The incorporation of increasing core counts in modern processors used to build state-of-the-art supercomputers is driving application development towards exploitation of thread parallelism, in addition to distributed memory parallelism, with the goal of delivering efficient high-performance codes. In this work we describe the exploitation of threading and our experiences with it with respect to a real-world ocean modeling application code, MPAS-Ocean. We present detailed performance analysis and comparisons of various approaches and configurations for threading on the Cray XC series supercomputers.