the more important effect causing the inductance of these low mass jets to be higher was probably the pinch to a very narrow jet radius that occurred att∼2–4µs (see Sec. 6.5.1).
of additional effort by working with the poloidal flux functionψ(r, z), as was done by Kumar and Bellan [39]; however, these authors ultimately showed thatBpol.had a negligible affect on the jet velocity for values of I and ψ relevant for the Caltech jet experiments, so the simplest possible model will be presented here.
In equilibrium, ther component of the MHD equation of motion (Eq. 1.10) is
∂P
∂r = (Jpol.×Btor.)·rˆ=− µ0
(2πr)2
∂
∂r I2
2
=−µ0I02r
2π2a4. (6.18) Integrating and applying the boundary condition P(r=a, z) = 0, we find
P(r, z) = µ0I02
(2πa(z))2 1− r
a(z) 2!
. (6.19)
We will approximate the jet as a steady state flow, in which case the axial component of the MHD equation of motion is
ρ(U· ∇U)·zˆ=ρ
Uz∂Uz
∂z +Ur∂Uz
∂r
= (Jpol.×Btor.)·zˆ−∂P
∂z. (6.20) Consider the situation along the jet axis (r = 0). HereBtor.= 0, so the jet motion must be driven entirely by the gradient in thermal pressure:
ρ(0, z) ∂
∂z
(Uz(0, z))2 2
!
=−∂P(0, z)
∂z =− ∂
∂z
µ0I02 (2πa(z))2
=− µ0I02 (2πa0)2
∂
∂z e−2κz . (6.21) Assuming that the jet is isothermal in the axial direction, soρ(0, z)∝P(0, z)∝1/a(z)2∝ e−2κz, this becomes
ρ0e−2κz ∂
∂z Uz2
= µ0I02κ
π2a20 e−2κz, (6.22)
whereρ0≡ρ(0,0). Integrating from z= 0 to z=Land using Uz(0,0)≈0, we find
Uz(0, L)≈ s
µ0I02κL ρ0π2a20 =
s Bφa2
0
µ0ρ0
4κL, (6.23)
where Bφa0 ≡ Bφ(a0,0) = µ0I0/2πa0. So indeed, the jet velocity scales as Bφ/√ µ0ρ as predicted by the simple energy balance argument, but this quantity is not really the local
1.3 µs 2.8 µs 4.3 µs 5.8 µs
Figure 6.10: Example of a jet velocity measurement using Imacon camera images. The jet shown was created with Vmain = 4 kV, Vgas,outer = 709 V, and Vgas,inner = 460 V. Since the jet front was not sharp, the identification of its location at each time, marked by the dashed lines, was somewhat subjective. Because the vacuum chamber walls were highly reflective, it was easiest to pinpoint the front location when the jet was passing in front of the window. The window was also useful for providing a sense of scale—with knowledge of its actual radius and of the distance from the camera to the jet and to the window, the distance in the plane of jet propagation corresponding to one horizontal pixel on the images could be calculated. We estimate that the uncertainty in the jet velocity measurement was about ±20%.
toroidal Alfv´en velocityvAφ, since it is evaluated usingρ measured at the jet center butBφ
measured at the jet edge.
6.4.2 Jet Velocity Measurements
The velocities of the MHD-driven jets were measured from fast camera movies: the Imacon camera was aligned perpendicular to the direction of jet propagation, and the jet front location at each time was identified, as illustrated in Fig. 6.10. When the experiment was operated without pre-ionization, with roughly the same amount of argon gas input as was typically used on the old version of the jet experiment, a fully collimated jet formed only after the gun current had already peaked and begun to decrease (compare Figs. 6.1 and 6.9)9. However, even at times before the plasma had fully evolved into a jet, it was possible to define an expansion velocity by tracking the leading edge of the spider leg loops as they were driven to expand and merge by the MHD hoop force.
The left panel of Fig. 6.11 shows the jet velocity as a function ofVgas,inner, withVgas,outer and Vgas,RF held fixed at 709 V and 550 V, respectively10. There was a modest trend
9The old jet experiment was able to follow the evolution of argon jets over a longer time interval because its main capacitor bank contained two 59µF capacitors in parallel, compared to a single 59µF capacitor in the experiment described here, and there was also a pulse forming network [43] available to further extend the current pulse duration when necessary.
10Values ofVgas,RFbelow 550 V were also tested, but the jet velocity did not measurably increase compared
400 450 500 550 600 650 700 750 0
5 10 15 20 25 30
Inner Electrode Fast Gas Valve Voltage (V)
Jet Velocity (km/s)
Lower limit for gas breakdown without pre−ionization
400 450 500 550 600 650 700 750
0 0.5 1 1.5 2 2.5 3
Inner Electrode Fast Gas Valve Voltage (V) vjet/ Igun,peak(arb. units)
Lower limit for gas breakdown without pre−ionization 21.3 km/s
13.4 km/s
Figure 6.11: Left: Jet velocity as a function of Vgas,inner measured from Imacon camera images with Vmain = 4 kV, Vbias = 80 V, Vgas,outer = 709 V, and Vgas,RF = 550 V. Each data point was the average of two shots. Right: Velocities from the same shots normalized to the peak gun current for each shot.
toward faster jet velocities asVgas,inner was decreased. By enabling plasma breakdown with Vgas,inner<575 V, the pre-ionization system allowed for the creation of faster jets than was previously possible withVmain= 4 kV.
In the right panel of Fig. 6.11, the jet velocities are plotted again, this time normalized to the peak discharge current for each shot, which was smaller for lower Vgas,inner (see Fig.
6.9). Let N be the ion density per unit length in the jet, so ρ0 ∝ N0/πa20. Eq. 6.23 then yields
Uz(0, L)∝ √I0
N0
√
κL. (6.24)
The jets became more collimated as they lengthened, such thatκLremained approximately constant. Therefore,vjet/Igun,peak was proportional toN0−1/2. Neglecting small differences in the lengths of the pre-ionized jets,N0was proportional to the total jet mass, so comparing the normalized jet velocities atVgas,inner= 575 V andVgas,inner = 425 V, we may infer that pre-ionization allowed the jet mass to be decreased by a factor of (2.6/1.5)2 ≈3.
0 1 2 3 4 5 6 7
−1 0 1 2 3 4 5 6
Time after breakdown (µs)
[Ar III 351.14 nm] / [Ar II 349.13 nm]
No pre−ionization Vgas,RF= 700 V Vgas,RF= 650 V Vgas,RF= 600 V Vgas,RF= 550 V Vgas,RF= 525 V
0 1 2 3 4 5 6 7
10−1 100 101
Time after breakdown (µs)
[Ar IV 291.30 nm] / [Ar II 294.29 nm]
No pre−ionization Vgas,RF= 700 V Vgas,RF= 650 V Vgas,RF= 600 V Vgas,RF= 550 V Vgas,RF= 525 V
Figure 6.12: Global emission line ratios measured with the optical fiber viewing the jet directly, without a lens. The ICCD exposure time was 1µs. Left: [Ar III 351.14 nm] / [Ar II 349.13 nm] intensity ratio, plotted with a linear y-axis scale. Right: [Ar IV 291.30 nm] / [Ar II 294.29 nm] intensity ratio, plotted with a logarithmic y-axis scale. Vgas,outer = 750 V was used for all shots. The shots labeled “No pre-ionization” had Vgas,inner = 700 V and Vgas,RF = 0 V, while the rest of the shots were pre-ionized (RF amplifier on) and had Vgas,inner= 0 V and variableVgas,RF (values labeled in the figure legend).
6.4.3 Jet Temperature and Ionization Balance
A coarse measure of the mean electron temperature and ionization balance in the jets was obtained by measuring global levels of Ar II, Ar III, and Ar IV emission as a function of time. The [Ar III 351.14 nm] / [Ar II 349.13 nm] and [Ar IV 291.30 nm] / [Ar II 294.29 nm] line intensity ratios are shown in Fig. 6.12 for pre-ionized jets with varying levels of gas input, as well as for jets formed without pre-ionization. The relative level of emission from the Ar III and Ar IV lines was significantly higher for the pre-ionized jets than it was for the non-pre-ionized jets, suggesting that the pre-ionized jets had a higher mean ionization state or higher electron temperature (probably both).
As discussed in Sec. 6.1.5.2, numerical values for Te and the relative populations of the different ionization stages generally cannot be derived from measured line ratios without complex models. Table 6.1 shows the electron temperatures that would be implied by the data if LTE were a valid approximation—although the Te values shown may not be
to the case with Vgas,RF = 550 V. As expected from the fast gas valve output measurements shown in Fig. 5.2, very little gas was released through the inner electrode gas inlets whenVgas,inner was 400 V or less, and the jets created with these low inner gas voltages were indistinguishable from those formed with Vgas,inner= 0 V.
Ratio \ ne 5×1021m−3 2×1022m−3 1×1023m−3 Pre-Ionized Jets
Ar III / Ar II 2.4 eV 2.7 eV 3.1 eV
Ar IV / Ar II 2.6 eV 2.9 eV 3.2 eV
Jets with No Pre-Ionization
Ar IV / Ar II 2.4 eV 2.6 eV 2.9 eV
Table 6.1: Electron temperatures calculated from measured emission line ratios by assuming LTE, for three different electron densities. LTE was not actually a valid approximation for the jets, so these values should be considered to be rough lower limits on Te rather than accurate measurements—see the discussion in the text. The ratios considered were [Ar IV 291.30 nm] / [Ar II 294.29 nm] (typical values: ∼5 for pre-ionized jets and ∼0.3 for non- pre-ionized jets, from Fig. 6.12) and [Ar III 350.36 nm] / [Ar II 347.67 nm] (typical value:
∼10 for pre-ionized jets, from Fig. 6.15). The NIST Saha-Boltzmann emission calculator [73] was used to construct this table.
accurate, the table is useful for demonstrating the extremely strong dependence of the equilibrium ionization balance on temperature. Because of this strong dependence, the order of magnitude increase in the [Ar IV] / [Ar II] emission ratio that was obtained with pre-ionization could have corresponded to Te being hotter by only a few tenths of an eV.
In fact, any LTE plasma in the relevant density range that has appreciable emission from both Ar II and Ar IV ions must have Te ∼2.5–3 eV. It must be stressed that we cannot actually conclude that the temperatures of the jets fell in this narrow range—Te may have been much hotter given that the dynamical timescale of the experiment was shorter than the time required for the plasma to reach ionization equilibrium11 (this will be discussed further in Sec. 6.5.1).
Increasing the amount of gas input through the RF source tube decreased the [Ar IV 291.30 nm] / [Ar II 294.29 nm] line ratio shown in the right panel of Fig. 6.12, suggesting that higher mass plasmas were cooler and had a lower mean ionization state, as expected.
However, even for the highest gas voltage tested (Vgas,RF = 700 V), the line ratio was roughly 10 times higher than the corresponding ratio in the “No pre-ionization” case, which
11The relationship between the temperature and ionization balance in a steady state laboratory discharge can also be modified by losses of ions to the walls or electrodes [75]. As one example of how using the Saha equation can be misleading, consider the RF discharge equilibrium discussed in Ch. 3. A typical measured value of the [Ar II 434.8 nm] / [Ar I 696.5 nm] line ratio was∼0.3 (see Figs. 3.14, 3.15, and 3.16), which forne≈5×1019m−3 givesTe≈0.9 eV if LTE is assumed. However, detailed modeling of the collisional, radiative, and diffusive processes showed thatTe was actually 2–3 eV (see Table 3.2).
3.7 µs Outline of inner 3.7 µs 3.7 µs 3.7 µs
electrode
gain = 5 gain = 4 gain = 3
gain = 6
Figure 6.13: Imacon camera images of pre-ionized argon jets with successively decreasing ICCD gains, demonstrating that the narrow collimated structure near the base of the jet was many times brighter than the front of the jet. The experiment parameters were: Vmain = 4 kV, Vbias = 80 V, Vgas,RF = 550 V, Vgas,inner = 460 V, andVgas,outer = 750 V.
had Vgas,RF = 0 V and Vgas,inner = 700 V. Although the gas bank voltages were the same in both cases, the initial gas pressure in front of the inner electrode may have been different because the gas delivery piping was not identical for the “inner” and “RF” gas valves12, and the inner valve was triggered earlier relative to the gas travel distance to the electrodes than the RF gas valve was (see Table 1.2). Nevertheless, it appears likely that the pre- ionized jets were in a fundamentally different regime, independent of the amount of gas input. This conclusion is supported by images of the plasma structure, which showed that the pre-ionized jets created with Vgas,inner = 0 V always pinched down to a very narrow radius that was approximately independent of Vgas,RF (see Fig. 6.21). This behavior will be discussed in the following sections.