Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

Linear Contributions of Different Time Scales to Teleconnectivity

The contributions of different time scales to extratropical teleconnections are examined. By applying empirical orthogonal functions and correlation analyses to reanalysis data, it is shown that eddies with periods shorter than 10 days have no linear contribution to teleconnectivity. Instead, synoptic variability follows wavelike patterns along the storm tracks, interpreted as propagating baroclinic disturbances. In agreement with preceding studies, it is found that teleconnections such as the North Atlantic Oscillation (NAO) and the Pacific–North America (PNA) pattern occur only at low frequencies, typically for periods more than 20 days. Low-frequency potential vorticity variability is shown to follow patterns analogous to known teleconnections but with shapes that differ considerably from them. It is concluded that the role, if any, of synoptic eddies in determining and forcing teleconnections needs to be sought in nonlinear interactions with the slower transients. The present results demonstrate that daily variability of teleconnection indices cannot be interpreted in terms of the teleconnection patterns, only the slow part of the variability.