There is an urgent demand in studying efficient methodologies to enable hybrid multi-and many-core accelerations in global climate simulations. The Model for Prediction Across Scales (MPAS) is a family of earth-system component models that receives increasingly more attention. Like many other models, MPAS, though features some emerging numerical algorithms, employs a pure MPI approach for parallel computing, which, to date, is in lack of support for multi-threaded parallelism, especially on many-core accelerated systems. In this work, we extend the shallow-water model in MPAS to demonstrate a pattern-driven approach for hybrid multi-and many-core accelerations of climate models. We first identify all basic computation patterns through a rigorous analysis of the MPAS code. Then for the whole model, we use the identified patterns as building blocks to draw a data-flow diagram, which serves as a perfect indicator to recognize data dependencies and exploit inherent parallelism. And finally, based on the data-flow diagram, a hybrid algorithm is designed to support concurrent computations done on both multi-core CPUs and many-core accelerators. We implement the algorithm and optimize it on an x86-based heterogeneous supercomputer equipped with both Intel Xeon CPUs and Intel Xeon Phi devices. Experiments show that our hybrid design is able to deliver an 8.35x speedup as compared to the original code and scales up to 64 processes with a nearly ideal parallel efficiency.