We provide a consistent theory of the tachocline confinement (or anisotropic momentum transport) within an hydrodynamical turbulence model. The goal is to explain helioseismological data, which show that the solar tachocline thickness is at most 5\% of the solar radius, despite the fact that, due to radiative spreading, this transition layer should have thickened to a much more significant value during the sun's evolution. Starting from the first principle with the physically plausible assumption that turbulence is driven externally (e.g. by plumes penetrating from the convection zone), we derive turbulent (eddy) viscosity in the radial (vertical) and azimuthal (horizontal) directions by incorporating the crucial effects of shearing due to radial and latitudinal differential rotations in the tachocline. We show that the simultaneous presence of both shears induces effectively a much more efficient momentum transport in the horizontal plane than in the radial direction. In particular, in the case of strong radial turbulence (driven by overshooting plumes from the convection zone), the ratio of the radial to horizontal eddy viscosity is proportional to ${\cal A}^{-1/3}$, where ${\cal A}$ is the strength of the shear due to radial differential rotation. In comparison, in the case of horizontally driven turbulence, this ratio becomes of order $-\epsilon^2$, with negative radial eddy viscosity. Here, $\epsilon$ ($\ll 1$) is the ratio of the radial to latitudinal shear. The resulting anisotropy in momentum transport could thus be sufficiently strong to operate as a mechanism for the tachocline confinement against spreading.
Self-consistent theory of turbulent transport in the solar tachocline. II. Tachocline confinement