CriticalSingleScatteringAlbedo
Critical single-scattering albedo at which an aerosol layer transitions from cooling to heating — Seinfeld & Pandis (2006) Ch. 24 §24.2, Eq. 24.15.
Description
Critical single-scattering albedo at which an aerosol layer transitions from cooling to heating — Seinfeld & Pandis (2006) Ch. 24 §24.2, Eq. 24.15.
Reference
Seinfeld, J. H., and Pandis, S. N. (2006). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd Edition, Chapter 24, Section 24.2, Equation 24.15.
_source_ref: Aerosol.jl@c8c640bb:src/aerosol_radiative_forcing.jl:106 Critical single-scattering albedo from the Charlson et al. two-stream forcing balance, ω_crit = 2·R_s / (2·R_s + β·(1−R_s)²). When ω > ω_crit the aerosol layer cools (scattering dominates); when ω < ω_crit it heats (absorption dominates). Authored as a closed-form algebraic .esm so that, after MTK structural_simplify, the compiled system has zero ODE states and the inline-test runner exercises ω_crit at its observed algebraic value.
Tolerance rationale (model-level rel=1e-8 / abs=1e-12): the closed-form expression is a small rational function of two dimensionless parameters, so 1e-8 captures float64 roundoff in the multi-step evaluation. abs=1e-12 floors the dark-surface limit (R_s=0.01) where ω_crit ≈ 0.066._
Parameters
| Name | Units | Default | Description |
|---|---|---|---|
R_s | 1 | 0.15 | Surface albedo (dimensionless) |
β | 1 | 0.29 | Upscatter fraction (dimensionless) |
Observed
| Name | Units | Default | Description |
|---|---|---|---|
ω_crit | 1 | Critical single-scattering albedo: ω_crit = 2·R_s / (2·R_s + β·(1−R_s)²). |
Observed expressions
$$ ω_{crit} = \frac{2 \cdot R_{s}}{2 \cdot R_{s} + β \cdot \left(1 + \left(-1\right) \cdot R_{s}\right)^{2}} $$Critical single-scattering albedo: ω_crit = 2·R_s / (2·R_s + β·(1−R_s)²).
Analyses
figure_24_2_omega_crit_vs_Rs
Reproduces Figure 24.2 from Seinfeld & Pandis (2006) — critical single-scattering albedo as a function of surface albedo at fixed upscatter fraction β = 0.29 (Table 24.2 land value). The curve rises from ~0 at R_s → 0 toward 1 at R_s → 1, crossing the canonical land value (R_s ≈ 0.15) near ω_crit ≈ 0.59.
Time span: 0.0 → 1.0
Swept over: R_s
figure_24_2_curve_family
Supplements Figure 24.2 with a sweep over β at fixed R_s — the published figure shows curve families for several β values. Larger β (more backscatter) suppresses ω_crit at every R_s, raising the threshold for net cooling.
Time span: 0.0 → 1.0
Swept over: β
Rs_beta_joint_sweep
Joint sensitivity of ω_crit to surface albedo and upscatter fraction. Heatmap visualizes the Figure 24.2 curve family as ω_crit(R_s, β). Confirms monotonicity: ω_crit increases with R_s and decreases with β.
Time span: 0.0 → 1.0
Swept over: R_s, β
Raw .esm
Raw .esm JSON (this component)
{
"reference": {
"citation": "Seinfeld, J. H., and Pandis, S. N. (2006). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd Edition, Chapter 24, Section 24.2, Equation 24.15.",
"notes": "source_ref: Aerosol.jl@c8c640bb:src/aerosol_radiative_forcing.jl:106\nCritical single-scattering albedo from the Charlson et al. two-stream forcing balance, ω_crit = 2·R_s / (2·R_s + β·(1−R_s)²). When ω > ω_crit the aerosol layer cools (scattering dominates); when ω < ω_crit it heats (absorption dominates). Authored as a closed-form algebraic .esm so that, after MTK structural_simplify, the compiled system has zero ODE states and the inline-test runner exercises ω_crit at its observed algebraic value.\n\nTolerance rationale (model-level rel=1e-8 / abs=1e-12): the closed-form expression is a small rational function of two dimensionless parameters, so 1e-8 captures float64 roundoff in the multi-step evaluation. abs=1e-12 floors the dark-surface limit (R_s=0.01) where ω_crit ≈ 0.066."
},
"variables": {
"R_s": {
"type": "parameter",
"units": "1",
"default": 0.15,
"description": "Surface albedo (dimensionless)"
},
"β": {
"type": "parameter",
"units": "1",
"default": 0.29,
"description": "Upscatter fraction (dimensionless)"
},
"ω_crit": {
"type": "unknown",
"units": "1",
"description": "Critical single-scattering albedo: ω_crit = 2·R_s / (2·R_s + β·(1−R_s)²)."
}
},
"equations": [
{
"lhs": "ω_crit",
"rhs": {
"op": "/",
"args": [
{
"op": "*",
"args": [
2,
"R_s"
]
},
{
"op": "+",
"args": [
{
"op": "*",
"args": [
2,
"R_s"
]
},
{
"op": "*",
"args": [
"β",
{
"op": "^",
"args": [
{
"op": "+",
"args": [
1,
{
"op": "*",
"args": [
-1,
"R_s"
]
}
]
},
2
]
}
]
}
]
}
]
}
}
],
"tolerance": {
"rel": 1e-08,
"abs": 1e-12
},
"tests": [
{
"id": "table_24_2_default_Rs015_beta029",
"description": "Reproduces the upstream Aerosol.jl `CriticalSingleScatteringAlbedo - Figure 24.2 Validation` testitem default case. Table 24.2 land-surface values R_s=0.15, β=0.29 give ω_crit ≈ 0.589 (upstream asserts ≈ 0.586 with rtol=0.01; we pin the exact analytic value).",
"parameter_overrides": {
"R_s": 0.15,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.5887836710661891
},
{
"variable": "ω_crit",
"time": 1.0,
"expected": 0.5887836710661891
}
]
},
{
"id": "figure_24_2_dark_surface_limit",
"description": "Dark-surface limit R_s → 0 implies ω_crit → 0. Upstream asserts `ω_crit(R_s=0.01, β=0.29) < 0.1`; we pin the exact analytic value.",
"parameter_overrides": {
"R_s": 0.01,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.06573995247001437
}
]
},
{
"id": "figure_24_2_bright_surface_limit",
"description": "Bright-surface limit R_s → 1 implies ω_crit → 1. Upstream asserts `ω_crit(R_s=0.99, β=0.29) > 0.99`; we pin the exact analytic value.",
"parameter_overrides": {
"R_s": 0.99,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9999853537498693
}
]
},
{
"id": "figure_24_2_monotonic_Rs_p1",
"description": "Monotonic R_s sweep point (upstream uses R_s ∈ {0.1, 0.3, 0.5, 0.7, 0.9} at β=0.29 and asserts `issorted(ω_crit_vals)`). R_s = 0.1.",
"parameter_overrides": {
"R_s": 0.1,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.45987583352494826
}
]
},
{
"id": "figure_24_2_monotonic_Rs_p3",
"description": "Monotonic R_s sweep point. R_s = 0.3.",
"parameter_overrides": {
"R_s": 0.3,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.8085163724565423
}
]
},
{
"id": "figure_24_2_monotonic_Rs_p5",
"description": "Monotonic R_s sweep point. R_s = 0.5.",
"parameter_overrides": {
"R_s": 0.5,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9324009324009324
}
]
},
{
"id": "figure_24_2_monotonic_Rs_p7",
"description": "Monotonic R_s sweep point. R_s = 0.7.",
"parameter_overrides": {
"R_s": 0.7,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9816983381249562
}
]
},
{
"id": "figure_24_2_monotonic_Rs_p9",
"description": "Monotonic R_s sweep point. R_s = 0.9.",
"parameter_overrides": {
"R_s": 0.9,
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9983914803927006
}
]
},
{
"id": "beta_sensitivity_low",
"description": "Sensitivity to upscatter fraction at fixed R_s = 0.5 (upstream asserts `issorted(ω_crit_vs_beta, rev=true)` over β ∈ {0.1, 0.2, 0.3, 0.4}). β = 0.1 — smaller upscatter → higher ω_crit.",
"parameter_overrides": {
"R_s": 0.5,
"β": 0.1
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9756097560975611
}
]
},
{
"id": "beta_sensitivity_high",
"description": "Sensitivity to upscatter fraction at fixed R_s = 0.5. β = 0.4 — larger upscatter → lower ω_crit.",
"parameter_overrides": {
"R_s": 0.5,
"β": 0.4
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"assertions": [
{
"variable": "ω_crit",
"time": 0.0,
"expected": 0.9090909090909091
}
]
}
],
"analyses": [
{
"id": "figure_24_2_omega_crit_vs_Rs",
"description": "Reproduces Figure 24.2 from Seinfeld & Pandis (2006) — critical single-scattering albedo as a function of surface albedo at fixed upscatter fraction β = 0.29 (Table 24.2 land value). The curve rises from ~0 at R_s → 0 toward 1 at R_s → 1, crossing the canonical land value (R_s ≈ 0.15) near ω_crit ≈ 0.59.",
"parameters": {
"β": 0.29
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "R_s",
"range": {
"start": 0.01,
"stop": 0.99,
"count": 60,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "omega_crit_vs_Rs",
"type": "line",
"description": "Critical single-scattering albedo ω_crit as a function of surface albedo R_s at β = 0.29.",
"x": {
"variable": "R_s",
"label": "Surface albedo R_s"
},
"y": {
"variable": "ω_crit",
"label": "Critical single-scattering albedo ω_crit"
}
}
]
},
{
"id": "figure_24_2_curve_family",
"description": "Supplements Figure 24.2 with a sweep over β at fixed R_s — the published figure shows curve families for several β values. Larger β (more backscatter) suppresses ω_crit at every R_s, raising the threshold for net cooling.",
"parameters": {
"R_s": 0.5
},
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "β",
"range": {
"start": 0.05,
"stop": 0.5,
"count": 40,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "omega_crit_vs_beta",
"type": "line",
"description": "Critical single-scattering albedo ω_crit as a function of upscatter fraction β at R_s = 0.5.",
"x": {
"variable": "β",
"label": "Upscatter fraction β"
},
"y": {
"variable": "ω_crit",
"label": "Critical single-scattering albedo ω_crit"
}
}
]
},
{
"id": "Rs_beta_joint_sweep",
"description": "Joint sensitivity of ω_crit to surface albedo and upscatter fraction. Heatmap visualizes the Figure 24.2 curve family as ω_crit(R_s, β). Confirms monotonicity: ω_crit increases with R_s and decreases with β.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "R_s",
"range": {
"start": 0.01,
"stop": 0.99,
"count": 30,
"scale": "linear"
}
},
{
"parameter": "β",
"range": {
"start": 0.05,
"stop": 0.5,
"count": 20,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "omega_crit_heatmap",
"type": "heatmap",
"description": "ω_crit(R_s, β) heatmap.",
"x": {
"variable": "R_s",
"label": "Surface albedo R_s"
},
"y": {
"variable": "β",
"label": "Upscatter fraction β"
},
"value": {
"variable": "ω_crit",
"reduce": "final"
}
}
]
}
]
}