Theory-based scaling laws of near and far scrape-off layer widths in single-null L-mode discharges
M. Giacomin, A. Stagni, P. Ricci, J. A. Boedo, J. Horacek, H. Reimerdes, C. K. Tsui
TTheory-based scaling laws of near and far scrape-offlayer widths in single-null L-mode discharges
M. Giacomin , A. Stagni , , P. Ricci , J. A. Boedo , J.Horacek , H. Reimerdes , and C. K. Tsui , Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Swiss Plasma Center (SPC),CH-1015 Lausanne, Switzerland Politecnico di Milano, Via Ponzio 34/3, 20133 Milan, Italy Center for Energy Research (CER), University of California (UCSD), San Diego, LaJolla, California 92093 Institute of Plasma Physics of the CAS, Za Slovankou 3, Prague, Czech RepublicE-mail: [email protected]
Abstract.
Theory-based scaling laws of the near and far scrape-off layer (SOL)widths are analytically derived for L-mode diverted tokamak discharges by using atwo-fluid model. The near SOL pressure and density decay lengths are obtained byleveraging a balance among the power source, perpendicular turbulent transport acrossthe separatrix, and parallel losses at the vessel wall, while the far SOL pressure anddensity decay lengths are derived by using a model of intermittent transport mediatedby filaments. The analytical estimates of the pressure decay length in the near SOLis then compared to the results of three-dimensional, flux-driven, global, two-fluidturbulence simulations of L-mode diverted tokamak plasmas, and validated againstexperimental measurements taken from an experimental multi-machine database ofdivertor heat flux profiles, showing in both cases a very good agreement. Analogously,the theoretical scaling law for the pressure decay length in the far SOL is comparedto simulation results and to experimental measurements in TCV L-mode discharges,pointing out the need of a large multi-machine database for the far SOL decay lengths.
Keywords: plasma turbulence, scrape-off layer width, GBS
1. Introduction
In ITER and future fusion reactors, a significant fraction of the fusion power is expectedto cross the separatrix and be transported along the magnetic field lines to the targetplates through the narrow region of the scrape-off layer (SOL). Due to technologicallimits imposed by materials, the peak of the heat flux reaching the tokamak wall must notexceed a value of the order of 10 MW/m [1,2]. Without high volumetric power radiationin the SOL and partial divertor detachment, this limit will be certainly exceeded inITER [3–7]. Therefore, understanding the mechanisms that regulate turbulent transportin the SOL and predicting the SOL power decay length is of fundamental importance to a r X i v : . [ phy s i c s . p l a s m - ph ] J a n determine the operational window for a divertor solution compatible with adeguate coreconfinement, not only for ITER but also for all future high-performance fusion devices.While ITER goal is to operate in H-mode where heat exhaust is most severe, the firstcampaigns, as well as the start-up and landing phase of the H-mode ITER discharges,will be in L-mode, thus prompting the need to provide theoretical scaling laws of theSOL power decay length in this regime. Moreover, understanding the mechanisms thatregulate the SOL width in L-mode discharges constitutes the first step towards a theory-based scaling law of the near SOL width in H-mode, which involves a more complexphysics.As observed experimentally (see e.g. Refs. [8, 9]), the SOL presents two densityand power decay lengths; a shorter one in the near SOL and a longer one in the farSOL, which is the result of different turbulence dynamics in these two regions. Indeed,as experimentally shown in Ref. [10], turbulence dynamics is wave-like in the near SOLand intermittent in the far SOL. The power fall-off length measured at the target platesis mainly related to the power fall-off length in the near SOL, while the heat load onthe main-chamber first wall depends mainly on the intermittent perpendicular transportoccurring in the far SOL.Significant experimental efforts have been put in the past few years to deriveexperimental scaling laws of the power fall-off length in the near SOL at the divertorplates in L-mode diverted plasmas (see, e.g., Refs. [11–15]). Recently, a nonlinearregression has been carried out on a set of power fall-off length measurements froma multi-machine database including the COMPASS, EAST, Alcator C-Mod, MAST,and JET tokamaks [15]. By combining five hundred L-mode outer and inner divertorheat flux profiles obtained by probes or IR camera, thirteen credible scaling laws havebeen derived. A scaling law in good agreement with the experimental data (describingR =92% of data variation) obtained by considering only outer divertor measurementsis [15] λ q = 2800 (cid:16) aR (cid:17) . f . Gw j − . p , (1)where λ q is the power fall-off length in units of mm, f Gw is the Greenwald fraction, a isthe tokamak minor radius, R is the tokamak major radius, and j p is the plasma currentdensity in units of MA/m .In parallel to the experimental effort, recent theoretical and numerical studies basedon two-fluid models, justified by the high plasma collisionality of this region, haveinvestigated the mechanisms that regulate the near SOL width in L-mode, leading toanalytical and numerical scaling laws of the SOL density and pressure gradient lengths inboth limited [16–19] and diverted [20–23] geometries. A direct comparison of theoreticalscaling laws to experimental data has been carried out in limited geometry [16,18,24,25],showing a good agreement with experimental measurements. However, no in-depthcomparison between first-principles theory-based scaling laws and experimental datataken from a multi-machine database has been carried out in L-mode diverted geometry.While turbulence in the near SOL is characterized by a wave-like dynamics,turbulent transport in the far SOL is dominated by intermittent events due to coherentplasma filaments, also known as blobs [26]. Filaments extend along the parallel directionwith a cross section spatially localized on the poloidal plane and their associated densityfluctuations have an amplitude even larger than the background density. Filamentsoriginate near the last-closed flux surface (LCFS) as the results of the nonlinearsaturation of interchange-like instabilities, with the density fluctuations sheared apartby the E × B velocity and detached from the main plasma [10, 26]. Then, the verticalcharge separation inside the filaments, caused by magnetic gradients and curvaturedrifts, generates a vertical electric field that, in turns, gives rise to a radial E × B drift that transports filaments outwards, contributing significantly to the perpendiculartransport in the far SOL, flattening the density and pressure profile, and increasing theplasma-wall contact, as experimentally observed in Ref. [27]. Plasma filaments have beenexperimentally studied in tokamaks [28–31], stellarators [32], reversed field pinch [33],and basic plasma devices [10, 34, 35].An analytical theory based on considering separately the SOL in the upstream (i.e.from the outboard midplane to the divertor entrance) and divertor regions, referredas two-region model, has been proposed to describe the propagation of filaments in theSOL. Four regimes of filament motion have been identified, depending on the mechanismresponsible for balancing the charge separation driven by the magnetic curvature andgradient drifts [36]: the sheath connected regime (C s ), where the curvature drive isbalanced by the current flowing to the sheath; the ideal interchange mode regime (C i ),where the ion polarization current due to fanning of the flux surfaces in the divertorregion damps the charge separation; the resistive ballooning regime (RB), where thedamping of upstream ion polarization current dominates; and the resistive X-pointregime (RX), where the parallel current flowing between the upstream and divertorregions is the key damping mechanism. Each damping mechanism results into a differentdependence of filament velocity on filament size. The two key parameters that determinethe filament regime are the collisionality parameter,Λ = ν ei L (cid:107) ρ s Ω ce L (cid:107) , (2)and the size parameter [36],Θ = (cid:16) a b a ∗ (cid:17) / , (3)where ν ei is the electron to ion collision frequency, ρ s = c s / Ω ci is the ion sound Larmorradius, with c s = (cid:112) T e /m i the sound speed and Ω ci = eB/m i the ion cyclotron frequency, L (cid:107) is the parallel connection length from upstream to the divertor region entrance, L (cid:107) is the parallel connection length from the divertor region entrance to the target plate,Ω ce = eB/m e is the electron cyclotron frequency, a b is the filament size in the poloidalplane, and a ∗ is the reference filament size [37], a ∗ = ρ s (cid:16) L (cid:107) ρ s R n b n (cid:48) (cid:17) / , (4)with n b the average of the filament density and n (cid:48) the density at the near-to-far SOLinterface. The two-region model for filament motion has been extensively validatedagainst experimental results (see, e.g., [31, 38, 39]) and verified through numericalsimulations. These include recent numerical nonlinear two-dimensional [40, 41], three-dimensional single-seed filament [42, 43], as well as three-dimensional self-consistentlygenerated filament simulations in realistic geometry [23,37,44], which have shown a goodagreement with the two-region model, also in H-mode plasmas [45]. Moreover, the workcarried out in Ref. [23] in double-null geometry has shown that the far SOL densitydecay length can be described as the result of the transport associated to filaments,whose velocity is described by using the two-region model.In the present work, we analytically derive first-principles based scaling laws of thenear and far density and pressure decay length for L-mode diverted plasmas. We exploitthe results of three-dimensional, flux-driven, global, two-fluid turbulence simulationscarried out by using the GBS code [46, 47] and we consider a balance among sources,perpendicular turbulent transport across the LCFS, and parallel losses at the vessel wall.We focus here on a single-null divertor geometry and we derive scaling laws in termsof engineering parameters, such as tokamak major and minor radii, plasma elongation,toroidal magnetic field at the tokamak axis, edge safety factor, and power crossingthe separatrix. The theoretical scaling laws are then compared to simulation resultsand against experimental data, showing a very good agreement in the near SOL, whilesuggesting the need of a large multi-machine database for the far SOL decay lengths.The paper is organized as follows. The physical model implemented in GBS andused to derive the near and far SOL pressure decay lengths is described in Sec. 2. Anoverview of simulation results is presented in Sec. 3. In Sec. 4, we derive theoreticalscaling laws of the near and far SOL density and pressure decay lengths. A comparisonbetween the analytical scaling laws and experimental results is reported in Sec. 5.Finally, the conclusions follow in Sec. 6.
2. Simulation model
The model considered in this work is based on the drift-reduced Braginskii two-fluidplasma model implemented in the GBS code [18, 46, 47]. A detailed description of themodel can be found in Ref. [22]. The use of a drift-reduced fluid model is justifiedby the high plasma collisionality in the SOL, λ e (cid:28) L (cid:107) ∼ πqR , with λ e the electronmean-free path, and by the large scale fluctuations, k ⊥ ρ i (cid:28)
1, with ρ i the ion Larmorradius, that dominate transport in this regime. The model is electrostatic, makes use ofthe Boussinesq approximation [46, 48], and neglects the interplay between plasma andneutrals, although this is implemented in GBS [49]. The use of the electrostatic limitis justified by the results of Ref. [50] that shows a negligible effect of electromagneticperturbations on equilibrium profiles and turbulent dynamics in the SOL at low andintermediate values of β , such as the one considered here. The effect of the Boussinesqapproximation is discussed in Refs. [48, 51], showing that it has a negligible effect onSOL turbulence and equilibrium profiles, although this approximation cannot be takenfor granted in general, as shown in Refs. [52, 53]. Since coupling with neutrals dynamicsis not considered in the present work, our analysis is restricted to low-density plasma inlow-recycling conditions. Within these approximations, the model equations are ∂n∂t = − ρ − ∗ B [ φ, n ] + 2 B (cid:104) C ( p e ) − nC ( φ ) (cid:105) − ∇ (cid:107) ( nv (cid:107) e ) + D n ∇ ⊥ n + s n , (5) ∂ω∂t = − ρ − ∗ B [ φ, ω ] − v (cid:107) i ∇ (cid:107) ω + B n ∇ (cid:107) j (cid:107) + 2 Bn C ( p e + τ p i ) + D ω ∇ ⊥ ω , (6) ∂v (cid:107) e ∂t = − ρ − ∗ B [ φ, v (cid:107) e ] − v (cid:107) e ∇ (cid:107) v (cid:107) e + m i m e (cid:16) νj (cid:107) + ∇ (cid:107) φ − n ∇ (cid:107) p e − . ∇ (cid:107) T e (cid:17) + 43 n m i m e η ,e ∇ (cid:107) v (cid:107) e + D v (cid:107) e ∇ ⊥ v (cid:107) e , (7) ∂v (cid:107) i ∂t = − ρ − ∗ B [ φ, v (cid:107) i ] − v (cid:107) i ∇ (cid:107) v (cid:107) i − n ∇ (cid:107) ( p e + τ p i ) + 43 n η ,i ∇ (cid:107) v (cid:107) i + D v (cid:107) i ∇ ⊥ v (cid:107) i , (8) ∂T e ∂t = − ρ − ∗ B [ φ, T e ] − v (cid:107) e ∇ (cid:107) T e + 23 T e (cid:104) . ∇ (cid:107) v (cid:107) i − . ∇ (cid:107) v (cid:107) e + 0 . v (cid:107) i − v (cid:107) e ) ∇ (cid:107) nn (cid:105) + 43 T e B (cid:104) C ( T e ) + T e n C ( n ) − C ( φ ) (cid:105) + χ (cid:107) e ∇ (cid:107) T e + D T e ∇ ⊥ T e + s T e , (9) ∂T i ∂t = − ρ − ∗ B [ φ, T i ] − v (cid:107) i ∇ (cid:107) T i + 43 T i B (cid:104) C ( T e ) + T e n C ( n ) − C ( φ ) (cid:105) − τ T i B C ( T i )+ 23 T i ( v (cid:107) i − v (cid:107) e ) ∇ (cid:107) nn − T i ∇ (cid:107) v (cid:107) e + χ (cid:107) i ∇ (cid:107) T i + D T i ∇ ⊥ T i + s T i , (10) ∇ ⊥ φ = ω − τ ∇ ⊥ T i . (11)In Eqs. (5-11) and in the following of the present paper (unless specified otherwise),the density, n , the electron temperature, T e , and the ion temperature, T i , are normalizedto the reference values n , T e , and T i . The electron and ion parallel velocities, v (cid:107) e and v (cid:107) i , are normalized to the reference sound speed c s = (cid:112) T e /m i . The norm ofthe magnetic field, B , is normalized to the toroidal magnetic field at the tokamakmagnetic axis, B T . Perpendicular lengths are normalized to the ion sound Larmorradius ρ s = c s / Ω ci and parallel lengths are normalized to the tokamak major radius R . Time is normalized to R /c s . The dimensionless parameters appearing in themodel equations are the normalized ion sound Larmor radius, ρ ∗ = ρ s /R , the ionto electron reference temperature ratio, τ = T i /T e , the normalized electron and ionviscosities, η ,e and η ,i , the normalized electron and ion parallel thermal conductivities, χ (cid:107) e and χ (cid:107) i , and the normalized Spitzer resistivity, ν = e n R / ( m i c s σ (cid:107) ) = ν T − / e ,with σ (cid:107) = (cid:16) . n e τ e m e (cid:17) n = (cid:16) . √ π (4 π(cid:15) ) e T / e λ √ m e (cid:17) T / e , (12) ν = 4 √ π . e (4 π(cid:15) ) √ m e R n λm i c s T / e , (13)where λ is the Coulomb logarithm.The dimensionless equilibrium magnetic field, B = ±∇ ϕ + ρ ∗ ∇ ψ × ∇ ϕ , is writtenin terms of the poloidal flux function ψ , which can be an analytical function or canbe obtained from an equilibrium reconstruction, with ϕ being the toroidal angle. Theplus (minus) sign refers to the direction of the toroidal magnetic field correspondingto the ion- ∇ B drift pointing upwards (downwards). The spatial operators appearingin Eqs. (5-11) are the E × B convective term [ g, f ] = b · ( ∇ g × ∇ f ), the curvatureoperator C ( f ) = B [ ∇ × ( b /B )] / · ∇ f , the parallel gradient operator ∇ (cid:107) f = b · ∇ f ,and the perpendicular Laplacian operator ∇ ⊥ f = ∇ · [( b × ∇ f ) × b ], where b = B /B is the unit vector of the magnetic field. The numerical diffusion terms, D f ∇ ⊥ f , areadded for numerical stability and they lead to significantly smaller transport thanthe turbulent processes described by the simulations. The differential operators arediscretized on a non-field-aligned ( R, φ, Z ) cylindrical grid, by means of a fourth-orderfinite difference scheme [47] ( R and Z are the radial, from the tokamak symmetryaxis, and vertical directions). Details on the numerical implementation of the spatialoperators are reported in Ref. [22].The source terms in the density and temperature equations, s n and s T , are addedto fuel and heat the plasma. The density and the temperature sources are analyticaland toroidally uniform functions of ψ ( R, Z ), s n = s n exp (cid:16) − ( ψ ( R, Z ) − ψ n ) ∆ n (cid:17) , (14) s T = s T (cid:104) tanh (cid:16) − ψ ( R, Z ) − ψ T ∆ T (cid:17) + 1 (cid:105) , (15)where ψ n and ψ T are flux surfaces located inside the LCFS. The density source islocalized around the flux surface ψ n , close to the separatrix, and mimics the ionizationprocess, while the temperature source extends through the entire core and mimics theohmic heating. Similarly to Ref. [22], we define S n and S T as the total density andtemperature source integrated over the area inside the LCFS, S n = (cid:90) A LCFS ρ ∗ s n ( R, Z ) d R d Z (16)and S T = (cid:90) A LCFS ρ ∗ s T ( R, Z ) d R d Z , (17)where the factor ρ ∗ appears from our normalization choices. Analogously, we define theelectron pressure source, proportional to the power source, as S p = (cid:82) A LCFS ρ ∗ s p d R d Z ,with s p = ns T e + T e s n and s T e the electron temperature source.The simulation domain we consider encompasses the whole tokamak plasma volumeto retain the core-edge-SOL interplay [54–56], as presented for the first time in Ref. [57].The poloidal cross section has a rectangular shape of radial and vertical extension L R and L Z , respectively. For the analysis of the near and far SOL decay lengths, we considerflux-coordinates ( ∇ ψ, ∇ χ, ∇ ϕ ), where ∇ ψ denotes the direction orthogonal to the fluxsurfaces, ∇ ϕ is the toroidal direction, and ∇ χ = ∇ ϕ × ∇ ψ .Magnetic pre-sheath boundary conditions, derived in Ref. [58], are applied at thetarget plates. Neglecting correction terms linked to radial derivatives of the density andpotential at the target plate, these boundary conditions can be expressed as v (cid:107) i = ± (cid:112) T e + τ T i , (18) v (cid:107) e = ± (cid:112) T e + τ T i exp (cid:16) Λ − φT e (cid:17) , (19) ∂ Z n = ∓ n √ T e + τ T i ∂ Z v (cid:107) i , (20) ∂ Z φ = ∓ T e √ T e + τ T i ∂ Z v (cid:107) i , (21) ∂ Z T e = ∂ Z T i = 0 , (22) ω = − T e T e + τ T i (cid:104) ( ∂ Z v (cid:107) i ) ± (cid:112) T e + τ T i ∂ ZZ v (cid:107) i (cid:105) , (23)where Λ = 3. The top (bottom) sign refers to the magnetic field pointing towards(away from) the target plate. Details on the numerical implementation can be found inRef. [47].
3. Overview of simulation results
We focus here on a set of simulations in the L-mode turbulent transport regime. Asdescribed in Ref. [22], where a dedicated analysis has been carried out to identify theturbulent transport regimes in the tokamak edge, a turbulent transport regime appearsin two-fluid simulations at low value of heat source and high value of collisionality, whereturbulent transport is driven by the resistive ballooning instability. This mode has beenassociated to the L-mode operational regime of tokamaks.The simulations are carried out with the following parameters: ρ − ∗ = 500, a/R (cid:39) . τ = 1, η ,e = 5 × − , η ,i = 1, χ (cid:107) e = χ (cid:107) i = 1, D f = 6 for all fields, L R = 600, L Z = 800, s n = 0 .
3, ∆ n = 800, ∆ T = 720, and different values of s T and ν (see Eq. (13)), with both favorable and unfavorable ion- ∇ B drift directionsbeing considered (see Tab. 1). The magnetic equilibrium, described in Ref. [22], isanalytically obtained by solving the Biot-Savart law in the infinite aspect-ratio limitfor a current density with a Gaussian distribution centered at the tokamak magneticaxis, which mimics the plasma current, and an additional current filament outside thesimulation domain to produce the X-point. The value of the plasma current and thewidth of its Gaussian distribution are chosen to have a safety factor q (cid:39) q (cid:39) R (cid:39) B T (cid:39) n (cid:39) m − and T e (cid:39)
20 eV, the size of the simulation domain inphysical units is L R (cid:39)
30 cm, L Z (cid:39)
40 cm and R (cid:39)
25 cm, which is approximatelya third of the TCV size. Regarding the numerical parameters, the numerical grid is N R × N Z × N ϕ = 240 × ×
80 and the time-step is 2 × − . We analyse the simulation s T0 ν ion- ∇ B drift
Table 1:
Dimensionless parameters (temperature source strength s T , normalizedresistivity ν , and ion- ∇ B drift direction) of the set of GBS simulations consideredin this work. results after an initial transient, when the simulations reach a global turbulent quasi-steady state resulting from the interplay between the sources in the closed flux surfaceregion, turbulence that transports plasma and heat from the core to the SOL, and thelosses at the vessel. In the following, we refer to the equilibrium of any quantity f as its time and toroidal average during the quasi-steady state, ¯ f = (cid:104) f (cid:105) ϕ,t , and to itsfluctuating component as ˜ f = f − ¯ f .As an example of typical simulation results, Fig. 1 shows the equilibrium density,the normalized standard deviation and the skewness of density fluctuations in the plasmaboundary. The simulation with s T =0.15 and ν =0.6 is considered. As a consequenceof turbulent transport being driven by a resistive ballooning mode [22], the normalizedstandard deviation of the density fluctuations peaks on the low-field side (LFS) andremains relatively large throughout the entire LFS SOL, as shown in Fig. 1 (b). Thenear and far SOL are characterized by large fluctuations with amplitude comparableto the equilibrium quantity, as experimentally observed in Refs. [59–61]. The skewnessis small in the LFS of the near SOL, suggesting wave-like turbulence, and increases inthe far SOL, hinting at the presence of intermittent turbulent events. A snapshot ofthe normalized density fluctuations for the same simulation (see Fig. 2 (a)) shows thatdensity fluctuations mainly develop across the separatrix, forming eddies that extendin the radial direction and detach from the main plasma. Detached eddies give rise tofilaments that radially propagate in the far SOL and are ultimately responsible for itsintermittent nature. Therefore, turbulence in the far SOL arises from the steep pressureand density gradients across the separatrix and not from the local equilibrium pressureand density profiles. The different nature of plasma turbulence in the near and far SOL,namely the highly intermittent and non-local character of turbulence in the far SOL in (a) (b) (c) Figure 1:
Equilibrium density, ¯ n , (a), normalized standard deviation, σ n , (b) andskewness, µ n , (c) of density fluctuations at the plasma boundary for the simulation with s T =0.15 and ν =0.6. The white line represents the separatrix. (a) time [ R /c s ] n Near SOLtime [ R /c s ] n Far SOL (b)
Figure 2:
Typical snapshot of normalized density fluctuations on a poloidal plane (a)and typical time traces of the density in the near and far SOL (b) for the simulationwith s T =0.15 and ν =0.6. contrast to wave-like turbulence dynamics in the near SOL, is highlighted in Fig. 2 (b),where two typical time traces of the density in the near and far SOL are shown.As a consequence of the different transport mechanisms taking place in the near andfar SOL, density and pressure show a different decay length in these two regions. This isshown in Fig. 3, where the equilibrium pressure and density radial profiles at the outermidplane are fitted by assuming only one or two distinct exponential decay lengths.One exponential overestimates the decay length in the near SOL and underestimatesthe one in the far SOL. On the other hand, the fit based on two distinct exponential0 r/r sep l og ( ¯ p e ) -1.5-1-0.500.5 (a) r/r sep l og ( ¯ n ) -0.8-0.6-0.4-0.200.20.4 (b) Figure 3:
Radial profile of the equilibrium electron pressure, ¯ p e , (a) and density, ¯ n ,(b) at the outer midplane (black dots) for the simulation with s T = 0 . and ν = 0 . ,and overposed the exponential fits based on one (blue line) or two (red lines) exponentialdecay lengths. The vertical dashed black line denotes the position of the separatrix. decay lengths agrees well with the equilibrium pressure and density radial profiles onthe entire plasma boundary. Two distinct exponential decay lengths have been observedalso in experiments (see, e.g., Refs. [8, 9]) as well as in other fluid simulations (see, e.g.,Refs. [23, 62]). We note that, as revealed by the fit based on two exponential functions,the decay lengths of density and pressure in the near SOL match the ones in the tokamakedge inside the LCFS. This is in agreement with experimental observations that showthe presence of one characteristic pressure decay length across the separatrix [63].Fig. 4 shows the near and far SOL pressure (density) decay lengths, denoted as L p, GBS ( L n, GBS ) and L (cid:48) p, GBS ( L (cid:48) n, GBS ), respectively, for the set of GBS simulations withinput parameters listed in table 1 (upwards ion- ∇ B drift direction). The near SOLpressure gradient length increases as the collisionality increases or the heat sourcedecreases, a feature that is associated with turbulent transport being driven by theresistive ballooning instability (see Eq. (27) in Sec. 4). The dependence of the far SOLdecay lengths on heat source and collisionality is similar to the one of the near SOL,although it shows a weaker dependency on the collisionality. Similar conclusions can bedrawn for the near and far SOL density decay lengths.
4. Scaling laws for the near and far SOL decay lengths
Plasma turbulence in the L-mode transport regime is mainly driven by resistiveballooning modes with the effect of shear flows being negligible, as shown in Ref. [22].Based on the analysis of the mechanisms that lead to the saturation of the ballooning1 ν s T L p,GBS (a) Pressure decay length in the near SOL. ν s T L ′ p,GBS (b) Pressure decay length in the far SOL. ν s T L n,GBS (c) Density decay length in the near SOL. ν s T L ′ n,GBS (d) Density decay length in the far SOL.
Figure 4:
Near and far SOL pressure ((a) and (b)) and density ((c) and (d)) decaylengths obtained from GBS simulations at the various values of s T and ν considered inthis work with the ion- ∇ B drift pointing upwards. Comparable values are obtained forthe simulations with downwards ion- ∇ B drift direction. mode, and a balance between heat sources and turbulent transport, Ref. [22] shows thatthe equilibrium pressure gradient length in the edge inside the LCFS of the consideredL-mode simulations agrees well with the analytical estimate, L p ∼ (cid:104) ρ ∗ νq ¯ n ) (cid:16) L χ S p ¯ p e (cid:17) (cid:105) / ¯ T e , (24)where L χ (cid:39) πa (cid:112) (1 + κ ) / κ being the plasma elongation, and ¯ n , ¯ T e , ¯ p e are the equilibrium density, electrontemperature, electron pressure, which are evaluated at the LCFS [22]. Based on theobservation (see Fig. 3) that the exponential decay length of the equilibrium pressure2profile in the near SOL and in the edge inside the LCFS correspond, Eq. (24) can alsobe used to estimate the near SOL pressure decay length. Indeed, a strong connectionbetween confined edge and near SOL physics has been experimentally observed inRefs. [64–66] across various confinement regimes.In the present work, we extend the result derived in Ref. [22] by expressing L p in terms of engineering parameters in order to facilitate both the comparison withexperimental results and its applicability to tokamak operation. Therefore, we makeexplicit in Eq. (24) the dependence of ¯ T e on S p and L p , by balancing S p with theparallel losses at the vessel walls. As an order of magnitude estimate, this balance canbe obtained by integrating the heat flux at the vessel wall over the SOL width, (cid:90) SOL ¯ p e ¯ c s d l ∼ S p . (25)In Eq. (25) we consider the low-recycling regime (i.e. no temperature drop in the divertorregion) and we assume that the plasma outflows at the divertor plate with the soundspeed. Moreover, by assuming that the electron pressure and electron temperature decayexponentially in the SOL on the scale L p and L T , respectively, where L T (cid:39) (1+ η e ) L p /η e ,with η e (cid:39) .
77 derived in Ref. [67] and in agreement with the simulations presented here,¯ T e at the LCFS becomes¯ T e ∼ (cid:104)(cid:16) η e η e ) (cid:17) S p ¯ nL p (cid:105) / . (26)The near SOL estimate of L p is then derived by replacing ¯ T e given by Eq. (26) intoEq. (24), L p ∼ (cid:104) (cid:16) η e η e ) (cid:17) ρ ∗ ν q L χ ¯ n S − p (cid:105) / . (27)We note that the equilibrium density gradient length in the near SOL can be directlyobtained from Eq. (27) and the relation L n (cid:39) (1 + η e ) L p [67], L n ∼ (1 + η e ) (cid:104) (cid:16) η e η e ) (cid:17) ρ ∗ ν q L χ ¯ n S − p (cid:105) / . (28)The theoretical estimates of the near SOL pressure and density decay lengths showa very good agreement with the simulation results for all the values of s T and ν considered in this work and both directions of the ion- ∇ B drift, as displayed in Fig. 5.Indeed, across the set of simulations performed, the difference between simulation resultsand theoretical predictions is below 20% for both L p and L n .In order to write the analytical scaling law of Eq. (27) in terms of engineeringparameters, such as the power entering into the SOL ( P SOL ), the tokamak major andminor radius, and the toroidal magnetic field, we substitute S p ∼ P SOL / (2 πR ) and ν given by Eq. (13) into Eq. (27) and, in physical units, we obtain L p (cid:39) . A / q / R / P − / a / (1 + κ ) / n / e B − / T , (29)where L p is in units of mm, A is the mass number of the main plasma ions, R and a are in units of m, P SOL is in units of MW, n e is the density at the LCFS in units of10 m − , and B T is in units of T.3 L p
10 20 30 40 50 60 70 L p , G B S (a) L n
30 40 50 60 70 80 90 100 110 L n , G B S (b) Figure 5:
Comparison between the analytical estimates of the near SOL pressure (a)and density (b) decay lengths and the corresponding ones obtained from GBS simulations.4.2. Far SOL
In order to estimate the far SOL density decay length, we consider here a similarapproach to the one presented for double-null geometry in Ref. [23]. With respectto Ref. [23], we provide also an estimate of the pressure decay length and we write boththeoretical scaling laws in terms of engineering parameters.As a first step, a pattern-recognition algorithm for filament detection/tracking,described in Refs. [37, 44], is applied to the GBS simulations considered in this work todetermine filament size, velocity, and collisionality parameter, allowing the identificationof the filament regime. A typical dispersion plot of the averaged Λ and Θ parametersof each detected filament in the simulation with s T = 0 .
15 and ν = 0 . v = v b /v ∗ of eachfilament, with v ∗ = c s (cid:104) (cid:16) πa ψ a χ (cid:17) n b ¯ n (cid:48) ρ s L (cid:107) ρ ∗ (cid:105) / (30)being the reference filament velocity (see Ref. [37]), is displayed for the same simulationas a function of the normalized size, ˆ a = a b /a ∗ . In Eq. (30), a ψ and a χ denote the averagesize of filaments along the ∇ ψ and ∇ χ direction, respectively. The normalized filamentvelocities are mainly scattered between zero and a maximum velocity that varies as afunction of size and collisionality in agreement with the analytical normalized velocitypredicted by the two-region model [36]. This numerical result agrees with experimental4 Θ -1 Λ -1 RB RXC i C s (a) ˆ a ˆ v -0.500.511.522.5 ˆ v = ˆ a / ˆ v = Λ ˆ a − -4-3-2-101234 log( Θ / Λ ) (b) Figure 6:
Dispersion plot in the phase space (Λ , Θ) of detected filaments in thesimulation with s T = 0 . and ν = 0 . (a). Black dashed lines are used to delimit thefour regimes. Normalized filament velocity as a function of filament size of each detectedfilament in the same simulation (b). The dashed black line represents the velocity scalingpredicted by the two-region model in the RB regime ( Θ < Λ ), while the dashed red linethe one in the RX regime ( Θ > Λ ) [36]. All quantities are obtained by averaging overthe filament life. observations that show that the theoretical predictions constitute an upper bound for thefilament velocities [31]. Indeed, some mechanisms responsible for decreasing the radialfilament velocity, such as the filament-filament interaction and the filament rotation, arenot included in the two-region model of Ref. [36].Since filament dynamics is responsible of the far SOL pressure and densitytransport, as shown in Ref. [23], we derive an analytical estimate of equilibrium pressuredecay length in the far SOL by balancing the perpendicular transport due to filamentmotion with parallel heat transport. For this purpose, we take the sum of Eq. (5),multiplied by T e , and Eq. (9), multiplied by n . Then, by time and toroidal averagingthe resulting equation, we obtain ρ − ∗ ∂ ψ ¯ q b,ψ + ∇ (cid:107) (¯ p e ¯ v (cid:107) e ) + 23 1 . p e ∇ (cid:107) ¯ v (cid:107) e + 23 0 .
71 ¯ T e ¯ v (cid:107) e ∇ (cid:107) ¯ n (cid:39) . (31)In Eq. (31), we identify the perpendicular heat transport with the one mediated byfilaments, ¯ q b,ψ . By assuming that the electron parallel velocity is of the order of c s , andapproximating ∂ ψ ∼ /L (cid:48) p and ∇ (cid:107) ∼ /L (cid:107) , Eq. (31) yields ρ − ∗ ¯ q (cid:48) b,ψ L (cid:48) p (cid:39) C ¯ p (cid:48) e ¯ c (cid:48) s L (cid:107) , (32)where C = 1 + 1 .
71 (2 /
3) + 0 .
71 (2 / (cid:39) .
6. In Eq. (32), the prime symbol appearingin ¯ p (cid:48) e , ¯ c (cid:48) s , and ¯ q (cid:48) b,ψ denotes that these quantities are evaluated at the near-far SOL5interface [23, 37]. We note that Eq. (32) only holds in case of negligible variation ofelectron temperature along the magnetic field lines, which is the case of the low-recyclingregime considered here. The far SOL pressure decay length can then be obtained fromEq. (32), L (cid:48) p ∼ ρ − ∗ C ¯ q (cid:48) b,ψ L (cid:107) ¯ p (cid:48) e ¯ c (cid:48) s , (33)which relates L (cid:48) p to the perpendicular heat flux associated to the filament motion.In order to estimate ¯ q (cid:48) b,ψ , we assume that a filament can be described on the poloidalplane as a coherent structure with Gaussian peak pressure p b,i and half width at halfmaximum a ψ,i , along the ∇ ψ direction, and a χ,i , along the ∇ χ direction ( i is the indexidentifying the i -th filament). The heat flux associated to the filament motion can beestimated by multiplying the pressure associated with a filament and the filament centerof mass radial velocity, v b,i , and summing over all the filaments. We obtain q (cid:48) b,ψ ( ψ, χ ) ∼ (cid:88) i p b,i v b,i exp (cid:16) − ( ψ − ψ b,i ) (2 a ψ,i ) − ( χ − χ b,i ) (2 a χ,i ) (cid:17) , (34)where ψ and χ denote coordinate variations along ∇ ψ and ∇ χ , and ( ψ b,i , χ b,i ) are the i -th filament center of mass coordinates. An estimate of the heat flux due to filamenttransport is then obtained by averaging q b,i over time and over the far SOL area [68],¯ q (cid:48) b,ψ = (cid:68) A SOL (cid:90) A SOL q b ( ψ, χ )d ψ d χ (cid:69) t = 2 πA SOL log 2 (cid:88) i (cid:104) a ψ,i a χ,i p b,i v b,i (cid:105) t , (35)where A SOL represents the total far SOL area. By neglecting possible correlation betweenfilaments [69] and defining N b as the average number of filaments such that the averagedpeak pressure is given by p b ∼ (cid:80) i (cid:104) p b,i (cid:105) t /N b , Eq. (35) can be approximated as¯ q (cid:48) b,ψ ∼ f b p b v b , (36)where f b = N b πa ψ a χ /A SOL is the blob packing fraction.In order to make further progress, since filament and background pressure are bothprogressively drained by the parallel heat flow as moving radially through the far SOL,we assume that the filament peak-to-background pressure ratio remains constant. InRef. [23], the density fluctuations in the far SOL were assumed to be three timeslarger than in the near SOL to account for turbulent transport being mainly due tolarge filaments. Here, as an order of magnitude estimate, we consider that pressurefluctuations in the near and far SOL have similar values, leading to p b ¯ p (cid:48) e ∼ ˜ p e ¯ p e ∼ L p k ψ . (37)The peak filament pressure at the near-far SOL interface can then be obtained, p b ∼ ¯ p (cid:48) e L p k ψ . (38)To estimate the blob packing fraction, we evaluate the average filament number bybalancing the filament generation and loss rates. As filaments are generated by the6nonlinear development of the ballooning instability appearing across the LCFS, thefilament generation rate R b, gen is given by the ballooning mode wavenumber along theLCFS, L χ k χ / (2 π ), divided by the filament generation time, which can be approximatedby the time that a streamer takes to travel its own extension, i.e. 4 a ψ /v b . We obtain R b, gen ∼ L χ k χ k ψ v b π . (39)The filament loss rate R b, loss is given by the average filament number on a poloidal planedivided by the time that a filament takes to cross the radial domain, R b, loss ∼ N b v b L ψ . (40)The average filament number is then obtained by equating Eqs. (39) and (40), N b ∼ A SOL k χ k ψ π , (41)where A SOL (cid:39) L χ L ψ . By using Eq. (41), the blob packing fraction becomes f b ∼ π/ , (42)where we have used a ψ ∼ π/ (2 k ψ ) and a χ ∼ π/ (2 k χ ). In all the simulations consideredin the present work, the value of f b is of the order of 0.1 and approximately the same,in agreement with Eq. (42) which predicts that f b is independent of SOL parameters,a feature also observed in experiments [70]. We note that the same estimate for f b isderived in double-null geometry in Ref. [23].The last quantity to estimate in Eq. (36) is the filament velocity, v b = ˆ vv ∗ , wherethe normalized filament velocity ˆ v depends on the filament motion regime. From thetwo-region model (see Refs. [36, 37] for details), we estimateˆ v RB ∼ ˆ a / (43)in the RB regime, andˆ v RX ∼ Λˆ a − (44)in the RX regime, with a b (cid:39) (cid:16) π a χ (cid:17) / a / ψ (45)and Λ = ν ¯ n (cid:48) L (cid:107) c s L (cid:107) . (46)By replacing the analytical estimates of p b , v b , and L p in Eq. (36), the far SOLpressure decay length of Eq. (33) in the RX and RB regimes becomes L (cid:48) p, RX ∼ / C log 2 (cid:16) η e η e ) (cid:17) − / (1 + η e ) − / f b ¯ n / ν / L (cid:107) L (cid:107) q / L / χ S / p ρ / ∗ , (47) L (cid:48) p, RB ∼ / √ πC log 2 (cid:16) η e η e ) (cid:17) − / (1 + η e ) − / f b ¯ n / q / ν / L (cid:107) S / p L / χ ρ / ∗ , (48)7where we approximate ¯ n (cid:48) and ¯ T (cid:48) e with ¯ n and ¯ T e at the LCFS, with ¯ T e given by Eq. (26).This approximation is justified by the weak dependence of the ratio ¯ q (cid:48) b,ψ / (¯ p (cid:48) e ¯ c (cid:48) s ) ∝ v b / ¯ c (cid:48) s ,appearing in Eq. (33), on the radial position of the near-to-far SOL interface.The equilibrium density decay length in the far SOL can be obtained by followingthe same procedure described above for the pressure decay length. We balance thefilament associated perpendicular particle flux, Γ b,ψ , and the parallel particle transportby considering the leading order terms in Eq. (5), ρ − ∗ ∂ ψ ¯Γ b,ψ + ∇ (cid:107) (¯ n ¯ v (cid:107) e ) (cid:39) , (49)and we obtain L (cid:48) n ∼ ρ − ∗ ¯Γ (cid:48) b,ψ L (cid:107) ¯ n (cid:48) ¯ c (cid:48) s , (50)where ¯Γ (cid:48) b,ψ ∼ n b f b v b , (51)with n b ∼ ¯ n (cid:48) / ( L n k ψ ). By replacing in Eq. (51) the analytical estimates of n b , v b , L p , and L n , the far SOL density decay length in the RX and RB regimes can then be obtained, L (cid:48) n, RX ∼ / log 2 (cid:16) η e η e ) (cid:17) − / (1 + η e ) − / f b ¯ n / ν / L (cid:107) L (cid:107) q / L / χ S / p ρ / ∗ , (52) L (cid:48) n, RB ∼ / √ π log 2 (cid:16) η e η e ) (cid:17) − / (1 + η e ) − / f b ¯ n / q / ν / L (cid:107) S / p L / χ ρ / ∗ . (53)Fig. 7 shows a comparison between the analytical prediction of the far SOL pressureand density decay lengths and the numerical results obtained from GBS simulations.The agreement is good for the pressure and density decay lengths, with differencesbetween theoretical and simulation results of the order of 20% for the pressure decaylength and up to 40% for the density decay length.Similarly to the near SOL decay length, we write the far SOL pressure decaylengths for the RB and RX regimes in terms of engineering parameters. We replace S p ∼ P SOL / (2 πR ) and ν given by Eq. (13) into Eqs. (47) and (48) and, in physicalunits, we obtain L (cid:48) p, RX (cid:39) . f b q − . R − . L (cid:107) L (cid:107) P − . a − . (1 + κ ) − . n . e A . B − . T , (54) L (cid:48) p, RB (cid:39) . f b q . R . L (cid:107) P − . a − . (1 + κ ) − . n . e A . B − . T , (55)where L (cid:48) p,RX , L (cid:48) p,RB are here in units of mm, R and a are the tokamak major and minorradii in units of m, L (cid:107) is the parallel connection length from upstream to the outer targetplate in units of m, L (cid:107) is the parallel connection length from upstream to the divertorregion entrance in units of m, n e is the density at LCFS in units of 10 m − , P SOL isthe power entering into the SOL in units of MW, and B T is the toroidal magnetic fieldat the magnetic axis in units of T.8 L ′ p
50 60 70 80 90 100 110 120 L ′ p , G B S (a) L ′ n
80 100 120 140 160 180 200 L ′ n , G B S (b) Figure 7:
Comparison between the analytical estimates of the far SOL pressure (a) anddensity (b) decay lengths and the corresponding ones obtained from GBS simulations.
5. Comparison with experimental data
We proceed first with the validation of the near SOL pressure decay length derived inSec. 4 against experimental data. For this purpose, we consider the multi-machinedatabase of Ref. [15] that contains a set of power fall-off lengths obtained from anonlinear regression of measurements of divertor heat flux profiles in attached conditionswith probes or IR cameras on different tokamaks. Both favorable and unfavorable ion- ∇ B drift directions are considered. We restrict our comparison to the outer target,considering data from JET, COMPASS, Alcator C-Mod, and MAST tokamaks. Weextend this database by including the TCV λ q measurements in attached conditionspresented in Ref. [71]. These values are obtained from heat flux profile measurementsat the TCV outer target by using an IR camera.In order to relate the analytical scaling of L p at the outboard midplane with λ q experimentally measured at the outer target, we first report λ q upstream accounting forthe flux expansion. We also assume that, being the considered discharges in attachedconditions, the pressure gradients along the magnetic field lines can be neglected. Thisallows for a direct comparison between L p in Eq. (29) and the experimental λ q , i.e. λ q ∝ L p , where the proportionality factor is determined from the best fit of experimentaland theoretical results, similarly to the procedure outlined in Ref. [18]. Since onlythe line-averaged density ¯ n e is available in the considered database, we assume theedge density contained in the analytical scaling to be proportional to the line-averageddensity, n e ∝ ¯ n e , where the proportionality factor is included in the unique fitting9 L p [mm] λ q [ mm ] λ q = 0 . L p R = 0 . JET by probes at Outer HorizontalJET by probes at Outer VerticalCOMPASS by probes at OuterMAST by IR at double-null OuterC-mod by probes at OuterTCV by IR cameras at Outer
Figure 8:
Comparison of the theoretical scaling law of Eq. (29) to experimental valuesof λ q taken from the multi-machine database of Ref. [15] extended including TCV datafrom Ref. [71]. The dashed black line represents the best fit λ q = αL p with α the uniquefitting parameter. parameter. This assumption is supported by experimental observations that show thepresence of an almost linear proportionality between n e and ¯ n e in low-density discharges(see, e.g., Ref. [12]). The quality of the fit is then expressed through the R parameter.The result of the fitting procedure is shown in Fig. 8. The theoretical scalingreproduces experimental data with a very high goodness parameter, R (cid:39) .
85. Wehighlight that the value of R obtained from the comparison between the theoreticalscaling and experimental data is even higher than some of the most credible scaling lawsderived in Ref. [15] from a direct nonlinear regression of experimental results. Indeed,as extensively discussed in Refs. [15, 72], the number of parameters that can be includedin a scaling based on the direct nonlinear regression of experimental measurements islimited by their mutual correlation. For instance, a very strong correlation is foundbetween R and P SOL /V [72], with the consequence that including both of them in thenonlinear regression leads to an ambiguity on their exponent. The mutual correlationbetween experimental input parameters limits the use of nonlinear regressions to findscaling laws directly from experimental databases, a limitation that is overcome bytheory-based first-principles scaling laws, such as the one derived in the present work.The proportionality constant returned by the fit is approximately 0.2. We note thatthis constant includes both the proportionality factor between λ q and λ p (we use here λ p to refer to the experimental value of the pressure decay length, while L p is used for the0theoretical prediction of Eq. (29)) and the one between n e and ¯ n e . By assuming that p e and T e decay exponentially in the SOL on the λ p and λ T ∼ λ p scales, respectively, thepower fall-off length can be written as λ q ∼ (cid:16) λ p + 12 λ T (cid:17) − ∼ λ p . (56)Moreover, from the experimental results shown in Ref. [12], we assume ¯ n e ∼ n e , whichleads to λ q L p ∼ λ q λ p (cid:16) n e ¯ n e (cid:17) / ∼ (cid:16) (cid:17) / ∼ . , (57)which is close to the proportionality factor returned by the best fit.Despite the very high value of R , we note a dispersion of the experimentalmeasurements around the best fit in Fig. 5. This may suggest incomplete or missingdependencies in the theoretical scaling law of Eq. (29). In particular, our theoreticalscaling law does not include the effect of plasma triangularity, which has been studiedwith GBS in Refs. [73, 74] for a limited configuration, showing that the near SOLwidth is enhanced (reduced) by positive (negative) values of triangularity, in agreementwith experimental observations [75]. In addition, interchange-like turbulence, whichcan develop along the divertor leg, can increase the power fall-off length at the targetplate [76]. This may be especially the case in TCV, where magnetic configurations witha long outer divertor leg are considered. This effect is not included in the present model.As a further comparison between theoretical and experimental results, we analysethe analogies and differences between the theoretical scaling law of Eq. (29) to the onebased on the fit of experimental results, reported in Eq. (1). For this purpose, we rewriteEq. (29) to make explicit the dependence on f Gw and j p , L p (cid:39) . A . (cid:16) n e ¯ n e (cid:17) . R − . (cid:16) aR (cid:17) . (1 + κ ) . κ − . j − . p (cid:16) P SOL S LCFS (cid:17) − . f . Gw , (58)where n e / ¯ n e is the ratio of the edge density to the line-averaged density that appearsfrom the definition of f Gw and S LCFS (cid:39) π aR (cid:112) (1 + κ ) / L p increases with the aspect ratio and the Greenwald fractionand decreases with the plasma current density, with exponents that are comparable tothe experimental ones (see Eq. (1)). According to the theoretical scaling, L p decreaseswith P SOL /S LCFS , a dependence that is not present in the experimental scaling of Eq. (1),although a similar dependence on P SOL /S LCFS has been retrieved in other credibleexperimental scaling laws derived from the same database in Ref. [15]. No dependenceon A is found in the experimental scaling of Eq. (1), in agreement with our theoreticalscaling that depends very weakly on A .As an aside, we note that the theoretical scaling in Eq. (29) depends on q , P SOL and B T with exponents that are comparable to the ones of the experimental scaling lawderived in Ref. [14] from a nonlinear regression performed on λ q measurements of L-modeASDEX discharges. This nonlinear regression has been carried out by considering thesame fitting quantities as the ones considered in the H-mode scaling of Ref. [4], providing1a link between the L-mode and the H-mode scaling laws. In particular, we note that,combining the dependence on q and B T , the theoretical scaling law of Eq. (29) inverselydepends on the poloidal magnetic field, a feature shared with the heuristic drift-basedH-mode scaling law derived in Ref. [77].The theoretical scaling of Eq. (29) with the proportionality constant given by thefitting procedure can be used to predict the SOL width for future tokamaks, such asITER, COMPASS Upgrade, JT-60SA, and DTT. Considering the baseline scenario justbefore the L-H transition, one obtains λ q, th (cid:39) . R = 6 . a = 2 m, q = 2, P SOL = 18 MW, κ = 1 .
4, ¯ n e = 4 · m − , and B T = 5 . λ q, th (cid:39) . R = 0 .
89 m, a = 0 .
27 m, q = 2 . P SOL = 3 . κ = 1 . n e = 2 · m − , and B T = 5 . λ q, th (cid:39) . R = 2 . a = 1 . q = 3, P SOL = 10 MW, κ = 1 .
9, ¯ n e = 6 . · m − , and B T = 2 . λ q, th (cid:39) . R = 2 . a = 0 . q = 3, P SOL = 15 MW, κ = 1 . n e = 1 . · m − , and B T = 6 . λ q for ITER L-mode is within the range of values predicted by the experimental scaling laws derivedin Ref. [15]. The absence of a multi-machine database or experimental scaling laws for the pressuredecay length in the far SOL strongly limits the possibility to carry out a completevalidation of our theoretical scaling. As a preliminary comparison with experimentaldata, we consider a set of measurements of the far SOL decay length taken at theoutboard midplane of TCV L-mode discharges in lower single-null configuration byusing a fast reciprocating probe [82]. Experimental far SOL decay lengths are measuredat fixed B T = 1 . L ′ p,RB [mm]
20 30 40 50 60 λ ′ p , e x p [ mm ] λ ′ p,exp = 0 . L ′ p,RB R = 0 . Figure 9:
Comparison of the theoretical scaling law of Eq. (55) to experimental values ofpressure decay length measured with a fast reciprocating probe at the outboard midplaneof TCV L-mode discharges in conduction regime. Experimental data are taken fromRef. [31]. The dashed black line represents the best fit λ (cid:48) p, exp = αL (cid:48) p, RB with α the uniquefitting parameter. absence of the neutral-plasma interaction processes that might affect the far SOL, asexperimentally observed in Refs. [85, 86]. For the comparison of the far SOL decaylength presented in Fig. 9, we choose to exclude the discharges that do not allow a clearidentification of the near and far SOL and consider high-density discharges, despite thequestions on the applicability of this model.The second difficulty emerges when fitting experimental data at high value of λ (cid:48) q . Indeed, a small variation of the fitting range produces a large variation of λ (cid:48) q .This is reflected on large experimental uncertainties that prevent us from an accuratecomparison with the theoretical prediction and potentially hide some dependencies. Infact, as shown by the error propagation, the relative uncertainties of λ (cid:48) p, exp inverselydepend on the radial derivative of the pressure profile, meaning that a particularly flatradial pressure profile leads to large uncertainties of λ (cid:48) p, exp .The subset of the database considered for this comparison includes discharges thatare mainly in the RB regime [31] and hence we fit experimental data by using thetheoretical RB scaling law in Eq. (55) with the unique fitting parameter being theproportionality constant between experimental measurements and L p, RB . The qualityof the fit is then expressed through the R parameter. As shown in Fig. 9, there is avery weak correlation between theoretical predictions and experimental data, being R only slightly positive.3
6. Conclusions
A theoretical scaling of the pressure and density decay lengths in the near SOL of L-mode diverted plasma discharges valid in low-recycling conditions is analytically derivedfrom an electrostatic two-fluid model by balancing the heat source in the core region,the perpendicular heat flux crossing the separatrix, and the parallel losses at the vesselwalls. Similarly, by balancing the perpendicular turbulent transport due to plasmafilament motion and the parallel flow, the far SOL pressure and density exponentialdecay lengths are analytically derived in the RB and RX filament regimes.The theoretical scaling laws for pressure and density decay lengths in the near andfar SOL are then compared to the results of flux-driven, global, two-fluid turbulentsimulations in a lower single-null geometry, carried out by using the GBS code. Inthe near SOL, there is a very good agreement between theoretical and numericalresults. Indeed, across the entire set of simulations considered in this work, thedifference between simulation results and theoretical predictions is below 20%. In the farSOL, a pattern-recognition algorithm for filament detection/tracking is applied to thesimulation results to determine filament size and velocity, and to identify the filamentmotion regime. Detected filaments in our simulations mainly belong to the RB and RXregimes. The theoretical estimates of the far SOL pressure and density decay lengthsin RB and RX regimes agree with simulation results within an error of 20 % for thepressure and 40 % for the density, pointing out that the model considered here containsthe main physics, although the dispersion of simulation results around the analyticalprediction suggests the need of future investigations with a more accurate model forthe filament velocity, which accounts for the filament-filament interaction and filamentrotation.A comparison between the theoretical scaling of the pressure decay length in thenear SOL and experimental measurements of the power fall-off length, taken from themulti-machine database presented in Ref. [15] and extended by adding TCV data fromRef. [71], is carried out with the only fitting parameter being the proportionality constantbetween the power fall-off length and the near SOL pressure decay length. Our modelreproduces experimental data with a very high value of the quality parameter, R (cid:39) . R (cid:39) . Acknowledgments
The authors thank C. F. Beadle and C. Theiler for useful discussions. The simulationspresented herein were carried out in part at the Swiss National Supercomputing Center(CSCS) under the project ID s882 and in part on the CINECA Marconi supercomputerunder the GBSedge project. This work, supported in part by the Swiss National ScienceFoundation, was carried out within the framework of the EUROfusion Consortium andhas received funding from the Euratom research and training programme 2014 - 2018 and2019 - 2020 under grant agreement No 633053. The views and opinions expressed hereindo not necessarily reflect those of the European Commission. This work was supportedin part by the US Department of Energy under Award Number DE-SC0010529 and byMEY S projects
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