We show that the differential rotation profile of the solar convection zone, apart from inner and outer boundary layers, can be reproduced with great accu- racy if the isorotation contours correspond to characteristics of the thermal wind equation. This requires that there be a formal quantitative relationship involving the entropy and the angular velocity. Earlier work has suggested that this could arise from magnetohydrodynamic stability constraints; here we argue that purely hydrodynamical processes could also lead to such a result. Of special importance to the hydrodynamical solution is the fact that the thermal wind equation is insensitive to radial entropy gradients. This allows a much more general class of solutions to fit the solar isorotation contours, beyond just those in which the entropy itself must be a function of the angular velocity. In particular, for this expanded class, the thermal wind solution of the solar rotation profile remains valid even when large radial entropy gradients are present. A clear and explicit example of this class of solution appears to be present in published numerical simulations of the solar convective zone. Though hydrodynamical in character, the theory is not sensitive to the presence of weak magnetic fields. Thus, the identification of solar isorotation contours with the characteristics of the thermal wind equation appears to be robust, accommodating, but by no means requiring, magnetic field dynamics.