Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are non-uniform and structured. Refinement is controlled by physics-based indicators that capture relevant physical processes and determine the areas of mesh refinement and coarsening. The proposed performance metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time. Using a number of test cases we demonstrate that correlating different quantities offers insight into computational overhead, the distribution of numerical error across various mesh resolutions as well as the evolution of numerical error and run-time per degree of freedom.