Potential vorticity dynamics in the framework of disk shallow-water theory: II. Mixed Barotropic-Baroclinic Instability

Abstract: We extend exploration of potential vorticity instabilities in cold astrophysical disks whose mean states are baroclinic. In particular, we seek to demonstrate the potential existence of traditional baroclinic instabilities of meteorological studies in a simplified two-layer Philips Disk Model. 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. 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. (For full abstract see article.)

• The paper demonstrates that mixed instability arises from resonant Rossby wave interactions between vertically sheared layers in a two-layer disk model.
• It uses a thin-disk shallow-water framework with spectral and Newton-Raphson methods to quantify how barotropic shear and density contrast govern growth rates and defect separations.
• The study implies that disks with weak stratification and moderate density contrasts are most prone to large-scale mixing, influencing vortex formation and angular momentum transport.

Mixed Barotropic-Baroclinic Instability in Disk Shallow-Water Theory

Introduction and Motivation

This study rigorously investigates the linear stability characteristics of midplane-symmetric, cold astrophysical disks with baroclinic mean states using a two-layer Phillips model. The motivation is to clarify the dynamical feasibility and physical structure of baroclinic and mixed barotropic-baroclinic instabilities in Keplerian disks, a long-standing and fundamental question in disk astrophysics. Traditional meteorological baroclinic instability relies on the misalignment of isobars and isopycnals in stratified, rotating sheared flows—a situation intrinsic to many disk systems but complicated by strong barotropic shear and the associated mathematical challenges.

Methodological Framework

The analysis employs a thin-disk shallow-water limit in which azimuthal length scales are much longer than vertical or radial scales, as justified by a systematic scaling analysis. Using this framework, a two-layer Phillips model is developed with incompressible, constant-density layers. The model captures the essential coupling of potential vorticity (PV) across the layers and allows for a tractable but nontrivial eigenmode analysis. Both the steady-state configuration and linearized disturbances are described, reducing to coupled equations for the layerwise enthalpy perturbations, discretized and solved via spectral (Chebyshev) methods and refined/verified with Newton-Raphson iterations.

Two scenarios are explored: a single-layer (barotropic) paradigm and the fully two-layer (baroclinic/barotropic) case. The PV profiles are chosen to be nontrivial (double-tanh for barotropic; single-jump per layer for baroclinic) to facilitate clear Rossby wave interpretations and minimize complications from critical layers.

Barotropic Model: Reference and Baseline

In the single-layer limit, the system exposes classic Rossby wave instability mechanisms. Instability, signaled by positive imaginary parts of eigenfrequencies, arises from mutual interaction and phase locking of azimuthally propagating Rossby waves localized at PV defects. The shear parameter $q$ strongly influences both the maximum growth rates and critical spacing required for instability: as $q$ decreases, instability requires greater defect separation and growth rates diminish, echoing the barotropic governor effect. The results are robust under variations in boundary placement and model symmetry, and reproduce key features delineated in previous meteorological and astrophysical literature.

Two-Layer Mixed Barotropic-Baroclinic Model

Transitioning to two layers, the mean state exhibits vertical shear and baroclinicity. Each layer contains a localized PV defect, with distinct radial locations. The resulting instability is of mixed character: while coupling between layers creates quintessential baroclinic resonance, the strong Keplerian barotropic shear means the instability qualitatively mirrors the barotropic scenario rather than classical midlatitude atmospheres.

The instability mechanism remains the resonant interaction of Rossby waves, but now those waves reside in different vertical layers. This mixed instability persists over a range of model parameters, but the instability becomes weaker (lower growth rate and reduced range of permitted PV defect separation) as the density contrast between the layers increases; in the limit where the upper layer vanishes, layers decouple and instability vanishes, as expected from energetic and modal arguments.

Crucially, unlike in traditional planetary atmospheres, instability does not occur when the PV jumps/defects in both layers are exactly radially aligned ($A = 0$), except as a boundary artifact when the radial domain is artificially restricted. Thus, generic traditional baroclinic instability is suppressed by the strong barotropic shear and midplane symmetry specific to disk geometries studied. The results suggest that disks with weak vertical stratification and moderate density contrasts are most susceptible to mixed barotropic-baroclinic instability, and that parameter dependence is subtle and sensitive to domain size and defect arrangement.

Reductions in the background shear parameter $q$ further suppress growth rates and push the instability to larger defect separations, consistent with both the barotropic governor and the understanding from modal resonance conditions.

Theoretical and Practical Implications

The essential implication is that baroclinic instability is feasible in cold, weakly stratified disks (such as protoplanetary or accretion disks), but its character is fundamentally governed by the coexistence of strong barotropic (Keplerian) shear and baroclinic structure. The instability is robust in mixed regimes, but its efficacy is limited by vertical density contrast and domain geometry.

For disk astrophysics, such mixed instabilities provide a plausible avenue for large-scale mixing and angular momentum transport, especially in regions where the disk is not strongly stratified. The scale selection properties and interaction structure of the instability could have direct consequences for vortex formation, turbulent cascades, or even planet formation scenarios.

From a theoretical viewpoint, the study extends the class of systems where PV-based, layer-coupling interpretations of instability provide accurate and insightful explanations, while also delineating the limitations of applying meteorological analogs to disk systems—most notably in the essential asymmetry introduced by strong barotropic flows and midplane symmetry.

Future Directions

Extension to more realistic multi-layer or continuously stratified disk models is warranted to capture the full spectrum of possible Rossby-type instabilities and their non-linear outcomes. Relaxing midplane symmetry and incorporating explicit radiative/chemical structuring could reveal additional modes and coupling scenarios. Magnetic stresses, which are ubiquitous in disks and known to produce effective baroclinic torquing in solar studies, remain a vital next step. Finally, the long-wavelength limit assumption, central to the disk shallow-water approximation, should be explicitly relaxed as in the followup studies promised, to capture short-wavelength and more localized instability forms.

Conclusion

The paper establishes that the two-layer Phillips disk model, in the disk-shallow-water regime, admits mixed barotropic-baroclinic instability rooted in the resonant coupling of localized Rossby waves across vertically sheared layers. The strength of this instability is controlled by background shear, density contrast, and radial arrangement of PV defects. For weak stratification and moderate density contrasts, such instabilities may play a significant role in the evolution of protoplanetary and accretion disks. However, the dynamical landscape remains distinct from classic meteorological baroclinic instability due to the dominant influence of Keplerian barotropic shear and midplane symmetry constraints. Further theoretical generalization and numerical simulation will be critical for delineating the astrophysical consequences and predictive use of these findings.

Reference: "Potential vorticity dynamics in the framework of disk shallow-water theory: II. Mixed Barotropic-Baroclinic Instability" (1203.1686)