Time-dependent probability density function analysis of H-mode transitions

Nuclear fusion has been studied as a potential energy source since the 1940s [1], but scientific and engineering progress has accelerated in the last few decades as the clear need for sustainable power has grown [2]. Crucial to achieving sustained magnetically confined fusion is the controlled access to a steady state high-confinement regime of operation known as H-mode [3,4]. This, in turn, requires an in-depth understanding and physics-based model of the low-confinement to high-confinement (L-H) and reverse high-confinement to low-confinement (H-L) transitions [5–7].

The current paradigm for sustained H-mode has been experimentally verified to be due to the suppression of edge plasma turbulence by sheared zonal and mean flows, leading to enhanced confinement [8,9]. The question of the trigger mechanism or spatio-temporal causality of the sheared flow, and correlation with turbulence flow energy transfer, remains an area of ongoing research; it has been suggested [6,10], and demonstrated experimentally [11], that turbulence itself is capable of driving these zonal flows in the first place.

In addition, it is not understood why some transitions rapidly reach H-mode through a sharp bifurcation of state and corresponding reduction in turbulence, while other transitions dither between H- and L-mode-like phases through so-called limit-cycle oscillations (LCO) [9,12,13] before the H-mode is reached.

Since H-mode transitions (L-H, H-L, L-LCO) are abrupt events that cause sudden changes in the plasma transport, the system is characterised by time-variability and large fluctuations in the causal variables. Conventional analysis approaches, which use assumed Gaussian distributions and associated mean and standard deviation values, result in the loss of valuable information over the very short L-H/H-L transition timescales. An alternative approach to studying very fast phenomena is the use of non-stationary, statistical methods such as time-dependent probability density functions (PDFs).

By analysing the PDFs of the measured experimental parameters thought to contribute to the H-mode transitions, information geometry can be used to quantify how the information and statistical states unfold over time [14]. This also enables key variables with different physical units to be compared in terms of dimensionless numbers (changes in statistical states), measuring to what degree they might be correlated. Furthermore, subtle changes in PDFs can be detected on time-scales relevant to the L-H and L-dithering-H transitions, allowing the dynamics of key linked variables to be explored.

The main aim of this letter is to report the first calculation of time-dependent PDFs from time-series edge plasma experimental data across H-mode transitions and the application of information geometry analysis. Our novel analysis technique is applied to a sharp and a dithering transition separately to improve our understanding of the evolution of edge plasma turbulence and flow velocities in the L-H transition under different experimental scenarios. This new method is shown to provide tools for capturing the complex evolution of the L-H transition, with promising potential for predicting the onset of the transitions.

The DIII-D Doppler Backscattering (DBS) diagnostic [15] is used to provide high temporal and spatial resolution measurements of the plasma turbulence in terms of relative electron density fluctuation, $\tilde{n_e}$, and perpendicular fluctuation velocity, $u_{\perp}$ [16]. This velocity is calculated from the backscattered signal, which has a doppler shift $\Delta\omega = k_{\perp}u_{\perp}$, where $k_{\perp}$ is the perpendicular wave number of the turbulence; $k_{\perp}$ ranges from $2\ \text{cm}^{-1}$ to $4.9\ \text{cm}^{-1}$. The fluctuation velocity, $u_{\perp}$, is the sum of the plasma mean $\textbf{E} \times \textbf{B}$ velocity, $v_{E\times B}$, and the turbulence phase velocity, v_ph , meaning that the $u_{\perp}$ measurements are sensitive to changes in $v_{E\times B}$ and zonal flows. The study presented in this letter focuses on these two measured variables, $\tilde{n_e}$ and $u_{\perp}$, which are understood to have important roles in the triggering and evolution of the L-H and H-L transitions.

Constructing time-dependent PDFs from time traces of experimental data requires good statistical samples, and thus data with high sampling frequency. This was achieved by using sliding time-windows of fixed duration to sample the DBS data and make a histogram, resulting in good-quality time-dependent PDFs. A full analysis of the uncertainties has been carried out and errors are provided for the PDF animations in the Supplementary Material Supplementarymaterial.pptx (SM).

The DBS diagnostic uses eight channels at different frequencies to probe the eight radial edge plasma locations shown in fig. 1 as a function of ρ, the normalized radius between the centre of the plasma and the last closed flux surface. In this study, results are presented for the single outermost channel (channel 1). The radial probing location of channel 1 remains within the narrow (few cm) confined plasma close to the separatrix and inside the pedestal electric field gradient layer where the LCO and L-H transition dynamics occur. The same analysis was performed on channel 2, which gives an independent measurement at a slightly lower radius, and provided very similar results for the PDF analysis.

Zoom In Zoom Out Reset image size

The DBS diagnostic has a temporal resolution of $0.2\,\mu\mathrm{s}$ and a typical spatial resolution ranging from 0.5 to 2 cm. The level of turbulence is proportional to the signal intensity. Values of $u_{\perp}$ have been determined from a 128-point Fourier transformation of the raw signal to evaluate the Doppler shift, meaning that the time resolution and available number of $u_{\perp}$ data points in a given time-window is 128 times smaller than for the $\tilde{n_e}$ measurements.

Sharp L-H and H-L transitions occur in under 1 ms, while dithering LCOs can have time periods of 0.3–10 ms; therefore, short time-windows of a few ms are essential for analysing and observing H-mode transition dynamics. However, generating PDFs from data using very short time-windows reduces the number of data points sampled, which can degrade the quality of the PDFs and the amount of information derived from them. The choice of time-window for the PDFs will depend on the temporal resolution of the diagnostic and the types of transitions being analysed; for this analysis a minimum of 50 data points per time-window was required for meaningful interpretation of the PDFs. For these data, time-windows of $\Delta t = 1\,\text{ms}$ for turbulence and $\Delta t = 5\ \text{ms}$ for $u_{\perp}$ were found to be optimal.

The time-dependent PDFs, p(x, t), of a stochastic variable x can be analysed by the time evolution of their moments, such as standard deviation, σ, and kurtosis, K. However, these low-order moments have limitations in understanding non-Gaussian statistical properties, particularly under non-equilibrium conditions involving large fluctuations. In this letter, a non-perturbative method is used to capture the evolution of the entire PDFs, from the perspective of geometry, by quantifying their temporal change in terms of a dimensionless distance (metric). This enables the interpretation of the evolution of different variables $(x=\tilde{n_e},\,u_{\perp})$ in terms of the same dimensionless numbers and provides a novel and promising methodology to study and compare different stochastic processes, such as phase transitions (for example, see [14,17]) with the variables, information length and rate.

Specifically, based on the relative change between two temporally adjacent PDFs, the square of information rate, $\mathcal{E}(t)$, and information length, $\mathcal{L}(t)$ [14,17], are calculated as follows:

$$\begin{eqnarray} \mathcal{E}(t) &\equiv \int \frac{1}{p(x, t)} \left( \frac{\partial p(x, t)}{\partial t} \right)^2 \mathrm{d}x = \int 4\left( \frac{\partial q(x, t)}{\partial t} \right)^2 \mathrm{d}x, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \mathcal{L}(t) &= \int_0^t \sqrt{\mathcal{E}(t_1)} \mathrm{d}t_1, \end{eqnarray} \tag{ 2 }$$

where $q\equiv \sqrt{p}$. To avoid numerical problems when calculating $\mathcal{E}(t)$ for very small values of p(x, t) in eq. (1), the second expression is used in these calculations.

Note that $\mathcal{L}(t)$ in eq. (2) gives a dimensionless measure of the total number of statistically different states that p(x, t) evolves through in the time interval (0, t). Information length, $\mathcal{L}(t)$, can be envisioned as a measure of the statistical distance between two time-dependent PDFs, which represents the system's trajectory in probability space [14,18,19], whereas $\mathcal{E}(t)$ measures the instantaneous change in statistical states. For a Gaussian PDF, a statistically different state can be defined as a change in the mean value, or due to a change in width, σ. Information length, $\mathcal{L}(t)$, is proportional to the time integral of the square root of the information rate, $\mathcal{E}(t)$, defined in eq. (1). To understand the L-H transition with and without dithering, $\mathcal{L}(\tilde{n_e},t)$ and $\mathcal{E}(\tilde{n_e},t)$ are calculated from the time-dependent PDF $p(\tilde{n_e},t)$, and similarly $\mathcal{L}(u_{\perp},t)$ and $\mathcal{E}(u_{\perp},t)$ are calculated from the PDF $p(u_{\perp},t)$.

Since information geometry tools were developed for continuous PDFs, care must be taken when applying these equations to discrete data. To minimize artificial spikes caused by the discrete derivative in eq. (1), $\mathcal{E}(t)$ is computed between PDFs 1 ms apart. These small increments of $1\ \text{ms}$ provide sufficient precision in the values of $\mathcal{E}(t)$ and $\mathcal{L}(t)$ to observe important L-H transition effects, whilst avoiding large discontinuities from discrete data.

The general parameters for the shots used in this study, one sharp transition and one dithering transition, are shown in fig. 2. These plasmas were part of a dedicated L-H transition investigation on DIII-D at plasma current, $I_p = 0.98\ \text{MA}$ and toroidal magnetic field, $B_T = 2.01\ \text{T}$. Both transitions took place in the initial NBI-only auxiliary heating phase, with shot 185461 operating with the small angle slot (SAS) divertor configuration [20], and 185498 with the conventional "partially open" divertor configuration.

Zoom In Zoom Out Reset image size

The L-H transition for shot 185461 occurred at $t = 2366.28\ \text{ms}$ at a line-averaged electron density $\bar{n_e} = 2.26\times 10^{19}\ \text{m}^{-3}$ and a power threshold $P_{th} = 1.4(\pm0.5)\ \text{MW}$. The power threshold is defined by the equation $P_{th} = P_{NBI} + P_{OH} - \frac{\mathrm{d}W}{\mathrm{d}t}$, where P_NBI is the NBI heating power, P_OH is the Ohmic power and $\frac{\mathrm{d}W}{\mathrm{d}t}$ is the rate of change of the stored plasma energy. The plasma transitioned directly to an ELM-free H-mode phase with an accompanying sharp drop in the divertor $D_\alpha$ emission and a rise in the edge electron density, n_e ^edge , and plasma confinement, H₉₈, shown by the time traces in fig. 2(a).

The second plasma, 185498, transitioned to a dithering H-mode or LCO phase at $t = 2108.13\ \text{ms}$, with $\bar{n_e} = 1.20\times 10^{19}\ \text{m}^{-3}$ and $P_{th} = 1.2 (\pm0.5)\ \text{MW}$. The dithering phase oscillations lasted for 50 ms before the shot entered an ELM-free H-mode, shown along with the other plasma parameters in fig. 2(b).

The radial profiles of n_e , electric field, E_r , and $-u_{\perp}$ in fig. 3 at times before and after the L-H transitions, show the formation of the n_e pedestal and E_r well for both plasmas, as well as the clear correlation between $u_{\perp}$ and E_r . For shot 185461, channel 1's probed radius (pink shading) remained in the steep pedestal gradient and E_r well region (ρ = 0.97–0.98). In shot 185498, channel 1 covered a wider range (ρ = 0.85–0.95), corresponding to the top and steep gradient regions of the pedestal, and the inner part of the E_r well.

Zoom In Zoom Out Reset image size

Examples of the PDFs $p(\tilde{n_e})$ and $p(u_{\perp})$ around the L-H transitions are shown in fig. 4(a) and fig. 5(a) for $\tilde{n_e}$ and fig. 4(b) and fig. 5(b) for $u_{\perp}$, along with animations of the time-dependent PDF evolution at every millisecond in the SM. The number of bins used to construct the PDFs for the two measurements was determined using the Rice rule [21], resulting in 11 and 34 bins for $u_{\perp}$ and $\tilde{n_e}$, respectively. A kernel density estimation method [22] for smoothing the PDFs was tested and found only to slow down the calculations without meaningfully altering the results; therefore, no filtering was applied. When comparing, $p(\tilde{n_e})$ appears smoother than $p(u_{\perp})$ because $\tilde{n_e}$ contains 128 times more data points.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The experimental parameters, such as the divertor magnetic configuration, L-mode $\bar{n_e}$ and P_th , are different for these two shots. A direct comparison of the stochastic model analysis of the transition dynamics would therefore provide limited insight and is not made here.

In the following sections, results are presented from the statistical analysis applied to two separate shots with distinct L-H transitions, sharp and dithering. These two shots provide a robust representation of the results from a comprehensive application of the time-dependent PDF analysis to all eight DBS channel data for all plasmas in the L-H transition investigation.

PDF analysis

Information geometry

PDF analysis

For shot 185498, the start of the dithering H-mode occurs at $t = 2108.13\ \text{ms}$ and is followed by a period of LCOs. These LCOs are useful because they prolong the L-H transition dynamics, allowing the behaviour (e.g., oscillations) of key variables to be observed more clearly over a longer time period [23]. As the dithering can result from self-regulation between turbulence and zonal flows, $u_{\perp}$ is referred to as a zonal flow for the dithering transitions [10].

Statistical analysis of the dithering H-mode phase for shot 185498 is shown in fig. 5, and an expanded trace is shown in fig. 6 with detailed features of the LCOs. Overall, the L-mode $\tilde{n_e}$ PDF (in dotted blue) in fig. 5(a) changes from a wide, even distribution to a more skewed, narrow distribution over the transition, with $\sigma({n_e}) = 0.38$ at $t = 2105\ \text{ms}$ in L-mode and $\sigma({n_e}) = 0.17$ at $2142\ \text{ms}$ well into the dithering phase. Correspondingly, $p(u_{\perp})$ becomes more positive and widths grow from $\sigma({u_{\perp}}) = 1800\ \text{ms}^{-1}$ to $3000\ \text{ms}^{-1}$ for the same times.

Zoom In Zoom Out Reset image size

Specifically, the transition to the dithering phase (blue to orange of $p(\tilde{n_e})$) involves the shortening of the right tail and a hint of dual peak formation around $\tilde{n_e} \approx 0.2, 0.4$, which then returns to a single peak in the green curve. The shortened right tail suggests that higher-amplitude turbulence $(\tilde{n_e} > 1)$ is suppressed by changes in the velocity prior to the transition, similar to what was observed in fig. 4 for the sharp transition. The narrowing of the $\tilde{n_e}$ PDF that was seen in fig. 4(a) is also observed here (from the green to the red line), indicating the perpendicular velocity suppression of both low $(\tilde{n_e} \approx 0.1)$ and high-amplitude turbulence $(\tilde{n_e} > 0.3)$, with a PDF peaking at a much lower $\tilde{n_e} \approx 0.1$, after the transition to the dithering phase.

Further time evolution of the PDFs in solid red, dotted purple, and solid brown lines reveal the oscillation of PDF itself, as observed in [14,17]; a wider and more skewed PDF (purple) with a right tail $\tilde{n_e}$ extended to $\approx 0.9$, is sandwiched between two narrow, low-amplitude $\tilde{n_e}\,\approx 0.1\text{--}0.4$ range PDFs (red and brown).

Remarkably, similar PDF oscillations are also observed for $p(u_{\perp})$ in the red, purple and brown curves in fig. 5(b), but in such a way that the narrow PDFs of $p(\tilde{n_e})$ with a shorter right tail (red and brown) correspond to the PDFs of $p(u_{\perp})$ peaking around $-700\ \text{ms}^{-1}$ while the wider PDF with a heavy right tail of $p(\tilde{n_e})$ (purple) corresponds to the $p(u_{\perp})$ peak at $0\ \text{ms}^{-1}$. This is a clear demonstration of self-regulation between turbulence and zonal flows. Furthermore, $p(u_{\perp})$ in the H-mode–like dithering phase shows a stretched PDF right tail in comparison with the L-mode and L-mode–like phase PDFs, suggesting that strong zonal flows play a crucial role in regulating turbulence [17].

The time-trace of mean $\tilde{n_e}$ in fig. 5(d), (e) shows no significant change until just before the transition, while the mean $u_{\perp}$ in fig. 5(e) undergoes a sudden increase in value 4 ms prior to the L-dithering phase transition (green vertical line). This increase, which is more clearly seen in the expanded trace of fig. 6(c), suggests that zonal flows trigger the initial transition from L-mode to the dithering phase. The PDF intermittency discussed above can also be seen in the time traces in fig. 5(f) and fig. 6(d); $K(\tilde{n_e})$ remains almost constant at $K(\tilde{n_e}) \approx 3$ over the transition to dithering, while $K(u_{\perp})$ in fig. 5(g) and fig. 6(e) increases sharply from 3 to 7 in the 7 ms prior to the L-dithering phase transition (green vertical line), again reflecting intermittency and the role of zonal flows in triggering the transition.

On the other hand, the mean $u_{\perp}$ rises to around $1000\ \text{ms}^{-1}$ at $t = 2115\ \text{ms}$ following the first transition and oscillates around that value for the remainder of the dithering phase. In comparison, the value of $K(u_{\perp})$ rises significantly after the first transition, peaking at 17.8 at $t = 2138\ \text{ms}$, again, reflecting a strong intermittency.

Information geometry

This letter presents the first application of a time-dependent PDF analysis to the sharp and dithering L-H transitions from DIII-D experimental data, using DBS measurements of edge plasma $u_{\perp}$ and turbulence $\tilde{n_e}$. Overall, the sharp and dithering L-H transitions exhibit complex temporal evolution involving non-Gaussian PDFs, long stretched tails and possible bi-modality, and PDF oscillations. In particular, the heavy right tails of $u_{\perp}$ suggest the presence of strong localised perpendicular (zonal) flows and their important role in turbulence regulation, as predicted theoretically [14,17]. Furthermore, evidence of two-step turbulence suppression near the transition to the H-mode and in the dithering phase has been demonstrated within the temporal resolution limits, with several cases of the localised perpendicular velocity followed by the reduction of larger amplitude turbulent structures prior to the transition, succeeded by a general suppression of turbulence. The novel information geometry diagnostics $\mathcal{L}$ and $\mathcal{E}$ provide a clear marker for the sharp and dithering L-H transitions. In addition, they have potential for predictive capability, with a sharp peak in $\mathcal{E}(u_{\perp})$ appearing to forecast the L-to-dithering and dithering-to-H transitions. Furthermore, the correlation between $u_{\perp}$ and $\tilde{n_e}$ through self-regulation is well captured by their similar $\mathcal{L}$ values, particularly in the dithering phase, providing valuable experimental evidence of the predator-prey model assumptions.

In conclusion, the time-dependent PDF analysis has been shown to be a sensitive tool for understanding the dynamic changes in the plasma across the L-H transition. This stochastic analysis approach has potential for predicting and controlling phase changes in magnetically confined plasma systems. It will be developed further, with plans for application to more L-H transition shots, higher-resolution data and other variables of interest. For example, the role of electromagnetic turbulence in H-mode transitions will be investigated along with an exploration of the cause of the H-mode quasi-oscillations observed in fig. 4.

The authors would like to acknowledge the DIII-D team, and give special thanks to Quinn Pratt and Kathreen Thome, for their assistance, support and illuminating discussions. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award(s) DE-FC02-04ER54698, DE-SC0019352, DE-SC-0020287, and DE-FG02-08ER54999. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclose or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily state of reflect those of the United States Government or any agency thereof.

Data availability statement: The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.