On the mean meridional circulation in the presence of a steady state, symmetric, circumpolar vortex

Abstract

The steady state equations governing the “forced” (in the sense of Kuo, 1956) mean meridional circulation in the presence of a circular vortex are presented, and it is shown that a constraining relation must exist between the sources of heat and momentum (eddy convergences, diabatic heating, and friction) and the parameters of the zonal mean state (zonal vertical wind shear, meridional temperature gradient, absolute vorticity, and static stability).

In the case of general heat and momentum forcing, solutions for vertical motion Ω are obtained explicitly. These solutions involve integrals along the characteristics of the equations, which are isolines of potential temperature and absolute angular momentum. In the special case of no heat or momentum forcing present, it is shown that a necessary but not sufficient condition for the existence of a “free” mean meridional circulation is that f = Z Ri (called resonance, after Kuo 1956) where Z is the absolute vorticity and Ri is the Richardson number. Equivalently, the characteristics of angular momentum and potential temperature must coincide. It is shown by means of the characteristics that the existence of a region in which the absolute vorticity, measured on an isobaric surface, is negative, may be a sufficient condition. The necessary condition implies that the static stability is also negative in such a region. The conclusion that such free circulations in the atmosphere cannot exist in the mean state agrees with Kuo, but the criterion is somewhat different.

In the general forcing case, a possible constraining relationship is derived, and its usefulness considered. Simpler formulae are presented which hold along lines on which one of the components of the meridional circulation vanishes. By considering also the ratio of the components of the meridional circulation along lines on which either the momentum or the heat source functions vanish, a consistency picture between the positioning of the mean meridional cells and the sources and sinks of momentum and heat is developed. In particular, it is shown that lines of vanishing heat or momentum source cannot, in general, be in phase. Such lines must cross at the “center” of each cell (where both components of the circulation are zero). The lines of zero momentum source must slope generally upward toward the equator, in general agreement with observation. For the constraints to be consistent with the observed counter-gradient transport of heat in the lower stratosphere, the lowest layer of cells must extend into the lower stratosphere. Equatorward counter-gradient transport of heat in the troposphere in very low and very high latitudes is also suggested.

The implications for observational and numerical model studies of the general circulation are briefly discussed.