In this work we discuss the equilibration of short Charney waves, emphasizing the role of the redistribution of momentum and the maintenance of the momentum balance for the equilibrated system. To this end, it is illustrative to compare the equilibration of the 2D problem (barotropic point jet) and 3D baroclinic problem.

A short Charney wave is a mode for which the mixing depth is externally constrained, so that there is a limited available PV gradient in the interior. Because in the 2D problem the column-integrated PV gradient is unchanged by the wave (including delta-function contributions), 2D short Charney modes lack the scale to eliminate the negative PV gradient. Essentially, the adjustment in shear over the limited depth requires the development of a curvature larger than beta, and an associated negative PV gradient in the interior. Short Charney modes are found to equilibrate in the 2D problem by eliminating the PV gradient at the steering level alone. This occurs as the modes transfer easterly momentum from the jet vertex into the interior, which results in a reduction of the mean shear.

However, things are more complicated in the 3D problem. The reason is that this problem cannot be interpreted in terms of the along-column redistribution of momentum alone, but there is also a net import of momentum into the column associated to the eddy momentum flux. Hence, it is no longer appropriate to look at the column-integrated PV gradient alone: rather, the full 2D structure should be considered. When this is taken into account, the concept of a short Charney wave is not so relevant. Idealized numerical simulations show that in this case the available interior PV gradient no longer limits the development of the vertical curvature. The reason is that the additional enhancement of the horizontal curvature prevents the development of a negative interior PV gradient in the 3D problem. Indeed, in our model the negative and positive contributions to the interior PV gradient associated to the the vertical and horizontal curvatures of the jet are comparable for the equilibrated system. This also seems to be the case in the real troposphere, even though the latter term is often neglected in the literature.

In the forced-dissipative system, the maintenance of the eddy enstrophy balance requires that the PV gradient changes sign (assuming downgradient PV fluxes). For the 2D problem, this implies that the PV gradient must be positive (negative) over regions with easterly (westerly) acceleration, so that the associated eddy drag locally balances the restoring term. On the other hand, in the 3D problem it is no longer possible to write such a local balance because there is an additional redistribution of momentum by the residual circulation. However, this contribution disappears when the vertical integral is considered. This leads to the important result that in the presence of friction, there must be a nonzero surface temperature gradient over the latitudes with surface westerlies.