  High-resolution very long baseline interferometry (VLBI) radio observations
have resolved the detailed emission structures of active galactic nucleus jets.
General relativistic magnetohydrodynamic (GRMHD) simulations have improved the
understanding of jet production physics, although theoretical studies still
have difficulties in constraining the origin and distribution of jetted matter.
We construct a new steady, axisymmetric GRMHD jet model to obtain approximate
solutions of black hole (BH) magnetospheres, and examine the matter density
distribution of jets. By assuming fixed poloidal magnetic field shapes that
mimic force-free analytic solutions and GRMHD simulation results and assuming
constant poloidal velocity at the separation surface, which divides the inflow
and outflow, we numerically solve the force-balance between the field lines at
the separation surface and analytically solve the distributions of matter
velocity and density along the field lines. We find that the densities at the
separation surface in our parabolic field models roughly follow $\propto
r_{ss}^{-2}$ in the far zone from the BH, where $r_{ss}$ is the radius of the
separation surface. When the BH spin is larger or the velocity at the
separation surface is smaller, the density at the separation surface becomes
concentrated more near the jet edge. Our semi-analytic model, combined with
radiative transfer calculations, may help interpret the high-resolution VLBI
observations and understand the origin of jetted matter.
