Local Pressure Maxima Research Articles

A Dust-trapping Ring in the Planet-hosting Disk of Elias 2-24

Rings and gaps are among the most widely observed forms of substructure in protoplanetary disks. A gap–ring pair may be formed when a planet carves a gap in the disk, which produces a local pressure maximum following the gap that traps inwardly drifting dust grains and appears as a bright ring owing to the enhanced dust density. A dust-trapping ring would provide a promising environment for solid growth and possibly planetesimal production via the streaming instability. We present evidence of dust trapping in the bright ring of the planet-hosting disk Elias 2-24, from the analysis of 1.3 and 3 mm Atacama Large Millimeter/submillimeter Array observations at high spatial resolution (0.″029, 4.0 au). We leverage the high spatial resolution to demonstrate that larger grains are more efficiently trapped and place constraints on the local turbulence (8 × 10−4 < α turb < 0.03) and the gas-to-dust ratio (Σ g /Σ d < 30) in the ring. Using a scattering-included marginal probability analysis, we measure a total dust disk mass of Mdust=13.8−0.5+0.7×10−4M⊙ . We also show that at the orbital radius of the proposed perturber the gap is cleared of material down to a flux contrast of 10−3 of the peak flux in the disk.

Shoulder of Dust Rings Formed by Planet–Disk Interactions

Recent analyses of millimeter-wavelength protoplanetary disk observations have revealed several emission excesses on the previously identified dust rings, referred to as dust shoulders. The prevalence of dust shoulders suggests that they trace a common but unclear mechanism. In this work, we combine 3D, multifluid hydrodynamic simulations with radiative transfer calculations to explain the formation of dust shoulders. We find that the ring-shoulder pairs can result from the 3D planet–disk interactions with massive, gap-opening planets. The key driver is the dust filtration effect at the local pressure maximum due to planet-driven outward gas flows. Our work provides a possible explanation for the outer dust shoulders in recent super-resolution analyses of Atacama Large Millimeter/submillimeter Array observations. It also provides insights into the formation of the inner dust shoulder in the PDS 70 disk and highlights the role of 3D effects in planet–disk interaction studies.

Equilibrium non-self-gravitating tori around black holes in parametrized spherically symmetric space–times

ABSTRACT Non-self-gravitating equilibrium tori orbiting around black holes have a long history and have been employed in numerous simulations of accretion flows on to black holes and other compact objects. We have revisited the problem of constructing such equilibria starting from spherically symmetric black-hole space–times expressed in terms of a fully generic and rapidly converging parametrization: the Rezzolla–Zhidenko metric. Within this framework, we have extended the definitions of all of the quantities characterizing these equilibria, starting from the concept of the von Zeipel cylinders and up to the possible ranges of the specific angular momenta that are employed to construct families of tori. Within the allowed space of parameters we have then encountered both standard ‘single-torus’ solutions and non-standard ‘double-tori’ solutions. While the properties of the first ones in terms of the presence of a single cusp, of a local pressure maximum and of a varying outer radius, are very similar to those encountered in general relativity, the properties of double-tori solutions are far richer and naturally allow for configurations having the same constant specific angular momentum and hence are potentially easier to produce in nature. The existence of these objects is at present very hypothetical, but if these equilibrium tori were to be observed, they would provide very valuable information on the properties of the space–time and on its deviation from general relativity.

On the Origin of Dust Structures in Protoplanetary Disks: Constraints from the Rossby Wave Instability

High-resolution submillimeter observations of protoplanetary disks with ALMA have revealed that dust rings are common in large, bright disks. The leading explanation for these structures is dust trapping in a local gas pressure maximum, caused by an embedded planet or other dynamical process. Independent of origin, such dust traps should be stable for many orbits to collect significant dust. However, ringlike perturbations in gas disks are also known to trigger the Rossby wave instability (RWI). We investigate whether axisymmetric pressure bumps can simultaneously trap dust and remain stable to the RWI. The answer depends on the thermodynamic properties of pressure bumps. For isothermal bumps, dust traps are RWI stable for widths from ∼1 to several gas scale heights. Adiabatic dust traps are stable over a smaller range of widths. For temperature bumps with no surface density component, however, all dust traps tend to be unstable. Smaller values of disk aspect ratio allow stable dust trapping at lower bump amplitudes and over a larger range of widths. We also report a new approximate criterion for RWI. Instability occurs when the radial oscillation frequency is ≲75% of the Keplerian frequency, which differs from the well-known Lovelace necessary (but not sufficient) criterion for instability. Our results can guide ALMA observations of molecular gas by constraining the resolution and sensitivity needed to identify the pressure bumps thought to be responsible for dust rings.

Dust Settling and Clumping in MRI-turbulent Outer Protoplanetary Disks

Abstract Planetesimal formation is a crucial yet poorly understood process in planet formation. It is widely believed that planetesimal formation is the outcome of dust clumping by the streaming instability (SI). However, recent analytical and numerical studies have shown that the SI can be damped or suppressed by external turbulence, and at least the outer regions of protoplanetary disks are likely weakly turbulent due to magneto-rotational instability (MRI). We conduct high-resolution local shearing-box simulations of hybrid particle-gas magnetohydrodynamics (MHD), incorporating ambipolar diffusion as the dominant nonideal MHD effect, applicable to outer disk regions. We first show that dust backreaction enhances dust settling toward the midplane by reducing turbulence correlation time. Under modest level of MRI turbulence, we find that dust clumping is in fact easier than the conventional SI case, in the sense that the threshold of solid abundance for clumping is lower. The key to dust clumping includes dust backreaction and the presence of local pressure maxima, which in our work is formed by the MRI zonal flows overcoming background pressure gradient. Overall, our results support planetesimal formation in the MRI-turbulent outer protoplanetary disks, especially in ring-like substructures.

MRI-active inner regions of protoplanetary discs – II. Dependence on dust, disc, and stellar parameters

ABSTRACT Close-in super-Earths are the most abundant exoplanets known. It has been hypothesized that they form in the inner regions of protoplanetary discs, out of the dust that may accumulate at the boundary between the inner region susceptible to the magneto-rotational instability (MRI) and an MRI-dead zone further out. In Paper I, we presented a model for the viscous inner disc which includes heating due to both irradiation and MRI-driven accretion; thermal and non-thermal ionization; dust opacities; and dust effects on ionization. Here, we examine how the inner disc structure varies with stellar, disc, and dust parameters. For high accretion rates and small dust grains, we find that: (1) the main sources of ionization are thermal ionization and thermionic and ion emission; (2) the disc features a hot, high-viscosity inner region, and a local gas pressure maximum at the outer edge of this region (in line with previous studies); and (3) an increase in the dust-to-gas ratio pushes the pressure maximum outwards. Consequently, dust can accumulate in such inner discs without suppressing the MRI, with the amount of accumulation depending on the viscosity in the MRI-dead regions. Conversely, for low accretion rates and large dust grains, there appears to be an additional steady-state solution in which: (1) stellar X-rays become the main source of ionization; (2) MRI-viscosity is high throughout the disc; and (3) the pressure maximum ceases to exist. Hence, if planets form in the inner disc, larger accretion rates (and thus younger discs) are favoured.

A dusty filament and turbulent CO spirals in HD 135344B - SAO 206462

ABSTRACT Planet–disc interactions build up local pressure maxima that may halt the radial drift of protoplanetary dust, and pile it up in rings and crescents. ALMA observations of the HD 135344B disc revealed two rings in the thermal continuum stemming from ∼mm-sized dust. At higher frequencies the inner ring is brighter relative to the outer ring, which is also shaped as a crescent rather than a full ring. In near-IR scattered light images, the disc is modulated by a two-armed grand-design spiral originating inside the ALMA inner ring. Such structures may be induced by a massive companion evacuating the central cavity, and by a giant planet in the gap separating both rings, that channels the accretion of small dust and gas through its filamentary wakes while stopping the larger dust from crossing the gap. Here we present ALMA observations in the J = (2 − 1) CO isotopologue lines and in the adjacent continuum, with up to 12 km baselines. Angular resolutions of ∼0${_{.}^{\prime\prime}}$03 reveal the tentative detection of a filament connecting both rings, and which coincides with a local discontinuity in the pitch angle of the IR spiral, proposed previously as the location of the protoplanet driving this spiral. Line diagnostics suggests that turbulence, or superposed velocity components, is particularly strong in the spirals. The 12CO(2-1) 3D rotation curve points at stellocentric accretion at radii within the inner dust ring, with a radial velocity of up to ${\sim}5{{\ \rm per\ cent}}\pm 0.5{{\ \rm per\ cent}}$ Keplerian, which corresponds to an excessively large accretion rate of ${\sim}2\times 10^{-6}\, M_\odot \,$yr−1 if all of the CO layer follows the 12CO(2-1) kinematics. This suggests that only the surface layers of the disc are undergoing accretion, and that the line broadening is due to superposed laminar flows.

The Effect of a Strong Pressure Bump in the Sun’s Natal Disk: Terrestrial Planet Formation via Planetesimal Accretion Rather than Pebble Accretion

Abstract Mass-independent isotopic anomalies of carbonaceous and noncarbonaceous meteorites show a clear dichotomy suggesting an efficient separation of the inner and outer solar system. Observations show that ring-like structures in the distribution of millimeter-sized pebbles in protoplanetary disks are common. These structures are often associated with drifting pebbles being trapped by local pressure maxima in the gas disk. Similar structures may also have existed in the Sun’s natal disk, which could naturally explain the meteorite/planetary isotopic dichotomy. Here, we test the effects of a strong pressure bump in the outer disk (e.g., ∼5 au) on the formation of the inner solar system. We model dust coagulation and evolution, planetesimal formation, as well as embryo growth via planetesimal and pebble accretion. Our results show that terrestrial embryos formed via planetesimal accretion rather than pebble accretion. In our model, the radial drift of pebbles fosters planetesimal formation. However, once a pressure bump forms, pebbles in the inner disk are lost via drift before they can be efficiently accreted by embryos growing at ⪆1 au. Embryos inside ∼0.5–1.0 au grow relatively faster and can accrete pebbles more efficiently. However, these same embryos grow to larger masses so they should migrate inwards substantially, which is inconsistent with the current solar system. Therefore, terrestrial planets most likely accreted from giant impacts of Moon to roughly Mars-mass planetary embryos formed around ⪆1.0 au. Finally, our simulations produce a steep radial mass distribution of planetesimals in the terrestrial region, which is qualitatively aligned with formation models suggesting that the asteroid belt was born low mass.

Properties of the pressure field in highly nonlinear free surface flows with critical jet

In this note we examine fluid violent kinematics in a plunging breaker. The fluid motion is computed in the frame of the potential flow theory. The fluid kinematics is basically generated by forcing the motion of a rectangular tank, starting from rest. When a sufficient level of energy is injected in the fluid, the free surface has highly nonlinear behavior, here a plunging breaker. In the vicinity of the main tip crest, a sharp corner (cusp) appears along the surface of the barrel. The appearance of this critical jet is described and discussed in terms of the spatial and temporal variations of the pressure field. In the present case a local pressure maximum is captured that follows a continuous decreasing pressure gradient in a region of positive Gaussian curvature of the pressure. It is also shown that, at the free surface before the appearance of the critical jet, there is a strong correlation between the change of sign of the Gaussian curvature of the pressure on the one hand and the radius of curvature of the free surface profile on the other hand.

Accretion bursts in magnetized gas-dust protoplanetary disks

Aims. Accretion bursts triggered by the magnetorotational instability (MRI) in the innermost disk regions were studied for protoplanetary gas-dust disks that formed from prestellar cores of a various mass Mcore and mass-to-magnetic flux ratio λ. Methods. Numerical magnetohydrodynamics simulations in the thin-disk limit were employed to study the long-term (~1.0 Myr) evolution of protoplanetary disks with an adaptive turbulent α-parameter, which explicitly depends on the strength of the magnetic field and ionization fraction in the disk. The numerical models also feature the co-evolution of gas and dust, including the back-reaction of dust on gas and dust growth. Results. A dead zone with a low ionization fraction of x≲10−13 and temperature on the order of several hundred Kelvin forms in the inner disk soon after its formation, extending from several to several tens of astronomical units depending on the model. The dead zone features pronounced dust rings that are formed due to the concentration of grown dust particles in the local pressure maxima. Thermal ionization of alkaline metals in the dead zone trigger the MRI and associated accretion burst, which is characterized by a sharp rise, small-scale variability in the active phase, and fast decline once the inner MRI-active region is depleted of matter. The burst occurrence frequency is highest in the initial stages of disk formation and is driven by gravitational instability (GI), but it declines with diminishing disk mass-loading from the infalling envelope. There is a causal link between the initial burst activity and the strength of GI in the disk fueled by mass infall from the envelope. We find that the MRI-driven burst phenomenon occurs for λ = 2–10, but diminishes in models with Mcore ≲ M⊙, suggesting a lower limit on the stellar mass for which the MRI-triggered burst can occur. Conclusions. The MRI-triggered bursts occur for a wide range of mass-to-magnetic flux ratios and initial cloud core masses. The burst occurrence frequency is highest in the initial disk formation stage and reduces as the disk evolves from a gravitationally unstable to a viscous-dominated state. The MRI-triggered bursts are intrinsically connected with the dust rings in the inner disk regions, and both can be a manifestation of the same phenomenon, that is to say the formation of a dead zone.

Observations of Protoplanetary Disk Structures

The disks that orbit young stars are the essential conduits and reservoirs of material for star and planet formation. Their structures, meaning the spatial variations of the disk physical conditions, reflect the underlying mechanisms that drive those formation processes. Observations of the solids and gas in these disks, particularly at high resolution, provide fundamental insights on their mass distributions, dynamical states, and evolutionary behaviors. Over the past decade, rapid developments in these areas have largely been driven by observations with the Atacama Large Millimeter/submillimeter Array (ALMA). This review highlights the state of observational research on disk structures, emphasizing the following three key conclusions that reflect the main branches of the field: ▪ Relationships among disk structure properties are also linked to the masses, environments, and evolutionary states of their stellar hosts. ▪ There is clear, qualitative evidence for the growth and migration of disk solids, although the implied evolutionary timescales suggest the classical assumption of a smooth gas disk is inappropriate. ▪ Small-scale substructures with a variety of morphologies, locations, scales, and amplitudes—presumably tracing local gas pressure maxima—broadly influence the physical and observational properties of disks. The last point especially is reshaping the field, with the recognition that these disk substructures likely trace active sites of planetesimal growth or are the hallmarks of planetary systems at their formation epoch.

Dual-wavelength ALMA Observations of Dust Rings in Protoplanetary Disks

Abstract We present new Atacama Large Millimeter/submillimeter Array observations for three protoplanetary disks in Taurus at 2.9 mm and comparisons with previous 1.3 mm data both at an angular resolution of ∼0.″1 (15 au for the distance of Taurus). In the single-ring disk DS Tau, double-ring disk GO Tau, and multiring disk DL Tau, the same rings are detected at both wavelengths, with radial locations spanning from 50 to 120 au. To quantify the dust emission morphology, the observed visibilities are modeled with a parametric prescription for the radial intensity profile. The disk outer radii, taken as 95% of the total flux encircled in the model intensity profiles, are consistent at both wavelengths for the three disks. Dust evolution models show that dust trapping in local pressure maxima in the outer disk could explain the observed patterns. Dust rings are mostly unresolved. The marginally resolved ring in DS Tau shows a tentatively narrower ring at the longer wavelength, an observational feature expected from efficient dust trapping. The spectral index (α mm) increases outward and exhibits local minima that correspond to the peaks of dust rings, indicative of the changes in grain properties across the disks. The low optical depths (τ ∼ 0.1–0.2 at 2.9 mm and 0.2–0.4 at 1.3 mm) in the dust rings suggest that grains in the rings may have grown to millimeter sizes. The ubiquitous dust rings in protoplanetary disks modify the overall dynamics and evolution of dust grains, likely paving the way toward the new generation of planet formation.

Hints of a Population of Solar System Analog Planets from ALMA

Abstract The recent Atacama Large Millimeter/submillimeter Array (ALMA) Disk Substructures at High Angular Resolution Project (DSHARP) survey provided illuminating results on the diversity of substructures in planet-forming disks. These substructures trace pebble-sized grains accumulated at local pressure maxima, possibly due to planet–disk interactions or other planet formation processes. DSHARP sources are heavily biased to large and massive disks that only represent the high (dust flux) tail end of the disk population. Thus it is unclear whether similar substructures and corresponding physical processes also occur in the majority of disks that are fainter and more compact. Here we explore the presence and characteristics of features in a compact disk around GQ Lup A, the effective radius of which is 1.5–10 times smaller than those of DSHARP disks. We present our analysis of ALMA 1.3 mm continuum observations of the GQ Lup system. By fitting visibility profiles of the continuum emission, we find substructures including a gap at ∼10 au. The compact disk around GQ Lup exhibits similar substructures to those in the DSHARP sample, suggesting that mechanisms of trapping pebble-sized grains are at work in small disks as well. Characteristics of the feature at ∼10 au, if due to a hidden planet, are evidence of planet formation at Saturnian distances. Our results hint at a rich world of substructures to be identified within the common population of compact disks, and subsequently a population of solar system analogs within these disks. Such study is critical to understanding the formation mechanisms and planet populations in the majority of protoplanetary disks.

Development of a design load patch for the consideration of ice loads

ABSTRACT Ships operating in ice-covered waters may experience structural damage due to ice-induced loads. This can occur even though the ship structure is designed according to rules considering ice loading. The basis of these design rules can be empirical, e.g. measured loads from full-scale trials and observed damages or energy based, which is known as the Popov Method. The rules propose to apply the design ice load as a uniformly loaded rectangular pressure patch. This simplification is, however, not in line with spatial variations of local ice loads and does not account for high pressure zones directly. Therefore, this paper proposes a method for identifying a representative ice load distribution under consideration of local pressure maxima based on a spatial and temporal varying ice load sequence. This identified load is applied to an example structure and the response is compared to the response using a current rule-based patch load. .

Inside-out Planet Formation. V. Structure of the Inner Disk as Implied by the MRI

Abstract The ubiquity of Earth- to super-Earth-sized planets found very close to their host stars has motivated in situ formation models. In particular, inside-out planet formation is a scenario in which planets coalesce sequentially in the disk, at the local gas pressure maximum near the inner boundary of the dead zone. The pressure maximum arises from a decline in viscosity, going from the active innermost disk (where thermal ionization yields high viscosities via the magnetorotational instability [MRI]) to the adjacent dead zone (where the MRI is quenched). Previous studies of the pressure maximum, based on α-disk models, have assumed ad hoc values for the viscosity parameter α in the active zone, ignoring the detailed MRI physics. Here we explicitly couple the MRI criteria to the α-disk equations, to find steady-state solutions for the disk structure. We consider both Ohmic and ambipolar resistivities, a range of disk accretion rates (10−10–10−8 M ⊙ yr−1), stellar masses (0.1–1 M ⊙), and fiducial values of the non-MRI α-viscosity in the dead zone (α DZ = 10−5 to 10−3). We find that (1) a midplane pressure maximum forms radially outside the dead zone inner boundary; (2) Hall resistivity dominates near the inner disk midplane, perhaps explaining why close-in planets do not form in ∼50% of systems; (3) X-ray ionization can compete with thermal ionization in the inner disk, because of the low steady-state surface density there; and (4) our inner disks are viscously unstable to surface density perturbations.

Dust-driven viscous ring-instability in protoplanetary disks

Protoplanetary disks often appear as multiple concentric rings in dust continuum emission maps and scattered light images. These features are often associated with possible young planets in these disks. Many non-planetary explanations have also been suggested, including snow lines, dead zones and secular gravitational instabilities in the dust. In this paper we suggest another potential origin. The presence of copious amounts of dust tends to strongly reduce the conductivity of the gas, thereby inhibiting the magneto-rotational instability, and thus reducing the turbulence in the disk. From viscous disk theory it is known that a disk tends to increase its surface density in regions where the viscosity (i.e. turbulence) is low. Local maxima in the gas pressure tend to attract dust through radial drift, increasing the dust content even more. We have investigated mathematically if this could potentially lead to a feedback loop in which a perturbation in the dust surface density could perturb the gas surface density, leading to increased dust drift and thus amplification of the dust perturbation and, as a consequence, the gas perturbation. We find that this is indeed possible, even for moderately small dust grain sizes, which drift less efficiently, but which are more likely to affect the gas ionization degree. We speculate that this instability could be triggered by the small dust population initially, and when the local pressure maxima are strong enough, the larger dust grains get trapped and lead to the familiar ring-like shapes. We also discuss the many uncertainties and limitations of this model.

Linear growth of streaming instability in pressure bumps

Streaming instability is a powerful mechanism which concentrates dust grains in pro- toplanetary discs, eventually up to the stage where they collapse gravitationally and form planetesimals. Previous studies inferred that it should be ineffective in viscous discs, too efficient in inviscid discs, and may not operate in local pressure maxima where solids accumulate. From a linear analysis of stability, we show that streaming instability behaves differently inside local pressure maxima. Under the action of the strong differential advection imposed by the bump, a novel unstable mode develops and grows even when gas viscosity is large. Hence, pressure bumps are found to be the only places where streaming instability occurs in viscous discs. This offers a promising way to conciliate models of planet formation with recent observations of young discs.

X-ray photoevaporation’s limited success in the formation of planetesimals by the streaming instability

The streaming instability is often invoked as solution to the fragmentation and drift barriers in planetesimal formation, catalyzing the aggregation of dust on kyr timescales to grow km-sized cores. However there remains a lack of consensus on the physical mechanism(s) responsible for initiating it. One potential avenue is disc photoevaporation, wherein the preferential removal of relatively dust-free gas increases the disc metallicity. Late in the disc lifetime, photoevaporation dominates viscous accretion, creating a gradient in the depleted gas surface density near the location of the gap. This induces a local pressure maximum that collects drifting dust particles, which may then become susceptible to the streaming instability. Using a one-dimensional viscous evolution model of a disc subject to internal X-ray photoevaporation, we explore the efficacy of this process to build planetestimals. Over a range of parameters we find that the amount of dust mass converted into planetesimals is often < 1 M_Earth and at most a few M_Earth spread across tens of AU. We conclude that photoevaporation may at best be relevant for the formation of debris discs, rather than a common mechanism for the formation of planetary cores. Our results are in contrast to a recent, similar investigation that considered an FUV-driven photoevaporation model and reported the formation of tens of M_Earth at large (> 100 AU) disc radii. The discrepancies are primarily a consequence of the different photoevaporation profiles assumed. Until observations more tightly constrain photoevaporation models, the relevance of this process to the formation of planets remains uncertain.

3D Radiation Nonideal Magnetohydrodynamical Simulations of the Inner Rim in Protoplanetary Disks

Abstract Many planets orbit within 1 au of their stars, raising questions about their origins. Particularly puzzling are the planets found near the silicate sublimation front. We investigate conditions near the front in the protostellar disk around a young intermediate-mass star, using the first global 3D radiation nonideal MHD simulations in this context. We treat the starlight heating; the silicate grains’ sublimation and deposition at the local, time-varying temperature and density; temperature-dependent ohmic dissipation; and various initial magnetic fields. The results show magnetorotational turbulence around the sublimation front at 0.5 au. The disk interior to 0.8 au is turbulent, with velocities exceeding 10% of the sound speed. Beyond 0.8 au is the dead zone, cooler than 1000 K and with turbulence orders of magnitude weaker. A local pressure maximum just inside the dead zone concentrates solid particles, favoring their growth. Over many orbits, a vortex develops at the dead zone’s inner edge, increasing the disk’s thickness locally by around 10%. We synthetically observe the results using Monte Carlo transfer calculations, finding that the sublimation front is near-infrared bright. The models with net vertical magnetic fields develop extended, magnetically supported atmospheres that reprocess extra starlight, raising the near-infrared flux 20%. The vortex throws a nonaxisymmetric shadow on the outer disk. At wavelengths , the flux varies several percent on monthly timescales. The variations are more regular when the vortex is present. The vortex is directly visible as an arc at ultraviolet through near-infrared wavelengths, given sub-au spatial resolution.

Pressure evolution in the shear layer of forming vortex rings

This study investigated the relationship between the pinch-off of axisymmetric vortex rings and the evolution of pressure in the shear layer being entrained into the vortex rings. A piston-cylinder apparatus was used to generate the vortex rings, and five cases of constant piston acceleration over distances ranging from zero (impulsive start) to eight piston diameters were investigated. It was determined that increasing the distance over which the piston accelerated increased the dimensionless formation time at which the vortex ring pinches off, consistent with previous observations. A limiting value of vortex ring formation number of approximately seven is approached when the piston is accelerated over more than six piston diameters. For each case, the evolution of pressure in the shear layer was calculated based on PIV measurements of the velocity field and spatial integration of the corresponding pressure gradients using a recently developed algorithm. Plots of the shear layer pressure in X-T diagrams aided in identifying key features of the pressure associated with the evolution of vortex rings, including a high-pressure region that forms behind the leading ring. By extrapolating the motion of this high-pressure region back to the nozzle exit plane in the X-T diagram, its time of first appearance can be estimated. It is found that the appearance of the extrapolated local pressure maximum in the shear layer at the nozzle exit plane coincides with vortex ring pinch-off, as conventionally quantified by the vortex ring formation number.