COALA Dust Evolution Module

• COALA Dust Evolution Module is a comprehensive framework that simulates dust physical processing, chemical evolution, and dynamics across astrophysical scales.
• It integrates dust microphysics with hydrodynamical, MHD, and chemical modules to enable high-resolution ISM, protoplanetary, and cosmological simulations.
• Efficiency is achieved through operator splitting, adaptive numerical closures, and seamless integration with simulation codes like SWIFT, RAMSES, and COLIBRE.

The COALA Dust Evolution Module is a comprehensive numerical framework for modeling the physical processing, chemical evolution, and dynamics of dust in astrophysical environments ranging from the interstellar medium (ISM) and protoplanetary disks to cosmological volumes. It is designed to efficiently evolve the grain size distribution (GSD) and corresponding dust moments by coupling dust microphysics with hydrodynamical, magnetohydrodynamical (MHD), and chemical modules. The module underpins several state-of-the-art simulations, including integration into the SWIFT, RAMSES, and COLIBRE frameworks, enabling both high-resolution and cosmological studies (Mattsson, 2016, Hirashita et al., 2017, Lombart et al., 11 Jan 2026, Marel et al., 2023, Trayford et al., 19 May 2025).

1. Underlying Physical Processes and Equation Systems

The COALA module encodes the evolution of dust via a set of coupled ordinary differential equations (ODEs) for moments of the grain size distribution $f(a,t)$, or directly via sectional (bin) methods in size or mass space. The fundamental processes treated are:

• Stellar Dust Injection and Astration: Dust is injected during star formation events according to effective yield parameters and is removed via astration when gas is formed into stars. Under instantaneous recycling, the moment update is given by

$$(dM_k/dt)_\star = (y_d/\zeta_\star)(K_k^\star/K_3^\star)\,\dot{\Sigma}_\star,$$

$$(dM_k/dt)_\mathrm{astr} = -M_k/\tau_\mathrm{sf}, \quad \tau_\mathrm{sf} = \Sigma_\mathrm{gas}/\dot{\Sigma}_\star.$$

• Growth by Condensation: In molecular clouds, dust grows by accretion of metals from the gas phase. The growth velocity is

$$\xi = \frac{da}{dt} = \alpha_s \langle v_\mathrm{mol} \rangle / (\rho_\mathrm{gr} \kappa) [D_\infty - D(t)],$$

leading to the moment source term

$(dM_k/dt)_\mathrm{cond} = k\xi M_{k-1}.$

• Coagulation (Smoluchowski Equation): Grain–grain mergers are described by the Smoluchowski equation with a collision kernel $C(a,a')$ (often a power-law or ballistic kernel). For moments,

$$(dM_k/dt)_\mathrm{coag} = \frac{1}{2}\int_0^\infty\!\!\int_0^\infty C(a,a')f(a)f(a')[ (a^3+a'^3)^{k/3} - a^k - a'^k ]\,da\,da',$$

with analytic closure for $k=0,3,6,\dots$ and cubic-spline interpolation for intermediate moments (Mattsson, 2016, Lombart et al., 11 Jan 2026).

• Fragmentation (Shattering): High-velocity impacts produce smaller fragments. The fragmentation is represented as

$$(dM_k/dt)_\mathrm{frag} = k_0\frac{1 - k/3}{k/3 + \nu + 1}M_{k+3\mu}.$$

• Destruction (Sputtering, SN shocks): Supernova-driven shocks erode grains at rates

$(dM_k/dt)_\mathrm{dest} = -M_k/\tau_{d,k},$

with $\tau_{d,k}$ depending on SN rate, gas density, and (optionally) grain size.

A summary table of processes and their corresponding equations appears below:

Process	Governing Equation/System	Notes
Injection	$(dM_k/dt)_\star$, $(dM_k/dt)_\mathrm{astr}$	Effective yield, astration
Condensation	$(dM_k/dt)_\mathrm{cond} = k\xi M_{k-1}$	Growth by gas accretion
Coagulation	$(dM_k/dt)_\mathrm{coag}$ (see above)	Smoluchowski, closed for $k=0,3,6,\dots$
Fragmentation	$(dM_k/dt)_\mathrm{frag}$	Power-law breakup
Destruction	$(dM_k/dt)_\mathrm{dest}$	Sputtering, SN rate, size-dependent

2. Numerical Methods and Algorithmic Structure

The COALA solver supports both method-of-moments (MoM) and sectional (bin) (size/mass binning) approaches:

• Method-of-Moments (MoM): The grain size distribution is reconstructed from a hierarchy of moments $M_k$. Evolution equations are solved for a selected set of moments ($k=0,3,6,\dots,k_\mathrm{max}$, typically $k_\mathrm{max}=15$ or 18). For intermediate $k$, a cubic-spline interpolative closure is used, ensuring numerical stability and accuracy to second order (Mattsson, 2016).
• Sectional/Bin Schemes: In protoplanetary disk and 3D envelope applications, the distribution is discretized into $N$ logarithmic bins in size or mass. The Smoluchowski equation is advanced with conservative finite-volume or Gaussian quadrature fluxes across bin interfaces. Time integration employs explicit or strong stability-preserving Runge–Kutta schemes (e.g., SSP RK3) (Lombart et al., 11 Jan 2026, Marel et al., 2023).
• Operator Splitting: Dust dynamics and microphysics are evolved in operator-split fashion after hydrodynamics/MHD updates. Dust transport (advection and drift) is handled as passive scalar or with the terminal-velocity approximation, particularly for N-fluid treatments in RAMSES or SWIFT (Lombart et al., 11 Jan 2026, Trayford et al., 19 May 2025).
• Closure and Output: After microphysics, intermediate moments or bin values are reconstructed using cubic-splines (MoM) or analytic fits for the GSD if required. For observational predictions, the module returns mass-weighted properties, dust extinction curves, or grain-size-dependent dust-to-gas ratios.
• Performance Optimization: Typical per-cell computational cost is $O(N_k)$ ($N_k \sim 16$ for MoM, $N \sim 40$ bins for sectional). Fast updates are achieved through precomputing binomial coefficients, lookup tables for kernel weights, and loop fusion. Single precision may be used in non-critical steps (Mattsson, 2016, Lombart et al., 11 Jan 2026).

3. Physical Parameterization and Calibration Protocols

Physical and numerical accuracy require robust parameterization of all dust processes and systematic calibration:

• Input Gas Properties: Each cell (or particle) requires $\rho_\mathrm{gas}$, $T_\mathrm{gas}$, metallicity ($Z$), star formation ($\dot{\Sigma}_\star$), and supernova rate ($R_\mathrm{SN}$).
• Dust Parameters: Includes sticking coefficients ($\alpha_s$, $S_\mathrm{acc}$), grain compositions/densities ($\rho_\mathrm{gr}$), grain bin boundaries, yield fractions ($y_d$, $\eta_G$), and collision/fragmentation kernel details.
• Molecular Cloud Enhancement: For dense gas, accretion and coagulation rates are scaled by an effective subgrid clumping factor $C(n_\mathrm{H})$ to account for unresolved density fluctuations. A canonical form is

$$n'_\mathrm{H} = C(n_\mathrm{H}) n_\mathrm{H}, \quad C(n_\mathrm{H}) = \begin{cases} 1, & n_\mathrm{H} \leq n_\mathrm{H,min} \ (n_\mathrm{H}/n_\mathrm{H,min})^m, & n_\mathrm{H,min} < n_\mathrm{H} \leq n_\mathrm{H,max} \ C_\mathrm{max}, & n_\mathrm{H} > n_\mathrm{H,max} \end{cases}$$

with $C_\mathrm{max}$, $m$ calibrated to match observed dust-to-gas and dust-to-metal ratios (Trayford et al., 19 May 2025).

• Calibration: Parameters are tuned against observed cosmic dust densities, dust-to-gas and dust-to-metallicity scaling relations, grain size distributions (e.g., S/L ratios), millimeter continuum data (ALMA), and CO/H$_2$ conversion factors. For disks, $\alpha$ (turbulence), $v_\mathrm{frag}$, and trap parameters ($A$, $r_b$, $\sigma_b$) are iteratively adjusted (Marel et al., 2023, Trayford et al., 19 May 2025).

4. Module Architecture and Integration in Simulation Codes

The COALA module is integrated into various astrophysical simulation platforms:

• SWIFT (SPH): Each gas particle carries a dust state vector for six components (three chemical species, each split into large/small size bins). The evolution loop includes grain injection (AGB/CCSN), accretion, sputtering, shattering, coagulation, SN-driven destruction, astration, and turbulent diffusion. Metal depletion is self-consistently tracked and propagates into cooling/heating routines ("live dust") (Trayford et al., 19 May 2025).
• RAMSES (AMR + MHD): In RAMSES, COALA operates as a stand-alone subgrid module for dust evolution, invoked after the monofluid N-dust + gas update at each step. Dust species are evolved under the terminal-velocity approximation for dynamics and SSP RK3 for growth in each cell. Full data parallelism is inherited from the parent MHD code (Lombart et al., 11 Jan 2026).
• Hydrodynamics + Chemistry Coupling: Dust evolution is operator-split with hydrodynamics, and the evolved dust state modifies chemical rates (e.g., H$_2$ formation, cooling) and extinction/shielding calculations. In some implementations (e.g., via CHIMES), a physical dust scaling $s_\mathrm{dust}$ is computed from model dust abundances and grain radii to interface with ISM/chemistry modules (Trayford et al., 19 May 2025).

5. Impact on Molecular Chemistry and Observational Diagnostics

The COALA framework enables joint modeling of dust and its chemical and observational consequences:

• Dust-to-Gas Ratio vs. Metallicity ($\mathcal{D}(Z)$): The module predicts a characteristic 'knee' behavior in $\mathcal{D}(Z)$, transitioning from stellar-injection dominance (at $Z\lesssim0.1\,Z_\odot$) to rapid accretion ("growth knee" at $Z\sim0.2\,Z_\odot$) and eventual SN-destruction control at higher $Z$ (Hirashita et al., 2017, Trayford et al., 19 May 2025). This is critical for interpreting extragalactic and cosmological dust scaling relations.
• Molecular Hydrogen and CO Chemistry: The module couples dust size and abundance to H$_2$ formation rates via enhanced surface area (dominated by small grains) and to CO photodissociation rates via dust extinction. The metallicity dependence of the CO–H$_2$ conversion factor ($X_\mathrm{CO}$) and the observed "CO-dark H$_2$" regime are emergent features of the coupled evolution (Hirashita et al., 2017).
• Protoplanetary Disks: In disk models, COALA tracks the evolution of the dust reservoir, including grain growth, fragmentation barriers, radial drift, and accumulation in pressure maxima ('traps'). The code supports calibration against ALMA continuum observations and tracks the partitioning of mass into pebble and planetesimal-forming sizes (Marel et al., 2023).

6. Known Limitations and Future Development Trajectories

Some features are incomplete or subject to refinement:

• Fragmentation Physics: In some implementations (e.g., (Lombart et al., 11 Jan 2026)), fragmentation is not modeled self-consistently. Current approaches may compare collision velocities to thresholds post facto, but break-up, radiative-torque disruption, and more complex grain–grain physics are active areas for extension.
• Radiative and Magnetic Coupling: Dust-size–dependent MHD resistivities (e.g., ambipolar, Ohmic, Hall) are not always updated on-the-fly. Coupling these with live dust populations is straightforward in principle but computationally intensive.
• Observational Uncertainties: The module's predictions for dust-to-gas and size distributions depend on metallicity calibrations, subgrid clumping choices, and ISM structure, which are sources of systematic uncertainty when benchmarking to extragalactic datasets (Trayford et al., 19 May 2025).

7. Observational and Simulation Benchmarks

Validation and physical reliability are established via:

• Reproducing Milky Way and Extragalactic Dust Relations: Cosmic dust densities, dust-to-metal ratios, and dust mass functions show strong agreement with Herschel, ALMA, Planck, and DustPedia surveys in fiducial runs (Trayford et al., 19 May 2025).
• Grain Size Distributions: Simulated S/L (small/large) ratios align with observational inferences in most environments, with COALA predicting diffuse ISM and warm-neutral medium to be small-grain dominated, in accord with Galactic cirrus (Trayford et al., 19 May 2025).
• Protoplanetary and Protostellar Applications: Early-stage envelope simulations reveal rapid micron-grain growth and strong radial/anisotropic trends consistent with expected spherically symmetric and turbulent infall scenarios (Lombart et al., 11 Jan 2026).
• CO and H$_2$ Modeling: The inclusion of a coupled dust microphysics and chemistry network is necessary for capturing the observed steep metallicity dependence of $X_\mathrm{CO}$ and the formation of “CO-dark” molecular gas at low $Z$ (Hirashita et al., 2017).

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References:

• (Mattsson, 2016)
• (Marel et al., 2023)
• (Lombart et al., 11 Jan 2026)
• (Hirashita et al., 2017)
• (Trayford et al., 19 May 2025)