arXiv subjects

Keaton J. Burns

Publications and source records attributed to Keaton J. Burns.

36 records · Page 2Linked to original sources

arXiv2022

Inverse cascade suppression and shear layer formation in MHD turbulence subject to a guide field and misaligned rotation

Astrophysical plasmas are often subject to both rotation and large-scale background magnetic fields. Individually, each is known to two-dimensionalize the flow in the perpendicular plane. In realistic flows, both of these effects are simultaneously present and, importantly, need not be aligned. In this work, we numerically investigate three-dimensional forced magnetohydrodynamic (MHD) turbulence subject to the competing effects of global rotation and a perpendicular background magnetic field. We focus on the case of a strong background field and find that increasing the rotation rate from zero produces significant changes in the structure of the turbulent flow. Starting with a two-dimensional inverse cascade at zero rotation, the flow first transitions to a forward cascade of kinetic energy, then to a shear-layer dominated regime, and finally to a second shear-layer regime where the kinetic energy flux is strongly suppressed and the energy transfer is mediated by the induced magnetic field. We show that the first two transitions occur at distinct values of the Rossby number, and the third occurs at a distinct value of the Lehnert number. More generally, our results demonstrate that, when considering the simultaneous limits of strong rotation and a strong guide field, the order in which those limits are taken matters in the misaligned case.

physics.flu-dyn↗

arXiv2021

A Validated Nonlinear 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.

astro-ph.IM↗

arXiv2020

Single-hemisphere dynamos in M-dwarf stars

M-dwarf stars below a certain mass are convective from their cores to their photospheres. These fully convective objects are extremely numerous, very magnetically active, and the likely hosts of many exoplanets. Here we study, for the first time, dynamo action in simulations of stratified, rotating fully convective M-dwarf stars. Importantly, we use new techniques to capture the correct full ball geometry down to the center of the star. We find surprising dynamo states in these systems, with the global-scale mean fields confined strongly to a single hemisphere, in contrast to prior stellar dynamo solutions. These hemispheric-dynamo stars are likely to have profoundly different interactions with their surroundings, with important implications for exoplanet habitability and stellar spindown.

astro-ph.SR↗

arXiv2020

Improved phase-field models of melting and dissolution in multi-component flows

We develop and analyse the first second-order phase-field model to combine melting and dissolution in multi-component flows. This provides a simple and accurate way to simulate challenging phase-change problems in existing codes. Phase-field models simplify computation by describing separate regions using a smoothed phase field. The phase field eliminates the need for complicated discretisations that track the moving phase boundary. However standard phase-field models are only first-order accurate. They often incur an error proportional to the thickness of the diffuse interface. We eliminate this dominant error by developing a general framework for asymptotic analysis of diffuse-interface methods in arbitrary geometries. With this framework we can consistently unify previous second-order phase-field models of melting and dissolution and the volume-penalty method for fluid-solid interaction. We finally validate second-order convergence of our model in two comprehensive benchmark problems using the open-source spectral code Dedalus.

physics.comp-ph↗

arXiv2020

Improving convergence of volume penalised fluid-solid interactions

We analyse and improve the volume-penalty method, a simple and versatile way to model objects in fluid flows. The volume-penalty method is a kind of fictitious-domain method that approximates no-slip boundary conditions with rapid linear damping inside the object. The method can then simulate complex, moving objects in general numerical solvers without specialised algorithms or boundary-conforming grids. Volume penalisation pays for this simplicity by introducing an equation-level error, the $\textit{model error}$, that is related to the damping time $η\ll 1$. While the model error has been proven to vanish as the damping time tends to zero, previous work suggests convergence at a slow rate of $\mathcal{O}(η^{1/2})$. The stiffness of the damping implies conventional volume penalisation only achieves first order numerical accuracy. We analyse the volume-penalty method using multiple-scales matched-asymptotics with a signed-distance coordinate system valid for arbitrary smooth geometries. We show the dominant model error stems from a displacement length that is proportional to a Reynolds number $\text{Re}$ dependent boundary layer of size $\mathcal{O}(η^{1/2}\text{Re}^{-1/2})$. The relative size of the displacement length and damping time leads to multiple error regimes. Our key finding derives a simple smoothing prescription for the damping that eliminates the displacement length and reduces the model error to $\mathcal{O}(η)$ in all regimes. This translates to second order numerical accuracy. We validate our findings in several comprehensive benchmark problems and finally combine Richardson extrapolation of the model error with our correction to further improve convergence to $\mathcal{O}(η^{2})$.

math.NA↗

arXiv2020

Linearly forced fluid flow on a rotating sphere

We investigate generalized Navier-Stokes (GNS) equations that couple nonlinear advection with a generic linear instability. This analytically tractable minimal model for fluid flows driven by internal active stresses has recently been shown to permit exact solutions on a stationary two-dimensional sphere. Here, we extend the analysis to linearly driven flows on rotating spheres. We derive exact solutions of the GNS equations corresponding to time-independent zonal jets and superposed westward-propagating Rossby waves, qualitatively similar to those seen in planetary atmospheres. Direct numerical simulations with large rotation rates obtain statistically stationary states close to these exact solutions. The measured phase speeds of waves in the GNS simulations agree with analytical predictions for Rossby waves.

physics.flu-dyn↗

arXiv2019

The Magnetorotational Instability Prefers Three Dimensions

The magnetorotational instability (MRI) occurs when a weak magnetic field destabilises a rotating, electrically conducting fluid with inwardly increasing angular velocity. The MRI is essential to astrophysical disk theory where the shear is typically Keplerian. Internal shear layers in stars may also be MRI unstable, and they take a wide range of profiles, including near-critical. We show that the fastest growing modes of an ideal magnetofluid are three-dimensional provided the shear rate, $S$, is near the two-dimensional onset value, $S_c$. For a Keplerian shear, three-dimensional modes are unstable above $S\approx0.10S_c$, and dominate the two-dimensional modes until $S\approx2.05S_{c}$. These three-dimensional modes dominate for shear profiles relevant to stars and at magnetic Prandtl numbers relevant to liquid-metal laboratory experiments. Significant numbers of rapidly growing three-dimensional modes remain well past $2.05S_{c}$. These finding are significant in three ways. First, weakly nonlinear theory suggests that the MRI saturates by pushing the shear rate to its critical value. This can happen for systems, like stars and laboratory experiments, that can rearrange their angular velocity profiles. Second, the non-normal character and large transient growth of MRI modes should be important whenever three-dimensionality exists. Finally, three-dimensional growth suggests direct dynamo action driven from the linear instability.

physics.plasm-ph↗

arXiv2019

Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods

Numerical solutions of partial differential equations enable a broad range of scientific research. The Dedalus Project is a flexible, open-source, parallelized computational framework for solving general partial differential equations using spectral methods. Dedalus translates plain-text strings describing partial differential equations into efficient solvers. This paper details the numerical method that enables this translation, describes the design and implementation of the codebase, and illustrates its capabilities with a variety of example problems. The numerical method is a first-order generalized tau formulation that discretizes equations into banded matrices. This method is implemented with an object-oriented design. Classes for spectral bases and domains manage the discretization and automatic parallel distribution of variables. Discretized fields and mathematical operators are symbolically manipulated with a basic computer algebra system. Initial value, boundary value, and eigenvalue problems are efficiently solved using high-performance linear algebra, transform, and parallel communication libraries. Custom analysis outputs can also be specified in plain text and stored in self-describing portable formats. The performance of the code is evaluated with a parallel scaling benchmark and a comparison to a finite-volume code. The features and flexibility of the codebase are illustrated by solving several examples: the nonlinear Schrodinger equation on a graph, a supersonic magnetohydrodynamic vortex, quasigeostrophic flow, Stokes flow in a cylindrical annulus, normal modes of a radiative atmosphere, and diamagnetic levitation. The Dedalus code and the example problems are available online at http://dedalus-project.org/.

astro-ph.IM↗

arXiv2019

Low-frequency variability in massive stars: Core generation or surface phenomenon?

Bowman et al. (2019) reported low-frequency photometric variability in 164 O- and B-type stars observed with K2 and TESS. They interpret these motions as internal gravity waves, which could be excited stochastically by convection in the cores of these stars. The detection of internal gravity waves in massive stars would help distinguish between massive stars with convective or radiative cores, determine core size, and would provide important constraints on massive star structure and evolution. In this work, we study the observational signature of internal gravity waves generated by core convection. We calculate the \textit{wave transfer function}, which links the internal gravity wave amplitude at the base of the radiative zone to the surface luminosity variation. This transfer function varies by many orders of magnitude for frequencies $\lesssim 1 \, {\rm d}^{-1}$, and has regularly-spaced peaks near $1 \, {\rm d}^{-1}$ due to standing modes. This is inconsistent with the observed spectra which have smooth ``red noise'' profiles, without the predicted regularly-spaced peaks. The wave transfer function is only meaningful if the waves stay predominately linear. We next show that this is the case: low frequency traveling waves do not break unless their luminosity exceeds the radiative luminosity of the star; and, the observed luminosity fluctuations at high frequencies are so small that standing modes would be stable to nonlinear instability. These simple calculations suggest that the observed low-frequency photometric variability in massive stars is not due to internal gravity waves generated in the core of these stars. We finish with a discussion of (sub)surface convection, which produces low-frequency variability in low-mass stars, very similar to that observed in Bowman et al. (2019) in higher mass stars.

astro-ph.SR↗

arXiv2019

The "Sphered Cube": A New Method for the Solution of Partial Differential Equations in Cubical Geometry

A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere. By choosing to embed the cube inside of the sphere, one obtains a discretization that is free of any sharp edges or corners. Taking full advantage of the simple geometry of the sphere, spectral bases based on spin-weighted spherical harmonics and Jacobi polynomials, which properly capture the regularity of scalar, vector and tensor components in spherical coordinates, can be applied to obtain moderately efficient and accurate numerical solutions of partial differential equations in the cube. This technique demonstrates the advantages of these bases over other methods for solving PDEs in spherical coordinates. We present results for a test case of incompressible hydrodynamics in cubical geometry: Rayleigh-Bénard convection with fully Dirichlet boundary conditions. Analysis of the simulations provides what is, to our knowledge, the first result on the scaling of the heat flux with the thermal forcing for this type of convection in a cube in a sphere.

physics.comp-ph↗

arXiv2019

Quantum hydrodynamics for supersolid crystals and quasicrystals

Supersolids are theoretically predicted quantum states that break the continuous rotational and translational symmetries of liquids while preserving superfluid transport properties. Over the last decade, much progress has been made in understanding and characterizing supersolid phases through numerical simulations for specific interaction potentials. The formulation of an analytically tractable framework for generic interactions still poses theoretical challenges. By going beyond the usually considered quadratic truncations, we derive a systematic higher-order generalization of the Gross-Pitaevskii mean field model in conceptual similarity with the Swift-Hohenberg theory of pattern formation. We demonstrate the tractability of this broadly applicable approach by determining the ground state phase diagram and the dispersion relations for the supersolid lattice vibrations in terms of the potential parameters. Our analytical predictions agree well with numerical results from direct hydrodynamic simulations and earlier quantum Monte-Carlo studies. The underlying framework is universal and can be extended to anisotropic pair potentials with complex Fourier-space structure.

cond-mat.quant-gas↗

arXiv2018

Tensor calculus in spherical coordinates using Jacobi polynomials, Part-II: Implementation and Examples

We present a simulation code which can solve broad ranges of partial differential equations in a full sphere. The code expands tensorial variables in a spectral series of spin-weighted spherical harmonics in the angular directions and a scaled Jacobi polynomial basis in the radial direction, as described in Part-I. Nonlinear terms are calculated by transforming from the coefficients in the spectral series to the value of each quantity on the physical grid, where it is easy to calculate products and perform other local operations. The expansion makes it straightforward to solve equations in tensor form (i.e., without decomposition into scalars). We propose and study several unit tests which demonstrate the code can accurately solve linear problems, implement boundary conditions, and transform between spectral and physical space. We then run a series of benchmark problems proposed in Marti et al (2014), implementing the hydrodynamic and magnetohydrodynamic equations. We are able to calculate more accurate solutions than reported in Marti et al 2014 by running at higher spatial resolution and using a higher-order timestepping scheme. We find the rotating convection and convective dynamo benchmark problems depend sensitively on details of timestepping and data analysis. We also demonstrate that in low resolution simulations of the dynamo problem, small changes in a numerical scheme can lead to large changes in the solution. To aid future comparison to these benchmarks, we include the source code used to generate the data, as well as the data and analysis scripts used to generate the figures.

astro-ph.IM↗

arXiv2018

Perspectives on Reproducibility and Sustainability of Open-Source Scientific Software from Seven Years of the Dedalus Project

As the Science Mission Directorate contemplates establishing an open code policy, we consider it timely to share our experiences as the developers of the open-source partial differential equation solver Dedalus. Dedalus is a flexible framework for solving partial differential equations. Its development team primarily uses it for studying stellar and planetary astrophysics. Dedalus was developed originally for astrophysical fluid dynamics (AFD), though it has found a much broader user base, including applied mathematicians, plasma physicists, and oceanographers. Here, we will focus on issues related to open-source software from the perspective of AFD. We use the term AFD with the understanding that astrophysics simulations are inherently multi-physics: fluid dynamics coupled with some combination of gravitational dynamics, radiation transfer, relativity, and magnetic fields. In practice, a few well-known open-source simulation packages represent a large fraction of published work in the field. However, we will argue that an open-code policy should encompass not just these large simulation codes, but also the input files and analysis scripts. It is our interest that NASA adopt an open-code policy because without it, reproducibility in computational science is needlessly hampered.

astro-ph.IM↗

arXiv2017

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 non-equilibrium 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 non-planar 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 anti-ferromagnetic 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.

physics.flu-dyn↗

arXiv2016

Turbulent Chemical Diffusion in Convectively Bounded Carbon Flames

It has been proposed that mixing induced by convective overshoot can disrupt the inward propagation of carbon deflagrations in super-asymptotic giant branch stars. To test this theory, we study an idealized model of convectively bounded carbon flames with 3D hydrodynamic simulations of the Boussinesq equations using the pseudospectral code Dedalus. Because the flame propagation timescale is much longer than the convection timescale, we approximate the flame as fixed in space, and only consider its effects on the buoyancy of the fluid. By evolving a passive scalar field, we derive a {\it turbulent} chemical diffusivity produced by the convection as a function of height, $D_{\rm t}(z)$. Convection can stall a flame if the chemical mixing timescale, set by the turbulent chemical diffusivity, $D_{\rm t}$, is shorter than the flame propagation timescale, set by the thermal diffusivity, $κ$, i.e., when $D_{\rm t}>κ$. However, we find $D_{\rm t}<κ$ for most of the flame because convective plumes are not dense enough to penetrate into the flame. Extrapolating to realistic stellar conditions, this implies that convective mixing cannot stall a carbon flame and that "hybrid carbon-oxygen-neon" white dwarfs are not a typical product of stellar evolution.

astro-ph.SR↗

arXiv2016

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.

math.NA↗

arXiv2015

Numerical Simulations of Internal Wave Generation by Convection in Water

Water's density maximum at 4C makes it well suited to study internal gravity wave excitation by convection: an increasing temperature profile is unstable to convection below 4C, but stably stratified above 4C. We present numerical simulations of a water-like fluid near its density maximum in a two dimensional domain. We successfully model the damping of waves in the simulations using linear theory, provided we do not take the weak damping limit typically used in the literature. In order to isolate the physical mechanism exciting internal waves, we use the novel spectral code Dedalus to run several simplified model simulations of our more detailed simulation. We use data from the full simulation as source terms in two simplified models of internal wave excitation by convection: bulk excitation by convective Reynolds stresses, and interface forcing via the mechanical oscillator effect. We find excellent agreement between the waves generated in the full simulation and the simplified simulation implementing the bulk excitation mechanism. The interface forcing simulations over excite high frequency waves because they assume the excitation is by the "impulsive" penetration of plumes, which spreads energy to high frequencies. However, we find the real excitation is instead by the "sweeping" motion of plumes parallel to the interface. Our results imply that the bulk excitation mechanism is a very accurate heuristic for internal wave generation by convection.

astro-ph.SR↗

arXiv2014

Conduction in low Mach number flows: Part I Linear & 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 regimes, 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.

astro-ph.SR↗