Two-fluid model of rf current condensation in magnetic islands
TTwo-fluid model of rf current condensation in magnetic islands
S. Jin,
1, 2, a) A. H. Reiman,
1, 2, b) and N. J. Fisch
1, 2, c) Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08543,USA Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540, USA (Dated: 24 February 2021)
The stabilization of tearing modes with rf waves is subject to a nonlinear effect, termed rf current condensation, thathas the potential to greatly enhance and localize current driven within magnetic islands. Here we extend previousinvestigations of this effect with a two fluid model that captures the balance of diffusive and thermal equilibrationprocesses within the island. We show that the effective power, and resulting strength of the condensation effect, can begreatly enhanced by avoiding collisional heat loss to the ions. The relative impact of collisions on the overall powerbalance within the island depends on the ratio of the characteristic diffusion time scale and the electron-ion equilibrationtime, rather than the latter alone. Although relative heat loss to ions increases with island size, the heating efficiencydoes as well. In particular, we show that the latter safely dominates for large deposition profiles, as is typically the casefor LHCD. This supports the possibility of passive stabilization of NTMs, without the precise aiming of the rf wavesrequired for ECCD stabilization.
I. INTRODUCTION
Perturbations to the nested flux surfaces in tokamaks canresult in magnetic islands, which greatly enhance radial trans-port and degrade confinement. In the absence of externalheating, the pressure profile within the island tends to flatten.This locally suppresses the bootstrap current, reinforcing theoriginal magnetic disturbance and driving the island unstable.These unstable islands, known as neoclassical tearing modes(NTMs), are a major cause of disruptions and set a principalperformance limit in tokamaks.
Stabilization via current drive by rf waves has long beenrecognized as the leading solution, and has been the subjectof much theoretical and experimental work. Only re-cently, however, has the presence of the island and its effecton rf deposition been accounted for.
The thermal insulation provided by the closed mag-netic topology and reduced cross field transport within theisland produces significant temperature perturbations rel-ative to the background plasma in the presence of rf heat-ing. Relative temperature increases as large as 20% have beenreported. Furthermore, the waves typically used, lower hy-brid (LHW) and electron cyclotron (ECW) , resonate withsuperthermal electron populations at the location of the is-land and are thus highly temperature sensitive. The powerthat these waves can deposit is proportional to the numberof resonant electrons ( P dep ∼ n res ), which is exponentiallysensitive to temperature perturbations ( n res ∼ exp ( − w ) ≈ exp ( − w ) exp ( w ∆ T e / T e ) ), where w : = v res / v th is the ratioof the resonant and thermal velocities. As w ≈ −
20 inpractice , even small increases in temperature can drasticallyenhance the power absorption.In combination, these properties create a positive feedbackbetween the island temperature and rf deposition, resulting in a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: fi[email protected] significant nonlinear enhancement of current driven within theisland. Since the temperature within the island is also gov-erned by diffusion, the temperature will tend to be hottest inthe center, thereby preferentially enhancing the driven currentwhere it is most stabilizing. This amplification and focusingare termed the current condensation effect. The identifica-tion of this nonlinear process naturally leads to the suggestionthat stabilization schemes should then be crafted to maximallyexploit it. To this end, accounting for energy coupling be-tween electrons and ions offers critical insights.Previous investigations of the rf-condensation effectin magnetic islands have been confined to single-fluidmodels.
Here we show that these are limiting cases ofa more general two fluid picture—rf power injected in to theelectrons may be lost in varying proportions through diffu-sive losses to the background plasma, or first through colli-sions with the ions. This energy balance is set not only by thestrength of energy coupling between electrons and ions, buton the relative thermal diffusivities as well.Typical experimental parameters span a broad range of cou-pling strengths, and although the reduction of the electronthermal diffusivity within the island by an order of magni-tude or more has been widely reported, similar results for theions are not as established. It is therefore of great value toestablish an understanding of rf condensation that covers thewide variety of regimes relevant in practice.We show here that the degree of energy coupling dependson the ratio of the characteristic electron diffusion time t D . e ∼ W i / χ e and electron-ion energy equilibration time t eq , ratherthan the latter alone. The extent to which heating and sta-bilization are impacted by coupling depends on the ratio ofdiffusivities γ : = χ e / χ i ; the effective power is reduced by afactor (1 + γ ) in the strongly coupled limit, compared to thedecoupled limit.We also show that although both parasitic energy loss to theions and effective power increase with the island width, thelatter safely dominates. This suggests the possibility of self-healing islands for tokamaks with a significant fraction of thetoroidal current sustained with lower hybrid waves (LHWs).We estimate the self-stabilization to occur at experimentally a r X i v : . [ phy s i c s . p l a s m - ph ] F e b feasible rf power densities and island widths.The paper is organized as followed. Section II introducesthe two fluid model for the island temperatures, establisheslimiting behaviors, and summarizes the essential features ofthe solutions. Section III characterizes the impact of energycoupling on the current condensation effect. Section IV dis-cusses the possibility of self healing islands under LHCD.Section V summarizes the main results and conclusions. II. TWO FLUID MODEL OF ISLAND TEMPERATURES
The energy transport equations for electrons and ions canbe written as:32 ∂ t n s T s − ∇ · ( n s χ s · ∇ T s ) = τ eq n s ( T r − T s ) + P s (1)where subscript s denotes either electrons or ions; subscript r denotes the other species; χ s is the heat diffusivity tensorof species s ; and τ eq = m i m e (cid:112) m e π ( kT e ) / ne λ is the electron ionequilibration time, where λ ≈
20 is the coulomb logarithm, m s is the mass of species s , µ is the ion mass in units of protonmasses. P s contains whatever species specific heat sourcesand sinks may be present, aside from the explicitly writtenelectron-ion equilibration term.Considering rf induced temperature perturbations to an oth-erwise flat pressure profile within the island , then P e = P r f and P i =
0. Alternatively, even if the ohmic and radiationterms do not balance , as long as they are negligible com-pared to the rf power, our selective choice of source termsis justified. As the time scales on which the island widthor background plasma parameters evolve are typically muchslower than the timescales of interest (i.e. the characteristicdiffusion time of either species, t D , s : = W i / χ ⊥ , s , and theelectron-ion equilibration time, t eq ), we may consider "steadystate" solutions for which ∂ t →
0. On these timescales, thefield lines will be approximately isothermal, so Eqs. (1) maybe written as coupled 1-D diffusion equations for the per-turbed temperatures u s : = w (cid:101) T s / T s , :ˆ Du e = P r f + c ( u i − u e ) (2) γ ˆ Du i = c ( u e − u i ) (3)with u s = c : = t D , e / t eq is theratio of the electron diffusion and electron-ion energy equili-bration times, and γ : = χ ⊥ , i / χ ⊥ , e is the ratio of the electronand ion diffusivities. The rf power term has been scaled to P scl : = nT e , / w t D , e . Further discussion of the boundary con-ditions and approximations used in arriving at Eqs. (2) and (3)can be found in Ref. 31, with the difference here being the re-laxed assumption of perfectly equilibrated ions and electrons.ˆ D is a diffusion operator that accounts for the geometry ofthe flux surfaces within the island :ˆ D : = − ρ K ( ρ ) dd ρ E ( ρ ) − ( − ρ ) K ( ρ ) ρ dd ρ (4) where ρ is a flux surface label that is 0 at the island center, and1 at the separatrix. The island topology is more often definedwith the alternate coordinate Ω : = ( r − r s ) / W i − cos ( m ξ ) = ρ −
1, where r s is the resonant radius, and ξ : = θ − nm φ isthe helical phase, θ ( φ ) and m ( n ) are the poloidal (toroidal)angle and mode number respectively. The current condensation effect enters through the nonlin-ear rf heating term, P r f . As discussed earlier, the exponentialincrease of superthermal resonant particles with small elec-tron temperature leads to the following form: P dep ∝ exp ( u e ) .In the case where the rf deposition profile is wide compared tothe island, the linear power deposition profile may be taken tobe constant across the island region. This is especially appro-priate for modeling lower hybrid current drive (LHCD) .For such a power bath model, the rf power term takes the form: P r f = P exp ( u e ) (5)This model isolates the physics of RF-condensation, but ne-glects the possibility of ray depletion.The two-fluid equations 2 and 3 capture the flow of energythrough the island system—in through rf heating of the elec-trons, and out through diffusive losses from both the electronsand ions. The total energy loss from the electron population isthe sum of two competing processes, direct diffusive losses tothe environment, and collisional heat exchange with the ions.The relative strength of each process is roughly character-ized by the dimensionless parameter c : = τ D , e / τ eq ∼ W i . Ofcourse, heat loss to the ions is also regulated by how quicklythe ions can dump this heat to the environment—this is char-acterized by the parameter γ : = χ i / χ e . A. Limiting cases
The two fluid model of the island temperature contains 3characteristic time scales: the electron diffusion time τ D , e ,the ion diffusion time τ D , i , and the electron-ion energy equi-libration time τ eq . The ratios of the first quantity to thelatter two give the two dimensionless coupling parameters: γ = : χ ⊥ , i / χ ⊥ , e = τ D , e / τ D , i describing the relative diffusivi-ties of the ions to the electrons, and c : = τ D , e / τ eq describ-ing the relative strength of collisional to diffusive processesfor the electrons. Familiarity with the limiting cases of thesecoupling parameters will build a foundation for understand-ing further two-fluid results, and contextualize the single fluidmodels that have been used thus far.
1. Single fluid reduction in limiting collisionality regimes
If there is significant scale separation of the electron diffu-sion time and equlibration time, the dynamics are essentiallythat of a single fluid. In the c → c → ∞ limit corresponds to the electrons and ions hav-ing identical temperatures and behaving as a single fluid withan averaged conductivity. This can be seen by writing equa-tion 3 in a more illuminating form: u i = u e − c ( γ ˆ Du i ) (6)so as c → ∞ , we can take u i ≈ u e = u .Adding the ion and electron equations then gives1 + γ Du = P r f / c = t scl → t scl / ( + γ ) , P scl → P scl ( + γ ) . We see that inthis limit, the system behaves as a single fluid with the powereffectively reduced by the factor 1 / ( + γ ) . Technical excep-tions to these limits will be discussed in the following section.
2. Influence of conductivity ratio γ The above single fluid cases can be further refined depend-ing on the relative ordering of the ion-diffusion time. The c → τ D , e << τ D , i << τ eq ( c ⇒ , γ →
0) and (2) τ D , i << τ D , e << τ eq ( c → , γ → ∞ ). When both parameters are go-ing to the same limit, the double arrow denotes the one thatapproaches that limit faster.For the special case of (3) τ D , e << τ eq << τ D , i ( c → , γ ⇒
0) , although c →
0, the ions will eventually also arrive at thesame temperature as the electrons. This heating occurs despitethe collisional heat source to the ions vanishing, as the rate atwhich they diffusively dissipate any input heat is vanishingeven faster. This is a particularly pathological case however,and only mentioned for completeness.For c → ∞ , the ions will generally be at the same tem-perature as the electrons. This holds true for the order-ings (4) τ eq << τ D , e << τ D , i ( c → ∞ , γ →
0) and (5) τ eq << τ D , i << τ D , e ( c ⇒ ∞ , γ → ∞ ), but case (6) τ D , i << τ eq << τ D , e ( c → ∞ , γ ⇒ ∞ ) is subtly different. Case (4) is the only oneof the c → ∞ cases that can accomplish efficient heating, since γ → c → + γ ). The subtle difference between the two cases is that therelative temperature difference in case (6) can be significant,as although the directly heated electrons are immediately ex-changing their heat with the ions, the ions are dumping thatheat even faster. B. Steady state solutions
Prior to discussing the impact of two fluid features in par-ticular, here we will highlight the basic properties of rf cur-rent condensation, common to all coupling regimes. As thereis typically a large separation between the time scales ofinterest , i.e. diffusion or energy equilibration, and the MHDtimescales on which the island evolves, we can discuss "steadystate" solutions for which the island width can be treated as aconstant. Eqs. (2) and (3) then yield two solution branches,joined at a bifurcation point, as shown in Fig. 1. The lower(upper) branch is stable (unstable), FIG. 1. Electron and ion temperatures vs scaled power P , for γ = As discussed in more detail in Ref. 30, the power bathmodel does not admit steady state solutions for rf powerspast the bifurcation point. At these higher powers, the is-land temperature will continue to grow until encountering ad-ditional physics not included here, such as ray depletion or stiffness. Note that power localization improves with in-creasing island temperature, further enhancing the stabiliza-tion efficiency. The bifurcation point therefore provides anestimate of the rf power required to access dramatic nonlinearenhancement, for a given set of plasma/island parameters. Thesudden temperature increase (relative to the MHD time scaleson which the island typically evolves) also serves to provide adistinctive experimental signature.It must be noted however, that the bifurcation point is by nomeans a requirement for stabilization, as will be discussed fur-ther in section IV. Since the power deposition is exponentiallyenhanced by the temperature perturbation, sufficient stabiliza-tion can occur at moderate electron temperatures, especiallywith the higher efficiency (larger w ) of LHWs. This couldbe beneficial for avoiding turbulent transport enhancement atlarger island temperatures . III. IMPACT OF ENERGY COUPLING ONSTABILIZATION
As shown in our discussion of limiting cases, the effectivepower is reduced by a factor of 1 + γ when the electrons arefully equilibrated with the ions, relative to when their ener-gies are decoupled. At minimum, γ ≈ , and could be much higher, as turbulence sup-pression and reduced electron diffusivities have been widelyreported in islands . In particular, if transport is predomi-nantly neoclassical, then γ ≈
10. As the power is nonlinearlyrelated to the island temperature, this can amount to an evenmore significant impact on stabilization. Fig. 2 summarizesthe effect of energy coupling on the effective power, using thebifurcation threshold as a marker.
FIG. 2. Scaled power to reach bifurcation point vs. coupling param-eter c . It can be seen that the dimensionless power correspond-ing to the bifurcation threshold scales roughly logarithmi-cally with coupling parameter c between the limiting regimes, c ≈ − − . Although Fig. 2 provides the most con-cise summary of the impact of energy coupling on effectivepower, it must be emphasized that P bi f and c are not the ac-tual power and equilibration rate, but are scaled to P scl = nT e / w t De and t − De respectively. To get a sense of the typ-ical values for magnetic islands in present day devices, with n ≈ − × cm − , T e ≈ . − keV , W i ≈ − cm , χ e ≈ . − m / s , this gives P scl ≈ . − KW / m and c ≈ . −
10. Lower effective power, both through collisionswith the ions, and through the increase of P scl can therefore bea concern for higher density plasmas. Although higher T e andsmaller W i reduce the relative heat loss to ions, the increase of P scl will dominate, leading to lower effective power.The stabilization efficiency also depends on the current pro-file, with current driven closer to the center being more stabi-lizing, while current driven closer to the periphery can evenbe destabilizing. Fortunately, the shape of the power deposi-tion profile is notably insensitive to the central heat sink pro-vided by the ions. It will require a larger effective power toachieve a certain u e ( ) , but for that u e ( ) the rest of the tem-perature (and resulting power) profile will not be appreciablydifferent. Note that this would not be the case for narrow,off center profiles which could result from imprecisely aimedECCD. Energy coupling can significantly impact the temper-ature profiles in such cases (Appendix C), exacerbating therisk of misalignment, on top of the reduced effective power. It should therefore be appreciated that there is one less thing toworry about for the broad current profiles considered here, asthe reduced effective power is decisively the dominant effectof energy coupling on stabilization. IV. SELF-HEALING UNDER LHCD
It has long been understood that localized current is essen-tial for stabilization . Only recently, however, has it beenshown that such localized deposition can be accomplishedeven with initially broad profiles, via the nonlinear enhance-ment of deposition with electron temperature. Traditional (lin-ear) calculations do not capture this natural focusing. As aresult, ECCD has received the vast majority of attention forNTM stabilization, while LHCD has been largely ruled outdue to its broader deposition profiles. This must be reconsid-ered in light of the current condensation effect.Although ECCD is valued for its ability to deposit powerin a steerable region smaller than the island width , the sta-bilization enhancement of this sharp localization is contingentupon precise aiming of the rf. This poses a significant chal-lenge in practice as it requires active control techniques. In stark contrast, rf condensation opens the possibility of fullypassive stabilization in steady state tokamaks where a signifi-cant fraction of the toroidal current is driven with LHWs.Significant theoretical progress has been made on this front.Simulations performed with the GENRAY ray tracing codecoupled to the CQL3D Fokker Planck solver demonstratedsignificant localization of LHCD for typical electron tempera-ture profiles within the island. A new code, OCCAMI, cou-pling GENRAY to a thermal diffusion equation solver withinthe island to provide higher fidelity simulations of the currentcondensation effect, is in development. This tool has alreadybeen used to show the feasibility of accessing the bifurcationpoint and related hysteresis phenomena in an ITER-like sce-nario with realistic parameters.The focused treatment of energy coupling presented hereallows us to further expand the presently single-fluid basedunderstanding of rf condensation. As will be shown shortly,energy coupling is a critical feature to take into account, es-pecially for the investigation of self-healing under LHCD. Wehave shown in this case that the increase of effective powerwith island width ( P ∼ W i ) dominates the increasing energylosses to the ions (which scales at most logarithmically with W i ). As a result, the stabilizing current naturally increases asthe island heats up and grows.At some island width, this may be enough to entirely sta-bilize the island. The width at which this occurs may be esti-mated as follows. The modified Rutherford equation (MRE)gives the island width growth rate as a function of differentdriving and stabilizing mechanisms :d w d t ∝ ∆ (cid:48) ( w ) − Σ i ∆ (cid:48) i ( δ j i ) (8)where ∆ (cid:48) is the classical stability index, and ∆ (cid:48) i ( δ j i ) are cor-rections to the classical tearing mode equation due to pertur-bations of the parallel current at the resonant surface. Thesecorrections have the form : ∆ i ∝ (cid:90) ∞ − ∞ d x (cid:73) d ξ cos ( ξ ) δ j i ( x , ξ )= (cid:90) − d Ω (cid:90) ˆ ξ − ˆ ξ d ξ cos ( ξ ) (cid:112) Ω + cos ( ξ ) δ j i ( Ω , ξ ) (9)where ˆ ξ = cos − ( − Ω ) , x : = r − r s is the radial displacementfrom the resonant surface, Ω is the flux surface label and ξ isthe helical phase, as defined in Section II.The dominant driving term for NTMs will be due to theperturbed bootstrap current δ j bs , while the stabilizing termof interest for our purposes comes from the perturbed LHcurrent δ j LH . Then we estimate stabilization to occur when0 ≈ ∆ (cid:48) bs + ∆ (cid:48) LH .We take the perturbed bootstrap current to be constant overthe island, δ bs = − J bs , where J bs is the bootstrap current den-sity at the resonant surface prior to the island formation: ∆ (cid:48) bs ∝ − J bs (cid:90) − d Ω F ( Ω ) (10)where F ( Ω ) : = (cid:82) ˆ ξ − ˆ ξ d ξ cos ( ξ ) √ Ω + cos ( ξ ) serves as a flux surfaceweighting function.The perturbed rf current δ LH is given by the nonlinear en-hancement to the LH current at the resonant surface, due tothe the heating of the island. The current driven is roughlyproportional to the power deposition, neglecting the depen-dence of the current drive efficiency on temperature, which isnegligible compared to the exponential enhancement factor : ∆ (cid:48) LH ∝ J LH (cid:90) − d Ω ( exp ( u e ( Ω )) − ) F ( Ω ) (11)where J LH is the LH current at the resonant surface prior to theisland formation. As before, the flux surface label Ω is relatedto the coordinate ρ in Eqs. (2) and (3) by ρ = ( Ω + ) / = R (cid:82) − d Ω ( exp ( u e ( Ω )) − ) F ( Ω ) (cid:82) − d Ω F ( Ω ) (12)where R : = J bs / J LH is the ratio of the bootstrap to LH currentsprior to island formation.As an example, Fig. 3 shows the island width at whichthe balance between the LH and bootstrap driving terms isreached as a function of ambient (pre-island) rf power density,with plasma parameters anticipated for ITER Scenario 2 at theq=2 surface, as studied in Ref. 53. Unsurprisingly, higherpowers will allow islands to be stabilized at smaller widths,by counteracting the decreased confinement within the island.Of course, Fig. 3 can just as easily also be interpreted as theamount of rf power required to reach the threshold for an is-land of a certain size. It should be noted that the island sizes FIG. 3. Island width W i at self-stabilization vs. rf power density P r f , calculated for R =
1. Dotted black line indicates c → w = χ e = . m / s , n = m − , T e = keV . and rf power densities corresponding to the threshold are in anexperimentally relevant range .It can be seen that at lower powers, due to the large is-land widths required to reach the threshold, there can be sig-nificant impact from energy losses to ions. In this regime,there is also a strong dependence on the ratio of diffusivi-ties γ . Due to the suppression of turbulent transport withinthe island, it may well be the case that γ is closer to its neo-classical value At higher powers, the stabilization width maybe reached at smaller island widths, and the effects of energycoupling are negligible. As typical experimental parametersspan both strong and weak coupling regimes, a two fluid anal-ysis is indispensable.Note that the island width at stabilization will take the form W i ∼ C / √ P r f in the high or low power limits, with the con-stant C differing by a factor of √ + γ . Consequently, therewill be diminishing gains from additional power input, interms of the island width at stabilization. Physically, this is aresult of smaller islands requiring steeper edge gradients, andthus more power input, to support a given central temperaturethat is needed for significant nonlinear enhancement.As can be seen in Fig. 4, the power required for stabi-lization at a target width is only appreciably sensitive to thebootstrap-LH current ratio for R (cid:47)
1, with the power require-ment barely increasing as the bifurcation power is approached.This is simply due to the rapid growth of the island tempera-ture close to the bifurcation point. For R (cid:39) .
5, stabilizationrequires island temperatures past the bifurcation point. If theisland grows past the bifurcation point, the island tempera-ture and resulting power deposition will rapidly increase untilreaching a hotter solution branch. A hysteresis effect may thenbe accessed—once on this upper branch, the rf stabilizationwill be greatly enhanced via the current condensation effect,and the island may then be suppressed to a smaller saturatedwidth than would otherwise be possible without reaching thebifurcation point.
FIG. 4. RF power density required for stabilization vs. bootstrap-LH fraction R. Dotted black line indicates bifurcation power for eachisland width. Calculated for w = χ e = . m / s , n = m − , T e = keV , γ = Therefore, a smaller bootstrap to LH ratio could be desir-able in the sense that the island will be stabilized at smallerwidths than the width at bifurcation. On the other hand, alarger bootstrap to LH ratio would allow the hysteresis effectto be accessed. Especially considering the insensitivity of thestabilization island width to higher powers, and weak depen-dence of the requisite stabilization power to the bootstrap-LHcurrent ratio, an interesting optimization issue is raised. Ifsmaller saturated island widths are made possible by passingthe bifurcation point and jumping to the hotter branch of so-lutions, it could potentially be more efficient to use a lowerpower such that the island is able to grow to the thresholdwidth and then experience enhanced absorption, rather than ahigher power that stabilizes the island below the threshold butat a larger final size. While this clearly depends on just howmuch better the upper branch performs (which would dependon saturation mechanisms not examined in this work), evenmore moving parts are introduced by the inter-dependenciesbetween P , w , and R . Altogether, this poses a fascinatingproblem that calls for additional modeling and experimentalinvestigations. V. SUMMARY
The impact of energy coupling between electrons and ionson the current condensation effect has been reported, usinga two temperature diffusive model. In the limit of perfectlyequilibrated electrons and ions, the effective power is reducedby a factor of ( + γ ) where γ : = χ i / χ e is the ratio of theion to electron thermal diffusivities, relative to the case wherethe electron and ion temperatures are entirely decoupled. Thestrength of energy coupling is characterized by the dimension-less quantity c : = t D , e / t eq where t D , e = W i / χ e and t eq are thecharacteristic electron diffusion time and electron-ion equili- bration time, respectively. Although energy coupling to theions increases with the island width, it is found that the result-ing reduction in effective power is overwhelmed by the im-proved heat confinement in larger islands. Thus, in the pres-ence of broad LHCD profiles, stabilization may be achievedpassively as an island grows, with higher rf powers triggeringthe condensation effect at smaller island widths. The populardismissal of LHWs for instability control must therefore bereconsidered. ACKNOWLEDGMENTS
This work was supported by Nos. U.S. DOE DE-AC02-09CH11466 and DE-SC0016072.
DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
Appendix A: Impact of energy coupling for narrow deposition
Here we examine the impact of energy coupling for the caseof narrow deposition profiles that are fully contained withinthe island, as would be typical of ECCD. Since there is no ad-ditional power absorbed as the island grows, it can be imme-diately anticipated that the relative impact of increased energycoupling will be greater in this case, than for broad deposi-tion profiles. A narrow deposition profile also introduces thepossibility of misalignment, which would further offset theimproved energy confinement provided by a larger island. Weexplore the impact of these effects with Eqs. (2) & (3), butin slab geometry ˆ D → ∂ x where x = ( r − r s ) / W i , and with ahighly simplified deposition profiles: P r f = P δ ( x ) (A1)The nonlinear factor of exp ( u e ) does not enter here, as thepower is already fully deposited and maximally localized. Thesource term Eq. (B1) admits the following solutions: u e = P ( + γ ) [ − | x | + γ k sinh ( k ( − | x | ) cosh ( k ) ] (A2) u i = P ( + γ ) [ − | x | − k sinh ( k ( − | x | )) cosh ( k ) ] (A3)where k : = (cid:112) c ( + γ − , and P is the linear power densityscaled to P scl = nT e W i / τ De . Note the additional factor of W i / c → c → ∞ reduce to u e ( ) → P / u e → P / ( + γ ) re-spectively. It should be noted however, that although W i → ∞ does correspond to c → ∞ , the limiting solution in this case FIG. 5. Central island temperature vs. island width for 20 MW of rfpower deposited at the q = n = m − , T e = keV , χ e = . m / s , r = . m ). Dashed line indicates single-fluidsolution neglecting collisions. c ≈ W i = cm . does not simply have the effective power reduced by the 1 + γ factor as in the broad deposition case, but there is an addi-tional offset as P ∼ W i and the second term of the solutions ∼ W − i . This can be seen in Fig. 5, which shows how the cen-tral electron temperature u e ( ) scales with island width forITER scenario II parameters. Appendix B: Narrow off-center deposition in intermediatecoupling regimes
Energy coupling has a uniquely detrimental effect for nar-row, off-center power deposition in the intermediate coupling(two fluid) regime. Steady state solutions of the single fluidequations satisfy an equation of the form − u (cid:48)(cid:48) = P , where P > = x and − x corresponding to the same flux sur-face, means that the maximum point in the temperature pro-file u ( x ) will be at x = c ( u i − u e ) exceedsthe power deposited at the center.Considering two u e profiles of equal area (so same totalamount of heating accomplished), the one with the central dipwill have larger edge gradients. Accordingly, this dipped pro-file will experience larger diffusive losses and cost more inputpower to maintain. A centrally dipped profile also means thatthe power deposition at the periphery will be enhanced at theexpense of the center. Thus we may anticipate that interme-diate collisionality regimes can perform poorly beyond whatmay be anticipated from simply sharing power with the ions.This can be simply demonstrated with the following toy profiles: P r f = P ( δ ( x − ∆ ) + δ ( x + δ )) / ∆ ∈ [ , ) represents the degree of misalignment. Thesymmetrized form is to account for the points at ± x belong-ing to the same flux surface. Again, the nonlinear factor ofexp ( u e ) does not enter here, as the power is already fully de-posited and maximally localized. The source term Eq. (B1)admits the following solutions: u e = P ( − ∆ ) ( + γ ) + x − ∆ + γ a sinh ( k ( + x )) sinh ( k ( − ∆ )) x < ∆ + γ a cosh ( kx ) cosh ( k ∆ ) | x | < ∆ − x − ∆ + γ a sinh ( k ( − x )) sinh ( k ( − ∆ )) x > ∆ (B2) u i = P ( − ∆ ) ( + γ ) + x − ∆ − a sinh ( k ( + x )) sinh ( k ( − ∆ )) x < ∆ − a cosh ( kx ) cosh ( k ∆ ) | x | ≤ ∆ − x − ∆ − a sinh ( k ( − x )) sinh ( k ( − ∆ )) x > ∆ (B3)where k : = (cid:112) c ( + γ − , a : = [ k ( − ∆ )( tanh ( k ∆ )+ coth ( k ( − ∆ ))] − , and P is the linear power density scaled to P scl = nT e W i / τ De . FIG. 6. Central island temperature (normalized for uncoupled impactof misalignment) vs. coupling parameter c for various ∆ . In the absence of coupling ( k → − ∆ . Fig. 6 shows the impact of coupling on the centralisland temperature for various degrees of misalignment, afternormalizing for the impact of misalignment in the uncoupledlimit by dividing out a factor of 1 − ∆ . 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