Size convergence of the $E × B$ staircase pattern in flux tube simulations of ion temperature gradient-driven turbulence

Ion temperature gradient-driven turbulence close to marginal stability exhibits zonal flow pattern formation on mesoscales, so-called $E × B$ staircase structures.¹ Such pattern formation has been observed in local gradient-driven flux-tube simulations,^2,3 including collisions⁴ and background $E × B$ shear,³ local flux-driven realizations including mean electric field shear,⁵ as well as global gradient-driven^6–8 and global flux-driven^1,9–12 studies. In global studies, spanning a larger fraction of the minor radius, multiple radial repetitions of staircase structures are usually observed, with a typical pattern size of several ten Larmor radii. By contrast, in the aforementioned local studies, the radial size of $E × B$ staircase structures is always found to converge to the radial box size of the flux tube domain. The above observations lead to the question: Does the basic pattern size always converges to the box size, or is there a typical mesoscale size inherent to staircase structures also in a local flux-tube description? The latter case would imply that it is not necessarily global physics, i.e., profile effects, that set (i) the radial size of the $E × B$ staircase pattern and (ii) the scale of avalanche-like transport events. These transport events are usually restricted to $E × B$ staircase structures and considered as a nonlocal transport mechanism.¹ In this brief communication, the above question is addressed through a box size convergence scan of the same cases close to the nonlinear threshold for turbulence generation as studied in Ref. 2.

The gyrokinetic simulations are performed with the non-linear flux tube version of Gyrokinetic Workshop (GKW)¹³ with adiabatic electron approximation. In agreement with Ref. 2, Cyclone Base Case (CBC) like parameters are chosen with an inverse background temperature gradient length $R / L T = 6.0$ and circular concentric flux surfaces. The numerical resolution is compliant to the “Standard resolution with 6th order (S6)” setup of the aforementioned reference, with a somewhat lowered number of parallel velocity grid points. It has been carefully verified that this modification preserves the same physical outcome as the original study. A summary of the numerical parameters is given in Table I, and for more details about the definition of individual quantities, the reader is referred to Refs. 2 and 13.

In the following, the box size is increased relative to the standard box size $( L x , L y ) = ( 76.3 , 89.8 ) ρ$ in the radial and binormal directions. Here, x is the radial coordinate that labels the flux surfaces normalized by the thermal Larmor radius ρ, and y labels the field lines and is an approximate binormal coordinate. Together with the coordinate s, which parameterizes the length along the field lines and is referred to as the parallel coordinate, these quantities form the Hamada coordinates.¹⁴ The increased box sizes are indicated by the real parameter $N R$ for radial and $N B$ for the binormal direction with the nomenclature $N R × N B$ throughout this work. Note that the number of modes in the respective direction, i.e., N_x and N_m, respectively, is always adapted accordingly to retain a spatial resolution compliant to the standard resolution [Table I] and standard box size.

In the first test, the radial box size is increased while the binormal box size is kept fixed to the standard size. The scan covers the realizations $N R × N B ∈ [ 1 × 1 , 2 × 1 , 3 × 1 , 4 × 1 ]$⁠. Each realization exhibits an initial quasi-stationary turbulent phase and a second final² phase with almost suppressed turbulence [Fig. 1(a)]. The latter state is indicative for the presence of a fully developed staircase pattern as depicted in Fig. 2. This type of structure is characterized by intervals of almost constant shear with alternating sign satisfying the Waltz criterion $| ω E × B | ≈ γ$^18,21 [γ is the growth rate of the most unstable linear ion temperature gradient (ITG) driven Eigenmode], connected by steep flanks where $ω E × B$ crosses zero. The negative gradient of the perturbed perpendicular and parallel ion pressure (not shown) exhibit positive corrugations in regions with maximum $| ω E × B |$ and negative corrugations at zero crossings of $ω E × B$⁠. A radial force balance analysis suggests that the structures in $ω E × B$ as depicted in Fig. 2 are not a consequence of the pressure gradient corrugations as discussed elsewhere.²² Rather, the corrugations in the pressure gradient have to be interpreted as a consequence of the staircase structure in $ω E × B$ due to the stabilizing zonal flow–turbulence interaction. Figure 2(a) shows a striking repetition of the staircase structure, with the number of repetitions equal to $N R$⁠. Hence, the basic size of the pattern not only converges with increasing radial box size, but the converged radial size also turns out to at least roughly agree with the standard radial box size of Ref. 2. Due to the lack of a substantial turbulent drive in the final suppressed state, no further zonal flow evolution is observed [Fig. 1(b)] and one might critically ask whether the structures shown in Fig. 2 represent the real converged pattern in a statistical sense. Note that in the 3 × 1 case the initial quasi-stationary turbulent state extends up to a few $∼ 10 4 R / v th$⁠. During this period, the zonal flow mode with $n ZF = 3$⁠, i.e., the mode that dominates the staircase pattern in final suppressed phase, undergoes a long-term evolution with a typical timescale of several $∼ 10 3 R / v th$⁠. Hence, several of such cycles are covered by the initial turbulent phase, which is evident from the occurrence of phases with reduced amplitude around $t ≈ 8000 R / v th$ and $t ≈ 18 000 R / v th$⁠. It is the $n ZF = 4$ zonal flow mode, i.e., the next shorter radial scale mode, that dominates the shear spectrum $| ω ̂ E × B | n ZF$ in the latter two phases (not shown). This demonstrates a competition between the $n ZF = 3$ and $n ZF = 4$ modes. Most importantly, no secular growth of the $n ZF = 1$ (box scale) zonal flow mode is observed during the entire quasi-stationary turbulent phase [Fig. 1(b), dotted line]. The above discussion indicates that although the $n ZF = 3 , 4$ zonal modes compete, the pattern scale does not converge to the radial box scale but rather to a mesoscale of $∼ 57 – 76 ρ$ (i.e., $n ZF = 4 , 3$ in the 3 × 1 case).

Since the radially elongated simulation domain might inhibit the development of isotropic turbulent structures, in the second test, the radial and binormal box size is increased simultaneously. This scan covers the realizations $N R × N B ∈ [ 1 × 1 , 1.5 × 1.5 , 2 × 2 , 2.5 × 2.5 , 3 × 3 ]$⁠. Interestingly, suppression of the turbulence by the emergence of a fully developed staircase pattern almost always occurs after $∼ 1000 R / v th$ [Fig. 3], i.e., significantly faster compared to the 3 × 1 and 4 × 1 realizations. As shown in Fig. 2(b), this test also confirms the convergence of the staircase pattern size to a typical mesoscale that is distinct from the radial box size in the $N R > 1$ realizations.

By contrast to the radial box size scan, the 3 × 3 realization shows a stationary pattern with four repetitions of the fully developed staircase structure, i.e., a somewhat smaller pattern size. Whether this is related to a possible pattern size dependence on the binormal box size or to the competition between patterns with the two sizes $λ ∈ [ 57 , 76 ] ρ$ as observed in the first test is addressed in the next paragraph. The scale of structures developing in the $1.5 × 1.5$ and $2.5 × 2.5$ realizations (not included in Fig. 2 to preserve the clarity of this figure) also lie within the range given above. Note that two additional realizations of the 3 × 3 case with different initial conditions and otherwise identical parameters confirm structure formation on scales within the range given above.

In a third test, the binormal box size is varied with the radial box size fixed to $N R = 3$⁠. This test covers the realizations $N R × N B ∈ [ 3 × 1.5 , 3 × 2.5 , 3 × 3 , 3 × 5 ]$⁠. As in the isotropic scan, the turbulence subdued and a fully developed staircase pattern forms after $∼ 2000 R / v th$ [Fig. 4]. The convergence of staircase pattern can be seen in Fig. 2(c) and confirms again a size of a typical mesoscale. Figure 2(c) also confirms that, indeed, a competition between patterns with two sizes $λ ∈ [ 57 , 76 ] ρ$ causing the different results for 3 × 1 and 3 × 3. The zonal flow mode number varies between $n ZF = 3 , 4$ which can be seen in Fig. 2(c) in the $3 × 2.5$ realization. The staircase structure has a pattern between three and four repetitions which get represented in the second repetition with no significant plateau at positive shear. Instead, the pattern returns immediately after reaching the maximum shear (⁠ $+ γ$⁠) to the minimum shear (⁠ $− γ$⁠) of the third repetition in a steep flank. The Fourier analysis of this case yields no definitely basic mode rather two dominating modes with $n ZF = 3 , 4$ with a fraction of the maximum amplitude $| ω ̂ E × B | n ZF$ each (not shown).

In the final test, the inverse background temperature gradient length $R / L T$ is varied at fixed 3 × 3 box size. Since suppression of turbulence usually occurs at later times when approaching the finite heat flux threshold from below,² the analysis aims to lengthen the phase during which the zonal flow varies in time due to turbulent Reynolds stresses. This scan covers realizations with $R / L T ∈ [ 6.0 , 6.2 , 6.4 ]$⁠. In the case of $R / L T = 6.2$⁠, turbulence suppression is observed for $t > 11 000 R / v th$⁠, while stationary turbulence during the entire simulation time trace of $12 000 R / v th$ is found for $R / L T = 6.4$⁠. The finite heat flux threshold, hence, is $R / L T | finite = 6.3 ± 0.1$ in accordance with Ref. 2. Although the initial quasi-stationary turbulence in the former case is significantly longer compared to the $R / L T = 6.2$ realization discussed in the second test, a stationary pattern with basic zonal flow mode $n ZF = 3$ establishes. Again, the $n ZF = 1$ (box scale) zonal flow mode does not grow secularly during the entire turbulent phase. Also, this test confirms the statistical soundness of the converged pattern size of $∼ 57 – 76 ρ$⁠.

Through careful tests this brief communication confirms the radial size convergence of the $E × B$ staircase pattern in local gyrokinetic flux tube simulations of ion temperature gradient (ITG)-driven turbulence. A mesoscale pattern size of $∼ 57 – 76 ρ$ is found to be intrinsic to ITG-driven turbulence for Cyclone Base Case parameters. This length scale is somewhat larger compared to results from global studies with finite $ρ ∗$⁠, which report of a few $10 ρ$⁠,¹ and has to be considered the proper mesoscale in the local limit $ρ ∗ → 0$⁠. The occurrence of this mesoscale implies that non-locality, in terms of Ref. 1, is inherent to ITG-driven turbulence, since avalanches are spatially organized by the $E × B$ staircase pattern.^1,2,6,15