Bjerknes pointed out for the first time that the evolution of the atmosphere is governed by a complete set of seven equations with seven unknowns. If we know both the equations and the initial conditions with sufficient accuracy, we can predict the weather. In this chapter, we first introduce the continuous equations that govern the atmosphere. From these equations, we then discuss several fundamental wave oscillations existing in the atmosphere and their filtering approximations. For comparison, we then introduce the primitive equations for the oceans and discuss the Kelvin and Equatorially trapped waves, a special kind of waves that can appear in both the atmosphere and the ocean.

Bjerknes pointed out for the first time that the evolution of the atmosphere is governed by a complete set of seven equations with seven unknowns. If we know both the equations and the initial conditions with sufficient accuracy, we can predict the weather. In this chapter, we first introduce the continuous equations that govern the atmosphere. From these equations, we then discuss several fundamental wave oscillations existing in the atmosphere and their filtering approximations. For comparison, we then introduce the primitive equations for the oceans and discuss the Kelvin and Equatorially trapped waves, a special kind of waves that can appear in both the atmosphere and the ocean.

Subgrid-scale processes refer to the processes that are vital for describing atmospheric motion but cannot be explicitly resolved due to insufficient model resolution. Although these processes occur at small scales, they depend on and, in turn, affect the larger-scale fields and processes that are explicitly resolved by a numerical model. Due to this two-way interaction, neglecting those subgrid-scale processes will degrade the quality of the weather forecast. To reproduce this two-way interaction, the subgrid-scale processes are “parameterized” by formulating their effects in terms of the resolved fields. Using the prognostic equation for water vapor as an example, we illustrate the general principle of parameterization. We then outline the crucial processes parameterized in today’s numerical weather prediction models. To facilitate the understanding about how parameterizations are implemented in a weather model, a simplified general circulation model with simple parameterizations, SPEEDY, is introduced to the readers.

NWP is an initial/boundary value problem: given an estimate of the present state of the atmosphere (initial conditions) and appropriate boundary conditions, the model simulates (forecasts) the atmospheric evolution. More accurate estimates of initial conditions lead to better forecasts. Currently, operational NWP centers produce initial conditions through a statistical combination of observations and short-range forecasts that account for the uncertainty associated with each source of information. This approach has become known as “data assimilation.” In this chapter, we review early attempts at data assimilation and then introduce the statistical estimation methods that provide a solid foundation for data assimilation. Examples using toy models are provided to illustrate the principles of data assimilation. We then discuss in detail all state-of-the-art data assimilation methods adopted in operational centers, including optimal interpolation, 3D-Var, 4D-Var, ensemble Kalman filter, and hybrid methods. Specifically, we discuss several improvements for the ensemble Kalman filter that make it competitive with 4D-Var. We also discuss Ensemble Forecast Sensitivity to Observations (EFSO), a powerful tool that can estimate the impact of any observations on short-range forecasts, and then we discuss the proactive quality control (PQC) built upon EFSO. We also briefly introduce the non-Gaussian assimilation method particle filters.

The governing equations for the atmosphere belong to partial differential equations (PDEs). For PDEs, the behavior of the solution, proper initial and/or boundary conditions, and the numerical methods to find the equation solutions depend essentially on the type of PDEs. We focus on three prototypes of PDEs that are representative of describing atmospheric motion: studying the numerical discretization that allows the numerical integrations of these equations on a computer. We then cover some topics unique for numerical weather prediction (NWP), including setting the proper lateral boundary conditions for regional models, non-hydrostatic models, and the need to replace the spectral global models. Many promising NWP model emulators based on machine learning and artificial intelligence have popped up in recent years. We suggest some approaches to validate those NWP model emulators.

Lorenz discovered that the atmosphere, like any dynamical system with instabilities, has a finite limit of predictability. In this chapter, we briefly review the fundamental concepts of chaotic systems, from which we introduce the concepts of global and local Lyapunov vectors. We then discuss singular and bred vectors, two concepts closely related to Lyapunov vectors. To represent the uncertainties associated with each deterministic forecasting, ensemble forecasting methods are developed. We review early studies of ensemble forecasting and operational ensemble forecasting methods. The growth rate of errors and predictability of the atmosphere in different regions are discussed. We then discuss the role of different earth components in predictability for different time-scale phenomena, highlighting the dominant role of humans in the coupled Earth-Human system. We also give an outlook on the potential application of data assimilation to the coupled Earth-Human systems. While the chaotic nature of the atmosphere reveals the intrinsic difficulty of making accurate long-range predictions, it also indicates the possibility of deploying an initial small control that can grow large enough to alternate the future flow trajectory. Under this context, we introduce the Control Simulation Experiment (CSE), where we insert small perturbations into a chaotic system to let it evolve as we expect, essentially "controlling" the weather.