Turbulence Trapping in Radial Electric Field Well at H-Mode Transport Barrier in JFT-2M Tokamak

Abstract

In this paper, an experimental consistency check for a theory describing the interplay between micro-scale turbulence and meso- and macro-scale flows is presented. The experimental data obtained in the JFT-2M tokamak are used for the consistency check. After L-H transition, a solitary radial electric field well, driving an E × B flow structure in the electron diamagnetic direction, forms. Turbulence is regulated in the flow shear regions, while it nearly remains essentially unchanged at the peak of the flow structure where the shear is zero. In a theory, this type of turbulence redistribution, called hill-trapping, is predicted to occur when the adiabatic parameter α is larger than unity. In the experiment, α ∼ 40 ≫ 1 is given, supporting the theory. An eigenmode model analysis for the turbulence system is performed for the experimental parameters, and the occurrence of the hill-trapping is found.

1.  Introduction

Interaction between turbulence and plasma flow is of great interest for understanding transport barrier formation physics in fusion plasmas. As a prototypical example, a key role of radially sheared poloidal flow on turbulent transport regulation across the L-H transition has been theoretically pointed out [1], and experimental evidences supporting that concept have been reported [2–4]. Those flows originate from the negative solitary radial electric field E_r structure at the edge, the so-called E_r-well, driving the $E\times B$ flow in the electron diamagnetic drift direction. Furthermore, the flow curvature effect on transport reduction has been discussed [5, 6], particularly for explaining the significant confinement improvement at the bottom of the E_r-well (peak of the $E\times B$ flow) where the shear is zero [7–9]. Aiming at consistent modeling, wave kinetic theory [10] was developed, and the concept of the ‘turbulence trapping’ was introduced. Afterwards, a new insight into the habitat segregation of turbulence was found in the presence of meso-scale flow structures, such as the geodesic acoustic mode (GAM) [11], the energetic particle driven GAM [12], and the zonal flow [13]. There the turbulence packet was found to be trapped at curvature peak regions in the flow structures. Extending the wave kinetic theory that described the adiabatic limit of the turbulence, the Hasegawa-Wakatani model [14] was analyzed to include the hydrodynamic property of turbulence [15]. The control parameter was the adiabatic coefficient $\alpha$ (see Eq. (1) for definition), showing how quickly electrons responded to the potential perturbation. In particular, $\alpha$ dependence of the turbulence trapping was clarified for elucidating the tokamak density limit maintained by the turbulence dynamics [15]. In that theoretical view, approaching the density limit while reducing $\alpha$, the position of the turbulence trapping changes from the positive (electron diamagnetic drift direction) to the negative flow region. This occurs in a bifurcative manner, and the two states were called the hill-trapping and the valley-trapping, respectively. Apart from the radially integrated zero-dimensional view [16] previously established and utilized in many experimental interpretations, e.g. in [2, 3], radial turbulence redistribution in the presence of meso- and macro-scale flow structures is compelling in elucidating how the profiles are determined in the presence of the transport barrier.

In this paper, the theoretical idea presented in [15] is examined using a set of experimental data obtained in the JFT-2M tokamak [17]. Turbulent density fluctuation and the E_r-well structure are directly measured by the heavy ion beam probe (HIBP) [18], allowing a qualitative test of the model prediction. The target plasma is low density and is classified as a nearly adiabatic limit with $\alpha$ ≫ 1. The turbulence is found to be trapped in the E_r-well (positive flow structure) inside the H-mode transport barrier as the theory predicts in such a regime. As an additional test, an eigenmode analysis for the Hasegawa-Wakatani model is presented, showing the turbulence trapping in the E_r-well region, being consistent with the observation.

2.  Experimental Observation

The JFT-2M is a medium tokamak with a major radius of R = 1.3 m and an averaged minor radius of a = 0.3 m. A single-null divertor configuration with the ion $\nabla B$ drift directed towards the X-point is employed to meet the marginal L-H transition threshold condition with a low line averaged density of $⟨n_{\mathrm{e}}⟩$ = 1.1 × 10¹⁹ m⁻³ and a neutral beam (NB) heating power of $P_{\mathrm{NB}}$ = 750 kW. Other experimental parameters are as follows: a toroidal magnetic field of $B_{\mathrm{t}}$ = 1.17 T, a plasma current of $I_{\mathrm{p}}$ = 190 kA, and the safety factor at the flux surface enclosing 95% of the total poloidal flux of $q_{95}$ = 2.9.

The heavy ion beam probe measures fluctuations on the electrostatic potential and the plasma density at four positions [18]. The channel separation is ∼ 2.5 mm when projected on the outer mid-plane, and the radial extent of the sampling volume of each channel is ∼ 6 mm. The temporal resolution is $\mathrm{\Delta }t$ = 1 μs. In a shot-to-shot basis, the edge region of −5 < r − a < 0 cm is measured, where r is the radial coordinate.

Figure 1 shows the radial profiles of the density fluctuation power spectrum and the $E\times B$ flow at the L-mode and H-mode periods. Here the density fluctuation is normalized by its time-averaged value. The positive direction of the $E\times B$ flow is defined in the electron diamagnetic drift direction. In the L-mode, a broadband turbulence is observed in a wide radial extent where the measurement is made. The turbulence intensity increases as approaching the last closed flux surface $r$ = a. Examining the density and potential fluctuation amplitude and phase difference, the Boltzmann relation is found to be satisfied, suggesting that the observed turbulence is the resistive drift wave. The poloidal propagation velocity of the turbulence agrees with the electron diamagnetic drift velocity, supporting the concept that the turbulence is the drift wave [19]. The poloidal wavenumber is measured in a dedicated shot where the HIBP sampling volumes are aligned in the polodial direction, and is found to be $k_{\theta }$ ∼ 0.75 cm⁻¹. The $E\times B$ flow is rather flat except for the scatter of points, which is due to imperfect reproducibility of plasmas in different shots (5 shots in total).

Fig. 1.  Radial profiles of (a, b) density fluctuation power spectrum and (c, d) $E\times B$ flow. Left and right columns show cases in L-mode and H-mode, respectively. Positive direction of $E\times B$ flow is electron diamagnetic drift direction.

After the H-mode transition, a solitary $E\times B$ flow structure is formed in the electron diamagnetic drift direction. The related E_r-well structure is identified to be predominantly driven by the loss-cone loss and neoclassical bulk viscosity processes [20, 21]. The turbulence power spectrum becomes narrower, likely because of the $E\times B$ flow shear turbulence suppression. At the location where the $E\times B$ flow structure peaks and the $E\times B$ flow shear is zero, the turbulence amplitude remains. This remaining turbulence has an up-shifted frequency at $f$ ∼ 100 kHz. The Doppler shift frequency of the turbulence in the rest frame is estimated as $k_{\theta }{V}_{\mathrm{E}\times \mathrm{B}}/2\mathrm{\pi }$ ∼ 120 kHz, which roughly agrees with the up-shift of the turbulence frequency. This situation corresponds to the turbulence trapping by the positive peak of the $E\times B$ flow structure, called the hill-trapping in [15].

3.  Test of Theory

According to a theory [15], hill-trapping is predicted to occur when the adiabatic parameter satisfies the $\alpha$ > 1 condition. In this section, $\alpha$ is computed for the present shot to examine this prediction and the possible eigenfunctions of turbulence are calculated for a fluid-type reduced turbulence model. The parameter $\alpha$ is defined as

  

$$\alpha =k_{||}^2{v}_{\mathrm{th},\mathrm{e}}^2/({\omega }_{\ast }{\nu }_{\mathrm{ei}}),$$(1)

where $k_{||}$ is the turbulence parallel wavenumber, $v_{\mathrm{th},\mathrm{e}}$ is the electron thermal velocity, ${\omega }_{\ast }=v_{\mathrm{d},\mathrm{e}}{k}_{⟂}$ is the drift frequency ($v_{\mathrm{d},\mathrm{e}}$ and $k_{⟂}$ being the electron diamagnetic drift velocity and turbulence perpendicular wavenumber, respectively), and ${\nu }_{\mathrm{ei}}$ is the electron-ion collision frequency. The parallel wavenumber is approximated as $k_{||}$ ∼ 1/qR, where q is the safety factor and R is the major radius. Those parameters are calculated according to the plasma parameter measurement [9]. Given $\alpha$ is 40, the theory predicts the hill-trapping [15], being consistent with the observation. Note that the value of $\alpha$ is not very precise because of, e.g., the assumption in $k_{||}$ and the diagnostic capability, but only gives an order of magnitude estimate. Nevertheless, it may be acceptable for the qualitative consistency check with experiment because the theory predicts the hill-trapping occuring in a wide $\alpha$ range in $\alpha$ > 1.

In order to discuss a possible turbulence profile in the presence of a positive solitary $E\times B$ flow structure in H-mode plasmas, the eigenmode problem is solved for the Hasegawa-Wakatani turbulence system [14]. For identifying the turbulence trapping condition, the eigenmode is analyzed under the prescribed background flow corresponding that in the H-mode with a given density gradient. In this framework, possible solutions of the Hasegawa-Wakatani model are given in the form

  

$$\tilde{\mathrm{\Psi }}(x,y,t)=\sum _{k_y}{\mathrm{\Psi }}_k(x)e^{-i\omega t+i{k}_{y}y},$$(2)

where $\mathrm{\Psi }$ denotes either the potential or density eigenfunctions, $x$ and $y$ correspond to the radial and poloidal coordinates, and $k_y$ and ω show the poloidal wavenumber and the complex frequency, respectively. The poloidal flow structure is defined as

  

$$V_y(x)=V_0\exp \left[{\kappa }_V\left\{\cos \left(2\mathrm{\pi }\frac{x-L_x/2}{L_x}\right)-1\right\}\right],$$(3)

where $L_x$ = $2\mathrm{\pi }$/0.3, ${\kappa }_V$ = 2 and $V_0$ = 2. Here the space and velocity is normalized by the sound Larmor radius and the diamagnetic drift velocity, respectively. The $E\times B$ flow shape is defined to mimic the experimentally observed one, where the specific radial width is comparable to the turbulence radial wavelength and the peak velocity in the order of the diamagnetic drift velocity. Figure 2 shows the spatial distribution of the density turbulence eigenfunctions $n_k(x)e^{i{k}_{y}y}$ having three largest growth rates, i.e., the imaginary part of the complex frequency. As in the case of the experimental observation, turbulence is trapped within the radial range with large positive $V_y$. By the present consistency check, the qualitative prediction capability of the theory [15], at least for the nearly adiabatic limit case for the hill-trapping, is found.

Fig. 2.  Results of eigenmode analysis. (a–c) Eigenfunctions having three largest growth rates, and (d) given poloidal flow structure mimicking H-mode transport barrier. Red and blue colors correspond to positive and negative local density variations, respectively. Since the absolute value of eigenmode does not provide meaning and colorbar is not shown. Space and velocity is normalized by sound Larmor radius and diamagnetic drift velocity, respectively.

4.  Summary

In this paper, a theoretical concept that elucidates the turbulence redistribution in the presence of larger scale sheared flows was tested using an experimental data set. The experiment was performed in a medium size tokamak, JFT-2M. Marginal H-mode discharges were analyzed to examine the turbulence amplitude profiles in L- and H-modes where the latter had an inward directed radial electric field, i.e., an E_r-well structure. The E_r-well corresponded to a solitary poloidal flow in the electron diamagnetic drift direction. The turbulence amplitude in the H-mode was found to accumulate at the flow peak position, while significantly suppressed in the flow shear region at the both sides of the flow peak. This signature was dubbed hill-trapping. In a theory, the hill-trapping was anticipated to occur when the adiabatic parameter $\alpha$ was larger than unity. In the experiment, $\alpha$ was calculated to be ∼ 40, supporting the theory. Eigenmode analysis for the Hasegawa-Wakatani system was performed for the experimental parameters, and the occurrence of the hill-trapping was found. A qualitative experimental consistency check in the high $\alpha$ case, namely the adiabatic limit, was achieved in this paper. More comprehensive examination with an $\alpha$ scan, either in experiments or high-fidelity simulations such as gyrokinetics, is highly desirable for completing the benchmark of the theory. Particularly, in a small $\alpha$ regime, the turbulence is predicted to survive in the negative peak of the $E\times B$ flow region, in contrast to the high $\alpha$ case. The turbulence localization across a wide $\alpha$ range, e.g., during the density lamp up phase in the density limit experiment, is of great interest for more quantitative examination of the theory.

Acknowledgments

Authors acknowledge T. Ido, K. Kamiya, the late S.-I. Itoh, K. Itoh, Y. Miura, K. Ida, the late H. Maeda, Y. Hamada, M. Mori, Y. Kamada, and K. Hoshino for strong support. This work is partly supported by the Grant-in-Aid for Scientific Research of JSPS (21K13902 and 25K00986).

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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