  We study an alternative model of infinitary term rewriting. Instead of a
metric on terms, a partial order on partial terms is employed to formalise
convergence of reductions. We consider both a weak and a strong notion of
convergence and show that the metric model of convergence coincides with the
partial order model restricted to total terms. Hence, partial order convergence
constitutes a conservative extension of metric convergence, which additionally
offers a fine-grained distinction between different levels of divergence. In
the second part, we focus our investigation on strong convergence of orthogonal
systems. The main result is that the gap between the metric model and the
partial order model can be bridged by extending the term rewriting system by
additional rules. These extensions are the well-known B\"ohm extensions. Based
on this result, we are able to establish that -- contrary to the metric setting
-- orthogonal systems are both infinitarily confluent and infinitarily
normalising in the partial order setting. The unique infinitary normal forms
that the partial order model admits are B\"ohm trees.
