The role of shear flow collapse and enhanced turbulence spreading in edge cooling approaching the density limit

Experimental studies of the dynamics of shear flow and turbulence spreading at the edge of tokamak plasmas are reported. Scans of line-averaged density and plasma current are carried out while approaching the Greenwald density limit on the J-TEXT tokamak. In all scans, when the Greenwald fraction fG=n¯/nG=n¯/(Ip/πa2) increases, a common feature of enhanced turbulence spreading and edge cooling is found. The result suggests that turbulence spreading is a good indicator of edge cooling, indeed better than turbulent particle transport is. The normalized turbulence spreading power increases significantly when the normalized E×B shearing rate decreases. This indicates that turbulence spreading becomes prominent when the shearing rate is weaker than the turbulence scattering rate. The asymmetry between positive/negative (blobs/holes) spreading events, turbulence spreading power and shear flow are discussed. These results elucidate the important effects of interaction between shear flow and turbulence spreading on plasma edge cooling.

Experimental studies of the dynamics of shear flow and turbulence spreading at the edge of tokamak plasmas are reported.Scans of line-averaged density and plasma current are carried out while approaching the Greenwald density limit on the J-TEXT tokamak.In all scans, when the Greenwald fraction f G = n/ n G = n/ ( I p / π a 2 ) increases, a common feature of enhanced turbulence spreading and edge cooling is found.The result suggests that turbulence spreading is a good indicator of edge cooling, indeed better than turbulent particle transport is.The normalized turbulence spreading power increases significantly when the normalized E × B shearing rate decreases.This indicates that turbulence spreading becomes prominent when the shearing rate is weaker than the turbulence scattering rate.The asymmetry between positive/negative (blobs/holes) spreading events, turbulence spreading power and shear flow are discussed.These results elucidate the important effects of interaction between shear flow and turbulence spreading on plasma edge cooling.Keywords: tokamak, density limit, edge cooling, turbulence spreading, shear flow (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Nuclear fusion reactors such as ITER and DEMO are expected to operate at high plasma density, since the fusion power is proportional to the square of density [1,2].However, there generally exists density limit in tokamaks, which limits highdensity operation [3,4].Thus, the physics behind the density limit is well worth exploring.The Greenwald limit is widely quoted as the empirical density limit for tokamaks, where the maximum plasma density is represented by the central lineaveraged density as n edge / n G [4].Here, I p is the plasma current and a is the minor radius.Discharges with deep fueling like pellet injection show that the Greenwald limit can be considerably exceeded with a centrally elevated density profile [5][6][7][8].Therefore, the underlying physical mechanism of the density limit probably lies at the plasma edge.It is pointed out that the edge Greenwald fraction n edge / n G could be more significant than the global Greenwald fraction f G = n/ n G in the density limit [9], while the global Greenwald fraction is still found to be a good indicator of density limit in many experiments on DIII, Alcator C, JET, PBX and TEXTOR-94 [4].
Beyond the controversy about density scaling, there is a general agreement that the density limit is associated with progressive cooling of the plasma edge [10].This may occur as a consequence of trying to increase density in the presence of strong particle transport.Several studies-both experimental and theoretical-have observed a possible power (or edge heat flux) dependence of the density limit [11][12][13][14][15][16][17].
The recent phase transition model of confinement points out that density limit can be linked to the breakdown of turbulence self-regulation at the tokamak edge [18].In this regime, the edge E × B shear flow drops, leading to increased levels of turbulence.This is reversed on the edge plasma state in Hmode, where strong shear flows and low turbulence levels are found.By way of contrast, for ordinary density L-mode, turbulence and shear flows coexist.The enhanced fluctuations and the induced transport can lead to cooling of the edge, and the onset of subsequent events related to the density limit, such as MARFE (multifaceted asymmetric radiation from the edge), shrinking of the plasma current channel and macroscopic MHD instabilities, approaching disruption [10].Experimental measurements indicates that, the edge shear layer indeed collapses and density fluctuation events increase as the central line-averaged density n increases to approach the density limit [19][20][21][22][23], as the Greenwald fraction , increases by scanning of density.Note that current dependence is the key content of the well-known Greenwald scaling of density limit.However, there is lack of experimental studies on the dynamics of edge E × B shear flow and turbulent transport for scans of current to increase f G to approach the density limit.
In this paper, we report comprehensive experimental measurements of edge E × B shear flow, turbulence and turbulence spreading in density (n) scan and current (I p ) scan experiments.It is known that turbulence spreading occurs because inhomogeneous turbulence tends to relax and entrain laminar or more weakly turbulent regions [24].Generally, turbulence spreading refers to the spatial propagation of turbulence intensity or energy due to nonlinear interactions [25].The turbulence spreading flux of density fluctuation intensity ⟨ ṽr ñ2 ⟩ /2 originates from the evolution equation of turbulence internal energy [26].Its negative divergence, i.e. spreading power −∂ r ⟨ ṽr ñ2 ⟩ /2, characterizes the turbulence internal energy increment due to spreading (rather than local production due to drive).To be expressed as the dimension of energy power, turbulence spreading power is written as 2 [20].The results of this paper show that, the interaction between E × B shear flow and turbulence spreading plays an important role on plasma edge cooling as the density limit is approached via either increasing density (n) or decreasing current (I p ). Turbulence spreading is found to be a key indicator of edge cooling, indeed a better one than turbulent particle transport.
It should be pointed out that since our studies are conducted in Ohmic plasmas, we are unable to separate current and power dependence.Therefore, we do not consider power dependence further.The remainder of the paper is organized as follows.Section 2 gives the experimental set up.Section 3 presents the turbulent transport and E × B flow velocity for different line-averaged densities and plasma currents.Section 4 shows the results of normalized flow shear and spreading power approaching the density limit, where the interaction between flow shearing and turbulence scattering is considered.Section 5 is the conclusion, and discussion of the different responses of positive/negative (blobs/holes) spreading events to shear flow.

Experimental set up
The experimental work in this paper was conducted on the J-TEXT device [27][28][29][30], which is a medium-size tokamak with major radius R ∼ 1.05 m and minor radius a ∼ 0.255 m.The Ohmic heated hydrogen plasmas were confined with a limiter configuration.The discharge numbers are 1070961 − 1070991 in the same day.The basic discharge parameters are as follows: the plasma current I p ∼ 125 − 187 KA, the toroidal field B t ∼ 1.6 − 2.2 T, the safety factor at the last closed flux surface (LCFS) q (a) ∼ 3.8, and the central line-averaged density n ∼ 2.8 − 4.9 × 10 19 m −3 .Plasma density limit disruptions occur at the Greenwald fraction (f G = n/ n G ) ∼ 0.7 with different currents.
The mean E × B poloidal flow, velocity fluctuations and density fluctuations are measured by a Langmuir probe array at the edge of J-TEXT.The layout and diagnostic principle of the probe array have been introduced in detail in previous works [20,23].The electron temperature and density (T e , n), the ion saturation current I sat , and the plasma potential ϕ p are measured by the triple probe.The mean E × B poloidal flow is obtained from the radial potential difference, v θ = ∇ r ⟨ϕ p ⟩ /B t .⟨•⟩ is the time averaged value.Here, it is assumed that the ion temperature T i ∼ T e .The density fluctuation ñ is estimated as Ĩsat ⟨n⟩/ ⟨I sat ⟩.The radial wavenumber-frequency spectra of density fluctuations S ñ (k r , f) can be obtained from the cross spectra of the two radially separated ion saturation current fluctuation ( Ĩsat ) channels [31,32].The radial velocity fluctuation is obtained from the poloidal potential difference of the floating probes, ṽr ∼ −∇ θ Ṽf /B t .The poloidal wavenumber-frequency spectra of potential fluctuations S φ (k θ , f) can be obtained from the cross spectra of the two poloidally separated floating potential fluctuation ( Ṽf ) channels [31,32].
The turbulent particle flux and the turbulence intensity flux (or the turbulence spreading flux of internal energy) can be synthesized as ⟨ṽ r ñ⟩ and ⟨ ṽr ñ2 ⟩ /2, respectively.We note that the total turbulence energy is the sum of kinetic energy and internal energy.Thus, the total turbulence spreading flux Γ ε is the sum of kinetic energy flux Γ K and internal energy flux Γ I [20].The expressions of kinetic energy flux ⟨ ṽr ṽ2 θ ⟩ and internal energy flux (actually the density fluctuation intensity flux) ⟨ ṽr ñ2 ⟩ /2 appear in the evolution equations of turbulence internal and kinetic energy [26,33] . To be expressed as the dimension of energy flux, the total turbulence spreading flux can be written as . However, the kinetic energy flux is observed to be about two orders of magnitude smaller than the internal energy flux in the experiments [20,34], and so is negligible.For simplicity, the turbulence spreading flux is calculated as ⟨ ṽr ñ2 ⟩ /2.This representation is also widely used in theory and simulations [25,35].The fluctuation signals (2 − 100 kHz) are obtained by applying a digital FIR filter to the sampled data of 2 MHz.

Edge turbulent transport and E × B flow
The radial profiles of the edge electron density n, the electron temperature T e and the relative density fluctuation σ (ñ) /n, for discharges of similar plasma current I p ∼ 125kA but different central line-averaged density n, are shown in figures 1(a1)-(a3).As n increases (2.8 × 10 19 m −3 → 3.6 × 10 19 m −3 → 3.9 × 10 19 m −3 ), the electron temperature decreases and the relative density fluctuation inside the LCFS increases.The edge density is almost unchanged.The same trend for edge density and electron temperature has also been shown in the previous paper [20].The increased central line-averaged density with the unchanged edge density corresponds to the density profile peaking.One possible explanation may be the reversal of particle convective velocity when the central line-averaged density increases.This can be associated with the TEM-ITG turbulence transition in r/a < 0.8 when the plasma transits from linear Ohmic confinement regime (LOC) in lower n to saturated Ohmic confinement regime (SOC) in higher n [36].The particle convection velocity driven by TEM turbulence is radially outward, while the particle convection velocity driven by ITG turbulence is radially inward [37,38].The enhanced inward turbulent particle flux might lead to the increased core density with the unchanged edge density.
The radial profiles of the edge electron density n, the electron temperature T e and the relative density fluctuation σ (ñ) /n, for discharges of similar n ∼ 4.0 × 10 19 m −3 but different I p , are shown in figures 1(b1)-(b3), respectively.We find that as I p decreases (187 KA → 152 KA → 126 KA), the electron temperature and density decrease, and the relative density fluctuation inside the LCFS increases.These results of both the density scan (n) and the current scan (I p ) show a common feature of enhanced density fluctuations and edge cooling when the Greenwald fraction (f G = n/ n G ) approaches its maximum value f G,max ∼ 0.7, prior to the density limit disruption of J-TEXT.
It is natural to examine and compare how turbulent particle flux and turbulence intensity flux evolve with changes in density and current.a2) and (b2).For discharges of similar I p but different n, in the region of r ∼ [24,25] cm on the inner side of edge shear layer, turbulent particle flux increases as n/ n G increases from 0.45 to 0.59, but decreases as n/ n G increases from 0.59 to 0.63, as shown by figure 2(a1).The same result has also been presented in the previously work [20].Note that there is no obvious change of turbulent particle flux in the region of the edge shear layer, as shown by figures 2(a1) and (b1).This indicates that turbulence spreading is a better indicator of edge cooling approaching the density limit, than turbulent particle transport is.

Normalized flow shear and spreading power
According to the Biglari-Diamond-Terry (BDT) model, turbulence suppression by shear flow becomes pronounced when the poloidal flow shear is larger than the turbulence (random) scattering rate [53].This in turn indicates that the turbulence becomes prominent when the flow shear is weaker than turbulence scattering.Following the idea of this model, the edge E × B poloidal flow shearing rate is normalized by the turbulence scattering rate in the shear layer region, i.e. ω N = ω s /ω t .Here, ω N is the normalized E × B shearing rate.ω s = k θ l cr |∂ r v θ | is the shearing rate [53], obtained from the radial gradient of E × B poloidal flow velocity.Here, k θ is the poloidal wave number and l cr is the radial correlation length.The turbulence scattering rate [53] τ ac is the turbulent radial diffusivity [54].The auto-correlation time τ ac is determined from the efolding width of the envelope of auto correlation function of ñ, which is close to the auto-correlation time of radial velocity fluctuations [54].The poloidal wave number k θ is obtained by the wavenumber spectral average ( k) from the poloidal wavenumber-frequency spectra [31,32], i.e. the first moment of S φ (k θ , f).The average poloidal wave number is calculated as The radial correlation length l cr is estimated by the reciprocal of wavenumber spectral width (⟨σ k ⟩ −1 ) from the radial wavenumberfrequency spectra [31,32], i.e. the second moment of S ñ (k r , f).The radial correlation length is calculated as  ).This is because the fluctuations or the turbulence exhibit broadband characteristic rather than being coherent or concentrated on a certain wave number or frequency.Shear flows interact with turbulence of all wave number components.The spectra at low wave number (k θ < 0.5 cm −1 ) look stronger but the contribution of the spectra at relatively higher wave number range (k θ > 0.5 cm −1 ) are not weak and thus not negligible, especially for the case with higher Greenwald fraction, as shown in the figures 4(b) and (c).We note here that the procedure to calculate the average k θ here is a simple one to obtain a usable figure of merit.One could derive a spectral evolution equation, compute saturation, and look for the dominant k θ or range of k θ .This is a laborious procedure, which would be worth a modelling paper in itself, and is beyond the scope of this paper.
For discharges of similar n, l cr at I p ∼ 187 kA is ∼0.32 cm, l cr at I p ∼ 152kA is ∼0.40 cm, and l cr at I p ∼ 152 kA is ∼0.40 cm.The radial correlation lengths of edge turbulence are reported to be ∼0.2-1.8 cm on JET, ∼1.0 cm on HL-2A and ∼0.3-1.0 cm on TJ-II [47,55,56].To the best of our knowledge, there are no experimental results for the turbulence radial correlation lengths under different currents as a comparison.Regarding magnetic dependencies (related to I p or q) of the radial correlation length, there are two possibilities: (i) poloidal ion gyro radius; (ii) magnetic shear ŝ.Poloidal gyro radius sets the screening length for zonal flows [13,57].Thus, larger I p → smaller ρ θ → smaller ρ screening → stronger zonal flow → smaller l cr .Magnetic shear can be expected to increase the localization of modes, and so decrease l cr [58,59].For discharges of similar I p , l cr at n ∼ 2.8 × 10 19 m −3 is ∼0.36 cm, l cr at n ∼ 3.6 × 10 19 m −3 is ∼0.42 cm, and l cr at n ∼ 3.9 × 10 19 m −3 is ∼0.40 cm.There is no simple monotonic relationship between l cr and n.The experiments on TJ-II stellarator show that turbulence radial correlation length decreases with line-averaged density [47], but how far the density is from density limit is unknown.The experiments on HL-2A tokamak show that there is no obvious variation in the radial cross correlation coefficient of turbulence (proportional to radial correlation length) when n increases [22].Therefore, no consensus can be drawn on the relationship between radial correlation length and density or current so far.
For the normalization of turbulence spreading, it is useful to consider the strength of turbulence spreading relative to local turbulence production.The normalized turbulence spreading power P N = P S /P I , defined as the ratio of turbulence spreading power serves as a measure of turbulence internal energy increment due to spreading, relative to local drive [20].Here, c s is the ion sound speed.The normalized spreading power is used to illustrate the change in the relative fraction of the turbulence energy coupled to transport for comparisons between different discharges with various parameters.Previous work regarding the analysis about the evolution of P S /P I elucidates that the turbulence energy is diverted from zonal flow drive to outward transport when the line-averaged density increases and electrons enter the hydrodynamic regime [20,50].From the theoretical model [53], it is assumed that turbulent transport level is approximate to turbulence intensity (ñ 2 /⟨n⟩ 2 , T2 /⟨T⟩ 2 , or p2 /⟨p⟩ 2 ).Thus, when the poloidal flow shear is smaller than the turbulence scattering rate, the enhancement of turbulence intensity can lead to higher turbulent transport level.In another word, shear flow can no longer quench turbulence, so that shear layer collapse coincides with enhanced turbulent transport.Of course, turbulence intensity and turbulence spreading are related (not identical).The experimental result here directly demonstrates that the turbulence spreading enhances when the ratio of poloidal flow shearing rate and turbulence scattering rate decreases.The experimental result is in agreement with the theoretical model to a certain extent.Indeed, it should be argued that enhancement of turbulence intensity and increased turbulence spreading are not exactly equivalent.The turbulence spreading flux ⟨ṽ r ñ2 ⟩/2 could be written as |ṽ r ||ñ 2 |C, where C is a coherence (or cross phase) factor between ṽr and ñ2 .Thus, for example, when fluctuation intensity increases, a drop in C may conceivably offset the increase, leaving spreading flux unchanged or even decreased.
Note that for discharges of similar n but different I p (shown by the data points of red diamond, green cross and blue inverted triangle) in figure 5(b), the normalized shearing rate decreases as I p decreases.Recall that the poloidal flow shearing |∂ r v θ | itself almost does not change as I p decreases in figure 2(b3).The tendency of the normalized shearing rate ω N = ω s /ω t to decrease with decreasing I p is due to the turbulence scattering dynamics being taken into account.The turbulence scattering rate ω t = 4D t /l 2 cr , estimated from the results in figures 3 and 4, is ∼1.4/2.6/4.9 × 10 5 s −1 for I p = 187/152/126 kA, respectively.In this case, the normalized shearing effect on turbulence is reduced when the turbulence scattering is enhanced, although the shearing rate remains unchanged.These results indicate that turbulence spreading dynamics depend on the relative magnitude of shearing rate to the turbulence scattering rate-i.e. the interaction between flow shearing and turbulence scattering-rather than only on the flow shear itself.The edge electron temperature decreases as the Greenwald fraction increases.Here, cooling can be both: (i) direct cooling via enhanced convection and conduction; (ii) as a result of increased spreading blob [60] out flow to the wall, resulting in increased recycling and thus enhanced influx of cold neutral gas [10].Probably both mechanisms contribute to edge cooling.

Conclusion and discussion
This paper reports a detailed experimental study of the role of enhanced turbulence spreading and shear flow collapse in the edge cooling approaching the density limit.Density (n) scan and current (I p ) scan experiments respectively are conducted to increase the Greenwald fraction, i.e. f G = n/ ( I p /π a 2 ) approaching the density limit.We see that the relative density fluctuations at the edge of plasma increase as n increases, and increase as I p decreases.A common feature of enhanced turbulence spreading and edge cooling is found when the Greenwald fraction increases.However, there is no obvious change of turbulent particle flux.This suggests that turbulence spreading is a better indicator of edge cooling approaching the density limit, than particle transport is.The normalized turbulence spreading power P N = P S /P I increases significantly when the normalized E × B shearing rate ω N = ω s /ω t decreases.This is in good agreement with a theoretical model (BDT), which predicts that the turbulence becomes stronger when the flow shear is weaker than the turbulence scattering.These results elucidate the important effects of the interaction between E × B shear flow and turbulence on plasma edge cooling as the density limit is approached.Two mechanisms can contribute to the edge cooling, which includes direct cooling via enhanced convection/conduction, and increased spreading blob out flow to wall, resulting in increased recycling and thus enhanced influx of cold neutral gas.
Figures 7(a /M for positive and negative density fluctuations, respectively.M is the number of samples.The results show that, turbulence spreading flux induced by positive density fluctuation events (i.e.blobs [60][61][62][63][64][65][66][67]) is positive, i.e. outward.Turbulence spreading flux induced by negative density fluctuation events (i.e.holes) is negative, i.e. inward.This is consistent with recent studies [68].Note that ⟨ ṽr ñ2 ⟩ /2 = ∑ A P spreading (A).Increased symmetry breaking between outgoing spreading fluxes and incoming spreading fluxes causes enhanced net total spreading flux, as the Greenwald fraction (f G = n/n G ) increases from ∼0.45 to ∼0.63.This trend is shown by figures 7(a) and (b).The asymmetrical contributions of blobs and holes with amplitudes |ñ| higher than 2σ (ñ) are particularly prominent.E × B flow shearing might be a mechanism causing the symmetry breaking.Therefore, the effects of shear flow on the turbulence spreading carried by positive and negative density fluctuations should be examined.
The turbulence spreading power contributed by positive/negative density fluctuations and their correlations to the E × B poloidal flow shear, are shown in figures 8(a) and (b), respectively.Note that the spreading power contributed by positive density fluctuations P S (ñ > 0)/P I is positive, which means the local turbulence internal energy increases due to spreading.The spreading power contributed by negative density fluctuations P S (ñ < 0)/P I is negative, which means the local turbulence internal energy decreases due to spreading.The total turbulence spreading power P S /P I = P S (ñ > 0)/P I + P S (ñ < 0)/P I is positive, as shown in figure 5(b).The normalized spreading power P S (ñ > 0)/P I decreases as the normalized E × B shearing rate ω N = ω s /ω t increases, as shown in figure 8(a).However, the normalized spreading power P S (ñ < 0)/P I is insensitive to ω N , as shown in figure 8(b).
The origin of the asymmetry between positive and negative spreading events is obviously very important.In the discharges of higher n with similar I p or lower I p with similar n, the asymmetrical contribution of blobs/holes to turbulence spreading is observed across the probe measurement region, including edge shear layer and no shear region on both sides of shear layer.Indeed, it is not sure whether the E × B flow shearing impact is completely local or covers the neighboring regions.Of course, the direction of the mean density gradient also merits consideration as a symmetry breaking mechanism.Another mechanism might be collisionality effect, which is related to the outgoing blob's electric connection to the sheath on material surface [65,69].Higher collisionality (high n e or low T e ) can lead to disconnection and thus lower damping of blobs.It is unknown how far this can affect the region inside LCFS.These will be explored in future studies.Also, scans of turbulence properties while heating power (i.e.edge heat flux) varies are a natural complement to the current scans reported in this paper.Recall that theoretical work and macroscopic scaling studies suggest a power dependence of the density limit.Negative triangularity plasmas, which do not undergo L → H transition, are an attractive venue for scanning a large dynamic range of power in L-mode.Recent experiments [12] indeed suggest a power dependence of the negative-triangularity density limit.
Zhongyong Chen  https://orcid.org/0000-0002-8934-0364Wei Chen  https://orcid.org/0000-0002-9382-6295 Figures 2(a1)-(a3) show the radial profiles of the edge particle flux ⟨ṽ r ñ⟩, the turbulence spreading flux ⟨ ṽr ñ2 ⟩ /2 and the E × B poloidal flow velocity, for discharges of similar I p ∼ 125kA but different n.Figures 2(b1)-(b3) show the radial profiles of the edge particle flux ⟨ṽ r ñ⟩, the turbulence spreading flux ⟨ ṽr ñ2 ⟩ /2 and the E × B poloidal flow velocity, for discharges of similar n but different I P .Results show that an edge E × B shear layer exists in the region of r ∼[25, 25.5] cm, i.e. several mm inside the LCFS, as shown in figures 2(a3) and (b3).Positive v θ, E×B corresponds to the electron diamagnetic drift direction, and negative v θ, E×B corresponds to the ion diamagnetic drift direction.The turbulence spreading flux grows as n increases or I p decreases-i.e. the Greenwald fraction f G = n/ n G increases-as presented in figures 2( . The decreased edge temperature, the weakened E × B shear flow, the enhanced relative density fluctuations and the increased turbulence spreading with increased n are also observed in other discharges of similar I P ∼ 150 KA (4.1 × 10 19 m −3 , I p ∼ 152kA, n/n G ∼ 0.55; 4.6 × 10 19 m −3 , I p ∼ 151kA, n/n G ∼ 0.62) and of the same I p ∼ 187kA (4.1 × 10 19 m −3 , I p ∼ 187kA, n/n G ∼ 0.45; 4.9 × 10 19 m −3 , I p ∼ 187kA, n/n G ∼ 0.54).However, for discharges of similar n ∼ 4.0 × 10 19 m −3 but different I P , the turbulence spreading flux varies significantly without notable

Figure 1 .
Figure 1.Radial profiles of edge electron density n, electron temperature Te, relative density fluctuation σ (ñ) /n for discharges of similar Ip but different n (left column) and for discharges of similar n but different Ip (right column).

Figure 3
presents the synthesized results of D t ∼ = ⟨ ṽ2 r ⟩ τ ac inside the edge shear layer (near r ∼25.25 cm) for discharges of similar n but different I p .Figure3(d)shows that the turbulent radial diffusivity D t increases as plasma current I p decreases.This is because both the radial velocity fluctuation intensity ⟨ ṽ2 r ⟩ and the auto-correlation time τ ac increase as I p decreases.As current decreases, the increased auto-correlation time can be observed from the broadening of the envelope of auto correlation (AC) function of density fluctuations, as shown in figures 3(c1)-(c3).In these figures, the colored dotted lines are the auto correlation functions and the colored solid lines are their envelopes.The wavenumber-frequency

Figure 2 .
Figure 2. Radial profiles of edge particle flux ⟨ṽrñ⟩, turbulence spreading flux ⟨ ṽr ñ2 ⟩ /2 and E × B poloidal flow velocity for discharges of similar Ip but different n (left column) and for discharges of similar n but different Ip (right column).

Figures 5 (
Figures 5(a) and (b) show the correlation between turbulence spreading and E × B poloidal flow shear, without normalization and with normalization respectively, as mentioned above.The data are from discharges of different plasma current I p and central line-averaged density n.Figures 5(a) shows the turbulence spreading power P S vs the E × B poloidal flow shearing rate ω s at r ∼ 25.25 cm, which are the mean values at the region of r ∼[25.15,25.35] cm inside the edge shear layer.There is no obvious trend of P S with ω s .No direct relationship is found between the absolute turbulence spreading power and the absolute E × B poloidal flow shear.However, as shown in figure 5(b), the normalized turbulence spreading power P N = P S /P I increases significantly when the normalized E × B shearing rate ω N = ω s /ω t decreases.The normalized turbulence spreading power P N = P S /P I is up to ∼0.6 when the normalized E × B shearing rate ω N = ω s /ω t is extremely small (∼0.1).The normalized E × B shearing rate ω N = ω s /ω t is smaller than 1.This is reasonable for Ohmic

Figure 4 .
Figure 4.The poloidal wavenumber-frequency spectra of potential fluctuations S φ (k θ , f) (left column) and the radial wavenumberfrequency spectra of density fluctuations Sñ (kr, f) (right column) for discharges of different Ip.

Figure 6 (
a) shows the normalized E × B shearing rate ω N = ω s /ω t vs the Greenwald fraction f G = n/n G .It shows that for

Figure 5 .
Figure 5. (a) Correlation between turbulence spreading and poloidal flow shear without normalization (i.e.turbulence spreading power P S vs E × B shearing rate ωs); (b) correlation between turbulence spreading and poloidal flow shear with normalization (i.e.normalized turbulence spreading power P N = P S /P I vs normalized E × B shearing rate ω N = ωs/ωt).(Data points of magenta left triangle, black right triangle, and blue inverted triangle correspond to the three discharges of n scan at similar Ip in figures 1 and 2. Data points of red diamond, green cross, and blue inverted triangle correspond to the three discharges of Ip scan at similar n in figures 1 and 2).
discharges of similar I p but different n, the normalized edge E × B shearing rate decreases as the line-averaged density increases.This trend is shown by the data points of magenta left triangle, black right triangle and blue inverted triangle.For discharges of similar n but different I p , the normalized edge E × B shearing rate decreases as the current decreases.This is shown by the data points of red diamond, green cross and blue inverted triangle.Taken together, when the Greenwald fraction increases, the edge normalized E × B shearing rate decreases.After taking the relative strength of flow shear and turbulence scattering into account, the apparent inconsistency of the shearing rate observed approaching the density limit by either current scan or density scan in figures 2(a3) and (b3) is resolved.

Figure 6 (
b) is same as figure 5(b), showing the normalized turbulence spreading power P N = P S /P I vs the normalized E × B shearing rate ω N = ω s /ω t .Figure 6(c) shows the edge electron temperature vs the normalized turbulence spreading power P N = P S /P I .The edge T e here refers to the average electron temperature at the region of r ∼[24.8,25.0] cm in the inner side of the edge shear

Figure 6 .
Figure 6.(a) The normalized E × B shearing rate ω N = ωs/ωt vs the Greenwald fraction f G = n/n G ; (b) the normalized turbulence spreading power P N = P S /P I vs the normalized E × B shearing rate ω N = ωs/ωt; (c) the edge electron temperature vs the normalized turbulence spreading power P N = P S /P I ; (d) the edge electron temperature vs the Greenwald fraction f G = n/n G .(Data points of magenta left triangle, black right triangle, and blue inverted triangle correspond to the three discharges of n scan at similar Ip in figures 1 and 2. Data points of red diamond, green cross, and blue inverted triangle correspond to the three discharges of Ip scan at similar n in figures 1 and 2.).
) and (b) present the distributions of turbulence spreading flux as a function of ñ amplitude, for discharges of different line-averaged density (n) and different plasma current (I p ) respectively, at r∼25.25 cm inside the edge shear layer.The amplitude of ñ is normalized to its standard deviation, as A = |ñ| /σ ñ.The distribution functions of turbulence spreading flux are P spreading (A) = [ ∑ (A−0.05)σñ<ñ<(A+0.05)σñ

Figure 7 .
Figure 7.The distribution function of turbulence spreading flux relative to ñ amplitude, (a) for discharges of similar Ip but different n and (b) for discharges of similar n but different Ip.

Figure 8 .
Figure 8.(a) Correlation between turbulence spreading power contributed by positive density fluctuations and E × B shear; (b) correlation between turbulence spreading power contributed by negative density fluctuations and E × B shear.(Data points of magenta left triangle, black right triangle, and blue inverted triangle correspond to the three discharges of n scan at similar Ip in figures 1 and 2. Data points of red diamond, green cross, and blue inverted triangle correspond to the three discharges of Ip scan at similar n in figures 1 and 2).