  Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster
tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$
is a $d$-torsion class if and only if it is closed under $d$-extensions and
$d$-quotients. This generalises an important result for classical torsion
classes. As an application, we prove that the $d$-torsion classes in
$\mathcal{M}$ form a complete lattice. Moreover, we use the characterisation to
classify the $d$-torsion classes associated to higher Auslander algebras of
type $\mathbb{A}$, and give an algorithm to compute them explicitly. The
classification is furthermore extended to the setup of higher Nakayama
algebras.
