Introduction to Climate Modelling

These Lecture Notes form the basis of a one-semester course taught at the Physics Institute and the Oeschger Centre of Climate Change Research of the University of Bern. The main goals of this course are: 1. To introduce the students to the physical basis and the mathematical description of the different components of the climate system. 2. To provide the students with a first approach to the numerical solution of ordinary and partial differential equations using examples from climate modelling.

There is no best climate model! Different models have different advantages which may be due to their complexity or the form of their implemented parameterisations. Table 2.1 gives an (incomplete) overview of the hierarchy of models used for climate simulations. They are ordered according to their spatial dimensions. Only model types are listed but each type may be formulated in different ways. For instance different resolutions are used, different grid structures, parameters and parameterisations are chosen in a different way, etc. There are, for example, more than a dozen different ocean circulation models, all of which basically solve the same conservation equations. For model development and progress the various Modelling Intercomparison Projects provide important insight: AMIP (Atmospheric Modelling Intercomparison Project), OMIP (Ocean …), OCMIP (Ocean Carbon-cycle …), CMIP (Coupled …), PMIP (Paleo …), C⁴MIP (Coupled Climate-Carbon Cycle Modelling Intercomparison Project), etc.

In nature the transport of energy and matter in fluids is determined by diffusion and advection. These processes induce fluxes of energy and matter, of which the mathematical description is derived by continuum mechanics. Diffusion is a random process taking place at all times and leading to a net transport only under certain conditions. Advection is caused by an ambient flow which transports energy and matter.

In the annual mean, the Earth takes up energy between 30^∘ S and 30^∘ N, while it has a negative energy balance towards the poles (Fig. 4.1). Since neither a continuous warming in the lower latitudes nor a cooling in the high latitudes are observed, a strong poleward transport of energy is required. The integration of the meridional radiation balance from the South Pole to the North Pole, as it is given in Fig. 4.1, yields the heat transport, required by the radiation balance (Fig. 4.2). In each hemisphere, about \(5 \cdot 1{0}^{15}{\textrm{ J\,s}}^{-1} = 5\textrm{ PW}\) (Petawatt) are transported polewards. This flux is split about evenly between ocean and atmosphere. The maximum heat transport in the northern hemisphere occurs around 45^∘ N in the atmosphere and around 20^∘ N in the ocean. This fact points to the different mechanisms and boundary conditions (continents) responsible for the meridional heat transport. The atmosphere transports heat in a way fundamentally different from that of the ocean. The most important mechanisms are briefly explained in the following sections.

The energy balance models by Sellers (1969) and Budyko (1969) result in a linear partial differential equation of 1st order in time and 2nd order in space, (4.9).

Every fluid parcel in the atmosphere and the ocean obeys the fundamental laws of fluid mechanics including the equation of motion and the continuity equation. In the following we will describe approximate forms of these two equations for large-scale circulations in the ocean. Analogous equations apply for large-scale circulations in the atmosphere, too. As a preparatory step, we consider a special time derivative.

In this chapter the general circulation in the atmosphere is presented in a simplified form. A comprehensive description of the dynamics of the atmosphere can be found in Holton (2004).

The most detailed information about past climate states of the last 800,000 years can be retrieved from polar ice cores (Jouzel et al. 2007). One example for the last 90,000 years is presented in Fig. 9.1. The Holocene, the present interglacial, has started after the abrupt end of the last glacial period, 11,650 years ago. The transition from the last ice age to the Holocene, called Termination I, started about 20,000 years ago. An increase in the concentrations of particular isotopes could be detected in Antarctic ice cores. Stable isotopes of the water molecule are a measure for the local temperature. The temperature indicators also show that the climate changed in an abrupt way 25 times in Greenland during the last glacial period. These abrupt warming events, numbered in Fig. 9.1, are now referred to as Dansgaard–Oeschger events (D/O events) in remembrance of the research of the two pioneers in ice core science Willy Dansgaard (1922–2011) and Hans Oeschger (1927–1998) from the University of Copenhagen and the University of Bern.