  The derivation and numerical implementation of a linearized version of the
gyrokinetic (GK) Coulomb collision operator (Jorge R. et al., J. Plasma Phys.
85, 905850604 (2019)) and of the widely-used linearized GK Sugama collision
operator (Sugama H. et al., Phys. Plasmas 16, 112503 (2009)) is reported. An
approach based on a Hermite-Laguerre moment expansion of the perturbed
gyrocenter distribution function is used, referred to as gyro-moment expansion.
This approach allows considering arbitrary perpendicular wavenumber and
expressing the two linearized GK operators as a linear combination of
gyro-moments where the expansion coefficients are given by closed analytical
expressions that depend on the perpendicular wavenumber and on the temperature
and mass ratios of the colliding species. The drift-kinetic (DK) limits of the
GK linearized Coulomb and Sugama operators are also obtained. Comparisons
between the gyro-moment approach with the GK continuum code GENE are reported
focusing on the ion-temperature-gradient (ITG) instability and zonal flow (ZF)
damping, finding an excellent agreement. In particular, we demonstrate that the
GK linearized Sugama yields a stronger collisional damping of the ZF residual
compared to the GK linearized Coulomb. Finally, we show that the numerical
efficiency of the gyro-moment approach increases with collisionality, a desired
property for boundary plasma applications.
