The rotating wall machine1 experimental geometry. Plasmas are illustrated as discrete flux-ropes, though measurements indicate that a fully merged axisymmetric profile is achieved by 1/3rd of the distance to the anode.
(a) Wall-induced electromagnetic torque () plotted vs. plasma rotation normalized to the wall time () for a variety of inter-wall spacings for equal walls (). (b) Modifications to for co- and counter-rotation for a variety wall speeds () utilizing experimental values for , and . is set by the direction of the natural frequency ().
(a) Floating potential () measured as a function of radius at different axial locations. (b) Calculated E × B rotation profiles () from (a). Measurements are made with a single-tip sweeping Langmuir probe using shot-to-shot reproducibility. Other probe measurements29 (not shown) indicate negligible radial current and uniform , justifying equating gradients in with . Values at small r are skipped to avoid numerical singularities. Measurements are from plasma that yielded a mode at kHz, faster than the kHz typical of this study.
(a)-(d) Net electromagnetic torque () as a function of plasma rotation () decomposed into electromagnetic () and restoring () contributions for various values of natural frequency (). Other variables in Eq. (3) are held constant. Arrows in (a)-(d) indicate the direction of net torque and thus point to the stable solutions. (e) This bifurcation diagram is made by plotting the torque balance equilibrium points (solutions of Eq. (3)) while varying , illustrating the locking and unlocking bifurcation points.
(a) Bifurcation diagram illustrating torque balance equilibrium points (solutions of Eq. (3)) as is varied for various values of differential wall rotation (). (b) Dependence of the locking and unlocking bifurcation frequencies () and natural frequencies () as is varied. Beyond the bifurcation is lost.
(a) Time-trace of radial magnetic field () as guide field ripple () is increased. (b) Measurements of mode frequency () for the discharges of (a). For Figs. 6–12, is fit (according to Eqs. (3) and (11)) such that matches the data, holding other parameters in Eq. (3) constant. (c) Bifurcation diagram (roots of Eq. (3)) calculated using the data from the fits of (b). (d)-(e) The calculated alteration of the guide field by the coil is shown, where panel (d) is for G while panel (e) is for G. The top, bottom outlines in (d) and (e) indicate the position of the segmented anode and plasma guns, respectively. t = 0 is when the bias voltage to drive is applied.
(a) Time-traces of radial magnetic field () as m = 1 error field () is increased, yielding mode locking. (b) Mode rotation () and resultant fits to the model of Eqs. (3) and (11). (c) Bifurcation diagrams (solutions of Eq. (3)) using the fit parameters of (b). (d) Expected locking frequencies () as is increased and comparison to the data. Arrows in (d) indicate that no locking bifurcation was observed, thus must be in the direction shown.
(a) Radial magnetic field () time-traces for two sequential discharges. (b) Mode rotation () and fits to Eqs. (3) and (11) for the discharges in (a). (c) Bifurcation diagram (solutions of Eq. (3)) using fits from (b).
(a) Radial magnetic field () for a discharge where locking occurred (at ms) after a longer final oscillation. (b) Enlargement of this final oscillation with fits to a single frequency model () and one in which .
(a) Time-trace of radial magnetic field () for a discharge illustrating mode-unlocking. (b) Mode rotation () and fits to Eqs. (3) and (11) for the same discharge. (c) Bifurcation diagram (solutions of Eq. (3)) using fits from (b), illustrating mode-locking and mode-unlocking bifurcations at constant , and .
(a) Radial magnetic field () traces illustrating mode-locking at higher frequency due to wall co-rotation (). (b) Mode rotation () traces and corresponding fits to Eqs. (3) and (11). (c) Bifurcation diagram (solutions of Eq. (3)) for fit parameters from (b). (d) Comparison of theoretical and experimental locking bifurcation frequency () as is varied for a range of values of .
(a) Radial magnetic field () traces illustrating mode-locking at lower frequency due to wall counter-rotation (). (b) Mode rotation () traces and corresponding fits to Eqs. (3) and (11). (c) Bifurcation diagram (solutions of Eq. (3)) for fit parameters from (b). (d) Comparison of theoretical and experimental locking bifurcation frequency () as is varied for a range of . As a lock is only observed for , only the upper bound of is known.
Time-traces of (a) amplitude (), (b) phase (), and (c) hodogram ( vs ) of the radial magnetic field as wall rotation () is increased, while holding error fields ( and ) constant. Dotted lines in (b) indicate the rate of wall rotation.
(a) amplitude diverges from at a critical which is raised as increases. (b) Phases of also vary as increases. (c) amplitude as a function of q as measured by the anode ring at r = 5 cm. (d) Comparison of the experimental data and theoretical predictions of the critical q () for instability. The squares in (d) have been offset such that for . Reprinted with permission from Paz-Soldan et al., Phys. Rev. Lett. 107, 245001 (2011). Copyright © 2011, American Physical Society.
Measured values of and calculated values of from the discharges of this study. Errors are estimated from the deviations between and within discharges.
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