  We investigate the linear theory of the ion-temperature-gradient (ITG) mode,
with the goal of developing a general understanding that may be applied to
stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and
kinetic models that follow closely from existing literature are reviewed and
two new first-principle models are presented and compared with results from
direct numerical simulation. One model investigates the effect of regions of
strong localized shear, which are generic to stellarator equilibria. These
"shear spikes" are found to have a potentially significant stabilizing affect
on the mode; however, the effect is strongest at short wavelengths
perpendicular to the magnetic field, and it is found to be significant only for
the fastest growing modes in W7-X. A second model investigates the
long-wavelength limit for the case of negligible global magnetic shear. The
analytic calculation reveals that the effect of the curvature drive enters at
second order in the drift frequency, confirming conventional wisdom that the
ITG mode is slab-like at long wavelengths. Using flux tube simulations of a
zero-shear W7-X configuration, we observe a close relationship to an
axisymmetric configuration at a similar parameter point. It is concluded that
scale lengths of the equilibrium gradients constitute a good parameter space to
characterize the ITG mode. Thus, to optimize the magnetic geometry for ITG mode
stability, it may be fruitful to focus on local parameters, such as the
magnitude of bad curvature, connection length, and local shear at locations of
bad curvature (where the ITG mode amplitude peaks).

Plasma Physics