensions in terms of electron skin depths do not match.
Scaling of the electron layer thickness in simulation compared with experiment. Electron layer thickness in units of , an election skin depth computed using the line-averaged density between the flux cores at the time of comparison, as a function of is plotted for three different drive times (top). All other quantities are held fixed in terms of ion units; relevant parameters are and . Data are averaged over the quasi-steady interval ; error bars in the simulation data represent the standard deviation from this averaging. In all three cases, the layer thickness scales approximately as ; exponents for each curve are given in the text. In the bottom panel, the same scaling is shown using and , parameters from the case that best matches a hydrogen discharge. Experimental data are shown with error bars from a fit of the layer width as a function of ; an extrapolation curve to realistic hydrogen mass ratio is shown. This extrapolation yields a layer width of .
Aspect ratio of the electron layer as a function of time for various (top) and drive times (bottom). Relevant parameters are and . In the top panel, ; in the bottom panel, . The boundary of the quasi-steady interval is marked on both panels by vertical dashed lines. Particles are absorbed at the flux core boundaries for all cases except for , which has a reflection coefficient of 0.1 and , which has a reflection coefficient of 0.2.
Layer length as a function of time for three different reflection coefficients. Reflection inhibits elongation of the layer in the quasi-steady-state region and prevents island formation. Simulation parameters are , , , and .
Sketch of the electron layer.
Layer width as a function of the meandering orbit width ; simulation data are averaged over ; error bars represent the standard deviation due to this averaging. Linear regime cases from Fig. 5 are shown. The layer width agrees with the meandering orbit calculation up to a factor of 1.6 in the simulation, but does not match in the experiment.
Balance of reconnection electric field near the X-point in the simulation (top) and experiment (reproduced from Ref. 3 by permission of American Geophysical Union) (bottom). Linear regime cases from Fig. 5 are shown; simulation data are averaged over ; error bars represent the standard deviation from this averaging. In the experiment, is the reconnection electric field at the center of the layer, is the portion of the field due to the classical Spitzer resistivity, and is the field estimated from Eq. (10). In the simulation, approximately balances the reconnecting electric field, but it is too small to do so in the experiment.
Scan of and with fixed . Other parameters are , , , and no reflection at the flux core surfaces.
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