  The secondary circulation of the tropical cyclone (TC) is related to its
formation and intensification, thus becomes very important in the studies. The
analytical solutions have both the primary and secondary circulation in a
three-dimensionally nonhydrostatic and adiabatic model. We prove that there are
three intrinsic radiuses for the axisymmetrically ideal incompressible flow.
The first one is the radius of maximum primary circular velocity $r_m$. The
second one is radius of the primary kernel $r_k>r_m$, across which the
vorticity of the primary circulation changes sign and the vertical velocity
changes direction. The last one is the radius of the maximum primary vorticity
$r_d$, at which the vertical flow of the secondary circulation approaches its
maximum, and across which the radius velocity changes sign. The first TC-like
vortex solution has universal inflow or outflow. The relations between the
intrinsic length scales are $r_k=\sqrt{2}r_m$ and $r_d=2r_m$. The second one is
a multi-planar solution, periodically in $z$-coordinate. Within each layer, the
solution is a convection vortex. The number of the secondary circulation might
be one, two, three, and even more. There are also three intrinsic radiuses
$r_m$, $r_k$ and $r_d$, but they have different values. It seems that the
relative stronger radius velocity could be easily found near boundaries. The
above solutions can be applied to study the radial structure of the tornados,
TCs and mesoscale eddies.
