 Unbounded real over a model

A function $f\in {\omega }^{\omega }$ is unbounded over a model $M$ if for every $g\in M\cap {\omega }^{\omega }$, infinitely many $k$ satisfy $g(k)<f(k)$. Equivalently, no ground-model function eventually dominates it. This is weaker than being a dominating real over $M$.