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Potential vorticity dynamics in the framework of disk shallow-water theory: II. Mixed Barotropic-Baroclinic Instability  [PDF]
O. M. Umurhan
Physics , 2012, DOI: 10.1051/0004-6361/201218803
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.)
Rossby Wave Instability of Keplerian Accretion Disks  [PDF]
R. V. E. Lovelace,H. Li,S. A. Colgate,A. F. Nelson
Physics , 1998, DOI: 10.1086/306900
Abstract: We find a linear instability of non-axisymmetric Rossby waves in a thin non-magnetized Keplerian disk when there is a local maximum in the radial profile of a key function ${\cal L}(r) \equiv {\cal F}(r) S^{2/\Gamma}(r)$, where ${\cal F}^{-1} = \hat {\bf z}\cdot ({\bf \nabla}\times {\bf v}) /\Sigma$ is the potential vorticity, $S = P/\Sigma^\Gamma$ is the entropy, $\Sigma$ is the surface mass density, $P$ is the vertically integrated pressure, and $\Gamma$ is the adiabatic index. We consider in detail the special case where there is a local maximum in the disk entropy profile $S(r)$. This maximum acts to trap the waves in its vicinity if its height to width ratio ${\rm max}(S)/\Delta r$ is larger than a threshold value. The pressure gradient derived from this entropy variation provides the restoring force for the wave growth. We show that the trapped waves act to transport angular momentum outward. A plausible way to produce an entropy variation is when an accretion disk is starting from negligible mass and temperature, therefore negligible entropy. As mass accumulates by either tidal torquing, magnetic torquing, or Roche-lobe overflow, confinement of heat will lead to an entropy maximum at the outer boundary of the disk. Possible nonlinear developments from this instability include the formation of Rossby vortices and the formation of spiral shocks. What remains to be determined from hydrodynamic simulations is whether or not Rossby wave packets (or vortices) ``hold together'' as they propagate radially inward.
Structure and evolution of 3D Rossby Vortices  [cached]
Richard S.,Barge P.
EPJ Web of Conferences , 2013, DOI: 10.1051/epjconf/20134603002
Abstract: Three dimensional compressible simulations of the Rossby Wave Instability are presented in a non-homentropic model of protoplanetary disk. The instability develops like in the two dimensional case, gradually coming to the formation of a single big vortex. This 3D vortex has a quasi-2D structure which looks like a vorticity column with only tiny vertical motions. The vortex survives hundred of rotations in a quasi-steady evolution and slowly migrates inward toward the star.
Structure, stability and evolution of 3D Rossby vortices in protoplanetary disks  [PDF]
Samuel Richard,Pierre Barge,Stephane Le Dizes
Physics , 2013, DOI: 10.1051/0004-6361/201322175
Abstract: Large-scale persistent vortices are known to form easily in 2D disks via the Rossby wave or the baroclinic instability. In 3D, however, their formation and stability is a complex issue and still a matter of debate. We study the formation of vortices by the Rossby wave instability in a stratified inviscid disk and describe their three dimensional structure, stability and long term evolution. Numerical simulations are performed using a fully compressible hydrodynamical code based on a second order finite volume method. We assume a perfect gas law and a non-homentropic adiabatic flow.The Rossby wave instability is found to proceed in 3D in a similar way as in 2D. Vortices produced by the instability look like columns of vorticity in the whole disk thickness; the small vertical motions are related to a weak inclination of the vortex axis appearing during the development of the RWI. Vortices with aspect ratios larger than 6 are unaffected by the elliptical instability. They relax to a quasi-steady columnar structure which survives hundred of rotations while slowly migrating inward toward the star at a rate that reduces with the vortex aspect ratio. Vortices with a smaller aspect ratio are by contrast affected by the elliptic instability. Short aspect ratio vortices are completely destroyed in a few orbital periods. Vortices with an intermediate aspect ratio are partially destroyed by the elliptical instability in a region away from the mid-plane where the disk stratification is sufficiently large. Elongated Rossby vortices can survive a large number of orbital periods in protoplanetary disks in the form of vorticity columns. They could play a significant role in the evolution of the gas and the gathering of the solid particles to form planetesimals or planetary cores, a possibility that receives a renewed interest with the recent discovery of a particle trap in the disk of Oph IRS48.
Rossby Wave Instability of Thin Accretion Disks -- II. Detailed Linear Theory  [PDF]
H. Li,J. M. Finn,R. V. E. Lovelace,S. A. Colgate
Physics , 1999, DOI: 10.1086/308693
Abstract: In earlier work we identified a global, non-axisymmetric instability associated with the presence of an extreme in the radial profile of the key function ${\cal L}(r) \equiv (\Sigma \Omega/\kappa^2) S^{2/\Gamma}$ in a thin, inviscid, nonmagnetized accretion disk. Here, $\Sigma(r)$ is the surface mass density of the disk, $\Omega(r)$ the angular rotation rate, $S(r)$ the specific entropy, $\Gamma$ the adiabatic index, and $\kappa(r)$ the radial epicyclic frequency. The dispersion relation of the instability was shown to be similar to that of Rossby waves in planetary atmospheres. In this paper, we present the detailed linear theory of this Rossby wave instability and show that it exists for a wider range of conditions, specifically, for the case where there is a ``jump'' over some range of $r$ in $\Sigma(r)$ or in the pressure $P(r)$. We elucidate the physical mechanism of this instability and its dependence on various parameters, including the magnitude of the ``bump'' or ``jump,'' the azimuthal mode number, and the sound speed in the disk. We find large parameter range where the disk is stable to axisymmetric perturbations, but unstable to the non-axisymmetric Rossby waves. We find that growth rates of the Rossby wave instability can be high, $\sim 0.2 \Omega_{\rm K}$ for relative small ``jumps'' or ``bumps''. We discuss possible conditions which can lead to this instability and the consequences of the instability.
Nonaxisymmetric Rossby Vortex Instability with Toroidal Magnetic Fields in Radially Structured Disks  [PDF]
Cong Yu,Hui Li
Physics , 2009, DOI: 10.1088/0004-637X/702/1/75
Abstract: We investigate the global nonaxisymmetric Rossby vortex instability in a differentially rotating, compressible magnetized accretion disk with radial density structures. Equilibrium magnetic fields are assumed to have only the toroidal component. Using linear theory analysis, we show that the density structure can be unstable to nonaxisymmetric modes. We find that, for the magnetic field profiles we have studied, magnetic fields always provide a stabilizing effect to the unstable Rossby vortex instability modes. We discuss the physical mechanism of this stabilizing effect. The threshold and properties of the unstable modes are also discussed in detail. In addition, we present linear stability results for the global magnetorotational instability when the disk is compressible.
Rossby wave instability at dead zone boundaries in 3D resistive magnetohydrodynamical global models of protoplanetary disks  [PDF]
Wladimir Lyra,Mordecai-Mark Mac Low
Physics , 2012, DOI: 10.1088/0004-637X/756/1/62
Abstract: It has been suggested that the transition between magnetorotationally active and dead zones in protoplanetary disks should be prone to the excitation of vortices via Rossby wave instability (RWI). However, the only numerical evidence for this has come from alpha disk models, where the magnetic field evolution is not followed, and the effect of turbulence is parametrized by Laplacian viscosity. We aim to establish the phenomenology of the flow in the transition in 3D resistive-magnetohydrodynamical models. We model the transition by a sharp jump in resistivity, as expected in the inner dead zone boundary, using the Pencil Code to simulate the flow. We find that vortices are readily excited in the dead side of the transition. We measure the mass accretion rate finding similar levels of Reynolds stress at the dead and active zones, at the $\alpha\approx 10^{-2}$ level. The vortex sits in a pressure maximum and does not migrate, surviving until the end of the simulation. A pressure maximum in the active zone also triggers the RWI. The magnetized vortex that results should be disrupted by parasitical magneto-elliptic instabilities, yet it subsists in high resolution. This suggests that either the parasitic modes are still numerically damped, or that the RWI supplies vorticity faster than they can destroy it. We conclude that the resistive transition between the active and dead zones in the inner regions of protoplanetary disks, if sharp enough, can indeed excite vortices via RWI. Our results lend credence to previous works that relied on the alpha-disk approximation, and caution against the use of overly reduced azimuthal coverage on modeling this transition.
Instability of combined gravity-inertial-Rossby waves in atmospheres and oceans
J. F. McKenzie
Annales Geophysicae (ANGEO) , 2011,
Abstract: The properties of the instability of combined gravity-inertial-Rossby waves on a β-plane are investigated. The wave-energy exchange equation shows that there is an exchange of energy with the background stratified medium. The energy source driving the instability lies in the background enthalpy released by the gravitational buoyancy force. It is shown that if the phase speed of the westward propagating low frequency-long wavelength Rossby wave exceeds the Poincaré-Kelvin (or "equivalent" shallow water) wave speed, instability arises from the merging of Rossby and Poincaré modes. There are two key parameters in this instability condition; namely, the equatorial/rotational Mach (or Froude) number M and the latitude θ0 of the β-plane. In general waves equatorward of a critical latitude for given M can be driven unstable, with corresponding growth rates of the order of a day or so. Although these conclusions may only be safely drawn for short wavelengths corresponding to a JWKB wave packet propagating internally and located far from boundaries, nevertheless such a local instability may play a significant role in atmosphere-ocean dynamics.
The Rossby wave instability in protoplanetary disks  [cached]
Meheut H.
EPJ Web of Conferences , 2013, DOI: 10.1051/epjconf/20134603001
Abstract: The Rossby wave instability has been proposed as a mechanism to transport angular momentum in the dead zone of protoplanetary disks and to form vortices. These vortices are of particular interest to concentrate solids in their centres and eventually to form planetesimals. Here we summarize some recent results concerning the growth and structure of this instability in radially and vertically stratified disks, its saturation and non-linear evolution. We also discuss the concentration of solids in the Rossby vortices including vertical settling.
Modulational Instability in the Whitham Equation with Surface Tension and Vorticity  [PDF]
Vera Mikyoung Hur,Mathew A. Johnson
Mathematics , 2015,
Abstract: We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions for the physical problem.
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