  We analyze the non-equilibrium steady states (NESS) of a one dimensional
harmonic chain of $N$ atoms with alternating masses connected to heat
reservoirs at unequal temperatures. We find that the temperature profile
defined through the local kinetic energy $T(j) \equiv {<p_j^2>}/{m_j}$,
oscillates with period two in the bulk of the system. Depending on boundary
conditions, either the heavier or the lighter particles in the bulk are hotter.
We obtain exact expressions for the bulk temperature profile and steady state
current in the limit $N \rightarrow \infty$. These depend on whether $N$ is odd
or even. We also study similar temperature oscillations in the NESS of systems
with noise in the dynamics. These die out as $N \rightarrow \infty$.
