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The microinstabilities that cause the confinement-limiting turbulence in modern fusion devices are driven by the radial gradients of density and temperature in the plasma. In multi‑scale gyrokinetic theory [1], these gradients act as a source of background free energy that feeds instabilities on the fluctuating micro scales, giving rise to turbulence, and ultimately, energy transport down the plasma gradients. Linear instability analyses, in which the unstable linear eigenmodes of the gyrokinetic system are studied, are central to our understanding of this process. However, in realistic magnetic geometry, these analyses often involve expensive gyrokinetic simulations.
Recent work has revealed that there is a thermodynamic upper bound on the rate at which free energy can be extracted by instabilities [2]. This bound is computed by constructing optimal modes, distribution functions that maximise the energetic growth on the micro scales. Due to the low-dimensionality of the optimal mode equations, these bounds can be computed very efficiently.
However, because the bound is valid in any confining magnetic field, the optimal modes are independent of the details of the magnetic geometry, which are often central to determining the growth rate of linear instabilities [3].
With the aim of capturing more of this geometric dependence in the upper bound, we develop a theory of constrained optimal modes: distribution functions that maximise free energy growth subject to a set of constraints that are also obeyed by the linear instabilities. We consider the linear gyrofluid equations as constraints, which restrict the moments of the distribution function in the variational principle, retaining some of the geometric dependence of the linear instabilities. The result is a system of gyrofluid equations to be solved for the upper bound on the linear growth rate, where the dimensionality of the system depends on the number of constraints considered; a tight bound is given in the limit of infinite constraints. We demonstrate, by comparison with gyrokinetic simulations and dispersion relations in simple limits, that the upper bounds capture some of the key geometry dependencies of the linear growth rate, even with a relatively small number of constraints. We then leverage this geometry dependence to perform a proof-of-principle optimisation of a quasi-isodynamic stellarator with reduced linear instability growth.
[1] I G Abel, G G Plunk, E Wang, M Barnes, S C Cowley, W Dorland, and A A Schekochihin. Rep. Prog. Phys., 76(11):
116201, November 2013. ISSN 0034-4885, 1361-6633. doi: 10.1088/0034-4885/76/11/116201.
[2] P. Helander and G. G. Plunk. Journal of
Plasma Physics, 88(2):905880207, April 2022. ISSN 0022-3778, 1469-7807. doi: 10.1017/S0022377822000277.
[3] L. Podavini, P. Helander, G. G. Plunk, and A. Zocco. Journal of Plasma Physics, 91(3):E79,
June 2025. ISSN 0022-3778, 1469-7807. doi: 10.1017/S0022377825000479.