Large power densities in gas turbines can be accompanied by combustion instabilities (Culick & Kuentzmann 2006; Lieuwen & Yang 2005) due to a coupling between the flames and acoustics, creating high pressure and heat release oscillations in the chamber. Such oscillations may destroy the whole propulsion system and combustion instabilities have been a key issue for aeronautics and propulsion systems (Candel 2002; Culick & Kuentzmann 2006; Lieuwen & Yang 2005) especially in high-performance engines (Harrje & Reardon 1972; Culick 1987) for a long time. Full-scale experiments in this field are difficult (Poinsot et al. 1987; Lee & Lieuwen 2003; Lee & Anderson 1999) and numerical simulations have been used heavily to replicate the complex mechanisms involved in combustion instabilities in full-scale geometries (Wolf et al. 2009; Staffelbach et al. 2009; Wolf et al. 2010). These simulations are not sufficient to understand or control unstable modes: low-order models and theory on simplified geometries (Dowling 1995, 1997; Kopitz et al. 2005; Nicoud et al. 2007) are needed to guide both large-eddy simulations (LES) and experiments. Annular chambers used in gas turbines sometimes exhibit a specific class of unstable modes: azimuthal modes (Figure 1) propagating along the azimuthal direction eθ and not only in the longitudinal direction ez (Candel 1992; Crighton et al. 1992; Lieuwen & Yang 2005; O’Connor et al. 2015). Mechanisms leading to azimuthal instabilities are more complex than those encountered in longitudinal configurations
