The concept of potential momentum has been introduced in a different session, and is also reviewed here. The potential momentum is essentially the momentum distribution that produces the same potential vorticity distribution as the stretching term in the full 3D problem. This definition allows us to establish some parallelisms between the 2D and 3D problem. Another important advantage of this formulation is that temperature and momentum appear on the same footing and can be directly compared. For instance, we can relate using this formulation the thermal imbalance at the surface to the net interior drag (dynamical plus frictional).
In terms of potential momentum the general circulation can be described as follows. Diabatic processes generate easterly potential momentum at the surface, which favors an unstable basic state. The equilibration of the baroclinic waves then produces a poleward residual circulation, which redistributes vertically the potential momentum initially locked at the surface. This poleward circulation has two important and related effects. On the one hand, it depletes the potential momentum at the surface (i.e., it reduces the surface temperature gradient). On the other, the conservation of the vertically integrated potential momentum demands that the depletion of the potential momentum at the surface be accompanied by a generation of an easterly potential momentum jet. This occurs through a conversion from (easterly) physical into potential momentum in the interior. Hence, the potential momentum framework makes most transparent the underlying connection between the reduction of the surface temperature gradient and the barotropic acceleration of the jet. This is illustrated with the equilibration of the 3D Charney problem (meridionally modulated by a Gaussian jet).
The thermal wind constraint requires that the potential and physical momentum jets have similar strength for this problem. This is ultimately enforced through the exchange between physical and potential momentum by the residual circulation. It can be shown that for convergent eddy momentum fluxes the dynamical drag (i.e., the eddy PV flux) is always insufficient to balance the conversion from potential to physical momentum. As a result, in the absence of friction there is no limit to the barotropic acceleration of the jet, and the flow can only equilibrate by depleting the easterly potential momentum at the surface.
To the extent that friction controls the strength of the physical momentum jet it should also control the strength of the interior potential momentum jet, since as discussed both must be comparable. This suggests that mechanical friction might be what ultimately limits the depletion of the surface temperature gradient. This was confirmed in our model by running a series of experiments in which only the frictional time scale was varied.We found that the frictional time scale constrained not just the acceleration of the jet, but also the potential momentum acceleration in the interior and the depletion of the surface temperature gradient.
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