PlanetaryEnergyBalance

{
  "reference": {
    "notes": "source_ref: GasChem.jl@8c12c048482b515fb8eb2110bf8ab4b4f4e71309:src/radiation_fundamentals.jl:211\nUpstream `PlanetaryEnergyBalance` is an MTK `@component` with one `@constants` block (sigma = 5.671e-8 W/(m^2*K^4) — Stefan-Boltzmann), two `@parameters` (S_0 = 1370 W/m^2 — solar constant; R_p = 0.3 — planetary albedo), three time-indexed `@variables` (F_S = 240 W/m^2 — absorbed solar flux; F_L = 240 W/m^2 — emitted longwave flux; T_e = 255 K — planetary equilibrium temperature), and three algebraic equations:\n  (1) F_S ~ S_0 / 4 * (1 - R_p)        — Eq. 4.5: factor of 4 = cross-section area / surface area of a sphere\n  (2) F_L ~ sigma * T_e^4               — Eq. 4.6: Stefan-Boltzmann law for emissive power\n  (3) F_S ~ F_L                          — Eq. 4.7 (rearranged): equilibrium condition\n\nUpstream structural test (test/radiation_fundamentals_test.jl:41-50, `Structural: PlanetaryEnergyBalance system`) asserts `length(equations(balance)) == 3` and `length(unknowns(balance)) == 3`. The migrated `.esm` preserves the 3-equation 3-state shape and the upstream variable names (F_S, F_L, T_e).\n\nThe upstream three-equation form is a closed nonlinear algebraic system whose unique physical solution can be written in closed form by composing equations (1) and (3) into (2): T_e = (F_L/sigma)^(1/4) = ((S_0/4)*(1-R_p)/sigma)^(1/4). The migrated `.esm` encodes this closed-form chain directly so the Python ESS runner (`earthsci_ast.simulation.simulate`) — which integrates an ODE/DAE problem rather than a true nonlinear problem — recovers the same physical solution as the upstream Julia `NonlinearProblem` solver without ambiguity in the fourth-root branch. The three migrated equations are:\n  (1) F_S ~ S_0 / 4 * (1 - R_p)         — bit-identical to upstream Eq. 4.5\n  (2) T_e ~ (F_S / sigma)^(1/4)         — closed-form rearrangement of Eqs. 4.6 and 4.7\n  (3) F_L ~ sigma * T_e^4               — bit-identical to upstream Eq. 4.6\n\nThe constraint F_S ~ F_L (upstream eq 3) is implicitly satisfied by substitution: F_L = sigma * T_e^4 = sigma * ((F_S/sigma)^(1/4))^4 = F_S. This identity is exact in real arithmetic; in IEEE-754 floating-point it agrees to ~1 ULP because the (.)^(1/4) followed by (.)^4 round-trip leaves a sub-ULP residue (the `equilibrium_F_S_equals_F_L_at_default` test below pins this residue at rtol=1e-12, comfortably above the ~2e-16 ULP floor for 240 W/m^2 magnitudes). Topologically the algebraic chain is F_S -> T_e -> F_L; the toolkit's topological-sort pass in `_flat_to_sympy_rhs` orders them this way and substitutes earlier-resolved bodies into later ones, so each output is expressed as a closed body in the upstream parameters (S_0, R_p, sigma).\n\nBranch-cut selection of the fourth root: T_e^4 = F_S/sigma has four complex solutions (±real, ±imaginary), and a generic `sp.solve` returns them in an unspecified order with the negative-imaginary root in the first position by default. The closed-form encoding here (Eq. 2 as `T_e ~ (F_S/sigma)^(1/4)`) selects the principal real positive root unambiguously through Python's `**` operator on a positive base, matching the physically meaningful branch (T_e > 0 K). Upstream MTK's `NonlinearProblem` solver uses an initial guess (the upstream test passes `Dict(balance.T_e => 250.0)`) to converge to the same positive root, but that mechanism requires a true nonlinear solver; the migrated form is robust without one.\n\nUpstream `@constants` sigma: encoded here as `type: parameter` with the upstream value (5.671e-8 W/(m^2*K^4)) as `default`, matching the convention used by sibling migrations in this rig (surface_energy_flux.esm encodes its sigma_SB this way; urban_radiation.esm encodes its sigma_SB the same way; snow_thermal.esm uses C_liq, C_ice, lambda_air the same way). This keeps the migrated `.esm`'s parameter-and-default surface identical to the upstream's symbolic `@constants` block — a downstream consumer that wanted to switch to a different Stefan-Boltzmann tabulation (e.g., CODATA 2018 5.670374419e-8 vs S&P's rounded 5.671e-8) reads sigma from `parameter_overrides` exactly as it would override S_0 or R_p. Greek `sigma` is renamed to ASCII `sigma` for parameter-name consistency with sibling temps files (heating_cooling_flux.esm uses alpha_CN; surface_energy_flux.esm uses sigma_SB, epsilon_g, lambdaE_g).\n\nDefault initial values: F_S = 240 W/m^2 and F_L = 240 W/m^2 carry the upstream Julia `@variables ... = 240.0` defaults (the convergence target near 239.75 W/m^2 at the canonical defaults); T_e = 255 K carries the upstream Julia `T_e(t) = 255.0` default (the convergence target near 254.99 K at the canonical defaults). These defaults are bit-identical to the upstream initial values used by MTK's NonlinearProblem solver as initial guesses. For the migrated closed-form algebraic chain the defaults are not needed for convergence (each output is a direct function of the parameters with no iteration), but preserving the upstream defaults keeps the migrated `.esm`'s variable surface identical to the upstream — any `bd ready` consumer that snapshots the variable defaults (e.g., for a parameter-override dashboard or for the docs renderer's default-value column) reads the same numbers as in the upstream Julia source.\n\nDomain of validity: physical R_p in [0, 1] (dimensionless albedo) and S_0 >= 0 (non-negative solar irradiance) keep F_S non-negative and the fourth root real. Outside this range (R_p > 1 or S_0 < 0) F_S becomes negative and `(F_S/sigma)^(1/4)` returns NaN under IEEE-754 (the principal complex fourth root of a negative real is complex); the migrated `.esm` does not guard against this because the upstream model does not — the upstream test `Limiting: Equilibrium temperature at extreme albedo` exercises R_p = 0 (T_max = 278.77 K) and R_p = 0.99 (T_min = 88.16 K but positive) without straying outside the physical range. The upstream `Qualitative: T_e decreases with increasing albedo` test exercises R_p in [0, 0.7], also within the physical range.\n\nObservability classification: this component is functionally observed-only — every variable is given by a single algebraic equation in terms of parameters and earlier-resolved algebraic states (the `(t)` time index has no role since there are no derivatives). Test scenarios use `time_span: [0, 1]` and sample at t=0 to mirror the convention used by sibling observed-only ESM files in this rig (urban_radiation.esm, surface_energy_flux.esm, heating_cooling_flux.esm). The upstream Julia tests use `NonlinearProblem` + `solve` because Julia distinguishes nonlinear-equilibrium and time-stepping solvers at the API level; the migrated `.esm` uses the ESS toolkit's algebraic-state path which substitutes the equations into closed form at compile time and returns the steady value at every sampled time.\n\nTolerance rationale (model-level rel=1.0e-9): observed-only algebraic system with one fourth-root composition (Eq. 2: `(F_S/sigma)^(1/4)`) and one fourth-power composition (Eq. 3: `sigma * T_e^4`). The round-trip composition F_L = sigma*((F_S/sigma)^(1/4))^4 = F_S agrees to ~1 ULP in IEEE-754 (~2e-16 relative for 240 W/m^2 magnitudes); F_S and T_e individually involve at most one multiplication, one subtraction, one division, and one `^` op against fixed parameters, with similar ULP-level round-off. rel=1.0e-9 leaves >5 orders of magnitude headroom above ULP and matches sibling temps files (surface_energy_flux.esm uses 1e-9 for additive flux balances; heating_cooling_flux.esm uses 1e-9). Tighter `abs: 1.0e-10` is asserted on outputs that are exactly zero by construction (e.g. R_p = 1 -> F_S = 0 -> T_e = 0 -> F_L = 0) so a stuck-at-1e-3 regression cannot silently pass.\n\nUpstream tolerance reproduction: upstream tests use rtol = 1.0e-6 on F_S at default conditions (because the expected value is the precisely-computed 239.75 and the upstream Tsit5/Rosenbrock23 solver returns it to ~6 digits), rtol = 1.0e-2 on T_e in the canonical case and the R_p = 0.15 case (because the expected values are book-quoted to 3 sig figs: 255 K and 268 K), and rtol = 1.0e-8 on the conservation test `F_S ~ F_L for multiple albedos`. Re-asserted here at comparable or tighter tolerances on the upstream-canonical tests; the migration computes the closed-form algebraically, so the achievable precision is sub-ULP relative to the parameter values."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "sigma": {
      "type": "parameter",
      "units": "W/(m^2*K^4)",
      "default": 5.671e-08,
      "description": "Stefan-Boltzmann constant (Seinfeld & Pandis 2006 Eq. 4.4 / Ch. 4 §4.1) — upstream `@constants` sigma; the proportionality constant in the Stefan-Boltzmann law for emissive power of a blackbody. Default 5.671e-8 W/(m^2*K^4) matches the upstream Julia source verbatim; CODATA 2018 value is 5.670374419e-8 W/(m^2*K^4), within ~1e-4 relative of the upstream rounded value."
    },
    "S_0": {
      "type": "parameter",
      "units": "W/m^2",
      "default": 1370.0,
      "description": "Solar constant — total radiative flux per unit area at the top of Earth's atmosphere measured normal to the Sun's direction at 1 AU. Default 1370 W/m^2 matches the upstream Julia source and Seinfeld & Pandis 2006 p. 101; modern observation-derived total solar irradiance is ~1361 W/m^2 (Kopp & Lean 2011). The factor of 4 in Eq. 4.5 (`S_0/4 * (1 - R_p)`) is the ratio of Earth's geometric cross-section (pi*R^2) to its surface area (4*pi*R^2) — averages the per-area solar flux from the cross-section that intercepts the beam onto the full surface area."
    },
    "R_p": {
      "type": "parameter",
      "units": "1",
      "default": 0.3,
      "description": "Planetary albedo (dimensionless) — fraction of incident solar radiation reflected back to space by clouds, atmosphere, and surface. Default 0.3 matches the upstream Julia source and Seinfeld & Pandis 2006 p. 101 (the standard Earth-with-clouds value). The same upstream test family exercises R_p = 0.15 (Earth without clouds, T_e ~ 268 K), R_p = 0 (perfect absorber, T_e ~ 278.77 K), and R_p = 0.99 (near-perfect reflector, T_e ~ 88.16 K)."
    },
    "F_S": {
      "type": "unknown",
      "units": "W/m^2",
      "default": 240.0,
      "description": "Absorbed solar flux at the top of atmosphere averaged over Earth's surface (Seinfeld & Pandis Eq. 4.5): F_S = S_0/4 * (1 - R_p). Default initial value 240 W/m^2 matches the upstream Julia source `F_S(t) = 240.0` — the convergence target near 239.75 W/m^2 at the canonical defaults (S_0 = 1370, R_p = 0.3). At equilibrium F_S = F_L (Eq. 4.7)."
    },
    "F_L": {
      "type": "unknown",
      "units": "W/m^2",
      "default": 240.0,
      "description": "Emitted longwave (terrestrial) radiative flux at the top of atmosphere (Seinfeld & Pandis Eq. 4.6, Stefan-Boltzmann law): F_L = sigma * T_e^4. Default initial value 240 W/m^2 matches the upstream Julia source `F_L(t) = 240.0`. At equilibrium F_L = F_S (Eq. 4.7); the migrated algebraic chain enforces this by composition through T_e."
    },
    "T_e": {
      "type": "unknown",
      "units": "K",
      "default": 255.0,
      "description": "Planetary equilibrium temperature (Seinfeld & Pandis Eq. 4.7): the temperature at which the emitted longwave flux F_L = sigma * T_e^4 balances the absorbed solar flux F_S = S_0/4 * (1 - R_p). Closed-form: T_e = (S_0/4 * (1 - R_p) / sigma)^(1/4). Default initial value 255 K matches the upstream Julia source `T_e(t) = 255.0` — the convergence target at the canonical defaults (S_0 = 1370, R_p = 0.3); upstream Julia `NonlinearProblem` uses 250 K as the initial guess and converges to ~254.99 K (book value 255 K)."
    }
  },
  "equations": [
    {
      "lhs": "F_S",
      "rhs": {
        "op": "*",
        "args": [
          {
            "op": "/",
            "args": [
              "S_0",
              4.0
            ]
          },
          {
            "op": "-",
            "args": [
              1.0,
              "R_p"
            ]
          }
        ]
      }
    },
    {
      "lhs": "T_e",
      "rhs": {
        "op": "^",
        "args": [
          {
            "op": "/",
            "args": [
              "F_S",
              "sigma"
            ]
          },
          0.25
        ]
      }
    },
    {
      "lhs": "F_L",
      "rhs": {
        "op": "*",
        "args": [
          "sigma",
          {
            "op": "^",
            "args": [
              "T_e",
              4.0
            ]
          }
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "upstream_canonical_default_R_p_0_3",
      "description": "Reproduces upstream `Equation: PlanetaryEnergyBalance (Eqs. 4.5-4.7)` test (GasChem.jl test/radiation_fundamentals_test.jl:167-182) at the canonical Earth defaults S_0 = 1370 W/m^2 and R_p = 0.3 (Seinfeld & Pandis 2006 p. 101). Expected: F_S = S_0/4 * (1-R_p) = 1370/4 * 0.7 = 239.75 W/m^2 exactly; T_e = (239.75 / 5.671e-8)^(1/4) = 254.9909... K (within 1.0e-2 relative of the book value 255 K); F_L = sigma * T_e^4 = 239.75 W/m^2 (= F_S by the equilibrium identity). Upstream asserts F_S at rtol = 1e-6 and T_e at rtol = 1e-2 against the book value; re-asserted here at rtol = 1e-6 for F_S, F_L (the precise computed values) and rtol = 1e-2 for T_e against the book value (compatible with the upstream test) plus a tighter rtol = 1e-7 for T_e against the high-precision computed value 254.9909354.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 239.75,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 254.9909354243224,
          "tolerance": {
            "rel": 1e-07
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 255.0,
          "tolerance": {
            "rel": 0.01
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 239.75,
          "tolerance": {
            "rel": 1e-12
          }
        }
      ]
    },
    {
      "id": "upstream_no_clouds_R_p_0_15",
      "description": "Reproduces upstream second-case assertion in `Equation: PlanetaryEnergyBalance (Eqs. 4.5-4.7)` (GasChem.jl test/radiation_fundamentals_test.jl:179-181) at R_p = 0.15 (Earth without clouds — only surface and atmosphere albedo, no cloud reflection). Expected: F_S = 1370/4 * 0.85 = 291.125 W/m^2; T_e = (291.125 / 5.671e-8)^(1/4) = 267.6732 K (within 1.0e-2 relative of the book value 268 K, S&P p. 101); F_L = 291.125 W/m^2 (= F_S). This is the canonical 'how much warmer Earth would be if clouds vanished' diagnostic — about +13 K relative to the cloudy case, consistent with the upstream test comment.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.15
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 291.125,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 267.6732447903574,
          "tolerance": {
            "rel": 1e-07
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 268.0,
          "tolerance": {
            "rel": 0.01
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 291.125,
          "tolerance": {
            "rel": 1e-12
          }
        }
      ]
    },
    {
      "id": "conservation_F_S_equals_F_L_at_R_p_0_0",
      "description": "Reproduces upstream `Conservation: Energy balance F_S = F_L at equilibrium` (GasChem.jl test/radiation_fundamentals_test.jl:224-235) for R_p = 0.0 (no reflection). Expected: F_S = 1370/4 * 1.0 = 342.5 W/m^2; T_e = (342.5 / 5.671e-8)^(1/4) = 278.7727 K; F_L = 342.5 W/m^2 (= F_S). Upstream asserts F_S ~ F_L at rtol = 1.0e-8 across R_p in {0.0, 0.15, 0.3, 0.5, 0.7}; this test and the four siblings below cover the same conservation invariant at the same R_p grid.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 342.5,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 342.5,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 278.77269406755795,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "conservation_F_S_equals_F_L_at_R_p_0_5",
      "description": "Reproduces the `Conservation: Energy balance F_S = F_L at equilibrium` invariant at R_p = 0.5 (50% reflection — between Earth and a high-albedo regime). Expected: F_S = 1370/4 * 0.5 = 171.25 W/m^2; T_e = (171.25 / 5.671e-8)^(1/4) = 234.4190 K; F_L = 171.25 W/m^2 (= F_S). Sibling of the R_p=0.0/0.15/0.3/0.7 entries — together they re-assert the upstream loop `for R_p in [0.0, 0.15, 0.3, 0.5, 0.7]` at test/radiation_fundamentals_test.jl:229.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 171.25,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 171.25,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 234.41895911202994,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "conservation_F_S_equals_F_L_at_R_p_0_7",
      "description": "Reproduces the `Conservation: Energy balance F_S = F_L at equilibrium` invariant at R_p = 0.7 (high-albedo regime — snowball-Earth/icy-moon territory). Expected: F_S = 1370/4 * 0.3 = 102.75 W/m^2; T_e = (102.75 / 5.671e-8)^(1/4) = 206.3149 K; F_L = 102.75 W/m^2 (= F_S). High-R_p sibling of the R_p=0.0/0.15/0.3/0.5 entries.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.7
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 102.75,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 102.75,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 206.31487724138813,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "limiting_zero_albedo_maximum_temperature",
      "description": "Reproduces upstream `Limiting: Equilibrium temperature at extreme albedo` test (GasChem.jl test/radiation_fundamentals_test.jl:286-303) at R_p = 0.0. Upstream asserts only T_e > 270 K; this test additionally pins T_e to the exact computed value 278.77 K and F_S/F_L to 342.5 W/m^2 each. The zero-albedo case is the upper bound on T_e at fixed S_0 (no fraction of solar input is reflected away).",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 278.77269406755795,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "limiting_near_perfect_albedo_minimum_temperature",
      "description": "Reproduces upstream `Limiting: Equilibrium temperature at extreme albedo` second sub-case (GasChem.jl test/radiation_fundamentals_test.jl:297-302) at R_p = 0.99 (near-perfect reflection). Upstream asserts T_e < 100 K but T_e > 0 K (the T^4 law guarantees strict positivity from any positive F_S). This test pins T_e to the exact computed value 88.1557 K and F_S/F_L to 3.425 W/m^2 each. Anchors that the migrated algebraic chain returns the correct physically meaningful root (the positive real fourth-root) under near-degenerate conditions where the cross-section of absorbed flux is two orders of magnitude smaller than the canonical case.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 0.99
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 3.425,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 88.15566627147928,
          "tolerance": {
            "rel": 1e-07
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 3.425,
          "tolerance": {
            "rel": 1e-12
          }
        }
      ]
    },
    {
      "id": "perfect_albedo_zero_temperature",
      "description": "Degenerate boundary: R_p = 1.0 (perfect mirror — zero absorbed solar flux). Expected F_S = 0 exactly; the algebraic chain composes as T_e = (0 / sigma)^(1/4) = 0^(0.25) = 0 exactly in IEEE-754 (no NaN/Inf branch for zero base, principal-root convention), and F_L = sigma * 0^4 = 0 exactly. Anchors the all-zero limit of the absorbed-emitted balance and exercises the runner against the F_S = 0 corner where the (.)^(1/4) and (.)^4 ops both reduce to zero by IEEE-754 contract.",
      "parameter_overrides": {
        "S_0": 1370.0,
        "R_p": 1.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-10
          }
        },
        {
          "variable": "F_L",
          "time": 0.0,
          "expected": 0.0,
          "tolerance": {
            "abs": 1e-12
          }
        }
      ]
    },
    {
      "id": "qualitative_higher_solar_constant_warmer",
      "description": "Reproduces upstream `Qualitative: T_e increases with increasing solar constant` test (GasChem.jl test/radiation_fundamentals_test.jl:355-374) by pinning T_e at S_0 = 1800 W/m^2 (the upper end of the upstream sweep `[1000, 1200, 1370, 1500, 1800]`). At R_p = 0.3, F_S = 1800/4 * 0.7 = 315 W/m^2 and T_e = (315 / 5.671e-8)^(1/4) = 273.0001 K — substantially warmer than the canonical 255 K at S_0 = 1370. Paired with `qualitative_lower_solar_constant_cooler` to surface the strictly monotonic increase of T_e with S_0 at fixed R_p (T_e proportional to S_0^(1/4)).",
      "parameter_overrides": {
        "S_0": 1800.0,
        "R_p": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 315.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 273.0000500276824,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "qualitative_lower_solar_constant_cooler",
      "description": "Reproduces upstream `Qualitative: T_e increases with increasing solar constant` test (GasChem.jl test/radiation_fundamentals_test.jl:355-374) at the lower end of the upstream sweep, S_0 = 1000 W/m^2. At R_p = 0.3, F_S = 1000/4 * 0.7 = 175 W/m^2 and T_e = (175 / 5.671e-8)^(1/4) = 235.69 K — substantially cooler than the canonical 255 K. Paired with `qualitative_higher_solar_constant_warmer` to anchor the T_e proportional to S_0^(1/4) scaling.",
      "parameter_overrides": {
        "S_0": 1000.0,
        "R_p": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 175.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 235.69186902503338,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    },
    {
      "id": "high_solar_no_albedo_venus_like",
      "description": "Extreme regime: S_0 = 2000 W/m^2 (approaching Venus-orbit irradiance ~2600 W/m^2 but lower) at R_p = 0.0. Expected: F_S = 500 W/m^2; T_e = (500 / 5.671e-8)^(1/4) = 306.43 K. This pairs with `limiting_near_perfect_albedo_minimum_temperature` to surface the full S_0 by R_p envelope the upstream tests explore: temperatures from ~88 K (high-albedo, default S_0) to ~306 K (no-albedo, near-Venus S_0). Confirms the closed-form algebraic chain handles a 2x larger absorbed flux without numerical issues.",
      "parameter_overrides": {
        "S_0": 2000.0,
        "R_p": 0.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "F_S",
          "time": 0.0,
          "expected": 500.0,
          "tolerance": {
            "rel": 1e-10
          }
        },
        {
          "variable": "T_e",
          "time": 0.0,
          "expected": 306.42739506038805,
          "tolerance": {
            "rel": 1e-07
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "albedo_sweep_T_e_at_S_0_1370",
      "description": "Sweep R_p across the upstream test grid [0.0, 0.7] at S_0 = 1370 W/m^2 to surface the (1 - R_p)^(1/4) scaling of equilibrium temperature with albedo (Seinfeld & Pandis Eq. 4.7). The curve descends from T_e ~ 279 K at R_p = 0 (no-albedo upper bound, the upstream `Limiting: Equilibrium temperature at extreme albedo` zero-albedo case) through the canonical T_e ~ 255 K at R_p = 0.3 (Earth-with-clouds) down to T_e ~ 206 K at R_p = 0.7 (high-albedo snowball-Earth regime). The 31-point linear sweep matches the resolution upstream uses for the `Qualitative: T_e decreases with increasing albedo` monotonicity check at test/radiation_fundamentals_test.jl:334-353. Paired plots for F_S vs R_p (a linear curve through (1, 0)) anchor the upstream Eq. 4.5 first-power dependence and provide visual cross-check against the cube-root behavior of T_e.",
      "initial_state": {},
      "parameters": {
        "S_0": 1370.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "R_p",
            "range": {
              "start": 0.0,
              "stop": 0.7,
              "count": 31,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "T_e_vs_R_p_curve",
          "type": "line",
          "description": "T_e vs R_p: monotonically decreasing curve scaling as (1 - R_p)^(1/4); descends from 279 K at R_p=0 through 255 K at R_p=0.3 to 206 K at R_p=0.7. The fourth-root convex shape (negative second derivative) anchors the Seinfeld & Pandis Eq. 4.7 algebraic structure.",
          "x": {
            "variable": "R_p",
            "label": "R_p (dimensionless)"
          },
          "y": {
            "variable": "T_e",
            "label": "T_e (K)"
          }
        },
        {
          "id": "F_S_vs_R_p_curve",
          "type": "line",
          "description": "F_S vs R_p: linear curve from 342.5 W/m^2 at R_p=0 down through 239.75 at R_p=0.3 to 102.75 at R_p=0.7 (slope = -S_0/4 = -342.5 W/m^2 per unit R_p). Anchors Eq. 4.5's linear dependence on (1 - R_p).",
          "x": {
            "variable": "R_p",
            "label": "R_p (dimensionless)"
          },
          "y": {
            "variable": "F_S",
            "label": "F_S (W/m^2)"
          }
        }
      ]
    },
    {
      "id": "solar_constant_sweep_T_e_at_R_p_0_3",
      "description": "Sweep S_0 across the upstream test grid [1000, 1800] W/m^2 at R_p = 0.3 (canonical Earth albedo) to surface the S_0^(1/4) scaling of equilibrium temperature with solar constant (Seinfeld & Pandis Eq. 4.7). The 17-point linear sweep matches the resolution upstream uses for the `Qualitative: T_e increases with increasing solar constant` monotonicity check at test/radiation_fundamentals_test.jl:355-374. The curve ascends from T_e ~ 236 K at S_0 = 1000 W/m^2 through the canonical T_e ~ 255 K at S_0 = 1370 W/m^2 to T_e ~ 273 K at S_0 = 1800 W/m^2 — a ~37 K shift across the ~80% S_0 range, consistent with the (1370/1000)^(1/4) ~ 1.082 and (1800/1370)^(1/4) ~ 1.071 ratios.",
      "initial_state": {},
      "parameters": {
        "R_p": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "S_0",
            "range": {
              "start": 1000.0,
              "stop": 1800.0,
              "count": 17,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "T_e_vs_S_0_curve",
          "type": "line",
          "description": "T_e vs S_0: monotonically increasing curve scaling as S_0^(1/4); ascends from 236 K at S_0=1000 through 255 K at S_0=1370 to 273 K at S_0=1800. The fourth-root concave shape (positive second derivative on this side of zero) anchors the upstream Eq. 4.7 algebraic structure.",
          "x": {
            "variable": "S_0",
            "label": "S_0 (W/m^2)"
          },
          "y": {
            "variable": "T_e",
            "label": "T_e (K)"
          }
        }
      ]
    },
    {
      "id": "two_d_sweep_T_e_in_S_0_R_p_space",
      "description": "Joint sweep over (S_0, R_p) at fixed sigma to surface the bilinear-quartic structure of the full algebraic chain (Seinfeld & Pandis Eqs. 4.5-4.7): T_e = (S_0/4 * (1 - R_p) / sigma)^(1/4). Iso-T_e contours follow curves of constant S_0 * (1 - R_p), i.e. constant F_S — at fixed T_e a larger S_0 must be compensated by a larger R_p to keep absorbed flux constant. The 13 x 11 grid samples cover S_0 in [800, 2000] W/m^2 (covering Earth ~1370 with bracketing values for stellar-evolution and Mars-orbit/Venus-orbit comparisons) and R_p in [0, 1] (the full physical albedo range), covering the full envelope a planetary-climate diagnostic would explore. The vertical strip at R_p = 1 collapses to T_e = 0 K (perfect reflector emits no longwave); the horizontal strip at S_0 = 0 (not sampled by the 800 start) would collapse similarly.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "S_0",
            "range": {
              "start": 800.0,
              "stop": 2000.0,
              "count": 13,
              "scale": "linear"
            }
          },
          {
            "parameter": "R_p",
            "range": {
              "start": 0.0,
              "stop": 1.0,
              "count": 11,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "T_e_S_0_R_p_heatmap",
          "type": "heatmap",
          "description": "T_e in (S_0, R_p) space: bilinear-quartic surface with iso-T_e contours along constant S_0 * (1 - R_p). The Earth operating point (S_0 = 1370, R_p = 0.3) sits in the middle of the grid at T_e ~ 255 K; the upper-right corner (R_p = 1) collapses to T_e = 0 K (perfect mirror); the lower-left bracket (S_0 = 800, R_p = 0) shows the modestly-cooler-Sun-with-no-albedo regime at T_e ~ 244 K.",
          "x": {
            "variable": "S_0",
            "label": "S_0 (W/m^2)"
          },
          "y": {
            "variable": "R_p",
            "label": "R_p (dimensionless)"
          },
          "value": {
            "variable": "T_e",
            "reduce": "final"
          }
        }
      ]
    }
  ]
}