  The question is addressed whether stellar differentially rotating radiative
zones (like the solar tachocline) excite nonaxisymmetric r-modes which can be
observed. To this end the hydrodynamical stability of latitudinal differential
rotation is studied. The amount of rotational shear required for the
instability is estimated in dependence of the character of radial
stratification and the flow patterns excited by the instability are found. The
eigenvalue equations for the nonaxisymmetric disturbances are formulated in 3D
and then solved numerically. Radial displacements and entropy disturbances are
included. The equations contain the 2D approximation of strictly horizontal
displacements as a special limit. The critical magnitude of the latitudinal
differential rotation for onset of the instability is considerably reduced in
the 3D theory compared to the 2D approximation. The instability requires a
subadiabatic stratification. It does not exist in the bulk of convection zone
with almost adiabatic stratification but may switch on near its base in the
region of penetrative convection. Growth rates and symmetry types of the modes
are computed in dependence on the rotation law parameters. The S1 mode with its
transequatorial toroidal vortices is predicted as the dominating instability
mode. The vortices show longitudinal drift rates retrograde to the basic
rotation which are close to that of the observed weak r-mode signatures at the
solar surface.
