TOMAS
TwO-Moment Aerosol Sectional (TOMAS) Brownian-coagulation model with five mass-doubling size bins. Each bin k carries two moments - particle number concentration N_k and mass concentration M_k - and the bins are coupled by Brownian coagulation: intra- and inter-bin collisions remove particles and transfer mass toward larger bins at rates set by a precomputed 5x5 coagulation-coefficient matrix. f_k and psi_k are the within-bin linear number-distribution reconstruction coefficients of the two-moment scheme, evaluated algebraically from N_k. Migrated from the module-level script Aerosol.jl src/TOMAS.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1; the upstream array-variable ODESystem was scalarized to 20 scalar unknowns and mtk2esm-scaffolded. All coagulation coefficients, bin masses and the bin-spacing factor xi are baked in as numeric literals, so the model has no free parameters.
Description
TwO-Moment Aerosol Sectional (TOMAS) Brownian-coagulation model with five mass-doubling size bins. Each bin k carries two moments - particle number concentration N_k and mass concentration M_k - and the bins are coupled by Brownian coagulation: intra- and inter-bin collisions remove particles and transfer mass toward larger bins at rates set by a precomputed 5x5 coagulation-coefficient matrix. f_k and psi_k are the within-bin linear number-distribution reconstruction coefficients of the two-moment scheme, evaluated algebraically from N_k. Migrated from the module-level script Aerosol.jl src/TOMAS.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1; the upstream array-variable ODESystem was scalarized to 20 scalar unknowns and mtk2esm-scaffolded. All coagulation coefficients, bin masses and the bin-spacing factor xi are baked in as numeric literals, so the model has no free parameters.
Reference
Adams, P. J. and Seinfeld, J. H.: Predicting global aerosol size distributions in general circulation models, J. Geophys. Res., 107(D19), 4370, 2002.
Source: Aerosol.jl src/TOMAS.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (migration bead esm-kxky). The upstream file is an undocumented, module-level (Pluto-style) script with no in-repo tests or docs (migration-tracker section 1.15 TOMAS.jl). It builds a five-bin TwO-Moment Aerosol Sectional (TOMAS) Brownian-coagulation ODESystem. The size grid is mass-doubling (m_upper = 2*m_lower, lowest bin m_lower = 1e-18 g), particle density 1.8 g/cm^3, so all five bins lie in the free-molecular regime (diameter < 0.1 um); the 5x5 coagulation-coefficient matrix is evaluated at T = 298 K and baked into the equations as numeric literals, together with the bin masses and the mass-doubling spectral factor xi = 1.0625. The model therefore has no free parameters. f_k and psi_k are the linear within-bin number-distribution reconstruction coefficients of the two-moment scheme; on the mass-doubling grid m_k/m_lower = 1.5 for every bin, so the f_k and psi_k expressions coincide identically (both reduce to N_k/m_lower_k) and the (psi_k - f_k) terms in the differential equations vanish. Migration: the upstream @named ODESystem is defined over array variables N(t)[1:5], M(t)[1:5], f(t)[1:5], psi(t)[1:5]; these were scalarized into 20 scalar unknowns (N1..N5, M1..M5, f1..f5, psi1..psi5) so the equations serialize without array-indexing ops, then scaffolded with mtk2esm. Inline-test expected values are an upstream reference run of the same scalarized ODESystem (OrdinaryDiffEq Tsit5, abstol=1e-14, reltol=1e-12). Tolerance rationale (model-level rel=2e-3): the Python inline-test gate re-integrates with SciPy LSODA (rtol=1e-10, atol=1e-12). All twenty state trajectories - the number concentrations N_k, the mass concentrations M_k, and the algebraic coefficients f_k/psi_k - were verified to agree between the Tsit5 reference and the LSODA gate to better than 2.1e-4 relative at every assertion point across all three test scenarios; rel=2e-3 is a ~10x margin over that observed Tsit5-vs-LSODA truncation difference. Mass-bin assertions are deliberately placed at t=0 and at early-to-mid times where M_k stays well above the gate’s atol=1e-12 integrator floor, so each remains a meaningful relative check.
- DOI: 10.1029/2001JD001010
- URL: https://doi.org/10.1029/2001JD001010
Variables
| Name | Units | Default | Description |
|---|---|---|---|
N1 | cm^-3 | 15830903.16595993 | Number concentration of aerosol particles in mass-doubling size bin 1 (of 5). |
N2 | cm^-3 | 27165938.46737124 | Number concentration of aerosol particles in mass-doubling size bin 2 (of 5). |
N3 | cm^-3 | 44789060.589685805 | Number concentration of aerosol particles in mass-doubling size bin 3 (of 5). |
N4 | cm^-3 | 70949185.69246289 | Number concentration of aerosol particles in mass-doubling size bin 4 (of 5). |
N5 | cm^-3 | 107981933.02637613 | Number concentration of aerosol particles in mass-doubling size bin 5 (of 5). |
M1 | g/cm^3 | 2.374635474893989e-11 | Mass concentration of aerosol particles in mass-doubling size bin 1 (of 5). |
M2 | g/cm^3 | 8.149781540211371e-11 | Mass concentration of aerosol particles in mass-doubling size bin 2 (of 5). |
M3 | g/cm^3 | 2.687343635381148e-10 | Mass concentration of aerosol particles in mass-doubling size bin 3 (of 5). |
M4 | g/cm^3 | 8.513902283095546e-10 | Mass concentration of aerosol particles in mass-doubling size bin 4 (of 5). |
M5 | g/cm^3 | 2.5915663926330265e-09 | Mass concentration of aerosol particles in mass-doubling size bin 5 (of 5). |
Observed
| Name | Units | Default | Description |
|---|---|---|---|
f1 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient f for size bin 1; evaluated algebraically as 2N1/m_lower(2 - m_k/m_lower). Equals psi1 on the mass-doubling grid (m_k/m_lower = 1.5). | |
f2 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient f for size bin 2; evaluated algebraically as 2N2/m_lower(2 - m_k/m_lower). Equals psi2 on the mass-doubling grid (m_k/m_lower = 1.5). | |
f3 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient f for size bin 3; evaluated algebraically as 2N3/m_lower(2 - m_k/m_lower). Equals psi3 on the mass-doubling grid (m_k/m_lower = 1.5). | |
f4 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient f for size bin 4; evaluated algebraically as 2N4/m_lower(2 - m_k/m_lower). Equals psi4 on the mass-doubling grid (m_k/m_lower = 1.5). | |
f5 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient f for size bin 5; evaluated algebraically as 2N5/m_lower(2 - m_k/m_lower). Equals psi5 on the mass-doubling grid (m_k/m_lower = 1.5). | |
psi1 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient psi for size bin 1; evaluated algebraically as 2N1/m_lower(m_k/m_lower - 1). Equals f1 on the mass-doubling grid (m_k/m_lower = 1.5). | |
psi2 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient psi for size bin 2; evaluated algebraically as 2N2/m_lower(m_k/m_lower - 1). Equals f2 on the mass-doubling grid (m_k/m_lower = 1.5). | |
psi3 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient psi for size bin 3; evaluated algebraically as 2N3/m_lower(m_k/m_lower - 1). Equals f3 on the mass-doubling grid (m_k/m_lower = 1.5). | |
psi4 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient psi for size bin 4; evaluated algebraically as 2N4/m_lower(m_k/m_lower - 1). Equals f4 on the mass-doubling grid (m_k/m_lower = 1.5). | |
psi5 | cm^-3/g | Within-bin linear number-distribution reconstruction coefficient psi for size bin 5; evaluated algebraically as 2N5/m_lower(m_k/m_lower - 1). Equals f5 on the mass-doubling grid (m_k/m_lower = 1.5). |
Observed expressions
$$ f1 = 1000000000000000128 \cdot N1 $$Within-bin linear number-distribution reconstruction coefficient f for size bin 1; evaluated algebraically as 2N1/m_lower(2 - m_k/m_lower). Equals psi1 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ f2 = 500000000000000064 \cdot N2 $$Within-bin linear number-distribution reconstruction coefficient f for size bin 2; evaluated algebraically as 2N2/m_lower(2 - m_k/m_lower). Equals psi2 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ f3 = 250000000000000032 \cdot N3 $$Within-bin linear number-distribution reconstruction coefficient f for size bin 3; evaluated algebraically as 2N3/m_lower(2 - m_k/m_lower). Equals psi3 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ f4 = 125000000000000016 \cdot N4 $$Within-bin linear number-distribution reconstruction coefficient f for size bin 4; evaluated algebraically as 2N4/m_lower(2 - m_k/m_lower). Equals psi4 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ f5 = 62500000000000008 \cdot N5 $$Within-bin linear number-distribution reconstruction coefficient f for size bin 5; evaluated algebraically as 2N5/m_lower(2 - m_k/m_lower). Equals psi5 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ psi1 = 1000000000000000128 \cdot N1 $$Within-bin linear number-distribution reconstruction coefficient psi for size bin 1; evaluated algebraically as 2N1/m_lower(m_k/m_lower - 1). Equals f1 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ psi2 = 500000000000000064 \cdot N2 $$Within-bin linear number-distribution reconstruction coefficient psi for size bin 2; evaluated algebraically as 2N2/m_lower(m_k/m_lower - 1). Equals f2 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ psi3 = 250000000000000032 \cdot N3 $$Within-bin linear number-distribution reconstruction coefficient psi for size bin 3; evaluated algebraically as 2N3/m_lower(m_k/m_lower - 1). Equals f3 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ psi4 = 125000000000000016 \cdot N4 $$Within-bin linear number-distribution reconstruction coefficient psi for size bin 4; evaluated algebraically as 2N4/m_lower(m_k/m_lower - 1). Equals f4 on the mass-doubling grid (m_k/m_lower = 1.5).
$$ psi5 = 62500000000000008 \cdot N5 $$Within-bin linear number-distribution reconstruction coefficient psi for size bin 5; evaluated algebraically as 2N5/m_lower(m_k/m_lower - 1). Equals f5 on the mass-doubling grid (m_k/m_lower = 1.5).
Equations
$$ \mathrm{D}\left(N1\right) = \left(-1.5339907267237098 \times 10^{-9}\right) \cdot N1^{2} + \left(-1\right) \cdot N1 \cdot \left(1.9984411888667577 \times 10^{-9} \cdot N3 + 3.304813445275614 \times 10^{-9} \cdot N5 + 1.685530410823016 \times 10^{-9} \cdot N2 + 2.515582965454013 \times 10^{-9} \cdot N4\right) $$ $$ \mathrm{D}\left(N2\right) = 6.715785230622954 \times 10^{-10} \cdot \left(\left(-1\right) \cdot f2 + psi2\right) \cdot M1 + \left(-1.6977404750594217 \times 10^{-9}\right) \cdot N2^{2} + 7.669953633618549 \times 10^{-10} \cdot N1^{2} + \left(-1\right) \cdot N2 \cdot \left(1.8564620188751522 \times 10^{-9} \cdot N3 + 2.7282326069301694 \times 10^{-9} \cdot N5 + 2.1865310959297857 \times 10^{-9} \cdot N4\right) + \left(-1.685530410823016 \times 10^{-9}\right) \cdot M1 \cdot psi2 $$ $$ \mathrm{D}\left(N3\right) = 6.715785230622954 \times 10^{-10} \cdot \left(f2 + \left(-1\right) \cdot psi2\right) \cdot M1 + 8.488702375297108 \times 10^{-10} \cdot N2^{2} + 132812500000000016 \cdot \left(\left(-1\right) \cdot f3 + psi3\right) \cdot \left(2.997661783300136 \times 10^{-27} \cdot M1 + 5.569386056625455 \times 10^{-27} \cdot M2\right) + \left(-1.8595125110708578 \times 10^{-9}\right) \cdot N3^{2} + \left(-1\right) \cdot N3 \cdot \left(2.35376992455032 \times 10^{-9} \cdot N5 + 2.0181985322458977 \times 10^{-9} \cdot N4\right) + 1.685530410823016 \times 10^{-9} \cdot M1 \cdot psi2 + \left(-1\right) \cdot \left(1.9984411888667577 \times 10^{-9} \cdot M1 + 1.8564620188751522 \times 10^{-9} \cdot M2\right) \cdot psi3 $$ $$ \mathrm{D}\left(N4\right) = \left(-2.0051076185437887 \times 10^{-9}\right) \cdot N4^{2} + 66406250000000008 \cdot \left(\left(-1\right) \cdot f4 + psi4\right) \cdot \left(1.2109191193475384 \times 10^{-26} \cdot M3 + 3.773374448181019 \times 10^{-27} \cdot M1 + 6.559593287789356 \times 10^{-27} \cdot M2\right) + 132812500000000016 \cdot \left(f3 + \left(-1\right) \cdot psi3\right) \cdot \left(2.997661783300136 \times 10^{-27} \cdot M1 + 5.569386056625455 \times 10^{-27} \cdot M2\right) + 9.297562555354289 \times 10^{-10} \cdot N3^{2} + \left(-2.153624885355299 \times 10^{-9}\right) \cdot N5 \cdot N4 + \left(-1\right) \cdot \left(2.0181985322458977 \times 10^{-9} \cdot M3 + 2.515582965454013 \times 10^{-9} \cdot M1 + 2.1865310959297857 \times 10^{-9} \cdot M2\right) \cdot psi4 + \left(1.9984411888667577 \times 10^{-9} \cdot M1 + 1.8564620188751522 \times 10^{-9} \cdot M2\right) \cdot psi3 $$ $$ \mathrm{D}\left(N5\right) = 1.0025538092718944 \times 10^{-9} \cdot N4^{2} + \left(-1\right) \cdot psi5 \cdot \left(2.35376992455032 \times 10^{-9} \cdot M3 + 2.153624885355299 \times 10^{-9} \cdot M4 + 3.304813445275614 \times 10^{-9} \cdot M1 + 2.7282326069301694 \times 10^{-9} \cdot M2\right) + 33203125000000004 \cdot \left(1.4122619547301916 \times 10^{-26} \cdot M3 + 2.5843498624263582 \times 10^{-26} \cdot M4 + 4.95722016791342 \times 10^{-27} \cdot M1 + 8.184697820790507 \times 10^{-27} \cdot M2\right) \cdot \left(psi5 + \left(-1\right) \cdot f5\right) + \left(-2.117154741395111 \times 10^{-9}\right) \cdot N5^{2} + 66406250000000008 \cdot \left(f4 + \left(-1\right) \cdot psi4\right) \cdot \left(1.2109191193475384 \times 10^{-26} \cdot M3 + 3.773374448181019 \times 10^{-27} \cdot M1 + 6.559593287789356 \times 10^{-27} \cdot M2\right) + \left(2.0181985322458977 \times 10^{-9} \cdot M3 + 2.515582965454013 \times 10^{-9} \cdot M1 + 2.1865310959297857 \times 10^{-9} \cdot M2\right) \cdot psi4 $$ $$ \mathrm{D}\left(M1\right) = \left(-1\right) \cdot M1 \cdot \left(1.9984411888667577 \times 10^{-9} \cdot N3 + 3.304813445275614 \times 10^{-9} \cdot N5 + 1.685530410823016 \times 10^{-9} \cdot N2 + 2.515582965454013 \times 10^{-9} \cdot N4\right) + \left(-1.5339907267237098 \times 10^{-9}\right) \cdot N1 \cdot M1 $$ $$ \mathrm{D}\left(M2\right) = 1.1372237880761914 \times 10^{-27} \cdot \left(\left(-1\right) \cdot f2 + psi2\right) \cdot M1 + \left(-1\right) \cdot \left(1.8564620188751522 \times 10^{-9} \cdot N3 + 2.7282326069301694 \times 10^{-9} \cdot N5 + 2.1865310959297857 \times 10^{-9} \cdot N4\right) \cdot M2 + 1.5339907267237098 \times 10^{-9} \cdot N1 \cdot M1 + \left(-1.6977404750594217 \times 10^{-9}\right) \cdot N2 \cdot M2 + \left(-1.3431570461245907 \times 10^{-27}\right) \cdot f2 \cdot M1 + \left(-6.742121643292064 \times 10^{-27}\right) \cdot M1 \cdot psi2 + 1.685530410823016 \times 10^{-9} \cdot N2 \cdot M1 $$ $$ \mathrm{D}\left(M3\right) = N3 \cdot \left(1.9984411888667577 \times 10^{-9} \cdot M1 + 1.8564620188751522 \times 10^{-9} \cdot M2\right) + \left(-7.999999999999999 \times 10^{-18}\right) \cdot \left(1.9984411888667577 \times 10^{-9} \cdot M1 + 1.8564620188751522 \times 10^{-9} \cdot M2\right) \cdot psi3 + \left(-1\right) \cdot M3 \cdot \left(2.35376992455032 \times 10^{-9} \cdot N5 + 2.0181985322458977 \times 10^{-9} \cdot N4\right) + 1.1372237880761914 \times 10^{-27} \cdot \left(f2 + \left(-1\right) \cdot psi2\right) \cdot M1 + 1.3431570461245907 \times 10^{-27} \cdot f2 \cdot M1 + 1.6977404750594217 \times 10^{-9} \cdot N2 \cdot M2 + \left(-0.53125\right) \cdot f3 \cdot \left(2.997661783300136 \times 10^{-27} \cdot M1 + 5.569386056625455 \times 10^{-27} \cdot M2\right) + 149932861328125024 \cdot \left(4.496492674950203 \times 10^{-45} \cdot M1 + 1.6708158169876362 \times 10^{-44} \cdot M2\right) \cdot \left(\left(-1\right) \cdot f3 + psi3\right) + \left(-1.8595125110708578 \times 10^{-9}\right) \cdot N3 \cdot M3 + 6.742121643292064 \times 10^{-27} \cdot M1 \cdot psi2 $$ $$ \mathrm{D}\left(M4\right) = \left(-2.0051076185437887 \times 10^{-9}\right) \cdot M4 \cdot N4 + \left(-0.53125\right) \cdot f4 \cdot \left(1.2109191193475384 \times 10^{-26} \cdot M3 + 3.773374448181019 \times 10^{-27} \cdot M1 + 6.559593287789356 \times 10^{-27} \cdot M2\right) + 74966430664062512 \cdot \left(7.265514716085229 \times 10^{-44} \cdot M3 + 5.660061672271527 \times 10^{-45} \cdot M1 + 1.9678779863368064 \times 10^{-44} \cdot M2\right) \cdot \left(\left(-1\right) \cdot f4 + psi4\right) + 149932861328125024 \cdot \left(f3 + \left(-1\right) \cdot psi3\right) \cdot \left(4.496492674950203 \times 10^{-45} \cdot M1 + 1.6708158169876362 \times 10^{-44} \cdot M2\right) + \left(2.0181985322458977 \times 10^{-9} \cdot M3 + 2.515582965454013 \times 10^{-9} \cdot M1 + 2.1865310959297857 \times 10^{-9} \cdot M2\right) \cdot N4 + \left(-2.153624885355299 \times 10^{-9}\right) \cdot N5 \cdot M4 + 0.53125 \cdot f3 \cdot \left(2.997661783300136 \times 10^{-27} \cdot M1 + 5.569386056625455 \times 10^{-27} \cdot M2\right) + \left(-1.5999999999999998 \times 10^{-17}\right) \cdot \left(2.0181985322458977 \times 10^{-9} \cdot M3 + 2.515582965454013 \times 10^{-9} \cdot M1 + 2.1865310959297857 \times 10^{-9} \cdot M2\right) \cdot psi4 + 1.8595125110708578 \times 10^{-9} \cdot N3 \cdot M3 + 7.999999999999999 \times 10^{-18} \cdot \left(1.9984411888667577 \times 10^{-9} \cdot M1 + 1.8564620188751522 \times 10^{-9} \cdot M2\right) \cdot psi3 $$ $$ \mathrm{D}\left(M5\right) = 2.0051076185437887 \times 10^{-9} \cdot M4 \cdot N4 + N5 \cdot \left(2.35376992455032 \times 10^{-9} \cdot M3 + 2.153624885355299 \times 10^{-9} \cdot M4 + 3.304813445275614 \times 10^{-9} \cdot M1 + 2.7282326069301694 \times 10^{-9} \cdot M2\right) + 0.53125 \cdot f4 \cdot \left(1.2109191193475384 \times 10^{-26} \cdot M3 + 3.773374448181019 \times 10^{-27} \cdot M1 + 6.559593287789356 \times 10^{-27} \cdot M2\right) + 74966430664062512 \cdot \left(f4 + \left(-1\right) \cdot psi4\right) \cdot \left(7.265514716085229 \times 10^{-44} \cdot M3 + 5.660061672271527 \times 10^{-45} \cdot M1 + 1.9678779863368064 \times 10^{-44} \cdot M2\right) + 37483215332031256 \cdot \left(psi5 + \left(-1\right) \cdot f5\right) \cdot \left(8.473571728381149 \times 10^{-44} \cdot M3 + 3.101219834911629 \times 10^{-43} \cdot M4 + 7.435830251870129 \times 10^{-45} \cdot M1 + 2.4554093462371517 \times 10^{-44} \cdot M2\right) + \left(-0.53125\right) \cdot \left(1.4122619547301916 \times 10^{-26} \cdot M3 + 2.5843498624263582 \times 10^{-26} \cdot M4 + 4.95722016791342 \times 10^{-27} \cdot M1 + 8.184697820790507 \times 10^{-27} \cdot M2\right) \cdot f5 + 1.5999999999999998 \times 10^{-17} \cdot \left(2.0181985322458977 \times 10^{-9} \cdot M3 + 2.515582965454013 \times 10^{-9} \cdot M1 + 2.1865310959297857 \times 10^{-9} \cdot M2\right) \cdot psi4 + \left(-2.117154741395111 \times 10^{-9}\right) \cdot N5 \cdot M5 $$Analyses
canonical_coagulation_day
The two figures produced by the upstream TOMAS.jl script: number concentration N_k and mass concentration M_k in each of the five size bins over a 24 h coagulation run, started from the default normal-weighted spectrum. Both decay monotonically as particles coagulate; plot on a log y-axis.
Time span: 0.0 → 86400.0
uniform_initial_collapse
Number-concentration decay from a flat initial spectrum (N_k = 5e7 cm^-3 in every bin). Illustrates that the model relaxes toward a self-preserving size distribution regardless of the initial shape.
Time span: 0.0 → 3600.0
Raw .esm
Raw .esm JSON (this component)
{
"reference": {
"doi": "10.1029/2001JD001010",
"citation": "Adams, P. J. and Seinfeld, J. H.: Predicting global aerosol size distributions in general circulation models, J. Geophys. Res., 107(D19), 4370, 2002.",
"url": "https://doi.org/10.1029/2001JD001010",
"notes": "Source: Aerosol.jl src/TOMAS.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (migration bead esm-kxky). The upstream file is an undocumented, module-level (Pluto-style) script with no in-repo tests or docs (migration-tracker section 1.15 TOMAS.jl). It builds a five-bin TwO-Moment Aerosol Sectional (TOMAS) Brownian-coagulation ODESystem. The size grid is mass-doubling (m_upper = 2*m_lower, lowest bin m_lower = 1e-18 g), particle density 1.8 g/cm^3, so all five bins lie in the free-molecular regime (diameter < 0.1 um); the 5x5 coagulation-coefficient matrix is evaluated at T = 298 K and baked into the equations as numeric literals, together with the bin masses and the mass-doubling spectral factor xi = 1.0625. The model therefore has no free parameters. f_k and psi_k are the linear within-bin number-distribution reconstruction coefficients of the two-moment scheme; on the mass-doubling grid m_k/m_lower = 1.5 for every bin, so the f_k and psi_k expressions coincide identically (both reduce to N_k/m_lower_k) and the (psi_k - f_k) terms in the differential equations vanish. Migration: the upstream @named ODESystem is defined over array variables N(t)[1:5], M(t)[1:5], f(t)[1:5], psi(t)[1:5]; these were scalarized into 20 scalar unknowns (N1..N5, M1..M5, f1..f5, psi1..psi5) so the equations serialize without array-indexing ops, then scaffolded with mtk2esm. Inline-test expected values are an upstream reference run of the same scalarized ODESystem (OrdinaryDiffEq Tsit5, abstol=1e-14, reltol=1e-12). Tolerance rationale (model-level rel=2e-3): the Python inline-test gate re-integrates with SciPy LSODA (rtol=1e-10, atol=1e-12). All twenty state trajectories - the number concentrations N_k, the mass concentrations M_k, and the algebraic coefficients f_k/psi_k - were verified to agree between the Tsit5 reference and the LSODA gate to better than 2.1e-4 relative at every assertion point across all three test scenarios; rel=2e-3 is a ~10x margin over that observed Tsit5-vs-LSODA truncation difference. Mass-bin assertions are deliberately placed at t=0 and at early-to-mid times where M_k stays well above the gate's atol=1e-12 integrator floor, so each remains a meaningful relative check."
},
"tolerance": {
"rel": 0.002
},
"variables": {
"N1": {
"type": "unknown",
"units": "cm^-3",
"default": 15830903.16595993,
"description": "Number concentration of aerosol particles in mass-doubling size bin 1 (of 5)."
},
"N2": {
"type": "unknown",
"units": "cm^-3",
"default": 27165938.46737124,
"description": "Number concentration of aerosol particles in mass-doubling size bin 2 (of 5)."
},
"N3": {
"type": "unknown",
"units": "cm^-3",
"default": 44789060.589685805,
"description": "Number concentration of aerosol particles in mass-doubling size bin 3 (of 5)."
},
"N4": {
"type": "unknown",
"units": "cm^-3",
"default": 70949185.69246289,
"description": "Number concentration of aerosol particles in mass-doubling size bin 4 (of 5)."
},
"N5": {
"type": "unknown",
"units": "cm^-3",
"default": 107981933.02637613,
"description": "Number concentration of aerosol particles in mass-doubling size bin 5 (of 5)."
},
"M1": {
"type": "unknown",
"units": "g/cm^3",
"default": 2.374635474893989e-11,
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"default": 8.149781540211371e-11,
"description": "Mass concentration of aerosol particles in mass-doubling size bin 2 (of 5)."
},
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"default": 2.687343635381148e-10,
"description": "Mass concentration of aerosol particles in mass-doubling size bin 3 (of 5)."
},
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"default": 8.513902283095546e-10,
"description": "Mass concentration of aerosol particles in mass-doubling size bin 4 (of 5)."
},
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"default": 2.5915663926330265e-09,
"description": "Mass concentration of aerosol particles in mass-doubling size bin 5 (of 5)."
},
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"description": "Within-bin linear number-distribution reconstruction coefficient f for size bin 1; evaluated algebraically as 2*N1/m_lower*(2 - m_k/m_lower). Equals psi1 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"f2": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient f for size bin 2; evaluated algebraically as 2*N2/m_lower*(2 - m_k/m_lower). Equals psi2 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"f3": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient f for size bin 3; evaluated algebraically as 2*N3/m_lower*(2 - m_k/m_lower). Equals psi3 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"f4": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient f for size bin 4; evaluated algebraically as 2*N4/m_lower*(2 - m_k/m_lower). Equals psi4 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"f5": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient f for size bin 5; evaluated algebraically as 2*N5/m_lower*(2 - m_k/m_lower). Equals psi5 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient psi for size bin 1; evaluated algebraically as 2*N1/m_lower*(m_k/m_lower - 1). Equals f1 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"psi2": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient psi for size bin 2; evaluated algebraically as 2*N2/m_lower*(m_k/m_lower - 1). Equals f2 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"psi3": {
"type": "unknown",
"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient psi for size bin 3; evaluated algebraically as 2*N3/m_lower*(m_k/m_lower - 1). Equals f3 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"psi4": {
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"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient psi for size bin 4; evaluated algebraically as 2*N4/m_lower*(m_k/m_lower - 1). Equals f4 on the mass-doubling grid (m_k/m_lower = 1.5)."
},
"psi5": {
"type": "unknown",
"units": "cm^-3/g",
"description": "Within-bin linear number-distribution reconstruction coefficient psi for size bin 5; evaluated algebraically as 2*N5/m_lower*(m_k/m_lower - 1). Equals f5 on the mass-doubling grid (m_k/m_lower = 1.5)."
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"id": "canonical_lognormal_day",
"description": "Upstream TOMAS.jl base case: a normal-distribution-weighted initial number spectrum (N_k proportional to pdf(Normal(15,5), k)) integrated over a full day. Brownian coagulation collapses the number concentration in every bin by 3-7 orders of magnitude over 86400 s. The N1 assertion at t=0 verifies initial-condition handling; N_k at 1 h and 24 h are the primary behavioral gate; the mass bins M3-M5 are checked at 5 min and M5 (slowest-decaying mass bin) again at 24 h.",
"initial_conditions": {
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"M1": 2.374635474893989e-11,
"N2": 27165938.46737124,
"M2": 8.149781540211371e-11,
"N3": 44789060.589685805,
"M3": 2.687343635381148e-10,
"N4": 70949185.69246289,
"M4": 8.513902283095546e-10,
"N5": 107981933.02637613,
"M5": 2.5915663926330265e-09
},
"time_span": {
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"end": 86400.0
},
"assertions": [
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"time": 0.0,
"expected": 15830903.16595993
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{
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"time": 3600.0,
"expected": 89.19930319142259
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{
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"time": 3600.0,
"expected": 1029.3536239398309
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{
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"expected": 6108.853146944454
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{
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"expected": 24748.96132511365
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{
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"time": 3600.0,
"expected": 110419.40805753833
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{
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"time": 86400.0,
"expected": 0.5899336964512746
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{
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"time": 86400.0,
"expected": 157.6329120425347
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{
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"time": 86400.0,
"expected": 4748.165756761327
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{
"variable": "M3",
"time": 300.0,
"expected": 6.846060673866528e-13
},
{
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"expected": 4.452831698407978e-12
},
{
"variable": "M5",
"time": 300.0,
"expected": 1.363178440218432e-10
},
{
"variable": "M5",
"time": 86400.0,
"expected": 1.2558915529539911e-12
}
]
},
{
"id": "canonical_fast_transient",
"description": "Same normal-weighted initial spectrum, first 10 minutes. Coagulation is fastest here: N1 falls ~250-fold in the first 60 s. Resolves the stiff initial transient that the 24 h test only samples at its endpoints. Mass bins M3-M5 are checked at 60 s.",
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"M1": 2.374635474893989e-11,
"N2": 27165938.46737124,
"M2": 8.149781540211371e-11,
"N3": 44789060.589685805,
"M3": 2.687343635381148e-10,
"N4": 70949185.69246289,
"M4": 8.513902283095546e-10,
"N5": 107981933.02637613,
"M5": 2.5915663926330265e-09
},
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"end": 600.0
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"time": 60.0,
"expected": 62841.472624463946
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{
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"time": 60.0,
"expected": 250965.44988886872
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{
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{
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{
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"time": 60.0,
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{
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{
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{
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{
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{
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{
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},
{
"id": "uniform_initial_distribution",
"description": "A flat initial number spectrum (N_k = 5e7 cm^-3 in every bin) coagulated for one hour. Exercises the model from an initial state unlike the upstream lognormal case and confirms it relaxes toward a similar self-preserving shape. The t=0 assertions verify initial-condition handling; the f3/psi3 assertions at 10 min check that the two algebraic reconstruction coefficients track N3 and remain identical.",
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"M1": 7.499999999999999e-11,
"N2": 50000000,
"M2": 1.4999999999999997e-10,
"N3": 50000000,
"M3": 2.9999999999999995e-10,
"N4": 50000000,
"M4": 5.999999999999999e-10,
"N5": 50000000,
"M5": 1.1999999999999998e-09
},
"time_span": {
"start": 0.0,
"end": 3600.0
},
"assertions": [
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"time": 0.0,
"expected": 50000000
},
{
"variable": "M1",
"time": 0.0,
"expected": 7.499999999999999e-11
},
{
"variable": "N1",
"time": 600.0,
"expected": 4967.89383163667
},
{
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"time": 600.0,
"expected": 21088.94133214904
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{
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"time": 600.0,
"expected": 67949.9528962513
},
{
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"time": 600.0,
"expected": 189928.1527596645
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{
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"time": 600.0,
"expected": 624149.7234928618
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{
"variable": "N1",
"time": 3600.0,
"expected": 274.29903359219924
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{
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"time": 3600.0,
"expected": 8072.106469110638
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{
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"time": 3600.0,
"expected": 108010.50164437758
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{
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"time": 3600.0,
"expected": 3.472817558310097e-13
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{
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"time": 3600.0,
"expected": 2.1354007089650603e-11
},
{
"variable": "f3",
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"expected": 1.6987488224062827e+22
},
{
"variable": "psi3",
"time": 600.0,
"expected": 1.6987488224062827e+22
}
]
},
{
"id": "descending_initial_distribution",
"description": "A descending initial number spectrum (most particles in the smallest bin: N = [2e8, 8e7, 2e7, 5e6, 1e6] cm^-3) coagulated for two hours. Exercises coagulation driven from the small-particle end. The t=0 assertions verify initial-condition handling; the f1/psi1 assertions at 15 min check the algebraic coefficients for the smallest bin.",
"initial_conditions": {
"N1": 200000000,
"M1": 2.9999999999999995e-10,
"N2": 80000000,
"M2": 2.3999999999999995e-10,
"N3": 20000000,
"M3": 1.1999999999999997e-10,
"N4": 5000000,
"M4": 5.999999999999999e-11,
"N5": 1000000,
"M5": 2.3999999999999995e-11
},
"time_span": {
"start": 0.0,
"end": 7200.0
},
"assertions": [
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"variable": "N1",
"time": 0.0,
"expected": 200000000
},
{
"variable": "N5",
"time": 0.0,
"expected": 1000000
},
{
"variable": "M1",
"time": 0.0,
"expected": 2.9999999999999995e-10
},
{
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"time": 3600.0,
"expected": 413.49249242033244
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{
"variable": "N2",
"time": 3600.0,
"expected": 2185.3912558754896
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{
"variable": "N3",
"time": 3600.0,
"expected": 8627.10600036097
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{
"variable": "N4",
"time": 3600.0,
"expected": 27896.963397602667
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{
"variable": "N5",
"time": 3600.0,
"expected": 113500.68716088931
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{
"variable": "N1",
"time": 7200.0,
"expected": 129.6738926321766
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{
"variable": "N3",
"time": 7200.0,
"expected": 3688.680555994787
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{
"variable": "N5",
"time": 7200.0,
"expected": 57496.815320082635
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{
"variable": "f1",
"time": 900.0,
"expected": 4.3661243090561317e+21
},
{
"variable": "psi1",
"time": 900.0,
"expected": 4.3661243090561317e+21
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],
"analyses": [
{
"id": "canonical_coagulation_day",
"description": "The two figures produced by the upstream TOMAS.jl script: number concentration N_k and mass concentration M_k in each of the five size bins over a 24 h coagulation run, started from the default normal-weighted spectrum. Both decay monotonically as particles coagulate; plot on a log y-axis.",
"time_span": {
"start": 0.0,
"end": 86400.0
},
"plots": [
{
"id": "number_concentration_decay",
"type": "line",
"description": "Number concentration N_k(t) for the five size bins (upstream TOMAS.jl figure 1).",
"x": {
"variable": "t",
"label": "time (s)"
},
"y": {
"variable": "N1",
"label": "number concentration (cm^-3)"
},
"series": [
{
"name": "N1",
"variable": "N1"
},
{
"name": "N2",
"variable": "N2"
},
{
"name": "N3",
"variable": "N3"
},
{
"name": "N4",
"variable": "N4"
},
{
"name": "N5",
"variable": "N5"
}
]
},
{
"id": "mass_concentration_decay",
"type": "line",
"description": "Mass concentration M_k(t) for the five size bins (upstream TOMAS.jl figure 2).",
"x": {
"variable": "t",
"label": "time (s)"
},
"y": {
"variable": "M1",
"label": "mass concentration (g cm^-3)"
},
"series": [
{
"name": "M1",
"variable": "M1"
},
{
"name": "M2",
"variable": "M2"
},
{
"name": "M3",
"variable": "M3"
},
{
"name": "M4",
"variable": "M4"
},
{
"name": "M5",
"variable": "M5"
}
]
}
]
},
{
"id": "uniform_initial_collapse",
"description": "Number-concentration decay from a flat initial spectrum (N_k = 5e7 cm^-3 in every bin). Illustrates that the model relaxes toward a self-preserving size distribution regardless of the initial shape.",
"initial_state": {
"N1": 50000000,
"M1": 7.499999999999999e-11,
"N2": 50000000,
"M2": 1.4999999999999997e-10,
"N3": 50000000,
"M3": 2.9999999999999995e-10,
"N4": 50000000,
"M4": 5.999999999999999e-10,
"N5": 50000000,
"M5": 1.1999999999999998e-09
},
"time_span": {
"start": 0.0,
"end": 3600.0
},
"plots": [
{
"id": "uniform_number_decay",
"type": "line",
"description": "Number concentration N_k(t) from a uniform initial spectrum.",
"x": {
"variable": "t",
"label": "time (s)"
},
"y": {
"variable": "N1",
"label": "number concentration (cm^-3)"
},
"series": [
{
"name": "N1",
"variable": "N1"
},
{
"name": "N2",
"variable": "N2"
},
{
"name": "N3",
"variable": "N3"
},
{
"name": "N4",
"variable": "N4"
},
{
"name": "N5",
"variable": "N5"
}
]
}
]
}
]
}