Figure B1.1: Illustration of the budget equation.

Note that the inputs and outputs have units of the quantity X divided by time. Here X 1 corresponds to the quantity X at time t 1 and X 2 correspond to the quantity X at time t 2. In practice the differentials can be replaced by finite differences (Δ) such that we getįinite differences can be easily calculated: Δ X = X 2 – X 1 and Δ t = t 2 – t 1. The left-hand-side of this equation is the rate-of-change of X. Where the differentials ∂ denote an infinitesimally small change, t is time, I is the input, and O is the output. Mathematically a budget equation can be written as It can be restricted to a specific part or component of the climate system, e.g. It could be energy, water, carbon, or something else that obeys a conservation law. To start, we need a well-defined quantity of interest.

Similarly, the total amounts of water and carbon on earth are conserved although they may change forms or flow from one component to another one. It is the first law of thermodynamics and states that energy cannot be destroyed or created it can only change between different forms or flow from one object to another. In physics, for example, we have the law of energy conservation. Scientists like to keep track of things: energy, water, carbon, anything really because it allows them to exploit conservation laws.

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