  Discs with a rotation profile depending on radius and height are subject to
an axisymmetric linear instability, the vertical shear instability. Here we
show that non-axisymmetric perturbations, while eventually stabilized, can
sustain huge exponential amplifications with growth rate close to the
axisymmetric one. Transient growths are therefore to all effects genuine
instabilities. The ensuing angular momentum transport is positive. These
growths occur when the product of the radial times the vertical wavenumbers
(both evolving with time) is positive for a positive local vertical shear, or
negative for a negative local vertical shear.
  We studied, as well, the interaction of these vertical shear induced growths
with a convective instability. The asymptotic behaviour depends on the relative
strength of the axisymmetric vertical shear (s_v) and convective (s_c) growth
rates. For s_v > s_c we observed the same type of behaviour described above -
large growths occur with asymptotic stabilization. When s_c > s_v the system is
asymptotically unstable, with a growth rate which can be slightly enhanced with
respect to s_c. The most interesting feature is the sign of the angular
momentum transport. This is always positive in the phase in which the vertical
shear driven transients growths occur, even in the case s_c > s_v . Thermal
diffusion has a stabilizing influence on the convective instability, specially
for short wavelengths.
