Applied axial magnetic field effects on laboratory plasma jets: Density hollowing, field compression, and azimuthal rotation

Magnetic fields can play a role in the dynamics of astrophysical plasma jet formation, collimation, and acceleration.^1–3 The research field of laboratory astrophysics seeks to elucidate astrophysical phenomena using laboratory experiments, provided the consistency of certain scaling relations and parameter regimes which are determined by dimensionless quantities.^4–6 Additionally, the laboratory experiments can help benchmark computer simulations that can model astrophysical processes.

Previous plasma jet research using pulsed power machines has looked at the effects of time-dependent magnetic fields on the jet dynamics.^7,8 Additional research has explored the effects of an applied magnetic field on plasma jets produced from laser ablated foils.^9–11 Rotating plasma jets have also been created with converging ablation streams using pulsed power machines.¹² This present work continues our previous magnetized, laboratory plasma jet research and comparisons of these results to simulations using the extended magnetohydrodynamics (XMHD) code, PERSEUS.^7,13–15 Our jets are produced by a pulsed power driven load in a radial foil configuration and are the result of magnetic and kinetic pressures that collimate the jet along the z-axis.^13,14 The addition of Thomson scattering, improved magnetic probe measurements, and inclusion of 3D and equation of state physics in PERSEUS has resulted in a better understanding of the dynamics of the magnetized laboratory plasma jet, as well as a better appreciation for the effects of the electron inertia and Hall effect terms in the XMHD (vs. MHD) simulations.

The COBRA (COrnell Beam Research Accelerator¹⁶) pulsed power generator (1 MA peak, 100 ns zero-to-peak rise time) sent current radially through a 15 μm thick aluminum or titanium disk (called a radial foil) and axially along a brass pin beneath the center of the foil [see Fig. 1(a)]. The COBRA current pulse has a calculated ∼40 μm Al skin depth and ∼150 μm Ti skin depth at room temperature. Here, we used a 36 mm diameter radial foil and center pins with diameters of 5 mm or 10 mm. These pins were solid cylinders with a flat surface of contact with the foil. The center pin was set as the cathode or anode corresponding with the outer annulus of the foil being the anode or cathode, respectively. In standard current polarity with a cathode pin [Fig. 1(a)], the current flowed radially inward through the foil and down the pin. In reverse current polarity with an anode pin, the current flowed up the pin and radially outward through the foil.

A Helmholtz coil, with a ∼150 μs rise time, applied a spatially uniform axial magnetic field (B_z) of up to 2 T over a ∼cm³ region above the foil that was occupied by the jet. As measured by B-dot probes before the experiment at 3 positions [see red dots in Fig. 1(b)], the B_z was uniform to within the probe measurement uncertainty (∼5%) for 1 mm above the foil on-axis, 8 mm above the foil on-axis, and 1 mm above the foil 9 mm radially away from the z-axis. This coil is pulsed ∼150 μs before the COBRA current pulse by a 200 μF, 8 kV capacitor bank.

Previous results using this hardware have shown the formation of well-collimated plasma jets that hollow and take on a conical configuration when an axial magnetic field is applied.¹³ Reversing the polarity resulted in wider jets with larger conical angles, as can be seen in Fig. 2.

PERSEUS simulations,¹⁷ also shown in Fig. 2, demonstrate remarkably similar magnetic field and polarity effects and also indicate that there is rotation of the jet and on-axis compression of the applied field. This compression and rotation are two major themes of this paper.

Three major diagnostics were used for the experimental measurements: interferometry, magnetic B-dot probes, and Thomson scattering.

We observed the jet structure and on-axis density hollowing using laser interferometry. The diagnostic laser produced a 532 nm, 148 ps, and 140 mJ pulse. For this wavelength, one interferogram fringe shift corresponded to an areal density of ∼4 × 10¹⁷ cm⁻², the plasma critical density was ∼4× 10²¹ cm⁻³, and our experimental setup geometric acceptance angle limited (by refraction) the maximum density we could view to be a few × 10¹⁹ cm⁻³. We used the IDEA software to get the areal density from the fringe shift phase data.^18,19 This diagnostic laser was also used for laser backlighting shadowgraphy.

In order to measure the compression of the axial magnetic field, we used B-dot probes placed in the central hollow region of the jet.^13,20 This central placement minimized any perturbation of the jet, limited any shock formed around the probe, and reduced any current in the plasma around the probe. The probe loop voltage is related to the B-dot (time derivative of the magnetic field) by Faraday's law. The loop areas were of order 0.5 mm², for which a 1 T change in the magnetic field over a time of 50 ns corresponded to an average loop voltage of 10 V. To convert a voltage to dB/dt (B-dot), before the experiment each B-dot probe area was calibrated using a well-characterized pulser. The loop area calibration had an estimated uncertainty for the B-dot measurement of ∼5% that we conservatively took to be ∼10% to account for some probe orientation shift in the experimental setup. An actual ∼10% change in the effective loop area corresponded to an orientation shift larger than a 25° angle, which is far larger than any realistic misalignment.

In the present research, for the aluminum jets, we used B-dot probes with two loops in each of the probes. In one case, in order to ensure that the probe signal was being generated by only the change in B_z, these two loops were counter-wound. The signals from these loops were then added and subtracted. The subtracted signal should have been from the magnetic field pick up and the added signal from electric field pickup. In another case, one of the loops was oriented to pick up the changing magnetic field parallel to the z-axis and the other loop sensitive to a changing field perpendicular to the z-axis. This was done to ensure that we were measuring an axial compression of just B_z. For the titanium jet experiments, we used simpler probes with a single loop in each probe that was oriented either parallel or antiparallel to the z-axis.

Our B-dot probes were made from 0.51 mm outer diameter, semi-rigid, coaxial cables. A loop orientation (normal to the plane of the loop) was either perpendicular to or parallel to the coaxial cable axis (z-axis). When the loop was oriented parallel to the z-axis, the coaxial cable outer conductor extended to the side and past the loop to prevent penetration of some of the non-z-directed fields. The probe loops were covered with Kapton tubing and the probe ends were filled with epoxy for electrical insulation. In the present work, for a probe with two loops, the overall tip diameter was ∼2.2 mm.

We measured the velocities along all 3 axes of the magnetized plasma jet using Thomson scattering. The Thomson scattering laser produced a maximum of 10 J, 3 ns, and 526.5 nm pulse, focused to a 340 μm spot diameter. These scattering measurements were made in the collective scattering regime, α = 1/kλ_De ≳ 1, where λ_De is the Debye length and k = k_s − k_L, with k_s being the scattered wave vector and k_L being the laser wave vector. The scattering is characterized by a theoretical spectral density function S(k, ω), where ω ≡ ω_s − ω_L.^21,22 Experimentally observed scattered spectra can be compared to S(k, ω) to determine several plasma parameters. The simplest measurement comes from a Doppler shift, Δω_d, of the entire spectrum. For the ion-acoustic waves being observed in our experiments, this Doppler shift gives the ion flow velocity, v, and Δω_d = v ⋅ k. In addition to this ion flow velocity, other parameters of the plasma, including the relative electron-ion drift velocity, ion temperature, and electron temperature, can be found by performing a best fit between the spectral density function, S(k, ω), and the experimentally observed scattered spectrum.^21,22

For the Thomson scattering measurements of our plasma jets, the 10 J laser was focused on the cylindrical axis of the jet at a height of 5 mm above the foil. Three linear optical fiber bundles were focused to look at scattered radiation perpendicular to the path of the laser. The individual fibers in the bundles had a diameter of 100 μm and were spaced linearly in each bundle with a center-to-center distance of 125 μm. In order to decouple the three components of the ion flow velocity, the 3 fiber bundles were set orthogonal to the laser vector, with the scattering direction for each fiber bundle characterized by k_s1, k_s2, and k_s3, as shown in Fig. 3. Each of the fiber bundles was set up to collect scattering in one of two planes, either a plane containing $r̂$ and $θ̂$ with z being 5 mm above the pin, which we will call the r–θ plane, and one containing $k̂L$ and $ẑ$⁠, which we will call the r–z plane. Two fiber bundles, with 10 fibers each, were located in the r–θ plane and observed scattering vectors k_s1 and k_s2, as shown in Fig. 3(a). Each of these fibers had a 340 μm diameter view of the laser, and all fibers in the bundle combined to capture a length of 4 mm along the laser's path. A third fiber bundle, with 20 fibers, was positioned to look at the scattered radiation in the k_s3 direction from both sides of the axis of the plasma jet, as shown in Fig. 3(b). Each of these fibers captured a spot size of 450 μm, slightly larger than the beam diameter, and all combined to captured a length of 10 mm along the laser's path.

These fiber bundles were taken to a 750 mm long Czerny-Turner spectrometer, part of the Andor Shamrock series. Using a 2400 ℓ/mm grating gave a spectral full width half max (FWHM) of 0.5 Å. The spectra from each fiber bundle were imaged onto a time gated intensified CCD (charge-coupled device) camera, Andor Istar, with a gating time of at least 5 ns in order to capture the full length of the laser pulse. On each image, 18 of the 20 fibers had a clearly resolved spectrum which, when fitted to the to the spectral density function, resulted in the simultaneous determination of the ion flow velocity at several spatial locations.

In order to collect scattered radiation from three angles, care needed to be taken in the choice of the polarization of the Thomson scattering laser, as an accelerated electron will not radiate in the polarization direction. The polarization of the laser was controlled by rotating a half wave plate prior to the focusing lens in order to minimize the reflection off a Brewster window set for the desired polarization. The polarization was set to an angle of 45° between k_s3 and k_s1. This dropped the power scattered to any individual fiber by a factor of 2, as the scattered power scales as $1−sin2(θ) cos2(ϕ)$⁠,²¹ with θ being the angle between k_L and k_s and ϕ being the angle between the polarization vector and the plane containing k_L and k_s. Although we lost this factor of 2 in power, it was the only way to observe simultaneous scattering in all 3 directions.

By using the Doppler shifts observed from these three scattering directions, all three components of the ion flow velocity were obtained. The flow velocity along k was obtained from the Doppler shift, $vk=v·k̂=Δωd/|k|$⁠. Since we were scattering from the ion-acoustic waves, which produce only small frequency shifts, the laser and scattered wavelength were approximately equal, $|kL|≈|ks|, |k|≈2kL sin (θ/2)$⁠, with θ being the angle between k_L and k_s. The directions and magnitudes of all the k_L, k_s, and k wave vectors for the three scattering directions are shown in Fig. 3. The two fiber bundles that lay in the r–θ plane, on opposite sides of the laser path, had paired fibers from each bundle looking at the same scattering volume, see Fig. 3(a). Since k₁ had equal radial and azimuthal components, $vk1=(vr+vθ)/2$ when $k̂1·r̂>0$⁠. Similarly, $vk2=(vr−vθ)/2$⁠. Consequently, the radial and azimuthal components of ion flow velocity are given by

The scattered radiation observed by the third fiber bundle, along the z-direction, can be analyzed by a similar method using the fact that k₃ had equal components in the z and r directions. By assuming azimuthal symmetry, and comparing two fibers on opposite sides of the jet, measurements of the axial ion flow velocity and a second measurement of the radial ion flow velocity were made. Consider two of the fibers from this bundle looking on either side of the center of the jet at – and + locations, with the sign being defined by the sign of $kL·r̂$ [see Fig. 3(b)]. Using the Doppler shifts observed from these two fibers, the radial and axial ion velocities are given by

With proper alignment of the scattering volumes for the three fiber bundles, these expressions can be used to completely resolve the ion flow velocity, v, into its three components, v_r, v_θ, and v_z.

In this section, we quantitatively discuss the effects of the applied axial magnetic field, B_z, on the plasm jet: on-axis density hollowing, on-axis magnetic field compression, and azimuthal rotation. The physical basis for these effects is as follows: as the plasma jet forms, the plasma compresses the B_z on-axis; magnetic pressure from this compression causes density hollowing; the J_rB_z component of the J × B force causes azimuthal rotation. In order to optimize the experimental measurements of these effects using interferometry, B-dot probes, and Thomson scattering, various experimental conditions were used: different machine current polarities, different magnetic field directions, different magnetic field strengths, different foil materials, different pin diameters, and different heights at which to measure magnetic field compression. Additionally, using these various parameters permitted further validation of the XMHD PERSEUS code.

The radial foil geometry used for producing the laboratory jets is shown in Fig. 1(a). The Helmholtz coil external to the foil applied an axial field of up to 2 T in the region occupied by the jet. Pin diameters of 5 mm or 10 mm were used. The pin diameter changed the maximum B_θ at the pin radius and the time for plasma to converge on-axis and form the plasma jet. At 1 MA peak current, the maximum B_θ beneath the foil at the pin radius corresponded to 80 and 40 T for the 5 and 10 mm pin diameters, respectively. The approximate jet radii were of order 1 mm at the jet base near the foil, and the self-generated B_θ from the axial current in the plasma jet (est. at 50 kA peak) was 10 T maximum at the approximate 1 mm jet radius.

Figure 2 shows that, for both the experiments and simulations, reverse current polarity (anode pin) jets had larger on-axis areal density hollowing (≲10¹⁷ cm⁻² relative to jet areal densities ≳10¹⁸ cm⁻²), larger conical angles, and larger base widths compared to standard current polarity (cathode pin) jets. For reference, taking a jet thickness of ∼1 mm and an inner radius of ∼1 mm corresponds with on-axis densities ≲5 × 10¹⁷ cm⁻³ and jet densities ≳3 × 10¹⁸ cm⁻³. The polarity difference, seen in the experimental and XMHD results, is not seen in MHD simulations that lack the electron inertia and Hall effect terms. We note that some of the features in the experimental images in Fig. 2—most notably the horizontal oscillations for the 10 mm cathode pin (SP) case—are artifacts from the IDEA algorithm used to convert the interferogram fringes to areal density measurements. For the B_z compression experiments, 10 mm pins were used, and for the azimuthal rotation measurements, 5 mm pins were used. The qualitative agreement between experiments and the XMHD simulations provides confidence that we can compare the experiments and simulations for both pin sizes in order to help understand the jet dynamics with an applied B_z.

The 10 mm pin, reverse current polarity case produced the jet with the largest (radial extent) on-axis density hollowing and conical angles. Therefore, it was the best candidate to make a B_z compression measurement with minimal perturbation of the jet. Our goal was to place a B-dot probe on-axis such that it resulted in a minimal perturbation to the jet density, azimuthal symmetry, and plasma current flow.

In Fig. 4, we show the raw B-dot data vs. time for two differential measurements using double loop probes: (a) a probe with loop orientations parallel and anti-parallel to the z-axis and (b) a probe with loop orientations parallel to the z-axis and parallel to the r-axis. In Fig. 4(a), we see that the two oppositely oriented probes led to two opposite voltage signals. The sum of these two signals should be proportional to the electric pickup of the probe and the difference proportional to the magnetic pickup. We see that for the first ∼60 ns, there was only an electric field signal, negative because of the polarity of COBRA. After 60 ns, the two traces diverge indicating the start of the B_z compression. We see that the peak voltages relating to field compression occurred at approximately the same time (125 ns after the start of current rise) for both probes. We see that the voltages returned to their baseline values near 200 ns, implying no further B_z compression. However, at these times, the measurement of no voltage on the loops may have been the result of mechanisms causing an inaccurate B-dot measurement, as further discussed in Sec. V.

In Fig. 4(b), we see that the probe parallel to the z-axis gave a signal, while the probe parallel to the r-axis gave essentially no signal, and these combined signals confirm that we were measuring compression of the applied axial magnetic field rather than some non-axial (radial or azimuthal) component. We again mention that a misalignment of the probe loop parallel to the r-axis can produce a small component that may have measured B_z. The loops for the Fig. 4(a) graph were placed at a lower height of 6.5 ± 0.5 mm above the foil compared to the loops for the Fig. 4(b) graph at 11.5 ± 0.5 mm above the foil. Comparing Figs. 4(a) and 4(b), it can be seen that there was larger B_z compression closer to the foil surface where there was larger plasma thermal pressure and dynamical ram pressure and that B_z compression started earlier in time closer to the aluminum foil surface.

Figure 5 shows the integrated signals that give the magnitude of the B_z, which was initially at 1.0 ± 0.1 T, as applied by the external Helmholtz coil. The experiment results are the solid lines, and the simulation predictions are the dashed lines. Compared to the simulation results, the experimental results show a smaller maximum B_z with compression starting later in time. Additionally, the simulation shows a decrease in the compressed B_z at around 200 ns, while in the experiments we do not observe a decrease in the compressed field at least until after 300 ns. At this time after 200 ns in the experiment, possible mechanisms may be influencing the measurement reliability, as further discussed in Sec. V A. The larger compression seen in the simulation results implies that there is more inward pressure (e.g., dynamical ram pressure and thermal pressure) in the simulation than was present in the experiment. Additionally, the simulation did not include a radiation package, which could reduce this inward pressure. The overall discrepancy between the simulation and experimental results is greater at 11.5 mm than at 6.5 mm. Assuming the experimental results to be accurate, this suggests the simulations may be missing some physics in these low plasma density regions near the top of the jet.

For the titanium jets, we looked at the effects on field compression of varying the initial applied magnetic field strength. In Fig. 6, we plot the total B_z as a function of time for these titanium experiments and compare them to the aluminum simulations. Again, we note the version of the code used in the present work did not include a radiation package, which should be more important for titanium than for aluminum, and we did not run simulations for titanium because we did not include this radiation. As in Fig. 5, the aluminum simulations show the B_z compression starting at ∼30 ns for all the initial applied field strengths 0.7–1.8 T. The titanium experiments showed B_z compression starting later in time than the aluminum simulations. For both experiments and simulations, we see the same trend of a larger change of B_z with a smaller initial applied magnetic field strength. This effect was most pronounced in the experimental results for the 0.7 T initial field strength. The larger peak B_z for the initial 0.7 T field case corresponded with a larger peak magnetic pressure that was supported by a larger dynamical ram pressure or pinching force (pressure from the B_θ field from the jet current). Furthermore, a larger initial B_z means more magnetic pressure to resist inward radial motion.

Experimentally, the rise time and peak current of the COBRA pulse influenced the plasma jet development and the corresponding B_z compression. COBRA can produce short pulses (nominally 100 ns and 1.0 MA) and long pulses (nominally 200 ns and 0.9 MA) by adjusting the main switch pressures. In Fig. 7, long and short COBRA pulses are shown along with the magnetic compression traces for the jets they produced. The shorter pulse with the larger peak current produced a higher peak value of the compression than the longer pulse with the lower peak current.

In the preceding paragraphs, the extended magnetohydrodynamics simulations were 2D (r–z azimuthally symmetric) with a 67 μm grid spatial resolution. The foil was initialized as an aluminum plasma with Spitzer resistivity. In Fig. 8, we compare B_z compression using a 2D simulation with a 3D simulation that has a 100 μm grid spatial resolution and includes modeling of the aluminum equation of state (EOS) and conductivity from the solid through plasma states. Here, we use a pin diameter of 5 mm (rather than 10 mm) and look at a region on-axis 5.0 mm above the foil. Because of computational time, the 3 D simulation was stopped at 83.5 ns. The 3 D simulation shows ∼20 ns delay in the onset of the compression, more similar to the experimental results, but with a higher B-dot signal than the 2D or experimental results.

In order to study the azimuthal rotation of the jets, we performed Thomson scattering measurements on Al jets created using a 5 mm pin and reverse current polarity. This setup was chosen as it produced a jet that (1) needed a less spatial resolution than a standard polarity jet because it had more conical divergence and (2) had a better signal to noise ratio than a jet produced by a 10 mm pin because it had a larger electron density. Previous work observed rotation of the background plasma, outside of the jet, using optical spectroscopy but was not able to resolve the jet rotation.¹⁴ 

We analyzed the Thomson scattering data by finding the best fit between the scattered spectrum from each fiber to the spectral density function, S(k, ω).²¹ The ion flow velocity was determined from the Doppler shift of the entire scattered spectrum and, therefore, was not greatly dependent on the spectrum shape. In addition to the ion flow velocity, the other fitting parameters are ion temperature, electron temperature, and the relative electron-ion drift velocity. An electron density of 5 × 10¹⁸ cm⁻³ was estimated from interferometry images, and the ionization state was found from the FLYCHK tables²³ for Al. Error bars have been calculated for the ion flow velocity values presented in this paper by employing a Monte Carlo simulation method that uses the variance between the best fit S(k, ω) and an experimental spectrum to generate 100 synthetic spectra.²⁴ When finding the fit to each synthetic spectrum, certain parameters are allowed to vary as shown in Table I. The error bars are then found by finding the standard deviation of the fitting parameters for 100 simulated spectra.

As discussed in Sec. III C, it is possible to simultaneously measure v_r and v_θ by looking at two different scattering vectors in the same r–θ plane. Figure 9 shows the raw spectra from pairs of fibers near and inside the jet. Some of the fiber pairs have significant differences in their Doppler shifts, for example the pair located at –0.67 mm is indicative of the rotational velocity. Using (1) and the measured velocities from each of the fibers, v_r and v_θ can be calculated. Figure 10(a) shows these measured velocities for a typical shot in reverse polarity, at 160 ns and with a −0.9 ± 0.1 T B_z (applied axial magnetic field antiparallel to the jet propagation direction). Figure 10(b) shows the calculated v_r and v_θ. In this figure, and for all following Thomson scattering data, r = 0 is the center of the jet, r < 0 identifies the laser incident side, and r > 0 identifies the laser exiting side. For any location, $r̂$⁠, the radial unit vector, points outward from the center of the jet, and $θ̂$ points in a counter clockwise direction, meaning that v_r and v_θ should have the same sign when measurements are compared on the opposite sides of the jet. Fibers near the dense region of the jet showed an azimuthal velocity of 15–20 km/s. Confirmation of the jet rotation can be seen in Fig. 11, as the direction of rotation switched by changing either the sign of B_z or the current polarity. The rotation of the standard polarity jet dropped off faster with radius than the reverse polarity jet. This is not surprising as the standard polarity jet was significantly narrower (see Fig. 2). The sign of v_θ was consistent with the direction of the J_rB_z component of the J × B force that caused the rotation.

The measured ion flow velocities for reverse polarity shot 4236 are shown in Fig. 12. We see that in this shot, the jet was rotating at 15–20 km/s. It was moving up at a velocity of between 40 and 90 km/s. This axial velocity was small inside the cone of the jet, increased in the conical shell of the jet and then decreased with the increasing radius outside of the jet. In addition, there was a radial inflow outside the jet at 10–25 km/s. Two measurements of this radial velocity component were made, one by the fiber bundle in the r–z plane and the other by the two bundles in the r–θ plane, that were remarkably consistent.

Figure 13 shows the simulation results generated for the same time and location that the experimental data was taken, 160 ns into the current pulse and 5 mm above the foil. Key differences include that the simulated jet peak density is at a larger radius (1.4 mm compared to 1.1 mm for shot 4236), and that the simulation has significant velocity in the low density hollow region of the jet. The peak azimuthal ion velocity of 82 km/s for the simulation occurs at a very low density region of plasma (∼10¹⁶ cm⁻³, approximately two orders of magnitude lower than the peak jet density) at a radius of 0.3 mm. The z-axis velocity also increases when getting close to the axis in simulation while experiments showed this velocity to be decreasing. The experiment and the simulation do show general agreements within the dense region and outside of the jet. In the simulation, the azimuthal ion velocity at the jet density peak is 14 km/s, the z-axis upward velocity is 60–70 km/s within and outside of the jet, and the radial inflow outside of the jet is 20 km/s, all comparable to the experimentally measured velocities.

Assuming these velocity peaks in the low density region in the simulation are physical, the experiments may not be able to resolve this low density region because the Thomson signal was so small. Since the power scattered is proportional to the electron density, ∼5 × 10¹⁶ cm⁻³ low density region should produce a 100 times less signal than that scattered by the peak electron density of the jet, ∼5 × 10¹⁸ cm⁻³. The simulation suggests the width of the low density region rotating at ≥40 km/s to be 500 μm (between r = 0.2 mm and r = 0.7 mm. This was not seen in the experimental data. Though this width should have been resolvable within our 350 μm field of view, if the region was any smaller than the simulation prediction it would have been hard to see because any scattering from more dense plasma would have dominated the signal.

On-axis compression of the applied axial magnetic field requires an azimuthal current to support the change in B_z across the plasma jet boundary. This current is a result of the relative electron-ion drift velocity. Thomson scattering measurements of the relative electron-ion drift velocity are based upon the intensity difference of the ion acoustic feature peaks. The difference between these peak intensities, however, is significantly dependent on the ratios between the relative electron-ion drift velocity and both the ion and electron thermal velocities. Although the measurement of the electron temperature can be performed relatively easily from peak separations, the ion temperature measurement is much more difficult. The ion temperature largely influences the peak broadening, but other more complicated factors, such as collisions,²⁵ velocity gradients,²⁴ and turbulence, also influence the broadening, so accurate measurements of ion temperature are challenging. In addition, peak intensities can vary due to finite optical effects creating the presence of multiple k_L and k_s vectors,²⁴ though this should have little effect here as the plasma jets had relatively low velocities. Consequently we were not able to obtain reliable values for the relative electron-ion drift velocities in our jets from the Thomson scattering measurements. However, experimental observations of the magnetic field compression and the jet rotation strongly indicate the presence of this azimuthal electrical current layer. Simulations predict the magnitude of the azimuthal relative electron-ion drift velocity to be on the order of 10–20 km/s in the dense region of the jet in reverse polarity, in agreement with the magnetic compression observed in the simulations. This probably gives an upper limit on the relative electron-ion drift velocities in our experiments because the magnetic compression was larger in the simulations than was observed in the experiments.

In this section, various issues of the experimental and simulation results will be presented. In general, there is a good agreement between the two. There are, however, some shortcomings of the experimental diagnostics and the computer code that should be pointed out.

There are a number of issues that must be considered in order to take meaningful data from the magnetic probe measurements. One is the formation of plasma around the tip of the probe. If this plasma becomes hot and conductive enough, the plasma can shield the loop and cause the loop to become insensitive to further magnetic field changes. This plasma can be formed by extreme ultraviolet (EUV) photoionization of the epoxy and Kapton surfaces. Early in the current pulse, this plasma was seen on our probe tip surfaces with interferometry but with no observable EUV self-emission until approximately 200 ns into the power pulse. As can be seen in Fig. 4(a), the two loops picked up changing magnetic field signals until approximately 200 ns, at which time the conductivity of the surface plasma may have been high enough to cause the loops to be insensitive to subsequent changes in B_z.

However, the plasma environment in the present experimental work was more benign than in other situations in which these B-dot probes have continued to reliably work.²⁰ If the flat B-dot signal after approximately 200 ns was not a failure of the probe measurement, then the B_z in the probe loop region was not changing very fast. In this case, the plasma configuration must also be changing very slowly. At this time in the current pulse, the foil driving voltage was going to zero and reversing. This feature of the current pulse could result in resistive decay of currents above the foil. If the resistive decay time is relatively long, the plasma dynamics and magnetic fields will change very slowly. This situation is an open issue, and understanding this slow change in the magnetic field is the subject of further study.

Another problem can occur if the tip or any part of the magnetic probe arcs to a high voltage surface (100 s kV) of the pulsed power machine. This connection produces a large voltage spike on the probe signal and is a clear indication of its failure. A similar voltage spike and failure can occur if energetic electrons from the power feed or load region strike the probe tip. None of these effects are seen in the traces presented in this paper.

The development of a shock around the probe tip also presents problems. When this happens, because of the jump conditions across the shock front, the magnetic field seen by the loop of the probe can be different from the ambient field. If a shock is strong enough, it can be seen using laser backlighting shadowgraphy. But even if the shock is weak and not visible to shadowgraphy, it can pose a problem for probe measurements. For our experiments, when a probe was placed in the high density flow of the jet, a shock was generated and could be seen in backlighting images. For this reason, to make the magnetic compression measurements, we placed the probe tip in the low density region on the z-axis and used magnetized jets with large cone angles.

There is always the question of the alignment of the loop(s) in order to be sure that only the desired component of the magnetic field is being measured and that the signal is not the result of the electric field pick up. When this was a concern, we used two types of double loop probes. For the first type, the two loops picked up the same component of the magnetic field but were counter wound. The signals from these two loops were subtracted so that any magnetic pickup doubled and any electric pickup canceled. In the second type, the two loops were aligned to pick up orthogonal components of the magnetic field. These independent signals would be a measure of the ability of a single loop to pick up the magnitude of the compressed field. A related problem occurs if a large current is carried on the outer conductor or in plasma on the outer insulation of the probe. This current can create unwanted components of the magnetic field that will be picked up by the loop. This problem can be minimized by proper inductive isolation of the probe.

With the proper construction of the B-dot probes and interpretation of the signals they produce, these problems can be minimized, and this B-dot probe technique for measuring magnetic fields is simple and very effective. One other measurement method is Faraday rotation from either the result of the birefringence of the magnetized plasma or an optical fiber.^26–28 For our purposes, because the density on axis was so small, the optical fiber method would probably be the best. Another method would be to use spectroscopy and look at Zeeman splitting or Zeeman broadening.^29–31 While these last two techniques have been used for magnetic field measurements, they are not as simple to implement as the B-dot probe approach with regard to decoupling the B_z and B_θ components.

Thomson scattering can be an extremely useful method for measuring a large number of plasma parameters like ion flow velocities, ion temperatures, electron temperatures, and relative electron-ion drift velocities. Our measurements were made in the collective regime with scattering from ion acoustic waves. While we were able to get reliable measurements of the ion flow velocities, there were problems with the electron temperature, ion temperature, and relative electron-ion drift velocity measurements. The electron temperature measurement was complicated by the fact that the laser was observed to heat the electrons, although this heating did not affect the velocity measurements. The ion temperature and electron-ion drift velocity measurements were uncertain because the features of S(k, ω) that give these parameters are also affected by uncertain plasma parameters like, collisionality, velocity gradients, and turbulence.

Experimental Thomson scattering measurements were not able to confirm the high azimuthal velocities of the jet in the low density region near the z-axis. Experimental measurements in this region were difficult because of the low signal to noise ratio and the size of the scattering volume. Both of these problems could be addressed by establishing a tighter focus for the laser beam and the collection optics. A tighter focus for the laser and collection optics will clearly improve the signal to noise ratio. The tighter focus for the collection optics will improve the spatial resolution which is particularly important for measurements near the z-axis, where the radial and azimuthal velocity components are changing rapidly with the position.

There was a general agreement between the computer simulations and experimental observations of the plasma jet dynamics. It was shown that the XMHD code, PERSEUS, that includes the electron inertia and Hall effect terms gives a better agreement between the experiments and simulations than a MHD code. However, there still are discrepancies between the PERSEUS simulation and our experimental results. For example, the onset of magnetic compression [Fig. 5(a)], the amount of magnetic compression [Figs. 5(a) and 5(b)], and the increase of the discrepancy with height above the foil [Fig. 5(b)] are examples of a noticeable difference between the experimental and computational results.

In addition to the Hall and electron inertia terms, additional physics is being added to PERSEUS with the idea of improving the agreement with the experimental results. Most of the simulation results presented in this paper were generated with a 2D version of PERSEUS that initialized the radial foil as a 1 eV plasma with Spitzer conductivity and a plasma equation of state (EOS) with a selected adiabatic index. As shown in Fig. 8, on going to 3D and a “cold start” for the foil using EOS data, the onset of magnetic compression is delayed, in better agreement with the experiment. It is likely the most important improvements to the physics in the code are in the EOS and conductivity models. In addition to 3D and a cold start capability, there are plans for additional physics to be added to PERSEUS. A radiation package and the inclusion of the Nernst effect are being worked on. The radiation package should be able to show differences with foil materials with different radiation properties like aluminum vs. titanium. Including the Nernst effect should result in more accurate modeling of the electron motion and, by extension, advection of the magnetic field in the dense surface plasma ablating from the foil. Additionally, there may be effects from the interface between the numerical floor density and the denser plasma related to what plasma density will advect and compress the B_z.

The effects of an applied magnetic field on laboratory plasma jets have been studied experimentally and the results have been compared to numerical simulations using PERSEUS, an extended magnetohydrodynamics code. On application of an axial magnetic field, on axis density hollowing was observed and the jet took on a conical configuration. These effects are seen both in the experiments and in the computer simulations. In both the experiment and simulation, this structure changed when the polarity of the current driver and pin diameter were changed. The formation of the jet was observed to compress the applied magnetic field and, during this formation and compression, the jet picked up angular momentum. This rotation was observed using Thomson scattering measurements of the ion flow velocity and was found to be in the J × B direction, again, in good agreement with the simulations. There are some areas where the experimental results and simulations are not in total agreement. For example, the experiments showed very little rotation near the axis of a magnetized jet, while the simulations show large rotation in this low density region [Fig. 10(b) vs. Fig. 13]. If this rotation is actually present, the problem may have been with the Thomson scattering measurements near the axis where the scattering signal was weak due to the low density, and there was the above mentioned problem with resolving ion flow velocity components near the cylindrical axis.

The current that must be present in the high density conical shell of the jet to support the ΔB_z associated with the magnetic compression is present in the simulation results. Simulations show that this current layer is moving in the direction of the J × B rotation for standard polarity and counter to the J × B rotation in reverse polarity. Experimentally, this current could be given by a measurement of the relative electron-ion drift velocity and the details of the Thomson scattered spectrum. Our scattered spectra did not reliably measure the relative electron-ion drift velocity. However, the presence of this current layer was indirectly confirmed in the experiments from the measured values of the applied B_z on axis, before and after compression.

The authors thank T. Blanchard, D. Hawkes, and H. Wilhelm for hardware fabrication and COBRA operation. This research was supported by the NNSA Stewardship Sciences Academic Programs under DOE Cooperative Agreement No. DE-NA0001836 and NSF Grant No. PHY-1102471.