One class of important problems involves diffusion in a single spatial dimension, for example, height profiles of reactive constituents in a turbulently mixing atmosphere, profiles of concentration as a function of depth in the ocean or other body of water, diffusion and diagenesis within sediments, and calculation of temperatures as a function of depth or position in a variety of media. The one-dimensional diffusion problem typically yields a chain of interacting reservoirs that exchange the species of interest only with the immediately adjacent reservoirs. In the mathematical formulation of the problem, each differential equation is coupled only to adjacent differential equations and not to more distant ones. Substantial economies of computation can therefore be achieved, making it possible to deal with a larger number of reservoirs and corresponding differential equations. In this chapter I shall explain how to solve a one-dimensional diffusion problem efficiently, performing only the necessary calculations. The example I shall use is the calculation of the zonally averaged temperature of the surface of the Earth (that is, the temperature averaged over all longitudes as a function of latitude). I first present an energy balance climate model that calculates zonally averaged temperatures as a function of latitude in terms of the absorption of solar energy, which is a function of latitude, the emission of long-wave planetary radiation to space, which is a function of temperature, and the transport of heat from one latitude to another. This heat transport is represented as a diffusive process, dependent on the temperature gradient or the difference between temperatures in adjacent latitude bands. I use the energy balance climate model first to calculate annual average temperature as a function of latitude, comparing the calculated results with observed values and tuning the simulation by adjusting the diffusion parameter that describes the transport of energy between latitudes. I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands.
Regularization effects of a noise propagating through a chain of differential equations: An almost sharp result