first years of the 20th century, a Norwegian physicist named Vilhelm Bjerknes developed the first set of what today would call “primitive equations” describing the dynamics of the atmosphere. Bjerknes’ equations, adapted primarily from the then-novel study of fluid dynamics, tracked four atmospheric variables--temperature, pressure, density, and humidity (water content)--along with three spatial variables, so that the state of the atmosphere could be represented in a realistic three-dimensional way. Bjerknes, that is, defined the first rigorous state space for atmospheric physics.
However, the nonlinearity and general ugliness of Bjerknes’ equations made their application prohibitively difficult. The differential equations coupling the variables together were far too messy to admit of an analytic solution in any but the most simplified circumstances. Richardson’s forecast factory, while never actually employed at the scale he envisioned, did contain a key methodological innovation that made Bjerknes’ equations practically tractable again: the conversion of differential equations to difference equations. While Bjerknes’ atmospheric physics equations were differential--that is, described infinitesimal variations in quantities over infinitesimal time-steps--Richardson’s converted equations tracked the same quantities as they varied by finite amounts over finite time-steps. Translating differential equations into difference equations opens the door to the possibility of generating numerical approximation of answers to otherwise intractable calculus problems. In cases like Bjerknes’ where we have a set of differential equations for which it’s impossible to discern any closed-form analytic solutions, numerical approximation by way of difference equations can be a godsend: it allows us to transform calculus into repeated arithmetic. More importantly, it allows
- Edwards (2010), pp. 93-98