  Let $G$ be a classical linear algebraic group over an algebraically closed
field, and let $\mathfrak{n}$ denote the subset of nilpotent elements in its
Lie algebra. In this paper we study a partial order on the $G$-orbits in
$\mathfrak{n}$ given by taking limits along cocharacters of $G$. This gives
rise to the so-called accessibility order on the nilpotent orbits. Our main
results show that for general and special linear algebras, this new order
coincides with the usual dominance order on nilpotent orbits, but for
symplectic and orthogonal algebras this is not the case.
