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Validation of single-fluid and two-fluid magnetohydrodynamic models of the helicity injected torus spheromak experiment with the NIMROD code
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Image of FIG. 1.
FIG. 1.

A cut-away view of the HIT-SI experiment. The bowtie cross-section with closed fitting walls stabilizes the spheromak against a tilt-mode. The shortened geometric axis increases the operational β.

Image of FIG. 2.
FIG. 2.

The simulation geometry of HIT-SI 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 . The mid-plane diagnostic gap on the outboard side is removed for computational convenience.

Image of FIG. 3.
FIG. 3.

(a) Total surface magnetic field on the annulus is helical as a result of the applied and . (b) A close-up of one of the injector mouths shows the tangential electric field (black vectors) required for the current injection and . induces a surface magnetic field (green, yellow, and orange arrows), which acts as the injector poloidal field on the annulus and in turn induces a normal current density (pseudo color) required to drive .

Image of FIG. 4.
FIG. 4.

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 ( = odd). 2MHD7.5 achieves  = 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 y-scale to magnify stages I and II.

Image of FIG. 5.
FIG. 5.

(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. 2fl-MHD 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).

Image of FIG. 6.
FIG. 6.

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 non-uniformities 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.

Image of FIG. 7.
FIG. 7.

(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) End-on view of the HIT-SI 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.

Image of FIG. 8.
FIG. 8.

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  = 0.74–1.8 ms, which excludes the pre-breakdown and decay periods.

Image of FIG. 9.
FIG. 9.

Time traces of the mid-plane poloidal magnetic field at five different radial locations from shot 122385 as measured by IMP and calculated from a 7.5-eV (a) 2fl-MHD 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 y-axis 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.

Image of FIG. 10.
FIG. 10.

Cycle-averaged mid-plane 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.

Image of FIG. 11.
FIG. 11.

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. is , not .

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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 HIT-SI 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.

Image of FIG. 15.
FIG. 15.

Comparison of 2nd toroidal topos from shot 122385 (left) with those of 2MHD7.5 (center), and rMHD7.5 (right). 2fl-MHD reproduces the prominent structures more accurately than rMHD. The 1st and 2nd topos from rMHD7.5 have been swapped as explained above.


Generic image for table
Table I.

Salient computational and experimental parameters for high-performance HIT-SI 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. stands for average electron density. We adhere to Izzo's definitions for and , which are calculated a posteriori after the completion of a simulation based. Quadrature is used for the experimental .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Validation of single-fluid and two-fluid magnetohydrodynamic models of the helicity injected torus spheromak experiment with the NIMROD code