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R. Alan Plumb and Raffaele Ferrari


A theoretical formalism for nongeostrophic eddy transport in zonal-mean flows, using a transformed Eulerian-mean (TEM) approach in z coordinates, is discussed. By using Andrews and McIntyre’s coordinate-independent definition of the “quasi-Stokes streamfunction,” it is argued that the surface boundary condition can be dealt with more readily than when the widely used quasigeostrophic definition is adopted. Along with the “residual mean circulation,” the concept of “residual eddy flux” arises naturally within the TEM framework, and it is argued that it is this residual eddy flux, and not the “raw” eddy flux, that might reasonably be expected to be downgradient. This distinction is shown to be especially important for Ertel potential vorticity (PV). The authors show how a closed set of transformed mean equations can be generated, and how the eddy forcing appears in the TEM momentum equations. Under adiabatic conditions, the “eddy drag” is just proportional to the residual eddy flux of PV along the mean isopycnals; in the diabatic layer close to the surface, it is more complicated, but becomes very simple for small Rossby number (without any assumption of small isopycnal slope).

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Allen Kuo, R. Alan Plumb, and John Marshall


The equilibrium of a modeled wind- and buoyancy-driven, baroclinically unstable, flow is analyzed using the transformed Eulerian-mean (TEM) approach described in Part I. Within the near-adiabatic interior of the flow, Ertel potential vorticity is homogenized along mean isopycnals—a finding readily explained using TEM theory, given the geometry of the domain. The equilibrium, zonal-mean buoyancy structure at the surface is determined entirely by a balance between imposed surface fluxes and residual mean and eddy buoyancy transport within a “surface diabatic layer.” Balance between these same processes and the wind stress determines the stratification, and hence potential vorticity, immediately below this layer. Ertel potential vorticity homogenization below then determines the mean buoyancy structure everywhere. Accordingly, the equilibrium structure of this flow can be described—and quantitatively reproduced—from knowledge of the eddy mixing rates within the surface diabatic zone and the depth of this zone, together with potential vorticity homogenization beneath. These results emphasize the need to include near-surface buoyancy transport, as well as interior PV transport, in eddy parameterization schemes. They also imply that, in more realistic models, the surface buoyancy balances may be impacted by processes in remote locations that allow diapycnal flow.

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Ivana Cerovečki, R. Alan Plumb, and William Heres


The baroclinically unstable wind- and buoyancy-driven flow in a zonally reentrant pie-shaped sector on a sphere is numerically modeled and then analyzed using the transformed Eulerian-mean (TEM) formalism. Mean fields are obtained by zonal and time averaging performed at fixed height. The very large latitudinal extent of the basin (50.7°S latitude to the equator) allows the latitude variation of the Coriolis parameter to strongly influence the flow. Persistent zonal jets are observed in the statistically steady state. Reynolds stress terms play an important role in redistributing zonal angular momentum: convergence of the lateral momentum flux gives rise to a strong eastward jet, with an adjacent westward jet equatorward and weaker multiple jets poleward. An equally prominent feature of the flow is a strong and persistent eddy that has the structure of a Kelvin cat’s eye and generally occupies the zonal width of the basin at latitudes 15°–10°S.

A strongly mixed surface diabatic zone overlies the near-adiabatic interior, within which Ertel potential vorticity (but not thickness) is homogenized along the mean isopycnals everywhere in the basin where eddies have developed (and thus is not homogenized equatorward of the most energetic eastward jet). A region of low potential vorticity (PV) is formed adjacent to the strong baroclinic front associated with that jet and subsequently maintained by strong convective events.

The eddy buoyancy flux is dominated by its skew component over large parts of the near-adiabatic interior, with cross-isopycnal components present only in the vicinity of the main jet and in the surface diabatic layer. Close to the main jet, the cross-isopycnal components are dominantly balanced by the triple correlation terms in the buoyancy variance budget, while the advection of buoyancy variance by the mean flow is not a dominant term in the eddy buoyancy variance budget.

Along-isopycnal mixing in the near-adiabatic interior is estimated by applying the effective diffusivity diagnostic of Nakamura. The effective diffusivity is large at the flanks of the mean jet and beneath it and small in the jet core. The apparent horizontal diffusivity for buoyancy obtained from the flux–gradient relationship is the same magnitude as the effective diffusivity, but the structures are rather different. The diapycnal diffusivity is strongest in the surface layer and also in a convectively unstable region that extends to depths of hundreds of meters beneath the equatorward flank of the main jet.

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