This book gives a research exposition of interdisciplinary topics at the cutting edge of the applied mathematics of climate change and long range weather forecasting through a hierarchy of models with contemporary applications to grand challenges such as intraseasonal weather prediction. The developments include recent physics constrained low-order models, new analog prediction models, and equation free methods to capture intermittency and low frequency variabilities in massive datasets through Nonlinear Laplacian Spectral Analysis (NLSA) which combines delayed embeddings, causal constraints, and machine learning. Applications to grand challenges such as tropical intraseasonal variability of the Madden-Julian Oscillation (MJO) and the Monsoon as well as sea ice re-emergence in the Arctic on yearly time scales. A highlight is the exposition and pedagogical development of recent intermediate stochastic skeleton models to capture the main features of the MJO through PDE ideas, stochastics, and physical reasoning and compared with observational data. The mathematical theory of model error and the use of information theory combined with linear statistical response theory in a calibration stage are applied to improve long range forecasting and multi-scale data assimilation with concrete examples.
