## References

Al-Mohy, A. H. & Higham, N. J.
2010
A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Applics.
31 (3), 970–989.

Anderson, F., Almagri, A. F., Anderson, D. T., Peter, G., Talmadge, J. N. & Shohet, J. L.
1995
The helically symmetric experiment, (HSX), goals, design and status. Fusion Technol.
27 (3T), 273–277.

Bañón Navarro, A., Morel, P., Albrecht-Marc, M., Carati, D., Merz, F., Gorler, T. & Jenko, F.
2011
Free energy balance in gyrokinetic turbulence. Phys. Plasmas
18 (9), 092303.

Baumgaertel, J. A., Belli, E. A., Dorland, W., Guttenfelder, W., Hammett, G. W., Mikkelsen, D. R., Rewoldt, G., Tang, W. M. & Xanthopoulos, P.
2011
Simulating gyrokinetic microinstabilities in stellarator geometry with GS2. Phys. Plasmas
18, 122301.

Beer, M. A., Cowley, S. C. & Hammett, G. W.
1995
Field-aligned coordinates for nonlinear simulations of tokamak turbulence. Phys. Plasmas
2 (7), 2687–2700.

Bhattacharjee, A., Sedlak, J. E., Similon, P. L., Rosenbluth, M. N. & Ross, D. W.
1983
Drift waves in a straight stellarator. Phys. Fluids
26 (1983), 880–882.

Blackford, L. S., Choi, J., Cleary, A., D’Azeuedo, E., Demmel, J., Dhillon, I., Hammarling, S., Henry, G., Petitet, A., Stanley, K.
et al.
1997
ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics.

Boozer, A. H.
1981
Plasma equilibrium with rational magnetic surfaces. Phys. Fluids
24 (11), 1999–2003.

Boozer, A. H.
1998
What is a stellarator?
Phys. Plasmas
5 (5), 1647–1655.

Candy, J. & Waltz, R. E.
2003
An Eulerian gyrokinetic-Maxwell solver. J. Comput. Phys.
186 (2), 545–581.

Candy, J., Waltz, R. E. & Rosenbluth, M. N.
2004
Smoothness of turbulent transport across a minimum-q surface. Phys. Plasmas
11 (5), 1879–1890.

Canik, J. M., Anderson, D. T., Anderson, F. S. B., Likin, K. M., Talmadge, J. N. & Zhai, K.
2007
Experimental demonstration of improved neoclassical transport with quasihelical symmetry. Phys. Rev. Lett.
98, 085002.

Chen, H. & Chen, L.
2018
On drift wave instabilities excited by strong plasma gradients in toroidal plasmas. Phys. Plasmas
25, 014502.

Chen, Y., Parker, S. E., Wan, W. & Bravenec, R.
2013
Benchmarking gyrokinetic simulations in a toroidal flux-tube. Phys. Plasmas
20 (9), 1–8.

Connor, J. & Hastie, R.
2004
Microstability in tokamaks with low magnetic shear. Plasma Phys. Control. Fusion
46 (10), 1501–1535.

Connor, J. W., Hastie, R. J. & Taylor, J. B.
1978
Shear, periodicity, and plasma ballooning modes. Phys. Rev. Lett.
40 (6), 396–399.

Connor, J. W., Hastie, R. J. & Taylor, J. B.
1979
High mode number stability of an axisymmetric toroidal plasma. Proc. R. Soc. Lond. A
365 (1720), 1–17.

Cuthbert, P. & Dewar, R. L.
2000
Anderson-localized ballooning modes in general toroidal plasmas. Phys. Plasmas
7 (6), 2302–2305.

Dewar, R. & Glasser, A.
1983
Ballooning mode spectrum in general toroidal systems. Phys. Fluids
26 (10), 3038–3052.

Dimits, A. M., Williams, T. J., Byers, J. A. & Cohen, B. I.
1996
Scalings of ion-temperature-gradient-driven anomalous transport in tokamaks. Phys. Rev. Lett.
77 (1), 71–74.

Dinklage, A., Beidler, C. D., Helander, P., Fuchert, G., Maaßberg, H., Rahbarnia, K., Sunn Pedersen, T., Turkin, Y., Wolf, R. C., Alonso, A.
et al. & the W7-X Team
2018
Magnetic configuration effects on the Wendelstein 7-X stellarator. Nat. Phys
14 (8), 855–860.

Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N.
2000
Electron temperature gradient turbulence. Phys. Rev. Lett.
85 (26), 5579–5582.

Eiermann, M. & Ernst, O. G.
2006
A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal.
44 (6), 2481–2504.

Faber, B. J., Pueschel, M. J., Proll, J. H. E., Xanthopoulos, P., Terry, P. W., Hegna, C. C., Weir, G. M., Likin, K. M. & Talmadge, J. N.
2015
Gyrokinetic studies of trapped electron mode turbulence in the helically symmetric experiment stellarator. Phys. Plasmas
22 (7), 072305.

Friedman, B. & Carter, T. A.
2014
Linear technique to understand non-normal turbulence applied to a magnetized plasma. Phys. Rev. Lett.
113, 025003.

Friedman, B., Carter, T. A., Umansky, M. V., Schaffner, D. & Joseph, I.
2013
Nonlinear instability in simulations of large plasma device turbulence. Phys. Plasmas
20, 055704.

Fulton, D. P., Lin, Z., Holod, I. & Xiao, Y.
2014
Microturbulence in DIII-D tokamak pedestal. I. Electrostatic instabilities. Phys. Plasmas
21, 042110.

Gates, D. A., Anderson, D., Anderson, S., Zarnstorff, M., Spong, D. A., Weitzner, H., Neilson, G. H., Ruzic, D., Andruczyk, D., Harris, J. H.
et al.
2018
Stellarator research opportunities: a report of the national stellarator coordinating committee. J. Fusion Energy
37 (1), 51–94.

Goldhirsch, I., Orszag, S. A. & Maulik, B. K.
1987
An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. J. Sci. Comput.
2 (1), 33–58.

Hatch, D. R., Jenko, F., Bañón Navarro, A., Bratanov, V., Terry, P. W. & Pueschel, M. J.
2016a
Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra. New J. Phys.
18, 075018.

Hatch, D. R., Kotschenreuther, M., Mahajan, S., Valanju, P., Jenko, F., Told, D., Gorler, T. & Saarelma, S.
2016b
Microtearing turbulence limiting the JET-ILW pedestal. Nucl. Fusion
56, 104003.

Hatch, D. R., Terry, P. W., Jenko, F., Merz, F. & Nevins, W. M.
2011a
Saturation of gyrokinetic turbulence through damped eigenmodes. Phys. Rev. Lett.
106, 115003.

Hatch, D. R., Terry, P. W., Jenko, F., Merz, F., Pueschel, M. J., Nevins, W. M. & Wang, E.
2011b
Role of subdominant stable modes in plasma microturbulence. Phys. Plasmas
18, 055706.

Hegna, C. C. & Hudson, S. R.
2001
Loss of second-ballooning stability in three-dimensional equilibria. Phys. Rev. Lett.
87 (3), 3–6.

Hegna, C. C., Terry, P. W. & Faber, B. J.
2018
Theory of ITG turbulent saturation in stellarators: identifying mechanisms to reduce turbulent transport. Phys. Plasmas
25, 022511.

Helander, P.
2014
Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys.
77, 087001.

Hernandez, V., Roman, J. E. & Vidal, V.
2005
SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw.
31 (3), 351–362.

Higham, N. J.
2008
Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics.

Jenko, F., Dorland, W., Kotschenreuther, M. & Rogers, B. N.
2000
Electron temperature gradient driven turbulence. Phys. Plasmas
7 (5), 1904–1910.

Martin, M. F., Landremann, M., Xanthopoulos, P., Mandell, N. R. & Dorland, W.
2018
The parallel boundary condition for turbulence simulations in low magnetic shear devices. Plasma Phys. Control. Fusion
60, 095008.

Meerbergen, K. & Sadkane, M.
1999
Using Krylov approximations to the matrix exponential operator in Davidson’s method. Appl. Numer. Maths
31 (3), 331–351.

Merz, F.2008 Gyrokinetic simulation of multimode plasma turbulence. PhD thesis.

Nadeem, M., Rafiq, T. & Persson, M.
2001
Local magnetic shear and drift waves in stellarators. Phys. Plasmas
8 (10), 4375–4385.

Nagaoka, K., Takahashi, H., Murakami, S., Nakano, H., Takeiri, Y., Tsuchiya, H., Osakabe, M., Ida, K., Yokoyama, M., Yoshinuma, M.
et al. & LHD Experimental Group
2015
Integrated discharge scenario for high-temperature helical plasma in LHD. Nucl. Fusion
55, 113020.

Pearlstein, L. D. & Berk, H. L.
1969
Universal eigenmode in a strongly sheared magnetic field. Phys. Rev. Lett.
23 (5), 220–222.

Peeters, A. G., Camenen, Y., Casson, F. J., Hornsby, W. A., Snodin, A. P., Strintzi, D. & Szepesi, G.
2009
The nonlinear gyro-kinetic flux tube code GKW. Comput. Phys. Commun.
180 (12), 2650–2672.

Plunk, G. G., Helander, P., Xanthopoulos, P. & Connor, J. W.
2014
Collisionless microinstabilities in stellarators. III. The ion-temperature-gradient mode. Phys. Plasmas
21, 032112.

Plunk, G. G., Xanthopoulos, P. & Helander, P.
2017
Distinct turbulence saturation regimes in stellarators. Phys. Rev. Lett.
118, 105002.

Proll, J. H. E., Xanthopoulos, P. & Helander, P.
2013
Collisionless microinstabilities in stellarators. II. Numerical simulations. Phys. Plasmas
20, 122506.

Pueschel, M. J., Faber, B. J., Citrin, J., Hegna, C. C., Terry, P. W. & Hatch, D. R.
2016
Stellarator turbulence: subdominant eigenmodes and quasilinear modeling. Phys. Rev. Lett.
116, 085001.

Sugama, H. & Watanabe, T. H.
2006
Collisionless damping of zonal flows in helical systems. Phys. Plasmas
13, 012501.

Terry, P. W., Baver, D. A. & Gupta, S.
2006
Role of stable eigenmodes in saturated local plasma turbulence. Phys. Plasmas
13, 022307.

Terry, P. W., Faber, B. J., Hegna, C. C., Mirnov, V. V., Pueschel, M. J. & Whelan, G. G.
2018
Saturation scalings of toroidal ion temperature gradient turbulence. Phys. Plasmas
25, 012308.

Waltz, R. E. & Boozer, A. H.
1993
Local shear in general magnetic stellarator geometry. Phys. Fluids B
5 (7), 2201–2205.

Whelan, G. G., Pueschel, M. J. & Terry, P. W.
2018
Nonlinear electromagnetic stabilization of plasma microturbulence. Phys. Rev. Lett.
120, 175002.

Xanthopoulos, P., Cooper, W., Jenko, F., Turkin, Y., Runov, A. & Geiger, J.
2009
A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations. Phys. Plasmas
16, 082303.

Xanthopoulos, P. & Jenko, F.
2007
Gyrokinetic analysis of linear microinstabilities for the stellarator Wendelstein 7-X. Phys. Plasmas
14, 042501.

Xanthopoulos, P., Merz, F., Görler, T. & Jenko, F.
2007
Nonlinear gyrokinetic simulations of ion-temperature-gradient turbulence for the optimized Wendelstein 7-X stellarator. Phys. Rev. Lett.
99, 035002.

Xanthopoulos, P., Plunk, G. G., Zocco, A. & Helander, P.
2016
Intrinsic turbulence stabilization in a stellarator. Phys. Rev. X
6, 021033.

Yoshimura, Y., Kubo, S., Shimozuma, T., Igami, H., Mutoh, T., Nakamura, Y., Ohkubo, K., Notake, T., Takita, Y., Kobayashi, S.
et al. & the LHD Experimental Group
2005
Achievement of one hour discharge with ECH on LHD. J. Phys.: Conf. Ser.
25, 189–197.

Zocco, A., Xanthopoulos, P., Doerk, H., Connor, J. W. & Helander, P.
2018
Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas. J. Plasma Phys.
84, 715840101.