A theoretical discussion of eddy-driven mean flows

Abstract

This paper gives fairly detailed account of our recent results on the induction of mean circulation by eddy processes {whether wave-like or turbulent). The relevance of the work to climatology is through the role of oceans in climate, as well as the need to understand and parameterize eddies in the atmospheric circulation itself. In Section 1 we discuss the vorticity constraints that are likely to dominate the wind-driven mean circulation, in the absence of strong eddy effects. In Section 2 we describe the principle formulation for Eulerian-mean flow, rewriting the eddy-induction effects in the spirit of early work by Taylor {1915). Both transient and statistically steady problems, wavelike and turbulent flows, and oceanic and a tmospher ic geometry can be treated, although the details are distinctly different in each case. The key quantities that express the eddy effects are the Lagrangian diffusivity of marked fluid particles (~) and the large-scale geometry of Q, where Q is the large-scale potential vorticity field. The non-dissipative theory stems from work of Taylor, but most problems are sensitive to the nature of potential vorticity dissipation by the enstrophy cascade. These new effects are summarized. The meanflow equation is solved by integrating along characteristics. The principal restriction is that the lateral scale of the Q-field is supposed to far exceed the displacement of fluid particles, over few eddy periods. The important role played by the eddy flux of potential vorticity, (q'u' }, is