The fundamental computational challenge for climate and weather models is to efficiently and accurately resolve the vast range of space and time scales that characterize atmosphere and ocean flows. Not only do these scales span many orders of magnitude, the minimum dynamically active scale is also highly intermittent in both time and space. In this talk we introduce an innovative wavelet-based approach to dynamically adjust the local grid resolution to maintain a uniform specified error tolerance. The wavelet multiscale method is used to make dynamically adaptive the TRiSK model (Ringler et al. 2010) for the rotating shallow water equations on the sphere. We have carefully designed the inter-scale restriction and prolongation operators to retain the mimetic properties that are the main strength of this model. The wavelet method is computationally efficient and allows for straightforward parallelization using MPI. We will show verification results from the suite of smooth test cases proposed by Williamson (1991), and a more recent nonlinear test case suggested by Galewsky (2004): an unstable mid-latitude zonal jet. To investigate the ability of the method to handle boundary layers in ocean flows, we will also show an example of flow past an island using penalized boundary conditions. This adaptive "dynamical core" serves as the foundation on which to build a complete climate or weather model.