A linearized approach to study stability and waveguidability of barotropic Rossby waves

The propagation and the characteristics of Rossby waves are influenced by the large-scale background flow where they occur: for instance, the role of upper-level jet streams in promoting Rossby wave propagation along preferred directions (so-called “waveguidability”) is a classic problem in climate dynamics. We propose a linear framework to study barotropic Rossby waves over a spherical domain for arbitrary orographic forcing and zonal background flow configurations, including cases with localised single and double jet streams. The approach allows to analytically obtain the steady-state linear flow response to orographic forcing without performing lengthy numerical integrations, together with the flow evolution as a combination of few modes composed by the various eigensolutions of the unforced problem (thus independent of the forcing). The connection between jet strength and waveguidability noticed by previous studies is confirmed. Background flow states featuring a strong jet stream are also prone to barotropic instability.

The eigenvalue analysis reveals the spatial structure of the associated Rossby modes and their growth rates, allowing to detect the presence of instabilities. We notice that, even in presence of a damping term, some background flow configurations allow wave instabilities to exist. According to the linear theory, the flow should diverge from the equilibrium state, since some waves are linearly unstable. Nonlinear simulations are performed to provide insights about the waves evolution in the unstable case. Such simulations reveal two interesting effects: 1) a damping effect operated by the nonlinear terms (i.e., the flow is unstable linearly but stable nonlinearly) for medium jet strengths; 2) a quasi-periodic behaviour around the unstable equilibrium state for the strongest jets, indicating the existence of a limit cycle. The linear analysis was still able to capture the unstable equilibrium state at the center of the limit cycle and to provide insights about the spatial structure of the dominant modes. These results indicate the usefulness of linearized approaches in the development of a reduced-order model to describe the barotropic instability mechanisms driving spherical Rossby waves.