How turbulent transport broadens the heat flux width: local SOL production or edge turbulence spreading?

This paper uses data from limited HL-2A Ohmic-plasma to answer the question of how turbulent transport broadens the heat flux width. A key issue in this study is the determination of the origin of scrape-off layer (SOL) turbulence. We develop the concept of the energy production ratio Ra , which compares the flux of turbulence energy across the last closed flux surface (LCFS) to the net, integrated energy production in the SOL. The flux of turbulence energy (i.e. spreading) is measured directly, using Langmuir probes. Experimental data is used to evaluate Ra . Results show that usually 1$?> Ra>1 , indicating that SOL turbulence is energized primarily by edge turbulence spreading. The exceptions—cases where Ra<1 —are those with relatively stronger E×B shear near the LCFS. The latter inhibits both turbulence spreading and local SOL production, but has greater effects on spreading. High blob fraction in the turbulence correlates with large values of Ra . The implications for heat flux width physics are discussed.

Traditionally, for Ohmically-heated and L-mode plasma, the heat flux width has been understood to be produced by the synergy between a local radial transport process in the scrape-off layer (SOL) and the flow of heat along the open field lines. So the heat flux width is defined as λ q ∼ χ ⊥ τ ∥ 1/2 [2], where χ ⊥ is a turbulent thermal diffusivity, and τ ∥ ∼ Rq/V th,i is ime. χ ⊥ is usually thought to result from some local instability process driven in the SOL. Meanwhile, intermittency and blob propagation also occur, though most studies of these have focused on the blob population [17][18][19] and have rarely addressed the impact of intermittency on the heat flux width. However, a moment's thought leads to the observation that the SOL is situated immediately adjacent to the edge or pedestal, which is always turbulent and usually strongly so, especially in Ohmically-heated or L-mode plasmas. The phenomenon that turbulent transport in the edge/pedestal broadens the heat flux width has been studied [12][13][14][15][16]. As an example of edge-SOL coupling, turbulence spreading-the radial propagation of turbulence energy and intensity-is now well established [20][21][22][23][24][25][26][27]. Thus, it is natural to explore the possibility of turbulence spreading from the edge to the SOL and its impact on λ q . Note that such spreading need not be a long range [27]. In this paper, then, we explore the relative contributions of turbulence spreading from the edge and local SOL production in setting SOL turbulence levels, and hence the heat flux widths. The aim here is to answer the question 'Given a turbulent SOL, how can we determine whether the turbulence is produced locally, or by spreading from the edge, i.e. inside the last closed flux surface (LCFS)?' To answer this question, we define the production ratio R a and evaluate it using experimental data. Here R a ∼ (Turbulence energy flux across LCFS)/ (Integrated SOL production is a dimensionless ratio which compares these two mechanisms. Analysis of distribution of, and the trends in R a is the major contribution of this paper. The effects of edge E × B shear and blob fraction are addressed. This study has intrinsic value in the interest of better understanding of the origin of SOL turbulence. In addition, more pragmatic motivation follows from the recent realization that SOL E × B shear in H-mode is strong enough to quench SOL turbulence, and thus 'uncover' residual neoclassical processes. These produce a pathetically small λ q . The neoclassical heuristic drift model (HD model) combines the magnetic drift velocity V D with the parallel transit time τ ∥ to predict λ q ∼ V D τ ∥ ∼ ϵρ θ,i [4]. Here ϵ is the inverse aspect ratio and ρ θ,i the poloidal ion gyro-radius. This pessimistic prediction is consistent with a wide range of measurements in H-mode plasma [4]. Given the unacceptably narrow λ q prediction of the HD model, interest in turbulent pedestal states has grown in recent years. Such states include I-mode [28], wide pedestal quiescent Hmode [29], etc. The aim here is to broaden λ q by possible turbulence spreading from the pedestal to the SOL. To this end, an understanding of the trends in edge-SOL turbulence spreading is obviously of great interest. This paper aims to explore these trends, and thus is of programmatic, as well as scientific, interest.
In this paper, we explore how local production in the SOL and turbulence spreading from the edge combine to broaden and even determine the heat flux width. A combined experimental and theoretical approach is used. Limited Ohmicplasma on HL-2A is studied, and data from Langmuir probe arrays is analyzed. The presence of edge geodesic acoustic mode (GAM) activity [30,31] in some discharges provides a natural range of variation of the edge E × B shear. Edge and SOL fluctuations are measured directly. Particle flux and radial flux of turbulence internal energy (i.e. spreading) are calculated from experimental data. A simple theory of the production ratio (i.e. R a ) is presented and calculated using the data. Correlations between λ q and edge turbulence intensity, edge and SOL particle flux, turbulence spreading and R a are all studied in detail. Special attention is focused on the effects of E × B shear and on the blob fraction of the turbulence. Results indicate a correlation between spreading across the LCFS and λ q . Furthermore, R a > 1 usually, and larger R a correlates with larger λ q . Overall, turbulence spreading emerges as a process which is important to the determination and broadening of the heat flux width in most cases.
The remainder of this paper is organized as follows. Section 2 presents the physics model for the origin of SOL turbulence. Section 3 describes the experimental arrangement and section 4 reports the plasma profiles and basic fluctuation properties. Section 5 presents the correlations of the heat flux width with relevant turbulence quantities. The impact of the turbulence spreading on the heat flux width is studied in section 6. Section 7 gives a discussion and section 8 presents conclusions and further study.

Physics model for the origin of SOL turbulence
We build a simplified model to investigate the origin of SOL turbulence, in order to distinguish the effects of turbulence spreading from the instabilities driven by local free energy sources in the SOL. The key element is the derivation of the energy production ratio R a , which gives the ratio of SOL turbulence production by spreading from the edge to the net local production of turbulence in the SOL. Here we start with a single fluid model, including an effective gravity force (ρ where ρ is the mass density, V is the velocity, p is the pressure, andr presents the radial direction. The fluctuation velocity equation is expressed as: Next, we multiply equation (2) byṼ and take the ensemble average. The edge turbulence frequency is low compared to that of the magnetosonic wave, and thus approximate perpendicular pressure balance applies, i.e.p + BB 4π ∼ = 0. This is equivalent to approximate incompressibility. Hence one obtains: Now we drop the magnetic bending term in the relevant limit of electrostatic turbulence. The turbulence kinetic energy (KE) in the SOL can then be written as ⟨KE⟩ SOL = ∫ ∞ LCFS Ṽ 2 dr ∼ = ∫ λq LCFS Ṽ 2 dr, so its evolution is given as: Performing the first integral shows that the evolution of turbulence KE can be rewritten as Here is the radial flux of turbulence KE, denoting the spreading flux across the LCFS into the SOL.
∂r is the Reynolds power [32,33], which represents the drive of E × B flow shear in the SOL by local turbulence. Γ E,∞ is then neglected by assuming there is no spreading downstream radially of the SOL. Hereafter, we take the limit of the SOL integral to be [LCFS, λ q ] and denote it as [0, λ q ].
Now we define the energy production ratio R a of turbulence spreading across the LCFS into the SOL to the integrated local production in the SOL as: In the expression of R a , the numerator Γ E,LCFS is the turbulence spreading, i.e. the flux of turbulence energy across the LCFS; the denominator is an integral-from the LCFS to λ qof the total SOL production of turbulence, which consists of local production via the interchange instability and the damping term due to Reynolds power. If the Reynolds power is smaller than the local interchange term and can be neglected, then R a is simplified to: Here Γ SOL = ⟨ V r n e ⟩ is the turbulent particle flux in the SOL. R a will be evaluated using experimental data. R a > 1 indicates that turbulence spreading is the dominant effect driving SOL turbulence, while R a < 1 suggests that local SOL production of interchange turbulence is dominant; if R a ∼ 1, then effects of edge turbulence spreading and local SOL production make comparable contributions.

Plasma discharges and database
The experiments were conducted in Ohmically-heated deuterium plasmas on the HL-2A tokamak [34]. HL-2A has major and minor radii of 1.65 m and 0.4 m, respectively. The crosssection is circular, with a limiter configuration. There are three limiters on HL-2A, including two fixed limiters at one toroidal location (one inner limiter on the high field side and one outer limiter on the low field side) and a movable limiter (which does not move in a discharge but can move between discharges) at another toroidal location. The effects of different limiter configurations on the heat flux width are studied in the [35,36]. The discharges discussed in this paper are limited by either the fixed inner limiter or the movable outer limiter. These are shown by the magnetohydrodynamic equilibrium configuration from equilibrium fitting (EFIT) in [15].
In order to investigate how turbulent transport broadens the heat flux width λ q , 37 shots that have similar discharge parameters on HL-2A are chosen and analyzed. The main plasma parameters are: plasma current I p ≈ 150 kA, line-average density n e ≈ 1.0 − 1.5 × 10 19 m −3 , the ratio of line-average density to the Greenwald density limit ne nG ≈ 0.33 − 0.47, toroidal magnetic field B t ≈ 1.4 T, loop voltage V L ≈ 1.8 V, safety factor q a ≈ 4.5, and stored energy W E ≈ 8 − 14 kJ. The electron collision frequency near the LCFS is about  the effective collisionality is ν e * = 6.921 × 10 −18 qRneZlnΛe The database includes four plasma scenarios that have similar discharge parameters but different edge turbulence levels and different values of λ q . The differences in edge turbulence levels and stored energy may be attributed to the variability in wall conditioning (since data was collected from 2015-2019), plasma control (an inner/outer limiter configuration) and the presence or absence of GAMs (as GAM can interact with edge turbulence non-linearly thus regulating fluctuation levels [30]).
It is worth noting that the average λ q predicted by the HD model [4] is λ HD q ≈ 2ϵρ θ,i = 8 mm. The HD model is based on neoclassical drift, and assumes there is no turbulence in the SOL region. Here, the heat flux widths listed in table 1 are larger than that predicted by the HD model. Note that for the scenarios 1-3, λ q ≫ λ HD q , while for the scenario 4 (with stronger E × B shear), λ q ⩾ λ HD q . Therefore, these results suggest that turbulence plays an important role in determining the measured λ q in all these discharges.

Diagnostic setup
The primary diagnostic system is the fast reciprocating Langmuir probe arrays on the low field side of the outer middle plane. The configuration of the probe arrays consists of two radial steps of 4-tip probes [15], which can provide information concerning the electron density, temperature, floating potential and so on [37]. In this study, the sampling rate of the probe measurements is 1 MHz. The fluctuation of poloidal electric field E θ is evaluated using the poloidal floating potential difference, when neglecting the effects of electron temperature fluctuations. Estimation of radial velocity fluctu- The assumption of using V f r instead of V p r [18,19,[38][39][40][41][42][43][44][45][46][47][48][49][50] is discussed in appendix A at the end of the paper. The turbulent particle flux is calculated as Γ r ≈ ⟨ n e V f r ⟩. The particle diffusion coefficient is calculated as D = −Γ r /∇ r n e . The radial flux of turbulence internal energy is estimated as C 2 s ⟨ V f r ( n e /n e ) 2 ⟩, with C s the ion sound velocity. Note that density fluctuation and radial velocity fluctuation are filtered in the frequency range of 20-150 kHz. This is a typical turbulence frequency band in HL-2A, and is used for the estimation of the above relevant turbulence quantities in this paper. Plasma potential is inferred by the expression V p = V f + αT e , with α = 2.8 as the sheath coefficient for deuterium plasma. The radial electric field is calculated by E r = −∇ r V p , and the shearing rate of the mean The parallel heat flux can be expressed as q ∥ = γJ s T e (here J s = I s /A eff ) since it is in the sheath-limited regime [1] of HL-2A Ohmic-plasma with a limiter configuration. Here γ stands for electron sheath heat transmission coefficient, which is usually taken to be 7 [1]. In this study, the radial profile of q ∥ is assumed to follow a single exponential decay in the SOL, i.e. q ∥ (r) = q ∥,LCFS exp (−r/λ q ), where q ∥,LCFS is the parallel heat flux at the LCFS, and r is the distance from the LCFS. Hence λ q is Similarly, the decay length of density λ ne is estimated as λ ne = − ∂ln(ne(r)) ∂r −1 . Both λ q and λ ne are estimated via the log-linear fit method [51].

Plasma profiles and fluctuations
This section provides information of basic radial profiles for two typical shots in the database, as shown in figures 1-3. The two shots (#32712 and #27156) have low/high edge turbulent transport and small/large heat flux width in the plasma boundary. Shot #32712 features a lower level of boundary turbulence and a smaller λ q (λ q ≈ 10 ± 0.2 mm, the black dotted curves). By comparison, shot #27156 is characterized by a higher level of turbulence and a larger λ q (λ q ≈ 23 ± 0.2 mm, the blue solid curves). Both shots have very similar discharge parameters, as can be seen in table 1. They have different stored energy and energy confinement times (10 kJ and 38 ms for shot #32712, and 12 kJ and 55 ms for shot #27156).   Comparing both shots in figure 1, the shot with a smaller λ q has larger electron density, and smaller electron temperature at the plasma boundary. Its parallel heat flux decays faster in the SOL, as compared to the other shot. This indicates a larger radial gradient and results in a smaller λ q . The profiles of the radial electric field are shown in figure 2. Each shot has an electric field well at the edge, but the well of the shot with a smaller λ q is deeper. Figure 3 shows the radial profiles of relative density fluctuations and the turbulent particle flux. Comparing these two shots, a lower/higher level of edge turbulence (relative density fluctuations) and transport (turbulent particle flux) corresponds to a smaller/larger λ q , implying an edge-SOL coupling effect. These trends are consistent with previous experimental results on HL-2A which suggest that edge turbulent transport plays a significant role in determining λ q [15]. States of large profile gradients existing in the presence of low levels of turbulence and transport are achieved via the feedback of E × B shear. The latter is well known to regulate turbulence and transport. Note that the error bars in figures 2 and 3 denote a confidence interval of 68%, rather than the standard deviation.

Heat flux widths vs E×B shear
The relation of λ q to the E × B shearing rate (ω E×B ) measured at LCFS is shown in figure 4. The four plasma scenarios in the database are marked with different colors and symbols in figure 4; each plasma scenario shares the same machine day; each symbol presents the data for one shot. The E × B shearing rate ranges from 1 × 10 4 s −1 to 4 × 10 5 s −1 , and λ q ranges from 8 mm to 24 mm. λ q decreases with the increase of E × B shearing rate.

Heat flux widths vs the levels of turbulence and transport
The relations of heat flux widths to the levels of turbulence and transport at the edge and in the SOL for the 4 scenarios in the database are shown in figures 5-8. Note that the location of LCFS, the edge, and the SOL region measured by the Langmuir probe in this paper are described in table 2. The legends-'Edge' or 'SOL' in figures 5-9 means the average values of quantities at the locations listed in table 2.
In figure 5, the relative density fluctuation is calculated as the root mean square of the electron density fluctuation divided by the average electron density. It is shown to rise from 0.05 to 0.25 at the edge, while it ranges from 0.25 to 0.55 in the SOL. Figures 5(a) and (b) show linear correlations between λ q and the relative density fluctuation at the edge and in the SOL, respectively. This result is consistent with simulation results from the SOLT code [52].
The turbulent particle flux and the particle diffusion coefficient are analyzed and illustrated in figures 6-8. In figure 6, the decay length of electron density λ ne increases with the turbulent particle flux at the edge and in the SOL. The tendency in figure 6(a) is linear and in figure 7(a) is non-linear. Figures 7(a) and (b) show that λ q tends to increase with turbulent particle flux at the edge and the SOL. The positive correlation of λ q with the particle diffusion coefficient is analogous to that for turbulent particle flux, shown in figures 7 and 8. It is reasonable since stronger turbulent transport in the SOL makes the profile gradients smaller, thus resulting in a larger λ q . In figures 6(b), 7 and 8, the nonlinear regression fit uses the Michaelis-Menten equation [53]. This fit features the trend that when the independent variable is small, the dependent variable shows a rapid increase; when the independent variable is larger, the dependent variable increases more slowly and approaches its maximal value. We use this fit because its trend is consistent with results appearing in those figures.
It is worth discussing why the plasma scenario 4 with black triangles has lower levels of turbulence and transport in the edge and SOL than the other 3 plasma scenarios. It turns out that the GAM is present at the plasma edge in scenario 4, while it is not seen in the other 3 scenarios (shown in table 1). The featured frequency is in the range 7-10 kHz, which is the characteristic frequency range of GAM on HL-2A [30,31]. In plasma scenario 4, the Langmuir probe arrays measure the GAM activity ranging from r − r LCFS ≈ −10 ∼ −13 mm to about r − r LCFS ≈ −15 mm (where the probe arrays stay). As reported in [30], GAM gains energy from background turbulence by non-linear coupling. In scenario 4 (with GAMs), the E × B shear is stronger as shown in figure 4, and the edge turbulence and transport levels are much lower as shown in figures 5-8. The presence of GAM plays an important role in determining the turbulence and transport levels at the plasma boundary.
Based on the above discussions, we conclude that the heat flux width increases with the levels of edge turbulence and transport but decreases with the E × B shearing rate. The mechanism for edge turbulent transport influencing the heat flux width will be studied in the next section.

Energy production ratio and the heat flux width
This section aims to understand the impact of turbulence spreading on the heat flux width by combining experimental data with the physics model described in section 2. As the   V θ cannot be measured by the two steps of 4-tip probes in the experiments, the radial flux of turbulence internal energy is used as a surrogate for the radial flux of turbulence KE. These two are expected to be closely related. In this section the turbulence spreading is quantified as the radial flux of turbulence internal energy C 2 s ⟨ V f r ( n e /n e ) 2 ⟩. Several other works have quantified the turbulence spreading by the measurement of ⟨ V f r ( n e /n e ) 2 ⟩ [15,25,26], which is directly proportional to the flux of internal energy density. Since this paper focuses on the energy production ratio R a , C 2 s ⟨ V f r ( n e /n e ) 2 ⟩ is chosen as the best available indicator of spreading. Figure 9 presents the correlation of λ q with turbulence spreading measured at the edge and the LCFS using a log-log scale. λ q increases with the radial flux of turbulence internal energy, and correlates positively with turbulence spreading. Combining figures 5 and 9, larger edge turbulence levels coincide with larger turbulence spreading at the LCFS and larger SOL fluctuation levels, suggesting that edge turbulence spreads into the SOL. Larger E × B shear tends to suppress edge turbulent transport as well as turbulence spreading. This is consistent with gyro-kinetic simulation in [23] and TJ-II stellarator results [26].    The energy production ratio R a , defined in section 2, is now evaluated using experimental data. As the Reynolds power could not be measured in these experiments, results from other similar Ohmic discharges on HL-2A indicate that the Reynolds power is typically one order of magnitude smaller than the local interchange production term (see appendix C). So the Reynolds power is neglected when calculating R a in this paper. We use Γ E,LCFS = C 2 s ⟨ V f r ( n e /n e ) 2 ⟩| LCFS to calculate R a by equation (8). Results are shown in the following figures. The relation between λ q and R a is shown in figure 10. First, R a ranges from −2 to 10, and 5 of the 37 shots have negative values of R a . Negative R a indicates inward spreading, from the SOL to the edge. A negative R a is possible since turbulence spreading is measured locally at the LCFS, where convective cells exist [54]. Nevertheless, the fact that R a is positive for most cases indicates turbulence usually spreads from the edge to the SOL. Second, as shown by the histogram in figure 10(b), there are 14 shots for |R a | < 1 and 23 shots for |R a | > 1. For weak edge turbulence (small λ q ), |R a | < 1 applies for most cases. This is shown by black triangles for plasma scenario 4 in figures 10(a) and (c), indicating that local interchange production of SOL turbulence is more important than spreading drive. This outcome is likely due to the GAM excited at the plasma edge, which drives E × B shear flow and so regulates and impedes turbulence spreading. For strong edge turbulence, |R a | > 1 applies for most cases, as is shown by the plasma scenarios with green diamonds, red circles and blue crosses. For theses cases, turbulence spreading from the edge into the SOL exceeds local SOL turbulence production. Third, when the turbulence level is moderate or high, λ q increases with |R a |, implying a positive correlation between turbulence spreading and |R a |. These experimental results suggest edge turbulence spreading has significant impact on SOL turbulence for increasing edge turbulence levels.
The impact of E × B shear on |R a | is shown in figures 11(a) and (b). In figure 11(a), when the E × B shearing rate ω E×B ⩽ 2 × 10 5 s −1 , |R a | is scattered and ranges from 1 to 10; when ω E×B > 2 × 10 5 s −1 , |R a | is clustered between 0 and 1. Large E × B shear tends to reduce |R a |, as shown in figure 11(b). E × B shear impacts |R a | through both its effects on edge turbulence spreading and on local SOL interchange production. It is useful to explore which one is more strongly affected by E × B shear. This is shown in figure 12, where the edge turbulence spreading and the local SOL interchange production decrease with increased E × B shear. We compare data in scenario 1 (red circles) and scenario 4 (black triangulars) in figures 12(a) and (b). With the increased E × B shear, the values of turbulence spreading measured at the LCFS decrease from 3.85 ×10 9 to 0.19 ×10 9 -reduced by a factor of 20, while the local SOL interchange production decreases from 0.84 ×10 9 to 0.35 ×10 9 , i.e. reduced by 1/2. Thus E × B shear has a stronger suppression effect on edge turbulence spreading than on local SOL interchange production. In addition, it seems that the data with blue crosses does not fit this tendency  well in figure 12. This outlier may relate to the very shallow E r well inside the LCFS, and requires further study.

Blob dynamics and turbulence spreading
In this study, the energy production ratio R a is used to characterize the contribution of the turbulence spreading. In previous studies, the turbulence spreading rate, defined by ω s = − 1 2 ∂⟨ ne 2 Vr⟩ ∂r 1 2 ⟨ ne 2 ⟩ [25], was introduced to quantify the effect of turbulence spreading. Figure 13 shows the correlations of turbulence spreading rate with the E × B shearing rate and the heat flux width. Absolute values of the average turbulence spreading rate are in the range of several kHz, for the edge region. The turbulence spreading rate decreases with increasing E × B shearing rate ( figure 13(a)), implying that E × B shear suppresses turbulence spreading. The turbulence spreading rate also exhibits a positive correlation with λ q in figure 13(b).
Blob dynamics is an essential part of transport at the plasma boundary [17][18][19]. The ratio of blob-induced particle flux to the total particle flux (Γ blob /Γ total ) in specific radial regions is used to quantify the contribution of blob transport to total turbulent transport. Here a 'blob' is identified when the amplitude of the fluctuation in ion saturation current exceeds twice its standard deviation [17][18][19]. The number of blobs in each time window (5000 points) is large enough for a statistical analysis. In this paper, Γ total = ∫ Γ (t) dt with Γ (t) = n e V f r , and Γ blob = i Γ(t i − τ /2 < t < t i + τ /2) (here i is the number of blobs; t i is the peak time point of each blob; τ is the life time of a blob estimated by the conditional average method).
The importance of blob transport to the origin of SOL turbulence and the heat flux width is presented in figure 14. Γ blob /Γ total ranges from about 0.3-0.5, revealing the important  role of blob transport in total turbulent transport at the edge. R a increases with Γ blob /Γ total , indicating that increased blob fraction at the edge enhances turbulence spreading. The heat flux width also increases with Γ blob /Γ total . Figures 13 and 14 demonstrate that more blob transport tends to broaden λ q .

Discussion-how turbulent transport broadens the heat flux width?-An overview
This paper addresses the impact of turbulence spreading on the heat flux width. Here, we overview the physical processes which determine the heat flux width. Fundamentally, λ q is determined by neoclassical drift and turbulent transport in the is the SOL turbulence intensity, V D is the neoclassical magnetic drift velocity, τ c is the turbulence correlation time, and τ ∥ is the parallel transit time. Note that in the HD model V D τ ∥ defines the λ q , and τ c /τ ∥ ⩽ 1, necessarily. Note also that cross field velocities, due to turbulence or drift, add in quadrature and that the duration of travel through the SOL is limited to τ ∥ . The key question, then, is what is the origin of the SOL turbulence intensity. There are two possibilities: (i) local production-in the SOL-by processes which tap free energy in the SOL region. Local SOL instabilities fall into this category. (ii) spreading of turbulence energy across the LCFS from the pedestal/edge. In this case, the free energy source is the relevant edge/pedestal gradient. E × B shear near LCFS can inhibit such spreading.
The energy production ratio seeks to quantify the ratio of (ii) spreading to (i) local SOL production. Figure 15 summarizes this discussion, and illustrates the edge-SOL coupling.

Conclusions and further study
In this study, we have examined the impact of turbulence spreading on the heat flux width. Detailed experiments have been performed on limited Ohmic-plasma in HL-2A tokamak. The principal results of this paper are: (i) The heat flux width λ q is usually much larger than neoclassical HD modal prediction (λ HD q ). An exception is for the case of stronger E × B shear at the LCFS. Then λ q ⩾ λ HD q . (ii) The measurements show that λ q correlates positively with edge fluctuation intensity and turbulent transport. (iii) Significantly, mostly outward turbulence spreading is measured at the LCFS. Here 'turbulence spreading' refers to the radial flux of turbulence internal energy C 2 s ⟨ V f r ( n e /n e ) 2 ⟩, which correlates positively with λ q . (iv) The energy production ratio R a is defined and calculated from experimental data. Here R a = (spreading flux across LCFS to SOL)/(integrated local production in SOL). R a characterizes the nature of the origin of SOL turbulence. (v) In most cases, R a > 1, so edge turbulence spreading determines SOL turbulence. These cases have weak edge E × B shear. Cases of R a < 1 correspond to those of stronger E × B shear near the LCFS. E × B shear has a stronger suppression effect on edge turbulence spreading than on local SOL interchange production.
Overall, these results indicate that without strong E × B shear near the LCFS, turbulence spreading from edge to SOL is a key element-indeed even a dominant element-of the dynamics of the SOL turbulence and the determination of λ q .
Several directions for future work are indicated. First, the energy production ratio R a merits further study and characterization. In particular, the value of R a deduced here is likely an lower estimate, since the ideal interchange instability is the most vigorous local instability and production mechanism appropriate. Other models would predict weaker local production, and thus larger values of R a . Second, the dependence of the heat flux width on the spreading Γ E should be characterized, i.e. the relation λ q = λ (Γ E , parameters) should be deduced. In particular, the penetration depth of turbulence spreading from a turbulent pedestal into a stable or weakly fluctuating SOL is of great interest. Third, the phase relation and coherence of turbulence spreading need to be investigated. These present a more severe challenge than the familiar 'cross phase' of the quadratic fluxes, but are likely the key to understanding the strong E × B shear sensitivity of spreading. Fourth, the structure of the fluxgradient relation for turbulence energy should be determined. Is the intensity flux Fickian, with a local diffusivity? This natural ansatz has long been known to be inadequate for describing turbulent fluid wakes [55]. It should be tested on magnetic confinement devices. The outcome is important to modeling turbulence spreading effects on the heat flux width.

Appendix C
This appendix aims to compare the local interchange production term and Reynolds power in the SOL. We have conducted experiments to measure both terms for discharge parameters very similar to those used here. We used another Langmuir probe array [56], which was able to measure the Reynolds power. Figure 16 shows the measurement results in the SOL. There are 12 shots in total, to show the reproducibility. The Reynolds power term ( has the range of 0.4 × 10 7 -3 × 10 7 m 3 s −3 (with an average value 0.9 × 10 7 m 3 s −3 ), while the local interchange term . The magnitude of Reynolds power term in the SOL is typically one order of magnitude smaller than the local interchange production term in similar Ohmic discharges on HL-2A. Thus we neglect the Reynolds power in this paper.