Turbulent, turbulent-like and complex systems are ubiquitous in the geosciences but there is a gap between classical models and reality. Advances in nonlinear dynamics, especially multifractal cascade models, allow us to close the gap, to investigate the weather, climate, earthquakes, geomorphology at unprecedented levels of accuracy, comparing theories, models and experiments over huge ranges of space-time scales.
Using new stochastic modelling and data analysis techniques, this short course provides an overview of the nonclassical, multifractal statistics. For example, by generalizing the classical turbulence laws, it is possible to obtain emergent laws of atmospheric dynamics and in other geosystems. These higher level laws are empirically validated from weather to lower-frequency macroweather scales to even lower frequency climate scales, and length-scales of millimetres to the size of the planet. By generalizing the notion of scale, complexity is reduced to a manageable scale-invariant hierarchy of processes, thus providing a new perspective for modelling and understanding. This new synthesis of state-of-the-art data and nonlinear dynamics is systematically compared with other analyses and model outputs. Applications of the theory are graphically demonstrated with multifractal simulations and examples are given of the application of freely available analysis and simulation software.