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In general, the orbit-averaged radial magnetic drift of trapped particles instellarators is non-zero due to the three-dimensional nature of the magneticfield. Stellarators in which the orbit-averaged radial magnetic drift vanishesare called omnigeneous, and they exhibit neoclassical transport levelscomparable to those of axisymmetric tokamaks. However, the effect of deviationsfrom omnigeneity cannot be neglected in practice. For sufficiently lowcollision frequencies (below the values that define the $1/\nu$ regime), thecomponents of the drifts tangential to the flux surface become relevant. Thisarticle focuses on the study of such collisionality regimes in stellaratorsclose to omnigeneity when the gradient of the non-omnigeneous perturbation issmall. First, it is proven that closeness to omnigeneity is required topreserve radial locality in the drift-kinetic equation for collisionalitiesbelow the $1/\nu$ regime. Then, it is shown that neoclassical transport isdetermined by two layers in phase space. One of the layers corresponds to the$\sqrt{\nu}$ regime and the other to the superbanana-plateau regime. Theimportance of the superbanana-plateau layer for the calculation of thetangential electric field is emphasized, as well as the relevance of the latterfor neoclassical transport in the collisionality regimes considered in thispaper. In particular, the tangential electric field is essential for theemergence of a new subregime of superbanana-plateau transport when the radialelectric field is small. A formula for the ion energy flux that includes the$\sqrt{\nu}$ regime and the superbanana-plateau regime is given. The energyflux scales with the square of the size of the deviation from omnigeneity.Finally, it is explained why below a certain collisionality value theformulation presented in this article ceases to be valid.

The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity