Convective-Reactive Nucleosynthesis in Stars

• Convective-reactive nucleosynthesis is a stellar burning regime where nuclear reactions and mixing occur on comparable timescales, typically characterized by a Damköhler number near unity.
• It drives diverse phenomena, from nova thermonuclear runaways and hydrogen-ingestion i-process events to massive-star shell mergers, thereby altering energy generation and nucleosynthetic yields.
• Modeling this regime challenges traditional 1D approaches, necessitating 3D hydrodynamic calibrations to accurately capture boundary mixing and non-local reaction effects.

Searching arXiv for relevant papers on convective-reactive nucleosynthesis in novae, H-ingestion, shell mergers, and related stellar contexts. Convective-reactive nucleosynthesis denotes a stellar burning regime in which nuclear transmutation and convective transport proceed on comparable timescales, so that burning cannot be treated as either purely local or passively mixed. In this regime, freshly entrained fuel reacts while being advected, energy release feeds back on the flow, and abundance evolution becomes non-local. The common diagnostic is the Damköhler number, $\mathrm{Da}$, with convective-reactive behavior occurring for $\mathrm{Da} \approx 1$. Across stellar environments, this condition appears in hydrogen ingestion into helium-burning convection zones, thermonuclear runaways in classical novae, interacting shell-burning regions in massive stars, and related high-entropy compact-object flows. The resulting nucleosynthesis includes hot-CNO cycling, i-process neutron bursts, odd-$Z$ element production, and, in rare cases, flows reaching the Fe peak or beyond (Glasner et al., 2011, Herwig et al., 2010).

1. Regime definition and physical diagnostics

The defining feature of convective-reactive nucleosynthesis is the competition between a mixing timescale and a nuclear timescale. In the notation used across the cited works, the nuclear timescale for species $i$ is written as

$\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$

or, equivalently in abundance form,

$$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$

A characteristic convective turnover or mixing time is written as

$$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$

with $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$ in mixing-length theory. The Damköhler number is then

$$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$

When $\mathrm{Da} \ll 1$, instantaneous mixing is a good approximation; when $\mathrm{Da} \approx 1$0, burning is locally faster than transport; when $\mathrm{Da} \approx 1$1, mixing and burning are tightly coupled and must be solved together (Denissenkov et al., 2012, Ritter et al., 2017).

In stellar evolution calculations, the onset of convection is typically described by the Schwarzschild criterion, $\mathrm{Da} \approx 1$2, with

$\mathrm{Da} \approx 1$3

or

$\mathrm{Da} \approx 1$4

Composition transport is then represented in 1D by diffusion-reaction equations of the generic form

$\mathrm{Da} \approx 1$5

or, in Lagrangian mass coordinate,

$\mathrm{Da} \approx 1$6

These forms formalize the central point that in convective-reactive layers transport and transmutation must be evolved simultaneously rather than sequentially (Glasner et al., 2011, Denissenkov et al., 2013).

A recurring limitation is that 1D diffusive mixing is only an effective model of intrinsically multidimensional entrainment, plume interaction, and intermittent burning. Several of the cited studies therefore compare 1D prescriptions with 2D or 3D hydrodynamics and show that the convective-reactive regime is especially sensitive to boundary mixing, delayed entropy-barrier formation, and anisotropic advection (Herwig et al., 2010, Stephens et al., 2020).

2. Classical novae as a canonical convective-reactive environment

Classical novae provide a particularly clear example of convective-reactive nucleosynthesis. In the standard picture, a white dwarf in a cataclysmic variable accretes hydrogen-rich material of approximately solar composition. Degeneracy at the base of the accreted envelope initially prevents thermal expansion from regulating proton-capture heating, so the envelope becomes unstable to convection days to weeks before the thermonuclear runaway and fully convective during peak burning. Processed material is transported to the outer envelope and later ejected, with typical ejected masses of $\mathrm{Da} \approx 1$7–$\mathrm{Da} \approx 1$8 and velocities of several $\mathrm{Da} \approx 1$9 (Glasner et al., 2011).

The convective-reactive character of novae follows from the comparison between hot-CNO $Z$0 lifetimes and envelope turnover times. The review by Glasner and Truran emphasizes that the relevant unstable isotopes, $Z$1, $Z$2, $Z$3, and $Z$4, have lifetimes of order $Z$5–$Z$6, whereas advection of newly synthesized nuclei through the envelope occurs in about $Z$7–$Z$8 and overall turnover times are tens of seconds. This places novae in the range $Z$9–$i$0, so $i$1-unstable nuclei are produced deep in the burning region and decay after transport into cooler layers (Glasner et al., 2011). In the MESA–NuGrid Nova Framework, the same conclusion is expressed in terms of hot-CNO $i$2 half-lives and convective mixing times of order $i$3–$i$4, again implying $i$5–$i$6 (Denissenkov et al., 2012).

At temperatures $i$7, proton captures on CNO nuclei outpace $i$8 decay, and energy generation becomes $i$9-limited. A representative loop is

$\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$0

If all stable CNO seeds are rapidly converted to $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$1-unstable species and no fresh $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$2, $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$3, or $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$4 is ingested, the burning becomes temperature-insensitive and is capped at

$\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$5

That cap is lifted when convective undershoot or convective boundary mixing entrains fresh white-dwarf material into the envelope during the runaway (Glasner et al., 2011).

Cross-boundary mixing is therefore central to both nova energetics and nova yields. In 1D MESA calculations it is represented as an exponentially decaying diffusion coefficient beneath the convective boundary,

$\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$6

with $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$7. The framework compares this physically motivated prescription with the older pre-mixed-envelope approximation. For ONe novae, exponential convective boundary mixing reproduces the temperature evolution and final abundances of pre-mixed models; for CO novae, the equivalence is more sensitive to the details of boundary mixing and thermal structure (Denissenkov et al., 2012). A later MESA–NuGrid study reports very good agreement between the exponential prescription with $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$8 and $\tau_{\rm nuc}^{(i)} \sim \frac{X_i}{|\dot{X}_i|},$9 pre-mixed models for both CO and ONe novae, including a $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$0 CO white dwarf with $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$1 and $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$2 (Denissenkov et al., 2013).

Novae also exhibit rare convective-reactive breakout from the traditional CNO cycle. In massive white dwarfs with cool cores and very low accretion rates, peak temperatures can exceed $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$3 for several hours, above a critical threshold

$$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$4

activating breakout channels such as

$$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$5

and

$$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$6

Under such conditions, synthesis can extend to the iron group, and the rate $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$7 exerts global sensitivity on the runaway energetics (Glasner et al., 2011).

3. Hydrogen ingestion, helium-shell convection, and the i-process

Hydrogen ingestion into helium-burning convective regions is another major class of convective-reactive nucleosynthesis. In Sakurai’s object, a very-late thermal pulse drove protons from an H-rich layer into a hot, C-rich He-shell flash convection zone, activating

$$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$8

Because the He intershell had a primary carbon mass fraction $$\tau_{\rm burn,i} = \left| \frac{Y_i}{(dY_i/dt)_{\rm nuc}} \right|.$$9, the proton-capture rate increased by roughly 12 orders of magnitude from the top to the bottom of the convection zone, creating a radius where $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$0 and thus $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$1. The convective turnover time was about $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$2, while the relevant $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$3 burn times were about $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$4–$$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$5 minutes at $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$6–$$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$7 (Herwig et al., 2010).

The 1D and 3D treatments of this event differ qualitatively. In 1D mixing-length models, a thin H-burning layer rapidly creates an entropy barrier and splits the convection zone, preventing efficient delivery of $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$8 into the hottest layers. This yields neutron densities of at most a few $$\tau_{\rm conv} \sim \frac{\ell}{v_{\rm conv}}, \qquad t_{\rm mix} \approx \frac{\ell^2}{D_{\rm MLT}} = \frac{3\ell}{v_{\rm conv}},$$9, too low to alter heavy-element abundances strongly. In contrast, 3D simulations show heterogeneous entrainment, Kelvin–Helmholtz instabilities at the upper boundary, and a broader distributed burning region. Calibrated delayed-split post-processing then delivers $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$0 to the bottom of the convection zone, where $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$1 acts on $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$2–$D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$3 timescales and produces neutron densities of a few $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$4, firmly in the i-process regime (Herwig et al., 2010).

That i-process interpretation is supported by the observed abundance pattern of Sakurai’s object: strong first-peak enhancement of Rb, Sr, and Y by about 2 dex relative to Ba and La, a low $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$5 ratio of about $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$6–$D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$7, and lithium production. The delayed-split convective-reactive framework reproduces these features, whereas the standard 1D immediate-split picture does not (Herwig et al., 2010).

Related H-ingestion i-process conditions also arise in rapidly accreting white dwarfs. In 3D PPMstar simulations of a He-shell flash in a rapidly accreting white dwarf at $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$8, hydrogen is entrained through the upper convective boundary and burns near mid-shell through the same $D_{\rm mix} \approx \frac{1}{3} v_{\rm conv}\ell$9 chain. The 3D runs give convective turnover times of $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$0, $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$1, and $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$2 minutes for the N15, N16, and N17 setups, comparable to the $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$3 half-life of about $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$4 minutes. This yields $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$5, with illustrative values of about $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$6 for N16 and $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$7 for N17 (Stephens et al., 2020).

A notable result of the 3D-calibrated post-processing is that an advective two-stream model resolves upstream–downstream asymmetry in the burning and in short-lived isotopes. In N16 and N17, isotopes such as $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$8 and $$\mathrm{Da} \equiv \frac{\tau_{\rm conv}}{\tau_{\rm nuc}} \quad\text{or}\quad \mathrm{Da} \equiv \frac{t_{\rm mix}}{t_{\rm burn}}.$$9 differ by factors of about $\mathrm{Da} \ll 1$0–$\mathrm{Da} \ll 1$1 between the two streams, consistent with $\mathrm{Da} \ll 1$2 for those half-lives. Yet in that specific rapidly accreting white-dwarf application, a diffusion-based model constrained by the same 3D data still reproduces the global i-process abundance pattern and yields good agreement with the CEMP-r/s star CS31062-050 (Stephens et al., 2020).

Super-AGB stars supply another H-ingestion channel. In these models, either overlap between the pulse-driven convection zone and the base of the convective envelope during thermal pulses, or dredge-out in the most massive super-AGB stars, transports protons into hot He-burning layers. With $\mathrm{Da} \ll 1$3 and $\mathrm{Da} \ll 1$4, the estimated convective time is about $\mathrm{Da} \ll 1$5, comparable to the $\mathrm{Da} \ll 1$6 $\mathrm{Da} \ll 1$7 half-life of about $\mathrm{Da} \ll 1$8, again implying $\mathrm{Da} \ll 1$9. Hydrogen-burning luminosities exceed $\mathrm{Da} \approx 1$00 and in some cases reach $\mathrm{Da} \approx 1$01, while neutron densities reach about $\mathrm{Da} \approx 1$02, making these events plausible i-process sites (Jones et al., 2015).

In extremely metal-poor AGB stars, the same general framework appears in two convective modes. Low-mass models undergo “convective $\mathrm{Da} \approx 1$03 burning” when H is engulfed into the He-flash convection; intermediate-mass models undergo “convective $\mathrm{Da} \approx 1$04 burning” when the He-flash base reaches $\mathrm{Da} \approx 1$05–$\mathrm{Da} \approx 1$06. The convective $\mathrm{Da} \approx 1$07 mode reaches maximum neutron densities from about $\mathrm{Da} \approx 1$08–$\mathrm{Da} \approx 1$09 for $\mathrm{Da} \approx 1$10 and up to about $\mathrm{Da} \approx 1$11–$\mathrm{Da} \approx 1$12 for $\mathrm{Da} \approx 1$13, while the radiative $\mathrm{Da} \approx 1$14-pocket mode remains at $\mathrm{Da} \approx 1$15 to $\mathrm{Da} \approx 1$16 and is explicitly non-convective-reactive. A central conclusion is that below $\mathrm{Da} \approx 1$17, oxygen in the He zone dominates the neutron-poison budget, so the $\mathrm{Da} \approx 1$18-process efficiency per Fe seed becomes nearly metallicity-independent (Yamada et al., 2023).

4. Shell mergers and convective-reactive burning in massive stars

Late shell interactions in massive stars provide a different but related manifestation of convective-reactive nucleosynthesis. In O–C shell interactions, an oxygen-burning convective shell ingests carbon-rich material from an adjacent C shell. The relevant 3D PPMstar simulation of the first O shell in a $\mathrm{Da} \approx 1$19, $\mathrm{Da} \approx 1$20 model gives a convective turnover time of about $\mathrm{Da} \approx 1$21. Moderate-ingestion run I2 remains approximately spherically symmetric and admits a 1D diffusion mapping, whereas more energetic configurations show greater deformation and, in the I11 case, a violent non-radial global oscillation that invalidates a spherical treatment (Ritter et al., 2017).

The nucleosynthesis consequences are significant. Multi-zone 1D post-processing based on the 3D-derived diffusion coefficient shows that O–C ingestion can produce the odd-$\mathrm{Da} \approx 1$22 elements P, Cl, K, and Sc. At an entrainment rate of $\mathrm{Da} \approx 1$23, the odd-$\mathrm{Da} \approx 1$24 suite reaches an overproduction factor $\mathrm{Da} \approx 1$25. Full O–C shell mergers in 1D stellar evolution models achieve $\mathrm{Da} \approx 1$26, with the $\mathrm{Da} \approx 1$27, $\mathrm{Da} \approx 1$28 case giving $\mathrm{Da} \approx 1$29 and p-process isotopes such as $\mathrm{Da} \approx 1$30 also exceeding $\mathrm{Da} \approx 1$31 (Ritter et al., 2017).

A distinct but related example is the Si–C shell merger identified in a $\mathrm{Da} \approx 1$32, $\mathrm{Da} \approx 1$33 massive-star model about an hour before collapse. There, mixing-length theory gives $\mathrm{Da} \approx 1$34 in the Si shell and a turnover time of about $\mathrm{Da} \approx 1$35. Carbon entrained into the hot Si shell burns far faster than a turnover time: at $\mathrm{Da} \approx 1$36 and $\mathrm{Da} \approx 1$37, the e-folding time of $\mathrm{Da} \approx 1$38 in a $\mathrm{Da} \approx 1$39 $\mathrm{Da} \approx 1$40 and $\mathrm{Da} \approx 1$41 $\mathrm{Da} \approx 1$42 mixture is about $\mathrm{Da} \approx 1$43. Thus $\mathrm{Da} \approx 1$44 near the base, while the lower third of the shell contains a region with $\mathrm{Da} \approx 1$45, interpreted as a distributed combustion zone and a possible site of a GOSH-like global oscillation (Côté et al., 2019).

The nucleosynthetic signature of that Si–C merger is the outward transport of incomplete Si-burning products, especially $\mathrm{Da} \approx 1$46 and $\mathrm{Da} \approx 1$47, to the edge of the CO core. In the adopted explosion model, a mass cut at $\mathrm{Da} \approx 1$48 ejects the outer $\mathrm{Da} \approx 1$49 of the CO core, including a significant fraction of this Cr-rich material. When this single progenitor is included in a galactic chemical-evolution interpolation, the resulting [Cr/Fe] near solar metallicity is overestimated by almost an order of magnitude relative to observations, indicating that either such mergers are rare or their ejecta are not representative (Côté et al., 2019).

A newer formulation of massive-star shell-merger reactivity is the “SPAr process” in C–O shell mergers. In a $\mathrm{Da} \approx 1$50, $\mathrm{Da} \approx 1$51 model, the merger raises the shell-base temperature from $\mathrm{Da} \approx 1$52 to $\mathrm{Da} \approx 1$53 and increases the proton abundance from $\mathrm{Da} \approx 1$54 to about $\mathrm{Da} \approx 1$55–$\mathrm{Da} \approx 1$56. Under these conditions, proton captures on $\mathrm{Da} \approx 1$57, $\mathrm{Da} \approx 1$58, and $\mathrm{Da} \approx 1$59 dominate the energy budget: $\mathrm{Da} \approx 1$60 At the beginning of the merger, these SPAr reactions account for about $\mathrm{Da} \approx 1$61 of the total nuclear energy generation, while $\mathrm{Da} \approx 1$62 contributes only about $\mathrm{Da} \approx 1$63–$\mathrm{Da} \approx 1$64. Their combined output is about $\mathrm{Da} \approx 1$65 times greater than that of classical C and O fusion under the same conditions, and neutron densities reach about $\mathrm{Da} \approx 1$66 (Roberti et al., 17 Sep 2025).

These massive-star cases show that convective-reactive nucleosynthesis is not limited to neutron-source episodes. It can also redirect the dominant energy generation away from the nominal fuel-burning channels, change pre-supernova structure, alter ejecta compositions, and produce strong galactic-chemical-evolution signatures in odd-$\mathrm{Da} \approx 1$67 and Fe-group elements. A plausible implication is that shell-merger nucleosynthesis is as much a hydrodynamic problem as a reaction-network problem, because the burning changes the convective shell itself (Ritter et al., 2017, Roberti et al., 17 Sep 2025).

5. Modeling strategies and the limits of 1D descriptions

Most quantitative studies of convective-reactive nucleosynthesis begin from 1D stellar-evolution models, but nearly all emphasize that 1D prescriptions become unreliable once nuclear heating is dynamically important on a turnover time. In novae, MESA-based calculations treat convection as diffusion and represent convective boundary mixing with an exponentially decaying coefficient beneath the envelope base, motivated by 3D velocity decay into adjacent stable layers (Denissenkov et al., 2012). In super-AGB stars, the same formalism is used with

$\mathrm{Da} \approx 1$68

but the predicted onset and strength of H ingestion depend sensitively on the chosen $\mathrm{Da} \approx 1$69 and $\mathrm{Da} \approx 1$70 parameters (Jones et al., 2015).

The most explicit critique of the 1D approximation appears in H-ingestion studies. In Sakurai’s object, the 1D split of the convection zone occurs too early because burning is confined to a thin layer; 3D hydrodynamics instead shows anisotropic entrainment, patchy energy release, and delayed entropy-barrier formation, which are necessary to reproduce the i-process abundance pattern (Herwig et al., 2010). In rapidly accreting white dwarfs, the 3D-calibrated advective two-stream model was introduced precisely because purely diffusive transport erases asymmetries between upflows and downflows. The model enforces zero net radial mass flux at each radius and couples radial advection to a calibrated horizontal exchange term, permitting separate nucleosynthesis in the two streams (Stephens et al., 2020).

The validity of spherically averaged transport depends strongly on the hydrodynamic regime. Moderate O-shell C-ingestion can remain close enough to spherical symmetry that a 1D diffusion coefficient extracted from 3D data captures the net entrainment and mixing profile. Once burning feedback is enhanced, however, boundary deformation grows and non-radial oscillatory behavior appears, at which point 1D post-processing becomes at best approximate (Ritter et al., 2017). The Si–C shell-merger analysis makes the same point from a different angle: mixing-length theory presumes many turnovers, but the merger occurs with roughly one turnover before collapse and with $\mathrm{Da} \approx 1$71 over much of the shell (Côté et al., 2019).

Even where 1D calculations remain useful, the transport–burning equations themselves show why instantaneous mixing is not appropriate. The advective-reactive or diffusive-reactive system must evolve abundance gradients that are continually regenerated by finite burning times, by differential decay of short-lived species, and by boundary injection of fresh fuel or ashes. This is why novae can preserve elevated $\mathrm{Da} \approx 1$72 and $\mathrm{Da} \approx 1$73 ratios in outer ejecta, why H-ingestion events can separate $\mathrm{Da} \approx 1$74 production from $\mathrm{Da} \approx 1$75 ignition, and why shell mergers can confine the most reactive burning to distributed subregions rather than the convective base itself (Glasner et al., 2011, Stephens et al., 2020).

A recurring methodological theme is therefore calibration rather than replacement: 3D simulations quantify entrainment rates, velocity spectra, boundary locations, and spatial burning morphology; 1D models then encode those results in effective coefficients or reduced mixing models. This suggests that the central modeling challenge is not merely larger networks, but transport closures that remain valid when $\mathrm{Da} \approx 1$76.

6. Nucleosynthetic signatures, observational constraints, and open problems

Convective-reactive nucleosynthesis produces several distinct observational signatures. In classical novae, rapid advection of $\mathrm{Da} \approx 1$77-unstable nuclei into cooler layers yields enhanced $\mathrm{Da} \approx 1$78 and $\mathrm{Da} \approx 1$79, strong $\mathrm{Da} \approx 1$80 production, and in ONe novae the radioactive isotopes $\mathrm{Da} \approx 1$81 and $\mathrm{Da} \approx 1$82, associated with $\mathrm{Da} \approx 1$83-ray lines at $\mathrm{Da} \approx 1$84 and $\mathrm{Da} \approx 1$85. Some CO novae produce $\mathrm{Da} \approx 1$86, leading to a $\mathrm{Da} \approx 1$87 signature from $\mathrm{Da} \approx 1$88, while rare breakout novae would be expected to show broader heavy-element enrichment up to the Fe group (Glasner et al., 2011).

Hydrogen-ingestion i-process events are constrained by elemental and isotopic abundance patterns rather than by explosive ejecta alone. Sakurai’s object displays the characteristic first-peak enhancement of Rb, Sr, and Y over Ba and La, low $\mathrm{Da} \approx 1$89, lithium, and high Sc/Ca, all pointing to neutron densities near the i-process regime and to a short, intense neutron burst before convective splitting (Herwig et al., 2010). In extremely metal-poor AGB stars, the comparison with CEMP stars indicates that convective $\mathrm{Da} \approx 1$90 burning can reproduce the full range of Sr, Ba, and Pb enrichment, while convective $\mathrm{Da} \approx 1$91 burning populates only the lower end of the observed distribution and radiative $\mathrm{Da} \approx 1$92-pocket burning cannot explain the highest enrichments (Yamada et al., 2023). Super-AGB H-ingestion has been proposed as a source of CEMP-s/r-like patterns and may also produce outbursts with rise and decay times of order 10 days and peak luminosities between novae and supernovae, although the transient treatment remains schematic (Jones et al., 2015).

In massive stars, the signatures are often indirect and population-level. O–C shell mergers can alleviate the underproduction of odd-$\mathrm{Da} \approx 1$93 elements in galactic chemical-evolution models if such events occur in a substantial fraction of massive stars (Ritter et al., 2017). Conversely, the Si–C shell-merger case demonstrates that pre-supernova convective-reactive episodes can overproduce [Cr/Fe] if their yields are adopted too broadly (Côté et al., 2019). The SPAr-process analysis strengthens this link between local reactive burning and integrated yields by showing that omission of proton captures on P, S, and Ar would underestimate shell-merger energy generation by factors of about $\mathrm{Da} \approx 1$94–$\mathrm{Da} \approx 1$95 and bias predictions for Cl and K (Roberti et al., 17 Sep 2025).

Several open problems recur across all sites. Nuclear-rate uncertainties remain central, including $\mathrm{Da} \approx 1$96 in nova breakout, $\mathrm{Da} \approx 1$97 and neutron-poison rates in i-process environments, and proton-capture and photodisintegration rates on P, S, Cl, and Ar in shell mergers (Glasner et al., 2011, Herwig et al., 2010, Roberti et al., 17 Sep 2025). Hydrodynamically, the unresolved issues are entrainment laws, delayed or suppressed convective splitting, intermittency, and the conditions under which reactive burning launches global non-spherical oscillations (Ritter et al., 2017, Jones et al., 2015). A consistent theme is that convective-reactive nucleosynthesis is not a marginal correction to otherwise standard burning. It is a regime in which transport regulates energy release, energy release reshapes transport, and the resulting element production depends on both with comparable strength.