On the paradox of convectively-coupled waves

Abstract

This thesis investigates the forced response of waves on a linearized, shallow water beta-plane in an attempt to develop some gross understanding of the properties of convectively coupled waves. Two concepts are central to this new theory: (i) a convectively coupled wave is a forced wave; (ii) a convectively coupled wave is a coupled feedback system, wherein the forcing and wave mutually reinforce one another. Furthermore, a convectively coupled wave exhibits phase-locked structures as it approaches steady state. The basic theory of forced waves is explored and it is shown that such forced waves are governed by two equations, one for the amplitude and one for the frequency of the wave. The difference between the speed of the forcing and the theoretical speed of the wave, δc, is an essential parameter of these solutions. It is shown that many of the observed features of convectively-coupled waves, for example the energy spectra and apparent paradox of slow propagation, can be explained by consideration of the parameter δc. Focus is then directed to the problem of the forced Kelvin wave, whose solutions show that a single wave produces two wave fronts that propagate at different phase speeds, one with a speed equal to the forced response and the other with a speed equal to the free response. Consideration of the feedback of the wave response to the forcing leads to the condition that δc < 0 is required to produce a positive feedback. Furthermore, using the derived definition of power and an alternate physical scaling argument leads to the same optimal condition, δc is proportional to – σ xε, where σx is the x Gaussian scale of the forcing and e is the dampening rate. This relationship serves as a predictive and diagnostic equation that correctly predicts the phase speed of convectively-coupled Kelvin waves and possibly the MJO as well adhering to the multi-scale view of these systems. The theory introduced demonstrates that the structure of the forcing is central to its application. Furthermore, the large scale forcing represents the aggregate effect of many processes operating on smaller scales. The role of shallow cumulus clouds in this process is also explored and it is demonstrated that shallow convection is fundamental to moistening as well as the low-level static stability, both of which have a significant effect on convection and indirectly, the forcing of convectively coupled waves.