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42nd AIAA /ASME/SAE/ASEE Joint Propulsion Conference 9–12 July 2006 Sacramento, California
Advanced Space Propulsion Based on the
Flow-Stabilized Z-Pinch Fusion Concept
U. Shumlak∗R. C. Lilly,†C. S. Adams,‡R. P. Golingo,§S. L. Jackson,¶
S. D. Knecht,and B. A. Nelson∗∗
Aerospace & Energetics Research Program, University of Washington, Seattle, Washington 98195-2250
A fusion space thruster based on the flow-stabilized Z-pinch may be possible in the
near-term and provide many advantages over other fusion-based thruster concepts. The Z-
pinch equilibrium is classically unstable to gross disruption modes according to theoretical,
numerical, and experimental evidence. However, a new stabilization mechanism has been
discovered that can stabilize these modes with plasma flow. The stabilizing mechanism
was developed for a Z-pinch plasma equilibrium which has an axial velocity profile that
is linear in radius. When the velocity shear exceeds a threshold, the plasma modes are
stabilized. The magnitude of the peak velocity is dependent on the mode wavelength but
is sub-Alfv´enic for the wavelengths of experimental interest, vmax >0.1VAka where VAis the
Alfv´en speed, kis th e axial wave vector, and ais the characteristic pinch radius. The flow
Z-pinch experiment ZaP has been built at the University of Washington to experimentally
verify the sheared flow stabilizing mechanism. The experiment has achieved plasma flow
velo cities of 105m/s and stability for almost 2000 growth times. For more information the
reader is encouraged to visit http://www.aa.washington.edu/AERP/ZaP. The extension of the
flow Z-pinch to a space thruster is straight forward. The plasma in a flow Z-pinch would
already be moving axially, fusing, and releasing a tremendous amount of nuclear energy.
The end of the Z-pinch can be left open to allow the escape of the energetic plasma. Specific
impulses in the range of 106sand thrust levels of 105Nare possible.
Nomenclature
rRadial coordinate, m
aCharacteristic pinch radius, mm
LPinch length, m
jCurrent density, A/m2
ICurrent, MA
BMagnetic flux density, T
pPressure, Pa
∗Associate Professor, Aeronautics & Astronautics, AIAA Senior Member.
†Graduate Student, Aeronautics & Astronautics, AIAA Student Member.
‡Graduate Student, Aeronautics & Astronautics, AIAA Student Member.
§Research Associate, Aeronautics & Astronautics.
¶National Research Council Associate, Naval Research Laboratory, AIAA Member.
Graduate Student, Aeronautics & Astronautics, AIAA Student Member.
∗∗Research Associate Professor, Electrical Engineering.
Copyright c
2006 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free
license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the
copyright owner.
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nNumber density, m−3
T,Te,T
iTotal, electron, ion temperatures, keV
kAxial wave vector, m−1
mAzimuthal mode number
Γ Ratio of specific heats
βRatio of plasma to magnetic pressures
vPlasma flow velocity, m/s
VAAlfv´en speed, m/s, (=B/√μoρ)
veJet exhaust velocity, m/s
ηResistivity, Ω-m
˙mMass flow rate, kg/s
Pin Input power, W
PfFus i on power, W
QFusion power gain ratio (=Pf/Pin)
τPlasma confinement time, s
μoPermeability of free space, (= 4π×10−7)
I. Introduction
Nuclear fusion energy holds the promise of enabling human exploration of the outer solar system, and
perhaps beyond. However, scientific and technical challenges must be surmounted before a fusion-based
space thruster will become reality. These challenges center on stable plasma configurations, and primarily
plasma configurations that have a sufficiently low mass for space applications.
This paper presents recent plasma research that shows a pure Z-pinch can be stabilized. The Z-pinch
discussed in this paper is a pure Z-pinch and not a screw pinch. No external magnetic fields are applied. A
stable Z-pinch has many advantages over other magnetic confinement configurations. External magnetic field
coils are not needed to provide plasma stability, and the compressing magnetic field only needs to compress
the plasma and not the stabilizing magnetic field. For space applications this represents a tremendous weight
and recirculating power savings.
II. Z-Pinch Equilibrium and Instabilities
The Z-pinch equilibrium is the simplest magnetic confinement configuration possible. It consists of a
plasma column with an axial electrical current which produces an azimuthal magnetic field. The Lorentz
force (j×B) confines and compresses the plasma. The equilibrium is described by the radial force balance,
(j×B)r=(∇p)r,(1)
or substituting Ampere’s law for the current gives
Bθ
μor
d(rBθ)
dr +dp
dr =0.(2)
The magnetic pressure balances the plasma pressure.
Z-pinch plasmas have been investigated since the beginning of magnetic confinement fusion research. The
Z-pinch magnetic confinement configuration has many overlapping research issues with the arcjet thruster.
They both use the same simple equilibrium and have the same stability issues. Unfortunately, the simple
equilibrium given by Eq.(2) is unstable to gross disruption modes which have been seen theoretically and
experimentally.1
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Figure 1. Schematic representation of the sausage (m=0) mode in a Z-pinch showing the axisymmetric per-
turbation which grows exponentially.
A. Sausage Instability
The plasma column can undergo a sausage (m=0) instability. The sausage instability is an axisymmetric
displacement of the plasma radius. See Fig. 1. Since the magnetic field varies like 1/r at the plasma/vacuum
interface, the magnetic force varies like 1/r2. The magnetic force is larger where the plasma radius has
decreased and smaller where the plasma radius has increased. The instability grows exponentially until the
axial plasma current is disrupted which quenches the plasma.
The sausage mode can be stabilized by placing a close-fitting conducting wall close to the pinch plasma.
When the instability tries to grow, the wall produces image currents which stabilize the mode. (The con-
strictor serves this role in arcjet thrusters.) A close-fitting wall is incompatible with fusion plasmas because
it precludes the high plasma temperatures required for fusion.
A stability condition against the sausage mode can be found by applying a functional minimization
method (or energy principle) to the linear MHD equations.2For the sausage mode the linear analysis gives
the stability condition
−dln p
dln r≤4Γ
2+Γβ(3)
where β=2μop/B2is a local measure of the ratio of plasma pressure to magnetic pressure. This condition
must be satisfied everywhere in the plasma for stability against the m= 0 mode. The sausage mode can be
stabilized if the pressure does not fall off too rapidly. However, tailoring the pressure profile cannot stabilize
the kink instability.
B. Kink Instability
The plasma column can undergo a kink (m=1) instability. The kink instabilityis an asymmetric displacement
of the plasma column. See Fig. 2. When the plasma kinks the magnetic field intensity increases on the inner
portion of the bend and decreases on the outer portion of the bend. The corresponding magnetic force causes
the instability to continue and grow exponentially.
The kink instability can also be stabilized by placing a conducting wall (constrictor) in close proximity
to the plasma column, but as stated, this method is unacceptable for fusion plasmas.
The kink instability can also be stabilized by embedding an axial magnetic field into the plasma. As the
plasma kinks, the axial field stretches and resists further kinking. The condition for stability against the
kink mode is found by applying an energy principle and is called the Kruskal-Shafranov limit.3, 4
Bθ
Bz
<2πa
L.(4)
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American Institute of Aeronautics and Astronautics Paper 2006-4805
Figure 2. Schematic representation of the kink (m=1) mode in a Z-pinch showing the asymmetric perturbation
which grows exponentially.
This condition limits the plasma current and the plasma pressure that can be stably achieved in a Z-pinch.
The equilibrium is now modified so that the radial force of the azimuthal magnetic field balances the plasma
pressure and the magnetic pressure of the axial field. A preferred approach would be to stabilize the z-
pinch without limiting the plasma current, thereby, allowing high plasma pressure. Additionally, the axial
magnetic field transforms the circular azimuthal field lines into helical field lines that thread the plasma and
the electrodes. The plasma can then lose heat by thermal conduction parallel to the magnetic field lines.
Parallel heat conduction is much larger than perpendicular heat conduction.
C. Flow Stabilization
A stabilization mechanism has been discovered that can stabilize the unstable modes of a Z-pinch by using
plasma flow.5–8
The stabilizing mechanism was developed for a Z-pinch plasma equilibrium which has an axial velocity
profile that is linear in radius. Magnetohydrodynamics (MHD) theory is applied and a linear stability analysis
is performed. The stability analysis confirms the known stabilizing effect of a conducting wall. However, an
additional result shows that a velocity shear threshold exists. When the velocity shear exceeds the threshold,
the plasma modes are stabilized. The magnitude of the peak velocity is dependent on the mode wavelength
but is sub-Alfv´enic for the wavelengths of experimental interest even in the no-wall limit.
vmax >0.1VAka (5)
where VAis the Alfv´en speed, kis the axial wave vector, and ais the characteristic pinch radius.5The
stability results are shown in Fig. 3.
The stabilizing mechanism can perhaps be best understood as an effective mode mixing that occurs when
the plasma flow is sheared. As either the sausage or kink instability begins to grow, the shear in the flow
mixes the axial locations of the mode. The mixing destructively interferes with the growth of the mode, and
the mode is stabilized. A similar stabilizing mechanism has been found when theoretically investigating the
classical Rayleigh-Taylor/Kelvin-Helmholtz instabilities.9
Nonlinear simulations have been performed and support the sheared flow stabilizing effect in Z-pinch
plasma equilibria. The simulations are performed with the Mach2 code10 which uses the time-dependent,
resistive, 2D MHD model. An equilibrium is initialized which has a sheared axial plasma flow and an
axially periodic density perturbation. The figure shows the pressure contours for the cases of no flow and
vz/a =0.2kVAat the same times in the simulation. Fig. 4 shows a well developed m=0 instability in a static
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American Institute of Aeronautics and Astronautics Paper 2006-4805
U
Z
D
Y
]
N9
$
8QVWDEOH
6WDEOH
Figure 3. Linear stability results for a Z-pinch with a sheared axial flow. The stabilizing effect of a conducting
wall is evident when the wall is close to the pinch radius. A stability threshold is seen at a normalized velocity
shear of approximately 0.1 in the no-wall limit.
Z pinch plasma and a substantially less developed m=0 instability in a Z pinch plasma with a sheared axial
flow.
III. The Flow Z-Pinch Experiment, ZaP
A flow Z-pinch experiment ZaP has been built at the University of Washington to experimentally verify
the sheared flow stabilizing mechanism.11 The experiment is designed to generate a Z-pinch plasma with
a large axial flow by coupling a coaxial acceleration region to an assembly region. A machine drawing of
the ZaP experiment is shown in Fig. 5 identifying the relevant features. The experiment is initiated by the
injection of neutral gas, usually hydrogen, with fast puff valves into the annular region between the coaxial
electrodes located in the middle of the 1 m coaxial acceleration region. An electrical potential is applied
across the coaxial electrodes, ionizing the neutral gas, and accelerating the plasma. When current flows
through the plasma, the Lorentz force (j×B) from the current and the self-field accelerates the plasma
axially. When the plasma reaches the end of the coaxial acceleration region, the plasma along the inner
electrode moves radially inward and assembles along the axis in the 1 m long assembly region. The plasma
along the outer electrode continues to move axially and radially inward during the assembly of the Z-pinch.
The plasma finally connects between the end of the inner electrode and the outer electrode end wall forming
a complete Z-pinch. Inertia maintains the plasma flow state, and plasma is continually exiting from the
coaxial accelerator and assembles into the pinch. The plasma parameters are shown in Table 1.
An azimuthal array of eight equally-spaced, surface-mounted magnetic probes are installed in the outer
electrode at the pinch midplane. The probes measure the azimuthal magnetic field at the surface of the outer
electrode. The magnetic field values from the probe array are Fourier analyzed to determine the evolution
of the low order azimuthal modes (m=1,2,3) of the Z-pinch plasma. Data are plotted in Fig. 6 showing the
time evolution of the normalized m=1,2,3 Fourier modes of the magnetic field. The figure also shows the
evolution of the plasma current for reference.
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American Institute of Aeronautics and Astronautics Paper 2006-4805
0
r
0
r
0
r
0
r
v’/ kVa = 0.2
v’/ kVa = 0.0
Figure 4. Nonlinear evolution of the sausage mode for a static Z-pinch equilibrium (top images) and one with
a sheared axial flow (lower images). Shown are pressure contours.
Table 1. Experimental Parameters of the ZaP Experiment
Entity Val ue
Accelerator Length 1m
Inner Electrode O.D. 0.1 m
Outer Electrode I.D. 0.2 m
Peak Current 0.3 MA
n1022–1023 m−3
a10 mm
L1m
Te+Ti0.15–0.25 keV
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American Institute of Aeronautics and Astronautics Paper 2006-4805
Figure 5. Side view machine drawing of the ZaP experiment showing the relevant features. The vertical ports
in the assembly region are used for spectroscopy. The horizontal ports are used for interferometry and visible
imaging with a fast framing camera. A “1 m” scale is included for reference.
B
m
/B
0
0.0
0.2
0.4
0.6
0.8
1.0
I
p
(kA)
0
50
100
150
200
Time (Ps)
0 20406080100
m=1
m=2
m=3
Ip
Pulse 40108045
Figure 6. Time evolution of m=1,2,3 Fourier components of the magnetic field fluctuation at z=0 showing the
quiescent period from 42 µsec to 79 µsec. The values are normalized to the average magnetic field value at the
pinch midplane. The evolution of the plasma current (dashed curve) is included for reference.
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American Institute of Aeronautics and Astronautics Paper 2006-4805
The magnetic fluctuation levels of the asymmetric modes are high when the Z-pinch plasma is assembling.
After the pinch has formed the fluctuation levels change character for approximately 37 μs, from 42 to 79
μs. The change in character is identified by lower levels and decreased frequency for the fluctuations.
After this quiescent period the fluctuation levels then again change character, increase in magnitude and
frequency, and stay high until the end of the plasma pulse. Fluctuations of the normalized m=1 component
as presented correspond to a current displacement of 2ξ/rw. Therefore, a value of B1/B0=0.2representsa
radial displacement of the current of 10 mm. The 0.2 value defines the quiescent period. These results are
consistent with other diagnostic measurements.
The axial velocity profiles are determined during the plasma pulse by measuring the Doppler shift of
impurity line radiation with an intensified CCD detector and an imaging spectrometer viewing the plasma
through the oblique view port.6The exposure times are typically 0.1 - 0.5 μs. The chord-integrated data are
then deconvolved to determine axial velocity profiles.12 By varying the recording time between pulses, an
evolution of the velocity profile can determine. These data are shown in Fig. 7 as a function of normalized
time, τ. Time is normalized to the quiescent period (defined as τ=[0,1]) to allow accurate comparison. The
magnetic field fluctuations from Fig. 6 are shown in the lower plot of Fig. 7 to provide comparison between
fluctuation levels and flow shear.
During the pinch assembly the magnetic fluctuation levels are high and the plasma axial velocity profile
is uniform with a value of approximately 105m/s. During the quiescent period the magnetic fluctuation
level is low and the plasma axial velocity is sheared with either a large velocity at the edge and a low velocity
in the plasma core or a low velocity at the edge and a large velocity in the plasma core. After the quiescent
period the magnetic fluctuation levels are high and the plasma axial velocity profile is uniform and low.
For the experimental plasma parameters the growth rate of the kink mode in a static plasma is 21 ns.
The experimental results show a stable period which is almost 2000 exponential growth times. Experimental
evidence suggests there may be an operational mode where the flow Z-pinch reaches a quasi steady-state.
For more information on the ZaP experiment the reader is encouraged to read the referenced articles and to
visit http://www.aa.washington.edu/AERP/ZaP.
IV. Flow-Stabilized Z-Pinch Fusion Space Thruster
The extension of the flow-stabilized Z-pinch to a space thruster is straight forward. See Fig. 8 for
a schematic. The plasma in a flow-stabilized Z-pinch is already moving axially, fusing, and releasing a
tremendous amount of nuclear energy. The end of the Z-pinch can be left open to allow the escape of the
energetic plasma composed of the fusion products.
The core of the plasma would be at fusion temperatures and should not contact any solid material.
The electrodes would flair at the ends to allow the plasma to cool before contact as shown in Fig. 8. The
plasma would expand from the fusion heat and propagate into the electrode. This component of the plasma
would be sent to a direct energy converter to capture the plasma velocity directly as electrical current. The
converted current could also supply power for a magnetic nozzle to improve the conversion of thermal to
directed kinetic energy of the fusion products. The power from the direct energy converter could also be
used to maintain the circulating plasma current and supply spacecraft power.
Since no external magnetic fields are required to provide plasma stability, the required mass is dramati-
cally reduced compared to other magnetic confinement fusion concepts.
A. Scaling Relations for the Flow-Stabilized Z-Pinch
Scaling relations are useful to determine the size of thruster required to match the requirements for any
mission. The total fusion power scales as
Pf∝I4L
a2T2(6)
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τ
UPP
(
(
(
(
(
(
(
(
(
(
(
(
(
τ
%
P
%
P
P
P
3XOVH
Figure 7. The upper plot shows the plasma axial velocity contours based on the C-III line at 229.7 nm as
a function of plasma radius and time. Velocity profiles are recorded once during each pulse. By varying the
recording time between pulses, an evolution of the velocity profile can determine. The times are normalized
to the quiescent period (defined as [0,1]) to allow accurate comparison. The magnetic field fluctuations from
Fig. 6 are shown in the lower plot to provide comparison between fluctuation levels and flow shear.
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American Institute of Aeronautics and Astronautics Paper 2006-4805
*
L
Fuel Exhaust
L
A
Direct Energy
Converter
Figure 8. Schematic representation of a flow-stabilized Z-pinch fusion space thruster.
for plasma temperatures high enough that the fusion cross-section is approximately constant.13, 14 Energy
must be provided to generate the stabilizing shear flow and the resistive dissipation within the plasma.
Pin =1/2˙mfuel v2
in +I2ηL/πa2(7)
A significant advantage of the Z-pinch is the high plasma density that is obtained. (In the ZaP experiment
peak densities of 1023 m−3and higher are measured.) The plasma density varies as
n∝I2
a2T.(8)
The high density avails the more attractive fusion reactions that generate fewer neutrons, like D–He3,
or are completely aneutronic, such as p–B11 . The reaction with the highest cross-section is D–T, but
it produces one neutron per reaction that must be shielded. The D–He3reaction does not produce any
neutrons, but two Dions can react and produce a neutron. Shielding against neutrons significantly increases
the required mass of the spacecraft.
B. Sample Thruster Parameters
Table 2 lists thruster parameters for a Z-pinch fusion space thruster that could use D–He3or p–B11
for fuels. Any temperature below approximately 50 keV effectively precludes the use of p–B11 because
the reaction cross-section decreases rapidly, though a lower temperature for D–He3will produce a more
optimized design because of the resulting higher density. Additional propellant mass may be added to the
exhaust plume to generate a higher thrust and lower specific impulse. The thruster parameters are more
restrictive for the p–B11 reaction because of the lower reaction cross-section.
The length of the pinch is set by satisfying a complete burn criteria.
L≥vinτ(9)
and nτ > 3×1020 s/m3for D–He3and nτ > 6×1021 s/m3for p–B11 . The volumetric power is much
lower for p–B11 than for D–He3which drives the length from 1.5 m to 18 m. Since most of this length is
empty space, the only real expense is the greater inductive energy which can be recovered by recirculating
the plasma current.
Since the Z-pinch equilibrium is internally balanced (no applied magnetic fields), the optimized design is
a smaller plasma radius. This can be seen in Figs. 9 and 10. The figures show the variation away from the
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American Institute of Aeronautics and Astronautics Paper 2006-4805
Table 2. Sample Thruster Parameters for a Flow-Stabilized Z-Pinch
Entity D–He3p–B11
I(MA) 510
L(m) 10 50
a(mm) 1 1
T(keV) 80 100
n(m−3) 1.5×1025 4.7×1025
Pf(W) 3.3×1012 9.9×1012
Pin (W) 1.8×1012 8.4×1012
˙mfuel (kg/s) 0.095 0.53
ve(m/s) 3.5×1061.3×106
Thrust (N) 3.3×1056.8×105
DPP
Y
H
PV
4
Y
H
4
'+H
, 0$/ P
Figure 9. Thruster parameters for a flow-stabilized Z-pinch operating on the D–He3reaction. Notice the
maximum permissible pinch radius and the improved performance for smaller pinch radii. The variation of
the fusion Qis also shown.
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American Institute of Aeronautics and Astronautics Paper 2006-4805
DPP
Y
H
PV
4
Y
H
4
S%
, 0$/ P
Figure 10. Thruster parameters for a flow-stabilized Z-pinch operating on the p–B11 reaction. The parameters
are more restrictive than the D–He3reaction. Notice the maximum permissible pinch radius and the improved
performance for smaller pinch radii. The variation of the fusion Qis also shown.
point designs in Table 2. For each design there is a maximum permissible pinch radius for the configuration
to operate. For each design there is also a maximum fusion Qwhich is the ratio of fusion power to input
power.
The thruster parameters for the flow-stabilized Z-pinch thruster are in line with the parameters derived
by C. H. Williams et al.15 for a fast transfer human mission to Saturn. An important difference is the
fusion power for the Z-pinch is larger by approximately 103due to its higher plasma density. Additionally,
no external field coils are required so the specific power would be much larger as well.
C. Fusion Burn Simulations of a Z-Pinch
The dynamics of a Z-pinch undergoing fusion burn have been simulated.?The simulations solve the time-
dependent ideal MHD equations including particle loss (due to fusion) and energy loss (due to fusion,
bremsstrahlung radiation, and synchrotron radiation). The plasma is assumed to be in axial equilibrium, or
more specifically, the axial gradient length is much larger than the radial gradient length.
The simulations shown here are for D + T reaction with nD=nTwhere it is assumed the alpha particles
are not confined by the magnetic field. For the simulation parameters, rLα≈a, and the assumption is only
marginally satisfied. The plasma current is constant in time. Figure 11 shows the evolution of the particle
number density. The number density decreases on axis as fusion burns up the plasma and it is removed
from the system. The plasma temperature increases to maintain radial force balance as seen in Fig. 12.
The plasma contracts slightly and the plasma pressure profile remains mostly constant. As a result of the
increased temperature and hollowed density profile, the fusion reaction rate becomes highest off-axis. See
Fig. 13. The primary power loss mechanism is synchrotron radiation which can be seen to maintain a cooler
plasma edge where the magnetic field is highest.
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r
0246810
n
0.0
0.2
0.4
0.6
0.8
1.0
1.2
t = 0
t = 25
t = 50
t = 75
t = 100
Figure 11. Particle number density evolution for a burning fusion Z-pinch. The number density decreases in
the plasma core as the fusion process burns up plasma in the core. Eventually the outer plasma also burns up.
r
0246810
T
0
2
4
6
8
10
12
14
t = 0
t = 25
t = 50
t = 75
t = 100
Figure 12. Temperature profile evolution for a burning fusion Z-pinch. Since the plasma current is constant,
the temp erature must increase as the number density decreases to maintain equilibrium.
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American Institute of Aeronautics and Astronautics Paper 2006-4805
r
0246810
Reaction Rate
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
t = 0
t = 25
t = 50
t = 75
t = 100
Figure 13. Fusion reaction rate profile evolution for a burning fusion Z-pinch. The combined effect of a
hollowing density profile and a rising temperature leads to a reaction rate that is peaked off-axis.
V. Conclusions
The Z-pinch has many of the desired features for a fusion space thruster: linear device, no external field
coils, high specific power, high plasma density (aneutronic fuels). The problem has always been the gross
stability of Z-pinch plasmas. However, using the stabilizing mechanism of sheared flows the Z-pinch may
finally have surmounted its most difficult challenge.
The ZaP experiment at the University of Washington may serve as a prototype thruster while it verifies
the flow stabilization effect. A fusion space thruster based on the flow-stabilized Z-pinch concept is a near-
term prospect that may make manned deep space exploration feasible.
References
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(2003).
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11This work is supported by U. S. Department of Energy under Grant DE-FG03-98ER54460.
12R. P. Golingo and U. Shumlak, Rev. Sci. Instrum. 74 (4), 2332 (2003).
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15C. H. Williams, S. K. Borowski, L. A. Dudzinski, A. J. Juhasz, AIAA 99-2704, June 1999.
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