Abstract
A timedependent helicity balance model applied to a spheromak helicityinjection experiment enables the measurement of the relaxation time during the sustainment phase of the spheromak. The experiment, the Helicity Injected Torus with Steady Inductive helicity injection (HITSI), studies spheromak formation and sustainment through inductive helicity injection. The model captures the dominant plasma behavior seen during helicity injection in HITSI by using an empirical helicitydecay rate, a timedependent helicityinjection rate, and a composite Taylor state to model both the helicity content of the system and to calculate the resulting spheromak current. During singleinjector operations, both the amplitude and the phase of the periodic rise and fall of the toroidal current are predicted by this model, with an exchange of helicity between the injector states and the spheromak state proposed as the causal mechanism. This phenomenon allows for the comparison of the delay between the current rises in the experiment and the numerical model, enabling a measurement of the relaxation time. The measured relaxation time of 4.8 μs ± 2.8 μs is shorter than the toroidal Alfvén timescale. These results validate Hall MHD calculations of the Geospace Environmental Modeling challenge.
I. INTRODUCTION
II. METHOD
A. Experiment
B. Measurement of toroidalcurrent
C. The helicity balance model
D. The Taylorstate helicity content
III. RESULTS
IV. DISCUSSION AND CONCLUSIONS
Key Topics
 Spheromaks
 35.0
 Helicity injection
 30.0
 Toroidal plasma confinement
 27.0
 Electric measurements
 22.0
 Relaxation times
 16.0
H05H1/02
Figures
Cutaway view of a surface magnetic probe, all of which are mounted in the copper wall of the HITSI vessel and separated from the vacuum by a stainless steel disc. Each probe is electrostatically isolated and shielded from the experiment and differential leads are brought back to the digitizer. Image adapted from Ref. ^{ 16 } .
Cutaway view of a surface magnetic probe, all of which are mounted in the copper wall of the HITSI vessel and separated from the vacuum by a stainless steel disc. Each probe is electrostatically isolated and shielded from the experiment and differential leads are brought back to the digitizer. Image adapted from Ref. ^{ 16 } .
Panel (a) depicts the locations of the surface magnetic probes on a cross section in the RZ plane of the HITSI device, taken at a toroidal angle that intersects two poloidal arrays (0° and 180°). The surface probe locations are shown in green along the edge of the bowtie crosssection. Panel b) depicts the locations of the four toroidal locations of the Amperian arrays, along with the gap array that consists of 16 probes.
Panel (a) depicts the locations of the surface magnetic probes on a cross section in the RZ plane of the HITSI device, taken at a toroidal angle that intersects two poloidal arrays (0° and 180°). The surface probe locations are shown in green along the edge of the bowtie crosssection. Panel b) depicts the locations of the four toroidal locations of the Amperian arrays, along with the gap array that consists of 16 probes.
The curved arrow depicts the integral path, and its approximate subdivision into segments, for accomplishing the discrete closedpath integrals. The geometric lengths are represented by the line segments between the hatch marks, and the actual path is taken directly upon the surface.
The curved arrow depicts the integral path, and its approximate subdivision into segments, for accomplishing the discrete closedpath integrals. The geometric lengths are represented by the line segments between the hatch marks, and the actual path is taken directly upon the surface.
From left to right, field line traces of the: Xinjector, spheromak, and Yinjector equilibria at 1 Amp, depicting how each is calculated independently; no field from one injector threads the other, and no field from the spheromak threads either injector. The set of images shows the three equilibria used to form the composite Taylor state, through scaling by measured currents and then superposition of the individual scaled states.
From left to right, field line traces of the: Xinjector, spheromak, and Yinjector equilibria at 1 Amp, depicting how each is calculated independently; no field from one injector threads the other, and no field from the spheromak threads either injector. The set of images shows the three equilibria used to form the composite Taylor state, through scaling by measured currents and then superposition of the individual scaled states.
The toroidal current and injector currents are shown at top for shot 117529, driven the Y injector at 5.8 kHz. The total helicity as calculated from the composite Taylor state, using the injector and spheromak currents, is shown center. The helicity decay time is shown at bottom, with the contributions from each injector summed (green) and then filtered (blue). The green vertical bar marks the formation of plasma and start of the model, and the black vertical bar corresponds to the time at which the constant helicity decay rate is calculated (3.6 ms) where the steady state period begins.
The toroidal current and injector currents are shown at top for shot 117529, driven the Y injector at 5.8 kHz. The total helicity as calculated from the composite Taylor state, using the injector and spheromak currents, is shown center. The helicity decay time is shown at bottom, with the contributions from each injector summed (green) and then filtered (blue). The green vertical bar marks the formation of plasma and start of the model, and the black vertical bar corresponds to the time at which the constant helicity decay rate is calculated (3.6 ms) where the steady state period begins.
Delay due to the finite relaxation rate in the measured toroidal current, versus the instantaneous relaxation in the model, for the 5.8 kHz Yinjectoronly shot 117529. The top traces show: the total measured helicity as scaled by the composite Taylor state (red), the model helicity as calculated through cumulative helicity injection and decay (blue), and the helicity content of the injector state (green). The middle traces show the absolute value of the toroidal current as measured (red) and as calculated by the model (blue) with vertical lines highlighting the peaks in each. The lower traces show the injector currents, with vertical black lines highlighting the zero crossings.
Delay due to the finite relaxation rate in the measured toroidal current, versus the instantaneous relaxation in the model, for the 5.8 kHz Yinjectoronly shot 117529. The top traces show: the total measured helicity as scaled by the composite Taylor state (red), the model helicity as calculated through cumulative helicity injection and decay (blue), and the helicity content of the injector state (green). The middle traces show the absolute value of the toroidal current as measured (red) and as calculated by the model (blue) with vertical lines highlighting the peaks in each. The lower traces show the injector currents, with vertical black lines highlighting the zero crossings.
Tables
Table of the fitting coefficients used to calculate the helicity of the composite Taylor state. The crosscoupling coefficients (C_{4} and C_{5}) between the injectors and the spheromak are five orders of magnitude lower than the other coefficients, allowing the helicity of the spheromak alone to be approximated by using only C_{1}. The coefficients are calculated to produce K_{rel} in SI units of Wb^{2}.
Table of the fitting coefficients used to calculate the helicity of the composite Taylor state. The crosscoupling coefficients (C_{4} and C_{5}) between the injectors and the spheromak are five orders of magnitude lower than the other coefficients, allowing the helicity of the spheromak alone to be approximated by using only C_{1}. The coefficients are calculated to produce K_{rel} in SI units of Wb^{2}.
The 5.8 kHz singleinjector shots used to calculate an average relaxation time; all shots are with helium plasmas, with an average of 37 zero crossings per shot.
The 5.8 kHz singleinjector shots used to calculate an average relaxation time; all shots are with helium plasmas, with an average of 37 zero crossings per shot.
Relevant time scales for comparison to relaxation rates in HITSI, as calculated or measured. The spread in the SweetParker time results from whether the empirical resistive decay time for τ_{L/R} is used, or if τ_{L/R} is calculated using other parameters, respectively.
Relevant time scales for comparison to relaxation rates in HITSI, as calculated or measured. The spread in the SweetParker time results from whether the empirical resistive decay time for τ_{L/R} is used, or if τ_{L/R} is calculated using other parameters, respectively.
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