2.1. The Rotating Plasma Experiment pulsed-power platform

The experiments were conducted on the Mega-Ampere Generator for Plasma Implosion Experiments (MAGPIE) pulsed-power generator which delivers $1.4$ MA peak electrical current with $240$ ns rise time (Mitchell et al. Reference Mitchell, Bayley, Chittenden, Worley, Dangor, Haines and Choi1996), as shown in figure 1(a). The load is a wire array Z pinch (Benjamin et al. Reference Benjamin, Pearlman, Chu and Riordan1981; Aivazov et al. Reference Aivazov, Vikharev, Volkov, Nikandrov, Smirnov and Tsarfin1987, Reference Aivazov, Vikharev, Volkov, Nikandrov, Smirnov and Tsarfi1988; Deeney et al. Reference Deeney, LePell, Failor, Wong, Apruzese, Whitney, Thornhill, Davis, Yadlowsky and Hazelton1995; Yadlowsky et al. Reference Yadlowsky, Moschella, Hazelton, Settersten, Spanjers, Deeney, Failor, LePell, Davis and Apruzese1996; Lebedev et al. Reference Lebedev, Beg, Bland, Chittenden, Dangor, Haines, Pikuz and Shelkovenko2000, Reference Lebedev, Beg, Bland, Chittenden, Dangor, Haines, Pikuz and Shelkovenko2001) consisting of a $R=8$ mm radius cylindrical arrangement of eight equally spaced aluminium wires ($40\,\mathrm {\mu }{\rm m}$ diameter each) that are $10$ mm long. The wire array is surrounded by eight 1 mm thick stainless steel return posts, offset by $\varphi _0 = 13^{\circ }$ and positioned on a $R_P=11$ mm radius cylinder, as shown in figure 1(a). The return posts are used to introduce angular momentum to the experiment (Bocchi et al. Reference Bocchi, Chittenden, Lebedev, Hall, Bennett, Frank and Blackman2013a; Valenzuela-Villaseca Reference Valenzuela-Villaseca2022; Valenzuela-Villaseca et al. Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a).

Figure 1. Schematic of experimental set-up. (a) MAGPIE's electrical current trace. (b) End-on view of the experimental components together with the plasma dynamics in the plane of rotation. (c) Side-on view. Top anode disc connects all wires and all return posts. Current path is split symmetrically through the load, but only one path through a wire and a return post is presented.

During the discharge, electrical current passes through the wires and return posts in opposite directions (figure 1b). In the first few nanoseconds, the wires heat up through Joule heating, creating a coronal plasma on their surface. The current path through the wires and return posts produces two magnetic field components $B_r$ and $B_\theta$ at each wire, which in combination with the coronal current density $\boldsymbol {J}$, accelerates the plasma radially inwards (by $\boldsymbol {J}\times B_\theta \boldsymbol {\hat {\theta }}$) and simultaneously introduces an azimuthal component of motion (by $\boldsymbol {J}\times B_r\boldsymbol {\hat {r}}$), producing ablation flows that follow slightly off-radial trajectories. As these ablation flows merge, they drive a rotating plasma on the axis (figure 1a).

Ablating wire array Z pinches go through distinct phases (see review by Lebedev et al. Reference Lebedev, Ampleford, Bland, Bott, Chittenden, Goyer, Jennings, Haines, Hall and Hammer2005) that are relevant to the RPX. In the ablation phase, the Lorentz force transports the coronal plasma inwards and the wires still act as a mass reservoir. When a significant fraction of the mass has been ablated (empirically, between $50\,\%$ and $80\,\%$), breakages along the wires appear, which quickly change the load's impedance, ending the ablation phase. During the following implosion phase, the current is suddenly redistributed and the remaining mass in the wires implodes in a shell-like implosion, snow ploughing through the ablated trailing mass. In the stagnation phase the load's mass reaches the axis, quickly decelerating and releasing an X-ray burst (e.g. Spielman et al. Reference Spielman, Deeney, Chandler, Douglas, Fehl, Matzen, McDaniel, Nash, Porter and Sanford1998). We find that RPX is able to drive rotation throughout the ablation and implosion phases. If there is a final X-ray burst from the final stagnation of the rotating plasma is currently unknown.

As shown in figure 1(b), the open (vacuum) boundaries and axial confining forces allow the plasma to expand upwards, producing an axial jet that rotates, transporting angular momentum.

2.2. Diagnostics

The plasma is probed by a suite of diagnostics which encompasses laser interferometry imaging, self-emission imaging and Thomson scattering that allow multiple measurements of relevant plasma parameters, such as electron density, temperature and velocity. In each experiment, the imaging diagnostics provide two orthogonal lines of sight, namely vertical (along the rotation axis, end on, $xy$-plane) and lateral (perpendicular to rotation axis, side on, $xz$-plane) simultaneously. Both the diagnostics probing direction and line of sight are shown schematically in figures 1(b) and 1(c), respectively. All diagnostics are timed relative to current start, which defines $t=0$.

An optical fast-framing camera (5 ns time resolution) images the plasma optical self-emission in the direction parallel to the rotation axis. On the same line-of-sight a two-colour Mach–Zehnder interferometry system (532 and 355 nm wavelengths, 0.5 ns time resolution) probes the plasma electron number density (Hutchinson Reference Hutchinson2002). The two wavelength beams are delayed by 20 ns to quantitatively probe the electron density evolution. The diagnostic provides a weighted average electron density measurement integrated along the line-of-sight $\langle n_e \rangle (x,y)Z_p = \int n_e(x,y,z)\,{\rm d}z$, where $Z_p$ is the characteristic plasma length scale along its axis.

A second Mach–Zehnder interferometer is aligned with the side-on field of view, perpendicular to the rotation axis. The probe beam passes above the top anode electrode to probe the axial outflows launched by the pinch. In addition, two pinhole, micro-channel plate cameras (MCP), with 5 ns time resolution, image the plasma extreme ultra-violet (XUV) self-emission. The cameras (labelled A and B) point towards the experimental axis from opposite locations. Pinhole sets are located 660 mm from the object plane and the photo-cathode front surface of the MCP at 200 mm from the pinholes. Each pinhole had a $100\,\mathrm {\mu }{\rm m}$ diameter aperture for a spatial resolution of ${\sim }300\,\mathrm {\mu }{\rm m}$. To reduce the fluence of low-energy photons ($h\nu < 60$ eV), a $1\,\mathrm {\mu }{\rm m}$ Mylar filter was placed in front of the pinholes, greatly improving the image contrast. The MCP cameras provide up to 7 plasma images in each experiment which are timed to study the expansion of the plasma along the rotation axis and estimate its axial velocity $u_z$.

Quantitative measurements of electron temperature $T_e$, ion temperature $T_i$, charge state $Z$ and bulk velocity in the plane of rotation $\boldsymbol {u} = (u_x, u_y)$ are obtained using an optical Thomson scattering (TS) diagnostic (Swadling et al. Reference Swadling, Lebedev, Hall, Patankar, Stewart, Smith, Harvey-Thompson, Burdiak, De Grouchy and Skidmore2014; Suttle et al. Reference Suttle, Hare, Halliday, Merlini, Russell, Tubman, Valenzuela-Villaseca, Rozmus, Bruulsema and Lebedev2021). A laser beam (532 nm wavelength, 3 J, 7 ns) is focused across the experimental axis with the incident wavevector $\boldsymbol {k_{{in}}}$ perpendicular to it. The beam passes in the horizontal direction approximately 5.5 mm above the upper surface of the top anode electrode, and therefore probes the axial outflows. The scattered spectrum is collected by two linear arrays of 14 optic fibres each located in the horizontal plane (perpendicular to the rotation axis) at ${\pm }90^{\circ }$ relative to $\boldsymbol {k_{{in}}}$, labelled bundles A and B. The fibre bundles are coupled to an imaging spectrometer to measure the TS spectra from the collection volumes. Thus, the combined TS configuration is sensitive to the orthogonal scattering vectors $\boldsymbol {k_{A(B)}} = \boldsymbol {k_{{out}, A(B)}} - \boldsymbol {k_{{in}}}$, where $\boldsymbol {k_{{out}, A(B)}}$ are the directions of collection for bundle A (and B, respectively).

The TS diagnostic probes local plasma conditions. A 532 nm wavelength, 7 ns, 3 J laser beam is focused through the plasma. The $200\,\mathrm {\mu }{\rm m}$ beam waist defines the spatial resolution along the collection direction. The spatial resolution along $\boldsymbol {k_{{in}}}$ is given by the product of the magnification ${\sim }0.54$ and the diameter of each optic fibre ($100\,\mathrm {\mu }{\rm m}$), yielding an imaged collection volume of diameter ${\sim }180\,\mathrm {\mu }{\rm m}$. Therefore, the diagnostic samples plasma conditions averaged over a collection region with volume ${\sim }200\times {\rm \pi}\times 180^2/4\,\mathrm {\mu }{\rm m}^3$. The characteristic collection diameter is much smaller than the characteristic plasma-gradient length scales. The spectral features can be fitted using the dynamical form factor $S(k,\omega )$ for ion-acoustic wave TS (Froula et al. Reference Froula, Glenzer, Luhmann and Sheffield2011; Hare Reference Hare2017). The fitting process was constrained by using the electron density measured by the side-on interferometry diagnostic to reduce the number of fitting variables.

2.3. Magnetic field configuration and direction of ablation flows

The ablation direction of the streams can be calculated analytically from a magneto-static model. In cylindrical coordinates and without loss of generality, the magnetic field components evaluated in the wire located at $r=R, \theta = 0$ are given by the Biot–Savart law (Felber & Rostoker Reference Felber and Rostoker1981)

(2.1)\begin{gather} B_r(R) ={-}\frac{\mu_0 I}{2{\rm \pi} N} \sum_{j=0}^{N-1}\frac{R\sin(\varphi_0 + 2{\rm \pi} j/N)}{R^2+R_P^2-2RR_P\cos(\varphi_0 + 2{\rm \pi} j/N)}, \end{gather}

(2.2)\begin{gather}B_\theta(R) = \frac{\mu_0 I}{2{\rm \pi} N} \left[ \frac{N-1}{2R} - \sum_{j=0}^{N-1}\frac{R-R_P\cos(\varphi_0 + 2{\rm \pi} j/N)}{R^2+R_P^2-2RR_P\cos(\varphi_0 + 2{\rm \pi} j/N)} \right], \end{gather}

where $R$ is the wire array radius, $R_P$ is the return post array radius, $\varphi _0$ is the wire-post off-set angle, $N$ is the number of wires (and return posts) and $I$ is the electrical current. The equations above describe the magnetic field components generated by all other wires together with all return posts.

For the cases $\varphi _0 = m{\rm \pi} /8$ ($m$ an integer), (2.1) becomes zero, and for $\varphi _0 = m{\rm \pi} /4$ (2.2) is reduced to the standard wire array Z pinch vacuum field. Hence, for any other value of $\varphi _0$, angular momentum will be introduced because the resulting $\boldsymbol {J}\times \boldsymbol {B}$ force has radially inwards and azimuthal components. The ablation flow is ejected along an off-radial trajectory parameterized by the direction angle $\theta \equiv \tan ^{-1}(B_r/2B_\theta )$. For the wire and return post array dimensions used in the experiment, figure 2(a) shows the vacuum fields calculated for parameters relevant to these experiments. They show that the radial magnetic field component is approximately $10\,\%$ of the azimuthal component. The result also shows that the azimuthal component changes slightly due to the return posts.

Figure 2. Magneto-static model calculated for relevant configuration: $R=8$ mm, $R_P=11$ mm and $I=1.4$ MA. (a) Magnetic field components at wire position. (b) Direction angle with respect to radial vector.

Figure 2(b) shows the off-radial direction angle $\theta$ as a function of $\varphi _0$. It shows that the maximum direction angle is obtained for $\varphi _0 = 11.25^{\circ } = 119.35$ mrad, as expected from geometric arguments presented above. The expected deflection in the experiment is in the range $\theta \sim 4^{\circ }\unicode{x2013}5^{\circ }$ for the array parameters described above.

3.1. Ablation dynamics and formation of rotating plasmas

Figure 3 shows a sequence of optical self-emission images of the formation and evolution of ablation flows and rotating plasma. Since the plasma contains aluminium ions which are not fully stripped (charge state $Z\sim 7$) and emit copious optical light through atomic transitions, the intensity of the images is a complicated function of the density and temperature of the plasma, but can be used for qualitative evaluation of the plasma structure. The images show elongated features oriented inwards. These are the ablation flows ejected from each of the aluminium wires. The analysis shows that their direction is not along a radial direction, but rather tilted azimuthally by $\theta = 4^{\circ }\pm 1^{\circ }$, which is consistent with the theoretical prediction of the vacuum magneto-static model presented in § 2. The self-emission of the ablation flows diminishes as they propagate inwards due to radiative cooling effects.

Figure 3. Sequence of end-on optical self-emission images obtained in a single experiment. Emission from the outer edges is likely produced by some residual ablation of the experimental hardware. The self-emission intensity along the ablation flows decreases due to the combination of radiative cooling effects and a decreasing density towards the axis (Valenzuela-Villaseca et al. Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a).

On the axis, a circular bright ring is formed at around $130$ ns, corresponding to the rotating plasma. This formation time can be used to estimate the ablation velocity $V_{ab}\sim 70\,{\rm km}\,{\rm s}^{-1}$, consistent with direct TS measurements by Harvey-Thompson et al. (Reference Harvey-Thompson, Lebedev, Patankar, Bland, Burdiak, Chittenden, Colaitis, De Grouchy, Doyle and Hall2012a) for standard wire array Z pinches. The self-emission structure exhibits reduced intensity on axis. This is consistent with a dense plasma shell surrounding a low-density core. This has been observed in previous experiments by Bennett et al. (Reference Bennett, Lebedev, Hall, Suttle, Burdiak, Suzuki-Vidal, Hare, Swadling, Patankar and Bocchi2015) also and is produced by the centrifugal barrier associated with dynamically significant angular momentum (Valenzuela-Villaseca et al. Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Russell, Merlini, Tubman, Hare, Chittenden and Koepke2024). Unfortunately, this sequence of images does not allow us to estimate the rotation velocity, since the smooth ring characterizing the plasma has no discernible features that co-rotate with the plasma which can be tracked with time. At ${\sim} 170$ ns, radially oriented features emerge from the ring. These are secondary shocks produced by the oblique interaction of the ablation flows undergoing thermal expansion (Swadling et al. Reference Swadling, Lebedev, Niasse, Chittenden, Hall, Suzuki-Vidal, Burdiak, Harvey-Thompson, Bland and De Grouchy2013).

Quantitative measurements of the density structure of the plasma at a given time can be obtained through laser interferometry. Figure 4(a) shows a laser interferogram obtained after the formation of the rotating plasma. The bright and dark fringes correspond to regions of constructive and destructive interference, respectively, between the probe beam (passing through the plasma) and a reference beam (passing around the plasma). The interferometer is aligned to have the reference interferogram (i.e. in the absence of plasma) with horizontal fringes. Thus, any bending or deformation of the interferogram is due to the added refractive index, which is proportional to the plasma electron density. Figure 4(b) shows the inferred line-integrated electron density map calculated using the MAGIC2 code (Swadling et al. Reference Swadling, Lebedev, Niasse, Chittenden, Hall, Suzuki-Vidal, Burdiak, Harvey-Thompson, Bland and De Grouchy2013; Hare et al. Reference Hare, MacDonald, Bland, Dranczewski, Halliday, Lebedev, Suttle, Tubman and Rozmus2019), which constructs a phase shift map by interpolating between the fringes and then comparing the reference and shot interferograms. The phase shift map is proportional to $\int n_e dz$, hence calculating a map of the electron density map integrated along the line of sight. Since in the image there is no region with zero fringe shift, the diagnostic can only probe relative changes in electron number density. For clarity, we have defined the region with minimum phase shift as having zero density (which coincides with the regions between the ablation flows). To unfold the (volumetric) electron density we must assume some symmetry. For the end-on view, the plasma can be assumed to be homogeneous along its length $L$ if we neglect contributions of the axial jet ejected from this region, which is significantly lower density than the plasma inside the wire array (Valenzuela-Villaseca et al. Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a). The plasma density is dominated by the region between the two electrodes (the axial jets are one order of magnitude less dense, which will be shown in the next section). Therefore, $L = 10$ mm, corresponding to the array's length.

Figure 4. End-on interferometry results obtained from the 355 nm wavelength probe. (a) Raw interferogram. (b) Processed line-integrated electron density map. The contour of two secondary shocks have been highlighted with orange dotted lines.

The data show the interaction between the inwards propagating ablation flows as they reach the rotating plasma column. The electron density decreases radially inwards by a factor of ${\sim} 10$, which may help explain features in the optical self-emission images. The ablation flows remain extremely collimated during the experiment, with divergence angles in the range $\alpha = 10^{\circ } \pm 3^{\circ }$. This can be used to calculate their sonic Mach number

(3.1)\begin{equation} M_s = \frac{u}{c_s} = \frac{1}{\sin{\alpha/2}} = 10\pm 3, \end{equation}

where $u$ is the flow velocity ($=V_{ab}$) and $c_s$ is the plasma sound speed. A characteristic plasma temperature $T=10$ eV can be inferred by considering the ion sound speed $c_s \sim \sqrt {k_BT/m_i}$, where $m_i$ is the ion mass and k_B is Boltzmann's constant. Our results are consistent with previous reports by Harvey-Thompson et al. (Reference Harvey-Thompson, Lebedev, Patankar, Bland, Burdiak, Chittenden, Colaitis, De Grouchy, Doyle and Hall2012a,Reference Harvey-Thompson, Lebedev, Patankar, Bland, Burdiak, Chittenden, Colaitis, De Grouchy, Hall and Khooryb) and Swadling et al. (Reference Swadling, Lebedev, Niasse, Chittenden, Hall, Suzuki-Vidal, Burdiak, Harvey-Thompson, Bland and De Grouchy2013). We note that TS measurements revealed $T_i$ and $T_e$ to be $>10$ eV, which indicates the ablation flows are magnetically collimated by diverting part of the driver's current through the streams (Harvey-Thompson et al. Reference Harvey-Thompson, Lebedev, Patankar, Bland, Burdiak, Chittenden, Colaitis, De Grouchy, Doyle and Hall2012a; Swadling et al. Reference Swadling, Lebedev, Niasse, Chittenden, Hall, Suzuki-Vidal, Burdiak, Harvey-Thompson, Bland and De Grouchy2013). Since our results exhibit similar structure, we conclude that the effect of the radial magnetic field applied mainly re-directs the plasma streams, rather than changing the overall ablation properties of the wire array Z pinch. Experimentally, we find that the ablation flows can sometimes break from axisymmetry, likely due to differences in the ablation azimuthal divergence that can be ascribed to discrepancies in the wire ablation rate due to uneven current distribution during the electrical discharge.

The interferogram also helps in visualizing the origin of the secondary shocks. As the ablation flows thermally expand in the azimuthal direction, they can interact with their immediate neighbours before merging with the rotating plasma on the axis. This produces oblique, standing shocks which remain stationary due to the steady-state ablation dynamics of the Z pinch.

The fringes can be seen in the inflows but are lost in the outer region of the rotating plasma. This is due to density gradients perpendicular to the line of sight deflecting the probe beam out of the optical system's acceptance angle, as pulsed-power plasmas have typical densities below critical density for 355 nm wavelength (Nd:YAG 3rd harmonic) and the effect of absorption is negligible. As a consequence, the electron density of the rotating plasma cannot be estimated from end-on probing. However, there are visible interferogram fringes in the region on axis. This is consistent with either a density distribution with an on-axis flat profile (i.e. no density gradients) or a hollow density structure wide enough to allow the probe to cross the plasma.

3.2. Propagation and density structure of axial jets

Figure 5(a–f) shows a sequence of XUV images of the launching and propagation of the axial jet. A $1\,\mathrm {\mu }{\rm m}$ thick Mylar filter was used to remove soft XUV emission and improve the contrast. In these images, dark regions correspond to high plasma XUV self-emission intensity, whereas the light-coloured regions correspond to non-emitting metallic hardware and vacuum regions. The $z$-axis is oriented vertically with the position $z = 0$ defined as the approximate location of the top disc's upper surface. Therefore, the wire array plasma is restricted to the region $z < 0$. Vertical light-coloured strips correspond to the shadows of the return posts. The results show that the posts remain static throughout the experiment, with very little visible ablation and no overall deformations. This robustness is key to sustaining a steady-state off-axis ablation.

Figure 5. Side-on XUV plasma self-emission data. (a–f) Sequence of pinhole images obtained in the same experiment. Features corresponding to hardware components are annotated in italics. Panel (d) corresponds to the image presented in Valenzuela-Villaseca et al. (Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a), reprinted with the authors’ permission. (e) Length of axial jet, measured using the full height at half-maximum intensity, as a function of time obtained as measured from the MCPs (labelled A and B) located on two opposing lines of sight. Image at $t=130$ ns is not presented and length corresponding to $t=240$ ns was not used in linear fit.

The sequence reveals the presence of plasma on the region $z > 0$. Since the initial experimental set-up does not accelerate plasma axially (as opposed to, e.g. Ampleford et al. Reference Ampleford, Lebedev, Ciardi, Bland, Bott, Hall, Naz, Jennings, Sherlock and Chittenden2008), any outflows reaching this region must have been launched upwards by the wire array plasma through gradients of the thermal and/or magnetic pressure. The outflows can be described as consisting of a well-collimated axial jet on the axis, and an unstructured plasma halo which surrounds the axial jet. The images show that the plasma remains in a quiescent state, with no large perturbations or significant changes in structure morphology up until $t = 270$ ns, both in the wire array and outflow regions. This sets a ${\sim }140$ ns time frame over which rotation can be reliably sustained, as the plasma is first formed at ${\sim }130$ ns after current start. This shows that rotation can be maintained through the ablation and implosion phases of the Z pinch.

The axial velocity of the flow can be estimated by following the upper end of the jet, which is representative of its length, through the sequence of images for the two XUV cameras fielded in the experiment. The result is presented in figure 5(b), with each XUV camera presented by a different colour. The horizontal uncertainty is given by the 5 ns of time integration, whereas the vertical error bars estimate the uncertainty in the position of the tip of the jet due to the smooth extinction of the plasma self-emission. The results show the linear trend of the position of the jet with time, from which a vertical velocity $u_z = 100 \pm 10\,{\rm km}\,{\rm s}^{-1}$ can be estimated.

The density structure of the axial jet was studied using laser interferometry measurements. Figure 6(a) shows a typical side-on raw interferogram. The coordinate system is consistent with figure 5, i.e. the line z = 0 corresponds to the upper surface of the top electrode. Figure 6(b) shows the line-integrated electron density map inferred from panel (a). The central elongated structure corresponds to the axial jet. This result confirms the high aspect ratio ($H/R \sim 10$, where $H$ is the jet length and $R$ its radius) and collimation degree of the jet (${\sim }3^{\circ }$ divergence angle). The jet's base exhibits the formation of vertical filaments. These are consistent with the secondary shocks formed by the oblique collision of adjacent ablation flows, which can expand axially upwards and become present in the outflow region.

Figure 6. Side-on laser interferometry results (532 nm, 0.5 ns time resolution) at 210 ns after current start. (a) Raw interferogram. (b) Line-integrated electron density map. The TS probe incident wave vector points from left to right as indicated by the white arrow and collection volumes are shown by white circles. (c) Density lineouts at three heights. Middle panel corresponds to a lineout at the height of TS probe. (d) Abel-inverted density profiles unfolded from lineouts in panel (c). Inversion algorithm assumes axisymmetry independently on both sides and the result is presented in solid black lines. An average between left- and right-hand side inversion is shown in a blue dashed arrow.

Lineouts of Abel-inverted (volumetric) electron density at constant height were calculated using a peeling-onion method (Dasch Reference Dasch1992; Valenzuela-Villaseca Reference Valenzuela-Villaseca2022). This technique requires the assumption of a cylindrical symmetry of the plasma, and therefore it is only applicable to the regions $z > 3$ mm, above the filamentary secondary shock structures that break the jet's symmetry.

Figure 6(c,d) summarizes the results obtained at three different heights. The line-integrated density lineouts (panel c) show that the jet is not exactly axisymmetric. To estimate the effect of the asymmetry in the result, we calculated the Abel inversion separately on the left- and the right-hand side about the axis. We show the results with black lines in panel (d). The two inversions are averaged and shown in blue. Hence, the difference between the one-sided inversion and the averaged inversion can be used to estimate the uncertainty introduced by the assumed axisymmetry. The results show that the effect is ${<}10\,\%$.

The measurement reveals the hollow density structure of the plasma, as it exhibits a steep density decrease towards the axis. This is consistent with the observed optical self-emission profile discussed above, i.e. a dense shell surrounding a lower-density core. This density profile is present at all values of $z$, and therefore we conclude that it is retained as the axial jet propagates upwards, although we observe a slight decrease in the average density gradient from $\langle {\rm d}n_e/{\rm d}r\rangle =0.5\times 10^{18}\,{\rm cm}^{-3}\,{\rm mm}^{-1}$ at $z = 5.5$ mm to $\langle {\rm d}n_e/{\rm d}r\rangle =0.3 \times 10^{18}\,{\rm cm}^{-3}\,{\rm mm}^{-1}$ at $z = 6.5$ mm. The maximal electron density of the axial jet gradually decreases with height at an average gradient $\langle {\rm d}n_e/{\rm d}z\rangle =0.1\times 10^{18}\,{\rm cm}^{-3}\,{\rm mm}^{-1}$, likely due to the ablation time history imprinted by the driver's discharge. The radial location of maximal electron density does not change on the lineouts, which further demonstrates the collimation of the jet, with negligible mass diffusion down the density gradients.

3.3. Thomson scattering measurements of plasma flow velocity and temperature profile

The absence of inwards mass diffusion is consistent with the axial jet's density profile being supported by a centrifugal barrier, which implies that it is spinning (Valenzuela-Villaseca et al. Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Russell, Merlini, Tubman, Hare, Chittenden and Koepke2024). In the experiments, direct measurements of the rotation and radial flow velocity components were obtained using an optical TS diagnostic. The TS probe beam passes horizontally through the rotating jet at $z = 5.5$ mm above the top disc's upper surface, passing approximately through the plasma rotation axis. We extend the analysis reported by Valenzuela-Villaseca et al. (Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a). In particular, we measure from the orthogonal interferometry line of sight the impact parameter which is important to interpret the flow velocity components as azimuthal and radial.

Figure 7 summarizes the plasma parameters measured from TS, together with the electron density lineout inferred from laser interferometry measurements (figure 8a) at the height of interest. The position of the plasma axis projected along the line of sight is shown in all panels. The plasma is inhomogeneous and can be separated into 3 distinct regions: the rotating plasma region ($- 1.2\,{\rm mm} < x < 1.2\,{\rm mm}$), a deceleration region ($1.2\,{\rm mm} < x < 2\,{\rm mm}$) and an acceleration region ($x > 2\,{\rm mm}$). The latter two correspond to sub-regions in the plasma halo indicated in figure 5.

Figure 7. Inferred plasma parameters obtained for plasma outflows (axial jet and halo). Regions of interest are highlighted in background colour: rotating plasma region in orange, and deceleration region in blue. (a) Electron density obtained from interferometry at the height of interest. (b) Horizontal flow velocity components along scattering vectors. Inset: characteristic Doppler shift error bar. Horizontal error bars given by TS collection volume diameter ($200\,\mathrm {\mu }{\rm m}$). (c) Plasma velocity component along laser wave vector $\boldsymbol {k_{{in}}}$ (flow radial velocity). (d) Plasma velocity component perpendicular to laser wave vector (flow rotation velocity). (e) Average charge state $Z$. (f) Electron temperature. Downwards-pointing arrow indicates value is an upper constraint (single-peaked ion-acoustic wave (IAW) spectra). (g) Ion temperature.

Figure 8. Horizontal flow velocity field on end-on interferometry imaging. (a) Full end-on laser interferometry (532 nm wavelength). Inner cathode diameter (dashed circular line), outer edge of rotating plasma (magenta dashed circle), plasma symmetry axis (solid magenta circle) and TS collection volumes (small white circles) are presented. (b) Close-up to the rotating plasma region. Flow velocity field obtained from TS Doppler is indicated using arrows to scale. (c) Calculated radial velocity for the inferred impact parameter ($b= 250\,\mathrm {\mu }{\rm m}$, blue circles) and zero impact parameter (same as figure 7 for reference). (d) Azimuthal velocity.

The TS spectrum is sensitive to the flow velocity components projected on the scattering vectors $(\boldsymbol {k_A}, \boldsymbol {k_B})$ defined in § 2. The Doppler shifted velocity components obtained from the fit on both fibre bundles are shown in figure 7(b). To interpret the measurement, we reconstruct the radial and azimuthal flow velocity components using the Cartesian $\boldsymbol {k_A}$ and $\boldsymbol {k_B}$ vectors. Since the observation points lie on the horizontal plane (perpendicular to the rotation axis), then the velocity components parallel and perpendicular to $\boldsymbol {k_{{in}}}$ are

(3.2a,b)\begin{equation} u_{{\parallel}} ={-}\frac{1}{\sqrt{2}}\left(u_A + u_B \right), \quad u_{{\perp}} = \frac{1}{\sqrt{2}}\left( u_A - u_B \right), \end{equation}

where $(u_{A},u_{B})$ are the flow velocity components projected on the scattering vector basis. When the probe beam passes through the rotation axis, the vector basis $(u_{\parallel },u_{\perp })$ corresponds to the radial and azimuthal flow velocity components.Footnote ¹ For clarity, in the following description we assume this is the case, but below we will apply a correction to a small offset between the probe beam and the rotation axis (§ 3.3.1).

Figure 7(c) shows the flow velocity component parallel to $\boldsymbol {k_{{in}}}$ (i.e. radial velocity). The data show that the halo flows inwards and merges into the axial jet. In the outermost (acceleration) region, the plasma accelerates from $40 \pm 3$ to $60 \pm 3\,{\rm km}\,{\rm s}^{-1}$. However, approximately 1 mm before reaching the axial jet (i.e. before the density jump associated with the rotating plasma shell), the inflow decelerates to $25 \pm 4\,{\rm km}\,{\rm s}^{-1}$. This deceleration can also be seen on the left-hand side of the lineout ($x < -1.2$ mm). Inside the rotating plasma region, the radial velocity is approximately zero, with a maximum value $5 \pm 3\, {\rm km}\,{\rm s}^{-1}$. This indicates that the plasma is in an approximate equilibrium state, with negligible mass radial transport.

The rotation of the axial jet is manifested in the consistent change of sign of $u_\perp$ about the axis, as shown in figure 7(d). In the rotating plasma region, the data points on the left of the axis have a consistent positive sign, whereas those on the right have a negative one. This is a direct measurement of rotation. The point on the axis has non-zero $u_\perp$ due to the slight misalignment of the probe beam with the rotation axis (see § 3.3.1). The maximum rotation velocity $23 \pm 3\, {\rm km}\,{\rm s}^{-1}$ is observed at $x=-0.4$ mm. In the deceleration region, $u_\perp$ does not exhibit a monotonic behaviour. This indicates that this velocity component is associated with the azimuthal expansion of the inflows (shown in figure 4b), rather than rotation. The acceleration region shows negligible flow perpendicular to the laser beam.

In the halo, the IAW spectral feature is single peaked, indicating that $3T_i > ZT_e$ (Froula et al. Reference Froula, Glenzer, Luhmann and Sheffield2011), which means that the fitting only provides an upper constraint on the electron temperature. The average charge state and electron temperature remain approximately constant in the halo (figure 7e,f), with $Z \sim 3$ and $T_e \leq 15$ eV. Inside the rotating plasma, the electron temperature increases towards the axis, from $T_e = 30 \pm 12$ to $45 \pm 15$ eV. The average charge state is calculated using a look-up table generated from the atomic kinetic code SpK (Hare Reference Hare2017; Crilly et al. Reference Crilly, Niasse, Fraser, Chapman, McLean, Rose and Chittenden2023) which calculates ionization states in the relevant conditions to decouple the product $ZT_e$. The result is a relatively constant charge state $Z \sim 7$ across the jet.

3.3.1. Effect of the probe impact parameter on the velocity measurements

In general, the mapping between the flow velocity components $(u_\parallel, u_\perp ) \longrightarrow (u_r, u_\theta )$ depends on the orthogonal distance between the probe beam and the rotation axis, mathematically defined by the impact parameter $b$. The transformation between flow velocity components accounting for the impact parameter is given by

(3.3a,b)\begin{equation} u_r(\chi_i) = \text{sign}(\chi_i)\frac{\chi_iu_\parallel{+} b u_\perp}{r_i},\quad u_\theta(\chi_i) = \text{sign}(\chi_i)\frac{\chi_iu_\perp{-} b u_\parallel}{r_i}, \end{equation}

where $\chi _i$ is the location of the collection volume $i$ along the laser probe path, $(u_\parallel, u_\perp )$ can be inferred from the Doppler shift using (3.2a,b) and

(3.4)\begin{equation} r_i(\chi_i) = \sqrt{\chi^2_i +b^2}, \end{equation}

is the distance from the collection volume to the rotation axis. It is evident that the particular case where $b = 0$ yields the result from the previous section, i.e. $u_r = \text {sign}(\chi _i)\cdot u_\parallel$ and $u_\theta = \text {sign}(\chi _i)\cdot u_\perp$. Moreover, the correction is important only in the region where $\chi _i \lesssim b$.

To experimentally correct the location of the TS probe beam relative to the rotation axis in the transverse direction, we map the measurements onto our end on interferometry data. Figure 8(a) shows a full shot interferogram with the TS collection volumes drawn. A close-up view to the region of interest is presented in figure 8(b). The region where interferometry fringes are lost define the shell-like density structure of the rotating plasma, and its outer boundary is indicated in a dashed magenta circle. Moreover, the axis of such a circle is indicated with a solid circle (indicated in panel a). Assuming the axis of plasma symmetry (i.e. where the centre of mass of the system should be) corresponds to the rotation axis, and noticing that the white circles indicate the path of the probe beam, a value of the impact parameter $b = 250 \pm 50\,\mathrm {\mu }{\rm m}$ can be estimated. In addition, the horizontal flow velocity field evaluated on the collection volumes can be obtained by combining $(u_A,u_B)$, which is shown as white arrows in figure 8(b).

The results of the correction on the radial and azimuthal flow velocity components is shown in figure 8 panels (c) and (d), respectively. The result was discussed only schematically by Valenzuela-Villaseca et al. (Reference Valenzuela-Villaseca, Suttle, Suzuki-Vidal, Halliday, Merlini, Russell, Tubman, Hare, Chittenden and Koepke2023a) to show that the identification of a quasi-Keplerian rotation curve is not dependent on the impact parameter. However, here, we present a direct measurement using a diagnostic with a line of sight orthogonal to the TS geometry. We find that, if this correction is ignored, then the result indicates that there is a constant sideways velocity, however, accounting for the impact parameter allows us to correctly interpret that, near the axis, $u_\parallel$ corresponds to $u_\theta$ rather than $u_r$. Although the magnitude of the rotation velocity changes, this correction may be important to correctly determine rotation curves close to the axis.