Solutions to the bernoulli integral of the funnel flow

The most significant feature of a magnetized accretion system is perhaps the formation of a funnel or a curtain flow. In the standard model, which is axisymmetric and steady, the Bernoulli integral for the funnel can be obtained. The existence of a solution to the Bernoulli integral has largely been taken for granted, but no single consistent solution has ever been provided. All evidence indicates that a steady and axisymmetrical magnetically funneled accretion could be rather difficult. We explored the topology of the Bernoulli integral and have found solutions only when the funnel flow velocity is close to (poloidal) Alfvén velocity. Such solutions are associated with strong toroidal magnetic fields at the funnel base, in which the toroidal magnetic pressure causes magnetic levitation of materials. Interestingly, they are the same toroidal magnetic fields that carry away the angular momentum from the accreting matter. The angular velocity of the funnel base can then be larger than the stellar rotation rate, contrary to the common wisdom that the funnel must be in corotation with the star. Our results put stringent constrains on the disk truncation radius.