I. Introduction: BackgroundThroughout the past century, “the growth of

the energy demand [has] strongly accelerated,following the spreading of the industrializationto underdeveloped countries and the global pop-ulation growth.” (11) With the dwindling sourceof energy, blooming human population, and ex-tension of the average lifespan, the demand forenergy in contemporary society has increased.It is imperative that, with the diminishing sup-ply of fossil fuel in addition to the repercussionsthe continual use of such has had upon the envi-ronment, having led to a substantial increase inatmospheric carbon dioxide concentration andlung cancer, alternative energy sources be as-sumed for continual future use (11). Recent re-search and modern scientific thought have re-vealed nuclear fusion as a potential and viablesource of electrical energy for future purposes,which, unlike contemporary alternative energysources, is not hindered by economic drawbacksor safety factors (12).

Despite its seemingly great potential, how-ever, nuclear fusion’s potential has yet to be re-alized: in its current state, nuclear fusion reac-tors require a greater expenditure of energy tooperate and initiate than is generated throughthe fusion process itself. Yet, precise theoreti-cal calculations have pointed to nuclear fusionas a promising candidate for future use afterovercoming these obstacles in efficiency peoplehave been researching for decades. Nuclear fu-sion energy revolves around one central tenet:the potential energy stored in nucleons increaseswith atomic mass up to a limit, portrayed in Fig1. This mass difference, translated to energy

as per E = mc2, yields far less energy in fis-sion than in nuclear fusion, in which two verylow mass particles combine into a more massiveparticle (1), (3). Fusion research consists of twomain approaches: inertial confinement and mag-netic confinement differing in, as their namesimply, how they confine the plasma.

Figure 1: Above illustrates the greater potential tobe harnessed with the exploitation of nuclear fusionas contrasted with that currently achieved, and ismaximally possible, with the use of fission (36).

Thus, much of modern research has prolifer-ated in determining how magnetic confinementreactors can be made to either operate using lessinput energy or generate greater energy. The lat-ter is currently being investigated through theITER project, an international effort costing $20billion and estimated to take 12 years to con-struct. Despite the great prospects of this devel-opment project, even if it achieves an efficiencyratio greater than one, the amount of moneyand time expended upon the project makes itinfeasible for future implementation. Withoutimproved efficiency, reactors cannot be imple-mented in impoverished areas where energy ismost needed.

Most of the energy expended in magnetic

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confinement is in one of two manners: the ini-tial heating process of the tokamak or the mag-netic confinement. Within a reactor, magneticconfinement serves two purposes, namely pre-venting contact with the interior surface of thereactor, known as the “tokamak,” and achievingfusion. More explicitly, when a fusion reactoris initially heated to ionize the neutral gas con-fined, typically on the order of 100 million ◦C,it is imperative that the ions do not make contactwith the tokamak surface, for the surface itselfdoes not achieve such temperatures (13), (14).

Therefore, the objective of fusion reactors isto maintain small distances between nucleonsfor long enough to yield the necessary strongforce interactions between them to overcomethe electrostatic barrier impinging upon fusion.However, there arise complications in the con-finement process, necessitating the use of threemagnetic fields to confine the plasma, namelythe toroidal, poloidal, and vertical fields, de-scribed with the following equations, respec-tively (12):

Bφ ≈BφR0

R≈ BφR

1− εcos(θ)(1)

Bθ ≈µ0Ip(r)

2πr(2)

Bvert =µ0Ip4πR0

ln(8R

a− 3

2+`

2+ βp) (3)

Where ε is the reverse aspect ratio of the toka-mak, a and b the minor radii of the horizon-tal and vertical portions of the tokamak, R0 themajor axis of the tokamak, and Ip the toroidalplasma current (18).

Combined, the three magnetic fields confine

Figure 2: These are three main magnetic fieldsapplied externally to a tokamak to ensure thatit achieves the confinement necessary for fusion,namely the poloidal, toroidal, and vertical fields (32).

the plasma and ensure that they neither imposemechanical stress upon the physical reactor norlead to instabilities in the plasma. However,this combination of three fields is inefficient inits energy usage, due to the overlap and can-cellation between them: each of the fields im-poses an undesired agitation in the plasma thatis resolved by introducing the other two fieldsimposed. As a result, herein, a new methodof achieving fusion confinement is explored us-ing a dual set of interlocked elliptical electro-magnets, one encompassing the reactor and onewithin the fusion chamber (39), (40). This par-ticular geometry was chosen for confinement as,in a prior fusion investigation, it was noted aboutthis geometry that “[this novel geometry is capa-ble of] confining positrons and electrons simul-taneously in the same volume, at any degree ofneutrality and at relatively high particle kineticenergies, operating steady state, and operatingat ultralow densities,” thus making it applicableand more efficient in various situations (4).

Heating of tokamaks, on the other hand, oc-curs via the injection of waves. The particularmechanism by which the heating takes place is

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electron cyclotron resonance heating; ECRH re-volves around the idea of resonance, wherebythe amplitude of the motion of the electrons,which move at a natural “cyclotron frequency”increases when the injected waves correspondsto said frequency (24). Though the mechanismsbehind electron cyclotron resonance are quitecomplex, simplifications led to the followingequations for the heating, in which the formerdictates how plasmas are heated as a whole andthe latter that which applies specifically to re-gions subjected to ECRH (16):

dW

dt=

e2E2⊥ν(ω2 + ω2

c )

m[(ω2 − ω2c )2 + 4ω2ν2]

+e2E2

‖ν

mω2(4)

dW

dt=

e2E2⊥ + e2E2

‖ν

mω2(5)

Where E corresponds to the electric field, ωthe injected wave frequency, ωc the electron cy-clotron frequency, and ν the collision frequency.Due to the evident contingency of the heatingon the electric field and, therefore, the magneticfield within the plasma, altering the shape ofthe confinement method too would have greatimplications on the heating efficiency, whichtoo was explored. Current heating methods of-ten fail to interact fully with regions subject toECRH. ECRH is solely exhibited within cer-tain regions of the plasma, and these are mostoften shielded by cutoff, or “reflective,” lay-ers in the plasma, preventing waves from pass-ing through and heating the plasma (23). Thenovel magnetic field shape, however, lacks suchself-intersecting cutoff boundaries, making theprospect of efficient heating appealing. Manystudies regarding heating use simulations, typ-

ically implementing the Vlasov Equation (11)coupled with Maxwell’s equations:

∂f

∂t+

dq̄

dt· ∂f

∂q̄+

dp̄

dt· ∂f

∂p̄= 0 (6)

∇ ·E =ρ

ε0,∇ ·B = 0 (7)

∇× E =∂B

∂t,∇× B = µ0I +

1

c2∂E

∂t(8)

In essence, the Vlasov Equation governs the mo-tion of the constituent electrons and ions, withina plasma, and how the evolution of the plasmadistribution function over time, represented asf in this scenario, can be simulated through aniterative process of electrostatic field and poten-tial calculations (12). Yet, its complexity, in fact,is so great that modern computers are simply un-able to perform all the necessary computationsand processes, which have forced researchers tosimplify studies by reducing the total numberof dimensions under consideration, from six toeither four or even two, which are two dimen-sional or one dimensional respectively due tothe multiplicity of the velocity and space dimen-sions. Through this investigation, the Vlasovequation was considered in a 1D simulation tostudy heating preliminarily.

Despite all the advancements in magneticconfinement fusion, there remains a dearth ofmethods by which the magnetic field orienta-tion internal to the plasma reactor can be de-termined: a necessary component for investi-gations regarding novel magnetic confinementschemes. Though there exist manners of detect-ing local external magnetic fields, the most com-mon method being the Langmuir Probe, such are

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not capable of detecting internal magnetic fieldof tokamaks (3). Solely rudimentary forms ofsuch inventions have been hitherto developed,and such technologies are incapable of beingable to adapt to a highly fluctuating magneticfield (17). Though useful for initial determina-tion of a magnetic field, its application to a fieldthat incessantly varies is impractical, which ledto the need to develop a new manner of probinga fluctuating magnetic field of tokamaks usingreflectometry (5).

Plasma waves, as with optical waves, suchas light, exhibit specific characteristics, such asdiffraction, refraction, reflection, interference,transmission, and absorption, to varying ex-tents as per the internal magnetic field strengthsand orientations in the reactor. Analyzing thiswave characteristic data thus allowed for ac-curate probing of internal magnetic propertiesof the plasma, hitherto undetectable (23) (24),which was herein preliminarily established andsubsequently implemented for the experimentalinvestigations.

When considering plasma and electromag-netic oscillations, phenomena not relevant in thecontext of mechanical vibrations become signif-icant, confounding heating investigations. The“dampening” that occurs at cutoffs results in thedissipation of energy, muddling the heating ofplasmas (17). As desired resonance regions areblocked off by undesirable cutoff regions, muchof the energy from the injected waves is unableto reach the desired locations and resonate withthe electrons to heat the plasma. It was there-fore the purpose of this investigation to deter-mine the ideal angle at which the waves should

be initially injected for maximal heating of theplasma through mode conversion, as to allow forthe propagation of waves past such cutoffs (37)(11) (43).

As with other barriers, those present in plas-mas are selectively obstructive for waves. Moreexplicitly, certain barriers are cutoffs for X-mode (extraordinary) propagation whereas oth-ers are for O-mode (ordinary) waves, which aretwo of forms of wave propagation in plasmas.Specifically, for waves injected at 2.45 GHz, re-gions containing the resonantly-responsive elec-trons are typically surrounded by cutoffs exclu-sively selective to O-modes. As a result, the ap-plicability of refined heating schemes is contin-gent upon overcoming the cutoff barriers usingOX conversion. In essence, mode conversionoccurs when an O mode, or any plasma wavemode, encounters a cutoff boundary. Typically,the wave’s amplitude attenuates until reachingan amplitude of zero, whereby the wave is con-sidered to then be nonexistent; however, sincewave modes are differing branches of dispersionrelation (Eq. 9), there are degenerate points inphase space where two modes coexist, as in theOX solution branch.

|b k

c ·ω× (

k

c ·ω× 1) + K(ω,

k

c ·ω)c| = 0 (9)

Where ω is the plasma wave oscillatory fre-quency, K is the wave number, and N is themagnitude of oscillation. Namely, this degen-erate solution means that, at particular points inphase space, the O and X modes coexist (25).By exploiting this degenerative solution of the

4

Figure 3: Though certain waves are unable to passthrough the cutoff region, as they have too great anattenuation to pass to the other end of the cutoff withsignificant amplitude or are reflected off of the cutoffsurface (purple), should the wave be injected at pre-cisely the desired velocity and angle, as is portrayedby the connection of the red and blue lines, the waveis capable of overcoming the cutoff(39).

wave dispersion relation, the wave can be madeto “convert” from an O mode to an X mode.Should the angle of injection of the wave bewithin a specific region of the cutoff layer, thewave is capable of bypassing the cutoff regionand reaching the other end with sufficiently highenergy/amplitude, albeit having attenuated sig-nificantly, such that it is able to incite the propa-gation of an X mode waves oscillation.Through said mechanism, the wave effectivelyovercomes the boundary by converting its modeto one not subject to the cutoff, specificallyan X mode. Modern fusion reactors, however,solely inject waves orthogonally to the centerconfine of the chamber, regardless of its geom-etry, which leads to further inefficiencies in the

heating scheme. Mathematically speaking theconversion efficiency follows the following re-lation:

exp{−2πk0L

√Y

2((1 + Y)(∆N2

z + N2y)} (10)

Where k0 is the vacuum wavelength, L is thelocal density scalelength, Y = ωe/ω, and N =√

Y/(Y + 1) (49). Thus, as with the lack ofself-intersecting cutoff boundaries in the novelelliptical design, the injection of waves at theoptimized angle could further lessen the neces-sary expenditure of energy to achieve the desiredtemperatures in the interior of the reactor.

Despite the ostensibly greater concentrationplaced upon magnetic confinement fusion, theretoo are great prospects for inertial confinement:a different perspective of approaching fusion en-ergy. Inertial confinement is the antithesis ofmagnetic confinement, in which both the timeand spatial scales are miniscule (51).

Thus, inertial confinement, rather than usinga reactor, uses a capsule containing a mixtureof deuterium and tritium typically enclosed ina cylindrical container, known as a hohlraum.When this hohlraum is superheated by lasers, itreleases electromagnetic radiation (25), directedtowards the capsule. After absorbing the radia-tion, the outer layers of the capsule vaporize dueto the excess of energy. As per Newton’s thirdlaw, the release of this excess energy absorbedfrom the hohlraum, causes an equal and oppositeimplosion of the rest of the capsule, containingthe fusion elements. In contemporary facilities,192 lasers fire with approximately 1.8 MJ of cu-

5

Figure 4: A contemporary hohlraum design: de-spite its seeming symmetry, however, its edges limitthe predictability in the radiation of energy used toheat the contained capsule. As a result, much of theenergy is distributed non-uniformly, leading to themanifestation of instabilities.

mulative power in a time scale of 10−11 s at acapsule with a diameter of 2-3 mm (48). The im-plosion of the capsule causes the propagation ofa heat front outward, achieving tremendous heatof 100,000,000 ◦C (27), which, in turn, yieldsthe ideal conditions to initiate fusion, ionizingthe surrounding gas (1). Thus, this heat wavegenerated through the implosion process beginsthe ignition process, which cascades into reac-tions amidst ions in further regions surroundingthe capsule, perpetuating nuclear fusion.

Despite the seeming prospects of the afore-mentioned method, tests in the National IgnitionFacility have been irrefutably negative, as “de-livering the driver energy and compressing thetarget uniformly without exciting instabilitiesthat compromise the compression [of inertialcapsules] requires high precision in space andtiming [which current technological prowessdoes not encompass]” (10), (20). Contrastedwith those in tokamaks, the instabilities in in-ertial capsules contribute far more significantlyto the implosion process, mostly due to thelarge contrast in the spatial scale; the Raleigh-Taylor (RT) Instability long has been cited as

the main impediment to achieving ignition (28),(29), (31). RT instabilities, in essence, manifestas a result of the mixture of hot and cold flu-ids, modeled with the use of the Navier-Stokesequations. With major simplifications, Eq. 15 isthe main result regarding the growth of capsuleinstabilities (26):

∂ρYi

∂t+∇ · (ρYiu + Ji) = 0 (11)

∂ρu

∂t+∇ · (ρuu + pδ − τ) = ρg (12)

∂E

∂t+∇ · ((E + p) · u− τu + q) = ρgu (13)

p = ρR(

√ρh − ρlρh + ρl

gα− 1)e

R0

2∑i=1

Yi

Wi

v

(14)

h ≈ h0eγt (15)

Where h is the “height” of the instabilitieson the capsule surface, thus making apparentthe exponential growth of these instabilities withtime (21). As such instabilities increase withplasma density, it is evident why they playmore significant roles in inertial confinementcontrasted with that which they play in mag-netic confinement. It was, therefore, further thepurpose of this herein research ordeal to de-velop a novel holhraum design that constrainsthe mixing of hot and cold fluids, in turn miti-gating instabilities (6). To accomplish such, thehohlraum design had to be symmetric. Thoughthe current cylindrical hohlraum device is seem-ingly symmetric, it suffers a loss in symmetrydue to edge effects: the novel design sought toeliminate such edge effects (33).

Furthermore, through a combination of these

6

two confinement methods, explored in a fieldcurrently in its infancy called magneto-inertialconfinement, great potential for increased lev-els of Q exists. Such potential manifestsfrom the beneficial repercussions arising fromthis coupling of confinement methods (10),(20). Namely, a phenomena known as abla-tion front stabilization occurs in inertial confine-ment: such a phenomena results in the mitiga-tion of instabilities upon the surface of the cap-sule when pressure is applied to the surface, for“the pressure Poisson equation are consideredas stabilized” (3). The use of magnets to con-fine the capsule, should the confinement meth-ods be combined, similarly leads to the applica-tion of external pressure, analogously reducingthe exhibited instabilities (26). Since the iner-tial confinement chamber would be located cen-tral to an external magnetic field, the capsulewould experience a magnetic pressure, in turnstabilizing its surface and mitigating turbulence.Moreover, in the inertial confinement process,highly energetic X-rays are released from thehohlraum, which, when coupled with the toka-mak chamber, could be used to instigate theheating/ionization of the neutral gas.

II. HypothesesThus, through the herein detailed research,

the following was investigated: determining amore efficient manner of magnetically confiningthe plasma within the tokamak; a manner of im-proving the efficiency of electron cyclotron res-onance heating; and studying the mitigation ofinstabilities that arise through the implosion ofa D-T capsule. The investigation, therefore, wascontingent upon the investigation of a set of hy-

potheses, all related in their efforts towards im-proving the efficiencies of fusion reactors:Magnetic Field Confinement: H0/H1: There isno/a significant difference between the magneticfield efficiency, defined as the ratio of confine-ment to the energy expenditure, of the three-field system as contrasted with that of the novelpolar magnetic confinement fields.Mode Conversion: Heating Efficiency: H0/H1:There is no/a significant difference between theheating achieved when the angle is injected or-thogonal to the center of the reactor as com-pared to when injected with the angle found tomostly closely resemble the optimized transmis-sion curve.Electron Resonance Heating (ECRH): H0/H1:There is no/a significant difference betweenthe temperature achieved through the ECRH ofmodern tokamaks as compared to that achievedin the novel confinement chamber (CNT).Inertial Confinement Instabilities: H0/H1:There is no/a significant difference between theamplitudes of the instabilities on the surfaceof the implosion capsule in the case of typicalhohlraum design versus that of the novel design.Magneto-Inertial Confinement: H0/H1: Thereis no/a significant difference between the energyefficiencies (Q), defined as the ratio betweenthe energy generated to that expended, achievedthrough solely the magnetic and magneto-inertial confinement chambers.

III. Experimental MethodologyPrior to conducting experimentation, a man-

ner by which the interior conditions of theplasma could be probed chamber had to be de-veloped, which implemented the reflectometry

7

method hereto referenced to detect the presenceof cutoffs and measure interior properties of thereactor. A simulation capable of modeling theVlasov equation and evolution of a plasma toohad to be developed. In the end, a program,dubbed “Gkeyll,” was developed, which con-sisted of nearly 60000 lines of code. Using thisdeveloped program, the plasma could be suc-cessfully investigated in simulations. For eachof the following tests, potentially confoundingvariables were kept constant, namely the mag-netic field at 500 Gauss (.5 T), the radiation in-jected as 2.45 GHz (microwaves), and the lo-cations of the components in the experimentalsetup.

A. Mode Conversion: Heating EfficiencyTo develop the diagnostic, data regarding the

reflection, transmission, and other optometricproperties of the plasma waves were collectedby a series of sensors that lined the interior ofthe plasma reactor and were responsive to thefrequency of the injected waves. As each prop-erty is contingent upon the internal magneticand pressure profiles, these profiles of the reac-tor could subsequently be extracted through re-verse calculation: a method later implementedin investigating the heating of tokamaks as a re-sult of an altered magnetic field geometry.

Subsequently, however, the heating schemewas developed to overcome cutoff boundariesusing OX mode conversion. With prelimi-nary investigation through simulations, O-modewaves were injected at oblique angles into theplasma, which, after encountering an O-modecutoff boundary, were to be partially reflectedand partially transmitted. Namely, those that

were “converted” to X-mode plasma oscillationswere able to bypass the cutoff location (25). Asthe extent of mode conversion greatly hingesupon the initial angle of incidence, the transmis-sion vs. angle of injection graph was expected toform a Gaussian curve with a central hole at theangle where the wave is fully converted from theO mode to X mode, indicative of the initial anglecapable of bypassing the cutoffs to the greatestextent. Due to the maximized transmission atthis initialization, this would be the optimal an-gle for injection so as to achieve heating in theplasma chamber.

Despite the contingency of the conversion be-tween the O and X modes upon both the fre-quency and angle of injection of the wave, thefrequency was preliminarily optimized at 10GHz, for, in the case of an elliptical reactorgeometry, this frequency demarcates the pointwhere additional energy in the wave would pro-duce unnecessary additional agitation to the sur-rounding medium and lower heating would pro-vide insufficient energy as to excite resonantelectrons (5) (19). As a result, simulations hadto be performed to measure the induced heatsolely respective to the angle of wave injection,with respect to the reactor center. Namely, aFEM (Finite-Element Method) COMSOL pro-gram was developed set up as per Fig 5. Thesimulation analyzed slab plasma, namely onewhich solely exhibits such characteristic wavepropagation in 2D rather than a fully 3D envi-ronment, which was too computationally expen-sive for these purposes, through which the re-sults were found to be sufficiently accurate as tobe implemented for further measurement.

8

As a result, to analyze the optimal heating an-gle, each of the curves was separated from Fig.11 and was normalized to a common height, toensure each could be compared to a commonsource without bias. Each was then overlaidupon the fully-optimized Gaussian-hole trans-mission graph and then analyzed to determinewhich minimized the sum of the residuals be-tween the two graphs or, in other words, ob-tained the one that most closely resembled thedesired curve indicative of pure transmission.For the optimal injection angle analysis, dθ waschosen to be 1◦.

As per the simulation results, the optimal an-gle was then applied to an experimental analy-sis to ascertain whether it improved heating ef-ficiency. Waves were injected at the optimal an-gle and the traditional orthogonal angle into theCNT reactor, as per the adjustments to the ex-ternal injector’s positioning. As the extent ofheating achieved translates to the temperatureachieved over the time span of the trial, the fi-nal temperatures, ascertained with a LangmuirProbe, were used to contrast the efficiencies ofinjections at varying angles.B. Magnetic Field Confinement: Simulation

Due to the great cost in conducting tokamakexperiments, prior to experimental verification,relevant simulations had to be conducted. Themagnetic confinement of plasmas using the typ-ical three-field confinement scheme was con-trasted with that of the elliptical field confine-ment in tokamaks initially using simulations, byapplying Comsol Multiphysics to a model of thetokamak developed in SolidWorks. The ellipti-cal reactor was chosen as, not only was there a

Figure 5: The above portrayal is that of the setup ofthe simulations conducted by which the cutoffs andtransmissions were observed for the varying frequen-cies under investigation. Specifically, as this investi-gation sought to solely determine the effects mani-fest in the context of a 2D phase space investigation,the thickness was made infinitesimally small, yield-ing an equivalent 2D workspace.

lesser overlap of the fields, but the lesser com-plexity in interactions between the multidimen-sional magnetic field vectors allowed for simplerdetermination of the location of cutoffs, thus al-lowing for ease in improving heating efficiencywithin the reactor.

Figure 6: The CNT reactor design is portrayed,making apparent the interlocked sets of ellipticalmagnetic rings constituting the reactor, one in thevacuum chamber and one encompassing the entirereactor. The red region is the predicted confinedshape of the plasma in the reactor (41).

9

Specifically, the simulation was used to deter-mine how the plasma waves, which propagatethroughout the reactor, respond to the mag-netic field, namely whether they remain withinthe reactor confines or drift towards the sides:Gkeyll was applied to observe the evolution ofthe plasma over time, through which the conver-gence or divergence of the plasma could be de-termined. Implementing the field sources intothe model, qualitative results for the confine-ment were determined, shown in Fig. 14, 13.

Figure 7: Above is delineated the conventionalspherical NSTX tokamak fusion reactor. Despite thecapabilities attained through NSTX, it has not yet at-tained breakthrough energy levels (52).

C. Magnetic Confinement: ExperimentalSubsequently, the confinement achieved was

verified with tokamaks. Namely, to ascertain thenecessary measurements pertaining to the mag-netic confinement of the plasma within the toka-mak, a Langmuir probe was implemented (Fig.8). Using this device, the interior plasma wavetemperature could be probed and so too couldthat of the plasma chamber wall be measured.As a result, by determining these diagnostics ofthe plasma chamber wall temperature, whose in-

crease is proportional to collisions due to parti-cles and is, therefore, inversely proportional tothe magnetic confinement achieved through thetokamak setup, the confinement could be quan-tified, though such measurements had to be nor-malized due to the varying interior surface ar-eas for the tokamaks under consideration. Thisprobe was positioned at the surface of the reac-tor, with an opening into the tokamak so as toprobe interior plasma conditions. In turn, theCNT and NSTX machines were implementedfor the purposes of magnetic confinement mea-surements, for the novel and traditional fieldorientations respectively. Prior to conductingthe tests, however, the theoretical confinementswere calculated for each of the two designs, des-ignating that achieved by the NSTX as τe andτp as that by the novel design. The standardconfinement time of a reactor (τe) is defined asthe ratio of the energy density to the rate of en-ergy loss to the surrounding system by a toka-mak (40):

Figure 8: The polar fields used for the magnetic con-finement of the plasma is shown in the left depiction,whereby the circular confinement devices simulatea Helmholtz Coil. The right portrays, for CNT, theLangmuir probe used for heating diagnostics of thereactor’s interior.

10

vD =E× B

B2+∇B× B

eB3(1

2mv2⊥ + mv2

‖) (16)

τp ≈ τea4

λ4D(17)

The first equation explores the forces experi-enced by the particles due to the magnetic fieldsand gradients. More important, however, is theconfinement determined, portrayed in Eq. 15,where λD is a fundamental constant in plasmaphysics known as the Debye length and a themajor radius; the confinement develops to agreater extent in the novel design.

The experimental setup involved the devel-opment of the polar tokamak reactor, as por-trayed in Fig 5. With such in place, the reac-tors were ignited with the injection of waves, re-sulting a temperature increase to approximately100,000,000 ◦C. However, solely the CNT toka-mak had to be tested experimentally, for theNSTX had been recently implemented, with thedesired specifications having already been mea-sured, whose results are delineated in Table II.

D. Cyclotron Heating: ExperimentalSubsequently, the extent of electron cyclotron

resonance heating was compared between thetwo reactor designs, namely through the tem-perature achieved within a specific time framewhen subjected to the same external conditions.In a similar vein, the ECRH was determinedand observed with the implementation of sim-ulations through Comsol Multiphysics and laterobserved in the context of experimental verifi-cation. Each of the two reactors was once againinvestigated, such that the amount of energy in-jected by the wave injector led to equivalent en-ergy per unit volume in the reactor. The temper-

atures in the interiors of the two were reactorswere then compared using the Langmuir probeto determine which design exhibited greater ef-ficiency. With equivalent exposure times, theefficiency of the heating mechanisms could bedetermined based on the final interior tempera-tures in terms of the heating achieved per unit ofinjected energy.

E. Hohlraum Instabilities: ExperimentalAfter magnetic confinement, inertial con-

finement too was investigated. The instabil-ities attained through prior inertial confine-ment hohlraum had to be contrasted withthat achieved through the novel design of thehohlraum, which too implemented the ellipti-cal containment design used for magnetic con-finement. This design evidently lacks the sharpedges associated with the cylindrical holhraumtypically employed; such edges often lead tounpredictable expulsion of waves and interac-tions with the surrounding medium. As a re-sult, there is often both unpredictable and non-uniform radiation that the capsule absorbs whenin a cylinder. For experimentation, the “heights”of the instabilities were verified using a novelmagnetic neutron spectrometer (27). In essence,a magnetic recoil neutron spectrometer is ca-pable of measuring the instabilities that arisewithin the D-T fuels through a mechanism in-volving “neutrono-deuteron (or proton) elasticscattering and magnetic dispersion of the recoildeuterons...The principle of the system is thata small fraction of the neutrons emitted fromthe implosion hit the CD2 (CH2) foil producingscattered recoil deuterons (protons). The energyrelationship between the recoil particle (Er) and

11

the neutron (En) is described by:

Er =4A

(1 + A)2Encos2θr (18)

Where A is the atomic mass number of therecoil particle, and θr is the angle between thedirection of the incoming neutron and the direc-tion of the outgoing recoil particle.” (20) Subse-quently, any neutron with a θr ≈ 0 is selectedthrough an apparatus positioned in front of thefoil that was struck by neutrons, which are ofprimary importance for the purposes of diagnos-tics. In turn, these particles are separated basedupon their varying momenta, as per:

Figure 9: Above is delineated how a magnetic recoilneutron spectrometer measures and determines prop-erties of the interior of inertial capsules by analyzingthe expulsion of neutrons (6).

Rg =p

qB=

√2mEr

qB(19)

Where p is the momentum, Er is the recoilparticle energy, q is the charge, and Rg is thegyro radius. As a result, through investigation ofthe resulting magnetic dispersion spectrum, crit-ical quantities for the purposes of inertial con-finement are determined, which can be adjustedto study the instabilities and turbulence that arise

Figure 10: The magnetic recoil neutron spectrom-eter is positioned with respect to the capsule underinvestigation, in an effort to determine the specificsof the interior of an inertial confinement capsule un-dergoing an implosion (6).

in an inertial confinement test. Specifically, theamplitudes of the instabilities that arise in thecapsule were measured by the displacement ofneutrons during the scattering process.

For the setup, each of the two holhraum de-signs was loaded with the D-T capsules, whichwere then fired, simultaneously, by 32 lasers,culminating in a 1.1 MJ of power. In turn, theamplitudes were compared to determine whichdesign exhibited greater turbulence based onthe spectrometer’s displacement readings. Thenovel design is shown as in Fig. 9, 10.F. Magneto-Inertial Reactor: Experimental

Subsequently, the magnetic and magneto-inertial reactors were ignited to verify and com-pare their respective efficiency (Q) values. Asthe NSTX has been extensively developed inthe past and is, alongside the DIII-D reactor,one of the most widely implemented nuclear fu-sion reactors in magnetic confinement fusion re-search, it is indicative of the extent of the capa-bilities achievable through the implementationof magnetic confinement. After constructing themagneto-inertial confinement reactors, shown in

12

19, each of these designs were tested in termsof the amount of electricity generated, as com-pared to that which was initially injected intothe system. Thus, equivalent amounts of energywere initially injected into the system for theheating, with a supply of 50kW power contin-uously supplied for 45 seconds, culminating in2.25 MJ of input energy. The total energy gener-ated was determined through a system of capac-itors, which was connected to the reactors fromthe trials; the capacitors were thereafter releasedthrough a circuit to determine the total energystored. For each design under investigation, theaforementioned research developments were in-corporated, namely in the heating, confinement,and turbulence mitigation mechanisms.

IV. ResultsFrom each of the conducted tests, the results

were compiled, in the numerical data obtained,graphical representations of such, and simula-tion results. The results of the test are portrayedbelow, though they are explicitly stated and ex-plored in the Discussions section (Section V).

A. Mode Conversion: Heating Efficiency

Through simulation of the proposed design im-plementing mode conversion scheme, the heat-ing scheme was confirmed in its efficiency, for

it exhibited the characteristic transmission curvepredicted prior to the simulation, shown in Fig.11. As per the analysis of the similarity of thedesired transmission graph profile and that ob-tained after normalization of each of the trials,the optimal angle was determined to be 37◦. Ap-plying this result to the final heating experimen-tal tests, the heating was evidently optimized us-ing the aforementioned angle of injection.

Figure 11: The electrical fields determined throughthe injection of the plasma wave is plotted as a func-tion of the major radius of the tokamak reactor forvarying angles of injection; thus, based upon theaforementioned parameters, the electrical field couldbe ascertained within bounds of error.

13

Figure 12: The heating achieved is plotted withrespect to the varying angles of injection, whichevidently exhibits a Gaussian curve and, therefore,achieved an optimal level of heating at the anglecoinciding with its peak, found to be 37◦. Herein,solely increments of five are shown, though the anal-ysis used to ascertain the optimal angle, in reality,incremented at intervals of 1◦.

B. Magnetic Field Confinement: SimulationThough the quantitative aspects of the confine-ment between the two reactors was comparedin the experimental verification, this providedqualitative confirmation of initial suspicions re-garding the confinement, whereby the ellipticalscheme reduced the instabilities and drift withinthe plasma to the central confines of the toka-mak, as was desired for fusion.

Figure 13: With the implementation of the Gkeyllprogram, the evolution of the plasmas when sub-jected to the NSTX confinement mechanism wascontrasted with that in the CNT reactor.

Figure 14: The plasma wave confinement simula-tion of the three-field vs. novel field, in which thecolors represent different intensities of the E-field.

C. Magnetic Confinement: Experimental

Through this test, therefore, the quantitative af-firmation of the previous trials became appar-ent, whereby the plasma was found to remainwithin the CNT plasma region with minimal in-teractions with the wall. By comparing the colli-sions between the inner tokamak surface and the

14

plasma of CNT to that of NSTX, the efficiencyof the confinement could be determined, affirm-ing the results of the simulations.

Figure 15: Images of the plasma in the CNT cham-ber subsequent to initial injection of waves and heat-ing. Such affirms the results obtained through thesimulation of the plasma confinement (4).

D. Cyclotron Heating: Experimental

After initially observing the expected heatingin the CNT model through simulations, the ex-perimental tests were conducted, through which

the temperatures attained with a given sourceof input energy were contrasted between NSTXwhich had previous test data and CNT. In turn,the efficiency of the cyclotron heating could bedetermined, as potentially confounding factorswere kept constant.

Figure 16: Once developed, the CNT model (left)was imported into the Comsol Multiphysics environ-ment for simulation work, and, through the use ofthe finite mesher, the previously continuous space ofthe reactor was broken into hundreds of millions oftetrahedrons. The heating simulation was conductedwith the model (right), whereby it was injected withwaves of frequency 2.45 GHz. The plasma was setso that only particular regions in the plasma resonatewith the frequency of the injected wave, as is the casein real reactors. Red indicates regions subject to thegreatest extent of heating and violet the lowest.

E. Hohlraum Instabilities: ExperimentalBy observing the simulations preliminarily, theunmitigated amplification of instabilities in thecylindrical capsule became apparent, as com-pared to the ones of finite amplitude in the noveldesign. Through the experimental data, sim-ilarly, the neutrons’ trajectories and displace-ments were measured from the initial restingpoint at the spectrometer. Determined basedupon the distance covered, the instabilities werecompared between the two designs.

15

Figure 17: Prior to experimentally verifying the mit-igation of instabilities on the capsule surface, the in-ertial hohlraums were developed into the Gkeyll pro-gram to contrast the manifestation of instabilities.

Figure 18: After performing simulation analysis,the inertial confinement hohlraum was constructed,which consisted of solely six copper enclosed rings,inspired by (42). The central space was where thecapsule was located during the conducted trials.

F. Magneto-Inertial Reactor: ExperimentalUnlike the previous cases, this trial pinpointedwhether the total efficiency of the reactor wasgreater in the novel design or in the traditionalmagnetic confinement scheme. Thus, by mea-suring and comparing the efficiencies, defined asthe energy generated per energy expended, thereactors could be compared to one another.

Figure 19: Portrayed is a combination of the novelinertial confinement chamber (the central compo-nent) and a surrounding tokamak. Excluded fromthis image are the elliptical magnetic loops that sur-round the tokamak reactor to confine the plasma forease of depiction. The blue capsule portrayed has al-ready undergone implosion and is now undergoingan expansion due to the released heat wave.

16

Figure 20: All the experimental results efficienciesdetermined are graphed, where the efficiencies aredefined as per the data in the tables and the reciprocalof the instability amplitudes for the inertial confine-ment. Too present are the standard deviation errorbars, portraying the significant differences presentbetween the sets of data.

F. Statistical AnalysisAs each of the hypotheses under investiga-

tion involved a comparison between two lev-els of investigation, each involved the usage oftwo-tailed, independent t-tests, for each test in-volved a sufficiently large data set to conduct t-tests, yet not large enough for a z-test. More-over, the usage of the phrase “significant dif-ference” as opposed to “significantly more/less”lent itself to the usage of a two-tailed test to de-termine whether there was indeed a significantdifference present. An independent test was ap-propriate for the investigation under considera-tion, for each of the hypotheses measured thedifference between two levels of investigationwith respect to different externally applied mag-netic fields and lasers, with the exception of themode conversion experimental trials, in whichthe tests were conducted on a single reactor and,thus, entailed a paired t-test. The p-values at-tained through this aforementioned t-test wereused to affirm the presence of a significant dif-ference or not, with respect to an α value of .05.

OX Mode: Heating p-value: 2.78 ×10−6

Cyclotron Heating p-value: 2.04 ×10−8

Magnetic Confinement p-value: 3.71 ×10−3

Inertial Instabilities p-value: 9.46 ×10−10

Magneto-Inertial p-value: 1.746 ×10−6

The p-values obtained through these testswere, evidently, less than the α of .05, lead-ing to a rejection of the null hypotheses (H0)and support of the alternatives (H1). Evidently,through the results of the statistical tests con-ducted, there was evidence of great statisticaldifferences amidst the tests under investigation.Namely, each of the aforementioned investiga-tions led to the conclusion that both magneticand inertial confinement have undergone im-provements in efficiency through the develop-ment of more efficient manners by which theplasma waves can be implemented for heatingcoupled with the development of a novel polarconfinement method; furthermore, such was de-veloped to reduce the instabilities on implosioncapsule surfaces.

V. DiscussionA. Mode Conversion: Heating EfficiencyAs a result, through the implementation of the

hitherto described mechanism, not only were theelectron and ion resonance heatings made moreefficient, but so too was a manner by which theinternal properties of the tokamak reactor couldbe ascertained developed. Through the simu-lations, it was found that the dispersion of thewaves exhibited reflection throughout its spec-trum, as is portrayed in Fig. 11, with the excep-tion of particular regions in space for specificangles, where the evanescent region was suffi-ciently small as to allow for the propagation of

17

further X waves. By injecting waves at theseoptimized angles, it was found that the heatingwas significantly more efficient and, therefore,achieved a greater temperature over the time in-terval as compared to that achieved in traditionalheating schemes, for it was capable of bypassingthe cutoff regions and solely interacting with theregions of the plasma sensitive to electron cy-clotron resonance.B. Magnetic Field Confinement: Simulation

Preliminarily, the Comsol Multiphysics sim-ulation was conducted, whose results are shownin Fig. 16, in which the latter (CNT confine-ment) portrays greater retention of the plasmawave development to the center confines of theplasma chamber, as desired, whereas the former(three-field scheme) leads to an irregular spreadin the wave dispersion, without any directionalconfinement. Evidently, there appears to be agreater potential for the confinement capabili-ties of the elliptical confinement fields. Suchis furthered by the results of the second Gkyelltest. The three-field approach, though it startedout well-confined, evolved into a free-streamingplasma with instabilities (dark ellipses stream-ing alongside the mainstream confined plasma).The CNT reactor, on the other hand, culminatedin a continual cycle of circular motion of theconstituent plasma, in which the plasma was re-tained in the central confines of the reactor, andthere was a lack of significant instabilities in theplasma evolution.

C. Magnetic Confinement: ExperimentalSubsequently, the results were carried out ex-

perimentally. Fig. 13 evidently portrays howthe plasma was confined to the central regions

of the reactor and did not, as in the case of thethree-field scheme, lead to great exhibition of in-stabilities and turbulence. Furthermore, the dataconfirm the theoretical predictions, in which theelliptical magnetic confinement scheme causedlesser contact with the reactor walls. As a re-sult, these findings portray the flaws of currentmagnetic field designs that have been hereto ac-cepted without much opposition, even thoughmore efficient reactor design geometries exist.Unlike previous research, this illustrates a com-pletely novel design and its efficacy as con-trasted with studying the theoretical effects orimprovement of the three-field design.

D. Cyclotron Heating: ExperimentalThrough simulations, only the CNT model

was investigated, as the heating of NSTX hashitherto been studied extensively. Evidently,the greatest heating was found to be concen-trated in the regions in which the plasma wasmost densely confined and dwindled with dis-tance from the central region being heated tothe greatest extent. Similarly, from the numer-ical data, the temperatures were found to haveincreased more efficiently when subjected tosolely the elliptical fields, caused by the lack ofself-intersecting cutoff regions in the reactor. Asprevious studies did not consider such confine-ment geometries, studies for heating have beenmostly limited to the context of a three-field re-actor, making the portrayal of heating unique.

E. Hohlraum Instabilities: ExperimentalContrasting the exhibition of instabilities and

turbulence in the typical cylindrical hohlraumwith that in the novel elliptical design in Fig.18, the evolution of the cylindrical hohlraum in-

18

dicates the lack of instability control, whereinthe amplitude grows exponentially within theshort time frame. On the other hand, the novelelliptical confinement constrains such instabili-ties to solely the confines of the capsule interior,thereby removing a major impediment in theimplementation of inertial confinement fusion.Though researchers have identified the sourcesof instabilities and turbulence as the hohlraum,they have failed to pinpoint the exact flaws in itsdesign, namely its lack of symmetry due to un-predictable effects along the cylinder edge, dis-tinguishing this study as one of the only onesregarding the development an alternative designto mitigate such an issue.F. Magneto-Inertial Reactor: Experimental

Each of the rings from which the magneto-inertial confinement magnet was constructedcomposed of a metal surface encompassing acopper loop interior, as shown in a cutaway forthe front ring. As an electrical circuit flowsabout the circular copper loops, they exhibitmagnetic fields, in turn confining the plasmathat manifests from the initial implosion of theconfined capsule. The emission of high-energyX-rays during the implosion causes ignition ofthe neutral gas enclosed within the encompass-ing tokamak. Due to the lack of a need to in-ject waves to ionize the tokamak, it achievedfar greater energy efficiencies, yielding levelsof Q > 1. Though physicists have attemptedto pursue magneto-inertial confinement in thepast, their efforts were unsuccessful, as they didnot exploit the complementary outputs of the in-ertial and magnetic systems to reduce the to-tal energy expenditure. By harnessing emitted

X-rays from the D-T capsule to heat the toka-mak and using the magnets to mitigate inertialturbulence, however, two of the main issues ofthe systems are resolved, unveiling the untappedpotential of magneto-inertial reactors on largescales.

VI. Conclusions/Future StudiesDespite such improvements made thus far,

however, neither of the two forms of confine-ment independently achieved the desired greateroutput of energy than that which was inputted.Combined, however, the Q achieved was ev-idently greater than one, a hallmark achieve-ment towards implementing fusion, due to theincrease in stability for the inertial capsule in ad-dition to the lessening of energy to be expendedfor the purposes of heating, due to the X-rayemission and ablation front stabilization.

Despite the prospects of global-scale projectssuch as ITER, researchers must couple thesewith smaller scale tokamaks that can be read-ily constructed and used for the purposes of en-ergy generation in the years to come, particu-larly in impoverished societies. Having deter-mined a manner by which the internal plasmaof a tokamak can be modeled accurately in 1D,so too must there be a development by whichthe heating simulation can account for toroidalripple, density and magnetic fluctuation effects,and be extended to fully three-dimensional mod-els. Having determined a more accurate modelfor plasma waves, the repercussions for plasmaphysics, specifically fusion research, are great,primarily due to the great importance the injec-tion of plasma waves plays in current drive andelectron cyclotron resonance heating. It is there-

19

fore of critical importance to optimize the anglewith which such waves are initially injected intothe reactor. Going forward, tests will be con-ducted regarding the implementation of such di-agnostics in larger scale reactors, though smallerscale testing has been hitherto conducted anddetermined to function sufficiently well. Fur-thermore, the mitigation of instabilities is in-complete, for, aside from the Raleigh-Taylor in-stabilities, the irregularities of the surface of thecapsule result in further turbulence. As withthe RT instabilities, despite their small scale,the effects of such irregularities becomes pro-nounced as the capsule expands outwards, tooproving to be a large impediment in achievingICF. Thus, research upon the development of amanner of controlling such surface irregularitieswill be conducted going forward.

Upon an experimental front, however, thereare many routes that emerge from the hithertopresented and conducted research. Namely, hav-ing determined the great prospects manifest-ing from the implementation of magneto-inertialconfinement, its efficiency can be explored inlarger-scale reactors. Furthermore, as ITER tooimplements the conventional three-field form ofmagnetic confinement, such projects can beginto implement the polar magnetic confinementmethod to determine whether its efficacy re-mains in large scale projects and to see whetherit significantly reduces the expenditure of en-ergy in confining of the plasma. Separately, an-other form of confinement exists, namely theZ-pinch fusion machine, demonstrated to ex-hibit levels of Q around 1000. The reason whysuch has not yet been implemented is in en-

ergy storage: due to the extremely short timespan during which such a tremendous amountof energy is generated, current energy systemsare unable to capture the full extent of the en-ergy. Yet, with further integration of graphene,which the student researcher is currently inves-tigating, the Z-pinch machine may well be im-plemented widespread soon. Therefore, despiteall the progress that has been hitherto accom-plished in the field of plasma physics, particu-larly in the case of nuclear fusion energy, thereremains much to explore and improve upon inboth theoretical and experimental studies.

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