Ideal Magnetohydrodynamic Simulations of Low Beta Compact Toroid Injection into a Hot Strongly Magnetized Plasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] J un Ideal Magnetohydrodynamic Simulations of LowBeta Compact Toroid Injection into a Hot StronglyMagnetized Plasma
Wei Liu , Scott C. Hsu , Hui Li Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA,87545 Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA, 87545E-mail: [email protected]
Abstract.
We present results from three-dimensional ideal magnetohydrodynamicsimulations of low β compact toroid (CT) injection into a hot strongly magnetizedplasma, with the aim of providing insight into CT fueling of a tokamak with parametersrelevant for ITER (International Thermonuclear Experimental Reactor). A regime isidentified in terms of CT injection speed and CT-to-background magnetic field ratiothat appears promising for precise core fueling. Shock-dominated regimes, which areprobably unfavorable for tokamak fueling, are also identified. The CT penetrationdepth is proportional to the CT injection speed and density. The entire CT evolutioncan be divided into three stages: (1) initial penetration, (2) compression in the directionof propagation, and reconnection with the background magnetic field, and (3) comingto rest and spreading in the direction perpendicular to injection. Tilting of the CT isnot observed due to the fast transit time of the CT across the background plasma.PACS numbers: 25.60.Pj, 28.52.Cx, 52.30.Cv,52.55.Fa,52.65.Kj Submitted to:
Nuclear Fusion ompact Toroid Fueling
1. Introduction
It is important to deliver fuel into the core of a tokamak fusion plasma to maintainsteady-state operation, achieve more efficient utilization of deuterium-tritium fuel,and optimize the energy confinement time [1]. Several fueling schemes have beenproposed, such as edge gas puffing, pellet injection [2], and compact toroid (CT)injection [3]. Among them, CT fueling is considered to be the most promising methodfor core fueling because the injection speed via this method is far higher than thoseof the other methods. Although extensive worldwide efforts have been devoted tostudy CT fueling theoretically [1, 3, 4], numerically [5, 6, 7, 8], and experimentally[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], the dynamics ofcore CT fueling of large devices like ITER (International Thermonuclear ExperimentalReactor) [28] is not well understood. CT injection has the potential to deposit fuel ina controlled manner at any point in the machine, from the edge to the core. Tangential(toroidal) injection can impart momentum for improving plasma β and stability [29]. Ina burning plasma device with only radio-frequency (rf) for auxiliary current drive, a CTinjection system may be the only internal profile control tool for optimizing bootstrapcurrent and maintaining optimized fusion burn conditions. CT fueling also provides agood chance to study core transport in present machines, helium ash removal, and EdgeLocalized Mode (ELM) [30] control.In this work, we employ a simple idealized model of a low β CT propagating intoa uniform slab plasma with a uniform magnetic field perpendicular to the CT injectiondirection, mimicking CT fueling into a tokamak with infinite aspect ratio. This modelhelps us identify different regimes of operation in terms of CT injection speed, density,and magnetic field strength, as well as understand the essential physics occurring duringCT injection. More realistic scenarios, including the use of realistic tokamak profilesand geometry in the background plasma, as well as high β CT’s and dense plasma jets,are planned for follow-on research. Compared to past work on CT injection simulations,we have investigated new regimes especially in terms of higher injection velocity anda more ITER-relevant ratio (at least for low β CT’s such as spheromaks) of CT-to-background magnetic field ( ∼ . ompact Toroid Fueling
2. Problem setup and numerical model
A low β CT with spherical radius r b = 1, centered initially at x b = 0, y b = 0and z b = z b, = −
12, is injected along the z axis into a lower density backgroundplasma with injection velocity v inj (see Figure 1). The basic model assumptions andnumerical treatments are briefly summarized here; they are essentially the same asthose in Li et al.[32] where more details are given. This code uses high-order Godunov-type finite-volume numerical methods. These methods conservatively update the zone-averaged fluid and magnetic field quantities based on estimated advective fluxes of mass,momentum, energy, and magnetic field at the zone interface [32]. The divergence-freecondition of the magnetic field is ensured by a constrained transport (CT) scheme[33]. All simulations were performed on the parallel Linux clusters at Los AlamosNational Laboratory. The nonlinear system of time-dependent ideal MHD equations in3D Cartesian coordinates ( x, y, z ) is given here: ∂ρ∂t + ∇ · ( ρ~v ) = 0 , (1) ∂ ( ρ~v ) ∂t + ∇ · (cid:18) ρ~v~v + ( p + B I − ~B ~B (cid:19) = 0 , (2) ∂E∂t + ∇ · (cid:20)(cid:18) E + p + B (cid:19) ~v − ~B ( ~v · ~B ) (cid:21) = 0 , (3) ∂ ~B∂t − ∇ × ( ~v × ~B ) = 0 , (4)in which ρ , p , ~v , ~B and E are the density, (gas) pressure, flow velocity, magneticfield, and total energy, respectively. I is the unit diagonal tensor. The total energyis E = p/ ( γ −
1) + ρv / B /
2, where γ is the ratio of the specific heats. Note thata factor of √ π has been absorbed into the scaling for both the magnetic field ~B andcurrent density ~j . It should be noted that the details of effects such as reconnection andheat evolution could not be addressed accurately due to the ideal MHD model and theuse of a simplified energy equation.It is well established empirically in coaxial gun spheromak experiments that, underproper conditions, a spheromak “magnetic bubble” (a low β CT) will be formed bythe gun discharge [34]. In our simulations, we do not model the CT formation processand instead start with a pre-formed CT moving toward the background slab plasma atspeed v inj . The stationary background plasma is composed of a slab plasma confined bya uniform background magnetic field B p in the x direction. For simplicity, we assumethe background plasma has uniform initial number density ρ p = 0 . T p = 0 . r b = 1 is given by ρ b ∝ r c exp[ − r c − ( z c − z b ) ] , up to a normalization coefficient and a uniform temperature T b , where r c = p x + y ompact Toroid Fueling z c = z (see Figure 1). The density profile used here has its peak shifted from thecenter of the CT, approximating a spheromak.The CT magnetic field is determined by three key quantities: the length scale ofthe bubble magnetic field r B = 2, the amount of poloidal flux Ψ p , and the index α ,which is the ratio of the CT toroidal to poloidal magnetic fields. For simplicity, the CTmagnetic field ~B b is also assumed to be axisymmetric. The poloidal flux function Ψ p isspecified as Ψ p ∝ r c exp[ − r c − ( z c − z b ) ] . (5)The poloidal fields, up to a normalization coefficient, are B b,r c = − r c ∂ Ψ p ∂z c , B b,z c = 1 r c ∂ Ψ p ∂r c , (6)while the toroidal magnetic field is B b,ϕ c = α Ψ p r c = αr c exp[ − r c − ( z c − z b ) ] . (7)The azimuthal component of the CT Lorentz force is zero, but the total azimuthalLorentz force due to the combined fields and currents of the CT may be non-zero.The CT also has uniform injection velocity v inj and uniform rotation angular speed ω . In this paper the ratio of the CT’s toroidal to poloidal magnetic fields α , therotation speed of the CT ω , and the specific heat γ are taken to be √
10, 0 and 5 / R = 10 cm, density ρ = 7 . × − g / cm − (corresponding to plasma numberdensity, which is the sum of electron and ion number density, of 18 . × cm − ), andvelocity V = 1 . × cm s − . Other quantities are normalized as: time t = 1 gives R /V = 5 . × − s, magnetic field B = 1 gives (4 πρ V ) / = 5 . × G, and energy E = 1 gives ρ V R = 2 . × ergs.The boundary conditions are all perfectly conducting in the y and z directionsexcept at the port where the CT is injected, while in the x direction outflow boundaryconditions are employed in order to mimic the toroidal geometry of a tokamak. Sincethe toroidal dimension is much larger than the poloidal dimension in a real tokamakand we focus on the early stage of the CT evolution before one toroidal propagationtime of the Alfv´en wave induced by the CT propagation, the out-flowing boundarycondition is more appropriate than the periodic boundary condition used in Suzukiet al. [7]. In order to minimize the influence of the entrance port, the port will beswitched on when the top of the CT reaches the bottom boundary at t = 2 /v inj andit will be switched off at t = 5 /v inj after the CT has fully entered the computationdomain ( t ∼ /v inj ). Suzuki et al. have pointed out that the boundary condition in thebackground magnetic field ( x ) direction is important, i.e. , magnetic reconnection hasmore influence on CT deceleration with perfectly conducting boundary conditions thanwith stress-free boundary conditions such as outflow and periodic boundary conditions[7]. The total computational domain is | x | ≤ | y | ≤
9, and | z | ≤
9, correspondingto a (180 cm) box in actual length units (assuming the physical dimension of the ompact Toroid Fueling r b = 10 cm). The numerical resolution used here is 400 × × x , y , and z directions. A cell δx (= δy = 2 δz = 0 . .
45 cm. Since the plasma skin depth and iongyroradius based on the parameters of ITER are no more than δx , the simulations basedon an MHD model are appropriate.
3. Results
In this section we present ideal MHD simulation results on the injection of a low β CT into a hot strongly magnetized plasma. We organize our results into three topics:(1) parameter regimes, in terms of v inj and ratio of CT-to-background magnetic field,of CT evolution including the identification of a promising regime for ITER-relevantprecise core fueling, (2) detailed description of the CT evolution for the ITER-relevantregime, and (3) dependence of the CT penetration depth on v inj and the initial CTdensity. Based on our simulation results, we find that the evolution of the injected CTdepends predominantly on the initial injection speed v inj and the initial ratio of CT-to-background magnetic field. As shown in Figure 2, there are several qualitatively distinctregimes of CT injection in terms of the above two parameters.First, below a threshold injection speed V L , the CT is unable to penetrate thebackground plasma at all (see left panel of Figure 3). The conducting sphere (CS) model[1] requires that the initial CT kinematic energy exceeds the background magnetic fieldenergy excluded by the CT volume,12 ρ b v > B p → v inj > V AC = B p √ ρ b . (8)Our results show that V L , which is inferred ( ∼ .
5) from extrapolation of the datashown in Fig. 9(a), is less than V AC . This is because of the compression of the CTduring the penetration, which is ignored by the CS model (see discussion of § i.e. ,the CT actually excludes less volume than its initial volume, therefore leading to asmaller initial injection speed threshold. And since this compression is related to thebackground plasma magnetic field (see discussion in § V AC derived fromthe CS model still gives a rough estimate of the lower limit of the injection speed neededfor penetration.Second, above a different threshold injection speed V AP , a strong shock andwavefront are observed to develop ahead of the CT, dominating the system evolution(see middle panel of Figure 3). This threshold is determined by the condition when v inj exceeds the Alfv´en speed of the background plasma, i.e. , v inj > V AP = B p / √ ρ p . Thisshock/wavefront-dominated regime is probably not favorable for CT fueling because it ompact Toroid Fueling middle panel of Figure 3). For comparison, the result ( v inj = 1 . right panel of Figure 3)( V AP = 3 .
16 in this case).Third, in the regime V L < v inj . V AP , CT evolution is further determined by theratio S r = B b /B p of the CT field B b to the background plasma magnetic field B p . If S r ≫
10, a large non-zero initial Lorentz force results in strong CT expansion thatalso leads to the development of a shock and a wavefront that dominate the systemevolution [35]. As stated above, this does not favor controlled plasma fueling. In theregime 1 < S r .
10, which is the regime Suzuki et al. have discussed extensively [5, 6, 7],the CT is decelerated by both the magnetic pressure and magnetic tension forces. Inthis regime, the CT tilts while reconnection occurs between the CT and backgroundplasma magnetic fields. Suzuki et al. proposed the Non-slipping Conducting Sphere(NS) model [5, 6, 7], which matches their simulation results of vertical injection withouta magnetic field gradient pretty well [7]. Our simulation results in this regime verifytheir conclusions. Because it is difficult for the CT field B b generated by a co-axialgun to be larger than the tokamak field B p of several Tesla or more, it is important toexplore the regime S r <
1, which is the primary focus of the remainder of this paper. Wehave identified this regime ( right panel of Figure 3) as a promising one for CT fuelingof ITER-relevant plasmas due to the precise spatial deposition of the CT and the deeppenetration that can be achieved for core fueling applications. In this paper, we havefocused on background plasmas with β p = 0 .
02 like in
ITER , while simulation resultswith β p = 0 . In this sub-section, we describe in detail the CT evolution for the ITER-relevant regimeof V L < v inj . V AP and S r <
1. In this regime, the CT evolution can be dividedinto three stages: (1) initial CT penetration, (2) CT compression in the propagationdirection ( z ) and reconnection, and (3) coming to rest and spreading in the x (toroidal)direction. Magnetic reconnection, starting late in the second stage in our simulations,arises due to numerical diffusion and leads to mixing between the CT and backgroundplasmas.In order to mimic CT injection with ITER-relevance, we adopt physical quantitiesas given in Table 1 and choose the injection speed v inj such that V AC = 1 . < v inj = 1 . 16 (this falls into the “ITER-relevant” case of Figure 2). Figure 4 displays thetime evolution of the plasma density (color contours in common logarithmic scale) inthe x - z plane with y = 0. The white solid contour lines indicate the magnetic pressure p B = B / 2. Fig. 5 displays the axial profile of x and y integrated density R x R y ρdxdy at different times corresponding to Fig. 4. The entire evolution can be described by thethree stages mentioned above. ompact Toroid Fueling B xz , (arrows) and current density j y (colorcontours) in the x - z plane at y = 0. During the first stage (initial penetration), the CTexperiences a very strong magnetic obstacle, and therefore the CT plasma is highlycompressed and the plasma density increases at the interface between the CT andbackground plasmas. A large plasma current also appears at the interface due to thecompression of the background magnetic field seen in Figure 6(a). This current sheetis bent and broken into two parts (Figure 6(b)) due to the magnetic field configurationof the CT field. Some reconnection takes place at the left part of the current sheet asshown in Figure 7. The CT is successively decelerated by the magnetic tension force ofthe background magnetic field.After the CT has fully entered the background plasma region (after t ∼ t = 4 . B x . From Figure 8, we can see that B x changessign from positive to negative and then to positive again. The transition from positiveto negative happens at larger z on the left hand side while at smaller z on the righthand side, which is the case in Figure 7( right ) (see two dashed lines in Figure 7( right )).From Figure 7, primary reconnection sites are at the upper left and lower right sectionsof the CT. The reconnection process allows the high-density CT plasma to escape fromthe CT and eventually flow outward along the background magnetic field horizontally(Figure 7). The CT plasma starts to contract in z and expand in x . The reconnectionis asymmetric about the CT axis, and this asymmetry results in CT plasma outflowin a direction that is not completely in the x direction, but rather obliquely (panel (d)of Figure 6). Eventually it will become more parallel to x ( right panel of Figure 3).Magnetic flux is being destroyed as well. During this process the MHD wave drag fromAlfv´en waves induced by horizontal plasma outflow [1, 4] might further slow down theCT plasma. Note that Suzuki et al.[7] showed that the CT penetration depth, basedon a model with magnetic tension force as the main deceleration mechanism, matchessimulation results very well, implying that MHD wave drag forces may not be importantin CT deceleration. The initially injected magnetic and perpendicular kinematic energiesare converted into parallel kinematic energy. Contrary to [1] and [5, 6, 7], CT tilting,the time scale of which is proportional to p B b B p /ρ b , is not observed in our case due tothe fast injection and short CT transit time as required for the ITER-relevant regime(Table 1).After the high-density CT plasma has been depleted during the compression stage(after t ∼ ompact Toroid Fueling z ). A narrow elongated structure along x results, as seen inthe lower middle and lower right panels of Figure 4. This line-shaped structure with aspread of only ∆ z ∼ . In this sub-section, we establish the dependence of CT penetration depth, an importantparameter for CT fueling of tokamaks, on the experimentally controllable parametersof CT injection speed v inj and initial density ρ b , which collectively determine the initialCT energy. The penetration depth S is defined as the axial ( z ) distance between thefinal mean position of the injected CT and the injection location on the boundary.We find that S is highly dependent on the initial injection speed v inj and CT density ρ b . Figure 9(a) displays the relationship between the S and v inj , showing that S isproportional to v inj if V L < v inj < V AP . Figure 9(b) displays the relationship between theCT density and penetration depth, which shows that the penetration depth S increaseswith the CT density. The CT spread in the z direction is around ∼ . ∼ 20. Thesetwo empirical relationships are important in the sense that they provide clues for howto choose the injection speed and CT density to get precise fuel deposition, thereforecontrolling the core plasma profile in a large tokamak such as ITER . It is very hard tomanage that by other methods such as pellet injection [26].For the case shown in Figure 9(b), the parameters given in Table 1 are adopted( v inj = 2 . 0) except CT density ρ b and CT field B b . We choose them such that theplasma parameter β = 0 . β constant since experimentally this is more reasonable thanchanging CT density or CT field strength independently. Thus, increasing CT densitymeans increasing both the initial kinematic energy E k = R / ρ b v dV , where dV isthe infinitesimal CT volume, and initial magnetic energy E m = R / B b dV . However,because of the limiting criterion given in Eq. 8, the initial CT kinematic energy E k should be much larger than the initial CT magnetic energy E m since the backgroundfield is much larger than the CT field in all simulation results presented in Figure 9(b)( B b /B p = 0 . j × B − ∇ p is proportional tothe CT density ∼ O ( ρ b ). Therefore, increasing the CT density ρ b would elevate theinitial residual force, which would result in shock/wavefronts. Thus, with too large adensity ratio ( ρ b /ρ p & 15, which is inferred from simulation results), the fueling is notas localized as the case with smaller CT density, while generating similar phenomena asthe shock/wavefront dominated cases.Results of Figure 9 remind us of the importance of initially injected energy uponthe penetration depth. As discussed in § ompact Toroid Fueling v inj (since plasma parameter β ≪ 1, the internal energy of the CT plasma can beignored). Certainly the larger injection speed would lead to deeper penetration.Figure 10 shows the time evolution of the net toroidal magnetic flux ψ t = R B y dS (only positive B y is selected) with v inj = 1 . ρ b /ρ p = 10. Before t = 6 . 25, the net CTtoroidal magnetic flux increases due to the initial penetration and compression of themagnetic field lines at the interface of the CT and background plasma. After t = 11 . t = 7 . t = 11 . 25, the only source of the destruction of the nettoroidal magnetic flux is magnetic reconnection due to the numerical diffusion. If wefit the decay as ψ t ( t ) /ψ t ( t = 0) ≡ exp( − t/τ res ), the “resistive” dissipation time due tonumerical diffusion is τ res ∼ . 9. Therefore the global resistive decay due to numericaldiffusion is not important on the time scales of CT transit time ( t ∼ 5) given injectionspeed v inj = 1 . 1. In the simulation, mixing arises from numerical diffusion inducedreconnection and occurs on the same time scale as the CT transit time. However, themixing should be less of a factor in reality due to the much smaller diffusion in a hightemperature tokamak. 4. Conclusions & discussion In this paper we presented nonlinear ideal MHD simulation results of a compact toroidinjected into a hot strongly magnetized plasma. The simulations are intended to provideinsights into CT fueling of an ITER-class tokamak. As a first step, we have investigatedthe problem of a high density low β CT injected into a slab background plasma withuniform magnetic field and density. We intend to investigate the injection of high β CT’s as well as unmagnetized dense plasma jets, and also incorporate more realisticbackground profiles in follow-on work.Our main findings are as follows. A regime is identified in terms of CT injectionspeed and CT-to-background magnetic field ratio that appears promising for precise corefueling. Shock-dominated regimes, which are probably unfavorable for tokamak fueling,are also identified. The CT penetration depth is proportional to the CT injection speedand density. For the regime identified as favorable for precise core fueling, the entire CTevolution can be divided into three stages: (1) initial penetration, (2) compression in thedirection of propagation, and reconnection with the background plasma, and (3) comingto rest and spreading in the direction perpendicular to injection. Tilting of the CT isnot observed due to the fast transit time of the CT across the background plasma.Reconnection occurring at the upper left and lower right portions of the CT fragmentsthe CT and leads to CT plasma outflow horizontally along the background magneticfield lines, forming a line-shaped structure almost parallel to the background magnetic ompact Toroid Fueling v inj ≪ V AP , ρ b /ρ p . B b /B p ≪ 1, then precise core deposition with minimal background equilibriumperturbation may be possible.Because V AC ∼ − for ITER (using parameters in Table 1), it will be achallenge for present CT injectors with v inj ∼ 400 km s − to achieve deep penetration. Apotential solution is to increase ρ b /ρ p to at least 100, which would reduce V AC to a morereasonable value ( ∼ 170 km s − ). Due to computational limitations, a simulation of sucha case (very dense CT) is beyond the scope of this study. However, to provide insightinto the penetration characteristics of a very high density injected plasma, we report heresome preliminary results on the injection of a dense ( ρ b /ρ p = 538 or ρ b ∼ cm − ),cool ( T b /T p = 3 . × − or T b = 2 . Acknowledgments The authors thank Shengtai Li for extensive advice on the code. The authors also thankRoger Raman and Xianzhu Tang for very useful discussions and constructive comments.This work was funded by DOE contract no.DE-AC52-06NA25396 under the Los AlamosLaboratory Directed Research and Development (LDRD) Program. [1] P. B. Parks. Phys. Rev. Lett. , 61:1364, 1988.[2] S. L. Milora et al. Phys. Rev. Lett. , 42, 1979.[3] L. J. Perkins et al. Nucl. Fusion , 28:1365, 1988.[4] W. A. Newcomb. Phys. Fluids B , 3:1818, 1991.[5] Y. Suzuki et al. Physics of Plasmas , 7:5033, 2000.[6] Y. Suzuki et al. Nuclear Fusion , 40:277, 2000.[7] Y. Suzuki et al. Nuclear Fusion , 41:873, 2001.[8] Y. Suzuki et al. Nuclear Fusion , 41:769, 2001.[9] M. R. 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Spheromaks . Imperial College Press, London, 2000.[35] W. Liu et al. Phys. Plasmas , 15:072905, 2008.[36] R. Raman et al. Fusion Technol. , 24:239, 1993.[37] F. D. Witherspoon et al. submitted to Rev. Sci. Instrum. , 2009. B x0 B x0 r cr b V injO(0,0,0) XZ Y ω Figure 1. Schematic of the simulation geometry showing the coordinate system. Inthe text, the direction along the z − axis is defined as the axial direction. ompact Toroid Fueling V inj B b B p >>B pAP >VV L Accurate Deposition Non−accurate DepositionLine−Shape Fueling No InjectionShock/Wavefront Dominated Figure 2. Qualitative behavior for CT injection in terms of CT injection speed andmagnetic field strength. Figure 3. (color) Density in the x - z plane at y = 0. Left panel: at t = 20 withinjection speed v inj = 0 . 3, which shows no penetration. Middle panel: at t = 2 . v inj = 10 . 0, which shows non-localized deposition. Right panel:at t = 20 with injection speed v inj = 1 . 1, which shows highly localized deposition. V AP = 3 . Table 1. Normalized physical quantities. Note that V AC = B p / √ ρ b = 1 . 0. Injectionspeed v inj is between V L and V AP . Plasma is assumed to be composed of half deuteriumand half tritium. The plasma density is the sum of electron and ion densities. CT BackgroundPhysical Quantities numerical physical numerical physicalMagnetic Field B b = 0 . . 53 T B p = 1 . . ρ b = 1 . . × cm − ρ p = 0 . . × cm − Temperature T b = 0 . 001 75 ev T p = 0 . . β = 2 ρT / < B > β b = 0 . β p = 0 . V A = B/ √ ρ V AB = 0 . − V AP = 3 . 16 5 . × km s − Sound Speed V c = √ γT V cB = 1 . × − . 89 km s − V cP = 0 . 41 7 . × km s − ompact Toroid Fueling Figure 4. (color) Density (logarithmic scale) in the x - z plane at y = 0 as a functionof time. The white lines are contours of magnetic pressure p B = B / 2. For t = 2 . p B ∈ [0 . , . t = 3 . p B ∈ [0 . , . t = 7 . p B ∈ [0 . , . t = 8 . p B ∈ [0 . , . t = 12 . p B ∈ [0 . , . t = 15, p B ∈ [0 . , . ompact Toroid Fueling −9 −8 −7 −6 −5 −4 −3 −23000400050006000700080009000 Z ∫ x ∫ y ρ d x d y t=2.5t=3.75t=7.5t=8.75t=12.5t=15 Figure 5. (color) x and y integrated density R x R y ρdxdy versus z at different timescorresponding to Fig. 4. Black dash: t = 2 . 5, black dash dot: t = 3 . 75; black solid: t = 7 . 5; red dash: t = 8 . 75; red dash dot: t = 12 . 5, red solid: t = 15. Figure 6. (color) Magnetic field and current density j y in the x - z plane as a functionof time at y = 0. The color contours show j y while arrows show B x and B z , whichare normalized to their maximum values. (a) t = 2 . 5; (b) t = 4 . t = 6 . 25; (d) t = 8 . 75. Please note that the scales for the abscissa and ordinate are not identical. ompact Toroid Fueling Figure 7. (color) Diagram of the magnetic configuration in the contraction stage.Red color indicates the high-density CT plasma. Yellow lines indicate magnetic fieldlines, while blue arrows indicate CT plasma flow patterns. See Figure 8 for dottedblack lines. Figure 8. Axial profiles of several quantities at t = 4 . 375 with ( x, y ) = ( − . , left ) and ( x, y ) = (0 . , 0) ( right ) . The density ρ and magnetic field strengthin the x -direction B x are shown for evaluating the magnetic configuration shownin Figure 7( right ). These two axial profiles correspond to the two dash lines inFigure 7( right ). ompact Toroid Fueling Figure 9. (a) Penetration depth S versus injection speed v inj with ρ b /ρ p = 10 and(b) Penetration depth S versus CT density ρ b with v inj = 2 . 0. Both panels assume B b /B p = 0 . 1. The penetration depth is defined as the distance between the final meanposition of the injected plasma and the bottom boundary. The error bar indicates thefinal size of the injected CT in the z -direction. In panel (a), the dash line indicates V L ∼ . V AP = 3 . Figure 10. Decay of the net toroidal magnetic flux ψ t = R B y dS , where onlypositive B y is selected with v inj = 1 . ρ b /ρ p = 10. The dashed line fits thedata between t = 7 . t = 11 . 25 with the formula ψ t ( t ) /ψ t ( t = 0) ≡ exp ( − t/τ res ),where τ res = 10 . ∼ v inj = 1 . ompact Toroid Fueling Figure 11. (color) Density in the x - z plane of y = 0 at t = 150 with injection speed v inj = 0 . ρ b /ρ p = 538, and B b /B p = 0 ..