  A. Moriwaki proved the following arithmetic analogue of the Bogomolov
unstability theorem. If a torsion-free hermitian coherent sheaf on an
arithmetic surface has negative discriminant then it admits an arithmetically
destabilising subsheaf. In the geometric situation it is known that such a
subsheaf can be found subject to an additional numerical constraint and here we
prove the arithmetic analogue. We then apply this result to slightly simplify a
part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.
