Episode 03 -- Terminology in ROMS

prognostic equation—Any equation governing a system that contains a time derivative of a quantity and therefore can be used to determine the value of that quantity at a later time when the other terms in the equation are known (e.g., vorticity equation)

diagnostic equation—Any equation governing a system that contains no time derivative and therefore specifies a balance of quantities in space at a moment of time (e.g., hydrostatic equation, balance equation)
http://amsglossary.allenpress.com/glossary

diagnostic variable —A quantity that is derived for the new timestep after the prognostic variables have been calculated, for example vertical wind speed. Further quantities are then computed via parameterizations
http://www.answers.com/topic