Turbulent-driven low-frequency sheared E × B flows as the trigger for the H-mode transition

When $0 \,{<}\, P_{{\rm LF}}\,{<}\, \gamma_{\rm ZF} \langle V_{\rm ZF}^{2} \rangle$ this system has a fixed-point solution given approximately as $V_{E\times B}^{\rm LF} = [\langle \tilde{v}_{\perp}^2 \rangle(\gamma_{\rm eff} -\gamma_{\rm decorr}^{\rm pl})/\gamma_{\rm LF}]^{1/2}$. A slow increase in the heat flux (e.g. slow enough so that the edge gradients evolution is slow compared with the confinement time) should then increase the density and temperature gradients and heat the edge, resulting in an increase in the turbulence amplitude and a decreased rate of zonal flow damping γ_ZF. As a result $\tilde{{v}}_{\bot}^{2}$, $V_{E\times B}^{{\rm 'LF}}$ and P_LF should grow with increased heating power under fixed-point L-mode conditions. We note that recent studies in L-mode discharges have confirmed this behaviour [8]. When $P_{{\rm LF}}>\gamma_{{\rm LF}} V_{E\times B}^{{\rm LF}^{2}}$, a growing solution $({\partial V_{E\times B}^{{\rm LF}^{2}}}/{\partial t})>0$ can now exist and thus $V_{E\times B}^{{\rm LF}^{2}}$ can begin to grow at the expense of the turbulent energy $\tilde{{v}}_{\bot}^{2}$, signalling the onset of the LCO regime. If the energy transfer rate becomes strong enough so that $P_{{\rm LF}} >(\gamma_{{\rm eff}} -\gamma_{{\rm decorr}} )\langle \tilde{{v}}_{\bot}^{2} \rangle$, then the turbulence amplitude (and thus the turbulent-driven cross-field transport) can collapse, resulting in a quenching of turbulent transport and an increase in the ion pressure gradient and the MSF. As noted above, in the predator–prey model the effective energy input rate is decreased by the mean shear flow, i.e. $\gamma_{{\rm eff}} ={\gamma_{{\rm l}}}/{(1+\alpha V_{{\rm MSF}}^{'2} )}$. As a result, as the MSF builds up during the LCO regime, the energy input rate into the turbulence gradually decreases. With sufficiently strong MSF, the turbulent energy never recovers after the peak in the turbulent-driven LF E × B shear flow energy. Instead the turbulence energy decays away and a regime of strong steady-state MSF with correspondingly large pressure gradient develops, signalling the onset of the H-mode regime.

When the heating power is sufficiently strong, the above sequence is compressed into a short (∼1 ms) transient event in which the gradients and associated mean shear flow increases. The turbulent-driven zonal flow then undergoes a rapid growth and for a short period (∼10⁻²a/C_S ∼ 1 ms) (here a denotes the minor radius and C_S is the ion acoustic speed) succeeds in transferring nearly all of the turbulent energy into the LF shear flow. As a result cross-field transport collapses, the gradients increase rapidly and a strong mean shear flow is then locked in. The details of these model dynamics have recently been published [9], and the interested reader is referred to that paper for further discussions. As shown below, experiments show these key signatures, providing support that the underlying predator–prey model captures the essential elements of the transition to H-mode.

In HL-2A a series of time-stationary inner-wall limited discharges with a variety of ECH heating powers are used to examine turbulence-ZF energy transfer [8] in steady-state L-mode discharges. The required multipoint turbulence measurements are obtained in the region just inside the LCFS in ohmically heated and ECH heated discharges, and the frequency-resolved energy transfer is then inferred using established techniques [10] yielding two-dimensional bispectral measurements of energy transfer across frequency scales. These bispectra can then be integrated over one frequency axis to yield the net energy transfer into/out of a particular frequency f. The frequency-resolved net energy transfer is shown in figure 1 for the case of a 700 kW ECH heated discharge. The results show that turbulent kinetic energy is transferred out of intermediate (20–50 kHz) frequencies and into both LF (<15 kHz) fluctuations previously identified as m, n = 0 sheared E × B flows, as well as into higher frequency (>50 kHz) ranges (figure 1). The cited work shows that this transfer process gets more pronounced with increased ECH heating. These observations support the notion that the two-scale power balance model described above (and which underpins the predator–prey model of the L–H transition) has a basis in experimental observation.

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Additional insight into this physics can be obtained by considering radial profiles of the Reynolds force $F_{\theta_{Rey}} =-({\partial \langle \tilde{{v}}_{r} \tilde{{v}}_{\theta} \rangle}/{\partial r})$, the low-frequency E × B flow $V_{E\times B}^{{\rm LF}}$ and the product of these two, P_Re, which is equal to the rate of work done by the turbulence on the low-frequency E × B flow which in a 0D model satisfies P_Re = P_LF. The Reynolds stress is computed using a time-domain high-pass digital filter to isolate velocity fluctuations with f > 20 kHz; the product of these fluctuations is then time averaged to produce the $F_{\theta_{Rey}}$ (figure 2 left panel). The low-frequency E × B shear flow is computed from the radial gradient of the time-averaged plasma potential (figure 2 centre panel) and P_Re is then computed from the product of the first two results (figure 2 right panel). We show results for ohmic (black curves) 380 kW ECH (blue curves) and 730 kW ECH (read curves) heated discharges. The results clearly show that in these time stationary L-mode discharges the Reynolds force acts to reinforce the E × B flow, and as a result the turbulence transfers energy into the large-scale shear flow consistent with the model expectations for fixed-point L-mode behaviour. Furthermore, the Reynolds force, the magnitude of the shear flow and the production term all increase substantially as the heating power is increased. These results show that the turbulence is acting to reinforce or amplify the shear flow at the boundary, and that this effect becomes stronger as the heating power is raised. The predator–prey model would then predict that when the transfer rate exceeds γ_ZF, then $V_{E\times B}^{{\rm LF}}$ can grow to much larger amplitudes and extract a significant fraction of energy from the turbulence. Without the knowledge of this damping rate, we cannot test this prediction in the HL-2A data. However, experiments in DIII-D (discussed next) do allow us to examine these expectations in a more quantitative manner.

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The transition to LCO behaviour is studied in DIII-D LSN discharges. The midplane fast scanning probe is inserted during the L-mode, is stationary approximately 1 cm inside the LCFS while it captures the L–I transition, and then is retracted in the early (∼10 ms) I-phase. The resulting time-resolved measurements just inside the LCFS (figure 3) then permit us to study the evolution of the power transfer into the m, n = 0 shear flow during the onset of the LCO regime. The results show that $V_{E\times B}^{{\rm LF}}$ begins to increase slightly (a few hundred µs) before the oscillations in divertor D_a light (which are characteristic of the LCO regime) begin (figures 3(a) and (b)). The time required for parallel plasma propagation along the field lines can introduce a slight (∼100–200 µs) delay between the $V_{E\times B}^{{\rm LF}}$ oscillations and the divertor D_a light oscillations, and thus this slight delay may reflect the time needed for parallel transport processes to begin to modulate plasma particle input into the divertor due to modulations of cross-field transport. A comparison of the rate of energy transfer, $P_{{\rm LF}} /\tilde{{v}}_{\bot}^{2}$ and the plasma-frame turbulent decorrelation rate $\gamma_{{\rm decorr}}^{{\rm pl}}$ provides important insights into the physics of the onset of the LCO or I-phase regime. Here the ratio $P_{{\rm LF}} /\tilde{{v}}_{\bot}^{2}$ provides a measure of the effective rate of energy transfer from the turbulent frequency range into the m, n = 0 shear flow. The rate $\gamma _{{\rm decorr}}^{{\rm pl}}$ is computed from the measured laboratory frame decorrelation rate γ_decorr|_lab, the measured poloidal decorrelation length $L_{\theta}^{{\rm corr}}$ and the measured low-frequency E × B drift velocity $V_{E\times B}^{{\rm LF}}$ via the relation $\gamma_{{\rm decorr}}^{{\rm pl}^{2}} ={\gamma_{{\rm decorr}}^{2}}|_{{\rm lab}} -({V_{E\times B}^{{\rm LF}}}/{L_{\theta}^{{\rm corr}}})^{2}$. The results (figure 3 lower panel) show that in the fixed-point L-mode regime, just before the transition to LCO regime, the turbulence has ${\gamma_{{\rm decorr}}^{{\rm pl}} \approx P_{{\rm LF}}}/{\tilde{{v}}_{\bot}^{2}}\approx (2\mbox{--}4)\times 10^{5}\,s^{-1}$, indicating that turbulent energy is dissipated to both LF E × B flows and to high frequency, high wavenumber (and thus presumably viscous) dissipation processes at comparable rates. Furthermore, because the L-mode state is time stationary, we can estimate that in L-mode just prior to entry into the LCO regime, the net rate of energy input into the turbulence must balance these combined dissipation processes, and thus we can estimate the effective energy input rate in L-mode as ${\gamma _{{\rm eff}}}|_{{\rm L}} =\gamma_{{\rm decorr}} +P_{{\rm LF}} /\langle \tilde{{v}}_{\bot}^{2} \rangle \approx (6\mbox{--}8)\times 10^{5}\,{\rm s}$. Examining the magnitudes of the stress and E × B flow in L-mode, we can also estimate γ_ZF ∼ 10⁵ s⁻¹.

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The transition to the LCO state is observed to occur at about 1.6062 s as seen by the onset of oscillations in the data in figure 3. In the short (a few hundered µs) time period of the onset of the LCO phase, it seems unlikely that the mean density and temperature profiles would be able to evolve. Thus, the free energy source driving the turbulence and the (m, n = 0) flow-damping rate γ_LF will remain roughly constant across the transition into the LCO state. Examining the results in figure 3(c), we note that at the onset of the LCO regime, the $P_{\rm ZF} /\tilde{{v}}_{\bot}^{2}$ channel increases by a factor of ∼2–3 to a value of about 10⁶ s⁻¹ or so at 1.606 s while the decorrelation rate shows no similar prompt jump. Clearly then $P_{\rm ZF} /\tilde{{v}}_{\bot}^{2}$ becomes the dominant turbulent energy dissipation channel as the LCO regime is entered. As a result, changes in $V_{E\times B}^{{\rm LF}}$ can have a significant impact on the turbulent energy balance via the model equations given above.

The above results are consistent with an interpretation that the time-varying shear flow in the LCO regime is in fact driven by the turbulence. Further supporting evidence can be found by an examination of the time-variation of the turbulent stress, turbulent energy and E × B shear flow during the LCO phase. Figure 4 presents a detailed examination of these quantities obtained from probe measurements taken ∼1 cm inside the LCFS during a DIII-D LCO discharge. The stress and turbulent energy both grow as the m, n = 0 E × B flow approaches its minimum value. Then, as the stress reaches its most negative value, the m, n = 0 E × B flow begins to accelerate and reaches its maximum acceleration either just at or very shortly after the peak in the turbulent stress. Since the stress is nearly zero outside the LCFS, we can take the Reynolds force as being proportional to the value of the stress; thus stress modulation represents a Reynolds force modulation. We therefore conclude that the stress can provide an acceleration which then modulates the m, n = 0 E × B flow and that the observed flow dynamics are consistent with this interpretation.

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We have also used BES turbulence imaging to study the power transfer evolution during the same L-mode to LCO transition. In particular, we have used velocimetry analysis [11, 12] of BES imaging data obtained in the same discharges to provide a similar calculation of the shearing rate and nonlinear power transfer rate during the L-mode to LCO transition. The results (figure 5) provide a qualitatively similar picture of the onset of strong nonlinear power transfer into the low-frequency shear flow at the moment of the LCO transition, and a subsequent modulation in this transfer rate during the LCO regime. Thus, these results do not seem to depend upon the use of probes to infer the results. We also note that radially resolved probe measurements show that these effects are localized to the region slightly (<1 cm) inside the LCFS (figure 6).

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The radial profiles of $V_{E\times B}^{{\rm LF}}$ and ${P_{{\rm LF}}}/{\langle \tilde{{v}}_{\bot}^{2} \rangle}$ obtained in L-mode, in early LCO phase, and in early (∼10 ms after onset) H-mode are shown in figure 6. The mean E × B velocity profile shows a weak shear layer in L-mode, and periods of significantly stronger flow in the LCO or I-phase regime. In the region inside the LCFS the E × B velocity shows large oscillations in the LCO regime, documented in detail above. In the H-mode, the E × B profile appears to 'lock-in' the peak values of $V_{E\times B}^{\rm LF}$ and $P/\langle \tilde{{v}}_{\bot}^{2}\rangle$ found in the LCO or I-phase regime. In the LCO or I-phase regime, the transfer rate ${P_{{\rm LF}}}/{\tilde{{v}}_{\bot}^{2}}$ increases markedly from L-mode values, and exhibits large variations as the m, n = 0 E × B flow and turbulence intensity oscillations occur. In H-mode, the power transfer rate then locks into values as large as those found in the intermediate LCO regime and equals or exceeds the L-mode effective energy input rate γ_eff from the free energy source.

The above results provide detailed insight into the L-LCO-H mode transition sequence. However, the question remains as to whether or not a similar physics picture is observed in a 'normal' L–H transition that does not exhibit the intermediate LCO phase. We have performed experiments on the EAST device which provide an answer to this question. We used a single discharge that exhibited an L-LCO-H mode transition, went back into L-mode, and then had a normal L–H transition. The macroscopic parameters of the plasma were quite similar during these two transitions, allowing us to use results from the LCO regime in the analysis of the L–H transition. A detailed discussion of these results can be found in a recent paper [13].

Measurements of the rate of turbulence kinetic energy recovery $(1/{\tilde{{v}}_{\bot}^{2}})({\partial \tilde{{v}}_{\bot}^{2}}/{\partial t})$ made in the LCO regime during periods when the zonal flow magnitude is negligible provide a direct experimental measurement of the net rate of energy input into the turbulence, i.e. of the quantity (γ_eff − γ_decorr) which gives the net rate of energy input into the turbulence after accounting for transfer to high wavenumber scales where viscous dissipation occurs. These data are taken under conditions in which the edge gradients during the LCO regime and the L–H transition are the same to within the uncertainty of the measurements. The plasma in these experiments is therefore sitting very close to the threshold for the transition to improved confinement. Multipoint probe measurements then provide a measure of P_LF during a 'normal' L–H transition that does not exhibit an LCO phase but which otherwise has edge plasma gradients are nearly identical to those found during the LCO phase. Because the edge pressure gradients are similar, then presumably the net rate of energy input into the turbulence, γ_eff − γ_decorr is unchanged going from the LCO to the L–H regimes. As a result, we can use the net rate of energy input γ_eff − γ_decorr during the LCO regime to the study of the L–H transition. The ratio ${P_{{\rm LF}}}/{(\gamma_{{\rm eff}} -\gamma_{{\rm decorr}} )\tilde{{v}}_{\bot}^{2}}$ then indicates the ratio of the power transfer into the ZF normalized by the net power input into the turbulence. Should this value exceed unity, then according to the model described above, the turbulent energy should drop significantly. The experimental results (figure 7) show that this ratio does in fact peak near unity just before or at the drop in D_α, which signifies the entrance into the H-mode regime. Thus, just before an L–H transition the power transfer into the m, n = 0 E × B flow becomes, momentarily at least, strong enough to transfer the turbulence energy to the flow at a rate faster than it can be input from the mean gradients, resulting in an observed turbulence collapse that is consistent with model expectations. Since cross field transport is caused by the turbulence, this collapse then leads to the build-up of the edge gradients and associated mean shear flow. A detailed discussion of these observations and a comparison with predator–prey model results is available in the literature [13].

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