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This is the first of two papers about collisionless, electrostaticmicro-instabilities in stellarators, with an emphasis on trapped-particlemodes. It is found that, in so-called maximum-$J$ configurations,trapped-particle instabilities are absent in large regions of parameter space.Quasi-isodynamic stellarators have this property (approximately), and thetheory predicts that trapped electrons are stabilizing to all eigenmodes withfrequencies below the electron bounce frequency. The physical reason is thatthe bounce-averaged curvature is favorable for all orbits, and that trappedelectrons precess in the direction opposite to that in which drift wavespropagate, thus precluding wave-particle resonance. These considerations onlydepend on the electrostatic energy balance, and are independent of allgeometric properties of the magnetic field other than the maximum-$J$condition. However, if the aspect ratio is large and the instability phasevelocity differs greatly from the electron and ion thermal speeds, it ispossible to derive a variational form for the frequency showing that stabilityprevails in a yet larger part of parameter space than what follows from theenergy argument. Collisionless trapped-electron modes should therefore be morestable in quasi-isodynamic stellarators than in tokamaks.

Collisionless microinstabilities in stellarators I - analytical theory of trapped-particle modes