Effects of plasma turbulence on the nonlinear evolution of magnetic island in tokamak
Minjun J. Choi Laszlo Bardoczi Jae-Min Kwon T. S. Hahm Hyeon K. Park Jayhyun Kim Minho Woo Byoung-Ho Park Gunsu S. Yun Eisung Yoon


1]Minjun J. Choi mjchoi@nfri.re.kr
2]Lászlo Bardóczi
1]Jae-Min Kwon
3]T. S. Hahm
4]Hyeon K. Park
1]Jayhyun Kim
1]Minho Woo
1]Byoung-Ho Park
5]Gunsu S. Yun
4]Eisung Yoon
[1]National Fusion Research Institute, Daejeon 34133, Korea, Republic of
[2]General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA
[3]Seoul National University, Seoul 08826, Korea, Republic of
[4]Ulsan National Institute of Science and Technology, Ulsan 44919, Korea, Republic of
[5]Pohang University of Science and Technology, Pohang, Gyungbuk 37673, Korea, Republic of
[ ]
[ ]mjchoi@nfri.re.kr

[
\begin@twocolumnfalse
Effects of plasma turbulence on the nonlinear evolution of magnetic island in tokamak
    [
    December 30, 2023
=====================================================================================


Magnetic islands (MIs), resulting from a magnetic field reconnection, are ubiquitous structures in magnetized plasmas.
In tokamak plasmas, recent researches suggested that the interaction between the MI and ambient turbulence can be important for the nonlinear MI evolution, but a lack of detailed experimental observations and analyses has prevented further understanding.
Here, we provide comprehensive two-dimensional observations that indicate various effects of the ambient turbulence on the nonlinear MI evolution.
It is shown that the modified plasma turbulence around the MI can lead to either destabilization or stabilization of the MI instability in tokamak plasmas. 
In particular, significantly enhanced turbulence at the X-point of the MI results in a violent disruption through the fast magnetic reconnection and magnetic field stochastization.



\end@twocolumnfalse
]

Magnetohydrodynamic instabilities accompanied by magnetic island (MI) are a serious concern in tokamak research since they destroy the nested structure of magnetic flux surfaces and lead to degradation of the plasma confinement.
MIs can grow spontaneously through the classical <cit.> and/or neoclassical <cit.> mechanisms in tokamak plasmas.
When MIs are driven by free energy from the gradient of the equilibrium plasma current, it is known as the classical tearing mode <cit.>.
On the other hand, the neoclassical mechanisms are related to perturbations of the non-inductive current such as the bootstrap current (J_BS).
The J_BS, which is driven by the pressure gradient, can be diminished inside the MI due to the pressure profile flattening.
It occurs when the transport along the radially reconnected field line dominates to equilibrate temperature across the MI.
The loss of J_BS can reinforce the initial deformation of the flux surfaces, accelerating the magnetic reconnection process.
This is called as the neoclassical tearing mode (NTM) <cit.>, driven dominantly by the J_BS loss over the classical mechanism. 
Observation of the NTM <cit.> demonstrates that the transport around the MI is important for the tearing mode stability in tokamak plasmas in addition to the other reconnection physics. 

Recent plasma experiments <cit.> and simulations <cit.> found that the MI affects the evolution of the ambient broadband fluctuations (or simply referred as `plasma turbulence' <cit.>). 
Plasma turbulence is significantly altered by the MI itself or a modification of the equilibrium profiles due to its presence.
In brief, it increases outside the MI and decreases inside the MI following the pressure gradient. 
This implies that the turbulent transport around the MI can be also modified, meaning that the evolution of the MI and the evolution of plasma turbulence are coupled.
However, there has been relatively little experimental research progress <cit.> in addressing the effects of the modified plasma turbulence on the MI evolution to date.

Here, we report on experimental observations explaining how the modified turbulence can affect the nonlinear MI evolution. 
Firstly, it is found that the increase of electron temperature (T_e) turbulence outside the MI can be limited and localized in the small region by the flow shear.
This could lead to the less turbulent transport into the MI from the inner (hotter) region, increasing the J_BS loss.
Combined with the dynamics of the flow shear development, it is shown that a positive feedback loop can exist for the NTM MI growth.
On the other hand, when the localized turbulence strength increases sufficiently to overcome the flow shear, the electron heat transport from the inner region into the MI is enhanced.
This leads to a beneficial T_e peaking inside the MI, which can decrease the MI width.
Finally, the further enhanced turbulence leads to a violent plasma disruption by expediting the magnetic reconnection and field line stochastization.
Compared to the case without the turbulence, a time scale of the turbulence-associated disruption is about one order smaller. 
These observations significantly extend our understanding of the nonlinear MI evolution in tokamak as well as provide general insights into the MI evolution physics in magnetized plasmas. 



§ INHOMOGENEOUS LOW-K T_E TURBULENCE


While turbulence around the MI has been limitedly observed in other experiments, the Korea Superconducting Tokamak Advanced Research (KSTAR) <cit.> experiment <cit.> clearly demonstrates that the increase of the low-k (kρ_i < 1 where k is the wavenumber and ρ_i is the ion Larmor radius) T_e turbulence can be localized in the inner region (r < r_s where r_s is the MI boundary) between the X-point and O-point poloidal angles (θ_X < θ < θ_O).
MIs in tokamak plasmas can grow either spontaneously through the TM <cit.> or NTM <cit.> instabilities, or be driven externally via the forced magnetic reconnection <cit.>. 
In the KSTAR experiment <cit.>, the m/n=2/1 MI was driven and held by the external n=1 magnetic field perturbation for an accurate measurement of small T_e fluctuations around the MI using the 2D local T_e diagnostics.
Here, m and n are the poloidal and toroidal mode number, respectively. 

The result of 2D measurements of the T_e fluctuation spectrum and flow shear in the inner region of the MI is summarized in figure <ref>.

Figures <ref>a and <ref>b show the normalized spectra (coherence) of the T_e fluctuation at θ_a = θ_O and θ_X < θ_b < θ_O, respectively.
Fluctuation components whose power exceeding the significance level (red dashed line) are not observed at the O-point angle, while the strong broadband fluctuations are observed in the region a little distant from the X-point. 
It means that the T_e turbulence does not increase in the inner region at the O-point angle where the temperature profile steepened most, even though its amplitude depends on the T_e gradient (see figure <ref>a).
This can be understood as an effect of the inhomogeneous flow shear around the MI. 
The strong flow shearing rate can suppress the turbulence growth <cit.>, and it was found that the flow shearing rate is strongly increasing towards the O-point angle in the inner region.
Figures <ref>c and <ref>d show radial two-point measurements of the local average dispersion relation (k_z(f)) at θ_c and θ_d, respectively, where θ_X < θ_d < θ_c < θ_O.
We can assume that the intrinsic phase velocity of turbulence is uniform in the small (Δ r ≪ r) measurement region.
Then, the radial variation of the laboratory frame phase velocity (v_L = 2 π f / k_z) is attributed to the radial variation of the background E × B flow (v_E× B). 
The radial shearing rate of the E × B flow (ω_E× B) can be obtained approximately using local two-point measurements of v_L, i.e. ω_E× B = Δ v_E × B / Δ r ≈Δ v_L / Δ r.

The measurements show that ω_E× B is strongly increasing towards the O-point angle (ω_E× B≈ 0.26 ± 0.15 × 10^5 [1/s] at θ_d and ω_E× B≈ 1.11 ± 0.48 × 10^5 [1/s] at θ_c) <cit.>.
This offers an explanation of the absence of the significant fluctuation in the inner region at the O-point angle since ω_E× B is expected to be larger than the typical auto-decorrelation rate (10^5 [1/s]) of tokamak plasma turbulence.

Though it is not shown in this paper, the flow shearing rate across the MI is also large enough (ω_E× B≥ 1.0 × 10^5 [1/s]) for preventing the increased turbulence near the X-point from spreading into the MI in this case <cit.>.
Note that these spatial patterns of the turbulence and the shear flow are in broad agreements with the results from the fluid and gyrokinetic simulations <cit.>.
In particular, reference <cit.> contains gyrokinetic simulation results validated against the experiments. 
In spite of some limitations, a main role of the increasing flow shear toward the O-point in suppressing the turbulence was confirmed.



§ A THRESHOLD BEHAVIOR AND POSITIVE FEEDBACK LOOP


The strong flow shear, developing around the MI, would suppress turbulence and affect the turbulent transport across the MI. 
With the suppressed turbulence and the reduced heat influx from the inner (hotter) region, the resulting profile would be sharper across the MI boundary and flatter inside the MI than the profile with the stronger turbulence. 
The flatter profile inside the MI is expected to increase the J_BS loss and enhance the NTM growth rate.

It is worth noting that the flow shear development by the MI has shown a threshold behavior.
In the experiment, as the external magnetic field perturbation is slowly increasing, the flow shear (or the difference between local two-point measurements of v_L) starts to develop above a certain value of the external field as shown in figure <ref>a.
It also seems to increase with the external field beyond the threshold value.
Note that error bars of v_L represent the standard deviation of measurements for many k_zs in the broadband local dispersion relation. 
Though the resolution of T_e profile measurements shown in figure <ref>b is not sufficient to detect the change of the MI width, the MI width is expected to slightly increase with the strength of the perturbation.
The flow shear development with a large magnetic island and its increasing trend with the island width are also observed in other experimental <cit.> and simulation <cit.> studies.


Comparing T_e profiles before and after the flow shear development, a sharper T_e profile across the inner boundary is found with a lower T_e value inside the MI, which can be the result of the reduced heat transport from the inner region with the increased flow shear. 
In the low-β_θ KSTAR experiment, the MI was driven by the external field and the effect of the reduced transport on the MI evolution may not be critical.
However, for the NTM in the high-β_θ plasma, the reduced electron heat transport would play a significant role via the dominant J_BS loss (∝β_θ) term <cit.> since its effect can be amplified through a positive feedback loop as presented in figure <ref>. 

As an MI width exceeds a threshold value, the flow shear develops across the MI and increases with the MI width. 
It quenches the turbulent influx of heat and particle into the MI, which results in the sharp boundary and a flatter pressure profile inside the MI. 
The J_BS loss is further enhanced, and the growth of the MI width is accelerated. 

The threshold behavior and proposed feedback loop suggested in the previous paragraph should be considered for a more complete understanding of the NTM growth.
There would be a width or a range of width in which the shearing rate (ω_E × B(W)) of the flow exceeds the turbulence growth rate driven by the local pressure gradient (Δω(W)).
The MI with ω_E × B(W) > Δω(W) can grow further via the suggested feedback loop.
In the recent DIII-D <cit.> experiment <cit.>, the density turbulence modification due to the NTM MI is observed in the particular period of the MI growing phase. 
Previous understanding on the modulation of density turbulence around the magnetic island relied on the flattening of the pressure profile inside the large island <cit.>.
However, this observation is limited for the particular range of the island width in the growing phase.
The suggested feedback loop associated with ω_E × B(W) > Δω(W) condition for turbulence suppression provides a reasonable explanation for this observation.
Note that, however, different patterns of the turbulence and the flow were also observed in other experiments, and even a transition between a strong and weak ω_E × B states (low and high accessibility states <cit.>, respectively) is observed. 









§ STABILIZING EFFECT OF THE LOW-K TURBULENCE VIA TURBULENCE SPREADING


While turbulence can be suppressed by the increased flow shear near the O-point angle, it remains significant near the X-point due to the finite T_e gradient and the insufficient flow shear there.
The increased turbulence would be localized in the inner region when its amplitude is too small to overcome the flow shear across the MI <cit.>. 
In other words, this turbulence outside the MI can spread into the MI if its amplitude is sufficiently large <cit.>.
In the recent DIII-D <cit.> and HL-2A <cit.> experiments, the spreading of the density and temperature turbulence into the O-point of the MI are observed, respectively.   

Turbulence spreading <cit.> is expected to play an important role for the MI evolution <cit.>, since the accompanying heat or particle flux can change the pressure profile inside the MI.
This can result in the smaller MI saturation width and the stabilization of the NTM via the Ohmic current perturbation <cit.> and the recovery of the pressure gradient (and J_BS) <cit.>, respectively.
Detailed observation of the dynamics of turbulence spreading would be helpful to understand its effect on the MI evolution.

An experimental observation, which can be interpreted as an intermittent turbulence spreading event, is introduced below with detailed measurements of its dynamics.
The nonlinear electrostatic gyrokinetic simulation based on the KSTAR plasma equilibrium and profiles showed that the turbulence and heat can spread into the MI which was initially stable region <cit.>.


Figure <ref>a shows evolution of the T_e gradient and low-k turbulence amplitude in the inner region of the MI near the X-point as the external magnetic field perturbation increases. 
Both the gradient and the turbulence amplitude increase in time following the external field.
Initially, the increased turbulence is localized in the outside of the MI due to the strong flow shear across the MI <cit.>.
When the turbulence amplitude is larger than a certain critical value, intermittent heat transport events are observed repeatedly (indicated by a dashed circle in figure <ref>a). 
The time traces of local T_e at b in the inner region, at c inside the MI, and at d in the outer region are also shown in figures <ref>b, <ref>c, and <ref>d, respectively. 
Bold arrows in figures <ref>b and <ref>c indicate the enhanced heat transport from the inner region (where T_e decreases) into the MI (where T_e increases) during one event, and a dashed arrow in figure <ref>d indicates the rapid and global exhaust of the heat accumulated inside the MI.

Four images (#1–#4) in figure <ref>e show the local change of the 2D T_e during a single event. 
From #1 to #3, T_e increases inside the MI spontaneously. 
This increase can be a result of the intermittent leakage of the turbulent heat flux from the inner region through a path near the X-point (weak flow shear).
In other words, the heat transport can be enhanced by turbulence spreading, which can carry the heat flux into the MI and result in a modest peaking of T_e inside the MI.
Note that the interior magnetic topology of the MI is known to have a good perpendicular confinement characteristics <cit.> to sustain the peaking.
Figure <ref> shows how the low-k turbulence amplitude varies during the event in higher temporal resolution.
Most events are found to be correlated with the RMS amplitude (|δ T_e / ⟨ T_e⟩ |) of the low-k broadband (30–70 kHz) turbulence.
|δ T_e / ⟨ T_e⟩ | in the inner region (shown in figure <ref>b) seems to start to decrease with the event and |δ T_e / ⟨ T_e⟩ | inside the island (shown in figure <ref>d) is peaked with the event, which is consistent with a picture of turbulence spreading. 
The former is relatively clear in the 1st, 2nd, 4th, and 6th event, and the later is in the 2nd, 3rd, 4th, 5th, and 6th event.
The rapid and global exhaust of the accumulated heat inside the MI, which is observed for #3–#4, can be attributed to the global enhancement of the heat transport with the increased turbulence inside the MI. 



Although this observation is made with the externally driven MI, what is shown in figure <ref> would represent a general feature of turbulence spreading dynamics across the MI.
It implies two potentially beneficial effects of turbulence spreading on TM/NTM stability.
Firstly, the T_e peaking inside the MI can be beneficial for the MI saturation through its perturbation on the Ohmic current <cit.> as demonstrated in the auxiliary heating experiment <cit.>.

In case of the MI driven by the NTM, the enhanced turbulent transport into the MI would eventually lead to a saturation at the smaller width or a stabilization, since it can recover the J_BS loss <cit.>.
Indeed, a partial stabilization of the NTM with the pellet injection was observed in the recent DIII-D and KSTAR experiments <cit.> in which the spreading of density turbulence <cit.> was regarded as a key mechanism for the stabilization. 




§ ROLES OF THE LOW-K TURBULENCE IN THE FAST DISRUPTION


Finally, we report an observation during the disruption phase of the MI, demonstrating a crucial role of the increased turbulence in the magnetic reconnection physics.
MIs in tokamak plasmas often lead to the disruption when it grows large enough. 
There are different kinds of the MI-associated disruption characterized by different temporal and spatial scales <cit.>.
The explosive and massive disruption involving the fast magnetic reconnection is the most dangerous one.  
Interestingly, a violent minor disruption is observed along with the significantly enhanced turbulence at the X-point. 


Figure <ref>a shows the process of the fast MI disruption captured by the 2D T_e diagnostics, and time trace of T_e measured at b in the inner region is shown in figure <ref>b. 
The constant external field is used to induce the MI in this experiment. 
The 2D T_e images represent the local change of T_e against the reference time (t=0) which is just before the disruption. 
The image at t=1.2 ms shows a local T_e collapse (indicated by a blue color) in the inner region with a structure reminiscent of the original MI (whose boundaries are indicated by purple dashed lines in the reference image), indicating an inward expansion of the MI width.
The following images show that T_e collapses more globally, probably due to the global field line stochastization and reconnection.
Figures <ref>c and <ref>d are the images of the T_e coherence sum (c_sum=∑_f c(f)) for periods of C (quasi-steady state) and D (just before the disruption), respectively. 
They represent the local strength of the low-k broadband T_e turbulence in each period.
In the period C, the turbulence is localized in a distant region from the X-point as shown in figure <ref>c. 
In the period D, the turbulence strength increases and it expands poloidally and reaches to the X-point as shown in figure <ref>d. 
It is noteworthy that this low-k T_e turbulence is only observed for the fast disruption whose time scale is a few milliseconds.
Other similar MI-associated disruptions occur in a longer timescale (5–10 times) without notable low-k turbulence <cit.>. 

The proximity of the increased turbulence to the X-point during the fast disruption indicates a role of turbulence in the disruption processes, i.e. the magnetic reconnection and the field line stochastization.
It is generally known that anomalous dissipations by turbulence can enhance the magnetic reconnection <cit.> and realignment of global magnetic fields. 
On the other hand, a fast disruption driven by successive formation and coalescence of plasmoid-like structures <cit.> was also observed in the KSTAR experiment <cit.>.
However, the low-k fluctuations in T_e discussed in this paper was not observed in the plasmoid-like-structure-associated disruption. 

The plasmoid-like structure might be attributed to the density turbulence. 

In summary, this paper reports on experimental observations which indicate that the modified turbulence around an MI can either accelerate the MI growth via the reduced influx of heat into the MI, or lead to a saturation or stabilization of the MI via turbulence spreading in tokamak plasmas. 
Turbulence can also lead to a violent plasma disruption by expediting the magnetic reconnection and forming the stochastic region at the X-point. 




§ ACKNOWLEDGEMENTS


The authors would appreciate all the supports from the KSTAR team.
This research was supported by Korean Ministry of Science and ICT under NFRI R&D programs (NFRI-EN1901-10 and NFRI-EN1941-5), and also by National Research Foundation of Korea under NRF-2019M1A7A1A03088462.




§ AUTHOR CONTRIBUTIONS


M.J.C. developed the central idea of the research based on discussion with L.B., J.-M.K., T.S.H., H.K.P. and E.Y. about the previous experiment conducted by J.K.. 
M.J.C. gathered the data and carried out the further experiment with general guidance from J.K., M.W. and B.-H.P.. 
L.B., J.-M.K., T.S.H., H.K.P. and E.Y. contributed to the interpretation of the experimental results and the revision of the draft.
G.S.Y. contributed to the acquisition of the diagnostics data presented in this article. 




§ COMPETING INTERESTS


The authors declare no competing interests.




§ METHODS




 §.§ The externally driven magnetic island


In tokamak plasmas, MIs can be driven at the rational q surface by the external magnetic field perturbation which has a resonant component to that rational surface.
The externally driven MI is locked in the position by the external field, which allows an accurate measurement of dynamics of the MI and ambient turbulence.
For the externally driven MI experiment on KSTAR, the n=1 external magnetic field perturbation was used to drive the m/n=2/1 tearing mode MI at the q=2 flux surface.
The KSTAR plasma of the driven MI experiment has the major radius R=180 cm, the minor radius a∼ 50 cm, the toroidal field B_T=2.0–2.2 T, the plasma current I_p=600–700 kA, the Spitzer resistivity η∼ 1.2×10^-7 Ohm·m, and β_θ∼ 0.5 %.



 §.§ The 2D T_e diagnostics


Tokamak core plasma can be optically thick for the fundamental O-mode or the second harmonic X-mode of the electron cyclotron emission (ECE). 
The measured intensity of the optically thick ECE depends linearly on the local electron temperature by Kirchhoff's law and Rayleigh-Jeans law. 
On the other hand, the ECE frequency in tokamak plasma follows 1/R dependence of the toroidal field.
A heterodyne detector can measure the radial profile of the electron temperature by measuring the ECE intensity selectively in frequency space. 
The electron cyclotron emission imaging (ECEI) diagnostics was developed to measure the local 2D T_e in (R,z) space using a vertical array of heterodyne detectors. 
The ECEI diagnostics on the KSTAR tokamak has 24 heterodyne detectors in vertical direction and each detector has 8 radial channels, i.e. total 192 channels <cit.>. 
It can measure the local 2D T_e with a high spatial (Δ R ≈Δ z ≤ 2 cm) and temporal (Δ t =0.5–2 μs) resolution.
This diagnostics has been utilized to study various tokamak plasma phenomena from the magnetohydrodynamic instabilities to the low-k broadband turbulence <cit.>.
For example, measurements of the local 2D electron temperature have revealed the change of the magnetic field topology successfully in various magnetic reconnection events of magnetohydrodynamic instabilities <cit.>. 



§ DATA AVAILABILITY


Raw data were generated at the KSTAR facility. 
Derived data are available from the corresponding author upon request. 




§ CODE AVAILABILITY


The data analysis codes used for figures of this article are available via the GitHub repository <https://github.com/minjunJchoi/fluctana> <cit.>. 



ieeetr


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