Abstract
We present a comparison study of 3D pressureless resistive MHD (rMHD) and 3D presureless twofluid MHD models of the Helicity Injected Torus with Steady Inductive helicity injection (HITSI). HITSI is a current drive experiment that uses two geometrically asymmetric helicity injectors to generate and sustain toroidal plasmas. The comparable size of the collisionless ion skin depth di to the resistive skin depth predicates the importance of the Hall term for HITSI. The simulations are run with NIMROD, an initialvalue, 3D extended MHD code. The modeled plasma density and temperature are assumed uniform and constant. The helicity injectors are modeled as oscillating normal magnetic and parallel electric field boundary conditions. The simulations use parameters that closely match those of the experiment. The simulation output is compared to the formation time, plasma current, and internal and surface magnetic fields. Results of the study indicate 2flMHD shows quantitative agreement with the experiment while rMHD only captures the qualitative features. The validity of each model is assessed based on how accurately it reproduces the global quantities as well as the temporal and spatial dependence of the measured magnetic fields. 2flMHD produces the current amplification and formation time demonstrated by HITSI with similar internal magnetic fields. rMHD underestimates and exhibits much a longer . Biorthogonal decomposition (BD), a powerful mathematical tool for reducing large data sets, is employed to quantify how well the simulations reproduce the measured surface magnetic fields without resorting to a probebyprobe comparison. BD shows that 2flMHD captures the dominant surface magnetic structures and the temporal behavior of these features better than rMHD.
We would like to thank Aaron Hossack and Jonathan Wrobel for editing this manuscript, Brian Nelson and Plasma Science and Innovation Center (PSICenter) for technical support, and Carl Sovinec for valuable discussions. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC0205CH11231.
This work was supported by a DOE Grant.
I. INTRODUCTION
II. THE NUMERICAL MODEL
A. Implementation of SIHI boundary conditions
B. The resistive edge layer
C. Injector operation
III. RESULTS
A. Evolution of the magnetic spectrum
1. Stage I
2. Stage II
3. Stage III
B. Discussion of the effect of the Hall term
IV. COMPARISONS TO THE EXPERIMENT
A. Comparison to global parameters
B. Comparison of internal magnetic fields
C. Comparison of surfacemagnetic fields using BD
V. CONCLUSIONS AND DISCUSSION
Key Topics
 Magnetic fields
 32.0
 Toroidal plasma confinement
 29.0
 Magnetohydrodynamics
 24.0
 Magnetic field measurements
 20.0
 Electrical resistivity
 18.0
H05H1/02
Figures
A cutaway view of the HITSI experiment. The bowtie crosssection with closed fitting walls stabilizes the spheromak against a tiltmode. The shortened geometric axis increases the operational β. 29
A cutaway view of the HITSI experiment. The bowtie crosssection with closed fitting walls stabilizes the spheromak against a tiltmode. The shortened geometric axis increases the operational β. 29
The simulation geometry of HITSI in NIMROD. The injectors are removed from the domain. SIHI BCs are applied at the upper and lower annular regions shown in green. The blue and red patches indicate the applied normal magnetic field Bn . The midplane diagnostic gap on the outboard side is removed for computational convenience.
The simulation geometry of HITSI in NIMROD. The injectors are removed from the domain. SIHI BCs are applied at the upper and lower annular regions shown in green. The blue and red patches indicate the applied normal magnetic field Bn . The midplane diagnostic gap on the outboard side is removed for computational convenience.
(a) Total surface magnetic field on the annulus is helical as a result of the applied Bn and . (b) A closeup of one of the injector mouths shows the tangential electric field (black vectors) required for the current injection and E z . induces a surface magnetic field Bt (green, yellow, and orange arrows), which acts as the injector poloidal field on the annulus and in turn induces a normal current density Jz (pseudo color) required to drive .
(a) Total surface magnetic field on the annulus is helical as a result of the applied Bn and . (b) A closeup of one of the injector mouths shows the tangential electric field (black vectors) required for the current injection and E z . induces a surface magnetic field Bt (green, yellow, and orange arrows), which acts as the injector poloidal field on the annulus and in turn induces a normal current density Jz (pseudo color) required to drive .
Natural logarithm of magnetic energy per toroidal mode from (a)rMHD7.5 and (b) 2MHD7.5. The spectrum is initially dominated by modes directly driven by the injectors (n = odd). 2MHD7.5 achieves n = 0 amplification and generation much earlier in time than rMHD7.5. The traces for 2MHD7.5 are shown over a shorter time period with a smaller yscale to magnify stages I and II.
Natural logarithm of magnetic energy per toroidal mode from (a)rMHD7.5 and (b) 2MHD7.5. The spectrum is initially dominated by modes directly driven by the injectors (n = odd). 2MHD7.5 achieves n = 0 amplification and generation much earlier in time than rMHD7.5. The traces for 2MHD7.5 are shown over a shorter time period with a smaller yscale to magnify stages I and II.
(a) and (b) total ME vs time (ms) from all four validation simulations. The black, red, blue, and green lines correspond to rMHD7.5, 2MHD7.5, rMHD12, and 2MHD12, respectively. 2flMHD yields more and total ME than rMHD because of greater injector impedance. The horizontal brown line in (a) corresponds to the amplitude of (20 kA).
(a) and (b) total ME vs time (ms) from all four validation simulations. The black, red, blue, and green lines correspond to rMHD7.5, 2MHD7.5, rMHD12, and 2MHD12, respectively. 2flMHD yields more and total ME than rMHD because of greater injector impedance. The horizontal brown line in (a) corresponds to the amplitude of (20 kA).
Toroidal plasma current as a function of the toroidal coordinate from 5 different times during the 10th full X injector cycle from (a) rMHD7.5 and (b) 2MHD7.5. The plasma current exhibits substantial nonuniformities with respect to . The black arrow indicates the direction of toroidal rotation. There are more data points in (a) than (b) because rMHD7.5 uses 43 toroidal Fourier modes, four times the toroidal resolution used in 2MHD7.5.
Toroidal plasma current as a function of the toroidal coordinate from 5 different times during the 10th full X injector cycle from (a) rMHD7.5 and (b) 2MHD7.5. The plasma current exhibits substantial nonuniformities with respect to . The black arrow indicates the direction of toroidal rotation. There are more data points in (a) than (b) because rMHD7.5 uses 43 toroidal Fourier modes, four times the toroidal resolution used in 2MHD7.5.
(Left) Poloidal layout of one of the four Ampérian surface probe (SP) arrays. SPs are shown as green dots. There are no synthetic probes in the NIMROD calculations that correspond to probes L5 and L6 as the diagnostic gap is excluded from the computational domain. (Right) Endon view of the HITSI flux conserver showing the toroidal layout of the 4 Ampérian SP arrays. Probes at and 180° bisect the X injector mouths. The green dots that encircle the flux conserver toroidally represent the gap probes, which have no synthetic counterparts in the calculations.
(Left) Poloidal layout of one of the four Ampérian surface probe (SP) arrays. SPs are shown as green dots. There are no synthetic probes in the NIMROD calculations that correspond to probes L5 and L6 as the diagnostic gap is excluded from the computational domain. (Right) Endon view of the HITSI flux conserver showing the toroidal layout of the 4 Ampérian SP arrays. Probes at and 180° bisect the X injector mouths. The green dots that encircle the flux conserver toroidally represent the gap probes, which have no synthetic counterparts in the calculations.
vs time (ms) traces from the 2MHD12 (solid red), 2MHD7.5 (dotted red), rMHD12 (solid black), and rMHD7.5 (dotted black) validation calculations compared with traces of five similar shots from the experiment (gray patch). The experimental traces are clipped to only include the interval between t = 0.74–1.8 ms, which excludes the prebreakdown and decay periods.
vs time (ms) traces from the 2MHD12 (solid red), 2MHD7.5 (dotted red), rMHD12 (solid black), and rMHD7.5 (dotted black) validation calculations compared with traces of five similar shots from the experiment (gray patch). The experimental traces are clipped to only include the interval between t = 0.74–1.8 ms, which excludes the prebreakdown and decay periods.
Time traces of the midplane poloidal magnetic field Bz at five different radial locations from shot 122385 as measured by IMP and calculated from a 7.5eV (a) 2flMHD and (b) rMHD simulation. (c) and (d) shows the same traces for . Measurements from only five locations are plotted although a total of 17 locations are available from the experiment. The yaxis is offset by 0.05 T in (a), 0.06 T in (b), 0.08 T in (c), and 0.08 T in (d) for each additional radial location. 2MHD7.5 produces internal magnetic fields that match the magnitude of the measured internal fields as well the amplitude of their oscillations.
Time traces of the midplane poloidal magnetic field Bz at five different radial locations from shot 122385 as measured by IMP and calculated from a 7.5eV (a) 2flMHD and (b) rMHD simulation. (c) and (d) shows the same traces for . Measurements from only five locations are plotted although a total of 17 locations are available from the experiment. The yaxis is offset by 0.05 T in (a), 0.06 T in (b), 0.08 T in (c), and 0.08 T in (d) for each additional radial location. 2MHD7.5 produces internal magnetic fields that match the magnitude of the measured internal fields as well the amplitude of their oscillations.
Cycleaveraged midplane magnetic field profiles from (a) 2MHD7.5 and (b) rMHD7.5 (all solid blue lines) simulations overlaid with measured profiles from shot 122385 (red dots) and those calculated from Taylor relaxation theory (green). The internal fields from both models have a lower magnitude than those from shot 122385. However, the models actually overestimate the internal fields relative to they yield, implying a more inductive plasma in the simulations than in the experiment.
Cycleaveraged midplane magnetic field profiles from (a) 2MHD7.5 and (b) rMHD7.5 (all solid blue lines) simulations overlaid with measured profiles from shot 122385 (red dots) and those calculated from Taylor relaxation theory (green). The internal fields from both models have a lower magnitude than those from shot 122385. However, the models actually overestimate the internal fields relative to they yield, implying a more inductive plasma in the simulations than in the experiment.
The first 5 weights from BD for (a) the 7.5 eV and (b) the 12 eV simulations. Weights from rMHD is shown in black, 2MHD in red, and shot 122385 in blue. Both 2MHD7.5 and 2MHD12 show remarkable agreement with shot 122385 in the first 4 weights. The dominant weights from rMHD deviate significantly from those of the experiment. Note what is plotted in Ref. 43 is , not Ak .
The first 5 weights from BD for (a) the 7.5 eV and (b) the 12 eV simulations. Weights from rMHD is shown in black, 2MHD in red, and shot 122385 in blue. Both 2MHD7.5 and 2MHD12 show remarkable agreement with shot 122385 in the first 4 weights. The dominant weights from rMHD deviate significantly from those of the experiment. Note what is plotted in Ref. 43 is , not Ak .
The first three poloidal chronos as a function of time from (a)2MHD7.5 and (b) rMHD7.5 simulations. The first chrono (black) is assumed to represent the zeroth order spheromak component because of its resemblance to the temporal trace of . The second and third chronos correspond to the injector fluctuations.
The first three poloidal chronos as a function of time from (a)2MHD7.5 and (b) rMHD7.5 simulations. The first chrono (black) is assumed to represent the zeroth order spheromak component because of its resemblance to the temporal trace of . The second and third chronos correspond to the injector fluctuations.
Correlations of the first five toposfrom shot 122385 with those of (a)2MHD7.5, (b) rMHD7.5 and the first five chronos from shot 122385 with those of (c) 2MHD7.5, and (d) rMHD7.5. The integers along the horizontal axis represent the component number from a simulation. Each integer contains five bars of assorted colors that indicate how well each of the first five topos/chronos from shot 122385 correlates with a particular topo/chrono from a simulation. The legend represents the first five components from each simulation.
Correlations of the first five toposfrom shot 122385 with those of (a)2MHD7.5, (b) rMHD7.5 and the first five chronos from shot 122385 with those of (c) 2MHD7.5, and (d) rMHD7.5. The integers along the horizontal axis represent the component number from a simulation. Each integer contains five bars of assorted colors that indicate how well each of the first five topos/chronos from shot 122385 correlates with a particular topo/chrono from a simulation. The legend represents the first five components from each simulation.
Comparison of 1st poloidal (top row) and toroidal (bottom row) topos from shot 122385 (left) with those of 2MHD7.5 (center) and rMHD7.5 (right). The square domain represents the unwrapped HITSI walls. The vertical and horizontal axes correspond to the poloidal and toroidal coordinates θ and . The black dots mark the locations of the real and synthetic SPs. Dead experimental probes are excluded. The gray diamonds mark the locations of the injector mouths. Toroidal topos 1 and 2 have been swapped for rMHD7.5 because the 1st toroidal topo from rMHD7.5 corresponds to the second, and not the most dominant component from shot 122385.
Comparison of 1st poloidal (top row) and toroidal (bottom row) topos from shot 122385 (left) with those of 2MHD7.5 (center) and rMHD7.5 (right). The square domain represents the unwrapped HITSI walls. The vertical and horizontal axes correspond to the poloidal and toroidal coordinates θ and . The black dots mark the locations of the real and synthetic SPs. Dead experimental probes are excluded. The gray diamonds mark the locations of the injector mouths. Toroidal topos 1 and 2 have been swapped for rMHD7.5 because the 1st toroidal topo from rMHD7.5 corresponds to the second, and not the most dominant component from shot 122385.
Comparison of 2nd toroidal topos from shot 122385 (left) with those of 2MHD7.5 (center), and rMHD7.5 (right). 2flMHD reproduces the prominent structures more accurately than rMHD. The 1st and 2nd topos from rMHD7.5 have been swapped as explained above.
Comparison of 2nd toroidal topos from shot 122385 (left) with those of 2MHD7.5 (center), and rMHD7.5 (right). 2flMHD reproduces the prominent structures more accurately than rMHD. The 1st and 2nd topos from rMHD7.5 have been swapped as explained above.
Tables
Salient computational and experimental parameters for highperformance HITSI shots. The first four parameters represent operational parameters related to SIHI. The plasma resistivity η and viscosity ν areexpressed as dissipative parameters and calculated using the Bragiinski formulae. ∼ indicates an approximate figure for a given physical quantityand means Spitzer resistivity. 33 ne stands for average electron density. We adhere to Izzo's definitions 20 for and , which are calculated a posteriori after the completion of a simulation based. Quadrature is used for the experimental .
Salient computational and experimental parameters for highperformance HITSI shots. The first four parameters represent operational parameters related to SIHI. The plasma resistivity η and viscosity ν areexpressed as dissipative parameters and calculated using the Bragiinski formulae. ∼ indicates an approximate figure for a given physical quantityand means Spitzer resistivity. 33 ne stands for average electron density. We adhere to Izzo's definitions 20 for and , which are calculated a posteriori after the completion of a simulation based. Quadrature is used for the experimental .
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