Magnetic flux (left) and number density (right) along the midplane, obtained with old and new implementation. Both old and new equilibria are calculated with a resolution of 256 points in each direction. Old and new results are to all practical purposes identical. The transonic discontinuity is at .
Poloidal field along the midplane, obtained with the old and new implementation. Four different resolutions are used for the new implementation. Old and new results are to all practical purposes identical. The only difference is found near the transonic discontinuity at (shown in the lower left corner pane).
Total pressure along the midplane, obtained with old and new implementation. Four different resolutions are used for the new implementation. Only the region near the transonic discontinuity at is shown, highlighting a minimal difference between the results of old and new implementation.
Magnetic flux (left) and poloidal velocity in km/s (right) along the midplane obtained with old and new implementation (method I). Results for are all very close (with some difference between old and new implementation). Poloidal velocities are also very close in most of the plasma, with a visible difference only around the transonic discontinuity. The box on the top left shows a zoom of the transonic discontinuity region.
Poloidal field along the midplane. All results are rather similar everywhere except in the supersonic zone at large major radius (i.e., in the area near and outside the MHD pedestal). The left pane shows the complete midplane profiles and the right pane a zoom of the transonic discontinuity.
Total pressure as a function of R along the midplane (left) and zoom of the transonic discontinuity region (right). All profiles are very similar in the part of the plasma inboard of the transonic discontinuity. A discrepancy between the new and old implementation is visible in the lower right hand corner of the figure. For consistency, all profiles are calculated in the same way, with centered differences for the poloidal field.
Toroidal current along the midplane. The total profile is shown on the left, and a zoom of the transonic region on the right. Large peaks are obtained across the discontinuity, with height increasing and width decreasing with increasing number of grid points. On the left, the transonic region is not plotted for the new implementation for clarity. Oscillations are present in and around the transonic region. The current is calculated with Eq. (16) (dashed lines) and with Eq. (17) (continuous lines).
Zoom of the discontinuity region for the poloidal field profile along the midplane. The sharp variation of in the new implementation, as opposed to the smooth profile of the old one, is visible. The variation of becomes steeper with increasing resolution.
Toroidal current along the midplane for the old and “Gauss-modified” algorithms. The appearance of a current density spike is observed. The spike height increases with increasing resolution. Some small numerical oscillations are still seen in the profiles obtained with the new algorithm, but disappear with high resolution. The current reversal by the transonic discontinuity is likely numerical. Observe that the spike is lower than the one obtained with the other new algorithm, but also wider. Results are calculated with Eq. (16) .
Total pressure along the midplane for the “Gauss-modified” algorithm (limited to the region of the transonic discontinuity). The profiles obtained with the new implementation are continuous and smooth (barring some slight oscillation in the one with lowest resolution). Also, the gradient of the total pressure changes across the discontinuity. This is consistent with the results of method I (compare with Fig. 6 ), but the plot is more clear due to the consistent representation for the total pressure used in the diagnostic and the internal evaluation of the total pressure gradient.
Density profile along the midplane for the old and “Gauss-modified” algorithms. All profiles are quite similar everywhere. Zoom of (density profile) restricted to the region of the discontinuity. The steepening of the profile with increasing resolution is visible. The position of the transonic discontinuity is slightly shifted with the new implementation. Comparing the new curves with the old one, it is seen that the new algorithm spreads the discontinuity over grid points (the slopes of the new curve with 1024 points is about the same as the old one with 256 points).
Mach number profile across the transonic region at the inner (a) and outer (b) midplane for the ST equilibrium in Sec. V with methods I and II. Both profiles are continuous at the inner midplane, while the profile obtained with method I is discontinuous at the outer midplane. Notice the different scale on the vertical axis in the two parts of the figure.
Safety factor profile for the high- ST equilibrium in Sec. V . The old (black curve) and new (red curve, diamonds) agree very well in most of the plasma, but the new curve shows a reversal of the magnetic shear in the transonic region.
Magnetic flux calculated with the 1D and 2D codes with different resolutions. All results show good agreement.
Plasma density (in arbitrary units) calculated with the 1D and 2D codes with different resolutions. All results show good agreement. The inset in the left-bottom corner of the plot shows details of the profile near the transonic surface.
Poloidal field calculated with the 1D and 2D codes with different resolutions. All results show good agreement, in particular, in the inner region of the plasma. Overshoots (of numerical origin) are seen in the transonic region. The inset in the right-bottom corner of the plot shows details of the profile near the transonic surface.
Total pressure (in arbitrary units) for 1D and 2D solvers. The agreement is excellent inward of the transonic surface, and very good outside the transonic surface, where some small difference appears between the one- and two-dimensional results.
Convergence of the two-dimensional solution to the one-dimensional one. All plots are in function of the grid size (inversely proportional to the number of grid points). The “0” point corresponds to the one-dimensional solution. Left: Current carried by the transonic surface singularity. Right, top: Error (in %) of the poloidal field on the two sides of the discontinuity; bottom: Value of on the two sides of the discontinuity.
Main parameters of equilibria in Secs. IV and V .
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