ClimateFeedback

{
  "reference": {
    "notes": "source_ref: GasChem.jl@8c12c048482b515fb8eb2110bf8ab4b4f4e71309:src/climate_forcing.jl:32\n\nUpstream `ClimateFeedback` is an MTK `@component` declaring two `@constants` (`ΔF_2xCO2` = 3.7 W/m² the CO₂-doubling radiative forcing per IPCC 2001 best estimate, and `ΔT0_2xCO2` = 1.25 K the no-feedback temperature change for CO₂ doubling per S&P p. 1037-1038), five `@parameters` (`ΔF` with no upstream default — the radiative forcing perturbation, `λ` = 0.8 K·m²/W the feedback-amplified climate sensitivity, `λ_CO2` = 0.8 K·m²/W the climate sensitivity for CO₂ doubling, `E_i` = 1.0 the efficacy of the forcing agent, `ΔT_r` = 0.7 K the realized warming to date), six time-indexed `@variables` (`ΔT_s`, `ΔT_0`, `λ_0`, `ΔF_e`, `ΔT_unrealized`, `ΔF_r` — all algebraic with no derivative), and six algebraic equations (`ΔT_s ~ λ·ΔF` Eq. 23.1, `λ_0 ~ ΔT0_2xCO2 / ΔF_2xCO2` Eq. 23.2 derivation, `ΔT_0 ~ λ_0·ΔF` Eq. 23.2, `ΔF_e ~ ΔF·E_i` Eq. 23.4 = Eq. 23.3 efficacy-weighted, `ΔF_r ~ ΔT_r / λ` inverse of Eq. 23.1 at realized warming, `ΔT_unrealized ~ (ΔF - ΔF_r)·λ` Eq. 23.7 warming commitment). Identical structure preserved here. Upstream structural test (`ClimateFeedback structural verification`, test/climate_forcing_test.jl:9-25) asserts `length(equations(sys)) == 6` and that the six state variables are in the unknowns list — matched here by the six entries in `equations` and the six state-typed variables.\n\nUpstream `λ_CO2` parameter preserved verbatim despite being declared-but-unused in the equation set. The upstream MTK system has `λ_CO2` in `@parameters` (with default 0.8) but no equation references it; this appears to be an upstream interface placeholder for a downstream consumer that might consume `λ_CO2` as a CO₂-doubling reference parameter through a future coupling.couple binding. The ESS Python runner and Julia MTK runner both accept declared-but-unused parameters as no-ops — they appear in `parameters(sys)` but never affect the solution. Preserving `λ_CO2` here means a downstream consumer can probe the upstream parameter signature without forking the migration. The unused-parameter convention is documented at the .esm schema level (ESS spec §6.2, parameter type description: 'Values set externally or held constant during integration' — silent on whether the parameter must be referenced by any equation, so unused-but-declared parameters are permitted).\n\nThe two upstream `@constants` (ΔF_2xCO2, ΔT0_2xCO2) are encoded here as `type: parameter` with the upstream default values, matching the rig-wide convention used by sibling .esm migrations (gwp.esm at components/gaschem/climate/gwp.esm encodes its τ_CO2 and a_CO2 `@constants` the same way; climate_sensitivity.esm at components/gaschem/radiation/climate_sensitivity.esm encodes its σ `@constants` the same way; chapman.esm at components/gaschem/stratospheric/chapman.esm encodes its four `@constants` k2_A/k4_A/C_k4/T_ref the same way). This makes the IPCC TAR no-feedback reference values overridable from a downstream consumer's `parameter_overrides` block — useful for sensitivity studies that probe alternative IPCC AR5 (ΔF_2xCO2 ≈ 3.71 W/m² with updated stratospheric adjustment) or AR6 (ΔF_2xCO2 ≈ 3.93 W/m² including rapid adjustments) reference values without forking the component.\n\nUpstream `ΔF` parameter has no default. The .esm migration defaults it to ΔF = 3.7 W/m² (= ΔF_2xCO2) so the default-parameter run gives the canonical 2×CO₂ scenario — at the upstream defaults (λ = 0.8, λ_CO2 = 0.8, E_i = 1.0, ΔT_r = 0.7, ΔF_2xCO2 = 3.7, ΔT0_2xCO2 = 1.25) and the migration's ΔF = 3.7 default, the model reproduces the canonical ΔT_s = 0.8·3.7 = 2.96 K equilibrium response (matching upstream `ClimateFeedback equation verification` test/climate_forcing_test.jl:27-42 at rtol=1e-6) and the canonical λ_0 = 1.25/3.7 = 0.3378... K·m²/W no-feedback Planck response (matching upstream `ClimateFeedback no-feedback response` test/climate_forcing_test.jl:44-62 at rtol=1e-6). This default choice differs from the upstream MTK model (which leaves ΔF unbound) but is the natural way to make the component runnable without overrides; downstream consumers will always override ΔF to the scenario-specific value (pre-industrial-to-present cumulative forcing ~1.7 W/m² per S&P p. 1045; the 4.6 W/m² alternative CO₂-doubling estimate from S&P p. 105; etc.).\n\nObservability classification: this component is observed-only — every output variable is given by a single algebraic equation in the inputs (no time derivative, the `(t)` time index has no role since there is no `D(·)` operator in any of the six equations). Test scenarios use `time_span: [0.0, 1.0]` and sample at t=0 to mirror the convention used by sibling observed-only ESM files in this rig (gwp.esm at components/gaschem/climate/gwp.esm, climate_sensitivity.esm at components/gaschem/radiation/climate_sensitivity.esm, wien.esm at components/gaschem/radiation/wien.esm, typical.esm at components/gaschem/combined/typical.esm) — algebraic identities have no time dependence and converge to the same fixed-point in one nonlinear-solver step regardless of the sampling time.\n\nPhysics structure: the six equations encode the linear-response climate-sensitivity framework with feedback amplification and warming commitment.\n  1. Eq. 23.1 `ΔT_s ~ λ·ΔF`: the feedback-amplified equilibrium temperature response to a forcing perturbation. λ is the feedback-amplified climate sensitivity that absorbs water-vapor, lapse-rate, albedo, and cloud feedbacks. The IPCC AR4 best estimate is λ ≈ 0.8 K·m²/W (= ΔT_2xCO2/ΔF_2xCO2 ≈ 3.0/3.7); the IPCC AR4 range 1.5-4.5 K for ΔT_2xCO2 corresponds to λ ∈ [0.4, 1.2] K·m²/W.\n  2. Eq. 23.2 derivation `λ_0 ~ ΔT0_2xCO2 / ΔF_2xCO2`: the no-feedback Planck response derived from the CO₂-doubling reference. At the IPCC TAR defaults ΔT0_2xCO2 = 1.25 K and ΔF_2xCO2 = 3.7 W/m², λ_0 = 1.25/3.7 ≈ 0.338 K·m²/W — close to the Stefan-Boltzmann analytic value 1/(4·σ·T_e³) ≈ 0.266 at T_e = 255 K (a ~27% difference reflecting the slightly different physical setup: the TAR's λ_0 is the surface-air response while the Stefan-Boltzmann λ_0 is the radiating-temperature response, with the difference absorbed into the surface-vs-radiating-temperature offset).\n  3. Eq. 23.2 `ΔT_0 ~ λ_0·ΔF`: the no-feedback temperature response — what ΔT_s would be in the absence of all feedback amplification. At the default ΔF = 3.7 W/m² and λ_0 = 1.25/3.7, ΔT_0 = 1.25 K exactly (by construction, since ΔF = ΔF_2xCO2 and ΔT_0 = ΔT0_2xCO2 by IEEE-754 division-then-multiplication identity). The feedback factor λ/λ_0 ≈ 0.8/0.338 ≈ 2.37 at the defaults is within the upstream `ClimateFeedback feedback factor` test's allowed range [1.2, 3.75] per S&P p. 1040.\n  4. Eq. 23.4 `ΔF_e ~ ΔF·E_i`: the efficacy-weighted effective forcing. E_i is the dimensionless efficacy that captures the different ΔT-per-unit-ΔF response of different forcing agents (Hansen 1984; Hansen et al. 2005). E_i = 1 by definition for CO₂; CH₄ has E_i ≈ 1.0-1.1; stratospheric water vapor has E_i ≈ 1.5; black carbon has E_i ≈ 0.5-0.7 (above-cloud bright BC) to 2-3 (below-cloud dark BC). The default E_i = 1.0 places the component in the CO₂-equivalent regime where ΔF_e = ΔF.\n  5. `ΔF_r ~ ΔT_r / λ`: the forcing equivalent of already-realized warming. Inverse of Eq. 23.1 evaluated at the realized warming ΔT_r. At the default ΔT_r = 0.7 K (rough estimate of pre-industrial-to-present warming circa 2000) and λ = 0.8, ΔF_r = 0.875 W/m² — the forcing the planet has 'paid down' in achieved warming so far.\n  6. Eq. 23.7 `ΔT_unrealized ~ (ΔF - ΔF_r)·λ`: the warming commitment. The remaining temperature change the planet will undergo if the current ΔF is held constant — represents the lag between current achieved (ΔT_r) and equilibrium (ΔT_s = λ·ΔF) warming. At the present-day S&P p. 1045 scenario (λ = 0.7, ΔF = 1.7, ΔT_r = 0.7), ΔT_unrealized = (1.7 - 1.0)·0.7 = 0.49 K — the textbook 'committed warming' figure.\n\nNumerical defaults (S&P p. 1037-1040 IPCC TAR baselines plus the migration's ΔF = 3.7 W/m² CO₂-doubling default):\n  ΔF_2xCO2    = 3.7  W/m²       (CO₂-doubling radiative forcing, IPCC TAR best estimate, S&P p. 1037)\n  ΔT0_2xCO2   = 1.25 K          (No-feedback temperature change for CO₂ doubling, S&P p. 1037-1038)\n  ΔF          = 3.7  W/m²       (Forcing perturbation, defaulted to ΔF_2xCO2 for the canonical 2×CO₂ scenario)\n  λ           = 0.8  K·m²/W     (Feedback-amplified climate sensitivity, IPCC AR4 central estimate)\n  λ_CO2       = 0.8  K·m²/W     (Climate sensitivity for CO₂ doubling — declared-but-unused upstream interface placeholder)\n  E_i         = 1.0             (Efficacy of forcing agent, CO₂-equivalent regime)\n  ΔT_r        = 0.7  K          (Realized warming to date, pre-industrial-to-present circa 2000)\nAt these defaults:\n  ΔT_s          = 0.8 · 3.7                                     = 2.96 K\n  λ_0           = 1.25 / 3.7                                    ≈ 0.33783783783783783 K·m²/W\n  ΔT_0          = (1.25/3.7) · 3.7                              = 1.25 K           (exact by IEEE-754)\n  ΔF_e          = 3.7 · 1.0                                     = 3.7 W/m²         (exact identity)\n  ΔF_r          = 0.7 / 0.8                                     ≈ 0.8749999999999999 W/m²\n  ΔT_unrealized = (3.7 - 0.8749999999999999) · 0.8              ≈ 2.2600000000000002 K\nThis matches the upstream `ClimateFeedback equation verification` test (test/climate_forcing_test.jl:27-42) which asserts ΔT_s ≈ 0.8·3.7 = 2.96 at rtol=1e-6, and the upstream `ClimateFeedback no-feedback response` test (test/climate_forcing_test.jl:44-62) which asserts λ_0 ≈ 1.25/3.7 and ΔT_0 ≈ λ_0·3.7 at rtol=1e-6.\n\nUpstream test reproduction:\n- The upstream `ClimateFeedback structural verification` test (test/climate_forcing_test.jl:9-25) asserts the system has 6 equations and that the six state variables (ΔT_s, ΔT_0, λ_0, ΔF_e, ΔT_unrealized, ΔF_r) are in the unknowns list — reproduced by the .esm structural shape (6 entries in `equations`, six state-typed variables).\n- The upstream `ClimateFeedback equation verification` test (test/climate_forcing_test.jl:27-42) asserts at λ=0.8, ΔF=3.7 that ΔT_s ≈ 0.8·3.7 = 2.96 at rtol=1e-6. Reproduced and tightened by `default_canonical_values` (pins ΔT_s = 2.9600000000000004 at rel=1e-10 along with the other five state variables — the rel=1e-10 tightening over the upstream rtol=1e-6 reflects the IEEE-754 single-multiplication precision at λ·ΔF ≈ 5 ULP ≈ 1.5e-15 relative, well below the 1e-10 tolerance bound).\n- The upstream `ClimateFeedback no-feedback response` test (test/climate_forcing_test.jl:44-62) asserts at ΔF=3.7 (defaults for the rest) that λ_0 ≈ 1.25/3.7 at rtol=1e-6 and ΔT_0 ≈ λ_0·3.7 at rtol=1e-6. Reproduced and tightened by `default_canonical_values` (covers the same parameters) plus `upstream_no_feedback_lambda_0_check` (pins λ_0 = 0.33783783783783783 at rel=1e-10 in isolation so the regression-detection signal is independent of other state variables) — the rel=1e-10 tightening over the upstream rtol=1e-6 reflects the IEEE-754 single-division precision at 1.25/3.7 ≈ 0.5 ULP ≈ 1.5e-16 relative.\n- The upstream `ClimateFeedback feedback factor` test (test/climate_forcing_test.jl:64-82) asserts at λ=0.8, ΔF=3.7 that the feedback factor λ/λ_0 lies in [1.2, 3.75] per S&P p. 1040. The .esm runner does not directly support cross-variable inequality assertions; reproduced by `default_canonical_values` (pins λ_0 = 0.33783783783783783) so the ratio 0.8/0.33783... = 2.3680000000000003 is exact by IEEE-754 division and lies in [1.2, 3.75] by construction. The bounds are not separately asserted because they reduce to checks on a single division of two exactly-pinned doubles.\n- The upstream `ClimateFeedback unrealized warming` test (test/climate_forcing_test.jl:84-114) asserts at λ=0.7, ΔF=1.7, ΔT_r=0.7 that ΔT_unrealized ≈ (ΔF - ΔT_r/λ)·λ = (1.7 - 1.0)·0.7 = 0.49 at rtol=1e-6, and qualitatively that ΔT_unrealized ≈ 0.5 K at rtol=0.05 per S&P p. 1045. Reproduced and tightened by `upstream_unrealized_warming_p1045_scenario` (pins ΔT_unrealized = 0.48999999999999994 at rel=1e-10 along with the intermediate ΔF_r = 1.0 — exact IEEE-754 0.7/0.7 division). The rel=1e-10 tightening over the upstream rtol=1e-6 reflects the IEEE-754 division-then-subtraction-then-multiplication precision at ~2 ULP ≈ 4.4e-16 relative.\n\nTolerance rationale (model-level rel=1e-10): observed-only algebraic system with a small number of arithmetic operations per equation. Each equation evaluates one of {multiplication, division, subtraction}; the worst-case error path is the Eq. 23.7 expression (ΔF - ΔT_r/λ)·λ which involves division-then-subtraction-then-multiplication for a total of ~3 IEEE-754 ULPs ≈ 1e-15 relative at any of the tested magnitudes. 1e-10 leaves 5 orders of magnitude of headroom over the IEEE-754 floor and matches the convention used by sibling algebraic .esm files in this rig (gwp.esm at rel=1e-10, climate_sensitivity.esm at rel=1e-10, wien.esm at rel=1e-10, typical.esm at rel=1e-10). The upstream Julia tests use rtol=1e-6 throughout — re-asserted here at the uniform model-level rel=1e-10 which is 4 orders of magnitude tighter.\n\nThe `zero_forcing_zero_response`, `equal_realized_warming_zero_commitment`, and `co2_2xCO2_E_i_one` tests use per-assertion `abs = 1.0e-12` overrides on exact-zero outputs because the `rel` tolerance is undefined when the expected value is zero — without the override, a stuck-at-1e-3 regression (e.g. an accidental constant offset added to ΔT_s) could silently pass at any nonzero `rel` bound. Pairs with the absolute-magnitude `default_canonical_values` test to cover both the zero-input and nonzero-input pathways.\n\nUnit balance (verified): all six equations balance dimensionally.\n  `ΔT_s ~ λ·ΔF`:                    (K·m²/W)·(W/m²)            = K           ✓ (matches ΔT_s unit u\"K\")\n  `λ_0 ~ ΔT0_2xCO2 / ΔF_2xCO2`:     K / (W/m²)                  = K·m²/W      ✓ (matches λ_0 unit u\"K*m^2/W\")\n  `ΔT_0 ~ λ_0·ΔF`:                  (K·m²/W)·(W/m²)            = K           ✓\n  `ΔF_e ~ ΔF·E_i`:                  (W/m²)·dimensionless        = W/m²         ✓\n  `ΔF_r ~ ΔT_r / λ`:                K / (K·m²/W)                = W/m²         ✓\n  `ΔT_unrealized ~ (ΔF - ΔF_r)·λ`:  (W/m² - W/m²)·(K·m²/W)     = K           ✓\nThe migrated `.esm` preserves the upstream units verbatim — all forcings (ΔF, ΔF_e, ΔF_r, ΔF_2xCO2) in W/m², all temperatures (ΔT_s, ΔT_0, ΔT_unrealized, ΔT_r, ΔT0_2xCO2) in K, all sensitivities (λ, λ_CO2, λ_0) in K·m²/W, E_i dimensionless. SI throughout; no unit conversion at parse or runtime."
  },
  "tolerance": {
    "rel": 1e-10
  },
  "variables": {
    "ΔF_2xCO2": {
      "type": "parameter",
      "units": "W/m^2",
      "default": 3.7,
      "description": "Radiative forcing for CO₂ doubling (IPCC 2001 best estimate, S&P p. 1037). The upstream default 3.7 W/m² is the canonical IPCC TAR value derived from line-by-line radiative transfer calculations with stratospheric temperature adjustment. Modern IPCC AR5 reports a slightly higher 3.71 W/m² (Myhre et al. 2013, Ch. 8) and IPCC AR6 reports 3.93 W/m² (Smith et al. 2020, including rapid adjustments) — the upstream choice is the legacy TAR value preserved for continuity with S&P Ch. 23's worked examples. Overridable to support sensitivity studies that probe the doubling-forcing assumption — a downstream consumer can bind ΔF_2xCO2 to AR5 or AR6 values to recompute λ_0 against the modern reference."
    },
    "ΔT0_2xCO2": {
      "type": "parameter",
      "units": "K",
      "default": 1.25,
      "description": "No-feedback temperature change for CO₂ doubling (S&P p. 1037-1038, taken as the midpoint of the 1.2-1.3 K range quoted in the text). The upstream default 1.25 K is the no-feedback Planck response to a CO₂ doubling — what the planet's surface would warm by if all feedbacks (water vapor, lapse rate, albedo, cloud) were inactive. The ratio ΔT0_2xCO2/ΔF_2xCO2 = 1.25/3.7 = 0.338 K·m²/W is the no-feedback climate sensitivity factor λ_0 — close to the Stefan-Boltzmann analytic value 1/(4·σ·T_e³) ≈ 0.266 at T_e = 255 K (the difference reflects the surface-vs-radiating-temperature offset; the sibling ClimateSensitivity.esm at components/gaschem/radiation/climate_sensitivity.esm uses the analytic Stefan-Boltzmann form for the no-feedback Planck response in the radiating-temperature framework). Overridable from a downstream consumer's `parameter_overrides` block for cross-framework comparisons."
    },
    "ΔF": {
      "type": "parameter",
      "units": "W/m^2",
      "default": 3.7,
      "description": "Radiative forcing perturbation (S&P §23.4 Eq. 23.1). The upstream MTK component leaves ΔF unbound (no default); the .esm migration defaults it to ΔF = 3.7 W/m² = ΔF_2xCO2 so the default-parameter run gives the canonical 2×CO₂ scenario (ΔT_s = 2.96 K with λ = 0.8 K·m²/W per the upstream `ClimateFeedback equation verification` test). Canonical scenario-specific values: ΔF ≈ 1.7 W/m² (pre-industrial-to-present cumulative anthropogenic forcing circa 2000 per S&P p. 1045), ΔF = 4.6 W/m² (alternative CO₂-doubling estimate from earlier studies per S&P p. 105), ΔF = -4 W/m² (large volcanic aerosol cooling such as Pinatubo 1991), ΔF = 8.4 W/m² (RCP8.5 end-of-century total anthropogenic forcing)."
    },
    "λ": {
      "type": "parameter",
      "units": "K*m^2/W",
      "default": 0.8,
      "description": "Climate sensitivity parameter with feedbacks (S&P Eq. 23.1). The feedback-amplified ΔT-per-unit-ΔF response that absorbs water vapor, lapse rate, albedo, and cloud feedbacks. The upstream default 0.8 K·m²/W is the IPCC AR4 central estimate (= ΔT_2xCO2/ΔF_2xCO2 ≈ 3.0/3.7). The IPCC AR4 likely range 1.5-4.5 K for ΔT_2xCO2 corresponds to λ ∈ [0.4, 1.2] K·m²/W; IPCC AR6 narrowed this to ΔT_2xCO2 ∈ [2.5, 4.0] K corresponding to λ ∈ [0.67, 1.08] K·m²/W. The feedback factor λ/λ_0 = 0.8/(1.25/3.7) ≈ 2.37 at the defaults — within the upstream `ClimateFeedback feedback factor` test's allowed range [1.2, 3.75] per S&P p. 1040."
    },
    "λ_CO2": {
      "type": "parameter",
      "units": "K*m^2/W",
      "default": 0.8,
      "description": "Climate sensitivity for CO₂ doubling (declared-but-unused upstream interface placeholder, see source_ref). The upstream MTK system declares λ_CO2 in `@parameters` with default 0.8 K·m²/W (matching λ) but no equation references it — it appears in `parameters(sys)` but never affects the solution. The .esm migration preserves the upstream parameter signature verbatim so downstream consumers can probe the parameter list without forking the migration. Conceptually λ_CO2 would represent the CO₂-doubling-specific feedback-amplified sensitivity in a multi-agent extension where each forcing agent might have its own λ_i; in the current single-agent equation set this is the same as λ, so the upstream default 0.8 is the natural choice."
    },
    "E_i": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Efficacy of forcing agent (S&P Eq. 23.3, dimensionless). E_i = λ_i / λ_CO2 where λ_i is the climate sensitivity for forcing agent i and λ_CO2 is the CO₂-doubling reference. E_i = 1 by definition for CO₂; CH₄ has E_i ≈ 1.0-1.1 (Hansen et al. 2005 Table 7); stratospheric water vapor from CH₄ oxidation has E_i ≈ 1.5; black carbon has E_i ≈ 0.5-0.7 for above-cloud bright BC (reduces brightness, weak ΔT response) to E_i ≈ 2-3 for below-cloud dark BC (snow albedo amplification). The default E_i = 1.0 places the component in the CO₂-equivalent regime where ΔF_e = ΔF. Used in Eq. 23.4 ΔF_e ~ ΔF · E_i to compute the effective forcing — the CO₂-equivalent forcing that produces the same ΔT response as the bare ΔF of agent i."
    },
    "ΔT_r": {
      "type": "parameter",
      "units": "K",
      "default": 0.7,
      "description": "Realized warming to date (S&P p. 1045 in the warming-commitment context). The temperature change the planet has already undergone in response to the cumulative forcing. The upstream default 0.7 K is the rough estimate of pre-industrial-to-present global mean surface warming circa 2000 (S&P p. 1045 quotes ΔT_r ≈ 0.7°C); modern (2020) estimates are higher at ~1.1 K per IPCC AR6 SPM. ΔT_r enters Eq. 23.7 ΔT_unrealized = (ΔF - ΔF_r)·λ through ΔF_r = ΔT_r/λ — the forcing equivalent of the already-realized warming, which when subtracted from the total ΔF gives the remaining 'committed' forcing that has not yet been expressed as warming."
    },
    "ΔT_s": {
      "type": "unknown",
      "units": "K",
      "description": "Equilibrium surface temperature change with feedbacks (S&P Eq. 23.1). ΔT_s = λ·ΔF is the linear-response equilibrium temperature change to a forcing perturbation, with feedback amplification absorbed into λ. At Earth's 2×CO₂ defaults (ΔF = 3.7 W/m², λ = 0.8), ΔT_s = 2.96 K — within the IPCC AR4 likely range [1.5, 4.5] K and matches the central estimate ΔT_2xCO2 ≈ 3.0 K (the 0.04 K shortfall reflects the round-trip 3.0/3.7·3.7 vs 3.0 which differs by ~0.01 in the third digit due to the rounded λ = 0.8 default). 'Equilibrium' here means the temperature the planet would reach if the forcing were held fixed indefinitely — distinct from the transient temperature ΔT_r the planet has actually achieved at any finite time."
    },
    "ΔT_0": {
      "type": "unknown",
      "units": "K",
      "description": "Surface temperature change without feedbacks (S&P Eq. 23.2). ΔT_0 = λ_0·ΔF is the no-feedback Planck-response temperature change — what ΔT_s would be if all feedbacks (water vapor, lapse rate, albedo, cloud) were inactive. At Earth's 2×CO₂ defaults, ΔT_0 = (1.25/3.7)·3.7 = 1.25 K (exactly, by IEEE-754 division-then-multiplication identity) — recovers ΔT0_2xCO2 by construction. The ratio ΔT_s/ΔT_0 = λ/λ_0 = 2.37 is the feedback amplification factor, capturing the ~2.4× boost feedbacks add to the Planck baseline."
    },
    "λ_0": {
      "type": "unknown",
      "units": "K*m^2/W",
      "description": "No-feedback climate sensitivity (S&P Eq. 23.2 derivation). λ_0 = ΔT0_2xCO2 / ΔF_2xCO2 is the no-feedback Planck response factor derived from the CO₂-doubling reference values. At the IPCC TAR defaults (ΔT0_2xCO2 = 1.25 K, ΔF_2xCO2 = 3.7 W/m²), λ_0 = 1.25/3.7 ≈ 0.338 K·m²/W — independent of ΔF (it depends only on the two reference constants). Comparable to the Stefan-Boltzmann analytic form λ_0_SB = 1/(4·σ·T_e³) ≈ 0.266 K·m²/W at T_e = 255 K used in the sibling ClimateSensitivity.esm — the difference reflects the surface-vs-radiating-temperature framing. λ_0 is the baseline against which the feedback-amplified λ is reported: typical IPCC feedback factors λ/λ_0 ∈ [1.2, 3.75] per S&P p. 1040."
    },
    "ΔF_e": {
      "type": "unknown",
      "units": "W/m^2",
      "description": "Effective radiative forcing incorporating efficacy (S&P Eq. 23.4, derived from Eq. 23.3). ΔF_e = ΔF·E_i is the CO₂-equivalent forcing — the bare forcing of agent i scaled by its efficacy E_i so that the same ΔT-per-unit-ΔF response (governed by the CO₂-doubling reference λ_CO2) can be applied. At the default E_i = 1.0, ΔF_e = ΔF identically (the CO₂-equivalent regime). For a non-CO₂ agent with E_i = 1.5 (stratospheric water vapor) and ΔF = 1.0, ΔF_e = 1.5 W/m² — equivalent to 1.5 W/m² of CO₂ forcing in terms of temperature response. The migration's Eq. 23.4 implementation differs subtly from the upstream §23.3 algebra: Hansen (1984) defines E_i = λ_i/λ_CO2, so ΔF_i_effective = ΔF_i·E_i has units W/m² but conceptually represents a 'CO₂-equivalent' forcing in a coupling to a downstream ΔT_s = λ_CO2·ΔF_total system."
    },
    "ΔT_unrealized": {
      "type": "unknown",
      "units": "K",
      "description": "Unrealized (committed) warming (S&P Eq. 23.7, p. 1045). ΔT_unrealized = (ΔF - ΔF_r)·λ is the remaining temperature change the planet will undergo if the current ΔF is held constant — the lag between current achieved (ΔT_r) and equilibrium (ΔT_s = λ·ΔF) warming. ΔT_unrealized = ΔT_s - ΔT_r by construction (substituting ΔF_r = ΔT_r/λ: (ΔF - ΔT_r/λ)·λ = λ·ΔF - ΔT_r = ΔT_s - ΔT_r). At the present-day S&P p. 1045 scenario (λ = 0.7, ΔF = 1.7, ΔT_r = 0.7), ΔT_unrealized = (1.7 - 1.0)·0.7 = 0.49 K — the textbook 'committed warming' figure. Positive ΔT_unrealized means the planet must still warm further; zero means the planet has already reached equilibrium (no warming commitment); negative means the planet has overshot equilibrium and must cool back (a transient artifact of the linear model — physically this only happens if ΔT_r is supplied externally as larger than the equilibrium ΔT_s)."
    },
    "ΔF_r": {
      "type": "unknown",
      "units": "W/m^2",
      "description": "Forcing corresponding to realized warming (inverse of S&P Eq. 23.1 evaluated at ΔT_r). ΔF_r = ΔT_r/λ is the forcing the planet has 'paid down' in achieved warming so far — the forcing that would have produced the realized ΔT_r at equilibrium. At the default ΔT_r = 0.7 K and λ = 0.8, ΔF_r = 0.875 W/m². The difference ΔF - ΔF_r is the remaining 'committed' forcing not yet expressed as warming; multiplying by λ gives the warming commitment ΔT_unrealized (Eq. 23.7). For the S&P p. 1045 scenario (λ = 0.7, ΔT_r = 0.7), ΔF_r = 1.0 W/m² exactly (IEEE-754 0.7/0.7 = 1.0)."
    }
  },
  "equations": [
    {
      "lhs": "ΔT_s",
      "rhs": {
        "op": "*",
        "args": [
          "λ",
          "ΔF"
        ]
      }
    },
    {
      "lhs": "λ_0",
      "rhs": {
        "op": "/",
        "args": [
          "ΔT0_2xCO2",
          "ΔF_2xCO2"
        ]
      }
    },
    {
      "lhs": "ΔT_0",
      "rhs": {
        "op": "*",
        "args": [
          "λ_0",
          "ΔF"
        ]
      }
    },
    {
      "lhs": "ΔF_e",
      "rhs": {
        "op": "*",
        "args": [
          "ΔF",
          "E_i"
        ]
      }
    },
    {
      "lhs": "ΔF_r",
      "rhs": {
        "op": "/",
        "args": [
          "ΔT_r",
          "λ"
        ]
      }
    },
    {
      "lhs": "ΔT_unrealized",
      "rhs": {
        "op": "*",
        "args": [
          {
            "op": "-",
            "args": [
              "ΔF",
              "ΔF_r"
            ]
          },
          "λ"
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_canonical_values",
      "description": "At defaults (ΔF = 3.7 W/m², λ = 0.8 K·m²/W, λ_CO2 = 0.8, E_i = 1.0, ΔT_r = 0.7 K, ΔF_2xCO2 = 3.7, ΔT0_2xCO2 = 1.25), every state variable evaluates to its analytic value. Reproduces and tightens the upstream `ClimateFeedback equation verification` test (GasChem.jl test/climate_forcing_test.jl:27-42): upstream asserts ΔT_s ≈ 0.8·3.7 at rtol=1e-6, this one pins ΔT_s = 2.9600000000000004 at rel=1e-10 along with the other five state variables. Also reproduces the upstream `ClimateFeedback no-feedback response` test (test/climate_forcing_test.jl:44-62) which asserts at the same defaults that λ_0 ≈ 1.25/3.7 at rtol=1e-6 and ΔT_0 ≈ λ_0·3.7 at rtol=1e-6 — covered here at rel=1e-10. The full state-variable pin anchors all six equations simultaneously: ΔT_s tests Eq. 23.1 (multiplication), λ_0 tests Eq. 23.2 derivation (division of two constants), ΔT_0 tests Eq. 23.2 (multiplication of derived sensitivity by forcing), ΔF_e tests Eq. 23.4 (efficacy passthrough at E_i=1), ΔF_r tests inverse-Eq.-23.1 (division), ΔT_unrealized tests Eq. 23.7 (subtraction-then-multiplication).",
      "parameter_overrides": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.33783783783783783,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 1.25,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": 3.7,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 0.8749999999999999,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 2.2600000000000002,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "upstream_no_feedback_lambda_0_check",
      "description": "Isolated check on λ_0 = ΔT0_2xCO2 / ΔF_2xCO2 = 1.25/3.7 = 0.33783... K·m²/W at the upstream defaults. Reproduces the upstream `ClimateFeedback no-feedback response` test (GasChem.jl test/climate_forcing_test.jl:44-62) in isolation so a regression in the Eq. 23.2 derivation surfaces independently of the other state variables. The upstream test asserts λ_0 ≈ 1.25/3.7 at rtol=1e-6; this one pins λ_0 = 0.33783783783783783 at rel=1e-10. Independence from ΔF: λ_0 depends only on the two reference constants ΔT0_2xCO2 and ΔF_2xCO2, not on the forcing perturbation ΔF — so changing ΔF (e.g. to 1.7 W/m² for the pre-industrial-to-present scenario) does NOT change λ_0. This is the only state variable in the system that is independent of ΔF.",
      "parameter_overrides": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.33783783783783783,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "upstream_unrealized_warming_p1045_scenario",
      "description": "Present-day warming-commitment scenario from S&P p. 1045: λ = 0.7 K·m²/W (slightly lower than the IPCC AR4 central, corresponding to ΔT_2xCO2 ≈ 2.6 K), ΔF = 1.7 W/m² (pre-industrial-to-present cumulative anthropogenic forcing circa 2000), ΔT_r = 0.7 K (realized warming circa 2000). Result: ΔT_unrealized = (1.7 - 1.0)·0.7 = 0.49 K — the textbook 'committed warming' figure. Reproduces and tightens the upstream `ClimateFeedback unrealized warming` test (GasChem.jl test/climate_forcing_test.jl:84-114): upstream asserts ΔT_unrealized ≈ (ΔF - ΔT_r/λ)·λ at rtol=1e-6 and qualitatively ΔT_unrealized ≈ 0.5 K at rtol=0.05, this one pins ΔT_unrealized = 0.48999999999999994 at rel=1e-10 along with the intermediate ΔF_r = 1.0 (exact IEEE-754 0.7/0.7 division) and ΔT_s = 1.19 K (λ·ΔF). Cross-check: ΔT_s - ΔT_r = 1.19 - 0.7 = 0.49 K — matches ΔT_unrealized by Eq. 23.7 identity (ΔT_unrealized = (ΔF - ΔT_r/λ)·λ = λ·ΔF - ΔT_r = ΔT_s - ΔT_r).",
      "parameter_overrides": {
        "λ": 0.7,
        "ΔF": 1.7,
        "ΔT_r": 0.7
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 1.19,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 1.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 0.48999999999999994,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "zero_forcing_zero_response",
      "description": "Zero forcing perturbation (ΔF = 0) at default λ = 0.8 and ΔT_r = 0.7. Results: ΔT_s = 0 (no equilibrium temperature change), ΔT_0 = 0 (no Planck response either), ΔF_e = 0 (zero efficacy-weighted forcing), but ΔF_r = 0.875 W/m² and ΔT_unrealized = (0 - 0.875)·0.8 = -0.7 K. The negative ΔT_unrealized at ΔF = 0 captures a transient pathology of the linear model: if the planet has already warmed by ΔT_r = 0.7 K but there is no longer any forcing to sustain it, the planet must cool back by ΔT_r to reach the zero-forcing equilibrium — ΔT_unrealized = ΔT_s - ΔT_r = 0 - 0.7 = -0.7. λ_0 = 1.25/3.7 = 0.3378 K·m²/W is independent of ΔF and stays at its default value. Uses per-assertion `abs = 1.0e-12` overrides on the exact-zero ΔT_s/ΔT_0/ΔF_e outputs because the `rel` tolerance is undefined when the expected value is zero — without the override, a stuck-at-1e-3 regression could silently pass at any nonzero `rel` bound.",
      "parameter_overrides": {
        "ΔF": 0.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.33783783783783783,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 0.8749999999999999,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": -0.7,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "negative_forcing_cooling",
      "description": "Negative forcing perturbation (ΔF = -4.0 W/m², e.g. large volcanic aerosol cooling such as Pinatubo 1991 → ~-3 W/m² peak) at default λ = 0.8 and ΔT_r = 0.7. Results: ΔT_s = 0.8·(-4.0) = -3.2 K (equilibrium cooling), ΔT_0 = 0.3378·(-4.0) = -1.351 K (no-feedback cooling), ΔF_e = -4.0 W/m² (E_i=1 passthrough), ΔF_r = 0.875 W/m² (unchanged from default), ΔT_unrealized = (-4.0 - 0.875)·0.8 = -3.9 K. Exercises the sign-correct propagation through Eqs. 23.1, 23.2, 23.4, and 23.7: a negative forcing produces a negative equilibrium ΔT_s and a negative warming commitment ΔT_unrealized — the planet must cool below the present-day ΔT_r to reach the new (cooled) equilibrium. This is the geoengineering / stratospheric aerosol injection scenario where -ΔF_S ≈ -ΔF_2xCO2 is meant to undo the historical greenhouse warming.",
      "parameter_overrides": {
        "ΔF": -4.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": -3.2,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": -1.3513513513513513,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": -4.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 0.8749999999999999,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": -3.9000000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "strong_forcing_8wm2",
      "description": "Strong forcing perturbation (ΔF = 8.0 W/m², roughly CO₂-quadrupling forcing or RCP8.5-end-of-century cumulative anthropogenic forcing) at default λ = 0.8 and ΔT_r = 0.7. Results: ΔT_s = 0.8·8.0 = 6.4 K (equilibrium warming), ΔT_0 = 0.3378·8.0 = 2.703 K, ΔF_e = 8.0 W/m², ΔF_r = 0.875 W/m², ΔT_unrealized = (8.0 - 0.875)·0.8 = 5.7 K (huge commitment — the planet has only realized 0.7/6.4 = 11% of its equilibrium response). Exercises Eqs. 23.1 and 23.7 at the upper end of the plausible forcing range, where the linear-response approximation may break down (water-vapor and ice-albedo feedbacks become nonlinear at large ΔT) but the linear model still gives the first-order estimate.",
      "parameter_overrides": {
        "ΔF": 8.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 6.4,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 2.7027027027027026,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": 8.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 5.7,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "efficacy_E_i_halved",
      "description": "Forcing-agent efficacy E_i = 0.5 (below-cloud bright black carbon or stratospheric sulfate aerosol — agents whose ΔT response per unit ΔF is half that of CO₂) at default ΔF = 3.7 W/m². Result: ΔF_e = 3.7·0.5 = 1.85 W/m² (the CO₂-equivalent forcing — what 1.85 W/m² of CO₂ would produce the same ΔT as 3.7 W/m² of this less-efficacious agent). The other five state variables are unchanged from defaults because E_i appears only in the Eq. 23.4 ΔF_e equation — Eqs. 23.1, 23.2, 23.7 use ΔF directly, not ΔF_e. Anchors the orthogonality between the efficacy channel (Eq. 23.4) and the temperature-response channel (Eqs. 23.1, 23.2, 23.7) — a regression that accidentally feeds ΔF_e back into Eq. 23.1 (giving ΔT_s = λ·ΔF·E_i instead of ΔT_s = λ·ΔF) would surface here as ΔT_s = 1.48 K instead of the correct 2.96 K.",
      "parameter_overrides": {
        "E_i": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": 1.85,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 1.25,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 2.2600000000000002,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "efficacy_E_i_doubled",
      "description": "Forcing-agent efficacy E_i = 2.0 (above-cloud dark black carbon or stratospheric water vapor from CH₄ oxidation — agents whose ΔT response per unit ΔF is twice that of CO₂) at default ΔF = 3.7 W/m². Result: ΔF_e = 3.7·2.0 = 7.4 W/m² (the CO₂-equivalent forcing). Same orthogonality check as `efficacy_E_i_halved` but on the other side of the E_i = 1 axis. Together with the halved-E_i test verifies that E_i acts linearly on ΔF_e at any sign of E_i and that no other state variable is contaminated.",
      "parameter_overrides": {
        "E_i": 2.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔF_e",
          "time": 0.0,
          "expected": 7.4,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "high_sensitivity_lambda_1p2",
      "description": "High-sensitivity scenario λ = 1.2 K·m²/W (= ΔT_2xCO2/ΔF_2xCO2 ≈ 4.44/3.7, top of the IPCC AR4 likely range [0.4, 1.2]) at default ΔF = 3.7. Results: ΔT_s = 1.2·3.7 = 4.44 K (top-of-range equilibrium warming), ΔF_r = 0.7/1.2 = 0.5833 W/m² (the planet has 'paid down' less forcing-equivalent at higher λ because each K of ΔT_r requires less forcing), ΔT_unrealized = (3.7 - 0.5833)·1.2 = 3.74 K (large commitment because ΔT_r is a smaller fraction of the equilibrium ΔT_s). λ_0 and ΔT_0 remain at their reference values (independent of λ). Feedback factor λ/λ_0 = 1.2/0.3378 = 3.55 — within the upstream `ClimateFeedback feedback factor` test's allowed range [1.2, 3.75] (and near the upper end, reflecting a 'high-feedback' planet with strong water-vapor + cloud amplification).",
      "parameter_overrides": {
        "λ": 1.2
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 4.44,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 1.25,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.33783783783783783,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 0.5833333333333334,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 3.7399999999999998,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "low_sensitivity_lambda_0p5",
      "description": "Low-sensitivity scenario λ = 0.5 K·m²/W (= ΔT_2xCO2/ΔF_2xCO2 ≈ 1.85/3.7, near the bottom of the IPCC AR4 likely range [0.4, 1.2]) at default ΔF = 3.7. Results: ΔT_s = 0.5·3.7 = 1.85 K (bottom-of-range equilibrium warming), ΔF_r = 0.7/0.5 = 1.4 W/m² (the planet has 'paid down' more forcing-equivalent at lower λ), ΔT_unrealized = (3.7 - 1.4)·0.5 = 1.15 K. Feedback factor λ/λ_0 = 0.5/0.3378 = 1.48 — within the upstream `ClimateFeedback feedback factor` test's allowed range [1.2, 3.75] (and near the lower end, reflecting a 'low-feedback' planet with weak water-vapor amplification and possibly negative cloud feedback).",
      "parameter_overrides": {
        "λ": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 1.85,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 1.4,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 1.1500000000000001,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "zero_realized_warming_full_commitment",
      "description": "Pre-industrial baseline scenario: ΔT_r = 0 K (no warming has been realized yet) at default ΔF = 3.7 and λ = 0.8. Results: ΔF_r = 0/0.8 = 0 W/m², ΔT_unrealized = (3.7 - 0)·0.8 = 2.96 K (= ΔT_s — the full equilibrium response remains 'committed' because nothing has been realized yet). Anchors the identity ΔT_unrealized = ΔT_s - ΔT_r at the ΔT_r = 0 limit. Uses per-assertion `abs = 1.0e-12` override on ΔF_r because the expected value is exactly zero.",
      "parameter_overrides": {
        "ΔT_r": 0.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "equal_realized_warming_zero_commitment",
      "description": "Equilibrium-reached scenario: ΔT_r = ΔT_s = λ·ΔF = 0.8·3.7 = 2.96 K (the planet has already warmed to its equilibrium response at this forcing). Results: ΔF_r = 2.96/0.8 = 3.7 W/m² (= ΔF), ΔT_unrealized = (3.7 - 3.7)·0.8 = 0 K (no commitment remains — the planet is at equilibrium). Anchors the identity ΔT_unrealized = ΔT_s - ΔT_r at the ΔT_r = ΔT_s limit. The expected ΔT_unrealized = 0 is exact only up to IEEE-754 ULP rounding because ΔT_r = 2.96 is not exact in double precision (ΔT_s = 2.9600000000000004 in IEEE-754); the computed ΔT_unrealized = 3.55e-16 reflects this single-ULP roundoff. Uses per-assertion `abs = 1.0e-12` override on ΔT_unrealized because the expected value is exactly zero up to roundoff.",
      "parameter_overrides": {
        "ΔT_r": 2.96
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "ΔF_r",
          "time": 0.0,
          "expected": 3.6999999999999997,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_unrealized",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        }
      ]
    },
    {
      "id": "DT0_2xCO2_constant_override",
      "description": "Override the upstream `@constants` ΔT0_2xCO2 to 1.2 K (S&P p. 1037-1038 quotes 1.2-1.3 K range; this exercises the lower bound vs the default midpoint 1.25 K) at the canonical 2×CO₂ scenario. Result: λ_0 = 1.2/3.7 = 0.3243 K·m²/W (vs the default 0.3378), ΔT_0 = 0.3243·3.7 = 1.2 K (exact by IEEE-754 division-then-multiplication identity). ΔT_s, ΔF_e, ΔF_r, ΔT_unrealized are unchanged (independent of ΔT0_2xCO2). Anchors the design choice that ΔT0_2xCO2 behaves as a parameter (not a hardcoded constant) at the .esm boundary: a downstream consumer can override the rounded midpoint to the lower-bound (1.2 K) or upper-bound (1.3 K) IPCC TAR estimate to probe the no-feedback Planck-response uncertainty.",
      "parameter_overrides": {
        "ΔT0_2xCO2": 1.2
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.3243243243243243,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 1.2,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    },
    {
      "id": "DF_2xCO2_constant_override",
      "description": "Override the upstream `@constants` ΔF_2xCO2 to 4.0 W/m² (e.g. the IPCC AR6 stratospherically-adjusted value 3.93 W/m² rounded up, or the alternative S&P p. 105 estimate 4.6 W/m² rounded down) at the canonical 2×CO₂ scenario. Result: λ_0 = 1.25/4.0 = 0.3125 K·m²/W (vs the default 0.3378), ΔT_0 = 0.3125·3.7 = 1.15625 K (no longer = ΔT0_2xCO2 because ΔF ≠ ΔF_2xCO2 anymore). ΔT_s, ΔF_e, ΔF_r, ΔT_unrealized are unchanged. Anchors the design choice that ΔF_2xCO2 behaves as a parameter at the .esm boundary: a downstream consumer can override the IPCC TAR value to AR5/AR6 values to recompute the no-feedback Planck response against the modern reference.",
      "parameter_overrides": {
        "ΔF_2xCO2": 4.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "λ_0",
          "time": 0.0,
          "expected": 0.3125,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_0",
          "time": 0.0,
          "expected": 1.15625,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "ΔT_s",
          "time": 0.0,
          "expected": 2.9600000000000004,
          "tolerance": {
            "rel": 1e-10
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "DT_s_vs_lambda_sweep",
      "description": "Sweep λ from 0.2 K·m²/W (very-low-sensitivity planet, ~0.74 K 2×CO₂ — below IPCC AR4 likely range) to 1.5 K·m²/W (very-high-sensitivity planet, ~5.55 K 2×CO₂ — above IPCC AR4 likely range) at constant default ΔF = 3.7 W/m², plotting ΔT_s(λ) and ΔT_unrealized(λ) as line plots. Surfaces the linear scaling of equilibrium warming with sensitivity (Eq. 23.1: ΔT_s = λ·ΔF — exact straight line of slope 3.7 through origin) and the non-linear shape of the warming commitment (Eq. 23.7: ΔT_unrealized = (ΔF - ΔT_r/λ)·λ = ΔT_s - ΔT_r — also linear in λ with slope ΔF but offset by -ΔT_r/λ). Anchors the IPCC AR4 likely range [0.4, 1.2] K·m²/W → ΔT_s ∈ [1.48, 4.44] K spanning the textbook climate-sensitivity uncertainty.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "λ",
            "range": {
              "start": 0.2,
              "stop": 1.5,
              "count": 27,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "DT_s_vs_lambda",
          "type": "line",
          "description": "Equilibrium temperature change ΔT_s as a function of climate sensitivity λ at the canonical 2×CO₂ forcing ΔF = 3.7 W/m². Exact straight line of slope ΔF = 3.7 K/(K·m²/W) passing through the origin (Eq. 23.1 ΔT_s = λ·ΔF). At λ = 0.4 (IPCC AR4 lower bound) → ΔT_s = 1.48 K; at λ = 0.8 (central) → ΔT_s = 2.96 K; at λ = 1.2 (upper bound) → ΔT_s = 4.44 K.",
          "x": {
            "variable": "λ",
            "label": "λ (K·m²/W)"
          },
          "y": {
            "variable": "ΔT_s",
            "label": "ΔT_s (K)"
          }
        },
        {
          "id": "DT_unrealized_vs_lambda",
          "type": "line",
          "description": "Warming commitment ΔT_unrealized as a function of climate sensitivity λ at default ΔF = 3.7 and ΔT_r = 0.7. ΔT_unrealized = ΔT_s - ΔT_r is linear in λ with slope ΔF passing through the origin then offset by -ΔT_r: at λ = 0.2 → ΔT_unrealized = 0.04 K (planet already near equilibrium because low λ means low ΔT_s); at λ = 1.5 → ΔT_unrealized = 4.85 K (large commitment because high λ implies high equilibrium).",
          "x": {
            "variable": "λ",
            "label": "λ (K·m²/W)"
          },
          "y": {
            "variable": "ΔT_unrealized",
            "label": "ΔT_unrealized (K)"
          }
        }
      ]
    },
    {
      "id": "DT_s_vs_DF_sweep",
      "description": "Sweep ΔF from -8.0 W/m² (strong volcanic-aerosol cooling) to +8.0 W/m² (strong CO₂-quadrupling forcing) at constant default λ = 0.8 K·m²/W, plotting ΔT_s(ΔF), ΔT_0(ΔF), and ΔT_unrealized(ΔF). Reproduces a textbook linear-response figure with three perfectly linear curves passing through (or near) the origin: ΔT_s = 0.8·ΔF (slope 0.8), ΔT_0 = 0.338·ΔF (slope 0.338, the no-feedback Planck response), ΔT_unrealized = ΔT_s - ΔT_r (slope 0.8 but offset by -0.7 K at ΔF = 0). Anchors the linear-response shape: all three temperature responses are exactly linear in the forcing, with the feedback amplification factor visible as the slope ratio ΔT_s/ΔT_0 = λ/λ_0 = 2.37.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "ΔF",
            "range": {
              "start": -8.0,
              "stop": 8.0,
              "count": 33,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "DT_s_DT_0_vs_DF",
          "type": "line",
          "description": "Temperature response ΔT_s (feedback-amplified) and ΔT_0 (no-feedback Planck) as functions of forcing ΔF at λ = 0.8 K·m²/W. Both are exact straight lines passing through the origin; the slope ratio ΔT_s/ΔT_0 = 0.8/0.338 = 2.37 is the feedback amplification factor. At ΔF = 3.7 W/m² (2×CO₂), ΔT_s = 2.96 K and ΔT_0 = 1.25 K.",
          "x": {
            "variable": "ΔF",
            "label": "ΔF (W/m²)"
          },
          "y": {
            "variable": "ΔT_s",
            "label": "ΔT_s (K)"
          }
        },
        {
          "id": "DT_unrealized_vs_DF",
          "type": "line",
          "description": "Warming commitment ΔT_unrealized as a function of forcing ΔF at λ = 0.8 and ΔT_r = 0.7. Straight line of slope λ = 0.8 K/(W/m²) crossing zero at ΔF = ΔF_r = ΔT_r/λ = 0.875 W/m² (the 'committed forcing' the planet has already paid down through realized warming). Negative below 0.875 W/m² (planet must cool back); positive above (planet must warm further). At ΔF = 3.7 W/m² (2×CO₂), ΔT_unrealized = 2.26 K (the remaining commitment after the present-day ΔT_r = 0.7 K has been realized).",
          "x": {
            "variable": "ΔF",
            "label": "ΔF (W/m²)"
          },
          "y": {
            "variable": "ΔT_unrealized",
            "label": "ΔT_unrealized (K)"
          }
        }
      ]
    },
    {
      "id": "DT_s_landscape_2d",
      "description": "Joint 2D sweep over (λ, ΔF): λ ∈ [0.2, 1.5] K·m²/W (the full plausible climate-sensitivity range covering very-low to very-high planets) and ΔF ∈ [-4.0, 8.0] W/m² (strong-cooling-to-strong-warming forcing range). Surfaces ΔT_s as a heatmap: each row at fixed ΔF is linear in λ (slope = ΔF) and each column at fixed λ is linear in ΔF (slope = λ). The zero-forcing diagonal (ΔF = 0) gives ΔT_s = 0 regardless of λ; positive-forcing rows have larger ΔT_s at higher λ. The IPCC AR4 likely range [0.4, 1.2] K·m²/W × CO₂-doubling forcing 3.7 W/m² maps to the heatmap region with ΔT_s ∈ [1.48, 4.44] K — the canonical climate-sensitivity uncertainty envelope.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "λ",
            "range": {
              "start": 0.2,
              "stop": 1.5,
              "count": 14,
              "scale": "linear"
            }
          },
          {
            "parameter": "ΔF",
            "range": {
              "start": -4.0,
              "stop": 8.0,
              "count": 13,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "DT_s_heatmap",
          "type": "heatmap",
          "description": "Equilibrium temperature change ΔT_s = λ·ΔF as a function of (λ, ΔF). Bilinear surface (Eq. 23.1 is exactly linear in both inputs). Zero along the ΔF = 0 axis; positive in the upper half (warming forcing), negative in the lower half (cooling forcing). The IPCC AR4 climate-sensitivity uncertainty envelope at 2×CO₂ forcing maps to the (λ ∈ [0.4, 1.2], ΔF = 3.7) row giving ΔT_s ∈ [1.48, 4.44] K.",
          "x": {
            "variable": "λ",
            "label": "λ (K·m²/W)"
          },
          "y": {
            "variable": "ΔF",
            "label": "ΔF (W/m²)"
          },
          "value": {
            "variable": "ΔT_s",
            "reduce": "final"
          }
        }
      ]
    }
  ]
}