  We propose a new ``helicity-pumping'' method for energizing coronal
equilibria that contain a magnetic flux rope (MFR) toward an eruption. We
achieve this in a sequence of magnetohydrodynamics relaxations of small
line-tied pulses of magnetic helicity, each of which is simulated by a suitable
rescaling of the current-carrying part of the field. The whole procedure is
``magnetogram-matching'' because it involves no changes to the normal component
of the field at the photospheric boundary. The method is illustrated by
applying it to an observed force-free configuration whose MFR is modeled with
our regularized Biot--Savart law method. We find that, in spite of the bipolar
character of the external field, the MFR eruption is sustained by two
reconnection processes. The first, which we refer to as breakthrough
reconnection, is analogous to breakout reconnection in quadrupolar
configurations. It occurs at a quasi-separator inside a current layer that
wraps around the erupting MFR and is caused by the photospheric line-tying
effect. The second process is the classical flare reconnection, which develops
at the second quasi-separator inside a vertical current layer that is formed
below the erupting MFR. Both reconnection processes work in tandem with the
magnetic forces of the unstable MFR to propel it through the overlying ambient
field, and their interplay may also be relevant for the thermal processes
occurring in the plasma of solar flares. The considered example suggests that
our method will be beneficial for both the modeling of observed eruptive events
and theoretical studies of eruptions in idealized magnetic configurations.
