Potential vorticity dynamics in the framework of disk shallow-water theory

Abstract

Context. We consider potential vorticity dynamics for astrophysical disks. Aims. The aim of this work is to extend exploration of shear instabilities in cold astrophysical disks that are three-dimensional and whose mean states are baroclinic. In particular, we seek to demonstrate the potential existence of traditional baroclinic instabilities in meteorological studies. We show this in a simplified two-layer Philips disk model of a midplane symmetric disk. Methods. We analyze the dynamical normal-mode response of thin annular disks with two strongly localized potential vorticity gradients, one in each of two disk layers. Each disk layer is of constant but differing densities. The resulting mean azimuthal velocity profile shows a variation in the vertical direction, implying that the system is baroclinic in the mean state. The stability of the system is treated in the context of disk shallow-water theory, wherein azimuthal disturbances are much longer than the corresponding radial or vertical scales. The normal-mode problem is solved numerically using two different methods. Results. The results of a symmetric single-layer barotropic model is considered and it is found that instability persists for models in which the potential vorticity profiles are not symmetric, consistent with previous results. The instaiblity is interpreted in terms of interacting Rossby waves. For a two-layer system, in which the flow is fundamentally baroclinic, we report here that instability takes on the form of mixed barotropic-baroclinic type: instability occurs, but it qualitatively follows the pattern of instability found in the barotropic models. Instability arises because of the phase locking and interaction of the Rossby waves between the two layers. The strength of the instability weakens as the density contrast between layers increases. Conclusions. Baroclinic instability is feasible for astrophysical disks but has the character of mixed barotropic-baroclinic type. The instability as explored in this study could be present in protoplanetary disks with weak vertical density stratification.

Role of Interaction between Magnetic Rossby Waves and Tachocline Differential Rotation in Producing Solar Seasons

“…HD and MHD shallow-water models have been extensively applied in various astrophysical contexts over the past decade (Gilman & Dikpati 2002;Dikpati et al 2003;Zaqarashvili et al 2007Zaqarashvili et al , 2009Zaqarashvili et al , 2010aUmurhan 2008Umurhan , 2012Dikpati 2012;Heng & Workman 2014;Klimachkov & Petrosyan 2017a, 2017b, and many details of MHD tachocline instability properties have been explored by many authors (Gilman & Dikpati 2002;Dikpati et al 2003;Arlt et al 2005;Dikpati & Gilman 2005). For toroidal field with latitude dependence included, the combination of differential rotation in latitude and toroidal field is unstable for almost any magnitude of differential rotation and for any profile and amplitude of toroidal field, as well as any subadiabaticity.…”

Effects of upper disc boundary conditions on the linear Rossby wave instability

“…However, failure to find the RWI for some conditions imply that whether or not the RWI can develop in protoplanetary discs depends on the disc vertical structure, which can be complicated (Terquem 2008). Calculating the RWI for protoplanetary discs will ultimately require multi-layer fluid models (Umurhan 2012).…”

On the mechanism of self gravitating Rossby interfacial waves in proto-stellar accretion discs

“…The essence of this resonance mechanism is illustrated in (Fig. 4(b)) and has been applied in order to explain the RWI in Umurhan (2010). When self-gravity is included, instability will manifest itself along similar lines with the same main requirement -that there are at least two separate interfaces able to resonate with one another.…”

“…The Atacama Large Millimeter/submillimeter Array (ALMA) has opened up a new window revealing the structure of proto-planet accretion discs (pp-discs hereafter). Most notable of the recently reported ALMA discoveries is the appearance of asymmetric "peanut" features in the outer (> 80AU) regions of several pp-disc systems including SAO 206462, SR 21, LKHα 330 OPH IRS 48 and HD 142527 (Casassus et al 2013, Fukagawa et al 2013, Isella et al 2013, van der Marel et al 2013, Pérez et al 2014. The ALMA website recently published a high resolution image of HL Tauri, a debris-disc phase T-Tauri system, showing a disc with alternating axisymmetric rings and gaps with unambiguous non-axisymmetric features set atop.…”

“…Marcus et al (2013)), instability can arise from the resonant interaction of two or more Rossby waves (Heifetz et al 1999). For example, in RWI unstable discs with a single axisymmetric pressure bump, there are two oppositely signed extrema in the radial gradient-PV which can each support Rossby waves with wave speeds boosted by the local flow of the disc (Umurhan 2010). Resonant interaction and subsequent instability can occur when the Rossby waves move with equal and opposite directions in some reference frame.…”