Plasma Turbulence in the Scrape-off Layer of the ISTTOK Tokamak
Rogerio Jorge, Paolo Ricci, Federico D. Halpern, Nuno F. Loureiro, Carlos Silva
PPlasma Turbulence in the Scrape-off Layer of the ISTTOKTokamak
Rog´erio Jorge,
1, 2, ∗ Paolo Ricci, Federico D. Halpern, Nuno F. Loureiro, and Carlos Silva ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL),Swiss Plasma Center (SPC), CH-1015 Lausanne, Switzerland Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico,Universidade de Lisboa, 1049-001 Lisboa, Portugal Plasma Science and Fusion Center,Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA
Abstract
The properties of plasma turbulence in a poloidally limited scrape-off layer (SOL) are addressed,with focus on ISTTOK, a large aspect ratio tokamak with a circular cross section. Theoreticalinvestigations based on the drift-reduced Braginskii equations are carried out through linearcalculations and non-linear simulations, in two- and three-dimensional geometries. The linearinstabilities driving turbulence and the mechanisms that set the amplitude of turbulence as well asthe SOL width are identified. A clear asymmetry is shown to exist between the low-field and thehigh-field sides of the machine. While the comparison between experimental measurements andsimulation results shows good agreement in the far SOL, large intermittent events in the near SOL,detected in the experiments, are not captured by the simulations. ∗ rogerio.jorge@epfl.ch a r X i v : . [ phy s i c s . p l a s m - ph ] O c t . INTRODUCTION In recent years, significant progress was made in the study of the plasma turbulenceproperties in the scrape-off layer (SOL) of tokamaks [1], the region that exhausts the tokamakpower, controls the plasma fueling and the impurity dynamics, and plays a major role indetermining the overall plasma confinement [2–5]. These theoretical investigations [6] focusedmainly on the toroidally limited SOL [7–9], a configuration that is relevant to the ITERstart-up and ramp-down phases during which the inner or the outer vessel wall will be used asthe limiting surface [10, 11]. In this scenario, using low-frequency fluid models, the turbulentregimes were identified. It was found that drift waves (DW) and ballooning modes (BM)drive the plasma turbulent dynamics, with the resistive BM being the main drive in typicalexisting tokamak conditions [12], a result in agreement with previous experimental results[13, 14]. Simulations and analytical estimates revealed that the fluctuations saturate dueto a local flattening of the plasma gradients and associated removal of the linear instabilitydrive [9]. By using a balance between turbulent transport and parallel losses at the vessel, ascaling of the pressure scale length was derived. A thorough comparison with experimentalmeasurements was carried out with significant success [15]. The question of how these findingscan be applied to other configurations remains open and is one of the main motivations ofthis work.The goal of the present paper is the study of turbulence properties in a poloidally limitedgeometry, such as the one of ISTTOK [16, 17], a large aspect ratio tokamak (
R/a ∼ . R and a are the major and minor radius respectively) with a circular cross section.By intercepting the magnetic field lines on a poloidal plane, a poloidal limiter avoids theconnection between the low- and the high-field sides of the machine. This allows the turbulentproperties, and therefore the pressure scale length and the SOL width, to retain a strongpoloidal dependence. The shorter connection length, with respect to the toroidally limitedcase, leads to enhanced parallel losses, steepening the gradients and, as we show, changingthe relative role of DW and BM in driving turbulence.We carry out our investigation by using linear and non-linear simulations, in two- andthree-dimensional geometries, that are based on the drift-reduced Braginskii equations [18].These are solved with GBS [19, 20], a numerical simulation code developed with the goalof simulating plasma SOL turbulence by evolving the full profiles of the various plasma2uantities with no separation between perturbations and equilibrium, and was validatedagainst experiments such as the TORPEX device [21] and several other machines [15], verifiedwith the method of manufactured solutions [22], and benchmarked against other major SOLsimulation codes, including BOUT++ [23], HESEL [24], and TOKAM3X [8]. The parametersof our study rely on the ones from ISTTOK, where a clear asymmetry between the low andthe high field sides was found [25]. We uncover the instabilities driving turbulence and theturbulent regimes in ISTTOK, and we quantitatively compare our simulation and theoreticalresults with some of the measurements taken in this device.This paper is organized as follows. Section II describes the model equations and theISTTOK simulation results. In Sec. III we investigate the nature of the instabilities drivingturbulence in a poloidally limited SOL. Sec. IV discusses the development of the linearinstabilities into non-linear turbulence and provides an estimate of the time-averaged pressuregradient scale length. Finally, in Sec. V, a comparison between ISTTOK experimentalmeasurements and simulations is reported. The conclusions are presented in Sec. VI. II. MODEL EQUATIONS AND ISTTOK SIMULATION RESULTS
In the ISTTOK SOL, the turbulent time scales (such as the one measured by Langmuirprobes < ∼ − s) are slower than the collisional time ( τ e ∼ − s), and the scale lengthsalong the (poloidally limited) magnetic field ( L (cid:107) = 2 πR ∼ λ mfp ∼ ω − ci ∼ − s), and the perpendicular scalelengths ( L p ∼ ρ i ∼ . α MHD = β e R/L p ∼ . × − in ISTTOK, we do not expect theideal ballooning mode to play a major role. We refer the reader to Ref. [27] for a detailedtreatment of electromagnetic effects in the SOL within the drift-reduced fluid descriptionand here we consider the electrostatic limit. The model equations are3 n∂t = − cB [ φ, n ] + 2 ceB (cid:104) C ( nT e ) − enC ( φ ) (cid:105) − ∇ (cid:107) (cid:0) nV (cid:107) e (cid:1) + D n ( n ) + S n , (1) ∂ Ω ∂t = − cB [ φ, Ω] − V (cid:107) i ∇ (cid:107) Ω + ω ci en C ( G i ) + D ω ( ω )+ m i ω ci en (cid:20) ∇ (cid:107) (cid:0) n ( V (cid:107) i − V (cid:107) e ) (cid:1) + 2 m i ω ci C ( n ( T i + T e )) (cid:21) , (2) m e ∂V (cid:107) e ∂t = − m e cB [ φ, V (cid:107) e ] − m e V (cid:107) e ∇ (cid:107) V (cid:107) e − . ∇ (cid:107) T e − n ∇ (cid:107) G e − . m e ν e ( V (cid:107) e − V (cid:107) i ) + e ∇ (cid:107) φ − T e ∇ (cid:107) ln n + D V (cid:107) e ( V (cid:107) e ) /n, (3) m i ∂V (cid:107) i ∂t = − m i cB [ φ, V (cid:107) i ] − m i V (cid:107) i ∇ (cid:107) V (cid:107) i − n ∇ (cid:107) G i − ∇ (cid:107) [ n ( T e + T i )] /n + D V (cid:107) i ( V (cid:107) i ) /n, (4) ∂T e ∂t = − cB [ φ, T e ] + 4 cT e eB (cid:20) C ( T e ) + T e n C ( n ) − eC ( φ ) (cid:21) + 0 .
71 2 T e (cid:2) ( V (cid:107) i − V (cid:107) e ) ∇ (cid:107) ln n + ∇ (cid:107) (cid:0) V (cid:107) i − . V (cid:107) e (cid:1)(cid:3) − V (cid:107) e ∇ (cid:107) T e + D T e ( T e ) + S T e , (5) ∂T i ∂t = − cB [ φ, T i ] + 4 c eB T i n [ C ( nT e ) − enC ( φ )] + 23 T i (cid:0) V (cid:107) i − V (cid:107) e (cid:1) ∇ (cid:107) ln n − T i ∇ (cid:107) V (cid:107) e − V (cid:107) i ∇ (cid:107) T i − cT i eB C ( T i ) + D T i ( T i ) + S T i . (6)where Ω = ω + ∇ ⊥ T i /e , with ω = ∇ ⊥ φ the vorticity and φ the electrostatic potential.In the density ( n ) and electron and ion temperature ( T e , T i ) equations, source terms S n,T = S n,T exp [ − ( x − x s ) /σ s ] are added to mimic the plasma outflow from the core intothe SOL. The diffusion operators for a generic field A , defined as D A ( A ) = χ A ∇ ⊥ A , arepresent for numerical reasons, i.e., to damp fluctuations at the grid scale. The gyroviscousterms G i,e are defined as G i,e = − η i,e (cid:110) ∇ (cid:107) V (cid:107) i,e + cenB [ enC ( φ ) ± C ( nT i,e )] (cid:111) , (7)with η i,e the Braginskii’s viscosity coefficients [26]. In Eqs. (1 - 6), we have also introducedthe magnetic field unit vector b = B /B , the curvature operator C ( f ) = ( B/ ∇ × ( b /B ) · ∇ f ,and the Poisson brackets operator [ φ, f ] = b · ( ∇ φ × ∇ f ). We use the Spitzer’s estimate ofthe electron-ion collision frequency, that is ν e = 2 . × − λnT − / e , with λ the Coulomblogarithm, T e in eV, and n in cm − .For simplicity, we consider a large aspect ratio geometry, and no magnetic shear. Anorthogonal coordinate system [ y, x, z ] is used, where x is the flux coordinate correspondingto the radial direction, z is a coordinate along the magnetic field B , and y is the coordinate4erpendicular to both x and z . Because of the considered large aspect ratio limit, theplane ( x, y ) coincides with the poloidal plane, which implies y = aθ , where θ is the poloidalangle ( − π < θ < π ), with θ = 0 corresponding to the low-field side (LFS) equatorialmidplane and θ = ± π to the high-field side (HFS). In the rest of the paper, we use θ and ϕ as the poloidal and toroidal coordinates respectively, with z = Rϕ/ cos (cid:15) , where (cid:15) is themagnetic field pitch angle (cid:15) = arctan( a/qR ) and q the safety factor. The parallel gradientis ∇ (cid:107) = ∂ z (cid:39) R − ( ∂ φ + q − ∂ θ ), and the perpendicular Laplacian is ∇ ⊥ = ∂ x + a − ∂ θ . Thepoloidal limiter is located at ϕ = 0 , π , where we impose the Bohm sheath conditions for theion and electron parallel velocities as V (cid:107) i = ± c s and V (cid:107) e = ± c s exp(Λ − eφ/T e ) respectively,with c s = (cid:112) ( T e + T i ) /m i and Λ = 0 . m i / (2 πm e )] (cid:39) R = 0 .
46 m, minor radius a = 0 .
085 m, and a toroidal magnetic field B T = 0 . T e = 20 eV, density n = 10 m − , magnetic field B = 0 . ρ s ≡ c s /ω ci (cid:39) . c s = (cid:112) T e /m i and ω ci = eB/ ( m i c )]. This results in R (cid:39) ρ s , a (cid:39) ρ s , dimensionless resistivity ν = e n R/ ( m i σ (cid:107) c s ) (cid:39) × − [where σ (cid:107) = 1 . ne / ( m e ν e ) is the parallel conductivity],mass ratio m i /m e (cid:39) × − , and safety factor q (cid:39)
8. As there are no detailed measures of theion temperature, we perform our non-linear simulations in the cold ion limit ( τ = T i /T e = 0),and analyze the effect of finite T i on the linear growth rate of the unstable modes and thetime-averaged pressure gradient length in Section IV.The simulation has a radial extension 0 < x < ρ s . The plasma and heat sources,located at x s = 10 ρ s , have a characteristic width of σ s = 2 . ρ s . Our analysis considersonly the physically meaningful region x > x s . We remark that ISTTOK’s radial distancebetween the last closed flux surface and the outer wall is approximately 16 ρ s , in practicecomparable to the experimental SOL width. Since a set of boundary conditions that properlydescribes the interaction of the plasma with the outer wall is not known, we consider a radialdomain extension larger than in the experiment, so that the plasma pressure decays to anegligible value at the outer wall, and the boundary conditions we impose at this location5ave a negligible impact on the turbulent properties. Specifically, at x = 0 and x = 50 ρ s ,Neumann boundary conditions are used for density, temperature, electric potential, whileDirichlet boundary conditions are used for the vorticity. By computing the power spectrumof the fluctuations, we observe that χ A ≥ c s / ( ρ s R ) properly damps fluctuations at thegrid scale. Moreover, the simulation results are not sensitive to the values of the diffusioncoefficients for the range of values 6 < χ A ρ s R/c s <
20, so the value of χ A = 12 c s / ( ρ s R ) isused for all fields. A spatial grid of 512 × ×
32 and a time step of 10 − R/c s is employed.A typical turbulent snapshot for the standard ISTTOK simulation is shown in Figs. 1and 2. Figure 1 shows the development of the plasma turbulence on the poloidal plane ϕ = π midway between the two sides of the limiter plate. We observe that n , T e and φ fluctuations are stronger on the LFS, θ = 0, compared to the HFS, θ = ± π , where the SOLwidth is narrower. Figure 2, taken at a toroidal plane x = x s + 5 ρ s , confirms that turbulentfluctuations tend to be aligned to the magnetic field lines. The ion parallel velocities V (cid:107) i are − c s and + c s at the limiter plates ϕ = 0 and 2 π respectively, and the V (cid:107) e fluctuations aremuch larger due to the small electron inertia. III. IDENTIFICATION OF DRIVING LINEAR INSTABILITIES
Previous studies on the drift-reduced Braginskii equations show that ballooning modes(BM) and drift waves (DW) are the instabilities that drive most of the transport in a toroidallylimited SOL [12, 29]. BM are driven unstable by magnetic field line curvature and plasmapressure gradients. They are characterized by a large ( ∼ π/
2) phase shift between n and φ [30], and their growth rate is maximum at the longest parallel wavelength allowed in thesystem. On the contrary, DW arise at finite k (cid:107) due to the E × B convection of the pressureprofile, and are driven unstable by finite resistivity and electron inertia, showing an adiabaticelectron response, and a small phase shift between n and φ [31]. Besides BM and DW, theKelvin-Helmholtz (KH) instability, driven by shear flows, and the sheath mode, driven by atemperature gradient when magnetic field lines terminate on a solid wall and sheath physicsplays a role, may also influence the SOL dynamics [32].The role of DW in the system is assessed by two different studies. First, we comparethe standard ISTTOK simulation with a two-dimensional simulation carried out with amodel that, having excluded k (cid:107) (cid:54) = 0 modes (and in particular DW), evolves the field-line6 IG. 1. Snapshots of plasma turbulence in the standard ISTTOK simulation on a poloidal crosssection halfway between the limiter plate ( ϕ = π ). We show: (a) plasma density n/n , (b) electrontemperature T e /T e , (c) electrostatic potential φ/eT e , (d) vorticity ω = ρ s ∇ ⊥ φ/eT e , (e) electron V (cid:107) e /c s , and (f) ion V (cid:107) i /c s parallel velocities. IG. 2. Snapshots of plasma turbulence for the standard ISTTOK simulation on a toroidal crosssection at x = x s + 5 ρ s . We show: (a) plasma density n/n , (b) electron temperature T e /T e ,(c) electrostatic potential φ/eT e , (d) vorticity ω = ρ s ∇ ⊥ φ/eT e , (e) electron V (cid:107) e /c s , and (f) ion V (cid:107) i /c s parallel velocities. averaged density, n ( r, θ ), potential, φ ( r, θ ), and temperature, T ( r, θ ) (see Ref. [33]). Second,we perform a three-dimensional simulation where we exclude DW dynamics by neglectingthe diamagnetic terms, T e ∇ (cid:107) ln n and 1 . ∇ (cid:107) T e , in Ohm’s law, Eq. (3). The results of these8 IG. 3. The equilibrium pressure scale length, L p , is plotted as a function of the poloidal angle, θ , from an exponential fit in the radial direction of the type p/p = e − x/L p ( θ ) of the two- (red)and three-dimensional simulations, with (blue) and without (purple) DW, and the prediction fromEq. (18) (green). experiments cast in terms of the averaged pressure gradient scale length L p (cid:39) | p/ ∇ p | (where p = nT e ), are compared in Fig. 3 with the result from the full 3D GBS simulations. Thisincludes the standard ISTTOK simulation (blue line), and the two- and three-dimensionalsimulations that exclude the DW dynamics (red and purple lines, respectively). Motivatedby the difference between the LFS and the HFS following the removal of DW in Fig. 3, weanalyse separately the different poloidal positions. We note that this is justified by the factthat the plasma rotates poloidally on a time scale 2 πa/V E × B ∼ πaL p ω ci / (Λ c s ) ∼ − s,which is much slower than the turbulent time scales ( ∼ − s).We start our analysis at the LFS. Here, curvature is unfavourable, BM are expected to beunstable and, comparing the standard GBS simulation with the one excluding DW in Fig. 3,it is observed that removing DW from the system leads to increasing values of L p , suggestingthat these may have a significant role. By linearizing the drift-reduced Braginskii system of9 IG. 4. Linear growth rate as a function of the poloidal angle, θ , and of the parallel wavenumbernormalized to the major radius k (cid:107) R . From left to right: the solution of the full dispersion relationthat couples inertial DW and BM, Eq. (8); the solution of Eq. (10) for the pure BM; and thesolution of Eq. (11) for the pure DW. equations (1 - 6) in the cold-ion limit, assuming background density and temperature profileswith radial scale lengths given by L n and L T e respectively, and a perturbation of the form e γt + ik y y + ik (cid:107) z , we obtain the following dispersion relation that captures DW and BM g γ ω ci k y k (cid:107) m e m i = i k y ρ s L n ω ci γ (1 + 1 . η e ) − . gk y ρ s − , (8)with g = 1 − θ (1 + η e )( ρ s /RL n )( ω ci /γ )1 + 4 . i cos θ ( k y ρ s /R )( ω ci /γ ) , (9)and η e = L n /L T e . We remark that, to deduce Eq. (8), we also take into account the fact that θ is almost constant along a field line due to a high q at the edge, we neglect both soundwave coupling and compressibility terms in the continuity (1) and temperature (5) equations,since γ (cid:29) k (cid:107) c s and L (cid:107) /R (cid:28) ν e in Ohm’s law (3). The inertial natureof the instabilities present in the system is confirmed in Sec. IV.The largest growth rate solution of Eq. (8) is plotted as a function of k (cid:107) and θ in the leftpanel of Fig. 4, having chosen L n , η e , and k y ρ s according to the results of the ISTTOKstandard simulation. This growth rate is compared with the maximum one resulting fromthe dispersion relation of the pure BM, 10 ω ci − θ (1 + η e ) ρ s RL n = − k (cid:107) k y m i m e , (10)and pure DW, γ ω ci k y k (cid:107) m e m i = i k y ρ s L n ω ci γ (1 + 1 . η e ) − . k y ρ s − . (11)One observes from Fig. 4 that pure BM are unstable for k (cid:107) R < .
15 and for k (cid:107) = 0 theyexhibit a strong growth rate at the LFS. However, as they are strongly stabilised by finite k (cid:107) , at the typical values of k (cid:107) ∼ φ in the [ φ, ω ] term of the vorticity equation (2) by its poloidallyaveraged counterpart. This simulation (not shown) exhibits an increase of L p from 18 ρ s to30 ρ s , revealing therefore that the KH instability does not drive turbulence, but it plays arole in regulating its saturation level, since it decreases the characteristic gradient lengths inthe SOL.We can therefore conclude that, at the LFS, finite k (cid:107) effects decrease the importance ofBM and lead to DW driven turbulence whose amplitude is partially regulated by the KHmode at the LFS. As a comparison, we remark that k (cid:107) is set by the ballooning character ofthe modes in a toroidally limited SOL. This leads to smaller values of k (cid:107) and, ultimately,enhances the importance of BM with respect to DW.We now focus on the HFS, where the DW removal in the nonlinear simulation of Fig. 3significantly decreases L p . This pinpoints the important role of DW as a turbulence drive atthis location and rules out BM and KH modes as the main drive of HFS turbulence. Theresidual turbulence in the DW-suppressed system is driven by the KH mode stabilized bythe favourable curvature. This is tested by removing the KH instability drive, and observingthat L p decreases even further to negligible values from approximately 5 ρ s to 2 ρ s .In addition, two-dimensional simulations (not shown) reveal that L p increases substantiallyat the HFS from L p (cid:39) ρ s to L p (cid:39) ρ s if the curvature term in the vorticity equationis removed, a value in agreement with the estimate in Ref. [34]. This shows that favorable11urvature has a stabilizing effect on KH. A study on the coupling between the KH instabilityand BM has been carried out in Ref. [35], where the same effect was noticed.In order to further justify our conclusions on the turbulent driving mechanisms, we analysethe simulation results by evaluating the cross-coherence and phase-shift between ˜ n and ˜ φ .Here, ˜ n denotes the density fluctuations, defined by ˜ n = n − ¯ n , with ¯ n the time averageddensity. An analogous definition is used for the other quantities. Figure 5 (top panels)displays the cross-coherence between ˜ n and ˜ φ for a standard ISTTOK simulation at theradial location x = x s + 5 ρ s and midway toroidally between the two limiter faces at ϕ = π .The fluctuations are normalized to their standard deviation. Since DW are characterized byan almost adiabatic electron response, a higher correlation between ˜ φ and ˜ n is expected inDW-driven turbulence with respect to BM-driven turbulence. Indeed, as shown in Fig. 5,the correlation is strong at the LFS, and even stronger at the HFS, which clearly points to aDW character of turbulence at this location, where the BM interchange drive is not present.We also perform a cross-coherence analysis for the three-dimensional simulations whereDW, and more specifically the diamagnetic terms T e ∇ (cid:107) ln n and 1 . ∇ (cid:107) T e in Ohm’s law (3),are removed from the system, and for three-dimensional simulations where the BM drive, thecurvature term in the vorticity equation (2), is neglected, yielding the middle and bottompanels of Fig. 5 respectively. One observes that BM removal does not affect the correlation atthe HFS, and increases it at the LFS (as compared with a standard simulation), as expectedfrom the DW nature of turbulence at the HFS and the mixed BM and DW nature at the LFS.On the other hand, removing DW has the effect of increasing the correlation at the HFS.As a matter of fact, the KH instability that drives transport at the HFS in DW-suppressedturbulent simulations leads to a high correlation between ˜ n and ˜ φ .We now turn our attention to the phase-shift − π < δ < π between ˜ n and ˜ φ , whichis expected to be large and close to π/ k (cid:107) = 0, temperature fluctuations, and KH effects, we have γ ∇ ⊥ ˜ φ ∼ ω ci T e + T i en C (˜ n ) , (12)and small in DW driven turbulence, where neglecting electron inertia, temperature fluctua-tions, and viscous G e terms we have instead in Eq. (3) ∇ (cid:107) ˜ φ ∼ T e en ∇ (cid:107) ˜ n. (13)12 IG. 5. Probability of correlation between density and electric potential fluctuations normalizedto their respective standard deviation, resulting from GBS simulations with standard ISTTOKparameters (top), and when DW (middle) or BM (bottom) are removed from the system. The HFS(left) and LFS (right) are shown.
In Fig. 6 we show δ at x = x s + 5 ρ s and ϕ = π (as in Fig. 5), by performing the Fouriertransform of ˜ φ and ˜ n along θ , on a domain with extension ∆ θ = π/ θ = 0for the LFS, and θ = ± π for the HFS, and computing the phase shift between these twoquantities as a function of k y . The phase shifts evaluated with a frequency of 10 c s /R ,during a time span of R/c s , are then binned as a function of k y with the proper weightgiven by the power spectral density of ˜ φ and ˜ n fluctuations. The results of this test, shownin Fig. 6, are not particularly clear. In fact, the phase-shift between ˜ φ and ˜ n is small both atLFS and HFS. Similarly small values are observed if BM and DW drive are removed fromthe simulation. In fact, Eq. (12) is too simplistic to study the phase shift between ˜ φ and˜ n . The short connection length of our configuration introduces finite k (cid:107) effects, that tend13 IG. 6. Phase-shift probability between density and electric potential fluctuations, resulting fromGBS simulations with standard ISTTOK parameters (top), and when DW (middle) or BM (bottom)are removed from the system. The HFS (left) and LFS (right) are shown. to reduce the phase shift. We have highlighted these effects by performing two-dimensionalsimulations (not shown) with an increasing connection length, and observing that δ tends tothe expected value of π/ IV. TURBULENCE SATURATION MECHANISMS
Having identified the nature of the linear turbulent drive at different locations, we nowturn to the investigation of the mechanisms that saturate the growth of the linearly unstablemodes. While a number of saturation mechanisms have been proposed (for a recent reviewsee Ref. [36]), it has been shown that the growth of a secondary KH instability and thegradient removal mechanism, i.e., the saturation of the linear mode due to the non-linear localflattening of the driving plasma gradients, are the main saturation mechanisms in the case ofDW and BM driven turbulence. Moreover, analytical estimates and numerical simulationssuggest that the gradient removal saturation mechanism is present when (cid:112) k y L p < ∼ (cid:112) k y L p (cid:39) (cid:112) k y L p (cid:39) . L p in the SOL can be derived by stating that the growth of thelinearly unstable modes saturates when the radial gradient of the perturbed pressure becomescomparable to the radial gradient of the background pressure dp/dx ∼ d ˜ p/dx , which can alsobe written as k x ˜ p ∼ pL p . (14)Following non-local linear theory as outlined in Refs [37, 38], for DW and BM respectively,we estimate the radial wavenumber as k x ∼ (cid:115) k y L p . (15)To estimate the balance between the pressure flux and the parallel losses at the limiterplates, we combine Eqs. (1) and (5), and ignore the curvature and diffusion terms, to derivethe leading order pressure equation ∂p∂t = − cB [ φ, p ] − ∇ (cid:107) ( pV (cid:107) e ) . (16)Writing [ φ, p ] = ∇ · Γ , we time average Eq. (16), integrate it along a magnetic field line,and neglect the pressure flux in the poloidal direction Γ y with respect to the turbulent radialflux Γ x = c ˜ p∂ y ˜ φ/B ∼ ck y ˜ φ ˜ p/B . In addition, estimating the parallel losses at the limiter as pV (cid:107) e (cid:12)(cid:12) limiter (cid:39) p c s , we obtain ∂ Γ x ∂x ∼ − p c s πR . (17)Finally, estimating the electrostatic potential ˜ φ by neglecting the k (cid:107) term in the pressureequation (16) as ˜ φ ∼ Bγ ˜ pL p / ( Rpk y c ), and with ∂ x Γ x ∼ Γ x /L p , we have L p = Rc s (cid:18) γk y (cid:19) max , (18)15here γ/k y is maximised over all possible instabilities present in the system. In practice,having fixed θ , the solution of Eq. (18) requires the evaluation of the linear growth rate γ asa function of k y and L p from the linear dispersion relation associated with the drift-reducedBraginskii system. We then seek the value of k y that yields the largest ratio γ/k y for each L p , and we obtain the value of L p that satisfies Eq. (18) using Muller’s secant method [39].A linear code was used to obtain γ and k y for the different unstable modes [12]. Here, aRobin boundary condition [40] is implemented that mimics the dynamics of the differentfields at the sheath entrance in the non-linear simulations.The L p solution of Eq. (18) for ISTTOK parameters is shown in Fig. 3 as a function of θ (green dashed line). The agreement with the simulation results is particularly good at theHFS, while at the LFS it overestimates L p by 25% (as expected from KH having a role insaturating turbulence).Using the result of Eq. (18), we also estimate L p as a function of the resistivity ν , ionto electron temperature ratio τ , and safety factor q in order to assess the dependence ofthe SOL radial pressure profile on these parameters. The results of this estimate are shownin Fig. 7, and reveal that L p depends weakly on the safety factor q , while it increases forincreasing values of ν and τ .Equation (18) allows us to further confirm the ISTTOK turbulent regimes identified inSection III, and extend this analysis to a wide parameter space. In fact, having estimated L p as a function of τ, ν , and q , one can evaluate the growth rate of the Resistive BM, InertialBM, Resistive DW, and Inertial DW instabilities. We note that the resistive branch of BMand DW is due to the presence of resistivity ( ν ) in Ohm’s law, Eq. (3), while an inertialbranch of BM and DW is made unstable by electron inertia ( m e ) effects. Therefore, thegrowth rate of the resistive BM and DW can be found by neglecting m e in Eq. (3), while theinertial instability is evaluated by neglecting ν e in Eq. (3). In order to identify the turbulentregimes we evaluate the growth rate of the four instabilities above at the k y and L p thatsolve Eq. (18). Turbulence is expected to be driven by the instability that has the largestlinear growth rate.The turbulence regimes are shown in Fig. 8, where Inertial DW drives turbulence at allpoloidal angles for typical ISTTOK parameters. An increase of the resistivity ν from thetypical ISTTOK standard simulation value, ν ∼ × − , to 1 × − leads to the ResistiveBM at the LFS, while for τ > θ = 0. For16 IG. 7. Equilibrium pressure scale-length, L p , as a function of the poloidal angle θ and normalisedresistivity, ν (left panel), ion to electron temperature ratio, τ (middle panel), and safety factor, q (right panel). The black lines represent the standard ISTTOK case.FIG. 8. Turbulent regimes as a function of the poloidal angle θ and normalised resistivity, ν (upperleft panel), ion to electron temperature ratio τ (middle panel), and safety factor q (right panel).The Resistive BM driven turbulence is in dark blue, Inertial BM in light blue, Resistive DW inyellow, and Inertial DW in red. The black lines represent the standard ISTTOK case. the case of 1 < τ <
2, Resistive DW drive turbulence at the HFS, and the turbulent regimeis not affected by the safety factor in a wide range of values (4 < q <
V. COMPARISON WITH EXPERIMENTAL RESULTS
To compare our numerical results with experimental measurements we consider an ISTTOKdischarge with density n = 4 × m − and q = 10 at the LCFS. The experimentalmeasurements were obtained with a multi-pin Langmuir probe measuring simultaneously thefloating potential V f and ion saturation current I sat . The probe was moved from shot-to-shotalong the radial direction and measurements were taken at r − a = 0 , IG. 9. Statistical moments of I sat from the experiment (red) and simulation (blue). From left toright: mean, standard deviation, skewness, and kurtosis. last-closed flux surface, − < r − a < r − a ≥
0. The experimental uncertaintywas estimated by performing three different discharges with the same parameters. From thesimulation results we evaluate I sat = enc s A ( A being the probe area) and V f = φ − Λ T e .First, we focus on the I sat statistical moments in Fig. 9. The temporal mean of I sat is monotonically decreasing for increasing radial locations, both in the simulations and inthe experiments. However, the large uncertainty does not allow us to compare reliablythe I sat gradient scale length. The standard deviation shows that fluctuations are large,approximately 50%, throughout the SOL both in the experiment and simulation, as it istypically observed in the SOL of fusion devices. The simulation results show a monotonicallyincreasing skewness, as expected from previous SOL studies [42–45]. On the other hand, inISTTOK, we find a rather large value ( (cid:39)
1) of the skewness at the LCFS. The skewness (aswell as the kurtosis) shows a better agreement between simulations and experiment in thefar SOL.These observations are confirmed by the comparison of the I sat probability distributionfunction (PDF) shown in Fig. 10. In all cases the I sat PDFs deviate strongly from a Gaussiandistribution and we observe that the I sat PDF is considerably more skewed in the experimentthan in the simulation at the LCFS. The level of agreement increases while moving towardsthe far SOL. The discrepancy between simulation and experimental results in the proximityof the LCFS might be due to intermittent events occurring in ISTTOK inside the LCFS.These events are not captured by the simulation that cannot properly describe the couplingwith core physics.As opposed to I sat , the V f PDFs show agreement with the simulation results within the18
IG. 10. I sat PDF from the experiment (red) and simulation (blue), at r − a = 0 mm (left), r − a = 5mm (center), and r − a = 10 mm (right).FIG. 11. V f PDF from the experiment (red) and simulation (blue), at r − a = 0 mm (left), r − a = 5mm (center), and r − a = 10 mm (right). error bars for the different radial locations (see Fig. 11). We remark that the V f PDFs arerather symmetric, possibly due to the bipolar nature of V f associated with the intermittentevents, and display Gaussian properties [46].We then consider the I sat and V f power spectral density, evaluated as the square of theabsolute value of the temporal Fourier transform. These are shown in Figs. 12 and 13 for I sat and V f respectively. In all cases, the power spectra are approximately flat for frequencies < ∼
20 kHz, a typical behavior observed in tokamak SOL turbulence [47]. At higher frequencies,we compare the spectrum decay index between ISTTOK and GBS profiles. Focusing on theregion 50 < f <
300 kHz, we assume a power-law of the form Af µ with A a constant, f the frequency, and µ the decay index. We find that the I sat power spectra show a sharperdecrease in the simulation, as compared to the experiment, while for V f , we find a sharperdecrease in the experimental values. Quantitatively, at r − a = 5 mm, the experiment andsimulation I sat spectral index are µ exp = − . ± .
32 and µ sim = − . ± .
03 respectively,while for V f we find µ exp = − . ± .
34 and µ sim = − . ± . IG. 12. I sat power spectra from the experiment (red) and simulations (blue), at r − a = 0 mm(left), r − a = 5 mm (center), and r − a = 10 mm (right).FIG. 13. V f power spectra from the experiment (red) and simulations (blue), at r − a = 0 mm(left), r − a = 5 mm (center), and r − a = 10 mm (right). Finally, in order to compare the experimental pressure gradient lengths L p with the onesdiscussed in Section IV, experimental measurements of n and T e were taken using sweepingLangmuir probes with 3 mm radial resolution. Experimental measurements suggest that L p is independent of q for a wide range of values ( L p = 4.8, 4.5, 4.3 ρ s for q = 7, 10, 13respectively), a behavior in agreement with simulation results (see Fig. 7). However, theexperimental value of L p (cid:39) . ρ s at the LFS differs from the one predicted in simulationresults by a factor larger than three. In fact, in the ISTTOK standard simulation we have L p (cid:39) ρ s at the LFS (see Fig. 3). This might be due to the presence of the outer wallin the experiment that acts effectively as a plasma sink and reduces L p . Its presence is notaccounted for in the GBS simulations, which considers a large radial domain extension.We remark that the longer pressure scale length observed in the experiment strengthensour theoretical observation that KH is not the driving mechanism of turbulence in ISTTOKSOL. A straightforward comparison of the linear growth rate of the KH instability, γ KH ∼ . c/B ) φ/L p , of the DW, γ DW ∼ . c s /L p , and of the BM, γ BM ∼ c s / (cid:112) RL p , (see Ref.[34]), shows that γ KH /γ DW ∼ γ KH /γ BM ∼ − . VI. CONCLUSIONS
The present paper addresses the study of plasma turbulence in a poloidally limited SOL,using linear calculations and non-linear simulations based on the drift-reduced Braginskiiequations. We focus our investigations on the parameters of the ISTTOK tokamak andcompare our theoretical results with experiments carried out there.Significant differences are found with respect to a toroidally limited SOL. Because of thepresence of the poloidal limiter that avoids the connection between the LFS and HFS, a clearpoloidal asymmetry is observed, with the time-averaged pressure scale length considerablyshorter at the LFS compared with the HFS. Due to the short connection length and relatedsteep pressure gradients, the role of DW is enhanced with respect to the toroidally limitedcase. In fact, for the typical ISTTOK parameters, we identify DW as the main linearinstability drive both at the LFS and HFS, where we also find KH to play a non-negligiblerole in saturating turbulence.The pressure scale length obtained from the non-linear simulations shows a remarkableagreement with estimates based on the saturation of the unstable linear modes due tothe non-linear local flattening of the driving plasma gradients at the HFS. The agreementdecreases at the LFS due to the aforementioned role of the KH instability in setting theturbulence amplitude.The comparison of the statistical properties of turbulence shows a good agreement betweenexperimental and numerical results particularly in the far SOL. Intermittent events observedin ISTTOK in the near SOL are not captured by the simulation. On the other hand, possiblybecause of the interaction of the plasma with the wall, the characteristic pressure scalegradient length found in the simulation is considerably larger than that measured in theexperiment. 21
II. ACKNOWLEDGMENTS
Part of the simulations presented herein were carried out using the HELIOS supercomputersystem at Computational Simulation Centre of International Fusion Energy Research Centre(IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between Euratomand Japan, implemented by Fusion for Energy and JAEA; and part were carried out at theSwiss National Supercomputing Centre (CSCS) under Project ID s549. This work has beencarried out within the framework of the EUROfusion Consortium and has received fundingfrom the Euratom research and training programme 2014-2018 under grant agreement No633053, and from Portuguese FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, under grantPD/BD/105979/2014. The views and opinions expressed herein do not necessarily reflectthose of the European Commission. [1] P. Ricci. Simulation of the scrape-off layer region of tokamak devices.
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