  Let R be a Henselian discrete valuation ring with field of fractions K. If X
is a smooth variety over K and G a torus over K, then we consider X-torsors
under G. If XX/R is a model of X then, using a result of Brahm, we show that
X-torsors under G extend to XX-torsors under a N\'eron model of G if G is split
by a tamely ramified extension of K. It follows that the evaluation map
associated to such a torsor factors through reduction to the special fibre. In
this way we can use the geometry of the special fibre to study the arithmetic
of X.
