  The temperature and field dependence of the effective magnetic penetration
depth in the vortex state of a d-wave superconductor, as measured by muon spin
rotation experiments, is calculated using a nonlocal London model. We show that
at temperatures below T^* \propto \sqrt{B}, the linear T-dependence of the
effective penetration depth crosses over to a T^3-dependence. This could
provide an explanation for the low temperature flattening of the effective
penetration depth curves observed in a recent muon spin rotation experiment.
