We solve the set of hydrodynamic equations for optically thin advection-dominated accretion flows by assuming a radially self-similar spherical coordinate system (r, q, f). The disk is considered to be in steady state and axisymmetric. We define the boundary conditions at the pole and the equator of the disk and, to avoid singularity at the rotation axis, the disk is taken to be symmetric with respect to this axis. Moreover, only the trf component of the viscous stress tensor is assumed, and we have set vq = 0. The main purpose of this study is to investigate the variation of dynamical quantities of the flow in the vertical direction by finding an analytical solution. As a consequence, we found that the advection parameter, f adv, varies along the θ direction and reaches its maximum near the rotation axis. Our results also show that, in terms of the no-outflow solution, thermal equilibrium still exists and consequently advection cooling can balance viscous heating.

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