Implementing convection into Lorenz's global cycle Part I. Gridscale averaging of the energy equations

Sub-gridscale processes take place throughout the global atmosphere. Yet they have been neglected in traditional estimates of the global energy cycle on the ground that they can be treated as molecular heat fluxes. This view may cause quantitative underestimates of the efficiency of the global circulation of the atmosphere. In Part I of this two-part study we revisit the classical theory, beginning with the local energy equations. Similar to Lorenz we introduce a barotropic reference pressure p(r) and define a generalized field equation for the integrand of available potential energy, without reference to hydrostasy. The emerging energy quantity is new in that it comprises not only the classical correlation between efficiency factor and enthalpy but also an additional potential that depends upon p(r). We then perform mass-averaging over the scale of contemporaneous global models (40-400 km) and come up with averaged field energy equations, valid at the gridscale. Additional global and time-averaging of these removes all divergences and tendencies and yields two equations for the global energy reservoirs. The available potential energy reservoir is fed by gridscale plus sub-gridscale generation. The kinetic energy reservoir is tapped by gridscale plus sub-gridscale dissipation. Exchange between the reservoirs is carried by both gridscale and sub-gridscale conversion terms (C(grid), C(sub)). Generation, conversion and dissipation fluxes are complete, as compared to the approximate quantities in the traditional formulation of the energy cycle. This approach allows to fully exploit Lorenz's original concept. The gridscale equations derived will be the basis for evaluating numerically the classical Lorenz terms plus a couple of new global conversion fluxes, notably C(sub), to be presented in Part II of this study.