Thickness diffusivity ($kappa$) according to the Gent and McWilliams parameterisation which accounts for eddy-driven advection in the ocean, is estimated using output from an eddy-resolving model of the Southern Ocean. A physically meaningful definition of rotational eddy fluxes leads almost everywhere to positive $kappa$. Zonally averaged near surface values of $kappa$ remain smaller than 200 m2/s poleward of the polar front, increases between 60--45¿S to about 600 m2/s and peak between 45--35¿ S at almost 3000 m2/s. $kappa$ stays high in the upper 500 m but decreases with depth and is essentially zero below 2500 m. In addition to the thickness diffusion ($kappa$) there is eddy-induced eastward (westward) advection of isopycnal thickness at the poleward (equatorward) flank of the ACC pointing toward strong anisotropic lateral mixing.