###### A book in preparation

by D. Mazza and E. Canuto, former faculty, Politecnico di Torino, Turin, Italy

Credits to Wallpaperaccess

###### Introduction to first Chapter

Earth climate is the results of energy exchanges between the outer space and the Earth surface (atmosphere, ocean, land and ice caps) and within the different phases of the Earth surface (gas, fluid and solid). The simplest model of energy exchange (the energy balance model, EBM) neglects the different phases, assumes that the Earth surface is a spherical shell subdivided between spherical elements defined by latitude and longitude range and depth. Each element is associated to an internal energy proportional to the temperature (playing the role of state variable). The energy variation is the imbalance result between the thermal power exchanged at the top of the atmosphere with the outer space, of the power exchanged between the different elements (horizontal energy transport) and the interior of the Earth (a negligible contribution). The model complexity (the number of elements) can be reduced by only considering the meridional energy transport (in other terms, by considering only spherical zones). The model is known as *one-dimensional EBM*, and has the peculiarity that the each zone receives a different average solar radiation due to Earth inclination and consequently has a different temperature which allows to account for ice formation. The latter is a critical surface condition to account for the Earth’s albedo, namely the reflected solar radiation (the short wavelength radiation, SW) which may vary between 30 and more than 70 % of the received radiation. The net received radiation should account for the re-radiated power (the outgoing longwave radiation, OLR) depending on the surface temperature and the atmosphere’s absorbance of infrared radiation. The atmosphere absorbance in turn depends on the volume of greenhouse gases (water vapor, carbon dioxide, methane, …) in the atmosphere.

The chapter simplifies the previous model down to the simplest EBM, the *zero dimensional **model*, in which the Earth’s spherical shell is associated with a single energy and only energy exchanges with the outer space and the Earth’s interior take place. We prefer to refer this model as *single-body *EBM, since Earth and atmosphere are treated as a single thermodynamic body. This is a critical assumption which implies uniform vertical temperature profile along the shell depth. The assumption is acceptable for thin layers of ocean and land surface, but not for the atmosphere due to its large vertical temperature gradient (the lapse rate). Therefore, the single-body model will be derived starting from a *multilayer model* of the atmosphere, which is simplified to the key *two-body model *of atmosphere and Earth’s surface, capable of providing a careful account of the exchanged power. Because of the smaller atmosphere’s thermal energy compared to ocean and land, the *singular perturbation principle* will be employed to obtain the single-body model. This procedure is believed to be necessary and essential for providing a neat and formal picture of the input and output power terms entering the model first-order state equation.

The resulting equation is highly nonlinear mainly because of the complex temperature dependence of albedo and OLR expressions, but it is possible to formulate and compute the equilibrium points and their stability properties, which are highly dependent on albedo and OLR profiles. The most interesting case is when three equilibrium points occur, two asymptotically stable and the intermediate one unstable. The result suggests that the global Earth’s climate may behave like a flip-flop. When the net received thermal power diminishes below a critical value the equilibrium becomes unstable and the Earth’s surface tends to become iced (the snow ball Earth, but it is uncertain whether the Earth ever reached this condition). On the opposite, if the net received thermal power increases above another critical value, the equilibrium becomes unstable and the Earth’s surface tends to become ice free, like in the actual epoch.