Turbulent Disk Research Articles

Magnetorotational dynamo chimeras

In Keplerian accretion disks, turbulence and magnetic fields may be jointly excited through a subcritical dynamo process involving the magnetorotational instability (MRI). High-resolution simulations exhibit a tendency towards statistical self-organization of MRI dynamo turbulence into large-scale cyclic dynamics. Understanding the physical origin of these structures, and whether they can be sustained and transport angular momentum efficiently in astrophysical conditions, represents a significant theoretical challenge. The discovery of simple periodic nonlinear MRI dynamo solutions has recently proven useful in this respect, and has notably served to highlight the role of turbulent magnetic diffusion in the seeming decay of the dynamics at low magnetic Prandtl number Pm (magnetic diffusivity larger than viscosity), a common regime in accretion disks. The connection between these simple structures and the statistical organization reported in turbulent simulations remained elusive, though. Here, we report the numerical discovery in moderate aspect ratio Keplerian shearing boxes of new periodic, incompressible, three-dimensional nonlinear MRI dynamo solutions with a larger dynamical complexity reminiscent of such simulations. These "chimera" cycles are characterized by multiple MRI-unstable dynamical stages, but their basic physical principles of self-sustainment are nevertheless identical to those of simpler cycles found in azimuthally elongated boxes. In particular, we find that they are not sustained at low Pm either due to subcritical turbulent magnetic diffusion. These solutions offer a new perspective into the transition from laminar to turbulent instability-driven dynamos, and may prove useful to devise improved statistical models of turbulent accretion disk dynamos.

Turbulent-diffusion Mediated CO Depletion in Weakly Turbulent Protoplanetary Disks

Abstract Volatiles, especially CO, are important gas tracers of protoplanetary disks (PPDs). Freeze-out and sublimation processes determine their division between gas and solid phases, which affects both which disk regions can be traced by which volatiles, and the formation and composition of planets. Recently, multiple lines of evidence have suggested that CO is substantially depleted from the gas in the outer regions of PPDs, i.e., more depleted than would be expected from a simple balance between freeze-out and sublimation. In this paper, we show that the gas dynamics in the outer PPDs facilitates volatile depletion through turbulent diffusion. Using a simple 1D model that incorporates dust settling, turbulent diffusion of dust and volatiles, as well as volatile freeze-out/sublimation processes, we find that as long as turbulence in the cold midplane is sufficiently weak to allow a majority of the small grains to settle, CO in the warm surface layer can diffuse into the midplane region and deplete by freeze-out. The level of depletion sensitively depends on the level of disk turbulence. Based on recent disk simulations that suggest a layered turbulence profile with very weak midplane turbulence and strong turbulence at the disk surface, CO and other volatiles can be efficiently depleted by up to an order of magnitude over Myr timescales.

The contribution of bulk Comptonization to the soft X-ray excess in AGN

Bulk velocities exceed thermal velocities for sufficiently radiation pressure dominated accretion flows. We model the contribution of bulk Comptonization to the soft X-ray excess in AGN. Bulk Comptonization is due to both turbulence and the background shear. We calculate spectra both taking into account and not taking into account bulk velocities using scaled data from radiation magnetohydrodynamic (MHD) shearing box simulations. We characterize our results with temperatures and optical depths to make contact with other warm Comptonization models of the soft excess. We chose our fiducial mass, $M = 2 \times 10^6 M_{\odot}$, and accretion rate, $L/L_{\rm Edd} = 2.5$, to correspond to those fit to the super-Eddington narrow line Seyfert 1 (NLS1) RE1034+396. The temperatures, optical depths, and Compton $y$ parameters we find broadly agree with those fit to RE1034+396. The effect of bulk Comptonization is to shift the Wien tail to higher energy and lower the gas temperature, broadening the spectrum. Observations of the soft excess in NLS1s can constrain the properties of disc turbulence if the bulk Comptonization contribution can be separated out from contributions from other physical effects, such as reflection and absorption.

Are the Formation and Abundances of Metal-poor Stars the Result of Dust Dynamics?

Abstract Large dust grains can fluctuate dramatically in their local density, relative to the gas, in neutral turbulent disks. Small, high-redshift galaxies (before reionization) represent ideal environments for this process. We show via simple arguments and simulations that order-of-magnitude fluctuations are expected in local abundances of large grains (>100 Å) under these conditions. This can have important consequences for star formation and stellar metal abundances in extremely metal-poor stars. Low-mass stars can form in dust-enhanced regions almost immediately after some dust forms even if the galaxy-average metallicity is too low for fragmentation to occur. We argue that the metal abundances of these “promoted” stars may contain interesting signatures as the CNO abundances (concentrated in large carbonaceous grains and ices) and Mg and Si (in large silicate grains) can be enhanced and/or fluctuate almost independently. Remarkably, the otherwise puzzling abundance patterns of some metal-poor stars can be well fit by standard IMF-averaged core-collapse SNe yields if we allow for fluctuating local dust-to-gas ratios. We also show that the observed log-normal distribution of enhancements in these species agrees with our simulations. Moreover, we confirm that Mg and Si are correlated in these stars; the abundance ratios are similar to those in local silicate grains. Meanwhile [Mg/Ca], predicted to be nearly invariant from pure SNe yields, shows very large enhancements and variations up to factors of ≳100 as expected in the dust-promoted model, preferentially in the [C/Fe]-enhanced metal-poor stars. Together, this suggests that (1) dust exists in second-generation star formation, (2) local dust-to-gas ratio fluctuations occur in protogalaxies and can be important for star formation, and (3) the light element abundances of these stars may be affected by the local chemistry of dust where they formed, rather than directly tracing nucleosynthesis from earlier populations.

Particle dynamics in discs with turbulence generated by the vertical shear instability

Among the candidates for generating turbulence in accretion discs in situations with low intrinsic ionization the vertical shear instability (VSI) has become an interesting candidate, as it relies purely on a vertical gradient in the angular velocity. Existing simulations have shown that $\alpha$-values a few times $10^{-4}$ can be generated. The particle growth in the early planet formation phase is determined by the dynamics of dust particles. Here, we address in particular the efficiency of VSI-turbulence in concentrating particles in order to generate overdensities and low collision velocities. We perform 3D numerical hydrodynamical simulations of accretion discs around young stars that include radiative transport and irradiation from the central star. The motion of particles within a size range of a fraction of mm up to several m is followed using standard drag formula. We confirm that under realistic conditions the VSI is able to generate turbulence in full 3D protoplanetary discs. The irradiated disc shows turbulence within 10 to 60au. The mean radial motion of the gas is such that it is directed inward near the midplane and outward in the surface layers. We find that large particles drift inward with the expected speed, while small particles can experience phases of outward drift. Additionally, the particles show bunching behaviour with overdensities reaching 5 times the average value, which is strongest for dimensionless stopping times around unity. Particles in a VSI-turbulent discs are concentrated in large scale turbulent eddies and show low relative speeds that allow for growing collisions. The reached overdensities will also allow for the onset streaming instabilities further enhancing particle growth. The outward drift for small particles at higher disk elevations allows for the transport of processed high temperature material in the Solar System to larger distances.

DYNAMO-HST survey: clumps in nearby massive turbulent discs and the effects of clump clustering on kiloparsec scale measurements of clumps

We present $\sim$100 pc resolution Hubble Space Telescope H$\alpha$ images of 10 galaxies from the DYnamics of Newly-Assembled Massive Objects (DYNAMO) survey of low-$z$ turbulent disk galaxies, and use these to undertake the first detailed systematic study of the effects of resolution and clump clustering on observations of clumps in turbulent disks. In the DYNAMO-{\em HST} sample we measure clump diameters spanning the range $d_{clump} \sim 100-800$~pc, and individual clump star formation rates as high as $\sim5$~M$_{\odot}$~yr$^{-1}$. DYNAMO clumps have very high SFR surface densities, $\Sigma_{SFR}\sim 15$~M$_{\odot}$~yr$^{-1}$~kpc$^{-2}$, $\sim100\times$ higher than in H{\sc ii} regions of nearby spirals. Indeed, SFR surface density provides a simple dividing line between massive star forming clumps and local star forming regions, where massive star forming clumps have $\Sigma_{SFR}> 0.5$~M$_{\odot}$~yr$^{-1}$~kpc$^{-2}$. When degraded to match the observations of galaxies in $z\sim 1-3$ surveys, DYNAMO galaxies are similar in morphology and measured clump properties to clumpy galaxies observed in the high-$z$ Universe. Emission peaks in the simulated high-redshift maps typically correspond to multiple clumps in full resolution images. This clustering of clumps systematically increases the apparent size and SFR of clumps in 1~kpc resolution maps, and decreases the measured SFR surface density of clumps by as much as a factor of 20$\times$. From these results we can infer that clump clustering is likely to strongly effect the measured properties of clumps in high-$z$ galaxies, which commonly have kiloparsec scale resolution.

On the maximum grain size entrained by photoevaporative winds

We model the behaviour of dust grains entrained by photoevaporation-driven winds from protoplanetary discs assuming a non-rotating, plane-parallel disc. We obtain an analytic expression for the maximum entrainable grain size in extreme-UV radiation-driven winds, which we demonstrate to be proportional to the mass-loss rate of the disc. When compared with our hydrodynamic simulations, the model reproduces almost all of the wind properties for the gas and dust. In typical turbulent discs, the entrained grain sizes in the wind are smaller than the theoretical maximum everywhere but the inner disc due to dust settling.

MASS TRANSPORT AND TURBULENCE IN GRAVITATIONALLY UNSTABLE DISK GALAXIES. II. THE EFFECTS OF STAR FORMATION FEEDBACK

ABSTRACT Self-gravity and stellar feedback are capable of driving turbulence and transporting mass and angular momentum in disk galaxies, but the balance between them is not well understood. In the previous paper in this series, we showed that gravity alone can drive turbulence in galactic disks, regulate their Toomre Q parameters to ∼1, and transport mass inwards at a rate sufficient to fuel star formation in the centers of present-day galaxies. In this paper we extend our models to include the effects of star formation feedback. We show that feedback suppresses galaxies’ star formation rates by a factor of ∼5 and leads to the formation of a multi-phase atomic and molecular interstellar medium. Both the star formation rate and the phase balance produced in our simulations agree well with observations of nearby spirals. After our galaxies reach steady state, we find that the inclusion of feedback actually lowers the gas velocity dispersion slightly compared to the case of pure self-gravity, and also slightly reduces the rate of inward mass transport. Nevertheless, we find that, even with feedback included, our galactic disks self-regulate to Q ∼ 1, and transport mass inwards at a rate sufficient to supply a substantial fraction of the inner disk star formation. We argue that gravitational instability is therefore likely to be the dominant source of turbulence and transport in galactic disks, and that it is responsible for fueling star formation in the inner parts of galactic disks over cosmological times.

PEBBLE ACCRETION AND THE DIVERSITY OF PLANETARY SYSTEMS

ABSTRACT This paper examines the standard model of planet formation, including pebble accretion, using numerical simulations. Planetary embryos that are large enough to become giant planets do not form beyond the ice line within a typical disk lifetime unless icy pebbles stick at higher speeds than in experiments using rocky pebbles. Systems like the solar system (small inner planets and giant outer planets) can form if icy pebbles are stickier than rocky pebbles, and if the planetesimal formation efficiency increases with pebble size, which prevents the formation of massive terrestrial planets. Growth beyond the ice line is dominated by pebble accretion. Most growth occurs early, when the surface density of the pebbles is high due to inward drift of the pebbles from the outer disk. Growth is much slower after the outer disk is depleted. The outcome is sensitive to the disk radius and turbulence level, which control the lifetime and maximum size of pebbles. The outcome is sensitive to the size of the largest planetesimals because there is a threshold mass for the onset of pebble accretion. The planetesimal formation rate is unimportant, provided that some large planetesimals form while the pebbles remain abundant. Two outcomes are seen, depending on whether pebble accretion begins while the pebbles are still abundant. Either multiple gas-giant planets form beyond the ice line, small planets form close to the star, and a Kuiper-belt-like disk of bodies is scattered outward by the giant planets; or no giants form and the bodies remain an Earth-mass or smaller.

ON THE FORMATION OF SUPER-EARTHS WITH IMPLICATIONS FOR THE SOLAR SYSTEM

ABSTRACT We first consider how the level of turbulence in a protoplanetary disk affects the formation locations for the observed close-in super-Earths in exosolar systems. We find that a protoplanetary disk that includes a dead zone (a region of low turbulence) has substantially more material in the inner parts of the disk, possibly allowing for in situ formation. For the dead zone to last the entire lifetime of the disk requires the active layer surface density to be sufficiently small, . Migration through a dead zone may be very slow and thus super-Earth formation followed by migration toward the star through the dead zone is less likely. For fully turbulent disks, there is not enough material for in situ formation. However, in this case, super-Earths can form farther out in the disk and migrate inward on a reasonable timescale. We suggest that both of these formation mechanisms operate in different planetary systems. This can help to explain the observed large range in densities of super-Earths because the formation location determines the composition. Furthermore, we speculate that super-Earths could have formed in the inner parts of our solar system and cleared the material in the region inside of Mercury’s orbit. The super-Earths could migrate through the gas disk and fall into the Sun if the disk was sufficiently cool during the final gas disk accretion process. While it is definitely possible to meet all of these requirements, we don’t expect them to occur in all systems, which may explain why the solar system is somewhat special in its lack of super-Earths.

The effects of a magnetic field on planetary migration in laminar and turbulent discs

We investigate the migration of low-mass planets ($5 M_{\oplus}$ and $20 M_{\oplus}$) in accretion discs threaded with a magnetic field using 2D MHD code in polar coordinates. We observed that, in the case of a strong azimuthal magnetic field where the plasma parameter is $\beta\sim 1-2$, density waves at the magnetic resonances exert a positive torque on the planet and may slow down or reverse its migration. However, when the magnetic field is weaker (i.e., the plasma parameter $\beta$ is relatively large), then non-axisymmetric density waves excited by the planet lead to growth of the radial component of the field and, subsequently, to development of the magneto-rotational instability, such that the disc becomes turbulent. Migration in a turbulent disc is stochastic, and the migration direction may change as such. To understand migration in a turbulent disc, both the interaction between a planet and individual turbulent cells, as well as the interaction between a planet and ordered density waves, have been investigated.

SINTERING-INDUCED DUST RING FORMATION IN PROTOPLANETARY DISKS: APPLICATION TO THE HL TAU DISK

ABSTRACT The latest observation of HL Tau by ALMA revealed spectacular concentric dust rings in its circumstellar disk. We attempt to explain the multiple ring structure as a consequence of aggregate sintering. Sintering is known to reduce the sticking efficiency of dust aggregates and occurs at temperatures slightly below the sublimation point of the constituent material. We present a dust growth model that incorporates sintering and use it to simulate global dust evolution due to sintering, coagulation, fragmentation, and radial inward drift in a modeled HL Tau disk. We show that aggregates consisting of multiple species of volatile ices experience sintering, collisionally disrupt, and pile up at multiple locations slightly outside the snow lines of the volatiles. At wavelengths of 0.87–1.3 mm, these sintering zones appear as bright, optically thick rings with a spectral slope of ≈ 2 , whereas the non-sintering zones appear as darker, optically thinner rings of a spectral slope of ≈ 2.3 –2.5. The observational features of the sintering and non-sintering zones are consistent with those of the major bright and dark rings found in the HL Tau disk, respectively. Radial pileup and vertical settling occur simultaneously if disk turbulence is weak and if monomers constituting the aggregates are ∼ 1 μ m in radius. For the radial gas temperature profile of T = 310 ( r / 1 au ) − 0.57 K , our model perfectly reproduces the brightness temperatures of the optically thick bright rings and reproduces their orbital distances to an accuracy of ≲ 30 % .

A ONE-DIMENSIONAL MODEL FOR RAYLEIGH–TAYLOR INSTABILITY IN SUPERNOVA REMNANTS

ABSTRACT This study presents a method for approximating the multidimensional effects of Rayleigh–Taylor (RT) instability as a modification of the one-dimensional (1D) hydro equations. This modification is similar to the Shakura–Sunyaev α prescription for modeling the coarse-grained effects of turbulence in astrophysical disks. The model introduces several dimensionless tunable parameters that are calibrated by comparing with high-resolution two-dimensional axisymmetric numerical calculations of RT unstable flows. A complete description of the model is presented, along with a handful of test problems that demonstrate the extent to which the 1D model is able to reproduce multidimensional effects.

Supernova feedback in a local vertically stratified medium: interstellar turbulence and galactic winds

We use local Cartesian simulations with a vertical gravitational potential to study how supernova (SN) feedback in stratified galactic discs drives turbulence and launches galactic winds. Our analysis includes three disc models with gas surface densities ranging from Milky Way-like galaxies to gas-rich ultra-luminous infrared galaxies (ULIRGs), and two different SN driving schemes (random and correlated with local gas density). In order to isolate the physics of SN feedback, we do not include additional feedback processes. We find that, in these local box calculations, SN feedback excites relatively low mass-weighted gas turbulent velocity dispersions ~3-7 km/s and low wind mass loading factors < 1 in all the cases we study. The low turbulent velocities and wind mass loading factors predicted by our local box calculations are significantly below those suggested by observations of gas-rich and rapidly star-forming galaxies; they are also in tension with global simulations of disc galaxies regulated by stellar feedback. Using a combination of numerical tests and analytic arguments, we argue that local Cartesian boxes cannot predict the properties of galactic winds because they do not capture the correct global geometry and gravitational potential of galaxies. The wind mass loading factors are in fact not well-defined in local simulations because they decline significantly with increasing box height. More physically realistic calculations (e.g., including a global galactic potential and disc rotation) will likely be needed to fully understand disc turbulence and galactic outflows, even for the idealized case of feedback by SNe alone.

Bulk Comptonization by turbulence in accretion discs

Radiation pressure dominated accretion discs around compact objects may have turbulent velocities that greatly exceed the electron thermal velocities within the disc. Bulk Comptonization by the turbulence may therefore dominate over thermal Comptonization in determining the emergent spectrum. Bulk Comptonization by divergenceless turbulence is due to radiation viscous dissipation only. It can be treated as thermal Comptonization by solving the Kompaneets equation with an equivalent "wave" temperature, which is a weighted sum over the power present at each scale in the turbulent cascade. Bulk Comptonization by turbulence with non-zero divergence is due to both pressure work and radiation viscous dissipation. Pressure work has negligible effect on photon spectra in the limit of optically thin turbulence, and in this limit radiation viscous dissipation alone can be treated as thermal Comptonization with a temperature equivalent to the full turbulent power. In the limit of extremely optically thick turbulence, radiation viscous dissipation is suppressed, and the evolution of local photon spectra can be understood in terms of compression and expansion of the strongly coupled photon and gas fluids. We discuss the consequences of these effects for self-consistently resolving and interpreting turbulent Comptonization in spectral calculations in radiation MHD simulations of high luminosity accretion flows.

On the water delivery to terrestrial embryos by ice pebble accretion

Standard accretion disk models suggest that the snow line in the solar nebula migrated interior to the Earth's orbit in a late stage of nebula evolution. In this late stage, a significant amount of ice could have been delivered to 1 AU from outer regions in the form of mm to dm-sized pebbles. This raises the question why the present Earth is so depleted of water (with the ocean mass being as small as 0.023% of the Earth mass). Here we quantify the amount of icy pebbles accreted by terrestrial embryos after the migration of the snow line assuming that no mechanism halts the pebble flow in outer disk regions. We use a simplified version of the coagulation equation to calculate the formation and radial inward drift of icy pebbles in a protoplanetary disk. The pebble accretion cross section of an embryo is calculated using analytic expressions presented by recent studies. We find that the final mass and water content of terrestrial embryos strongly depends on the radial extent of the gas disk, the strength of disk turbulence, and the time at which the snow lines arrives at 1 AU. The disk's radial extent sets the lifetime of the pebble flow, while turbulence determines the density of pebbles at the midplane where the embryos reside. We find that the final water content of the embryos falls below 0.023 wt% only if the disk is compact (< 100 AU), turbulence is strong at 1 AU, and the snow line arrives at 1 AU later than 2-4 Myr after disk formation. If the solar nebula extended to 300 AU, initially rocky embryos would have evolved into icy planets of 1-10 Earth masses unless the snow-line migration was slow. If the proto-Earth contained water of ~ 1 wt% as might be suggested by the density deficit of the Earth's outer core, the formation of the proto-Earth was possible with weaker turbulence and with earlier (> 0.5-2 Myr) snow-line migration.

On the technique for the recovery of the spectrum of turbulence in astrophysical discs

We present a method that can be used to recover the spectrum of turbulence from observations of optically thin emission lines formed in astrophysical disks. Within this method we analyze how line intensity fluctuations depend on the angular resolution of the instrument, used for the observations. The method allows us to restore the slope of the power spectrum of velocity turbulent pulsations and estimate the upper boundary of the turbulence scale.

Falling outer rotation curves of star-forming galaxies at 0.7 &lt; z &lt; 2.6 probed with KMOS3D and SINS/zC-SINF

AbstractWe exploit the deep Hα IFU kinematic data from the KMOS3D and SINS/zC-SINF surveys to explore the so far unconstrained outer rotation curves of star-forming disk galaxies at high redshift. Through stacking the signal of ~ 100 massive disks at 0.7 &lt; z &lt; 2.6, we construct a representative rotation curve reaching out to several effective radii. Our stacked rotation curve exhibits a turnover with a steep falloff in the outer regions, significantly strengthening the tantalizing evidence previously hinted at in a handful only of individual disks among the sample with the deepest data.This finding confirms the high baryon fractions found by comparing the stellar, gas and dynamical masses of high redshift galaxies independently of assumptions on the light-to-mass conversion and Initial stellar Mass Function (IMF). The rapid falloff of the stacked rotation curve is most naturally explained by the effects of pressure gradients, which are significant in the gas-rich, turbulent high-z disks and which would imply a possible pressure-driven truncation of the outer disk.

The KMOS Redshift One Spectroscopic Survey (KROSS): dynamical properties, gas and dark matter fractions of typicalz∼ 1 star-forming galaxies

The KMOS Redshift One Spectroscopic Survey (KROSS) is an ESO guaranteed time survey of 795 typical star-forming galaxies in the redshift range z=0.8-1.0 with the KMOS instrument on the VLT. In this paper we present resolved kinematics and star formation rates for 584 z~1 galaxies. This constitutes the largest near-infrared Integral Field Unit survey of galaxies at z~1 to date. We demonstrate the success of our selection criteria with 90% of our targets found to be Halpha emitters, of which 81% are spatially resolved. The fraction of the resolved KROSS sample with dynamics dominated by ordered rotation is found to be 83$\pm$5%. However, when compared with local samples these are turbulent discs with high gas to baryonic mass fractions, ~35%, and the majority are consistent with being marginally unstable (Toomre Q~1). There is no strong correlation between galaxy averaged velocity dispersion and the total star formation rate, suggesting that feedback from star formation is not the origin of the elevated turbulence. We postulate that it is the ubiquity of high (likely molecular) gas fractions and the associated gravitational instabilities that drive the elevated star-formation rates in these typical z~1 galaxies, leading to the ten-fold enhanced star-formation rate density. Finally, by comparing the gas masses obtained from inverting the star-formation law with the dynamical and stellar masses, we infer an average dark matter to total mass fraction within 2.2$r_e$ (9.5kpc) of 65$\pm$12%, in agreement with the results from hydrodynamic simulations of galaxy formation.

FROM PLANETESIMALS TO PLANETS IN TURBULENT PROTOPLANETARY DISKS. I. ONSET OF RUNAWAY GROWTH

ABSTRACT When planetesimals grow via collisions in a turbulent disk, stirring through density fluctuation caused by turbulence effectively increases the relative velocities between planetesimals, which suppresses the onset of runaway growth. We investigate the onset of runaway growth in a turbulent disk through simulations that calculate the mass and velocity evolution of planetesimals. When planetesimals are small, the average relative velocity between planetesimals, v r ?> , is much greater than their surface escape velocity, v esc ?> , so that runaway growth does not occur. As planetesimals become large via collisional growth, v r ?> approaches v esc ?> . When v r &approx; 1.5 v esc ?> , runaway growth of the planetesimals occurs. During the oligarchic growth subsequent to runaway growth, a small number of planetary embryos produced via runaway growth become massive through collisions with planetesimals with radii of that at the onset of runaway growth, r p,run ?> . We analytically derive r p,run ?> as a function of the turbulent strength. Growing &sim; 10 M &oplus; ?> embryos that are suitable to become the cores of Jupiter and Saturn requires r p,run &sim; 100 ?> km, which is similar to the proposed fossil feature in the size distribution of main belt asteroids. In contrast, the formation of Mars as quickly as suggested from Hf-W isotope studies requires small planetesimals at the onset of runaway growth. Thus, the conditions required to form Mars, Jupiter, and Saturn and the size distribution of the main-belt asteroids indicate that the turbulence increased in amplitude relative to the sound speed with increasing distance from the young Sun.