Mapping Diagnostic Profiles

SNAP divides the plasma into a set of radial zones. Most local plasma parameters (T_i, T_e, n_e, q_bi, q_be, q_ei, etc.) are stored on zone centers which correspond to minor radii.

For several of its calculations (e.g., visible Bremsstrahlung intensity along the VB sightline, and neutron emission along several chords viewed by the neutron collimator) SNAP does perform chord integrals in real space, but the local plasma parameters it uses in these calculations are the values it finds on its internal minor radius grid, mapped to real space using SNAP's interpretation of the magnetic geometry, i.e., the Shafranov shift as determined from a solution of the Grad-Shafranov equation.

Most profiles measured by plasma diagnostics measure quantities in real space, i.e., as a function of major radius on the horizontal midplane. So one of the first, and most important tasks, performed by SNAP is to map the diagnostic data from major radius to minor radius. Several different mapping techiques are available, and these can be specified by the user in the appropriate SNAPIN profile menu. These are: Slice and Stack, Outside, Inside, Both, and Partial Slice and Stack.

The mapping of a measured velocity profile to minor radius differs slightly from that of the other profiles; we assume that the angular velocity is a constant on a flux surface, rather than the rotational speed itself.

Note: If the profile data does not cover the entire interval (R₀ - a) to (R₀ + a), SNAP adds ficticious data points at x=R₀ - a and x=R₀ + a with y equal to a user-specified edge value.

If SNAP smoothing is requested, SNAP first smooths each profile (see Section 3.7.4) and then maps the profiles to minor radius according to the selected mapping algorithms.

Slice and Stack

SNAP will draw a set of horizontal lines through the profile and determine where each one intersects the data. The length of the line is interpreted as , where r_minor is the minor radius to which the intersection value y will be assigned. See Figure 4.

This procedure effectively assumes that the profile quantity being mapped should be constant on a flux surface. It is most appropriate when the diagnostic may have a small absolute error in its position calibration, since such errors do not affect the mapping to minor radius.

This procedure works only for monotonically decreasing profiles! e.g., inverted post-pellet profiles would be grossly mis-mapped by this procedure.

Note: be sure to trim away all spurious data points that lie well outside the plasma (R<R₀-a or R>R₀+a) to avoid mapping problems with slice and stack.

Outside

Since SNAP solves the Grad-Shafranov equation, it determines the Shafranov shift as a function of minor radius. Based on this calculation SNAP can therefore assign a minor radius to each major radius, i.e., where R₀ is the center of the outermost flux surface, shift(r) is the shift of the center of the flux surface of radius r relative to R₀. R(r) = R₀ + shift(r) + r indicates data points that lie outside of the magnetic axis, and R(r) = R₀ + shift(r) - r indicates data points that lie inside the magnetic axis.

If you select outside mapping, then SNAP will map the profile to minor radius, using its internal Shafranov shift, and using only the input profile data that lie outside the magnetic axis. See Figure 5.

This mapping technique is normally used when profile data exists over only part of the plasma, e.g., CHERS T_i(R) and measurements, or ECE T_e(R) measurements.

Inside

Essentially the same as outside mapping, but only the profile data inside the major radius of the magnetic axis is used in the map. See Figure 5.

Both

SNAP determines two major radii, one inside and one outside the magnetic axis, which correspond to each minor radius, using its internal Shafranov shift array. It averages the values of the profile at these two major radii to assign the value of the profile corresponding to the minor radius. See Figure 5 and also the description of the outside mapping.

Partial Slice and Stack

This algorithm is most useful if the profile covers the full minor radius on the outside part, but only a portion of the minor radius on the inside part, i.e., the profile data doesn't go all the way to R = R₀ - a. This mapping will use the slice and stack from r_minor = 0 out to the largest minor radius for which profile data exists on both sides. Beyond this point, it uses the outside mapping technique. See Figure 6.

There is an obvious problem with this algorithm, which arises from the consideration that the slice and stack algorithm effectively determines its own Shafranov shift versus minor radius, wherein the mid-point of each horizontal line that slices the profile represents the center of the corresponding flux surface. There is no reason to expect that this Shafranov shift array, determined from a single diagnostic profile, will agree exactly with the Shafranov shift array determined from a solution of the Grad-Shafranov equation. This would cause a discontinuity in the profile mapped to minor radius, at the point where we switch from slice and stack to outside mapping. To remedy this problem, the algorithm normalizes the magnitude of the Shafranov shift array, as determined by the solution to the Grad-Shafranov equation, to that determined from the profile, at the minor radius where we switch from slice and stack to outside mapping. This ensures continuity of the mapped profile, although practice has shown that it typically results in a discontinuity in the radial derivative of the profile.

If the physical quantity under consideration is constant on a flux surface, if the diagnostic is functioning perfectly, and if SNAP\ calculates the correct total stored energy (which is essential if it is to get the correct Shafranov shift), then we expect the raw profile to overlay precisely with the mapped profile. To the extent that there are differences one of these assumptions is wrong.

Examining the differences between raw profiles and SNAP's mapping of them to r_minor and then back to R_major is one of the important ways to verify the fidelity of SNAP's modeling and to identify potential errors. Of course, this really provides a check only when the original profile has data on both sides of the magnetic axis. If data is present on only one side of the profile, then we are forced to use either the outside or (very rarely) the inside map, and by construction the mapped profile will correspond nearly exactly to the raw profile. There may be small differences due to smoothing of the data.

Note that with outside mapping the peak values of profiles will be cut off at values less than their true maxima if SNAP calculates a Shafranov shift larger than the actual Shafranov shift.

  
Figure 4: Mapping Techniques: Slice and Stack

  
Figure 5: Mapping Techniques: Inside, Outside, or Both

  
Figure 6: Mapping Techniques: Partial Slice and Stack