General circulation, potential vorticity transport, and surface heat transport

The zonal average circulation in potential-temperature coordinate S is known to represent well the Lagrangian general circulation in the atmosphere. We show that with the additional use of mean absolute angular momentum L as the meridional coordinate in the extratropics, the zonal average equations take on a simple form, where potential vorticity (PV) is the primary prognostic variable. In particular, if PV is interpreted as the mixing ratio of a PV-substance (PVS) --- following Haynes and McIntyre 1987 & 1990 --- then the concentration of PVS is constant in (L,S)-coordinates. This result implies that PV changes only because of dilution/concentration of PVS, which arises from mass convergence/divergence in (L,S)-coordinates. The zero net transport of PVS also means that the mean and eddy transport of PVS must always exactly cancel in these coordinates, even when the atmosphere is evolving in time. This important result implies that equatorward eddy PVS flux must force a poleward mean flow. Assuming gradient-wind and hydrostatic balance, the PV distribution can be inverted under suitable boundary conditions, to give geographical latitude y and zonal mean pressure p. Thus, a mapping to the more familiar (y,p)-coordinates is achieved.

Because of zonal variations in potential temperature of air at the Earth's surface, there exists a "surface zone" in which latitude circles on isentropes are interupted by the Earth's surface, even in the absence of topography. Judicious definition of the zonal averages in this region is necessary. From angular momentum considerations, we show that the mean meridional flow in the (potentially) colder half of the surface zone is equatorward, because it must generate a Coriolis force to balance the eastward eddy form drag from the Earth's undulating surface in isentropic coordinate. In steady-state, we show that this equatorward flow is determined by the surface heating tendency, for a given meridional (potential) temperature gradient. As this flow closes the mass circulation in the atmosphere, we obtain clear relations between (1) equatorward eddy PVS flux in the atmosphere, and (2) eastward eddy form drag in the surface zone or (2') positive heating tendency on the Earth's surface. (2) and (2') are both related to eddy potential temperature advection on the Earth's surface.