Instability of Long, Low Frequency Rossby Waves in Zonally Varying Shear Flow: Marginally Stable versus Deeply Stable Flow Regimes

The instability of zonally varying shear flow to long, low frequency Rossby waves is among the mechanisms shown to play an important role in the genesis and regional structure of atmospheric low frequency variability. Fundamental questions remain, however, regarding the connection between the local structure of the zonally varying flow and the low frequency instabilities that develop. In particular, are there differences in the low frequency instabilities that develop on zonally varying flow that is deeply stable versus flow that is marginally stable? Is the instability mechanism that traditionally operates in zonally uniform flow, i.e., wave-resonance, also operating in zonally varying flow? If not, are there other, possibly new, instability mechanisms operating?

To address these questions, we employ both barotropic and two-layer baroclinic models and consider both Hamiltonian and non-Hamiltonian zonally varying shear flows. For the two-layer baroclinic model the stability problem is couched within the framework of coupled wave instability (CWI). The CWI is shown to be mediated by a `physical' wave and a `virtual' wave. The CWI framework concisely describes the differences between the instabilities that develop on the different flow regimes and facilitates the development of new necessary conditions for instability. We show that all exponentially amplifying disturbances that develop on zonally varying flow that is locally marginally stable must have oscillatory structure. In sharp contrast, exponentially amplifying disturbances that develop on zonally varying flow that is locally deeply stable can have either oscillatory or non-oscillatory structures. The non-oscillatory disturbances appear as smooth envelope structures and represent a novel means for the development of coherent structures.

We define the stability of zonally varying flow by its (local) zonally uniform stability properties. Zonally varying flows are deemed locally supercritical if they satisfy, at any streamwise location, the necessary condition for instability of zonally uniform flow. Zonally varying flows are deemed globally subcritical if they do not satisfy, at any streamwise location, the necessary condition for instability of zonally uniform flow. For the marginally stable problem, we show that globally subcritical Hamiltonian flows are stable, whereas non-Hamiltonian flows may be unstable in both locally supercritical and globally subcritical regimes. In locally supercritical flow, the CWI grows via wave-resonance between the physical/virtual wave pair and is highly localized to the supercritical region. In contrast, in globally subcritical flow, the CWI grows via pseudomomentum extraction from the background flow and either radiates away or remains trapped to the streamwise variation in the flow. The growth mechanism for subcritical instabilities is shown to be fundamentally different than that for supercritical instabilities, but identical to that for deeply stable instabilities. These results show that the stability criteria for zonally uniform flow is a good predictor of instability for zonally varying flow that is locally supercritical but not for zonally varying flow that is globally subcritical. The application of these results to observed atmospheric low frequency variability is discussed.