A mathematical model is developed of the geophysical vortex flow using an order-of-magnitude analysis based on a laminar, steady axisymmetric vortex motion in a cylindrical frame of reference. A similarity method is adopted. The classical solution of Long (J. Fluid. Mech., 11, p. 611, 1961; Rossby number > 1) is reexamined. It is shown that true similarity solutions for the intermediate case of the Rossby number ∼ 1 do not exist, since this implies the physically impossible vortex flow wherein the fluxes of radial momentum and angular momentum are simultaneously zero. Numerical solutions are presented for our model using a shooting method with graphs depicting the variation of pressure and also axial, azimuthal, and radial velocities with non-dimensional radius parameter. The results are discussed with applications to tornado swirl and compared to the earlier studies by Long and Herbert.