Non-axisymmetric instabilities in discs with imposed zonal flows

We conduct a linear stability calculation of an ideal Keplerian flow on which a sinusoidal zonal flow is imposed. The analysis uses the shearing sheet model and is carried out both in isothermal and adiabatic conditions, with and without self-gravity (SG). In the non-SG regime, a structure in the potential vorticity (PV) leads to a non-axisymmetric Kelvin–Helmholtz (KH) instability; in the short-wavelength limit its growth rate agrees with the incompressible calculation by Lithwick, which only considers perturbations elongated in the streamwise direction. The instability’s strength is analysed as a function of the structure’s properties, and zonal flows are found to be stable if their wavelength is $\gtrsim$ 8 $\small{H}$, where $\small{H}$ is the disc’s scaleheight, regardless of the value of the adiabatic index $\gamma$. The non-axisymmetric KH instability can operate in Rayleigh-stable conditions, and it therefore represents the limiting factor to the structure’s properties. Introducing SG triggers a second non-axisymmetric instability, which is found to be located around a PV maximum, while the KH instability is linked to a PV minimum, as expected. In the adiabatic regime, the same gravitational instability is detected even when the structure is present only in the entropy (not in the PV) and the instability spreads to weaker SG conditions as the entropy structure’s amplitude is increased. This eventually yields a non-axisymmetric instability in the non-SG regime, albeit of weak strength, localized around an entropy maximum.Science and Technology Facilities CouncilThis is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/mnras/stw223

Applications of a Cole-Hopf transform to the 3D Navier-Stokes equations

The Navier-Stokes equations written in the vector potential can be recast as the nonlinear Schr¨odinger equations at imaginary times, i.e. the heat equations with a potential term, using the Cole-Hopf transform introduced in Ohkitani(2017). On this basis, we study two kinds of Navier-Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection is taking place, whereas the signature of the nonlinear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magnetohydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions

Zonal flow evolution and overstability in accretion discs

This work presents a linear analytical calculation on the stability and evolution of a compressible, viscous self-gravitating (SG) Keplerian disc with both horizontal thermal diffusion and a constant cooling time-scale when an axisymmetric structure is present and freely evolving. The calculation makes use of the shearing sheet model and is carried out for a range of cooling times. Although the solutions to the inviscid problem with no cooling or diffusion are well known, it is non-trivial to predict the effect caused by the introduction of cooling and of small diffusivities; this work focuses on perturbations of intermediate wavelengths, therefore representing an extension to the classical stability analysis on thermal and viscous instabilities. For density wave modes, the analysis can be simplified by means of a regular perturbation analysis; considering both shear and thermal diffusivities, the system is found to be overstable for intermediate and long wavelengths for values of the Toomre parameter Q ≲ 2; a non-SG instability is also detected for wavelengths ≳18H, where H is the disc scale-height, as long as γ ≲ 1.305. The regular perturbation analysis does not, however, hold for the entropy and potential vorticity slow modes as their ideal growth rates are degenerate. To understand their evolution, equations for the axisymmetric structure's amplitudes in these two quantities are analytically derived and their instability regions obtained. The instability appears boosted by increasing the value of the adiabatic index and of the Prandtl number, while it is quenched by efficient cooling.The research was conducted thanks to the funding received by the Science and Technology Facilities Council (STFC)

Three-Dimensional Simulations of Massive Stars: II. Age Dependence

We present 3D full star simulations, reaching up to 90% of the total stellar radius, for three $7M_\odot$ stars of different ages (ZAMS, midMS and TAMS). A comparison with several theoretical prescriptions shows the generation spectra for all three ages are dominated by convective plumes. Two distinct overshooting layers are observed, with most plumes stopped within the layer situated directly above the convective boundary (CB); overshooting to the second, deeper layer becomes increasingly more infrequent with stellar age. Internal gravity wave (IGW) propagation is significantly impacted in the midMS and TAMS models as a result of some IGWs getting trapped within their Brunt-V\"{a}is\"{a}l\"{a} frequency spikes. A fundamental change in the wave structure across radius is also observed, driven by the effect of density stratification on IGW propagation causing waves to become evanescent within the radiative zone, with older stars being affected more strongly. We find that the steepness of the frequency spectrum at the surface increases from ZAMS to the older models, with older stars also showing more modes in their spectra.Comment: 24 pages, 14 figures / Accepted at Ap

Three-dimensional Simulations of Massive Stars. II. Age Dependence

We present 3D full star simulations, reaching up to 90% of the total stellar radius, for three 7 M _⊙ stars of different ages: zero-age main sequence (ZAMS), mid–main sequence (midMS), and terminal-age main sequence (TAMS). A comparison with several theoretical prescriptions shows that the generation spectra for all three ages are dominated by convective plumes. Two distinct overshooting layers are observed, with most plumes stopped within the layer situated directly above the convective boundary; overshooting to the second, deeper layer becomes progressively more infrequent with increasing stellar age. Internal gravity wave (IGW) propagation is significantly impacted in the midMS and TAMS models as a result of some IGWs getting trapped within their Brunt–Väisälä frequency spikes. A fundamental change in the wave structure across radius is also observed, driven by the effect of density stratification on IGW propagation causing waves to become evanescent within the radiative zone, with older stars being affected more strongly. We find that the steepness of the frequency spectrum at the surface increases from ZAMS to the older models, with older stars also showing more modes in their spectra

Chemical Mixing Induced by Internal Gravity Waves in Intermediate-mass Stars

Internal gravity waves can cause mixing in the radiative interiors of stars. We study this mixing by introducing tracer particles into 2D hydrodynamic simulations. Following the work of Rogers & McElwaine, we extend our study to different masses (3, 7, and 20 M _⊙ ) and ages (ZAMS, midMS, and TAMS). The diffusion profiles of these models are influenced by various parameters such as the Brunt–Väisälä frequency, density, thermal damping, the geometric effect, and the frequencies of waves contributing to these mixing profiles. We find that the mixing profile changes dramatically across age. In younger stars, we noted that the diffusion coefficient increases toward the surface, whereas in older stars the initial increase in the diffusion profile is followed by a decreasing trend. We also find that mixing is stronger in more massive stars. Hence, future stellar evolution models should include this variation. In order to aid the inclusion of this mixing in 1D stellar evolution models, we determine the dominant waves contributing to these mixing profiles and present a prescription that can be included in 1D models