Microtearing turbulence saturation via electron temperature flattening at low-order rational surfaces

Microtearing instability is one of the major sources of turbulent transport in high-$\beta$ tokamaks. These modes lead to very localized transport at low-order rational magnetic field lines, and we show that flattening of the local electron temperature gradient at these rational surfaces plays an important role in setting the saturated flux level in microtearing turbulence. This process depends crucially on the density of rational surfaces, and thus the system-size, and gives rise to a worse-than-gyro-Bohm transport scaling for system-sizes typical of existing tokamaks and simulations.

Confinement in tokamaks is enabled by magnetic field lines that trace out nested toroidal surfaces. On rational surfaces, the field lines connect back to themselves after integer numbers of poloidal and toroidal turns; certain electromagnetic plasma instabilities, such as the microtearing modes of interest here, are localized near these rational surfaces, and break the nested topology by forming magnetic islands [1][2][3]. Therefore, microtearing modes have a significant impact on confinement in high-β spherical tokamaks [4][5][6] and understanding microtearing transport is crucial for designing large spherical tokamak reactors such as STEP [7]. In general, gyro-Bohm [8] transport scaling may be applied when the turbulence is primarily electrostatic and allows extrapolation of predictions and observations to reactor-scale, but microtearing turbulence is less well-understood -for instance, first-principles gyrokinetic simulations often fail to saturate. We report a microtearing saturation mechanism that clarifies certain computational difficulties and provides insight into system-size scaling.
Microtearing modes are characterized by radiallynarrow parallel electron current layers that are driven resonantly at the rational surface and associated magnetic islands. While various branches of microtearing modes have been identified, including those driven by the time-dependent thermal force [9] or by curvature [10,11], and are present in various collisionality regimes [12][13][14][15], the electron temperature gradient remains a necessary condition for instability in all cases. Previous works have reported various microtearing saturation mechanisms: by background shear flow [5], zonal flows [16] or zonal fields [17]. However, despite these advances, predicting saturation levels remains a challenging task. We explore a microtearing turbulence saturation mechanism where the magnetic islands associated with the resonant current layers flatten the electron temperature gradient, thereby reducing the linear drive at the rational surfaces.
The radial width of the resonant region at the rational surfaces is generally of the order of few ion Larmor radii and is set by the parallel correlation length in linear theory which scales with the square root of the mass ratio between ions and electrons [18]. Nonlinearly, the flux associated with these modes, although slightly broadened, is still localized at the rational surfaces. This is already know to be important in setting global flux levels in the pedestal [19]. In the most extreme case, if turbulent diffusivity is sufficiently large and localized near low-order rational surfaces, the system will remove the local driving gradients, increase the gradients away from the low-order rational surfaces, and saturate in a zero-flux state. In this work, we find a less extreme version of this process occurring in a standard microtearing regime. We make a scaling argument to quantify this effect and suggest that future reactor-size devices subject to microtearing turbulence may perform worse that expected.
We proceed by demonstrating the strong electrontemperature-gradient flattening at low-order rationals in gyrokinetic simulations, and showing that this allows saturation by reducing mode drive. We test the impact of the various saturation mechanisms by suppressing zonal modulations and show the dominance of the temperature corrugations. Lastly, we consider system-size scaling and explain the origin of a non-gyro-Bohm scaling.
Simulation set-up.-Our numerical investigation uses Gene flux-tube gyrokinetic simulations [20] with a fieldaligned coordinate system [21] where x is the radial coordinate, y the binormal coordinate and z the parallel coordinate. The physical and numerical parameters used in this study are taken from Ref. [22]. Concentric circular flux-surface geometry [23] is considered with an inverse aspect ratio ǫ = 0.18, safety factor q 0 = 3, magnetic shearŝ = 1, mass ratio m i /m e = 1836, temperature ratio T i,0 /T e,0 = 1 and normalized pressure β = 0.4%. The inverse of the density, ion temperature and electron temperature background gradient scale lengths, normalized to the major radius R, are R/L n = 1, R/L Ti = 0 and R/L Te = 4.5, respectively. To model collisions, the linearised Landau operator is used with an electron-ion collision frequency ν ei /(v th,e /R) = 0.02, where v th,s = (T s,0 /m s ) 1/2 is the thermal velocity of species s.
The standard nonlinear simulation considered in this work, with a minimum binormal wavenumber of k y,min ρ i = 0.02 and run until normalized time tv th,i /R = 2000, saturates to give a gyro-Bohm normalized electron electromagnetic heat flux of Q e,em /Q GB = 7.9 (see Fig. 5). The corresponding k y spectrum of Q e,em peaks at k y ρ i = 0.04. T e flattening at low-order rational surfaces.-Modes at a specific toroidal mode number k y create magnetic islands around the resonantly driven current layers at their respective mode rational surfaces (MRSs). Note that the distance between MRSs for a given k y is 1/(ŝk y ). The MRS of all k y radially align at the lowest-order mode rational surface (LMRS), where the magnetic islands can persist even in the turbulent phase. For the standard nonlinear simulation, this can be seen at the LMRSs at x/ρ i = −50, 0 and 50 in the Poincaré plot in Fig. 1. The Poincaré plot records the positions where each magnetic field line crosses the outboard midplane on successive poloidal turns [24,25]. Each color denotes an individual field line. Away from the LMRSs, the MRSs of each k y are radially misaligned and the overlapping magnetic islands give rise to ergodic regions.
As the electrons move swiftly along the parallel direction following the perturbed magnetic field associated with the islands at the low-order MRSs, they also undergo periodic radial excursions. This leads to a shortcircuit of the perturbed T e profile, leading to its flattening. This can be seen in Fig. 2, where the green curve denoting the time-averaged effective temperature gradient ω eff Te is plotted as a function of the radial coordinate for the standard nonlinear simulation. ω eff Te is defined as the sum of the contributions from the background temperature gradient and the time-averaged zonal per-turbed temperature gradient, i.e. ω eff Te = R/L T e − ∂δT e /∂x yzt /(T 0,e /R). The flux-surface average, denoted by · yz , extracts the zonal contribution. This temperature flattening is a microturbulence analogue of profile flattening resulting from the large-scale neoclassical tearing modes (NTMs).
One can also understand the temperature flattening as a consequence of the turbulence self-interaction -a mechanism where modes that are significantly extended along the field line 'bite their tails' at the rational surfaces [26,27]. In the case of microtearing modes, the parallel electron heat current density q e, = v 3 δf e d 3 v which is extended along the field line, interacts with the A of the same eigenmode, to drive zonal parallel electron temperature perturbations δT e, yz , leading to its flattening at MRSs. Here, δf e is the perturbed electron distribution function and δT e, = (m e /n 0 ) v 2 δf e d 3 v.
Taking the v 2 moment and the flux-surface average of the gyrokinetic Vlasov equation, one arrives at an equation for the time evolution of the zonal δT e, . Considering only the electromagnetic (∝ A ) nonlinear term and ignoring the gyro-average over A , one obtains where the constant C = B 0 /|∇x × ∇y|. The linear structures ofq e, ,ky andÂ ,ky for k y ρ i = 0.04 are plotted with dashed lines in Fig. 3. The product of the two, proportional to a linear heat flux contribution, drives a zonal δT e, that leads to the flattening of the parallel electron temperature at each MRS. The same process repeats for the perpendicular electron temperature. However, note a significant broadening of the timeaveragedq e, ,ky in nonlinear simulation, also shown in    3 with a solid red line. A detailed description of this nonlinear broadening mechanism is given in Refs. [27,28] and can be summarized as follows. The radially narrow linear eigenmode structures lead to extended tails in k x −Fourier space and in ballooning representation (called 'giant tails' [29]). However, in a nonlinear simulation, only the first few linearly coupled k x -Fourier modes starting from k x = 0 of the eigenmode are able to retain their linear characteristics, i.e., their high amplitudes and relative phase differences with the k x = 0 mode, whereas the Fourier modes further away in the tail undergo a significant reduction in their amplitudes as a result of dominant nonlinear interactions, implying a broadening in real space. The width of the flattened electron temperature is therefore also broadened.
Microtearing stability with corrugated background gradients.-Now, we consider the linear stability of microtearing modes when the effective electron temperature gradient ω eff Te has local flattenings at LMRSs; the profiles are plotted in Fig. 4(a). This is equivalent to the tertiary instability analysis of zonal flows [30], except with a fixed temperature corrugation rather than a zonal flow pattern.
Since the resonant current drive leading to the microtearing instability is also localized at the MRS, we expect the growth rate of the k y ρ i = 0.02 modes considered in these tertiary instability simulations to be set mostly by the effective gradient ω eff Te,MRS at the MRS, i.e., the temperature gradient away from MRS is of little significance. This is verified in Fig. 4(b) by the close match between the growth rate obtained from the tertiary instability simulations plotted as a function of ω eff Te,MRS (magenta) and the growth rate obtained from standard linear simulations plotted as a function of R/L Te (blue). The figure also suggests that the time averaged ω eff Te,MRS in standard nonlinear simulation is set by the critical gradient of the instability. That is, the system may saturate by reducing the local drive of microtearing modes, almost stabilizing them in this case.
Removing zonal modulations.-To further investigate the role of electron temperature flattening on saturation, a nonlinear simulation is run while eliminating any local modifications to the temperature gradient. This is achieved by redefining the zonal component of the electron distribution function as δf e yz = δf e yz − K[v 2 − 1.5] f M yz . At each time-step, K is set such that δT e yz = 0, so that ω eff Te = R/L T e throughout the simulation. The heat flux Q e,em /Q GB = 28.1 in this simulation is many times higher than Q e,em /Q GB = 7.9 in the original standard nonlinear simulation, as shown in Fig. 5, confirming that electron temperature flattening indeed plays a significant role in saturation.
Another way to reduce the electron temperature flattening is by weakening the self-interaction process by increasing the parallel length L z = 2πN pol of the simulation volume [26,27,31], where N pol indicates the number of times the flux-tube wraps around poloidally before connecting back to itself. Increasing N pol weakens the temperature flattening, as shown in Fig. 2(b), and increases the flux level as shown in Fig. 5.
While these results confirm that the local flattening of electron temperature is crucial for correctly predicting the saturated turbulent state, the fact that these simulations, either with fully eliminated or weakened electron temperature flattenings, did saturate, indicates the presence of other saturation mechanism(s). Deleting the zonal electrostatic potential Φ or the zonal A in simulations changes the flux levels at most by 12%, implying that zonal flows and fields do not play a significant role in the saturation in the case considered. Although it is unknown at the moment what the other saturation mechanism(s) is, the free-energy analysis in Ref. [32] may provide a starting point.
Effect of system-size.-Given that microtearing turbulence saturation via temperature flattening happens pri-marily at LMRSs, the separation distance 1/(ŝk y,min ) between the LMRSs is crucial, and we thus scan k y,min = 2π/L y . Capturing the full toroidal domain requires k y,min ρ i = q 0 (a/r 0 )ρ ⋆ , where a is the minor radius and r 0 is the radial position of the flux-tube, and therefore scanning k y,min ρ i can also be interpreted as a scan in tokamak size measured by ρ ⋆ = ρ i /a [26,27]. For a typical MAST equilibrium [4] with q 0 = 1.35 at r 0 = 0.31m, k y,min ρ i ≃ 0.03.
As k y,min is decreased, the radial density of regions with flattened electron temperature (see Fig. 2(a)), and hence weaker linear drive, at low-order MRSs decreases. Concurrently, flux increases, as shown by the blue asterisks in Fig. 5. That is, the temperature flattening mechanism becomes less effective in large systems. For the k y,min ρ i = 0.02 case, the k y ρ i = 0.04 mode contributing most to the flux has six MRSs, three of which at LMRSs experience ∼ 70% flattening, and the other three at second-order MRSs experience ∼10% flattening. Whereas for k y,min ρ i = 0.04, the k y ρ i = 0.04 mode sees a ∼70% temperature flattening at every MRS, so mode stabilization is much more effective. When the electron temperature flattenings are eliminated, there is still some non-gyro-Bohm scaling (black markers in Fig. 5), but this is less consistent.
We suggest a crude model to understand the increase in flux with increasing system-size. In the turbulent steady state, when the electron heat flux Q e becomes radially constant, one defines the pointwise diffusivity via χ e ≡ Q e / (dT e /dx) and the radial average Boundary conditions impose zero average temperature fluctuation, thus Q e = dT e,0 /dx 1/χ e −1 x , where 1/χ e −1 x is the effective average diffusivity. Microtearing modes are modelled to lead to regions of high diffusivity near each MRS, which reinforce at LMRSs, resulting in the temperature-gradient corrugations seen in simulations.
As the harmonic mean of diffusivity sets flux levels, concentrating the diffusivity at widely-spaced MRSs (large k y,min , small system-size) leads to lower flux than distributing it more evenly at a larger number of closelyspaced MRSs (low k y,min , large system-size). In an extreme limit, microtearing creates infinite local diffusivity and completely flattens gradients near each MRS, but elsewhere the diffusivity is a small constant χ b . The effective average diffusivity, crudely assuming no overlap between flattened regions, is χ b /(1 − W N ), where W is the proportion of the radius flattened by each toroidal mode and N is the number of toroidal modes. This leads to a scaling Q e ∝ 1/(1 − w/ρ ⋆ ) with w a small parameter; note that the flux rises sharply at small ρ ⋆ . This is analogous to the avalanche transport arising when trans- Qe,em/QGB as a function of ky,minρi in simulations with N pol = 1 (blue asterisks) and N pol = 2 (violet squares). Black markers indicate simulations whose electron temperature flattenings are eliminated such that ω eff Te = R/LT e . Up-arrow indicates that the simulation saturates only transiently. port windows caused by fast-particle-driven modes overlap across much of the tokamak [33].
The width W of the high-transport region was assumed fixed in this simple picture, but actually may increase with higher flux, and the larger overlap may cause a runaway situation; this may be tied to failure to reach saturation in certain microtearing simulations.
Apart from system-size scaling, another possible consequence of electron temperature flattenings and magnetic islands at low-order rational surfaces is the potential to seed the growth of NTMs [34]. The possibility for microturbulence to excite NTMs via nonlinear coupling has been demonstrated in the past [35]. Furthermore, to experimentally verify our results, one may measure the electron temperature and look for flattenings near rational surfaces, similar to previous investigations of ITG turbulence [36]. One may also be able to measure low toroidal mode number magnetic perturbations associated with the microtearing islands in external magnetic coils; for instance, the radial magnetic perturbation associated with the islands is (δB x /B 0 )/(ρ i /R) ≃ 0.14 in the standard nonlinear simulation.
In conclusion, the fast motion of electrons across the magnetic islands at the LMRSs short-circuit the electron temperature, resulting in local electron temperature flattening, which then decreases the local linear drive of microtearing modes and allows lower saturated transport levels. The spacing and width of the low-confinement regions near low-order rationals are crucial, and this provides a pathway to understand microtearing saturation (or lack thereof); one direct consequence is that microtearing turbulent transport and its study are more important in larger future devices than previously thought.
We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support.