Physical mechanism of centrifugal-gravity wave resonant instability in azimuthally symmetric swirling flows

Abstract

We present an explicit analysis of wave-resonant instability of swirling flows inside fast rotating cylindrical containers. The linear dynamics are decomposed into the interaction between the horizontal inner centrifugal edge waves, the outer vertical gravity waves with the aim of understanding the dynamics of the centrifugal waves. We show how the far field velocity induced respectively by the centrifugal and the gravity waves affect each other's propagation rates and amplitude growth. We follow this with an analysis of the instability in terms of a four wave interaction, two centrifugal and two gravity ones, and explain why the resonant instability can be obtained only between a pair of two counter-propagating waves, one centrifugal and one gravity. Furthermore, a near resonant regime which does not yield instability is shown to result from a phase-locking configuration between a pair of a counter-propagating centrifugal wave and a pro-propagating gravity one, where the interaction affects the waves' propagation rates but not the amplitude growth.

A generalized action-angle representation of wave interaction in stratified shear flows

“…Indeed, Rabinovich et al (2011) analyzed the Taylor-Caulfield instability (Caulfield 1994) and showed that for a large range of Richardson numbers the growth rates, corresponding to the most unstable modes, are practically unaffected when the pro-propagating waves are neglected. Similar results have been obtained by for the more complex non-Boussinesq dynamics of stratified shear flows , for swirling flow instability in rotating cylinders (Yellin-Bergovoy et al 2017), for gravity-capillary waves (Biancofiore et al (In Press)), and even for Alfvén waves in magneto-hydrodynamic shear flows . Therefore, next we consider the subset dynamics of the counter propagating interfacial vorticity waves.…”

“…It has been implemented for swirling flow instability in rotating cylinders (Yellin-Bergovoy et al 2017), and even for Alfvén waves in magneto-hydrodynamic shear flows . Therefore, it is our plan to generalize this generalized action-angle formulation further to such more complex setups of shear flows.…”

The Initial Conditions for Planet Formation: Turbulence Driven by Hydrodynamical Instabilities in Disks around Young Stars

“…Rossby waves are also prevalent in protoplanetary disks and this has been extensively written about (Lovelace et al 1999;Li et al 2000;Umurhan 2010). There are notable differences, the main one being: while accretion disks exhibit gradients in their mean PV simply owing to the mean Keplerian flow, Rossby waves are mostly supported where there are anomalous changes in the mean Keplerian flow, like locations where the disk material supports pressure extrema (Lovelace et al 1999;Li et al 2000;Meheut et al 2010;Lin 2014) or where there are sharp edges like near disk gaps (Koller et al 2003;de Val-Borro et al 2007;Lyra et al 2009;Yellin-Bergovoy et al 2017), or potentially in places where the disk goes from being Ohmic to significantly magnetized (Varnière & Tagger 2006;Lyra & Mac Low 2012). However, the Rossby wave frequency response estimated based on our simplified rotating sphere model, Eq.…”

Parametric Study of the Rossby Wave Instability in a Two-Dimensional Barotropic Disk

“…Lin (2014) shows that the RWI is largely unaffected by viscosity and occurs in the linear regime. The effect of self-gravity is discussed by some studies (Lovelace & Hohlfeld 2013;Yellin-Bergovoy et al 2015). Lovelace & Hohlfeld (2013) shows that the onset of the RWI is depressed for mQH 0 /r n π/2, where Q is the Toomre Q parameter of the disks (Toomre 1964).…”