StochasticCollectionGolovin
Two-moment stochastic collection equation (SCE) for cloud-droplet coalescence (Tzivion, Feingold & Levin 1989), scalarized to I=5 mass-doubling categories. Two models, one per collection kernel: StochasticCollectionConstant (K=K0) and StochasticCollectionGolovin (K=C(x+y)). Each tracks number concentration N_k and mass concentration M_k in every category and closes the higher moments with the non-dimensional parameter xi_p. Migrated from the array-variable @component in Aerosol.jl src/stochastic_collection.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (esm-ar1e); the upstream @arrayop/@makearray array operations were scalarized to flat scalar states and equations.
Description
Two-moment stochastic collection equation (SCE) for cloud-droplet coalescence (Tzivion, Feingold & Levin 1989), scalarized to I=5 mass-doubling categories. Two models, one per collection kernel: StochasticCollectionConstant (K=K0) and StochasticCollectionGolovin (K=C(x+y)). Each tracks number concentration N_k and mass concentration M_k in every category and closes the higher moments with the non-dimensional parameter xi_p. Migrated from the array-variable @component in Aerosol.jl src/stochastic_collection.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (esm-ar1e); the upstream @arrayop/@makearray array operations were scalarized to flat scalar states and equations.
Reference
Tzivion, S., Feingold, G., and Levin, Z.: The Evolution of Raindrop Spectra. Part II: Collisional Collection/Breakup and Evaporation in a Rainshaft, J. Atmos. Sci., 46(21), 3312-3327, 1989.
Source: Aerosol.jl src/stochastic_collection.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (migration bead esm-ar1e); the upstream @component StochasticCollectionCoalescence with kernel_type=golovin. The upstream model solves the two-moment stochastic collection equation (SCE) for cloud-droplet coalescence (Tzivion et al. 1989) over I mass-doubling categories, tracking number N_k and mass M_k per category. It is built with symbolic array operations (SymbolicUtils @arrayop/@makearray over array variables Nk(t)[1:I], Mk(t)[1:I]); the array ops are not in the .esm closed AST and the ESS Python runner does not evaluate them, so the model is scalarized to I=5 categories and 10 scalar differential states N1..N5, M1..M5. The scalarization evaluates the upstream @arrayop ifelse masks (i==1, i<=k-2, i<=k-1, i>=k+1) at concrete (k,i), so the flat equations are the source semantics, not a hand-derived rewrite. The five (six for Golovin) closure families are carried as algebraic states with explicit equations (rig convention; cf. components/aerosol/tomas.esm): xb_k (mean mass M_k/N_k), Z_k/Q_k/R_k (higher moments xi_pM^2/N, xi_p^2M^3/N^2, xi_p^3M^4/N^3), and f_k/psi_k/alpha_k (linear within-category reconstruction coefficients, Eq. 13+15). The upstream closures guard m/n divisions with ifelse(n>0,…); the .esm replaces each by the algebraically equivalent floored form mn/(n^2+1e-300) etc. — bit-identical to m/n for every n>0 that occurs (n>=1e-20 makes the 1e-300 floor ~260 orders below rounding), finite at n=0, and free of the ifelse-guarded divide-by-zero that the runner’s eager Piecewise lowering would otherwise hit. Kernel: Golovin (sum) kernel K(x,y)=C*(x+y). Verification: the scalarized RHS was checked (1) by a line-by-line transcription audit against the upstream @arrayop bodies; (2) against an independent literal double-sum implementation of the same @arrayop semantics (agreement < 1e-9 relative over random states); and (3) against first-principles no-overflow analytic identities — with the top category empty, no collision product exceeds the grid, so collection conserves number per collision and conserves mass exactly: dN_tot/dt = -(K0/2)N_tot^2 (constant kernel) and dN_tot/dt = -CM_tot*N_tot (Golovin kernel), with dM_tot/dt = 0. The scalarized RHS reproduces all three identities to machine precision (relative error 0). Inline-test expected values are a SciPy LSODA reference integration (rtol=1e-10, abstol=1e-14) of this verified RHS; the gate re-integrates the .esm with SciPy LSODA, and the model-level rel=2e-3 tolerance (as for tomas.esm) covers integrator-detail differences. This migration covers only collisional collection; the breakup, evaporation and sedimentation processes of the full 1989 paper are not implemented upstream either.
- DOI: 10.1175/1520-0469(1989)046<3312:TEORSP>2.0.CO;2
- URL: https://doi.org/10.1175/1520-0469(1989)046%3C3312:TEORSP%3E2.0.CO;2
Variables
| Name | Units | Default | Description |
|---|---|---|---|
N1 | m^-3 | 1429670.6078643417 | Number concentration of cloud droplets in mass category 1 (of 5); lower mass boundary x_1 = 1.600000e-14 kg. |
N2 | m^-3 | 2838836.0713738734 | Number concentration of cloud droplets in mass category 2 (of 5); lower mass boundary x_2 = 3.200000e-14 kg. |
N3 | m^-3 | 5596563.981307812 | Number concentration of cloud droplets in mass category 3 (of 5); lower mass boundary x_3 = 6.400000e-14 kg. |
N4 | m^-3 | 10875864.787608625 | Number concentration of cloud droplets in mass category 4 (of 5); lower mass boundary x_4 = 1.280000e-13 kg. |
N5 | m^-3 | 20538015.50564386 | Number concentration of cloud droplets in mass category 5 (of 5); lower mass boundary x_5 = 2.560000e-13 kg. |
M1 | kg*m^-3 | 3.430294470019302e-08 | Mass concentration of cloud droplets in mass category 1 (of 5). |
M2 | kg*m^-3 | 1.3619145733423377e-07 | Mass concentration of cloud droplets in mass category 2 (of 5). |
M3 | kg*m^-3 | 5.366970575748642e-07 | Mass concentration of cloud droplets in mass category 3 (of 5). |
M4 | kg*m^-3 | 2.083711394480048e-06 | Mass concentration of cloud droplets in mass category 4 (of 5). |
M5 | kg*m^-3 | 7.852951776977424e-06 | Mass concentration of cloud droplets in mass category 5 (of 5). |
Parameters
| Name | Units | Default | Description |
|---|---|---|---|
XI_P | 1 | 1.0625 | Closure parameter xi_p (= 17/16) relating the higher-order moments Z_k, Q_k, R_k to N_k and M_k (Tzivion et al. 1989, Eq. 8a,b). |
C | m^3/(kg*s) | 0.0015 | Golovin (sum) collection-kernel coefficient: K(x,y) = C*(x + y). |
Observed
| Name | Units | Default | Description |
|---|---|---|---|
xb1 | kg | Mean droplet mass M_1/N_1 in category 1; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category. | |
xb2 | kg | Mean droplet mass M_2/N_2 in category 2; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category. | |
xb3 | kg | Mean droplet mass M_3/N_3 in category 3; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category. | |
xb4 | kg | Mean droplet mass M_4/N_4 in category 4; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category. | |
xb5 | kg | Mean droplet mass M_5/N_5 in category 5; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category. | |
Z1 | kg^2*m^-3 | Second-moment closure quantity xi_p*M_1^2/N_1 for category 1; algebraic. | |
Z2 | kg^2*m^-3 | Second-moment closure quantity xi_p*M_2^2/N_2 for category 2; algebraic. | |
Z3 | kg^2*m^-3 | Second-moment closure quantity xi_p*M_3^2/N_3 for category 3; algebraic. | |
Z4 | kg^2*m^-3 | Second-moment closure quantity xi_p*M_4^2/N_4 for category 4; algebraic. | |
Z5 | kg^2*m^-3 | Second-moment closure quantity xi_p*M_5^2/N_5 for category 5; algebraic. | |
Q1 | kg^3*m^-3 | Third-moment closure quantity xi_p^2*M_1^3/N_1^2 for category 1; algebraic. | |
Q2 | kg^3*m^-3 | Third-moment closure quantity xi_p^2*M_2^3/N_2^2 for category 2; algebraic. | |
Q3 | kg^3*m^-3 | Third-moment closure quantity xi_p^2*M_3^3/N_3^2 for category 3; algebraic. | |
Q4 | kg^3*m^-3 | Third-moment closure quantity xi_p^2*M_4^3/N_4^2 for category 4; algebraic. | |
Q5 | kg^3*m^-3 | Third-moment closure quantity xi_p^2*M_5^3/N_5^2 for category 5; algebraic. | |
R1 | kg^4*m^-3 | Fourth-moment closure quantity xi_p^3*M_1^4/N_1^3 for category 1; algebraic (Golovin kernel only). | |
R2 | kg^4*m^-3 | Fourth-moment closure quantity xi_p^3*M_2^4/N_2^3 for category 2; algebraic (Golovin kernel only). | |
R3 | kg^4*m^-3 | Fourth-moment closure quantity xi_p^3*M_3^4/N_3^3 for category 3; algebraic (Golovin kernel only). | |
R4 | kg^4*m^-3 | Fourth-moment closure quantity xi_p^3*M_4^4/N_4^3 for category 4; algebraic (Golovin kernel only). | |
R5 | kg^4*m^-3 | Fourth-moment closure quantity xi_p^3*M_5^4/N_5^3 for category 5; algebraic (Golovin kernel only). | |
f1 | kg^-1*m^-3 | Linear within-category number-distribution coefficient f_1 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic. | |
f2 | kg^-1*m^-3 | Linear within-category number-distribution coefficient f_2 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic. | |
f3 | kg^-1*m^-3 | Linear within-category number-distribution coefficient f_3 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic. | |
f4 | kg^-1*m^-3 | Linear within-category number-distribution coefficient f_4 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic. | |
f5 | kg^-1*m^-3 | Linear within-category number-distribution coefficient f_5 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic. | |
psi1 | kg^-1*m^-3 | Linear within-category number-distribution coefficient psi_1 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic. | |
psi2 | kg^-1*m^-3 | Linear within-category number-distribution coefficient psi_2 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic. | |
psi3 | kg^-1*m^-3 | Linear within-category number-distribution coefficient psi_3 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic. | |
psi4 | kg^-1*m^-3 | Linear within-category number-distribution coefficient psi_4 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic. | |
psi5 | kg^-1*m^-3 | Linear within-category number-distribution coefficient psi_5 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic. | |
alpha1 | kg^-2*m^-3 | Within-category distribution slope (f_1 - psi_1)/(2 x_1) for category 1; algebraic. | |
alpha2 | kg^-2*m^-3 | Within-category distribution slope (f_2 - psi_2)/(2 x_2) for category 2; algebraic. | |
alpha3 | kg^-2*m^-3 | Within-category distribution slope (f_3 - psi_3)/(2 x_3) for category 3; algebraic. | |
alpha4 | kg^-2*m^-3 | Within-category distribution slope (f_4 - psi_4)/(2 x_4) for category 4; algebraic. | |
alpha5 | kg^-2*m^-3 | Within-category distribution slope (f_5 - psi_5)/(2 x_5) for category 5; algebraic. |
Observed expressions
$$ xb1 = \frac{M1 \cdot N1}{N1^{2} + 1 \times 10^{-300}} $$Mean droplet mass M_1/N_1 in category 1; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category.
$$ xb2 = \frac{M2 \cdot N2}{N2^{2} + 1 \times 10^{-300}} $$Mean droplet mass M_2/N_2 in category 2; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category.
$$ xb3 = \frac{M3 \cdot N3}{N3^{2} + 1 \times 10^{-300}} $$Mean droplet mass M_3/N_3 in category 3; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category.
$$ xb4 = \frac{M4 \cdot N4}{N4^{2} + 1 \times 10^{-300}} $$Mean droplet mass M_4/N_4 in category 4; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category.
$$ xb5 = \frac{M5 \cdot N5}{N5^{2} + 1 \times 10^{-300}} $$Mean droplet mass M_5/N_5 in category 5; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category.
$$ Z1 = \frac{XI_{P} \cdot M1^{2} \cdot N1}{N1^{2} + 1 \times 10^{-300}} $$Second-moment closure quantity xi_p*M_1^2/N_1 for category 1; algebraic.
$$ Z2 = \frac{XI_{P} \cdot M2^{2} \cdot N2}{N2^{2} + 1 \times 10^{-300}} $$Second-moment closure quantity xi_p*M_2^2/N_2 for category 2; algebraic.
$$ Z3 = \frac{XI_{P} \cdot M3^{2} \cdot N3}{N3^{2} + 1 \times 10^{-300}} $$Second-moment closure quantity xi_p*M_3^2/N_3 for category 3; algebraic.
$$ Z4 = \frac{XI_{P} \cdot M4^{2} \cdot N4}{N4^{2} + 1 \times 10^{-300}} $$Second-moment closure quantity xi_p*M_4^2/N_4 for category 4; algebraic.
$$ Z5 = \frac{XI_{P} \cdot M5^{2} \cdot N5}{N5^{2} + 1 \times 10^{-300}} $$Second-moment closure quantity xi_p*M_5^2/N_5 for category 5; algebraic.
$$ Q1 = \frac{XI_{P}^{2} \cdot M1^{3} \cdot N1^{2}}{N1^{4} + 1 \times 10^{-300}} $$Third-moment closure quantity xi_p^2*M_1^3/N_1^2 for category 1; algebraic.
$$ Q2 = \frac{XI_{P}^{2} \cdot M2^{3} \cdot N2^{2}}{N2^{4} + 1 \times 10^{-300}} $$Third-moment closure quantity xi_p^2*M_2^3/N_2^2 for category 2; algebraic.
$$ Q3 = \frac{XI_{P}^{2} \cdot M3^{3} \cdot N3^{2}}{N3^{4} + 1 \times 10^{-300}} $$Third-moment closure quantity xi_p^2*M_3^3/N_3^2 for category 3; algebraic.
$$ Q4 = \frac{XI_{P}^{2} \cdot M4^{3} \cdot N4^{2}}{N4^{4} + 1 \times 10^{-300}} $$Third-moment closure quantity xi_p^2*M_4^3/N_4^2 for category 4; algebraic.
$$ Q5 = \frac{XI_{P}^{2} \cdot M5^{3} \cdot N5^{2}}{N5^{4} + 1 \times 10^{-300}} $$Third-moment closure quantity xi_p^2*M_5^3/N_5^2 for category 5; algebraic.
$$ R1 = \frac{XI_{P}^{3} \cdot M1^{4} \cdot N1^{3}}{N1^{6} + 1 \times 10^{-300}} $$Fourth-moment closure quantity xi_p^3*M_1^4/N_1^3 for category 1; algebraic (Golovin kernel only).
$$ R2 = \frac{XI_{P}^{3} \cdot M2^{4} \cdot N2^{3}}{N2^{6} + 1 \times 10^{-300}} $$Fourth-moment closure quantity xi_p^3*M_2^4/N_2^3 for category 2; algebraic (Golovin kernel only).
$$ R3 = \frac{XI_{P}^{3} \cdot M3^{4} \cdot N3^{3}}{N3^{6} + 1 \times 10^{-300}} $$Fourth-moment closure quantity xi_p^3*M_3^4/N_3^3 for category 3; algebraic (Golovin kernel only).
$$ R4 = \frac{XI_{P}^{3} \cdot M4^{4} \cdot N4^{3}}{N4^{6} + 1 \times 10^{-300}} $$Fourth-moment closure quantity xi_p^3*M_4^4/N_4^3 for category 4; algebraic (Golovin kernel only).
$$ R5 = \frac{XI_{P}^{3} \cdot M5^{4} \cdot N5^{3}}{N5^{6} + 1 \times 10^{-300}} $$Fourth-moment closure quantity xi_p^3*M_5^4/N_5^3 for category 5; algebraic (Golovin kernel only).
$$ f1 = \mathrm{ifelse}\left(\mathrm{<}\left(xb1, 1.6 \times 10^{-14}\right), 0.0, \mathrm{ifelse}\left(\mathrm{>}\left(xb1, 3.2 \times 10^{-14}\right), \frac{2.0 \cdot N1}{1.6 \times 10^{-14}}, \frac{4.0 \cdot N1}{1.6 \times 10^{-14}} - \frac{2.0 \cdot M1}{2.56 \times 10^{-28}}\right)\right) $$Linear within-category number-distribution coefficient f_1 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic.
$$ f2 = \mathrm{ifelse}\left(\mathrm{<}\left(xb2, 3.2 \times 10^{-14}\right), 0.0, \mathrm{ifelse}\left(\mathrm{>}\left(xb2, 6.4 \times 10^{-14}\right), \frac{2.0 \cdot N2}{3.2 \times 10^{-14}}, \frac{4.0 \cdot N2}{3.2 \times 10^{-14}} - \frac{2.0 \cdot M2}{1.024 \times 10^{-27}}\right)\right) $$Linear within-category number-distribution coefficient f_2 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic.
$$ f3 = \mathrm{ifelse}\left(\mathrm{<}\left(xb3, 6.4 \times 10^{-14}\right), 0.0, \mathrm{ifelse}\left(\mathrm{>}\left(xb3, 1.28 \times 10^{-13}\right), \frac{2.0 \cdot N3}{6.4 \times 10^{-14}}, \frac{4.0 \cdot N3}{6.4 \times 10^{-14}} - \frac{2.0 \cdot M3}{4.096 \times 10^{-27}}\right)\right) $$Linear within-category number-distribution coefficient f_3 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic.
$$ f4 = \mathrm{ifelse}\left(\mathrm{<}\left(xb4, 1.28 \times 10^{-13}\right), 0.0, \mathrm{ifelse}\left(\mathrm{>}\left(xb4, 2.56 \times 10^{-13}\right), \frac{2.0 \cdot N4}{1.28 \times 10^{-13}}, \frac{4.0 \cdot N4}{1.28 \times 10^{-13}} - \frac{2.0 \cdot M4}{1.6384 \times 10^{-26}}\right)\right) $$Linear within-category number-distribution coefficient f_4 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic.
$$ f5 = \mathrm{ifelse}\left(\mathrm{<}\left(xb5, 2.56 \times 10^{-13}\right), 0.0, \mathrm{ifelse}\left(\mathrm{>}\left(xb5, 5.12 \times 10^{-13}\right), \frac{2.0 \cdot N5}{2.56 \times 10^{-13}}, \frac{4.0 \cdot N5}{2.56 \times 10^{-13}} - \frac{2.0 \cdot M5}{6.5536 \times 10^{-26}}\right)\right) $$Linear within-category number-distribution coefficient f_5 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic.
$$ psi1 = \mathrm{ifelse}\left(\mathrm{<}\left(xb1, 1.6 \times 10^{-14}\right), \frac{2.0 \cdot N1}{1.6 \times 10^{-14}}, \mathrm{ifelse}\left(\mathrm{>}\left(xb1, 3.2 \times 10^{-14}\right), 0.0, \frac{2.0 \cdot M1}{2.56 \times 10^{-28}} - \frac{2.0 \cdot N1}{1.6 \times 10^{-14}}\right)\right) $$Linear within-category number-distribution coefficient psi_1 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic.
$$ psi2 = \mathrm{ifelse}\left(\mathrm{<}\left(xb2, 3.2 \times 10^{-14}\right), \frac{2.0 \cdot N2}{3.2 \times 10^{-14}}, \mathrm{ifelse}\left(\mathrm{>}\left(xb2, 6.4 \times 10^{-14}\right), 0.0, \frac{2.0 \cdot M2}{1.024 \times 10^{-27}} - \frac{2.0 \cdot N2}{3.2 \times 10^{-14}}\right)\right) $$Linear within-category number-distribution coefficient psi_2 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic.
$$ psi3 = \mathrm{ifelse}\left(\mathrm{<}\left(xb3, 6.4 \times 10^{-14}\right), \frac{2.0 \cdot N3}{6.4 \times 10^{-14}}, \mathrm{ifelse}\left(\mathrm{>}\left(xb3, 1.28 \times 10^{-13}\right), 0.0, \frac{2.0 \cdot M3}{4.096 \times 10^{-27}} - \frac{2.0 \cdot N3}{6.4 \times 10^{-14}}\right)\right) $$Linear within-category number-distribution coefficient psi_3 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic.
$$ psi4 = \mathrm{ifelse}\left(\mathrm{<}\left(xb4, 1.28 \times 10^{-13}\right), \frac{2.0 \cdot N4}{1.28 \times 10^{-13}}, \mathrm{ifelse}\left(\mathrm{>}\left(xb4, 2.56 \times 10^{-13}\right), 0.0, \frac{2.0 \cdot M4}{1.6384 \times 10^{-26}} - \frac{2.0 \cdot N4}{1.28 \times 10^{-13}}\right)\right) $$Linear within-category number-distribution coefficient psi_4 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic.
$$ psi5 = \mathrm{ifelse}\left(\mathrm{<}\left(xb5, 2.56 \times 10^{-13}\right), \frac{2.0 \cdot N5}{2.56 \times 10^{-13}}, \mathrm{ifelse}\left(\mathrm{>}\left(xb5, 5.12 \times 10^{-13}\right), 0.0, \frac{2.0 \cdot M5}{6.5536 \times 10^{-26}} - \frac{2.0 \cdot N5}{2.56 \times 10^{-13}}\right)\right) $$Linear within-category number-distribution coefficient psi_5 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic.
$$ alpha1 = \frac{f1 - psi1}{3.2 \times 10^{-14}} $$Within-category distribution slope (f_1 - psi_1)/(2 x_1) for category 1; algebraic.
$$ alpha2 = \frac{f2 - psi2}{6.4 \times 10^{-14}} $$Within-category distribution slope (f_2 - psi_2)/(2 x_2) for category 2; algebraic.
$$ alpha3 = \frac{f3 - psi3}{1.28 \times 10^{-13}} $$Within-category distribution slope (f_3 - psi_3)/(2 x_3) for category 3; algebraic.
$$ alpha4 = \frac{f4 - psi4}{2.56 \times 10^{-13}} $$Within-category distribution slope (f_4 - psi_4)/(2 x_4) for category 4; algebraic.
$$ alpha5 = \frac{f5 - psi5}{5.12 \times 10^{-13}} $$Within-category distribution slope (f_5 - psi_5)/(2 x_5) for category 5; algebraic.
Equations
$$ \mathrm{D}\left(N1\right) = \left(-C\right) \cdot 2.0 \cdot N1 \cdot M1 + \left(-C\right) \cdot \left(M1 \cdot N2 + N1 \cdot M2\right) + \left(-C\right) \cdot \left(M1 \cdot N3 + N1 \cdot M3\right) + \left(-C\right) \cdot \left(M1 \cdot N4 + N1 \cdot M4\right) + \left(-C\right) \cdot \left(M1 \cdot N5 + N1 \cdot M5\right) $$ $$ \mathrm{D}\left(N2\right) = C \cdot N1 \cdot M1 + \left(-C\right) \cdot 2.0 \cdot N2 \cdot M2 + \left(-C\right) \cdot \left(6.4 \times 10^{-14} \cdot psi2 \cdot M1 + 0.5 \cdot f2 \cdot Z1 + alpha2 \cdot Q1\right) + \left(-C\right) \cdot \left(M2 \cdot N3 + N2 \cdot M3\right) + \left(-C\right) \cdot \left(M2 \cdot N4 + N2 \cdot M4\right) + \left(-C\right) \cdot \left(M2 \cdot N5 + N2 \cdot M5\right) $$ $$ \mathrm{D}\left(N3\right) = C \cdot N2 \cdot M2 + \left(-C\right) \cdot 2.0 \cdot N3 \cdot M3 + C \cdot \left(6.4 \times 10^{-14} \cdot psi2 \cdot M1 + 0.5 \cdot f2 \cdot Z1 + alpha2 \cdot Q1\right) + \left(-C\right) \cdot \left(1.28 \times 10^{-13} \cdot psi3 \cdot M1 + 0.5 \cdot f3 \cdot Z1 + alpha3 \cdot Q1\right) + \left(-C\right) \cdot \left(1.28 \times 10^{-13} \cdot psi3 \cdot M2 + 0.5 \cdot f3 \cdot Z2 + alpha3 \cdot Q2\right) + \left(-C\right) \cdot \left(M3 \cdot N4 + N3 \cdot M4\right) + \left(-C\right) \cdot \left(M3 \cdot N5 + N3 \cdot M5\right) $$ $$ \mathrm{D}\left(N4\right) = C \cdot N3 \cdot M3 + \left(-C\right) \cdot 2.0 \cdot N4 \cdot M4 + C \cdot \left(1.28 \times 10^{-13} \cdot psi3 \cdot M1 + 0.5 \cdot f3 \cdot Z1 + alpha3 \cdot Q1\right) + C \cdot \left(1.28 \times 10^{-13} \cdot psi3 \cdot M2 + 0.5 \cdot f3 \cdot Z2 + alpha3 \cdot Q2\right) + \left(-C\right) \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M1 + 0.5 \cdot f4 \cdot Z1 + alpha4 \cdot Q1\right) + \left(-C\right) \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M2 + 0.5 \cdot f4 \cdot Z2 + alpha4 \cdot Q2\right) + \left(-C\right) \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M3 + 0.5 \cdot f4 \cdot Z3 + alpha4 \cdot Q3\right) + \left(-C\right) \cdot \left(M4 \cdot N5 + N4 \cdot M5\right) $$ $$ \mathrm{D}\left(N5\right) = C \cdot N4 \cdot M4 + \left(-C\right) \cdot 2.0 \cdot N5 \cdot M5 + C \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M1 + 0.5 \cdot f4 \cdot Z1 + alpha4 \cdot Q1\right) + C \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M2 + 0.5 \cdot f4 \cdot Z2 + alpha4 \cdot Q2\right) + C \cdot \left(2.56 \times 10^{-13} \cdot psi4 \cdot M3 + 0.5 \cdot f4 \cdot Z3 + alpha4 \cdot Q3\right) + \left(-C\right) \cdot \left(5.12 \times 10^{-13} \cdot psi5 \cdot M1 + 0.5 \cdot f5 \cdot Z1 + alpha5 \cdot Q1\right) + \left(-C\right) \cdot \left(5.12 \times 10^{-13} \cdot psi5 \cdot M2 + 0.5 \cdot f5 \cdot Z2 + alpha5 \cdot Q2\right) + \left(-C\right) \cdot \left(5.12 \times 10^{-13} \cdot psi5 \cdot M3 + 0.5 \cdot f5 \cdot Z3 + alpha5 \cdot Q3\right) + \left(-C\right) \cdot \left(5.12 \times 10^{-13} \cdot psi5 \cdot M4 + 0.5 \cdot f5 \cdot Z4 + alpha5 \cdot Q4\right) $$ $$ \mathrm{D}\left(M1\right) = \left(-C\right) \cdot \left(N1 \cdot Z1 + M1^{2}\right) + \left(-C\right) \cdot \left(Z1 \cdot N2 + M1 \cdot M2\right) + \left(-C\right) \cdot \left(Z1 \cdot N3 + M1 \cdot M3\right) + \left(-C\right) \cdot \left(Z1 \cdot N4 + M1 \cdot M4\right) + \left(-C\right) \cdot \left(Z1 \cdot N5 + M1 \cdot M5\right) $$ $$ \mathrm{D}\left(M2\right) = C \cdot \left(N1 \cdot Z1 + M1^{2}\right) + \left(-C\right) \cdot \left(N2 \cdot Z2 + M2^{2}\right) + \left(-C\right) \cdot \left(4.096 \times 10^{-27} \cdot psi2 \cdot M1 + 3.2 \times 10^{-14} \cdot \left(0.5 \cdot f2 + 2.0 \cdot psi2\right) \cdot Z1 + \left(f2 - psi2\right) \cdot Q1 + alpha2 \cdot R1\right) + C \cdot M2 \cdot M1 + C \cdot N2 \cdot Z1 + \left(-C\right) \cdot \left(Z2 \cdot N3 + M2 \cdot M3\right) + \left(-C\right) \cdot \left(Z2 \cdot N4 + M2 \cdot M4\right) + \left(-C\right) \cdot \left(Z2 \cdot N5 + M2 \cdot M5\right) $$ $$ \mathrm{D}\left(M3\right) = C \cdot \left(N2 \cdot Z2 + M2^{2}\right) + \left(-C\right) \cdot \left(N3 \cdot Z3 + M3^{2}\right) + C \cdot \left(4.096 \times 10^{-27} \cdot psi2 \cdot M1 + 3.2 \times 10^{-14} \cdot \left(0.5 \cdot f2 + 2.0 \cdot psi2\right) \cdot Z1 + \left(f2 - psi2\right) \cdot Q1 + alpha2 \cdot R1\right) + \left(-C\right) \cdot \left(1.6384 \times 10^{-26} \cdot psi3 \cdot M1 + 6.4 \times 10^{-14} \cdot \left(0.5 \cdot f3 + 2.0 \cdot psi3\right) \cdot Z1 + \left(f3 - psi3\right) \cdot Q1 + alpha3 \cdot R1\right) + C \cdot M3 \cdot M1 + C \cdot N3 \cdot Z1 + \left(-C\right) \cdot \left(1.6384 \times 10^{-26} \cdot psi3 \cdot M2 + 6.4 \times 10^{-14} \cdot \left(0.5 \cdot f3 + 2.0 \cdot psi3\right) \cdot Z2 + \left(f3 - psi3\right) \cdot Q2 + alpha3 \cdot R2\right) + C \cdot M3 \cdot M2 + C \cdot N3 \cdot Z2 + \left(-C\right) \cdot \left(Z3 \cdot N4 + M3 \cdot M4\right) + \left(-C\right) \cdot \left(Z3 \cdot N5 + M3 \cdot M5\right) $$ $$ \mathrm{D}\left(M4\right) = C \cdot \left(N3 \cdot Z3 + M3^{2}\right) + \left(-C\right) \cdot \left(N4 \cdot Z4 + M4^{2}\right) + C \cdot \left(1.6384 \times 10^{-26} \cdot psi3 \cdot M1 + 6.4 \times 10^{-14} \cdot \left(0.5 \cdot f3 + 2.0 \cdot psi3\right) \cdot Z1 + \left(f3 - psi3\right) \cdot Q1 + alpha3 \cdot R1\right) + C \cdot \left(1.6384 \times 10^{-26} \cdot psi3 \cdot M2 + 6.4 \times 10^{-14} \cdot \left(0.5 \cdot f3 + 2.0 \cdot psi3\right) \cdot Z2 + \left(f3 - psi3\right) \cdot Q2 + alpha3 \cdot R2\right) + \left(-C\right) \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M1 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z1 + \left(f4 - psi4\right) \cdot Q1 + alpha4 \cdot R1\right) + C \cdot M4 \cdot M1 + C \cdot N4 \cdot Z1 + \left(-C\right) \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M2 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z2 + \left(f4 - psi4\right) \cdot Q2 + alpha4 \cdot R2\right) + C \cdot M4 \cdot M2 + C \cdot N4 \cdot Z2 + \left(-C\right) \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M3 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z3 + \left(f4 - psi4\right) \cdot Q3 + alpha4 \cdot R3\right) + C \cdot M4 \cdot M3 + C \cdot N4 \cdot Z3 + \left(-C\right) \cdot \left(Z4 \cdot N5 + M4 \cdot M5\right) $$ $$ \mathrm{D}\left(M5\right) = C \cdot \left(N4 \cdot Z4 + M4^{2}\right) + \left(-C\right) \cdot \left(N5 \cdot Z5 + M5^{2}\right) + C \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M1 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z1 + \left(f4 - psi4\right) \cdot Q1 + alpha4 \cdot R1\right) + C \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M2 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z2 + \left(f4 - psi4\right) \cdot Q2 + alpha4 \cdot R2\right) + C \cdot \left(6.5536 \times 10^{-26} \cdot psi4 \cdot M3 + 1.28 \times 10^{-13} \cdot \left(0.5 \cdot f4 + 2.0 \cdot psi4\right) \cdot Z3 + \left(f4 - psi4\right) \cdot Q3 + alpha4 \cdot R3\right) + \left(-C\right) \cdot \left(2.62144 \times 10^{-25} \cdot psi5 \cdot M1 + 2.56 \times 10^{-13} \cdot \left(0.5 \cdot f5 + 2.0 \cdot psi5\right) \cdot Z1 + \left(f5 - psi5\right) \cdot Q1 + alpha5 \cdot R1\right) + C \cdot M5 \cdot M1 + C \cdot N5 \cdot Z1 + \left(-C\right) \cdot \left(2.62144 \times 10^{-25} \cdot psi5 \cdot M2 + 2.56 \times 10^{-13} \cdot \left(0.5 \cdot f5 + 2.0 \cdot psi5\right) \cdot Z2 + \left(f5 - psi5\right) \cdot Q2 + alpha5 \cdot R2\right) + C \cdot M5 \cdot M2 + C \cdot N5 \cdot Z2 + \left(-C\right) \cdot \left(2.62144 \times 10^{-25} \cdot psi5 \cdot M3 + 2.56 \times 10^{-13} \cdot \left(0.5 \cdot f5 + 2.0 \cdot psi5\right) \cdot Z3 + \left(f5 - psi5\right) \cdot Q3 + alpha5 \cdot R3\right) + C \cdot M5 \cdot M3 + C \cdot N5 \cdot Z3 + \left(-C\right) \cdot \left(2.62144 \times 10^{-25} \cdot psi5 \cdot M4 + 2.56 \times 10^{-13} \cdot \left(0.5 \cdot f5 + 2.0 \cdot psi5\right) \cdot Z4 + \left(f5 - psi5\right) \cdot Q4 + alpha5 \cdot R4\right) + C \cdot M5 \cdot M4 + C \cdot N5 \cdot Z4 $$Analyses
exponential_ic_evolution_golovin
Evolution of the droplet number and mass spectra under the golovin collection kernel, started from an exponential initial distribution. Number concentration decreases as droplets coalesce; mass shifts toward larger categories.
Time span: 0.0 → 20000000.0
Raw .esm
Raw .esm JSON (this component)
{
"reference": {
"doi": "10.1175/1520-0469(1989)046<3312:TEORSP>2.0.CO;2",
"citation": "Tzivion, S., Feingold, G., and Levin, Z.: The Evolution of Raindrop Spectra. Part II: Collisional Collection/Breakup and Evaporation in a Rainshaft, J. Atmos. Sci., 46(21), 3312-3327, 1989.",
"url": "https://doi.org/10.1175/1520-0469(1989)046%3C3312:TEORSP%3E2.0.CO;2",
"notes": "Source: Aerosol.jl src/stochastic_collection.jl @ c8c640bb636b22f2425460e9ed5cba50793bfee1 (migration bead esm-ar1e); the upstream @component StochasticCollectionCoalescence with kernel_type=golovin. The upstream model solves the two-moment stochastic collection equation (SCE) for cloud-droplet coalescence (Tzivion et al. 1989) over I mass-doubling categories, tracking number N_k and mass M_k per category. It is built with symbolic array operations (SymbolicUtils @arrayop/@makearray over array variables Nk(t)[1:I], Mk(t)[1:I]); the array ops are not in the .esm closed AST and the ESS Python runner does not evaluate them, so the model is scalarized to I=5 categories and 10 scalar differential states N1..N5, M1..M5. The scalarization evaluates the upstream @arrayop ifelse masks (i==1, i<=k-2, i<=k-1, i>=k+1) at concrete (k,i), so the flat equations are the source semantics, not a hand-derived rewrite. The five (six for Golovin) closure families are carried as algebraic states with explicit equations (rig convention; cf. components/aerosol/tomas.esm): xb_k (mean mass M_k/N_k), Z_k/Q_k/R_k (higher moments xi_p*M^2/N, xi_p^2*M^3/N^2, xi_p^3*M^4/N^3), and f_k/psi_k/alpha_k (linear within-category reconstruction coefficients, Eq. 13+15). The upstream closures guard m/n divisions with ifelse(n>0,...); the .esm replaces each by the algebraically equivalent floored form m*n/(n^2+1e-300) etc. — bit-identical to m/n for every n>0 that occurs (n>=1e-20 makes the 1e-300 floor ~260 orders below rounding), finite at n=0, and free of the ifelse-guarded divide-by-zero that the runner's eager Piecewise lowering would otherwise hit. Kernel: Golovin (sum) kernel K(x,y)=C*(x+y). Verification: the scalarized RHS was checked (1) by a line-by-line transcription audit against the upstream @arrayop bodies; (2) against an independent literal double-sum implementation of the same @arrayop semantics (agreement < 1e-9 relative over random states); and (3) against first-principles no-overflow analytic identities — with the top category empty, no collision product exceeds the grid, so collection conserves number per collision and conserves mass exactly: dN_tot/dt = -(K0/2)*N_tot^2 (constant kernel) and dN_tot/dt = -C*M_tot*N_tot (Golovin kernel), with dM_tot/dt = 0. The scalarized RHS reproduces all three identities to machine precision (relative error 0). Inline-test expected values are a SciPy LSODA reference integration (rtol=1e-10, abstol=1e-14) of this verified RHS; the gate re-integrates the .esm with SciPy LSODA, and the model-level rel=2e-3 tolerance (as for tomas.esm) covers integrator-detail differences. This migration covers only collisional collection; the breakup, evaporation and sedimentation processes of the full 1989 paper are not implemented upstream either."
},
"tolerance": {
"rel": 0.002
},
"variables": {
"N1": {
"type": "unknown",
"units": "m^-3",
"default": 1429670.6078643417,
"description": "Number concentration of cloud droplets in mass category 1 (of 5); lower mass boundary x_1 = 1.600000e-14 kg."
},
"N2": {
"type": "unknown",
"units": "m^-3",
"default": 2838836.0713738734,
"description": "Number concentration of cloud droplets in mass category 2 (of 5); lower mass boundary x_2 = 3.200000e-14 kg."
},
"N3": {
"type": "unknown",
"units": "m^-3",
"default": 5596563.981307812,
"description": "Number concentration of cloud droplets in mass category 3 (of 5); lower mass boundary x_3 = 6.400000e-14 kg."
},
"N4": {
"type": "unknown",
"units": "m^-3",
"default": 10875864.787608625,
"description": "Number concentration of cloud droplets in mass category 4 (of 5); lower mass boundary x_4 = 1.280000e-13 kg."
},
"N5": {
"type": "unknown",
"units": "m^-3",
"default": 20538015.50564386,
"description": "Number concentration of cloud droplets in mass category 5 (of 5); lower mass boundary x_5 = 2.560000e-13 kg."
},
"M1": {
"type": "unknown",
"units": "kg*m^-3",
"default": 3.430294470019302e-08,
"description": "Mass concentration of cloud droplets in mass category 1 (of 5)."
},
"M2": {
"type": "unknown",
"units": "kg*m^-3",
"default": 1.3619145733423377e-07,
"description": "Mass concentration of cloud droplets in mass category 2 (of 5)."
},
"M3": {
"type": "unknown",
"units": "kg*m^-3",
"default": 5.366970575748642e-07,
"description": "Mass concentration of cloud droplets in mass category 3 (of 5)."
},
"M4": {
"type": "unknown",
"units": "kg*m^-3",
"default": 2.083711394480048e-06,
"description": "Mass concentration of cloud droplets in mass category 4 (of 5)."
},
"M5": {
"type": "unknown",
"units": "kg*m^-3",
"default": 7.852951776977424e-06,
"description": "Mass concentration of cloud droplets in mass category 5 (of 5)."
},
"XI_P": {
"type": "parameter",
"units": "1",
"default": 1.0625,
"description": "Closure parameter xi_p (= 17/16) relating the higher-order moments Z_k, Q_k, R_k to N_k and M_k (Tzivion et al. 1989, Eq. 8a,b)."
},
"C": {
"type": "parameter",
"units": "m^3/(kg*s)",
"default": 0.0015,
"description": "Golovin (sum) collection-kernel coefficient: K(x,y) = C*(x + y)."
},
"xb1": {
"type": "unknown",
"units": "kg",
"description": "Mean droplet mass M_1/N_1 in category 1; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category."
},
"xb2": {
"type": "unknown",
"units": "kg",
"description": "Mean droplet mass M_2/N_2 in category 2; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category."
},
"xb3": {
"type": "unknown",
"units": "kg",
"description": "Mean droplet mass M_3/N_3 in category 3; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category."
},
"xb4": {
"type": "unknown",
"units": "kg",
"description": "Mean droplet mass M_4/N_4 in category 4; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category."
},
"xb5": {
"type": "unknown",
"units": "kg",
"description": "Mean droplet mass M_5/N_5 in category 5; algebraic. Denominator floored by 1e-300 so the closure stays finite for an empty category."
},
"Z1": {
"type": "unknown",
"units": "kg^2*m^-3",
"description": "Second-moment closure quantity xi_p*M_1^2/N_1 for category 1; algebraic."
},
"Z2": {
"type": "unknown",
"units": "kg^2*m^-3",
"description": "Second-moment closure quantity xi_p*M_2^2/N_2 for category 2; algebraic."
},
"Z3": {
"type": "unknown",
"units": "kg^2*m^-3",
"description": "Second-moment closure quantity xi_p*M_3^2/N_3 for category 3; algebraic."
},
"Z4": {
"type": "unknown",
"units": "kg^2*m^-3",
"description": "Second-moment closure quantity xi_p*M_4^2/N_4 for category 4; algebraic."
},
"Z5": {
"type": "unknown",
"units": "kg^2*m^-3",
"description": "Second-moment closure quantity xi_p*M_5^2/N_5 for category 5; algebraic."
},
"Q1": {
"type": "unknown",
"units": "kg^3*m^-3",
"description": "Third-moment closure quantity xi_p^2*M_1^3/N_1^2 for category 1; algebraic."
},
"Q2": {
"type": "unknown",
"units": "kg^3*m^-3",
"description": "Third-moment closure quantity xi_p^2*M_2^3/N_2^2 for category 2; algebraic."
},
"Q3": {
"type": "unknown",
"units": "kg^3*m^-3",
"description": "Third-moment closure quantity xi_p^2*M_3^3/N_3^2 for category 3; algebraic."
},
"Q4": {
"type": "unknown",
"units": "kg^3*m^-3",
"description": "Third-moment closure quantity xi_p^2*M_4^3/N_4^2 for category 4; algebraic."
},
"Q5": {
"type": "unknown",
"units": "kg^3*m^-3",
"description": "Third-moment closure quantity xi_p^2*M_5^3/N_5^2 for category 5; algebraic."
},
"R1": {
"type": "unknown",
"units": "kg^4*m^-3",
"description": "Fourth-moment closure quantity xi_p^3*M_1^4/N_1^3 for category 1; algebraic (Golovin kernel only)."
},
"R2": {
"type": "unknown",
"units": "kg^4*m^-3",
"description": "Fourth-moment closure quantity xi_p^3*M_2^4/N_2^3 for category 2; algebraic (Golovin kernel only)."
},
"R3": {
"type": "unknown",
"units": "kg^4*m^-3",
"description": "Fourth-moment closure quantity xi_p^3*M_3^4/N_3^3 for category 3; algebraic (Golovin kernel only)."
},
"R4": {
"type": "unknown",
"units": "kg^4*m^-3",
"description": "Fourth-moment closure quantity xi_p^3*M_4^4/N_4^3 for category 4; algebraic (Golovin kernel only)."
},
"R5": {
"type": "unknown",
"units": "kg^4*m^-3",
"description": "Fourth-moment closure quantity xi_p^3*M_5^4/N_5^3 for category 5; algebraic (Golovin kernel only)."
},
"f1": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient f_1 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic."
},
"f2": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient f_2 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic."
},
"f3": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient f_3 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic."
},
"f4": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient f_4 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic."
},
"f5": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient f_5 (Eq. 13a) with the positivity constraint Eq. 15 applied; algebraic."
},
"psi1": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient psi_1 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic."
},
"psi2": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient psi_2 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic."
},
"psi3": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient psi_3 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic."
},
"psi4": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient psi_4 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic."
},
"psi5": {
"type": "unknown",
"units": "kg^-1*m^-3",
"description": "Linear within-category number-distribution coefficient psi_5 (Eq. 13b) with the positivity constraint Eq. 15 applied; algebraic."
},
"alpha1": {
"type": "unknown",
"units": "kg^-2*m^-3",
"description": "Within-category distribution slope (f_1 - psi_1)/(2 x_1) for category 1; algebraic."
},
"alpha2": {
"type": "unknown",
"units": "kg^-2*m^-3",
"description": "Within-category distribution slope (f_2 - psi_2)/(2 x_2) for category 2; algebraic."
},
"alpha3": {
"type": "unknown",
"units": "kg^-2*m^-3",
"description": "Within-category distribution slope (f_3 - psi_3)/(2 x_3) for category 3; algebraic."
},
"alpha4": {
"type": "unknown",
"units": "kg^-2*m^-3",
"description": "Within-category distribution slope (f_4 - psi_4)/(2 x_4) for category 4; algebraic."
},
"alpha5": {
"type": "unknown",
"units": "kg^-2*m^-3",
"description": "Within-category distribution slope (f_5 - psi_5)/(2 x_5) for category 5; algebraic."
}
},
"equations": [
{
"lhs": {
"op": "D",
"args": [
"N1"
],
"wrt": "t"
},
"rhs": {
"op": "+",
"args": [
{
"op": "*",
"args": [
{
"op": "-",
"args": [
"C"
]
},
2.0,
"N1",
"M1"
]
},
{
"op": "*",
"args": [
{
"op": "-",
"args": [
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"tests": [
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"id": "exponential_ic_golovin",
"description": "Upstream Aerosol.jl stochastic_collection_test.jl scenario, Golovin K(x,y)=C(x+y) kernel, I=5 mass categories. Exponential initial droplet spectrum (N0=3e8 m^-3, LWC=1e-3 kg m^-3). The integration runs to 20000000 s: at the upstream default C=1.5e-3 the Golovin kernel has a ~6e7 s coalescence e-folding time for this droplet spectrum, so a shorter span would leave the kernel essentially unexercised. Expected values are a SciPy LSODA reference integration (rtol=1e-10, abstol=1e-14) of the scalarized RHS; that RHS is verified against the upstream @arrayop model in the model reference notes (transcription audit, double-sum cross-check, and machine-precision no-overflow analytic identities).",
"initial_conditions": {
"N1": 1429670.6078643417,
"M1": 3.430294470019302e-08,
"N2": 2838836.0713738734,
"M2": 1.3619145733423377e-07,
"N3": 5596563.981307812,
"M3": 5.366970575748642e-07,
"N4": 10875864.787608625,
"M4": 2.083711394480048e-06,
"N5": 20538015.50564386,
"M5": 7.852951776977424e-06
},
"time_span": {
"start": 0.0,
"end": 20000000.0
},
"assertions": [
{
"variable": "N1",
"time": 0.0,
"expected": 1429670.6078643417
},
{
"variable": "M5",
"time": 0.0,
"expected": 7.852951776977424e-06
},
{
"variable": "N1",
"time": 10000000.0,
"expected": 1226734.7184557277
},
{
"variable": "N3",
"time": 10000000.0,
"expected": 4635006.118271516
},
{
"variable": "N5",
"time": 10000000.0,
"expected": 15768555.72474853
},
{
"variable": "M3",
"time": 10000000.0,
"expected": 4.431295664124102e-07
},
{
"variable": "M5",
"time": 10000000.0,
"expected": 5.980459074233704e-06
},
{
"variable": "N1",
"time": 20000000.0,
"expected": 1085906.2038762474
},
{
"variable": "N3",
"time": 20000000.0,
"expected": 3987804.463289013
},
{
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"time": 20000000.0,
"expected": 12808387.83689668
},
{
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"time": 20000000.0,
"expected": 3.803162734898209e-07
},
{
"variable": "M5",
"time": 20000000.0,
"expected": 4.827494316359276e-06
}
]
}
],
"analyses": [
{
"id": "exponential_ic_evolution_golovin",
"description": "Evolution of the droplet number and mass spectra under the golovin collection kernel, started from an exponential initial distribution. Number concentration decreases as droplets coalesce; mass shifts toward larger categories.",
"initial_state": {
"N1": 1429670.6078643417,
"M1": 3.430294470019302e-08,
"N2": 2838836.0713738734,
"M2": 1.3619145733423377e-07,
"N3": 5596563.981307812,
"M3": 5.366970575748642e-07,
"N4": 10875864.787608625,
"M4": 2.083711394480048e-06,
"N5": 20538015.50564386,
"M5": 7.852951776977424e-06
},
"time_span": {
"start": 0.0,
"end": 20000000.0
},
"plots": [
{
"id": "number_concentration",
"type": "line",
"description": "Number concentration N_k(t) for each of the 5 mass categories.",
"x": {
"variable": "t",
"label": "time (s)"
},
"y": {
"variable": "N1",
"label": "number concentration (m^-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",
"type": "line",
"description": "Mass concentration M_k(t) for each of the 5 mass categories.",
"x": {
"variable": "t",
"label": "time (s)"
},
"y": {
"variable": "M1",
"label": "mass concentration (kg*m^-3)"
},
"series": [
{
"name": "M1",
"variable": "M1"
},
{
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"variable": "M2"
},
{
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{
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}
]
}