arXiv subjects

Bradley W. Hindman

Publications and source records attributed to Bradley W. Hindman.

At least 19 recordsLinked to original sources

arXiv2026

Effects of density stratification on Rossby waves in deep atmospheres

Though Rossby waves have been observed on the Sun, their radial eigenfunctions remain a mystery. The prior theoretical work either considers quasi-2D systems, which do not apply to the solar interior, or only considers fully radiative or fully convective atmospheres. This project calculates the radial eigenfunctions for Rossby waves in a deep atmosphere for a general stratification. Here, we use the $β$-plane approximation to derive a vertical equation in terms of the Lagrangian pressure fluctuation $δP$, and we then calculate radial eigenfunctions for Rossby waves in a standard solar model, Model S. We find that working in the Lagrangian pressure fluctuation results in cleaner wave equations that lack internal singularities that have been encountered in prior work. The resulting radial wave equation makes it abundantly clear that there are two wave cavities in the solar interior, one in the radiative interior and another in the convection zone. Surprisingly, our calculated radial vorticity eigenfunctions for the radiative interior modes are nearly constant throughout the convection zone, raising the possibility that they may be observable at the solar surface.

astro-ph.SR↗

arXiv2025

Diffusive Braking of Penetrative Convection in Stably-Stratified Fluids

Mixing at the interface between a convection zone and an adjacent, stably-stratified layer plays a crucial role in shaping the structure and evolution of stars and planets. In this work, we present a suite of 2D and 3D Boussinesq simulations that explore how bottom-driven convection penetrates into a compositionally stratified region. Our results reveal two distinct regimes: a penetrative regime, where the convection zone steadily grows by entraining fluid from above, and a stalled regime, where growth halts and transitions to overshooting convection. We extend classical entrainment theory by incorporating thermal and compositional diffusion and by deriving a modified entrainment law that predicts interface speeds in the weak-diffusion limit. We show that convection stalls when the interface speed becomes comparable to the compositional diffusion speed and validate the transition between behaviors across a wide parameter space of Richardson and Lewis numbers. Such diffusively-controlled stalling is unlikely to occur in stellar and planetary interiors, where the Lewis number is typically large and compositional diffusion is extremely slow. In these environments, compositional diffusion will merely slow the growth of the convection zone and convective boundaries are expected to stall only in the presence of other curtailing mechanisms such as strong radiative diffusion or rapid rotation.

physics.flu-dyn↗

arXiv2024

Evolution of Semi-convective Staircases in Rotating Flows: Consequences for Fuzzy Cores in Giant Planets

Recent observational constraints on the internal structure of Jupiter and Saturn suggest that these planets have ``fuzzy" cores, i.e., gradients of the concentration of heavy elements that might span a large fraction of the planet's radius. These cores could be composed of a semi-convective staircase, i.e., multiple convective layers separated by diffusive interfaces arising from double-diffusive instabilities. However, to date, no study has demonstrated how such staircases can avoid layer mergers and persist over evolutionary time scales. In fact, previous work has found that these mergers occur rapidly, leading to only a single convective layer. Using 3D simulations, we demonstrate that rotation prolongs the lifetime of a convective staircase by increasing the timescale for both layer merger and erosion of the interface between the final two layers. We present an analytic model for the erosion phase, predicting that rotation increases the erosion time by a factor of approximately $\mathrm{Ro}^{-1/2}$, where $\mathrm{Ro}$ is the Rossby number of the convective flows (the ratio of the rotation period to the convective turnover time). For Jovian conditions at early times after formation (when convection is vigorous enough to mix a large fraction of the planet), we find the erosion time to be roughly $10^{9}~\mathrm{yrs}$ in the non-rotating case and $10^{11}~\mathrm{yrs}$ in the rotating case. If these timescales are confirmed with a larger suite of numerical simulations, the existence of convective staircases within the deep interiors of giant planets is a strong possibility, and rotation could be an important factor in the preservation of their fuzzy cores.

astro-ph.EP↗

arXiv2024

Confinement of the Solar Tachocline by a Non-Axisymmetric Dynamo

We recently presented the first 3D numerical simulation of the solar interior for which tachocline confinement was achieved by a dynamo-generated magnetic field. In this followup study, we analyze the degree of confinement as the magnetic field strength changes (controlled by varying the magnetic Prandtl number) in a coupled radiative zone (RZ) and convection zone (CZ) system. We broadly find three solution regimes, corresponding to weak, medium, and strong dynamo magnetic field strengths. In the weak-field regime, the large-scale magnetic field is mostly axisymmetric with regular, periodic polarity reversals (reminiscent of the observed solar cycle), but fails to create a confined tachocline. In the strong-field regime, the large-scale field is mostly non-axisymmetric with irregular, quasi-periodic polarity reversals, and creates a confined tachocline. In the medium-field regime, the large-scale field resembles a strong-field dynamo for extended intervals, but intermittently weakens to allow temporary epochs of strong differential rotation. In all regimes, the amplitude of poloidal field strength in the RZ is very well explained by skin-depth arguments, wherein the oscillating field that gives rise to the skin depth (in the medium- and strong-field cases) is a non-axisymmetric field structure rotating with respect to the RZ. These new simulations reaffirm that tachocline confinement by the solar dynamo (the so-called fast magnetic confinement scenario) is possible, but suggest a new picture in which non-axisymmetric field components rotating with respect to the RZ play the primary role, instead of the regularly reversing axisymmetic field associated with the 22-year cycle.

astro-ph.SR↗

arXiv2023

Inertial Waves in a Nonlinear Simulation of the Sun's Convection Zone and Radiative Interior

Recent observations of Rossby waves and other more exotic forms of inertial oscillations in the Sun's convection zone have kindled the hope that such waves might be used as a seismic probe of the Sun's interior. Here we present a 3D numerical simulation in spherical geometry that models the Sun's convection zone and upper radiative interior. This model features a wide variety of inertial oscillations, including both sectoral and tesseral equatorial Rossby waves, retrograde mixed inertial modes, prograde thermal Rossby waves, the recently observed high-frequency retrograde (HFR) vorticity modes, and what may be latitudinal overtones of these HFR modes. With this model, we demonstrate that sectoral and tesseral Rossby waves are ubiquitous within the radiative interior as well as within the convection zone. We suggest that there are two different Rossby-wave families in this simulation that live in different wave cavities: one in the radiative interior and one in the convection zone. Finally, we suggest that many of the retrograde inertial waves that appear in the convection zone, including the HFR modes, are in fact all related, being latitudinal overtones that are mixed modes with the prograde thermal Rossby waves.

astro-ph.SR↗

arXiv2023

Assessing the Observability of Deep Meridional Flow Cells in the Solar Interior

Meridional circulation regulates the Sun's interior dynamics and magnetism. While it is well accepted that meridional flows are poleward at the Sun's surface, helioseismic observations have yet to provide a definitive answer for the depth at which those flows return to the equator, or the number of circulation cells in depth. Here, we explore the observability of multiple circulation cells stacked in radius. Specifically, we examine the seismic signature of several meridional flow profiles by convolving time-distance averaging kernels with mean flows obtained from a suite of 3D hydrodynamic simulations. At mid and high latitudes, we find that weak flow structures in the deep convection zone can be obscured by signals from the much stronger surface flows. This contamination of 1--2 m s$^{-1}$ is caused by extended side lobes in the averaging kernels, which produce a spurious equatorward signal with flow speeds that are one order of magnitude stronger than the original flow speeds in the simulations. At low latitudes, the flows in the deep layers of the simulations are stronger ($> 2$ m s$^{-1}$) and multiple cells across the convection zone can produce a sufficiently strong signal to survive the convolution process. Now that meridional flows can be measured over two decades of data, the uncertainties arising from convective noise have fallen to a level where they are comparable in magnitude to the systematic biases caused by non-local features in the averaging kernels. Hence, these systematic errors are beginning to influence current helioseismic deductions and need broader consideration.

astro-ph.SR↗

arXiv2023

Dwindling Surface Cooling of a Rotating Jovian Planet Leads to a Convection Zone that Grows to a Finite Depth

Recent measurements of Jupiter's gravitational field (by Juno) and seismology of Saturn's rings (by Cassini) strongly suggest that both planets have a stably-stratified core that still possesses a primordial gradient in the concentration of heavy elements. The existence of such a "diffusely" stratified core has been a surprise as it was long expected that the Jovian planets should be fully convective and hence fully mixed. A vigorous zone of convection, driven by surface cooling, forms at the surface and deepens through entrainment of fluid from underneath. In fact, it was believed that this convection zone should grow so rapidly that the entire planet would be consumed in less than a million years. Here we suggest that two processes, acting in concert, present a solution to this puzzle. All of the giant planets are rapidly rotating and have a cooling rate that declines with time. Both of these effects reduce the rate of fluid entrainment into the convection zone. Through the use of an analytic prescription of entrainment in giant planets, we demonstrate that these two effects, rotation and dwindling surface cooling, result in a convection zone which initially grows but eventually stalls. The depth to which the convective interface asymptotes depends on the rotation rate and on the stratification of the stable interior. Conversely, in a nonrotating planet, or in a planet that maintains a higher level of cooling than current models suggest, the convection zone deepens forever, eventually spanning the entire planet.

astro-ph.EP↗

arXiv2023

Latitudinal Propagation of Thermal Rossby Waves in Stellar Convection Zones

Using an analytic model, we derive the eigenfrequencies for thermal Rossby waves that are trapped radially and latitudinally in an isentropically stratified atmosphere. We ignore the star's curvature and work in an equatorial f-plane geometry. The propagation of inertial waves is found to be sensitive to the relative direction of the wave vector to the zonal direction. Prograde propagating thermal Rossby waves are naturally trapped in the radial direction for frequencies above a critical threshold, which depends on the angle of propagation. Below the threshold frequency, there exists a continuous spectrum of prograde and retrograde inertial waves that are untrapped in an isentropic atmosphere, but can be trapped by gradients in the specific entropy density such as occurs in a stellar convection zone. Finally, we discuss the implications of these waves on recent observations of inertial oscillations in the Sun, as well as in numerical simulations.

astro-ph.SR↗

arXiv2023

Low-Frequency Internal Gravity Waves are Pseudo-incompressible

Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequency $ω$ is much less than that of surface gravity waves, i.e., $ω\ll \sqrt{g k_h}$ where $g$ is the gravitational acceleration and $k_h$ is the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency $\varepsilon = ω/ \sqrt{g k_h}$. Further, we show that in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz-Braginsky-Roberts (LBR) formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation. Whereas, in the anelastic approximation Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. But, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.

physics.flu-dyn↗

arXiv2023

Rotation reduces convective mixing in Jupiter and other gas giants

Recent measurements of Jupiter's gravitational moments by the Juno spacecraft and seismology of Saturn's rings suggest that the primordial composition gradients in the deep interior of these planets have persisted since their formation. One possible explanation is the presence of a double-diffusive staircase below the planet's outer convection zone, which inhibits mixing across the deeper layers. However, hydrodynamic simulations have shown that these staircases are not long-lasting and can be disrupted by overshooting convection. In this paper we suggests that planetary rotation could be another factor for the longevity of primordial composition gradients. Using rotational mixing-length theory and 3D hydrodynamic simulations, we demonstrate that rotation significantly reduces both the convective velocity and the mixing of primordial composition gradients. In particular, for Jovian conditions at $t\sim10^{8}~\mathrm{yrs}$ after formation, rotation reduces the convective velocity by a factor of 6, and in turn, the kinetic energy flux available for mixing gets reduced by a factor of $6^3\sim 200$. This leads to an entrainment timescale that is more than two orders of magnitude longer than without rotation. We encourage future hydrodynamic models of Jupiter and other gas giants to include rapid rotation, because the decrease in the mixing efficiency could explain why Jupiter and Saturn are not fully mixed.

astro-ph.EP↗

arXiv2023

The Puzzling Structure of Solar Convection: Window into the Dynamo

The operation of the solar dynamo, with all of its remarkable spatio-temporal ordering, remains an outstanding problem of modern solar physics. A number of mechanisms that might plausibly contribute to its operation have been proposed, but the relative role played by each remains unclear. This uncertainty stems from continuing questions concerning the speed and structure of deep-seated convective flows. Those flows are in-turn thought to sustain both the Sun's turbulent EMF and the large-scale flows of differential rotation and meridional circulation suspected of influencing the dynamo's organization and timing. Continued progress in this area is complicated by (i) inconsistencies between helioseismic measurements of convective and meridional flow made with different techniques and instruments, and (ii) a lack of high-latitude data for convection, differential rotation, and meridional flow. We suggest that the path forward to resolving these difficulties is twofold. First, the acquisition of long-term helioseismic and emissivity measurements obtained from a polar vantage point is vital to complete our picture of the Sun's outer convection zone. Second, sustained and expanded investment in theory-oriented and combined theory/observational research initiatives will be crucial to fully exploit these new observations and to resolve inconsistencies between existing measurements.

astro-ph.SR↗

arXiv2023

Overstable Convective Modes in a Polytropic Stellar Atmosphere

Within the convection zone of a rotating star, the presence of the Coriolis force stabilizes long-wavelength convective modes. These modes, which would have been unstable if the star lacked rotation, are called overstable convective modes or thermal Rossby waves. We demonstrate that the Sun's rotation rate is sufficiently rapid that the lower half of its convection zone could possess overstable modes. Further, we present an analytic solution for atmospheric waves that reside within a polytropic stratification. We explore in detail the properties of the overstable and unstable wave modes that exist when the polytrope is weakly unstable to convective overturning. Finally, we discuss how the thermal Rossby waves that reside within the convection zone of a star might couple with the prograde branch of the $g$ modes that are trapped within the star's radiative zone. We suggest that such coupling might enhance the photospheric visibility of a subset of the Sun's $g$ modes.

astro-ph.SR↗

arXiv2022

Confinement of the Solar Tachocline by Dynamo Action in the Radiative Interior

A major outstanding problem in solar physics is the confinement of the solar tachocline, the thin shear layer that separates nearly solid-body rotation in the radiative interior from strong differential rotation in the convection zone. Here, we present the first 3-D, global solar simulation in which a tachocline is confined by a self-excited dynamo. The non-axisymmetric magnetism is initially built in the convection zone and then diffusively imprints downward. Additionally, the field is locally amplified throughout the radiative interior by vigorous horizontal motions that arise from equatorial Rossby waves and possibly shear instabilities. Our work thus challenges the long-held notion that the Sun's dynamo magnetic field is amplified only as deep as the tachocline and stored in a quiescent radiative interior.

astro-ph.SR↗

arXiv2022

Radial Trapping of Thermal Rossby Waves within the Convection Zones of Low-Mass Stars

We explore how thermal Rossby waves propagate within the gravitationally stratified atmosphere of a low-mass star with an outer convective envelope. Under the conditions of slow, rotationally constrained dynamics, we derive a local dispersion relation for atmospheric waves in a fully compressible stratified fluid. This dispersion relation describes the zonal and radial propagation of acoustic waves and gravito-inertial waves. Thermal Rossby waves are just one class of prograde-propagating gravito-inertial wave that manifests when the buoyancy frequency is small compared to the rotation rate of the star. From this dispersion relation, we identify the radii at which waves naturally reflect and demonstrate how thermal Rossby waves can be trapped radially in a waveguide that permits free propagation in the longitudinal direction. We explore this trapping further by presenting analytic solutions for thermal Rossby waves within an isentropically stratified atmosphere that models a zone of efficient convective heat transport. We find that within such an atmosphere, waves of short zonal wavelength have a wave cavity that is radially thin and confined within the outer reaches of the convection zone near the star's equator. The same behavior is evinced by the thermal Rossby waves that appear at convective onset in numerical simulations of convection within rotating spheres. Finally, we suggest that stable thermal Rossby waves could exist in the lower portion of the Sun's convection zone, despite that region's unstable stratification. For long wavelengths, the Sun's rotation rate is sufficiently rapid to stabilize convective motions and the resulting overstable convective modes are identical to thermal Rossby waves.

astro-ph.SR↗

arXiv2022

Constraining Global Solar Models through Helioseismic Analysis

Global hydrodynamic simulations of internal solar dynamics have focused on replicating the conditions for solar-like differential rotation and meridional circulation using the results of helioseismic inversions as a constraint. Inferences of meridional circulation, however, have provided controversial results showing the possibility of one, two, or multiple cells along the radius. To resolve this controversy and develop a more robust understanding of global flow regimes in the solar interior, we apply a "forward-modeling" approach to the analysis of helioseismic signatures of meridional circulation profiles obtained from numerical simulations. We employ the global acoustic modeling code GALE to simulate the propagation of acoustic waves through regimes of mean mass flows generated by global hydrodynamic and magnetohydrodynamic models: EULAG, the Pencil Code, and the Rayleigh code. These models are used to create synthetic dopplergram data products, used as inputs for local time-distance helioseismology techniques. Helioseismic travel-time signals from solutions obtained through global numerical simulations are compared directly with inferences from solar observations, in order to set additional constraints on global model parameters in a direct way. We show that even though these models are able to replicate solar-like differential rotation, the resulting rotationally-constrained convection develops a multi-cell global meridional circulation profile that is measurably inconsistent with local time-distance inferences of solar observations. However, we find that the development of rotationally-unconstrained convection close to the model surface is able to maintain solar-like differential rotation, while having a significant impact on the helioseismic travel-time signal, replicating solar observations within one standard deviation of the error due to noise.

astro-ph.SR↗

arXiv2021

Do Coronal Loops Oscillate in Isolation?

Images of the solar corona by extreme-ultraviolet (EUV) telescopes reveal elegant arches of glowing plasma that trace the corona's magnetic field. Typically, these loops are preferentially illuminated segments of an arcade of vaulted field lines and such loops are often observed to sway in response to nearby solar flares. A flurry of observational and theoretical effort has been devoted to the exploitation of these oscillations with the grand hope that seismic techniques might be used as probes of the strength and structure of the corona's magnetic field. The commonly accepted viewpoint is that each visible loop oscillates as an independent entity and acts as a one-dimensional (1D) wave cavity for magnetohydrodynamic (MHD) kink waves. We argue that for many events, this generally accepted model for the wave cavity is fundamentally flawed. In particular, the 3D magnetic arcade in which the bright loop resides participates in the oscillation. Thus, the true wave cavity is larger than the individual loop and inherently multidimensional. We derive the skin depth of the near-field response for an oscillating loop and demonstrate that most loops are too close to other magnetic structures to oscillate in isolation. Further, we present a simple model of a loop embedded within an arcade and explore how the eigenmodes of the arcade and the eigenmodes of the loop become coupled. In particular, we discuss how distinguishing between these two types of modes can be difficult when the motions within the arcade are often invisible.

astro-ph.SR↗

arXiv2020

Revisiting the Sun's Strong Differential Rotation along Radial Lines

Current state-of-the-art models of the solar convection zone consist of solutions to the Navier-Stokes equations in rotating, 3D spherical shells. Such models are highly sensitive to the choice of boundary conditions. Here, we present two suites of simulations differing only in their outer thermal boundary condition, which is either one of fixed-entropy or fixed-entropy-gradient. We find that the resulting differential rotation is markedly different between the two sets. The fixed-entropy-gradient simulations have strong differential rotation contrast and isocontours tilted along radial lines (in good agreement with the Sun's interior rotation revealed by helioseismology), whereas the fixed-entropy simulations have weaker contrast and contours tilted in the opposite sense. We examine in detail the force balances in our models and find that the poleward transport of heat by Busse columns drives a thermal wind responsible for the different rotation profiles. We conclude that the Sun's strong differential rotation along radial lines may result from the solar emissivity being invariant with latitude (which is similar to the fixed-entropy-gradient condition in our models) and the poleward transport of heat by Busse columns. In future work on convection in the solar context, we strongly advise modelers to use a fixed-gradient outer boundary condition.

astro-ph.SR↗

arXiv2019

An improved multi-ridge fitting method for ring-diagram helioseismic analysis

Context: There is a wide discrepancy in current estimates of the strength of convection flows in the solar interior obtained using different helioseismic methods applied to observations from SDO/HMI. The cause for these disparities is not known. Aims: As one step in the effort to resolve this discrepancy, we aim to characterize the multi-ridge fitting code for ring-diagram helioseismic analysis that is used to obtain flow estimates from local power spectra of solar oscillations. Methods: We updated the multi-ridge fitting code developed by Greer et al.(2014) to solve several problems we identified through our inspection of the code. In particular, we changed the merit function to account for the smoothing of the power spectra, model for the power spectrum, and noise estimates. We used Monte Carlo simulations to generate synthetic data and to characterize the noise and bias of the updated code by fitting these synthetic data. Results: The bias in the output fit parameters, apart from the parameter describing the amplitude of the p-mode resonances in the power spectrum, is below what can be measured from the Monte-Carlo simulations. The amplitude parameters are underestimated; this is a consequence of choosing to fit the logarithm of the averaged power. We defer fixing this problem as it is well understood and not significant for measuring flows in the solar interior. The scatter in the fit parameters from the Monte-Carlo simulations is well-modeled by the formal error estimates from the code. Conclusions: We document and demonstrate a reliable multi-ridge fitting method for ring-diagram analysis. The differences between the updated fitting results and the original results are less than one order of magnitude and therefore we suspect that the changes will not eliminate the aforementioned orders-of-magnitude discrepancy in the amplitude of convective flows in the solar interior.

astro-ph.SR↗