Know Your Rank!

We study definable ranks of ordered fields, ordered abelian groups, and linear orders. For an arbitrary linear order $\mathrm{\Gamma }$, we construct an ordered abelian group $G$ with archimedian spine $\mathrm{\Gamma }$ and an ordered field $K$ with natural value group $G$ such that the definable ranks of $K$, $G$ and $\mathrm{\Gamma }$ are all isomorphic. This answers a question of Krapp, Kuhlmann, and the second author.