Anomalous edge plasma transport, neutrals, and divertor plasma detachment
Yanzeng Zhang, Sergei I. Krasheninnikov, Rebecca Masline, Roman D. Smirnov
AAnomalous edge plasma transport, neutrals, and divertor plasmadetachment
Yanzeng Zhang, Sergei I. Krasheninnikov, Rebecca Masline, and Roman D. Smirnov Mechanical and Aerospace Engineering Department,University of California San Diego, La Jolla, CA 92093, USA
Abstract
An impact of neutrals on anomalous edge plasma transport and zonal flow (ZF) is considered.As an example, it is assumed that edge plasma turbulence is driven by the resistive drift wave(RDW) instability. It is found that the actual effect of neutrals is not related to a suppression ofthe instability per se , but due to an impact on the ZF. Particularly, it is shown that, whereas theneutrals make very little impact on the linear growth rate of the RDW instability, they can largelyreduce the zonal flow generation in the nonlinear stage, which results in an enhancement of theoverall anomalous plasma transport. Even though only RDW instability is considered, it seemsthat such an impact of neutrals on anomalous edge plasma transport has a very generic feature. Itis conceivable that such neutral induced enhancement of anomalous plasma transport is observedexperimentally in a detached divertor regime, which is accompanied by a strong increase of neutraldensity. a r X i v : . [ phy s i c s . p l a s m - ph ] M a y . INTRODUCTION Due to plasma neutralization on plasma facing components and volumetric recombinationprocesses, edge plasma in magnetic fusion devices contains significant amount of neutrals.These neutrals play crucial role in both plasma fueling (resulting in so-called plasma “recy-cling”) and in establishing the regime of divertor plasma detachment [1]. In addition, thereis a significant amount of papers, both experimental [2–6] and theoretical [7–12], discussingthe role of neutrals in plasma anomalous cross-field transport and in the transition fromL-mode to H-mode confinement regime.Whereas experimental data demonstrate some controversial/indecisive conclusions withregard to neutral effects in edge plasma turbulence (e.g. see [5, 6] and the references therein),early, most analytical and theoretical estimates were showing that the role of neutrals couldbe important. Recently, the results of numerical simulations of edge plasma turbulencehave demonstrated that the incorporation of neutral effects is causing significant and, insome cases, very strong impact [13–16]. Although, to deduce physics behind this impactadditional post processing of 3D plasma turbulence simulation results should be performed,which in most cases is not available.Potentially strong impact of neutrals on the processes governing plasma turbulence couldbe seen from simplified fluid neutral model, which is based on a strong coupling of plasma ionsand neutral atoms caused by ion-neutral collisions (e.g. see [17] and the references therein).Indeed, in this case the contribution of neutrals, which are not magnetized, to combinedplasma/neutral cross-field viscosity coefficient, η N , is given by the following expression η N ≈ Nn T i /Mν Ni , (1)where N and n are the neutral and plasma density, T i is the ion temperature, M is the ionmass, ν Ni = K Ni · n is the neutral-ion collision frequency, and K Ni is the ion-neutral collisionrate constant. For T i ≈ eV , K Ni ≈ · − cm /s , and n ≈ · cm − , from Eq. (1) wefind η N ≈ ( N/n )[ cm /s ]. As a result, for rather typical edge plasma ratio N/n ˜ > − , η N exceeds characteristic anomalous edge plasma diffusivities ∼ [ cm /s ].Such simplified fluid description of plasma-neutral coupling allowing to study the neutralimpact on plasma dynamics (e.g., plasma instability) only holds for the case where: i) elasticneutral-ion collisions dominate over the rate of electron impact ionization of neutrals, and2i) the characteristic spatial scale (e.g., wavelength λ ) and the frequency, ω , of the problemunder consideration are, respectively, larger and smaller than the mean-free path of neutralswith respect to neutral-ion collisions, λ Ni = (cid:112) T i /M /ν Ni , and ν Ni .However, in practice, the situation is more complex. First of all, edge plasma parametersvary strongly and whereas in divertor region of current tokamaks plasma temperature couldbe below 10 eV and plasma density could exceed 10 cm − , in the vicinity of the separatrixat the midplane of the scrape-off layer (SOL) temperature could reach 100 eV and densitycould be below 10 cm − and falls even more at the main chamber wall. This shows thatthe fluid approximation for atomic hydrogen transport is not valid in the entire edge plasmavolume. Next, in addition to neutral-ion elastic collisions, neutrals undergo the electronimpact ionization process accompanied by a significant amount of the radiation loss. Forelectron temperature above ∼ eV , the neutral ionization rate constant becomes compa-rable with the rate constant of elastic (charge-exchange) collisions of neutral atoms withplasma protons/deuterons (e.g. see [1] and the references therein). Finally, apart from hy-drogen atoms, a large fraction ( ∼
50% ) of hydrogen comes from the walls as molecules,which have a much lower rate of elastic collisions with plasma ions (protons/deuterons).Therefore, a fluid approximation for hydrogen molecule transport is even more problematicthan that for hydrogen atoms. For the case where either of inequalities, λ Ni < λ , ω < ν Ni ,needed to ensure the application of the fluid neutral model is violated, taking into accountthat elastic neutral-ion collisions in a ballpark dominate, one could consider neutrals asstationary “scatters” of ions, causing ion momentum loss.As a result, rigorous consideration of the impact of neutrals on the linear stage of edgeplasma instabilities becomes a rather complex problem. On the other hand, the most impor-tant practical issue is the impact of neutrals on anomalous edge plasma transport, associatedwith the nonlinear stage of instabilities. However, the nonlinear stage is strongly impactedby the generation of plasma zonal flows (ZF) (e.g., see [18, 19] and the references therein).Therefore, it is plausible that the actual effect of neutrals on edge plasma anomalous trans-port is not related to the suppression of instability per se , but due to an impact on ZF,which has a strong impact on both nonlinear stage of instability and anomalous transport.In this case, the most crucial part is an appropriate description of an impact of neutrals onZF.In this paper, we examine the impact of neutrals on resistive drift-wave (RDW) turbu-3ence and plasma transport described with modified [20] Hasegawa-Wakatani [21] (MHW)equations. We find that indeed, whereas neutrals make very little impact on the growthrate of the RDW instability, they have a pronounced effect on ZF and, therefore, on overallanomalous plasma transport.We notice that the transition to the regime of divertor plasma detachment results in astrong increase of neutral density in the divertor volume [1]. Therefore, it is conceivablethe neutral density at the core-edge interface is also increasing during the transition intodetachment. It could reduce the amplitude of ZF and provoke an enhancement of anomalouscross-field plasma transport. Interestingly, the correlation between the divertor plasmadetachment and the increase of cross-field plasma transport, causing the broadening of thewidth of the scrape-off layer, was observed in experiments [22]. To see how strongly neutraldensity at the core-edge interface increases in the course of the transition into divertorplasma detachment, we performed corresponding simulations of the edge plasma transportwith the code UEDGE [23]. We have found that indeed neutral density just inside theseparatrix is strongly increased after the transition into the detached regime and, therefore,could significantly alter the ZF, and thus the anomalous plasma transport.The rest of the paper is organized as follows. In Section II we present the results of ouranalytic consideration of an impact of neutrals on both RDW instability and ZF generation.In section III we discuss the results of our numerical simulation of an impact of neutrals onthe RDW turbulence (including ZFs) performed within the framework of modified Hasegawa-Wakatani (MHW) model. In section IV we present the results of our simulations of an impactof the transition into detached divertor regime on neutral density at the core-edge interface.The main results are summarized in section V. II. IMPACT OF NEUTRALS ON RDW INSTABILITY AND ZF GENERATION
We consider the RDW instability and ZF generation assuming cold ions and constantelectron temperature. To account for an impact of neutrals on ion dynamics we add theterm S neut = M ˆ s neut V i into ion equation of motion: M d V i dt = − e ∇ φ + ec ( V i × B ) − S neut , (2)4here the operator ˆ s neut is defined asˆ s neut = − η N ∇ , for fluid model( N/n ) ν Ni , for “scattering” neutrals , (3) B = e z B is the magnetic field, φ is the fluctuating electrostatic potential, and V i is the ionvelocity. Then by employing the standard drift-orders from Eq. (2), we obtain a modificationof the MHW model to account for the neutral impact on the RDW and ZF ρ s (cid:20) ∂∂t + V · ∇ + ˆ s neut (cid:21) ∇ eφT e = ν (cid:107) (cid:32) e ˜ φT e − ˜ N n (cid:33) + νρ s ∇ ∇ eφT e , (4) ∂∂t N n + V · ∇ N n = ν (cid:107) (cid:32) e ˜ φT e − ˜ N n (cid:33) − V ∗ ∂∂y eφT e + D ∇ N n , (5)where ρ s = C s / Ω Bi , C s = T e /M , Ω Bi = eB /cM , V = c e z × ∇ φ/B , V ∗ = cT e /eB L n , L − n = − d(cid:96)n ( n ) /dx , T e is electron temperature, N is the plasma density fluctuations, ν (cid:107) = k z T e / ( mν ei ), m is electron mass, and ν ei is electron collision frequency. Here theresistive coupling term between the electrostatic potential and plasma density fluctuationsis determined only by the non-zonal components ˜ f = f − (cid:104) f (cid:105) y , where (cid:104) f (cid:105) y ≡ (cid:82) L y f dy/L y denotes the integration along the poloidal line at a given radial location. The last terms onthe right hand side of Eqs. (4, 5) with constant coefficients D and ν are dissipation termsfor the purpose of numerical stability and will be ignored in the following analysis.We first consider the neutral impact on the RDW by deriving its dispersion equation.For this purpose, we ignore the ZF and adopt the eikonal approximation (and thus ∇ influid ˆ s neut is replaced by − k ⊥ ). Moreover, we assume ν (cid:107) to be larger than both the RDWfrequency ω and ω ∗ ≡ k y V ∗ . As a result, we arrive to the following dispersion equation ω ∗ ω = 1 + ρ s k ⊥ + i (cid:18) ρ s k ⊥ ˆ s neut ω + ω − ω ∗ ν (cid:107) (cid:19) . (6)From Eq. (6) we find that an impact of neutrals stabilizes the RDW instability forˆ s neut > ˆ s dwth ≡ ω ∗ /ν (cid:107) (1 + ρ s k ⊥ ) , (7)where ω ≈ ω ∗ / (1 + ρ s k ⊥ ). Assuming that ρ s k ⊥ ∼
1, for ω ∗ /ν (cid:107) ∼ . ω ∗ ∼ × s − ,this inequality shows that even for “scattering” neutral model the RDW is stabilized for theneutral density N ˜ > cm − , which is too high for the SOL plasma in most of tokamaksand, therefore, the stabilization of the RDW instability is only possible in divertor volumewhere neutral density is significantly higher.5ext, we examine an impact of neutrals on the generation of ZF by averaging Eq. (4)over the poloidal direction. As a result, we obtain the evolution equation for the mean flow ∂∂t (cid:104) φ (cid:105) y = cB (cid:42) ∂ ˜ φ∂x ∂ ˜ φ∂y (cid:43) y − ˆ s zfneut (cid:104) φ (cid:105) y . (8)Note that the large-scale mean flow varies on a longer time scale compared to the small-scaleRDW fluctuations and we assume that the mean flow is one dimensional so that ∇ y → φ = ˜ φ ( x, y ). As a result, for the fluidmodel, ˆ s neut defined in Eq. (3) should be averaged along the poloidal direction and we haveˆ s zfneut = − η N d /dx , whereas, for the scattering neutrals, ˆ s zfneut = ˆ s neut = ( N/n ) ν Ni .To quantify (cid:68) ∂ x ˜ φ∂ y ˜ φ (cid:69) y , considering that Eqs. (4, 5) reduce to the modified Hasegawa-Mima [24] equations in the adiabatic limit, ν (cid:107) > ω, ω ∗ , if we ignore the contribution ofneutrals to the RDW when ˆ s neut < ω ∗ /ν (cid:107) as shown in Eq. (7), it can be convenientlycomputed from the kinetic equation for the drift wave action [25]. As a result, if we assumethat the ZF fluctuations are described by the frequency Ω < ω and radial wave number q = − id/dx (and thus d/dx in the fluid description of ˆ s zfneut is replaced by − q ), which isassumed to be smaller than the width of the drift wave spectrum N k = (1 + ρ s k ⊥ ) | e ˜ φ k /T e | in k x , we find − i Ω = − (cid:18) cB (cid:19) q (cid:90) R (Ω , q, ∆ ω k ) k y k x (1 + ρ s k ⊥ ) ∂N k ∂x d k ⊥ − ˆ s zfneut , (9)where R (Ω , q, ∆ ω k ) = i/ (Ω − qV g + i ∆ ω k ) is the response function with ∆ ω k being thenonlinear broadening increment and V g = ∂ω/∂k x . Eq. (9) shows that the neutrals causesuppression of the ZF generation.Here we consider a generation of the ZF resulting from a monochromatic spectrum N k = N δ ( k ⊥ − k ), where the growth rate of ZF generation is larger compared to theresonant instability corresponding to a broad spectrum of drift waves [26]. This assump-tion is consistent with the simulation observation in the next section, where ρ s k y ≈ ρ s k x <
1. As a result, from Eq. (9) we obtain1 + σ (Ω − qV g ) = − i ˆ s zfneut Ω , (10)where σ = qC s ρ s | k y | N / (1 + ρ s k − ρ s k x ) / (1 + ρ s k ) − / characterizes the growth rateof ZF in the absence of neutrals. Given that V g ∝ ρ s k x <
1, we can ignore qV g in Eq. (10).6s a result, we obtain the growth rate of ZF: γ zf = (cid:113) σ + (ˆ s zfneut ) / − ˆ s zfneut / . (11)It follows that the neutrals can largely reduce the ZF growth rate whenˆ s zfneut > ˆ s zfth ≡ σ ∼ κρ s qω ∗ , (12)where we take N / ∼ | e ˜ φ/T e | ≈ κ/k y L n , and κ < κρ s q < ω ∗ /ν (cid:107) (ˆ s zfth < ˆ s dwth ) andthus it is plausible that the neutrals will enhance the RDW turbulence by reducing the ZFrather than affect the RDW instability. From the numerical simulations in the next sectionwe will see that the neutrals with small density ( N < cm − ) can indeed reduce the ZFintensity resulting in an enhancement of the RDW turbulence transport, but this occursmainly in the nonlinear regime while in the linear state the impact of neutrals is negligibleas ˆ s neut < ˆ s dwth ∼ ˆ s zfth .So far we consider an impact of neutrals on both development of the RDW turbulenceand ZF generation and neglect an impact of ions. Such approximation is justified for thecase where ˆ s neut > ˆ s ion , here ˆ s ion is the analog of ˆ s neut caused by ions. Assuming that both ρ i k ⊥ and qρ i are below unity, we conclude that ˆ s ion is due to cross-field ion viscosity, whereasˆ s neut could be determined by neutrals in a scattering regime. In the latter case, using theexpression for cross-field ion viscosity [27] and the ion-neutral collision frequency from [17],we find that for ρ i k ⊥ ∼ q ρ i ∼ ∼ eV , ˆ s neut > ˆ s ion for N/n > − .In our further considerations we assume that the latter inequality holds. III. NUMERICAL SIMULATION OF AN IMPACT OF NEUTRALS ON RDWTURBULENCE AND PLASMA TRANSPORT
In this section, Eqs. (4, 5) will be numerically solved to examine the impact of neutrals onthe RDW turbulence and plasma transport. The numerical scheme used is a pseudo-spectralFourier code by employing Dedalus[28], where the computation domain is a square box withsize L x = L y = 20 πρ s so that the lowest wavenumber is ρ s ∆ k = 0 .
2. We employ thedoubly periodic boundary conditions and the number of the modes are chosen as 256 × t = 10 − ρ s /V ∗ , where ρ s /V ∗ ∼ ω − ∗ is the characteristic temporal scale of RDWturbulence. In fact, we can rescale eφ/T e and N /n by V ∗ /C s = ρ s /L n and, as a result,the system is controlled by two free parameters α = ν (cid:107) ρ s /V ∗ and ˆ s = ˆ s neut ρ s /V ∗ (here weconsider the “scattering” neutrals only and thus ˆ s neut = ˆ s zfneut is a constant coefficient),where the dissipation coefficients ν = D = 10 − V ∗ ρ s are fixed. In the simulations, we alsofix α = 2 but vary ˆ s = (0 , , × − corresponding to N ∼ (0 , , × cm − for V ∗ /ρ s = 3 × s − . We are particularly interested in the impact of the neutrals on thenormalized particle flux in the radial directionΓ n ≡ − (cid:90) (cid:32) L n ρ s ˜ N n (cid:33) × ρ s ∂∂y (cid:32) L n ρ s e ˜ φT e (cid:33) d x , (13)where (cid:82) f d x ≡ (cid:82) L x (cid:82) L y f dxdy/L x L y .In all the cases, we start the simulations from the same small amplitude perturbationwith N /n , eφ/T e ∝ exp ( − x / ∆ ), where ∆ = 4 ρ s . We observe that, the perturbations willfirst grow due to the RDW instability, e.g., see Fig. 1(a) for the root mean square (RMS) ofthe normalized electrostatic potential fluctuations (cid:68) ˜ φ (cid:69) / ≡ [ (cid:82) ( L n e ˜ φ/ρ s T e ) d x ] / . In thelate stage of the linear regime, the dominant mode of ρ s k y ≈ ρ s k x ≈ . γ ≈ k y V ∗ /ν (cid:107) (1+ ρ s k ⊥ ) ≈ . V ∗ /ρ s .Whereas, taking into account ˆ s dwth ρ s /V ∗ = 0 .
12 from Eq. (7), the neutrals even with ˆ s = 10 − can only reduce the growth rate γ by approximately 10%, which agrees with the numericalsimulations. Meanwhile, these neutrals reduce the linear growth rate of the ZF by ∼ s neut /σ ∼ . κ = L n e ˜ φ/T e ρ s ∼ . qρ s ∼ . IG. 1. Time evolution plots of root mean square of the normalized (a) electrostatic potentialfluctuations and (b) zonal flow for α = 2, where the insets plot results in the linear stage in thelogarithmic scale. The black, blue and red curves are for ˆ s = 0 , − and 10 − . flux is enhanced as shown in Fig. 2, where the time-averaged Γ n at the saturated state from V ∗ t/ρ s = 400 to V ∗ ρ s = 1000 are Γ n = 7 . × − , . × − and 2 . × − for ˆ s = 0 , − and 10 − , respectively. FIG. 2. Time evolution plots of the normalized particle flux for α = 2, where the black, blue andred curves are for ˆ s = 0 , − and 10 − . V. NEUTRAL DENSITY AT THE CORE-EDGE INTERFACE DURING THETRANSITION INTO DETACHED DIVERTOR PLASMA
To address the issue of neutral density variation at the core-edge interface during thetransition into detached divertor regime we perform a series of the UEDGE simulationsof edge plasma for DIII-D-like geometry and magnetic configuration (see [30] for details).Self-consistent modeling of plasma detachment in a high confinement mode (H-mode) ofthe operation of a tokamak and an impact of neutrals on anomalous cross-field plasmatransport goes beyond the scope of this paper. Therefore, here we just illustrate an impactof the transition into detached divertor regime on neutral density at the core-edge interfaceassuming that cross-field plasma transport coefficients are fixed and equal to 1 m /s . It isknown that the transition to divertor plasma detachment could be reached by the increaseof impurity fraction or plasma density, or both [1]. In our simulations we take fixed plasmadensity at the core-edge interface equal to 2 × cm − , assume constant power flux of4 M W into edge plasma from the core (split evenly between electron and ion components),and gradually increase nitrogen impurity fraction in the simulation domain (for simplicitywe used so-called “fixed impurity fraction model”). We monitor detachment process by themagnitude of plasma flux onto divertor targets, Γdiv, since one of the signature of divertorplasma detachment is the reduction of Γdiv [1].In Fig. 3 we present both Γdiv and the neutral density averaged over the flux tube justinside the separatrix, ¯ N sep, as the functions of impurity fraction. One can clearly see thatthe transition into divertor detachment, manifested as the reduction of Γdiv, is accompaniedby a strong increase of ¯ N sep. There are two reasons accounting for this increase of neutraldensity: i) an overall increase of neutral density in a low temperature detached divertorplasma and ii) the expansion of this region (detachment front) toward the X-point[1], whichreduces an effective opacity of the scrape-off layer plasma for neutral penetration throughthe separatrix.We notice that the neutral density exceeding 10 cm − found in the most detached casein our series of runs, corresponds to the parameter ˆ s > − , and, according to out sim-ulations of the RDW turbulence (recall Fig. 2), can make a profound impact on ZF andanomalous plasma transport. We also notice that, in our UEDGE simulations, we considerthe simplest model of edge plasma transport. More complete/complicated models show that10 IG. 3. Plasma flux to the divertor targets (black) and neutral density just inside the separatrix(red) as the functions of impurity fraction. the transition to detachment can have the bifurcation-like character [31, 32], which couldbe accompanied by corresponding bifurcation of neutral density inside the separatrix andanomalous transport.
V. CONCLUSIONS
In this paper, the impact of neutrals on the RDW and turbulence transport is examinedfor the tokamak edge plasmas by employing the MHW model, which characterizes the basicphysics of the RDW-ZF system. The neutrals effect on the ion dynamics are modeled bothin the fluid regime, when the wavelength and frequency of the RDW are, respectively, largerand smaller than the mean-free path of neutrals with respect to neutral-ion collisions andthe collision frequency, and the “scattering” regime for the opposite conditions. From theanalysis, we find that the neutrals make very little impact on the RDW instability in thelinear regime, which agrees well with the numerical simulations. This is also true for thelinear ZF, where the fraction of the reduced growth rate due to the neutrals is comparableto that of the RDW. However, the neutrals can largely reduce the ZF in the nonlinear stageas shown in Fig. 1(b), which, in turn, leads to an enhancement of the radial particle flux11n Fig. 2. Even though only RDW instability is considered, it seems that such an impactof neutrals on anomalous edge plasma transport has very generic feature (e.g., the neutralsimpact on the ITG turbulence was simulated in [14]).Our numerical simulations of the transition of edge plasma into detached divertor regimewith the UEDGE transport code show that such transition is accompanied by a strong in-crease of neutral density at the core-edge interface, which, according to our turbulence simu-lations, results in significant reduction of the amplitude of ZF and an increase of anomalouscross-field plasma transport. This could explain experimental data from [22], showing thewidening of the scrape-off layer width after transition into detached divertor regime.However, we notice that in our model the neutrals are simplified as a uniform backgroundaffecting only the ion momentum balance. Therefore, an ultimate conclusion of the role ofneutrals in the anomalous cross-field edge plasma transport and the transition to plasmadetachment requires a more comprehensive self-consistent evolution of neutrals, which isbeyond the scope of this paper.
ACKNOWLEDGMENTS
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