Note on bunching of field lines in black hole magnetospheres

Numerical simulations of Blandford-Znajek energy extraction at high spin have revealed that field lines tend to bunch near the poles of the event horizon. We show that this behavior can be derived analytically from the assumption of fixed functional dependence of current and field line rotation on magnetic flux. The argument relies crucially on the existence of the Znajek condition, which offers nontrivial information about the fields on the horizon without requiring a full force-free solution. We also provide some new analytic expressions for the parabolic field configuration.