  Behrstock and Dru\c{t}u raised a question about the existence of Morse
geodesics in $CAT(0)$ spaces with divergence function strictly greater than
$r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than
$1$. In this paper, we answer the question of Behrstock and Dru\c{t}u by
showing that for each real number $s\geq 2$, there is a $CAT(0)$ space $X$ with
a proper and cocompact action of some finitely generated group such that $X$
contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.
