PLASMA ADIABATICITY IN
A DIVERGING MAGNETIC NOZZLE
J. P. Sheehan and Benjamin W. Longmier University of Michigan
Edgar A. Bering University of Houston
Christopher S. Olsen, Jared P. Squire, Mark D. Carter,
Franklin R. Chang Díaz, Timothy W. Glover,
Andrew V. Ilin, and Leonard D. Cassady Ad Astra Rocket Company
ABSTRACT
We propose a fluid model for ambipolar ion acceleration in a magnetic
nozzle that preserves the adiabaticity of the plasma. This adiabatic theory
predicts that the change in average electron energy depends linearly on
the change in plasma potential, providing an important design metric for
electric propulsion devices which employ magnetic nozzles. The fluid
theory predictions were compared to measurements made in the VASIMR
VX-200 experiment which has conditions conducive to ambipolar ion
acceleration. A planar Langmuir probe was used to measure the plasma
potential, electron density, and electron temperature for a range of mass
flow rates (50 – 140 mg / s) and power levels (12 – 30 kW). The linear
relationship between electron temperature and plasma potential was
observed as predicted. The adiabatic theory relies on collisions to
rethermalize the electrons and establish a temperature gradient. Coulomb
collisions cannot account for the high collisionality but an ion acoustic
instability may enhance the collision frequency.
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HELICONS AND MAGNETIC NOZZLES FOR
APPLICATIONS IN PROCESSING AND PROPULSION
Helicons
Radio frequency
High ionizing efficiency
Electron heating
Magnetic nozzle
Functions like physical
nozzle
Accelerates ions
Converts thermal energy
into directed kinetic
energy
Applications
Materials processing
Electric propulsion
CHI KUNG
VASIMR
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CURRENT FREE DOUBLE LAYERS
IN HELICON EXPERIMENTS
Narrow layer (10s of λd) of large
potential jump (several Te/e)
Isothermal
Occurs downstream of nozzle
Current free, expanding
Accelerates ions
May be thrust mechanism in helicon
thrusters
Open question of how/why current
free double layers form
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ION ACCELERATION IN VASIMR
VASIMR: < 200 kW helicon + ICH
thruster
Went looking for double layers, but
found none!
Vp, ne, and Te derivatives coincide
Long length scales: 10,000s of λd
λd ~ 10 μm
Corroborated with RPA
B. W. Longmier et al., Plasma
Sources Sci. Technol. 20, 015007
(2011).
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PLASMA IN NOZZLE IS ADIABATIC
Assume plasma is adiabatic
Energy loss to surfaces small
Radiation loss small
Geometry dictates degrees of freedom (N)
Expanding plasma sphere: N = 3
Magnetic nozzle: N = 2
Electrons remain Maxwellian, though
temperature can change
Relies on collisions
Conservation of momentum
Relationship between average electron energy
loss and potential (→ion energy gain)
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HELICON EXPERIMENT
USING VASIMR HARDWARE
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• Bmax = 20,000 G
• P = 15 – 30 kW
• ṁ = 50 – 140 mg/s
• Only helicon coupler used, no ICH
• Superconducting magnets generate converging/diverging
magnetic field
MEASUREMENTS IN PLUME WERE MADE
WITH PLANAR LANGMUIR PROBE
Vp < 20 V
Te < 15 eV
ne = 1010 – 1012 cm-3
Planar tungsten probe
No RF compensation
needed
Guard ring reduces
sheath expansion
effects
Parameters extracted
from I-V traces
Vp: knee
Te: semilog fit
ne: saturation
current
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PARAMETRIC STUDY
OF OPERATING PARAMETERS
Measured at fixed position
50 cm from throat
Highest density where probe
could survive
Lower mass flow rate
Higher Te, Vp
Lower ne
Power flow density ∝ input
energy per ion
Optimize energy deposition
for given flow rate
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Vp, Te, ne DECAY DOWNSTREAM
Axial measurements
Lowest and highest mass flow
rates are shown
Low flow rates → high
temperature, larger
gradients
Temperature decay: plasma is
not isothermal
Length scale: 10,000s of λd
No double layer, but
significant density and
temperature gradients
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DATA WERE CONSISTENT
WITH ADIABATIC THEORY
Plasma potential decays
proportionally to electron
temperature
Only parallel electron temperature
was measured with planar probe
Some electron energy may be lost
to other sinks
Collisions
Instabilities
Radiation
J. P. Sheehan et al., Plasma Sources
Sci. Technol., (submitted).
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ALTERNATIVE THEORY:
RAREFACTION WAVE THEORY
Nonlocal theory
Far downstream,
rarefaction waves create
potential structures which
confine electrons
Electrons lose energy on
wave, cooling in
downstream region
Two regions
Steady state: ambipolar
ion acceleration
Rarefaction wave: further
acceleration
Steady-state part
Rarefaction wave part
Arefiev and Breizman, Phys. Plasmas 16, 055707 (2009).
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COMPARISON BETWEEN ADIABATIC THEORY
AND RAREFACTION WAVE THEORY
Fundamental difference: local
theory or non-local theory
Both predict same relationship
between φ and Te
Different relationship between φ
and ne
Suggests some discrepancy in
number of degrees of freedom
No observed acceleration in low-
field region
Factor of 3 increase in ion
velocity from rarefaction
wave
Importance of collisions
Rarefaction wave theory
Experiment
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ADIABATIC THEORY RELIES ON
COLLISIONS, RETHERMALIZATION
VASIMR has >95% ionization
fraction → Coulomb collisions
dominate
Electron-electron collisions cause
rethermalization of EVDF
Collisions necessary to reduce
thermal conductivity along field
lines
Classical coulomb collision are
insufficient to explain
rethermalization
Collisional mean free path only
becomes long enough after major
temperature gradient
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PLASMA IS DETACHED FROM
MAGNETIC FIELD LINES
In a fully attached nozzle, density
should follow field lines
ne/B > 1
Plasma beam more focused
Detachment occurs
Possible detachment mechanisms
Particle collisions
Anomalous transport
Demagnetization
Frozen flow
Electron inertia
C. S. Olsen et al., IEEE Trans. Plasma
Sci. (accepted, 2014).
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INSTABILITIES INCREASE
THE COLLISION FREQUENCY
In the kinetic equation, collisions represented by collision operator
which has depends on the collisional kernel
Collision frequency affected by both the stable and unstable parts
S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Rev. Lett. 102, 245005 (2009).
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AMBIPOLAR ION ACCELERATION SCALES
ARE SIMILAR TO PRESHEATH SCALES
LE, LB, λcx >> λd
vi ~ cs
Presheath and sheath
Ti << Te
Δφ ~ Te
Quasineutral
J = 0
Ambipolar acceleration
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CONVECTIVE ION ACOUSTIC INSTABILITY
INCREASES DOWNSTREAM COLLISION FREQUENCY
Electron-electron collision frequency
in magnetic nozzle without
considering parameter gradients
In presheath:
Te constant
ne ~ constant
In magnetic nozzle:
dTe/dx ~ Te/LB
dne/dx ~ ne/LB
Instability grows as ions propagate
Collisions rethermalize electrons
Ignoring gradients (see figure)
overestimates collision frequency
Complicating factors
Ion temperature, i-n collisions
Magnetic field divergence
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CONCLUSIONS
Ambipolar ion acceleration
observed in VX-200 experiment
Longer length scale than in
CFDLs
Adiabatic theory describes
experimental results
Potential drop depends on
temperature, not density
Local vs. non-local theory—
unresolved questions
Unknown cause of
rethermalization
Proposed collision mechanism:
ion acoustic instability
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