Three-dimensional unmagnetized Mach probe analysis and initial flow measurements in reversed-field pinch experiments

TL;DR

This work develops a matrix-based method to extract local 3D plasma velocity from data collected by unmagnetized Mach probe arrays in reversed-field pinch experiments on MST. By formulating a general linear system $\mathbf{A}\frac{\mathbf{v}}{v_s} = -\frac{2}{K}\mathbf{J}$, the authors accommodate 3D tip geometries, including a six-tip octahedral array and a four-tip tetrahedral array, enabling true 3D velocity measurements without requiring direct opposite-tip pairs. They derive uncertainty estimates for velocity from machining and saturation-current errors, predicting relative uncertainties of a few percent under typical conditions. Initial applications to octahedral and tetrahedral probes show velocity magnitudes in MST-compatible ranges but reveal direction- and magnitude-specific discrepancies likely tied to probe conditioning and fast-electron effects, motivating further investigations and hardware refinements. The study demonstrates the viability of compact 3D Mach probes for local flow and helicity studies in laboratory plasmas, with future work aimed at cross helicity and vorticity measurements using multi-probe datasets.

Abstract

A novel matrix method of analyzing ion saturation current data from a general three-dimensional (3D) array of unmagnetized Mach probe tips is developed and used with data sets from two 3D Mach probes to make initial measurements of local plasma flow velocity in reversed-field pinch (RFP) experiments in the Madison Symmetric Torus (MST). The two 3D Mach probes are composed of regular polyhedral arrays of six and four tips, respectively, with the six-tip array composed of three orthogonal pairs of mutually opposite tips at the vertices of a regular octahedron and the four-tip array composed of non-opposite tips at the vertices of a regular tetrahedron, the analysis of which is specifically facilitated by the matrix method. Velocity measurement uncertainties for the Mach probes are derived based on uncertainties in probe machining and ion saturation current measurements, and typical relative uncertainties for the probes are estimated to be of order several percent, likely smaller than systematic uncertainties related to the Mach probe calibration constant and experimental uncertainties related to plasma and probe conditioning. Initial results for the octahedral probe show flow speeds of roughly the expected magnitudes based on previous MST measurements but with somes differences in flow direction, while those for the tetrahedron probe show similar flow directions to some previous measurements but also some larger than expected speeds. We consider possible causes for the unexpected results of these initial tests, with a focus on probe conditioning and fast electron issues.

Figures (5)

• Figure 1: Octahedral probe example.
• Figure 2: Tetrahedral probe example.
• Figure 3: RFP operational signals vs. time for the example shot with the octahedral probe in blue (see FIG. \ref{['fig:mst_octa']}), tetrahedral probe in orange (see FIG. \ref{['fig:mst_tetra']}). (a) Plasma current $I_\text{p}$. (b) Reversal parameter $F$. (c) Electron density $n_\text{e}$.
• Figure 4: Initial measurements of 3D plasma velocity components vs. time from the octahedral Mach probe inserted 4.5 cm deep into the plasma for the blue shot in FIG. \ref{['fig:mst_shots']}. Radial component in blue; positive means outward. Poloidal component in orange; positive means toward outboard on top. Toroidal component in green; positive means counterclockwise (CCW) from above, with $B_\text{t}$, against $I_\text{p}$. White time window indicates the plasma flattop; shaded time windows indicate plasma startup and rampdown. Data is boxcar-averaged over a running time window of 250 $\mu$s.
• Figure 5: Initial measurements of 3D plasma velocity components vs. time from the tetrahedral Mach probe inserted 7 cm deep into the plasma for the orange shot in FIG. \ref{['fig:mst_shots']}. Radial component in blue; positive means outward. Poloidal component in orange; positive means toward outboard on top. Toroidal component in green; positive means CCW from above, with $B_\text{t}$, against $I_\text{p}$. White time window indicates the plasma flattop; shaded time windows indicate plasma startup and rampdown. Data is boxcar-averaged over a running time window of 250 $\mu$s.