  Current sheets thinner than the ion inertial length are unstable to the
tearing instability and will develop magnetic islands that grow due to magnetic
reconnection. We investigate whether the growth of magnetic islands in a
current sheet can continue indefinitely, or in the case of the heliosheath
until reaching a neighboring current sheet, and at what rate the islands grow.
We investigate the development and growth of magnetic islands using a
particle-in-cell code, starting from particle noise. Performing a scaling of
the growth of magnetic islands versus the system size, we find that the growth
rate is independent of the system size up to the largest simulation we were
able to complete. The islands are able to continue growing as long as they
merge with each other and maintain a high aspect ratio. Otherwise there is not
enough magnetic tension to sustain reconnection. When applied to the sectored
magnetic fields in the heliosheath, we show that the islands can continue
growing until they reach the sector width and do so in much less time than it
takes for the islands to convect through the heliosheath.
