Fluctuation-induced inward particle flux during L–I–H transition on HL-2A tokamak

In HL-2A tokamak, H-mode is often achieved by NBI. In this discharge, the L–I–H transition is distinctly distinguished from the D_α signal representing the recycling state, as shown in figures 1(c) and 2(a). The reciprocating probe array inserts the plasma via a port located on the low field side vacuum vessel midplane and stays steady at r = 35.5 cm during the entire L–I–H transition [27]. As a typical H-mode discharge, the electron density increases, and the radial electric field becomes sharply negative at the edge when the plasma enters the H-mode state, as shown in figures 2(d) and (e). A coherent mode with an approximate peak frequency of 10 kHz is generated at 630 ms in L-mode, as shown in the core SXR and edge Mirnov coil signals, in figures 2(f) and (g), respectively. Further, density and electric field fluctuations are measured by the edge Langmuir probe, as shown in figures 2(h)–(j). This coherent mode has been identified as the LLM, a kind magnetohydrodynamic (MHD) mode, on the HL-2A tokamak [19, 32]. According to amplitude profiles of SXR fluctuation for this coherent mode, the LLM is generated at the core plasma (r = 16.2 cm) which reconfirms that this coherent mode is indeed LLM [27].

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The fluctuation-induced radial particle flux remains radially outward (Γ_r,fluc > 0) during the L-mode. But it decreases sharply when the plasma enters the I-phase state, even changes its direction to radially inward (Γ_r,fluc < 0) and the direction is kept to H-mode [27]. However, the change in the transport direction of each component cannot be distinguished from the time evolution of radial particle flux. Therefore, to elucidate the detailed variations of fluctuation-induced particle flux Γ_r,fluc during the L–I–H transition, the frequency-resolved particle flux ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left(f\right)$ and its fluctuating components in the frequency domain are calculated. For comparison, the values of ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left(f\right)$ during different plasma states are shown in figures 3(a) and (b).

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The fluctuation-induced particle flux ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left(f\right)$ is mainly induced by the coherent LLM and broadband fluctuation (30–100 kHz), and it takes the form of ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\simeq {{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\text{LLM}}\right)+{{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{B}\mathrm{F}}\right)$. Here, the broadband fluctuation-induced particle flux during the L-mode and I-phase is mainly due to the ambient turbulence, which can be denoted as ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{B}\mathrm{F}}\right)\approx {{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{A}\mathrm{T}}\right)$ (in L-mode and I-phase). The direction of turbulent transport ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{A}\mathrm{T}}\right)$ from the inner radial position of probe array A is radially outward during the L-mode (Γ_r,fluc(f_AT) > 0), as shown in figure 3(a). Besides, the particle fluxes in the LLM ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\text{LLM}}\right)$ from the two different radial positions also remain in radially outward direction during the L-mode (Γ_r,fluc(f_LLM) > 0). However, ambient turbulence is highly suppressed by the increased shear flow after the L–H transition, turbulent transport Γ_r,fluc(f_AT) exhibits an obvious decrease. Moreover, due to the generation of the high-frequency coherent mode (HCM) at the end of the I-phase state [33], the HCM induced particle flux will dominant the broadband fluctuation-induced particle flux ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{B}\mathrm{F}}\right)\approx {{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\text{HCM}}\right)$ (in H-mode). The direction of Γ_r,fluc(f_LLM) from the inner position changes to radially inward after L–H transition, but Γ_r,fluc(f_LLM) from the outer position still remains radially outward, as shown in figure 3(b). Inward flux only happens in a certain radial position, and only the direction of particle flux associated with the global LLM becomes radially inward. It is quite different from that observed in previous experiments [7, 11], where the inward particle flux is usually caused by broadband fluctuations.

To explore the origin of inward particle flux in the LLM Γ_r,fluc(f_LLM), we investigate the phase difference between different fluctuating components during the L–H transition. According to each fluctuating component in the frequency domain, it can be confirmed that the cross-phase term in the LLM becomes negative, i.e., cos α_nV (f_LLM) < 0, which directly leads to the radially inward particle flux. The correlation analyses between fluctuating components ($\tilde {n}$ and ${\tilde {V}}_{\mathrm{r}}$) for LLM and poloidal magnetic fluctuation ${\tilde {B}}_{\theta }$ are shown in figures 3(c) and (d) to figure out the primary cause of the phase shift.

There is high coherency for LLM during the entire L–H transition as expected from the global nature of LLM. The exact phase shifts of fluctuating components during L–H transition can be found by comparing the difference of the cross-phase between poloidal magnetic fluctuation ${\tilde {B}}_{\theta }$ and fluctuating components ($\tilde {n}$ and ${\tilde {V}}_{\mathrm{r}}$) from different plasma states, as shown in figures 3(e) and (f). The cross-phase between ${\tilde {B}}_{\theta }\left({f}_{\text{LLM}}\right)$ and $\tilde {n}\left({f}_{\text{LLM}}\right)$ during the L–H transition remains nearly unchanged ${\Delta}{\alpha }_{n,{B}_{\theta }}\left({f}_{\text{LLM}}\right){< }0.10\enspace \pi$, as shown in figure 3(e). However, the cross-phase between ${\tilde {B}}_{\theta }\left({f}_{\text{LLM}}\right)$ and ${\tilde {V}}_{\mathrm{r}}\left({f}_{\text{LLM}}\right)$ changes to nearly ${\Delta}{\alpha }_{V,{B}_{\theta }}\left({f}_{\text{LLM}}\right)\sim 1.45\enspace \pi$ after the L–H transition, as shown in figure 3(f). Consequently, the considerable change in ${\alpha }_{V,{B}_{\theta }}\left({f}_{\text{LLM}}\right)$ indicates that the phase of ${\tilde {V}}_{\mathrm{r}}\left({f}_{\text{LLM}}\right)$ regards to magnetic fluctuations shifts significantly after the L–H transition and directly leads to the transport direction change. Besides, the negligible change in ${\alpha }_{n,{B}_{\theta }}\left({f}_{\text{LLM}}\right)$ indicates that ${\tilde {n}}_{\mathrm{e}}\left({f}_{\text{LLM}}\right)$ retains its phase during the L–H transition, which indicates density fluctuations largely come from field lines advection due to finite density gradient [34].

This result can also be confirmed by using localized electron cyclotron emission (ECE) signal as the reference signal instead of the edge Mirnov signal. The cross-phase between ECE fluctuation and ${\tilde {n}}_{\mathrm{e}}\left({f}_{\text{LLM}}\right)$ remains unchanged during the L–H transition (Δα_n,ECE(f_LLM) < 0.10 π, not shown in this paper), but the cross-phase between ECE fluctuation and ${\tilde {V}}_{\mathrm{r}}\left({f}_{\text{LLM}}\right)$ shifts significantly during L–H transition (Δα_V,ECE(f_LLM) ∼ 0.50 π, not shown in this paper). The correlations between SXR signal and fluctuating components also confirm this conclusion [27]. Therefore, the phase shift of radial velocity fluctuation with magnetic fluctuations for the LLM becomes a crucial factor, which leads to the change in the sign of cross-phase term as well as the generation of inward particle flux in the LLM. For a linear MHD mode, it usually retains the phase of density and velocity fluctuation. So, there must be another mechanism that leads to the phase shift of the radial velocity fluctuation.

Over recent years, the study of the fundamental mechanisms for inward particle flux has garnered considerable research interest. It has been found that the cross-phase term can change the transport direction when the shear effect reaches a certain threshold [35]. The amplitude of the inward particle flux also strongly depends on the radial electric field shear [7] or the negative total Reynolds work [15]. Numerical simulations also confirm that the cross-phase in the strong shearing regime is more sensitive than that in the weak shearing regime [36]. All these results indicate that the fundamental tendency of strong shear can change the transport direction by altering the cross-phase.

To explore the flow shear effect on particle flux, the particle flux Γ_r,fluc(f_LLM) vs poloidal velocity shear ∂V_θ /∂r and radial velocity gradient dV_r/dr is plotted in figure 4 in the time domain. During the L-mode, the radial velocity gradient dV_rdr remains relatively small, and the direction of Γ_r,fluc(f_LLM) remains in radially outward. As dV_r/dr increases, Γ_r,fluc(f_LLM) begins to decrease to zero and even changes its direction to radially inward at 637 ms, as shown in figures 4(a) and (f). However, the poloidal velocity shear ∂V_θ /∂r shows no obvious change while particle flux Γ_r,fluc(f_LLM) varies. This becomes clear from figures 4(a) and (e), ∂V_θ /∂r begins to increase at 640 ms which is much later than the inward flux occurs. This implies the radial velocity gradient shows a strong correlation with inward particle flux in the LLM rather than the poloidal flow shear.

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The particle flux in the frequency range of 30–100 kHz is mainly due to the broadband turbulence (${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{B}\mathrm{F}}\right)\approx {{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{A}\mathrm{T}}\right)$, in L-mode and I-phase). The fluctuation amplitude of density and radial velocity in the frequency range of 30–100 kHz (${P}_{n}\left({f}_{\mathrm{B}\mathrm{F}}\right)$ and ${P}_{V}\left({f}_{\mathrm{B}\mathrm{F}}\right)$) do not show any obvious change during L–I–H transition, while a remarkable six-fold increase in ∂V_θ /∂r, as shown in figures 4(b), (c) and (e). This indicates that the strong shear flow can suppress the turbulent transport by modulating the cross-phase between the fluctuations of density and radial velocity instead of amplitude reduction. The poloidal shear flow can strongly affect the small-scale turbulence, but its effect on the large-scale mode like the LLM is not evident. At the end of the I-phase state, the direction of Γ_r,fluc(f_BF) changes to radially inward. This is mainly caused by the suppression of turbulent transport and the generation of HCM (${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\mathrm{B}\mathrm{F}}\right)\approx {{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\text{HCM}}\right)$, in H-mode). The HCM can also drive inward particle flux, but inward particle flux for HCM starts much later than that for LLM [33], and we will discuss this phenomenon elsewhere.

To explore the effects of fluctuations on particle flux, the direct measurements of fluctuation terms of equation (1) are shown in figure 5. The particle source can be estimated roughly by D_α signal since it reflects hydrogen recycling. The transport in the LLM begins to change its direction during the L–I transition (shown in figure 5(a)). However, the D_α signal shows no evident variation. When the inward flux in the LLM increases significantly at the H-mode state, but D_α signal decreases sharply which is proportional to neutral particle density. Therefore, we do not expect the particle source has obvious effect on the inward particle transport in the LLM.

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Due to the incompressibility of the flow ($\nabla \cdot \overrightarrow {V}\simeq 0$), the linear term $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}\nabla \cdot \tilde {V}\right\rangle {n}_{0}$ and $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle {\tilde {n}}^{2}\right\rangle \nabla \cdot V$ can be ignored, but their radial gradient components of radial velocity are not necessarily zero, as shown in figures 5(b) and (c). During the entire L–I–H transition, the radial velocity gradient dV_r/dr increases a factor of two (see figure 4(f)), accompanied by the reversal of particle flux Γ_r,fluc(f_LLM) direction (see figure 4(a)), which implies that the radial velocity gradient may play a role in the generation of inward flux for the LLM. But the linear term induced by the radial velocity gradient $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle {\tilde {n}}^{2}\right\rangle {\partial }_{\mathrm{r}}{V}_{\mathrm{r}}$ only contributes to radially outward flux during the I–H transition, which cannot explain inward flux explicitly, as shown in figure 5(c). The measured fluctuation term $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle {V}_{\mathrm{r}}$ proportional to radial velocity indeed contributes to the radially inward flux during the I–H transition. However, this term only accounts for inward flux partially during I-phase, as shown in figure 5(d). The mean radial velocity V_r remains radially outward during the L–I–H transition, as shown in figure 4(d). So, the phase shift of $\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle$ directly leads to the sign change of $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle {V}_{\mathrm{r}}$. The LLM density fluctuations at the outer position have been highly suppressed after the beginning of inward flux (at 637.5 ms) as shown in figure 5(e). This directly leads to the density fluctuation gradient ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ become negative, and it will reverse the direction of $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle {V}_{\mathrm{r}}$.

As we mentioned before, the poloidal velocity shear shows no obvious effect on inward flux but the radial velocity gradient does. To clearly show the effect by radial velocity gradient dV_r/dr, the entire L–I–H transition data (630–644.5 ms) of the LLM density fluctuation gradient ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ and radial velocity gradient dV_r/dr is shown in figure 5(g). It is clear that ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ become negative when dV_r/dr increases beyond a certain threshold (red vertical dashed line in figure 5(g)). This indicates that the radial velocity gradient dV_r/dr can modulate the particle flux Γ_r,fluc(f_LLM) by modifying the LLM density fluctuation gradient. In addition, the relation between dV_r/dr and ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ in the different plasma state is quite different. It has been confirmed that ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ increased gradually with the increase of dV_r/dr during the I-phase state. When plasma enters the H-mode state, ${\partial }_{\mathrm{r}}\tilde {n}\left({f}_{\text{LLM}}\right)$ remains steady as dV_r/dr increases. Even though the radial velocity gradient may not directly contribute to inward particle flux yet it can affect density fluctuation profile implicitly via redistribution of flow, which requires further investigation.

In the I-phase state, the linear term $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle {V}_{\mathrm{r}}$ only contributes half of the measured particle flux Γ_r,fluc(f_LLM) (see in figure 5), which indicates the nonlinear term of equation (2) also contributes to the inward particle flux. Otherwise, equation (1) cannot balance. However, the nonlinear term cannot be directly measured at the moment, which requires absolute phase measurements among three waves. Instead, we can evaluate the nonlinear three-wave interaction of LLM as long as the phases of three waves are aligned.

The nonlinear coupling is usually detected with bispectrum technique [37, 38]. The squared bicoherence of the three physical quantities, $x\left(t\right)$, $y\left(t\right)$, and $z\left(t\right)$, are given by ${\hat{b}}_{XYZ}^{2}\left({f}_{1},{f}_{2}\right)=\frac{{\left\vert {\hat{B}}_{XYZ}\left({f}_{1},{f}_{2}\right)\right\vert }^{2}}{\left\langle {\left\vert X\left({f}_{1}\right)Y\left({f}_{2}\right)\right\vert }^{2}\right\rangle \left\langle {\left\vert Z\left(f={f}_{1}+{f}_{2}\right)\right\vert }^{2}\right\rangle }$. However, the cross-bicoherence between $\tilde {n}$, ${\tilde {V}}_{\mathrm{r}}$, and $\nabla \tilde {n}$ is not very clear due to the phases are not aligned. This leads to the nonlinear term that cannot be directly measured. However, the nonlinear interaction between LLM and broadband turbulence can be confirmed by the auto-bicoherence ${\hat{b}}_{{V}_{f}{V}_{f}{V}_{f}}^{2}\left({f}_{1},{f}_{2}\right)$. Considering the electromagnetic characteristics of LLM, the magnetic fluctuation can also be used to verify the nonlinear interaction due to the electromagnetic characteristic of LLM. Figures 6(a) and (b) show the contour plots of squared auto-bicoherence ${\hat{b}}_{{B}_{\theta }{B}_{\theta }{B}_{\theta }}^{2}\left({f}_{1},{f}_{2}\right)$, which denotes the total nonlinear interactions of the poloidal magnetic fluctuation at f₁ ± f₂, during the L-mode and I-phase, respectively. It clearly shows that the bicoherence values around the frequencies of f₂ = ±11.7 kHz and f = f₁ + f₂ = 11.7 kHz (as the purple dash lines shown in figure 6(b)) are larger than those at other frequencies, but there are no such clear lines in figure 6(a). It indicates a significant level of nonlinear interaction concentrated at the LLM frequency during the I-phase, which suggests that the LLM has strong nonlinear coupling with the broadband turbulence. However, when the plasma enters the H-mode state, the ambient turbulence has been highly suppressed. Due to the disappearance of turbulence, the nonlinear interaction between turbulence and LLM has gone off, and it may directly lead to a sharp decrease of ${{\Gamma}}_{\mathrm{r},\mathrm{fl}\mathrm{u}\mathrm{c}}\left({f}_{\text{LLM}}\right)$ at 642 ms. Lastly, the inward particle flux is mainly comes from the linear term $-\frac{1}{{\partial }_{\mathrm{r}}{n}_{0}}\left\langle \tilde {n}{\partial }_{\mathrm{r}}\tilde {n}\right\rangle {V}_{\mathrm{r}}$ during the H-mode state.

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