6.5 Structure and Dynamics of Pre-Ionized Jets
6.5.2 Radial Jet Structure
IV line that has minimal Stark broadening.
A surprising feature of the emission spectrum at the time of the dense pinch was a sudden increase in the continuum brightness, previously undetectable by our spectroscopy system, by a factor of 100 or more (see Fig. 6.17). The line profile shown in the right panel of Fig. 6.6 is one example of this effect; unlike the constant offset in the profile fits due to re-absorption discussed above, this was a true continuum, visible across all wavelengths in the full spectrum. The most obvious source of continuum radiation from the jet was thermal Bremsstrahlung; the emissivity in W/m3 is [138, p. 162]
f f = 1.5×10−38
TeV1/2neniZ2, (6.25)
whereZeis the typical ion charge. Assumingne= 1×1023m−3,ni = 5×1022m−3,Z = 2, T = 5 eV, andV = 10−5 m3 for the dense portion of the jet, we find a total radiated power Pf f = 7×103 W. Based on measurements by Perkins [67, Figure 4.2] of visible radiated power exceeding 105 W from hydrogen plasmas withne≈1021m−3 in the Caltech Arched Flux Rope Experiment, we expect that the visible radiated power from line emission in the denser argon jet plasmas was Pvis. & 106 W. Therefore, the thermal Bremsstrahlung might have been too small to be detectable by comparison. Non-thermal Bremsstrahlung arising from the sudden deceleration of the pinching plasma at the jet axis is an intriguing possibility, but further measurements and analysis are needed to determine whether it could reasonably produce sufficient optical power.
2.3 μs 2.8 μs 3.3 μs 3.8 μs
5.8 μs 5.3 μs
4.8 μs 4.3 μs
Figure 6.18: Imacon camera images of a pre-ionized argon jet formed with Vmain = 4 kV, Vbias= 80 V,Vgas,RF = 550 V,tgas,RF =−6.75 ms,Vgas,inner = 0 V, andVgas,outer = 709 V.
The camera gain was set higher for this image than for most others shown in this chapter, so much of the jet is saturated, obscuring some features but highlighting others.
plasma axially toward the region where the flux tube radius is large, and the local radial force balance condition leads to an axial pressure gradient that drives plasma in the same direction. Toroidal magnetic flux that is frozen into the plasma is convected along with the flow. If the flow stagnates for some reason, then this flux will pile up, increasing the radial MHD pinch force (see Fig. 1.1), which decreases the radius of the wide part of the flux tube until it is collimated. This MHD pumping process has been studied extensively in the Bellan plasma lab, where the ingestion of plasma by flared flux tubes has been tracked using spectroscopy [130] and color-coded imaging [40], and it may be responsible for the observed collimation of astrophysical jets as well.
A related and surprisingly subtle issue is what determines the final jet radius. Consider an isothermal axisymmetric cylinder of plasma in radial equilibrium, with current flowing only in the axial (ˆz) direction. Setting∂P/∂r=−JzBφ(see Eq. 1.10) and integrating from r= 0 to r =a, we can derive the Bennett relation [4, p. 317]:
I2= 8πN kBT µ0
, (6.26)
where N = Ra
0 (ni+ne) 2πrdr is the total number of particles per unit length. Eq. 6.26
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r (cm)
Enclosed Poloidal Current (kA)
z = 3.0 cm z = 5.0 cm z = 7.0 cm z = 11.0 cm t = 2.3 µs
z = 3.0 cm
z = 11.0 cm
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z = 11.0 cm
z = 3.0 cm
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z = 3.0 cm z = 5.0 cm z = 7.0 cm z = 11.0 cm t = 5.3 µs
z = 3.0 cm
z = 11.0 cm
Figure 6.19: Poloidal current profile I(r) for pre-ionized argon jets formed with Vmain = 4 kV, Vbias = 80 V, Vgas,RF = 550 V, Vgas,inner = 460 V, and Vgas,outer = 750 V. As in Fig. 6.5, axisymmetry was assumed in order to calculate I(r) from the measuredBtor.(r).
This assumption may have started to break down byt= 5.3µs, when many of the jets had already begun to kink.
specifies the axial current I needed to confine N particles per unit length at temperature T, but it places no restriction on the details of the current profile Jz(r), nor on the radius of the plasma column. It is clear that a collimated jet that is in radial force balance will not expand or contract its radius, but the choice of that equilibrium radius must depend on the boundary conditions or on the past history of the jet evolution.
To better understand the radial structure of the pre-ionized jets, the magnetic probe array was used to measure the axial current distribution as a function of radius at four different axial locations. Results for jets with Vgas,RF = 550 V, Vgas,inner = 460 V, and Vgas,outer = 750 V are shown in Fig. 6.19. The jets were highly flared at the probe cluster locations at early times (the current channel radius was larger further from the electrodes),
but at later times collimation was evident once the flared jet front had moved past the clusters at z= 3.0–7.0 cm. The current channel radius, defined as the radius inside which 70% of the total poloidal current was enclosed, was acur. ∼ 3 cm near the electrodes.
This was much larger than the final equilibrium visible jet radius, avis. ≈ 0.5 cm (see Figs. 6.14 and 6.18). A similar but proportionally smaller discrepancy was observed for hydrogen jets formed without pre-ionization (see Fig. 6.5)—the visible radius of these jets was avis. ≈2–3 cm, while the current channel radius near the electrodes was acur. ∼5 cm.
There must have been some plasma in the outer portion of these jets in order to carry the measured current, but its density was low enough that the visible brightness of the outer envelope was negligible compared to the brightness of the dense center of the jets.
If we assume that the current profile Jz(r) in the pre-ionized argon jets was uniform (Fig. 6.19 suggests that it was actually somewhat centrally peaked) and that the electron density in the outer envelope of the jet wasne= 1021m−3, then the electron drift velocity necessary to carry the measured current (I ∼25 kA) was
ue= Jz
nee = I
πa2cur.nee ≈6×104 m/s. (6.27) This is a physically plausible fluid velocity (as discussed in Sec. I.1, kinetic instabilities would have been expected to develop for ue&vT e∼106 m/s). Since the brightness of the fully ionized jets scaled approximately asneni [28], it is likely that a region of the plasma with ne = 1021 m−3 would have indeed been invisible next to the central column, which had ne1022m−3 (see the left panel of Fig. 6.16).
Now consider the radial force balance requirement in the outer part of the jet. A uniform axial current density leads to a parabolic pressure profile, P(r) =P0
1−(r/a)2
(we are temporarily ignoring the dense central region of the jet in order to roughly analyze the outer region), so ∂P/∂r=−2P0r/a2. The radial pinch force is (J×B)r=−(J) (µ0I(r)/2πr) =
−µ0J2r/2. Therefore, in equilibrium we have
2P0r
a2 = µ0J2r
2 (6.28)
P0= (ne+ni)kT = µ0J2a2
4 = µ0I2
4π2a2. (6.29)
With ne+ni = 1.5×1021 m−3, I = 25 kA, and a= 3 cm, Eq. 6.29 implies T ≈ 90 eV.
This is an unrealistically high temperature, so it appears that radial force balance could not have been satisfied with the calculatedJ×B force. The density would have needed to be an order of magnitude higher in order for Eq. 6.29 to give a plausible temperature, but then the plasma at 0.5 cm< r <3 cm surely would have been visible.
The solution to this dilemma is to conclude that the magnetic field in the outer region of the jet must have been force free—i.e., it had a helical structure such that Jk B. This interpretation was previously put forward by Kumar [134, p.83] to explain the aforemen- tioned mismatch between the visible radius and current channel radius in non-pre-ionized hydrogen jets. It is well known that MHD plasmas with β ≡ 2µ0P/B2 1, including spheromaks and many astrophysical plasmas, naturally evolve toward a force-free state [4, p. 394]. However, it is not clear why a force-free field structure would have developed in the outer region of the jet, rather than the whole current channel pinching down to form a high density plasma column withβtor.∼1. An important project for future research is to make detailed measurements of bothBφ(r) and Bz(r) in order to confirm the existence of a force-free field geometry in both pre-ionized and non-pre-ionized jets.
We may repeat the radial force balance calculation for the dense central part of the jet. Estimating from Fig. 6.19 that the current within the visible jet (avis. ≈0.5 cm) was I = 4 kA and setting ne+ni = 5×1022 m−3, we find that Eq. 6.29 required T = 2.5 eV, which agrees well with the rough estimates in Table 6.1.
The analysis thus far has ignored the bias magnetic field, which exerted a radially outward force FB = −∂/∂r B2z/2µ0
when it was compressed by the pinching of the jet.
Because of this magnetic pressure force, which added to the thermal pressure force−∂P/∂r, applying a stronger bias field was expected to impede the pinching behavior. The measured current channel radius acur. and the minimum visible jet radius avis. during the pinch are shown as a function ofVbias in Fig. 6.20. Large values ofVbiasled to a measurable increase in the minimumavis., as expected, but the effect was too weak to be resolved atVbias ≤120 V, and no clear trend was visible in the current channel radii.
Increasing the initial neutral gas density (and consequently the initial plasma density, since the jets were nearly fully ionized) was likewise expected to cause the minimum radius
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Minimum visible jet radius (cm)
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Current channel radius (cm) z = 3.0 cm
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Figure 6.20: Left: Minimum visible jet radius avis. during the bright pinch as a function of bias field strength. The radius was the defined as half of the visible plasma column width, measured by identifying the locations at which the brightness had fallen to 10% of the central peak value in unsaturated Imacon camera images. It was averaged over an axial length of ∼0.5 cm near the electrodes. Right: Current channel radiusacur., defined as the location inside which 70% of the total poloidal current flowed, as a function of bias field strength att= 3.5µs, roughly the time at which the visible plasma column radius typically reached its minimum.
attained by the pinched jet to increase: for a given initial temperature, a plasma with high initial density would not need to be compressed as much in order for the thermal pressure to balance the magnetic pressure. However, Fig. 6.21 shows that neitheravis. noracur.became larger asVgas,RF was increased (in fact, the smallest visible pinch radius of all was obtained with Vgas,RF = 700 V). This surprising result implies that the extreme narrowness of the pre-ionized jets was probably not caused by the low level of neutral gas input enabled by pre-ionization. Instead, the jet morphology differed from that of non-pre-ionized jets such as the one shown in Fig. 6.1 because the gas from which the jet formed was injected at a different location—along thezaxis rather than atr = 4.8 cm. The non-pre-ionized jets also started out of equilibrium and pinched down to a smaller radius, but the final equilibrium values of avis. and acur. were much larger than those for pre-ionized jets. One possible explanation for the difference is that the non-pre-ionized jets were composed of 8 individual
“spider leg” loops (see Fig. 1.5) that had to merge along the z-axis in order for the jet radius to pinch down. In the limiting case of perfectly conducting, flux conserving plasma, this merging would not occur—once the flux tubes got close together, currents induced on
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Minimum visible jet radius (cm)
likely overestimated due to saturated image
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Current channel radius (cm) z = 3.0 cm
z = 5.0 cm z = 7.0 cm t = 3.5 µs
Figure 6.21: Left: Minimum visible jet radius avis. during the bright pinch as a function of RF source fast gas valve voltage. Right: Current channel radius as a function ofVgas,RF at t = 3.5 µs, roughly the time at which the visible plasma column radius typically reached its minimum.
their surfaces would cancel out the attractive force from the parallel poloidal currents. The actual spider legs had finite resistivity and thus could undergo magnetic reconnection and merge into a central jet column, but incomplete merging may have limited the pinching of the jet radius.