Matter density distribution of general relativistic highly magnetized jets driven by black holes

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.