# 2018 - Quaternion structure of azimuthal instabilities

Abstract: Rotationally symmetric systems can exhibit acoustic fluctuations in the azimuthal direction. In experimental works the nature (standing or spinning) of these fluctuations is often described by a set of indicators. These indicators either depend on the chosen frame of reference or are not state space variables for the acoustic field. Conversely, in theoretical works the field is projected on two orthogonal modes, and the system is characterized in terms of two amplitudes and one phase difference. Also in these works the nature of the field is not a state space variable but a derived quantity. Moreover the phase difference between the two modes is undetermined when one of the amplitudes of the two modes is zero, making the phase space ill-posed. We present a solution to these limitations, and we show how the acoustic field can be embedded in quaternion algebra, by calculating a suitable analytic signal of the complex-valued embedding of the acoustic field. This allows us to map the state of the system to a point that moves as function of time on a two-dimensional sphere in three-dimensional (3D) space, the Poincar{'{e}}-Bloch sphere. To each state of the system corresponds just one point in this 3D space, which is then a well-posed phase space. We term the spherical coordinates of the point the amplitude of oscillation, the nature angle, and the orientation angle. The amplitude of oscillation of the system is the radius of the sphere. The nature angle is the latitude angle and positions the point closer to the equator (pure standing mode states) or closer to the poles (pure spinning mode states). The orientation angle is the longitude angle and describes the position of the pressure antinodes of the part of the acoustic field that is standing. These coordinates have a straightforward physical interpretation, can be easily calculated from experimental data, and are at the same time state space variables of a simple and elegant ansatz that can be used in low-order models. We also present an example of characterization of an experimental azimuthal thermoacoustic instability.