Keaton Burns

A validated non-linear Kelvin–Helmholtz benchmark for numerical hydrodynamics

The nonlinear evolution of the Kelvin-Helmholtz instability is a popular test for code verification. To date, most Kelvin-Helmholtz problems discussed in the literature are ill-posed: they do not converge to any single solution with increasing resolution. This precludes comparisons among different codes and severely limits the utility of the Kelvin-Helmholtz instability as a test problem. The lack of a reference solution has led various authors to assert the accuracy of their simulations based on ad-hoc proxies, e.g., the existence of small-scale structures. This paper proposes well-posed Kelvin-Helmholtz problems with smooth initial conditions and explicit diffusion. We show that in many cases numerical errors/noise can seed spurious small-scale structure in Kelvin-Helmholtz problems. We demonstrate convergence to a reference solution using both Athena, a Godunov code, and Dedalus, a pseudo-spectral code. Problems with constant initial density throughout the domain are relatively straightforward for both codes. However, problems with an initial density jump (which are the norm in astrophysical systems) exhibit rich behavior and are more computationally challenging. In the latter case, Athena simulations are prone to an instability of the inner rolled-up vortex; this instability is seeded by grid-scale errors introduced by the algorithm, and disappears as resolution increases. Both Athena and Dedalus exhibit late-time chaos. Inviscid simulations are riddled with extremely vigorous secondary instabilities which induce more mixing than simulations with explicit diffusion. Our results highlight the importance of running well-posed test problems with demonstrated convergence to a reference solution. To facilitate future comparisons, we include the resolved, converged solutions to the Kelvin-Helmholtz problems in this paper in machine-readable form.

CONDUCTION IN LOW MACH NUMBER FLOWS. I. LINEAR AND WEAKLY NONLINEAR REGIMES

Thermal conduction is an important energy transfer and damping mechanism in astrophysical flows. Fourier’s law—the heat flux is proportional to the negative temperature gradient, leading to temperature diffusion—is a well-known empirical model of thermal conduction. However, entropy diffusion has emerged as an alternative thermal conduction model, despite not ensuring the monotonicity of entropy. This paper investigates the differences between temperature and entropy diffusion for both linear internal gravity waves and weakly nonlinear convection. In addition to simulating the two thermal conduction models with the fully compressible Navier–Stokes equations, we also study their effects in the reduced, “sound-proof” anelastic and pseudo-incompressible equations. We find that in the linear and weakly nonlinear regime, temperature and entropy diffusion give quantitatively similar results, although there are some larger errors in the pseudo-incompressible equations with temperature diffusion due to inaccuracies in the equation of state. Extrapolating our weakly nonlinear results, we speculate that differences between temperature and entropy diffusion might become more important for strongly turbulent convection.

Tensor calculus in polar coordinates using Jacobi polynomials

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Conversion of internal gravity waves into magnetic waves

Asteroseismology probes the interiors of stars by studying oscillation modes at a stars surface. Although pulsation spectra are well understood for solar-like oscillators, a substantial fraction of red giant stars observed by Kepler exhibit abnormally low-amplitude dipole oscillation modes. Fuller et al. (2015) suggest this effect is produced by strong core magnetic fields that scatter dipole internal gravity waves (IGWs) into higher multipole IGWs or magnetic waves. In this paper, we study the interaction of IGWs with a magnetic field to test this mechanism. We consider two background stellar structures: one with a uniform magnetic field, and another with a magnetic field that varies both horizontally and vertically. We derive analytic solutions to the wave propagation problem and validate them with numerical simulations. In both cases, we find perfect conversion from IGWs into magnetic waves when the IGWs propagate into a region exceeding a critical magnetic field strength. Downward propagating IGWs cannot reflect into upward propagating IGWs because their vertical wavenumber never approaches zero. Instead, they are converted into upward propagating slow (Alfvenic) waves, and we show they will likely dissipate as they propagate back into weakly magnetized regions. Therefore, strong internal magnetic fields can produce dipole mode suppression in red giants, and gravity modes will likely be totally absent from the pulsation spectra of sufficiently magnetized stars.

Anomalous Chained Turbulence in Actively Driven Flows on Spheres

Recent experiments demonstrate the importance of substrate curvature for actively forced fluid dynamics. Yet, the covariant formulation and analysis of continuum models for nonequilibrium flows on curved surfaces still poses theoretical challenges. Here, we introduce and study a generalized covariant Navier-Stokes model for fluid flows driven by active stresses in nonplanar geometries. The analytical tractability of the theory is demonstrated through exact stationary solutions for the case of a spherical bubble geometry. Direct numerical simulations reveal a curvature-induced transition from a burst phase to an anomalous turbulent phase that differs distinctly from externally forced classical 2D Kolmogorov turbulence. This new type of active turbulence is characterized by the self-assembly of finite-size vortices into linked chains of antiferromagnetic order, which percolate through the entire fluid domain, forming an active dynamic network. The coherent motion of the vortex chain network provides an efficient mechanism for upward energy transfer from smaller to larger scales, presenting an alternative to the conventional energy cascade in classical 2D turbulence.

Rolling resistance of shallow granular deformation

Experiments are conducted to measure the resistance experienced by light cylinders rolling over flat beds of granular media. Sand and glass spheres are used for the beds. The trajectories of the rolling cylinders are determined through optical tracking, and velocity and acceleration data are inferred through fits to these trajectories. The rolling resistance is dominated by a velocity-independent component, but a velocity-dependent drag exceeding the expected strength of air drag is also observed. The results are compared to a theoretical model based on a cohesionless Mohr–Coulomb rheology for a granular medium in the presence of gravity. The model idealizes the flow pattern underneath the rolling cylinder as a plastically deforming zone in front of a rigidly rotating plug attached to the cylinder, as proposed previously for cylinders rolling on perfectly cohesive plastic media. The leading-order, rate-independent rolling resistance observed experimentally is well reproduced by the model predictions.