The differential equations describing the global climate system represent the conservation of momentum
(differential equations for horizontal and vertical velocity of air movement in the atmosphere),
conservation of energy (differential equation for temperature), conservation of mass (an auxiliary
differential equation relating velocity gradients to mass) and a differential equation for water vapor
(also equations for water droplets and ice crystals). These equations describe changes from some initial
value to some future values at each point on a spatial grid. For instance, the temperature equation gives
the change at a particular location (e.g., Dallas, Texas) at successive time increments between the initial
time and some time in the future (e.g., the change between 6 AM this morning and 6 PM tomorrow). The
temperature equation looks deceptively simple but actually requires information about the winds speeds,
solar heating, cloud cover, turbulent mixing, surface evaporation, cooling by infrared radiation, and
numerous other processes, most of which are described by other equations (i.e., we say that the
differential equations are "coupled").
Initial conditions represent the collection of initial temperatures, windspeeds, pressures, and water vapor
(relative humidity) values at all grid points. In this analogy, the positions A, +/-B....+/-E
represent the collection of all initial values.
