Abstract
The effect of rotating conducting walls on modelocking is studied in a linetied, linear screw pinch experiment and then compared to a torque balance model which has been extended to include differential wall rotation. Wall rotation is predicted to asymmetrically affect the modeunlocking threshold, with fast rotation eliminating the locking bifurcation. Static error fields are observed to lock the resistive wall mode(RWM) variant of the current driven kink instability by modifying the electromagnetictorque. Using locked modes, the stabilizing effect of wall rotation on the RWM is experimentally demonstrated by illustrating a reduction of the RWM growth rate and an extension of the RWMstable operation window.
This work is supported by the Department of Energy Grant Nos. DEFG0200ER54603 and DEFG0286ER53218, the National Science Foundation Grant No. 0903900, and the National Science and Engineering Research Council, Canada. We also wish to acknowledge several individuals from the University of Wisconsin Plasma Physics Group and the Center for Magnetic SelfOrganization (CMSO), such as M. Clark, A. T. Eckhart, J. M. Finn, J. JaraAlmonte, N. Katz, E. S. Mueller, J. S. Sarff, J. Wallace, and W. Zimmerman, who have provided their guidance, support, and time in support of our work.
I. INTRODUCTION
II. MODELOCKING MODEL AND FREE PARAMETERS
A. Derivation of twowall electromagnetictorque
1. Properties of twowall electromagnetictorque
B. Derivation of error field torque
C. Estimation of phenomenological restoring torque
D. Walllocking: Bifurcations in torque balance
E. Bifurcations with differential rotation
III. STATIC WALL MODELOCKING
A. Slowing of plasma rotation by guide field ripple
B. Modelocking by vertical error fields
C. Nearthreshold effects
1. Sensitivity to input conditions
2. Nonuniform rotation
3. Hysteresis in modeunlocking
IV. MODELOCKING WITH DIFFERENTIAL WALL ROTATION
V. STABILIZATION OF LOCKED MODES BY WALL ROTATION
A. Growth rate reduction and lockedmode rotation
B. Extension of stability window to higher current
VI. DISCUSSION
Key Topics
 Resistive wall mode
 49.0
 Torque
 40.0
 Bifurcations
 27.0
 Magnetohydrodynamics
 9.0
 Electrical resistivity
 7.0
H05H1/02
Figures
The rotating wall machine^{1} experimental geometry. Plasmas are illustrated as discrete fluxropes, though measurements indicate that a fully merged axisymmetric profile is achieved by 1/3rd of the distance to the anode.
The rotating wall machine^{1} experimental geometry. Plasmas are illustrated as discrete fluxropes, though measurements indicate that a fully merged axisymmetric profile is achieved by 1/3rd of the distance to the anode.
(a) Wallinduced electromagnetic torque () plotted vs. plasma rotation normalized to the wall time () for a variety of interwall spacings for equal walls (). (b) Modifications to for co and counterrotation for a variety wall speeds () utilizing experimental values for , and . is set by the direction of the natural frequency ().
(a) Wallinduced electromagnetic torque () plotted vs. plasma rotation normalized to the wall time () for a variety of interwall spacings for equal walls (). (b) Modifications to for co and counterrotation 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 singletip sweeping Langmuir probe using shottoshot reproducibility. Other probe measurements^{29} (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) 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 singletip sweeping Langmuir probe using shottoshot reproducibility. Other probe measurements^{29} (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)(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) 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) Timetrace 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) Timetrace 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) Timetraces 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) Timetraces 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 () timetraces 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 () timetraces 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) 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) Timetrace of radial magnetic field () for a discharge illustrating modeunlocking. (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 modelocking and modeunlocking bifurcations at constant , and .
(a) Timetrace of radial magnetic field () for a discharge illustrating modeunlocking. (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 modelocking and modeunlocking bifurcations at constant , and .
(a) Radial magnetic field () traces illustrating modelocking at higher frequency due to wall corotation (). (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 modelocking at higher frequency due to wall corotation (). (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 modelocking at lower frequency due to wall counterrotation (). (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.
(a) Radial magnetic field () traces illustrating modelocking at lower frequency due to wall counterrotation (). (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.
Timetraces 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.
Timetraces 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 PazSoldan et al., Phys. Rev. Lett. 107, 245001 (2011). Copyright © 2011, American Physical Society.
(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 PazSoldan et al., Phys. Rev. Lett. 107, 245001 (2011). Copyright © 2011, American Physical Society.
Tables
Measured values of and calculated values of from the discharges of this study. Errors are estimated from the deviations between and within discharges.
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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