Investigation into Fusion Feasibility of aMagnetized Target Fusion Reactor: APreliminary Numerical Framework

Michael Lindstrom, Sandra Barsky, Brian Wetton

August 3, 2014

Abstract

The efforts to engineer devices to produces conditions suitable fornuclear fusion on earth have made significant leaps and bounds inrecent years due to improved technology and engineering methods.Magnetized Target Fusion, or magneto-inertial fusion, involves the in-ertial compression of a pre-magnetized plasma, using high magneticfields to insulate the plasma, thereby allowing slower and lower conver-gence compression than achievable by inertial techniques alone. Oneparticular form of MTF (suggested by General Fusion) involves us-ing an intense pressure wave transmitted through a dense medium,growing in intensity due to focusing, and finally reaching a plasmaand compressing it. Our model consists of a spherically symmetricdomain with moving boundaries that we solve numerically througha coordinate transformation and a flux-limited finite volume method.Our work ventures towards a proof of concept, both of a mathematicaltechnique to solve nonlinear conservation laws with moving boundariesand of the notion that current designs being considered show potentialfor successful fusion conditions. Our work also includes a sensitivityanalysis to estimate the optimal conditions for such a reactor.

1 Introduction

This paper provides a proof of concept for numerical techniques to solvenonlinear conservation laws with moving boundaries which could be of use in

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fusion reactor design, and through an elementary model, we explore the sys-tem response to various design parameters. The physical model we developis quite basic, but it demonstrates the applicability of our numerical frame-work and is one step in a greater modelling and numerical study of fusionreactors. The paper is divided into 5 sections. In this section, we providean introduction to the engineering requirements and obstacles in producingfusion and introduce the fusion reactor design we have chosen to model. Thisserves as a motivation for the numerical work we do. In section 2, we presenta simplified model and set of equations, and present relevant experimentaldata and some very basic plasma physics details. Following this, section 3provides an overview of the numerical method we used and we validate itsaccuracy. Section 4 presents our results and predictions of the numericalwork, and we conclude and discuss future work in section 5.

1.1 Fusion Challenges and Requirements

There are a number of ongoing large-scale efforts to acheive nuclear fusion,including NIF[15] and ITER. This paper reports on an effort by a Canadiannuclear fusion research company General Fusion [5]. The main idea is tocompress a D-T plasma located in a cylindrical cavity by an imploding shellof lead-lithium. With sufficient compression, the plasma will heat to igni-tion, and a recovery system will convert the heat and radiation generated toelectricity. There are a number of factors to consider in the design of such asystem. This paper focuses on developing a basic model of the compressionso that we can apply finite volume numerical schemes for nonlinear hyper-bolic conservation laws with moving boundaries to make predictions of thereactor being designed, and thereby do a parameter search to optimize theconditions under which compression, and thereby fusion, can occur. Sincethis is a first attempt at applying hyperbolic numerical methods to the spe-cific application of magnetized target fusion, we consider a highly simplifiedmodel as the first step to developing a more complete physical picture.

Nuclear fusion is a complicated process in which lighter particles are fusedtogether, often at very high temperatures and pressures, to produce heavierparticles. Through a change in rest mass between the reacting and resul-tant particles, energy is released through radiation. The major challenge inproducing fusion is forcing the lighter particles to interact enough with eachother so that they can fuse. Modern ideas of fusion are often based on us-ing deuterium (2 H) and tritium (3 H) plasmas as fuel and fusing them to

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produce helium and free neutrons.One engineering obstacle is that heating the fuel to a high temperature

greatly increases convective cooling, and any contact a plasma makes withthe walls of its confinement yields significant energy losses [1]. Fortunately, itis possible to suspend a plasma with magnetic fields and minimize its contactwith its enclosure. This insulation helps to reduce energy losses.

Beyond the mere concern of energy losses is the necessary particle densityand pressure necessary for fusion to occur. One useful criterion is the Lawsoncondition for fusion (whereby more energy is released than goes into heat-ing) states that during a compression the product of number density andtime should exceed 1014 s/cm3 for deuterium-tritium fusion and the tem-perature should exceed 6− 10 keV [7]. This is sometimes generalized into atriple product where temperature times density times confinement time mustexceed ∼ 4×1015 s keV / cm3 [8]. The product is even higher for deuterium-deuterium fusion. In the work we present, we compare our predictions ofplasma densities and pressures with this triple product by integrating theproduct of number density and temperature over the time the plasma isbeing compressed.

The design we consider consists of a 3-m diameter spherical region ofmolten lead-lithium, rotating about a 0.4-m diameter central vertical axis athigh velocity such that a near vacuum, vertical cylindrical region is formeddown the central axis. Within this axis, at a very precise time, a toroidalplasma of deterium and tritium is released and reaches the very centre ofthe spherical region (we assume the plasma can survive for hundreds of mi-croseconds). Also at a precise time, pistons, delivering impact pressures of2 GPa hit the outside of the sphere, and a pressure wave travels throughthe lead-lithium until reaching the central axis at the precise moment theplasma arrives. The intense pressure and radially inward particle velocitycompress the plasma. The initial total pressure (gas plus magnetic pressure)of the plasma is 5 MPa with particle density 3.2×1016 particles per cm3 andtemperature of 100 eV. The intense pressure pulse is modelled by a Gaussianwith characteristic time 45 µs. A diagram illustrating the reactor is given infigure 1.

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Figure 1: Magnetized Target Fusion Reactor Design

2 The Model

2.1 Physical Assumptions

Geometrically, we adopt a spherically symmetric model in which the plasmais treated as a spherical gas, immersed in the lead-lithium whose outer bound-ary is a sphere. The pistons hit the lead-lithium from all sides, generating aspherically symmetric pressure pulse that moves radially inward. We neglectthe rotation of the lead-lithium in this setting. Although the physical geom-etry is more complicated than this, a simple model like this is a necessarystep to reaching more complex models. Given the fact that the empty re-gion of the cylinder will, in actual operation, become partially filled in by thelead-lithium in compression, the real world system is closer to being sphericalthan it may seem.

We treat the plasma as a spatially homogeneous spherical region, with apressure given by the sum of a gas pressure and a magnetic pressure, withthe initial ratio of the gas to magnetic pressure of β0 = 0.1. We assumeadiabatic (reversible) conditions. Initially the pressure of the plasma andthe lead-lithium is 5 MPa, and the pressure pulse hits discontinuously, suchthat the pressure on the outer walls of the lead-lithium jumps to 2 GPaat time t = 0. The motion of the outer boundary between the lead-lithiumand pistons, and the inner boundary between the lead-lithium and plasmahave velocities prescribed by the local fluid velocity of the lead-lithium. Thisis equivalent to the plasma and lead-lithium being immiscible and no lead-lithium leaving the confinement. We additionally neglect radiation losses dueto material impurities.

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Table 1: Data for Quadratic Equation of StateDensity Value11340 kg m−3 Pressure = 101325 N m−2

11340 kg m−3 Sound Speed = 2090 m s−1

16017 kg m−3 Pressure = 4.06185× 1010 N m−2

2.2 Plasma Equation of State

Because the plasma consists of charged particles moving in a magnetic field,the pressure of the plasma has two components: a gas pressure and a mag-netic pressure due to the conservation of magnetic flux. We take the particledensity to be uniform within the sphere. Under an adiabatic assumption,the pressure Pg of a gas scales inversely with its volume V according to thelaw PgV

γ = constant where γ = 5/3 for a monatomic gas.In the toroidal situation, as the torus major radius decreases, the magnetic

pressure Pm of the plasma increases according to 1/r4L where rL denotes the

torus major radius. This arises from the magnetic field increasing accordingto 1/r2

L [16] and the magnetic pressure being proportional to the square ofthe magnetic field.

We take the total pressure PL as the sum of the two component pressuresPL = Pg + Pm. A more detailed description of the plasma would be animportant improvement to add in future models of this technology.

2.3 Lead-Lithium Equation of State

Based on the experimental data for lead [11], we fit a quadratic equation

of state for lead-lithium using the density ρ0 and sound speed√

∂P∂ρ

at 1

atm, and the density at a pressure of 4.06185 × 1010 Pa. This gave us twoprescribed points in the density-pressure plane and a slope at a point, whichdefines a quadratic equation of state P = A(ρ− ρ0)2 + B(ρ− ρ0) + C. Thedata we used in this quadratic fit are given in table 1. The fit is plotted infigure 2.

It is important to note that such an equation of state does yield negativeabsolute pressures for density values less than those at standard conditions.Some experiments have suggested that for some liquids, a negative pressurecan be sustained for fast time scales [13]. For this reason, we allow forthe negative pressures. We remark further that as can be seen in table 4,

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Figure 2: Quadratic fit to experimental equation of state data.

the fit parameters (such as the sound speed used) do influence the modelpredictions. Further experimental investigation into the equation of state oflead-lithium would be useful in improving the modelling.

2.4 Governing Equations

In this subsection, we present the equations used in our model. The numericalvalues of the parameters are tabulated in the following subsection.

Within the lead-lithium, we use the compressible Euler equations forconservation of mass and conservation of momentum, equations (1) and (2)respectively. The variable ρ denotes mass density, v denotes the local fluidvelocity, P denotes pressure, and t is the time. Note that there is a tensorproduct ⊗.

ρt +∇ • (ρv) = 0 (1)

(ρv)t +∇ • (ρv ⊗ v) +∇P = 0 (2)

In a spherically symmetric coordinate system, equations (1) and (2) re-duce to the equations used in our model, (3) and (4) respectively.

ρt + (ρv)r +2

r(ρv) = 0 (3)

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(ρv)t + (P + ρv2)r +2

r(ρv2) = 0 (4)

The lead-lithium pressure is given by equation (5)

P = A(ρ− ρ0)2 +B(ρ− ρ0) + C (5)

and the total pressure (magnetic plus gas pressure) is governed by equation(6)

PL(t) = P (rL(t), t) =κ1

rL(t)4+

κ2

rL(t)5. (6)

The temperature is given as

Θ = kT =κ2/rL(t)5

n(t)(7)

with k the Boltzmann constant for a particle density

n(t) =n0

(rL(t)/rL(0))3. (8)

The pressure of the outer wall is given by equation (9)

PR(t) = P (rR(t), t) = Patm + (Pmax − Patm)e−t2/t20 (9)

The position of the lead-lithium and plasma interface rL and the outer wall ofthe lead-lithium rR are described by equations (10) and (11). As discussedpreviously, these equations correspond to the boundaries moving with thelocal fluid velocity.

drLdt

= v(rL(t), t) (10)

drRdt

= v(rR(t), t) (11)

2.5 Parameters, Initializations, and Setup

The baseline parameters used in our study are given in table 2. We remarkthat our numerical computations explained in the following section were donein a dimensionless framework. A figure illustrating our simplified geometryis given in figure 3.

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Figure 3: Spherically Symmetric Model

Table 2: Parameters in SimulationsParameter ValuePiston impact pressure PR(0) = Pmax 2 GPaInitial total pressure PL(0) 5 MPaInitial plasma particle density n0 3.2× 1016 particles per cm3

Initial plasma temperature Θ0 100 eVInitial lead-lithium radius rR(0) 1.5 mInitial plasma radius rL(0) 0.2 mPulse time scale t0 45 µsInitial gas/magnetic pressure ratio β0 0.1Magnetic pressure coefficient κ1 7.273× 103 kg m3 s−2

Gas pressure coefficient κ2 145.5 kg m4 s−2

Lead-lithium parameter A 922.9 m5 kg−1 s−2

Lead-lithium parameter B 4368100 m2 s−2

Lead-lithium parameter C = Patm 101325 kg m−1 s−2

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3 Numerical Computations

3.1 Change of Variables to Map Problem to Fixed Do-main

The equations themselves are challenging to solve numerically due to themoving boundaries, the stiff equation of state, and the geometric source termsarising from the spherical geometry [10]. One strategy involves interpolationand extrapolation while keeping the computational domain in the radial co-ordinate fixed [12]. We choose another approach here by transforming thecoordinate system so that there is an underlying, constant computationaldomain, without using interpolation or extrapolation. We perform a changeof variables to deal with the moving boundaries, setting τ = t and y = r−rL(t)

∆(t)

with ∆(t) = rR(t)− rL(t). We will denote Γ(t) by vR(t)− vL(t).Under this transformation, our new equations for mass and momentum

conservation are given by equations (12) and (13)

ρτ + { 1

∆[−(vL + Γy)ρ+ ρv]}y =

−2

rL + ∆yρv − Γ

∆ρ (12)

(ρv)τ + { 1

∆[−(vL + Γy)(ρv) + P + ρv2]}y =

−2

rL + ∆yρv2 − Γ

∆ρv (13)

in the fixed computational domain 0 ≤ y ≤ 1 and τ ≥ 0. Equations (12)and (13) are expressed in conservation form with time-dependent geometricsource terms on the right-hand sides.

3.2 Finite Volume Discretization

After the coordinate transformation, we use a finite volume discretizationfor the numerical simulations. The fundamental ideas we present here arepresented with excellent clarity and full detail in [10].

We discretize the computational domain as follows. First, we choose anN and corresponding spatial mesh size h = 1/N , defining the coordinatesyi = ih. We denote ρji to be numerical approximation to the average cell

value∫ yi+h/2yi−h/2 ρ(y, tj)dy at time tj. The index j denotes our time-stepping.

Similarly we define (ρv)ji .The pseudocode of what is done at each time step is provided below:

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- Use constant extrapolation to define ρ and ρv at i = −1, 0 and i =N + 1, N + 2.

- Compute the eigenvalues λ± =v−(vL+Γy)±

√dP/dρ

∆of the linearized sys-

tems of 12 and 13.

- Obtain a time step k = 0.8h/λ∗ where λ∗ is the largest value |λ±| takes.

- Using λ±i and λ±i+1, determine the direction of information propagationof the corresponding eigencomponents of the system by taking theiraverage.

- Define flux limiters based on how rapidly changing the information isin the different directions.

- Solve the homogeneous Riemann problem at each interface using thedirection information from the eigenvalues, but analytically precise nu-merical flux.

- Compute a second-order flux correction, and use the flux limiters todefine an updated set of values for ρ and ρv, again without the right-hand-side terms.

- Advance forward once more in time using the nonzero right-hand sideas update information (a split step).

- Update the time, boundary positions, and boundary pressures/densities.

In the end, our method is first-order, despite the second-order correctionterm, due to the split stepping and the time-varying source terms. How-ever, this approach does accurately capture many aspects of the solutions ofconservation laws with shocks [10].

3.3 Validation

3.3.1 Linear Equation of State and Acoustic Limit

In the case of a linear equation of state for the lead-lithium A = 0, anda small-amplitude pressure pulse, the mass and momentum equations caneffectively be reduced to the acoustic equations. Under this reduction, the

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Figure 4: Profiles of velocity (left) and density (right) at various time valuescompared to the 1/rs growth rate predicted in the acoustic limit.

amplitude of the shock front in both the local fluid velocity and the pressureshould grow inversely with the shock position rs. See [6] for details.

In figure 4, we plot the pressure and velocity profiles at various timesalong with the curves predicted by a 1/rs growth rate. Our numerics arevalidated in this acoustic limit.

3.3.2 Qualitative Convergence with N →∞

We choose a fixed time of t = 1.243× 10−4 s and plot the profiles of velocity,density, and pressure for N = 500 and N = 4000. The plots as displayed infigure 5 confirm convergence.

3.3.3 First Order Convergence

We choose N = 250000 and define the predictions of this model as the“numerically exact” solution. Then we compare the errors of the velocityand density (normalized by their characteristic scales) in the L1−norms forN = 1000, N = 2000, N = 4000, N = 8000 and N = 16000 at the fixedtime t = 1.243× 10−4 s. In order to make this comparison, the N = 250000solution was projected onto the coarser mesh by a linear interpolation. Atthese same N -values, we computed the maximum error in total mass by

numerically integrating∫ rR(t)

rL(t)r2ρ(r, t)dr and finding the maximum deviation

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Figure 5: Profiles of velocity (left) and density (right) at t = 7.03×10−4s forN = 500 and N = 4000. The plots are very similar.

Table 3: Convergence of Numerical Scheme at t = 1.243× 10−4 sN Velocity Error Density Error Mass Error1000 5.68× 10−4 1.11× 10−3 4.79× 10−4

2000 2.95× 10−4 5.58× 10−4 2.39× 10−4

4000 1.39× 10−4 2.77× 10−4 1.20× 10−4

8000 6.97× 10−5 1.39× 10−4 5.97× 10−5

16000 3.51× 10−5 6.93× 10−5 2.99× 10−5

Rate 1.01 1.00 1.00

from t = 0 for all times up to the fixed time. The errors are tabulated intable 3 and there is a very clear first-order trend. The convergence ratesbased on the slopes of the lines of best fit of the errors and N -values on alog-log plot are indicated in the table.

4 Model Predictions

4.1 Plasma Compression and Fusion Possibility

The figures 6, 7, and 8 depict the position of the inner wall, its velocity, andthe pressure of the plasma as a function of time given the baseline param-eters in table 2. With these baseline parameters, our triple product value

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Figure 6: Motion of the plasma wall.

is 5.2 × 1014 keV s cm−3, which is smaller than the Lawson triple productrequired for yielding more energy output than input. Under a modified pa-rameter regime, as given in second last row of table 4, we obtain a tripleproduct that exceeds 4× 1015 keV s cm−3. We also make a brief remark onthe plasma survival time as surival times are an important aspect of the en-gineering. With our simulations, the plasma is compressed over the scale ofhundreds of microscends, which is within the survival times of plasmas suchas spheromaks [17]. We thus infer that survival time should not be an issue.With these observations, it seems there is promise to yielding fusion energythrough such a design, although we cannot overemphasize the rudimentarynature of the physical modelling considered.

4.2 Sensitivity Analysis

Here we analyze what results are obtained in varying some of the physicalparameters of the system. We concern ourselves with five aspects of thereactor performance: firstly, the minimal plasma radius rm; secondly, themaximum total plasma pressure PM; thirdly, the duration over which theplasma is compressed - the time over which the inner wall velocity is negativetc; fourthly, the maximum plasma temperature ΘM; and finally the Lawsontriple product computed as

ΠL =

∫t:vL<0

n(t)Θ(t)dt. (14)

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Figure 7: Velocity of the plasma wall.

Figure 8: Plasma pressure.

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Table 4 provides the baseline value of the predictions, and subsequently variesonly one of the parameters, displaying the results, except for the last linewhere four parameters were varied in their optimal direction to yield ourmost optimistic operating condition of a reactor.

While many of the parameters have little effect upon the compression,some parameters cause drastic changes in the reactor behaviour, even whenperturbed by 10%, most notably the impulse pressure of the pistons, theinitial size of the plasma, the initial size of the lead-lithium sphere, and thetimescale of the impulse. In interpreting these results, some care is needed,as not all the variables are physically independent. For example, the pressureintensity of the pistons is primarily governed by their impact velocity andthe sound speed of the lead-lithium. Increasing the pressure could come fromincreasing sound speed, perhaps by changing the lead-lithium to a differentmaterial, or by increasing the impact velocity. Observe, though, that purelyby increasing the sound speed at a fixed impulse pressure, the compressionis less effective.

With a fixed initial plasma pressure, if the initial plasma radius is smallerthen there is more focusing and the compression is much more effective, justas it is with a higher impulse pressure or a larger lead-lithium initial radius.The time scale is also important; the more drawn out the pulse is, the moreenergy that can be added to the system to drive compression.

As a point of interest, we consider the engineering parameters that canmost readily be controlled and the aspects of the model with the largest pos-sible error and perform larger perturbations. We consider the performance-enhancing prospects of doubling the pressure at the outer wall, doubling thepulse time scale, and doubling the initial radius of the lead sphere. Thedoubling of the pressure could in principle be achieved with faster pistonsand doubling the radius of the lead-lithium sphere is a possibility. There isuncertainty as to the precise form of the pressure imparted as a function oftime and doubling the time-scale, which essentially amounts to consideringa symmetric instead of one-sided Guassian wave, is well within the realm ofmodelling uncertainty.

Under optimal conditions, we obtain a Lawson triple product of 1.6×1016

keV s cm−3, giving promise of net energy gain, which occurs when the outerradius is doubled. Doubling the pulse time or the impulse pressure also leadto large triple products, which are close to 4× 1015 keV s cm−3.

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Table 4: A sensitivity analysis of the reactor performance upon variations inthe model parameters. We denote the minimum radius by rm, the maximumtotal pressure by PM, the compression time by tc, the maximum temperatureby θM, and the Lawson triple product by ΠL. The final three rows of thetable illustrate the most promising conditions for fusion.Parameters rm (cm) PM (GPa) tc (ms) ΘM (keV) ΠL (1015 keV s cm−3)Baseline 3.6 6.7 0.43 2.7 0.52β0 ↓ 10% 3.6 6.8 0.43 2.5 0.49β0 ↑ 10% 3.6 6.6 0.43 2.9 0.55c∗ ↓ 10% 3.3 9.5 0.33 3.2 0.56c∗ ↑ 10% 3.8 5.3 0.52 2.5 0.46PR(0) ↓ 10% 4.4 2.8 0.63 1.8 0.37PR(0) ↑ 10% 3.0 15. 0.30 4.0 0.64PL(0) ↓ 10% 3.4 7.5 0.43 2.7 0.53PL(0) ↑ 10% 3.8 6.0 0.43 2.7 0.48rL(0) ↓ 10% 1.9 71. 0.20 8.0 1.4rL(0) ↑ 10% 5.5 1.7 0.85 1.4 0.25rR(0) ↓ 10% 4.3 3.0 0.61 1.9 0.27rR(0) ↑ 10% 3.0 13. 0.31 3.8 0.92t0 ↓ 10% 4.3 3.2 0.60 1.9 0.39t0 ↑ 10% 3.1 13. 0.32 3.7 0.61t0 ↑ 100% 1.3 570 0.15 20. 1.9PR(0) ↑ 100% 1.2 690 0.14 22 16rR(0) ↑ 100% 0.84 4900 0.12 24 2.5

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5 Summary and Future Work

Our work here has developed the basis of a numerical framework that could beuseful in fusion modelling, and with this numerical framework, we have madea preliminary examination into the operational conditions and feasibility ofmagnetized target fusion reactors. Within the model assumptions, currentdesigns for magnetized target fusion reactors are within reach of reachingfusion conditions. There is, however, a high degree of sensitivity of physicalparameters upon the resultant behaviour. Having a higher impulse pressure,a large confining sphere, and a long impulse are key factors in achieving ahighly efficient reactor.

There are many avenues for future work. Firstly, the some of the modelassumptions such as isentropic conditions could be relaxed, and we couldinclude more material properties of the plasma instead of it being homoge-neous, and consider the possible effects of turbulence. We could also considera geometry that is not spherically symmetric and examine how much the re-sults differ. Another natural question is how deviations from a sphericalsymmetry behave, and if there is a growth in such deviations, such as whathas been investigated in [14]. It would also be of great interest if an analyticestimate for the plasma compression could be obtained. We also note thatthis model’s predictions about the collapse differ somewhat from previousresults [4], except for some of the more promising results of the sensitivityanalysis. This model represents success in numerically solving conservationlaws with free boundaries with a constant computational domain, but at thesame time we realize that the physics has been greatly simplified. Furtherinvestigation, including tuning the mathematical model to more accuratelyrepresent the underlying physics, is warranted.

References

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[3] V. Kedrinskii, Hydrodynamics of Explosion: Experiment and Models,Springer: 2005.

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[4] M. Laberge “An Acoustically Driven Magnetized Target Fusion Reac-tor”, Journal of Fusion Energy 2007, 27.

[5] M. Laberge “Experimental Results for an Acoustic Driver for MTF”,Journal of Fusion Energy 2008, 28.

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[7] Lawson, J., “Some Criteria for a Power Producing Thermonuclear Re-actor”, Proceedings of the Physics Society, Section B 1957, 70.

[8] Mansfield, D. at al., “Enhancement of Tokamak Fusion Test Reactorperformance by lithium conditioning”, Physics of Plasmas 1996, 3.

[9] Miller, R. L. and Krakowski, R. A. “Assessment of the Slowly-ImplodingLiner (LINUS) Fusion Reactor Concept” 4th ANS Topical Meeting onthe Technology of Nuclear Fusion 1980

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[11] Rothman, S. D.; David, J-P; Maw, J; Robinson, C. M.; Parker, K.;and Palmer, J. “Measurement of the Principal Isentropes of Lead andLead-Antimony Alloy to ∼ 400 kbar by Quasi-Isentropic Compression”,Journal of Physics D: Applied Physics 2005, 38.

[12] Tan, S., and Shu, C. “Inverse Lax-Wendroff Procedure for NumericalBoundary Conditions of Hyperbolic Equations: Survey and New Devel-opments” (pre-print)

[13] Herbeert, E., Balibar, S., and Caupin, F., “Cavitation pressure in wa-ter”: 2006 (pre-preint)

[14] Suponitsky, V., Froese, A., and Barsky, S. “Richtmyer-Meshkov Instabil-ity of a Liquid-Gas Interface Driven by a Cylindrical Imploding PressureWave ” Computers and Fluids 2014, 89C, pp1-19.

[15] Edwards et al. Phys. Plasmas 20, 070501 2013

[16] Woodruff, S., Macnab, A., Mattor, N. “Adiabatic Compression of aDoublet Field Reversed Configuration (FRC)”, Journal of Fusion En-ergy 2007.

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[17] Gray, T., Brown, M. R., Cothran, C.D., Marklin, G., Schaffer, M.J.“Stable spheromack formation by merging in an oblate flux conserver”,Physics of Plasmas 2010.

Acknowledgments

M.L. would like to thank Westcoast Energy Inc. for a scholarship during thisresearch. B.W. would like to thank NSERC for a research grant.

M.L. is grateful to Michael Ward and David Seal for insightful discus-sions, and would also like to express a huge thanks to Randy LeVeque for anice discussion and for writing such a clear book on the complex subject ofnumerical methods for hyperbolic conservation laws.

The authors would like to thank Aaron Froese of General Fusion forhelpful literature references and clarifications, and the reviewers for their ex-tremely helpful and constructive comments in making this manuscript clearerand more accurate; as lead author, not being a plasma physicist, M.L. is mostappreciative for the insights shared by the reviewers to make the discussionsmore physically accurate and application-relevant.

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