1 EX/P3-32

Results from the Sheared-Flow StabilizedZ-Pinch and Scaling to Fusion Conditions

U. Shumlak1, H.S. McLean2, B.A. Nelson1, A.E. Schmidt2, R.P. Golingo1, E.L. Claveau1,D.P. Higginson2, M.C. Hughes1, M.P. Ross1, J.R. Barhydt1, D.L. Caudle1, E.G. Forbes1,B. Kim1, L.L. Su1, R.M. Townsend1, K.K. Tummel2 and T.R. Weber1

1Aerospace & Energetics Research Program, University of Washington, Seattle, WA, USA2Lawrence Livermore National Laboratory, Livermore, CA, USA

Corresponding Author: shumlak@uw.edu

Abstract:

The sheared-flow stabilized Z-pinch has been experimentally demonstrated to produce long-lived plasmas that satisfy radial force balance and are stable for thousands of exponentialgrowth times. The sheared-flow stabilized Z-pinch has the potential to lead to a com-pact plasma confinement device that scales to fusion conditions. The stabilizing effect of asheared axial flow on the m = 1 kink instability in Z-pinches has been studied using idealMHD theory to reveal that a sheared axial flow stabilizes the kink mode when the shearexceeds a threshold value. Following these theoretical results, the ZaP Flow Z-Pinch groupat the University of Washington has been experimentally investigating the connection be-tween flow shear and gross plasma stability. Plasma stability is diagnosed with azimuthalarrays of magnetic probes that measure the plasma’s magnetic structure. Large magneticfluctuations occur during pinch assembly, after which the amplitude and frequency of themagnetic fluctuations diminish. This stable behavior continues for an extended quiescentperiod. Plasma flow profiles are measured from the Doppler shift of plasma impurity lines.The experimental flow shear exceeds the theoretical threshold during the quiescent period.Scaling relations suggest that high energy density plasma and fusion conditions are possiblein a compact design. Recent experiments with the upgraded ZaP-HD device have demon-strated the ability to increase the plasma parameters by compressing the plasma radius tosmaller values than achieved with the ZaP device. Plasma parameters of ne = 1018 cm−3,Te = 200 eV, and Ba = 4 T have been experimentally achieved. Based on the successfulresults of ZaP and ZaP-HD, a new experiment FuZE is designed to scale the plasma pa-rameters to fusion conditions. The project will focus on furthering our understanding ofthe physics with specific emphasis on the limitations of sheared flow stabilization and onthe importance of kinetic effects at large drift speeds.

1 Introduction

The Z-pinch is a plasma confinement configuration with many favorable properties: per-pendicular heat conduction, simple geometry, high plasma beta, and compact design.The cylindrical equilibrium is described by a radial force balance between the azimuthal

EX/P3-32 2

magnetic field and the plasma pressure,

Bθ

µ0r

drBθ

dr= −dp

dr. (1)

The azimuthal magnetic field is generated by the axial current driven through the plasma.No other magnetic fields exist, obviating the need for magnetic field coils. Electrodessupply the axial current to the plasma, and except for the magnetic null at the axis,thermal conductivity to the electrodes is perpendicular to the magnetic field. Averageplasma beta is unity, an ideal in confinement efficiency. As described by Eq. (1), increasingthe current and the resulting magnetic field compresses the plasma to a smaller radiusand towards fusion conditions in a compact device. Fusion power increases as the plasmaradius decreases.

The Z-pinch configuration described by Eq. (1) is classically unstable to large-scaleinstabilities, such as the m = 0 sausage and m = 1 kink modes, where m representsthe azimuthal mode number. Conventional stabilization approaches include limiting thepressure gradient [1], introducing an axial magnetic field [2,3], or installing a close-fittingconducting wall [4]. However, these approaches degrade the favorable properties of theZ-pinch configuration and are generally incompatible with a compact fusion device.

In spite of the well-known instabilities, previous experiments have generated Z-pinchplasmas that existed for many times longer than theoretically predicted for a static plasma[5, 6]. These experiments generated Z-pinch plasmas that inherently contained an axialflow.

2 Sheared-Flow Stabilized Z-Pinch

The stabilizing effect of a sheared axial flow on the m = 1 kink instability in Z-pincheshas been studied using ideal MHD theory to reveal that a sheared axial flow stabilizesthe kink mode when the shear exceeds a threshold value, dVz/dr > 0.1kVA [7]. Followingthese theoretical results, the ZaP Flow Z-Pinch group at the University of Washingtonhas been experimentally and computationally investigating the connection between flowshear and gross plasma stability [4, 8–12]. The ZaP experiment initiates a plasma witha coaxial accelerator, similar to a Marshall gun [13]. The plasma is accelerated to largeaxial velocities. The plasma exits the accelerator and forms long-lived, Z-pinch plasmasthat are over 100 cm long with 1–1.5 cm radii. Current in the accelerator continuesto accelerate plasma into the Z-pinch assembly replenishing plasma as it exits from theZ-pinch. Inertia maintains the axial flow within the Z-pinch [10]. Plasma stability isdiagnosed with azimuthal arrays of magnetic probes that measure the plasma’s magneticstructure through Fourier analysis to determine the time-dependent evolution of the low-order azimuthal modes (m = 1, 2, 3). Large magnetic fluctuations occur during pinchassembly, after which the amplitude and frequency of the magnetic fluctuations diminish.This stable behavior continues for an extended quiescent period, lasting 20–50 µs. Atthe end of the quiescent period, the fluctuation levels again change character, increase inmagnitude and frequency, and remain until the end of the plasma pulse.

3 EX/P3-32

FIG. 1: Machine drawing of the ZaP-HD experimental device, identifying the coaxial ac-celeration region and the Z-pinch assembly region. A 1 meter scale is provide for reference.The three-electrode design provides independent control of the plasma compression (usingthe gold and blue electrodes) and of the plasma formation and acceleration (using the goldand red electrodes).

Plasma flow profiles are determined by measuring the Doppler shift of plasma impu-rity line emission using an imaging spectrometer with an intensified CCD camera (ICCD)operated with a 100 ns gate. The spectrometer images 20 spatial chords through theplasma pinch at a 35◦ angle to the plasma axis providing a measurement of the instan-taneous, axial plasma velocity. The chord-integrated data are deconvolved to determinethe axial-velocity profile [14]. Varying the ICCD trigger time between pulses provides thetime-dependent evolution of the flow profile throughout the plasma pulse.

The measurements from the ZaP experiment are compared to the theoretical shearthreshold. A two-chord He-Ne interferometer measures a peaked electron density profileduring the quiescent period in the range of 1016–1017 cm−3, which is combined with themeasured magnetic field to calculate the Alfven speed, the theoretical growth time, andthe theoretical flow-shear required for stability. The theoretical growth time for the kinkmode in a static Z-pinch is approximately 20 ns for the experimental plasma parameters,while the experimentally observed quiescent period is 20–50 µs. The experimental flowshear exceeds the theoretical threshold during the quiescent period and the flow shearis lower than the theoretical threshold at other times [11]. The flow profile is consistentwith that expected from viscosity calculations based on the local magnetized plasmaparameters. MHD simulations have shown that the sheared-flow stabilizing effect existseven when the shear is nonuniform. These results are consistent with other research thatindicates local shear determines stability [15].

Recent experiments with the ZaP-HD device have demonstrated the ability to increasethe plasma parameters by compressing the plasma to smaller radii. The ZaP-HD device,shown in Fig. 1, improves on the ZaP device by providing separate control of the plasmacompression and of the plasma formation and acceleration. The ZaP-HD experiment hasan extensive diagnostic suite to measure the smaller plasmas.

A digital holographic interferometer (DHI) [16] has been developed for the ZaP-HD de-vice to measure the plasma density with high spatial and temporal resolution. A Nd:YAG

EX/P3-32 4

7.8 8.2z [cm]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y[cm]

-6

-5

-4

-3

-2

-1

0

(a)

0 0.2 0.4 0.6 0.8 1r [cm]

0

2

4

6

8

ne[cm

−3]

×1017

Pulse 141209015

(b)

FIG. 2: (a) Unwrapped phase map from digital holographic interferogram imaging a centralportion of the ZaP-HD Z-pinch plasma. (b) Density profile at z = 8.0 cm computed froman Abel inversion of the phase map.

laser generates a 2 ns pulse to produce a hologram recorded by a digital SLR camera with24-megapixel resolution. Figure 2(a) shows the two-dimensional unwrapped phase mapfrom the DHI obtained for a ZaP-HD plasma pulse. The phase map yields the density pro-file shown in Fig. 2(b) after performing an Abel inversion. The profile reveals a plasmaradius that is approximately 0.4 cm and a peak density that is 6 × 1017 cm−3. Theseplasma parameters are significantly smaller and denser than the values obtained from theZaP device, which had ranges of 1–1.5 cm radii and 1016–1017 cm−3 peak densities.

The sheared-flow stabilizing effect has been studied using plasma fluid models [7, 17].However, kinetic effects may become important as the plasma dimensions decrease andhigher currents may lead to drift speeds that are greater than the ion thermal velocity. Ini-tial kinetic modeling has been performed using the particle-in-cell (PIC) code LSP, whichhas been used to model dense plasma focus Z-pinches [18]. The simulation results areshown in Fig. 3 for plasma parameters similar to those produced in the ZaP experiment.The figure compares the structure of the ion number density at 642 ns for two differentinitializations, one where the plasma initially has no axial flow and another where theplasma is initialized with a sheared axial flow set to a uniform value, dVz/dr = 0.16kVA.The stabilizing effect of a sheared flow is clearly evident in the kinetic simulations, andconsistent with the plasma fluid modeling simulations and experimental results.

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(a) (b)

FIG. 3: Axisymmetric kinetic simulations of the sheared-flow stabilization in a Z-pinchwith ZaP plasma parameters: ne = 1017 cm−3, Te = Ti = 100 eV, and Ip = 100 kA.Shown is the structure of the ion number density at t = 642 ns. (a) Static plasma withVz initialized to zero rapidly develops instabilities that grow linearly. (b) Axially flowingplasma with the shear, dVz/dr, initially set to a uniform value of 0.16kVA remains stablewith no degradation of the equilibrium. The simulations used 256 million particles.

3 Scaling Relations for the Z-pinch

Theoretical scaling relations [19] have been derived for the sheared-flow stabilized Z-pinchsatisfying the radial force balance given by Eq. (1), the Bennett relation [20]

(1 + Z)NkT =µ0I

2

8π, (2)

and assuming adiabatic compression

0 =d

dt

( pnγ

)=

d

dt

((1 + Z) kT

nγ−1

). (3)

From these governing equations, a set of scaling relations can be derived which describethe change in pinch radius, magnetic field, and pressure as the plasma current and lineardensity are modified.

a2a1

=

√n1

n2

N2

N1

=

(I1I2

) 1γ−1(N2

N1

) γ2(γ−1)

(4)

B2

B1

=a1a2

I2I1

=

(I2I1

) γγ−1(N1

N2

) γ2(γ−1)

(5)

p2p1

=n2

n1

T2T1

=

(I2I1

) 2γγ−1(N1

N2

) γγ−1

(6)

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FIG. 4: Plasma and fusion parameters scaled from an initial plasma described by a 50 kAplasma current Ip, 20 eV temperature Te = Ti, 1 cm pinch radius a, and 6 × 1016 cm−3

ion number density.

Provided stability can be maintained, the scaling relations indicate a path to achieve fusionconditions in a Z-pinch. For a deuterium–tritium reaction, the fusion power production iscomputed assuming a 50/50 mixture of deuterium/tritium. The input power includes thethermal, flow, and radiative power requirements, Pin = Pth + Pflow + Prad . The thermalpower encompasses the power required to heat and compress the reactants to fusionconditions. The plasma and its energy are assumed to be confined for the duration of theaxial flow-through time, τflow . The sheared-flow stabilized Z-pinch requires power to drivethe axial flow of the plasma, Pflow , which must be externally supplied. Stability requiresVz ≥ 0.1VA for uniform shear [7]. The radiative power loss for a D–T fusion plasma isprimarily due to bremsstrahlung radiation. Fusion gain is defined as Q ≡ Pf/Pin andprovides a measure of the amount of recirculating power.

Starting with plasma parameters equivalent to those produced in the ZaP experimentand assuming a plasma lifetime equal to five flow-through times, Fig. 4 presents the in-crease in Q and decrease in pinch size as the current is increased. The linear density isassumed to remain constant. The scaling relations demonstrate that the plasma densityand temperature increase rapidly as the plasma pinch radius is decreased, which is ac-complished by increasing the plasma current or decreasing the linear plasma density. Thistrend illustrates the naturally compact size of a fusion Z-pinch. This favorable scalingsuggests that high energy density and fusion conditions are possible in a compact design.

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4 Conclusions

The sheared-flow stabilized Z-pinch has been experimentally demonstrated to producelong-lived plasmas that satisfy radial force balance and are stable for thousands of ex-ponential growth times [11]. The sheared-flow stabilized Z-pinch confines plasma in asimple configuration that has the potential to lead to a compact fusion device. Plasmaparameters of ne = 1018 cm−3, Te = 200 eV, and Ba = 4 T have been experimentallyachieved. The successful results of ZaP and ZaP-HD led to support from ARPA-E for anew project, FuZE, to scale these parameters to fusion conditions. The project focuseson furthering our understanding of the physics with specific emphasis on determining thelimitations of sheared flow stabilization and on evaluating the importance of kinetic effectsat large drift speeds.

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