Where $\underline{a}$ is a Turing degree and $\xi$ is an ordinal $< (\aleph_1)^{L^\underline{a}}$, the result of performing $\xi$ jumps on $\underline{a},\underline{a}^{(\xi)}$, is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.