DropletGrowth

{
  "variables": {
    "p_sat": {
      "type": "unknown",
      "units": "Pa",
      "description": "Saturation vapor pressure (Magnus)"
    },
    "one_val": {
      "default": 1.0,
      "type": "parameter",
      "units": "1",
      "description": "Unit value (dimensionless)"
    },
    "α_T": {
      "default": 1.0,
      "type": "parameter",
      "units": "1",
      "description": "Thermal accommodation coefficient (dimensionless)"
    },
    "D_v": {
      "type": "unknown",
      "units": "m^2/s",
      "description": "Diffusivity of water vapor in air"
    },
    "T": {
      "default": 293.15,
      "type": "parameter",
      "units": "K",
      "description": "Temperature"
    },
    "Hv_exp_const": {
      "default": 0.167,
      "type": "parameter",
      "units": "1",
      "description": "Latent-heat exponent constant (dimensionless)"
    },
    "d_u": {
      "default": 0.0,
      "type": "parameter",
      "units": "m",
      "description": "Insoluble core diameter"
    },
    "π_val": {
      "default": 3.141592653589793,
      "type": "parameter",
      "units": "1",
      "description": "Pi (dimensionless)"
    },
    "M_w": {
      "default": 0.018015,
      "type": "parameter",
      "units": "kg/mol",
      "description": "Molecular weight of water"
    },
    "p_sat_ref": {
      "default": 610.94,
      "type": "parameter",
      "units": "Pa",
      "description": "Reference saturation pressure (Magnus)"
    },
    "D_v_prime": {
      "type": "unknown",
      "units": "m^2/s",
      "description": "Modified diffusivity of water vapor"
    },
    "ρ_air": {
      "type": "unknown",
      "units": "kg/m^3",
      "description": "Air density from ideal gas law"
    },
    "σ_w": {
      "default": 0.0728,
      "type": "parameter",
      "units": "N/m",
      "description": "Surface tension of water"
    },
    "Dv_ref": {
      "default": 2.11e-05,
      "type": "parameter",
      "units": "m^2/s",
      "description": "Reference water-vapor diffusivity at 273 K, 1 atm"
    },
    "vol_eps": {
      "default": 1e-60,
      "type": "parameter",
      "units": "m^3",
      "description": "Small volume for numerical stability"
    },
    "p_ref": {
      "default": 101325.0,
      "type": "parameter",
      "units": "Pa",
      "description": "Reference pressure (1 atm)"
    },
    "magnus_b": {
      "default": 243.04,
      "type": "parameter",
      "units": "K",
      "description": "Magnus coefficient b"
    },
    "S_eq": {
      "type": "unknown",
      "units": "1",
      "description": "Equilibrium saturation ratio over droplet (dimensionless)"
    },
    "n_s": {
      "default": 0.0,
      "type": "parameter",
      "units": "mol",
      "description": "Moles of solute in droplet"
    },
    "D_p": {
      "default": 1e-06,
      "type": "parameter",
      "units": "m",
      "description": "Droplet diameter"
    },
    "T_Dv_ref": {
      "default": 273.0,
      "type": "parameter",
      "units": "K",
      "description": "Reference temperature for D_v"
    },
    "Hv_exp_coeff": {
      "default": 0.000367,
      "type": "parameter",
      "units": "1/K",
      "description": "Latent-heat exponent coefficient"
    },
    "ρ_w": {
      "default": 997.0,
      "type": "parameter",
      "units": "kg/m^3",
      "description": "Density of liquid water"
    },
    "c_p_air": {
      "default": 1005.0,
      "type": "parameter",
      "units": "J/kg/K",
      "description": "Specific heat of air"
    },
    "M_a": {
      "default": 0.02897,
      "type": "parameter",
      "units": "kg/mol",
      "description": "Molecular weight of air"
    },
    "p_a": {
      "default": 101325.0,
      "type": "parameter",
      "units": "Pa",
      "description": "Atmospheric pressure"
    },
    "dDp_dt": {
      "type": "unknown",
      "units": "m/s",
      "description": "Droplet diameter growth rate"
    },
    "Hv_ref_kg": {
      "default": 2500000.0,
      "type": "parameter",
      "units": "J/kg",
      "description": "Reference latent heat of vaporization"
    },
    "k_a": {
      "type": "unknown",
      "units": "J/m/s/K",
      "description": "Thermal conductivity of air"
    },
    "ΔH_v": {
      "type": "unknown",
      "units": "J/kg",
      "description": "Latent heat of vaporization"
    },
    "T_Hv_ref": {
      "default": 273.15,
      "type": "parameter",
      "units": "K",
      "description": "Reference temperature for latent heat"
    },
    "T_magnus_offset": {
      "default": 273.15,
      "type": "parameter",
      "units": "K",
      "description": "Magnus temperature offset"
    },
    "S_v": {
      "default": 1.01,
      "type": "parameter",
      "units": "1",
      "description": "Environmental saturation ratio (dimensionless)"
    },
    "α_c": {
      "default": 1.0,
      "type": "parameter",
      "units": "1",
      "description": "Mass accommodation coefficient (dimensionless)"
    },
    "ka_c1": {
      "default": 7.1e-05,
      "type": "parameter",
      "units": "J/m/s/K^2",
      "description": "Thermal conductivity slope"
    },
    "R": {
      "default": 8.314,
      "type": "parameter",
      "units": "J/mol/K",
      "description": "Universal gas constant"
    },
    "ka_c0": {
      "default": 0.00439,
      "type": "parameter",
      "units": "J/m/s/K",
      "description": "Thermal conductivity constant"
    },
    "k_a_prime": {
      "type": "unknown",
      "units": "J/m/s/K",
      "description": "Modified thermal conductivity of air"
    },
    "magnus_a": {
      "default": 17.625,
      "type": "parameter",
      "units": "1",
      "description": "Magnus coefficient a (dimensionless)"
    }
  },
  "equations": [
    {
      "rhs": {
        "args": [
          {
            "args": [
              "p_ref",
              {
                "args": [
                  {
                    "args": [
                      "T",
                      "T_Dv_ref"
                    ],
                    "op": "/"
                  },
                  1.94
                ],
                "op": "^"
              },
              "Dv_ref"
            ],
            "op": "*"
          },
          "p_a"
        ],
        "op": "/"
      },
      "lhs": "D_v"
    },
    {
      "rhs": {
        "args": [
          "D_v",
          {
            "args": [
              "one_val",
              {
                "args": [
                  {
                    "args": [
                      2,
                      {
                        "args": [
                          {
                            "args": [
                              {
                                "args": [
                                  2,
                                  "M_w",
                                  "π_val"
                                ],
                                "op": "*"
                              },
                              {
                                "args": [
                                  "T",
                                  "R"
                                ],
                                "op": "*"
                              }
                            ],
                            "op": "/"
                          }
                        ],
                        "op": "sqrt"
                      },
                      "D_v"
                    ],
                    "op": "*"
                  },
                  {
                    "args": [
                      "D_p",
                      "α_c"
                    ],
                    "op": "*"
                  }
                ],
                "op": "/"
              }
            ],
            "op": "+"
          }
        ],
        "op": "/"
      },
      "lhs": "D_v_prime"
    },
    {
      "rhs": {
        "args": [
          {
            "args": [
              "T",
              "ka_c1"
            ],
            "op": "*"
          },
          "ka_c0"
        ],
        "op": "+"
      },
      "lhs": "k_a"
    },
    {
      "rhs": {
        "args": [
          {
            "args": [
              "M_a",
              "p_a"
            ],
            "op": "*"
          },
          {
            "args": [
              "T",
              "R"
            ],
            "op": "*"
          }
        ],
        "op": "/"
      },
      "lhs": "ρ_air"
    },
    {
      "rhs": {
        "args": [
          "k_a",
          {
            "args": [
              "one_val",
              {
                "args": [
                  {
                    "args": [
                      2,
                      {
                        "args": [
                          {
                            "args": [
                              {
                                "args": [
                                  2,
                                  "M_a",
                                  "π_val"
                                ],
                                "op": "*"
                              },
                              {
                                "args": [
                                  "T",
                                  "R"
                                ],
                                "op": "*"
                              }
                            ],
                            "op": "/"
                          }
                        ],
                        "op": "sqrt"
                      },
                      "k_a"
                    ],
                    "op": "*"
                  },
                  {
                    "args": [
                      "c_p_air",
                      "α_T",
                      "D_p",
                      "ρ_air"
                    ],
                    "op": "*"
                  }
                ],
                "op": "/"
              }
            ],
            "op": "+"
          }
        ],
        "op": "/"
      },
      "lhs": "k_a_prime"
    },
    {
      "rhs": {
        "args": [
          "p_sat_ref",
          {
            "args": [
              {
                "args": [
                  {
                    "args": [
                      {
                        "args": [
                          "T",
                          {
                            "args": [
                              -1,
                              "T_magnus_offset"
                            ],
                            "op": "*"
                          }
                        ],
                        "op": "+"
                      },
                      "magnus_a"
                    ],
                    "op": "*"
                  },
                  {
                    "args": [
                      "T",
                      {
                        "args": [
                          -1,
                          "T_magnus_offset"
                        ],
                        "op": "*"
                      },
                      "magnus_b"
                    ],
                    "op": "+"
                  }
                ],
                "op": "/"
              }
            ],
            "op": "exp"
          }
        ],
        "op": "*"
      },
      "lhs": "p_sat"
    },
    {
      "rhs": {
        "args": [
          "Hv_ref_kg",
          {
            "args": [
              {
                "args": [
                  "T_Hv_ref",
                  "T"
                ],
                "op": "/"
              },
              {
                "args": [
                  "Hv_exp_const",
                  {
                    "args": [
                      "T",
                      "Hv_exp_coeff"
                    ],
                    "op": "*"
                  }
                ],
                "op": "+"
              }
            ],
            "op": "^"
          }
        ],
        "op": "*"
      },
      "lhs": "ΔH_v"
    },
    {
      "rhs": {
        "args": [
          {
            "args": [
              {
                "args": [
                  {
                    "args": [
                      4,
                      "M_w",
                      "σ_w"
                    ],
                    "op": "*"
                  },
                  {
                    "args": [
                      "T",
                      "R",
                      "D_p",
                      "ρ_w"
                    ],
                    "op": "*"
                  }
                ],
                "op": "/"
              },
              {
                "args": [
                  {
                    "args": [
                      -6,
                      "M_w",
                      "n_s"
                    ],
                    "op": "*"
                  },
                  {
                    "args": [
                      {
                        "args": [
                          {
                            "args": [
                              "D_p",
                              3
                            ],
                            "op": "^"
                          },
                          {
                            "args": [
                              -1,
                              {
                                "args": [
                                  "d_u",
                                  3
                                ],
                                "op": "^"
                              }
                            ],
                            "op": "*"
                          },
                          "vol_eps"
                        ],
                        "op": "+"
                      },
                      "π_val",
                      "ρ_w"
                    ],
                    "op": "*"
                  }
                ],
                "op": "/"
              }
            ],
            "op": "+"
          }
        ],
        "op": "exp"
      },
      "lhs": "S_eq"
    },
    {
      "rhs": {
        "args": [
          {
            "args": [
              "one_val",
              {
                "args": [
                  {
                    "args": [
                      -1,
                      "S_eq"
                    ],
                    "op": "*"
                  },
                  "S_v"
                ],
                "op": "+"
              }
            ],
            "op": "*"
          },
          {
            "args": [
              {
                "args": [
                  {
                    "args": [
                      {
                        "args": [
                          {
                            "args": [
                              {
                                "args": [
                                  -1,
                                  "one_val"
                                ],
                                "op": "*"
                              },
                              {
                                "args": [
                                  {
                                    "args": [
                                      "M_w",
                                      "ΔH_v"
                                    ],
                                    "op": "*"
                                  },
                                  {
                                    "args": [
                                      "T",
                                      "R"
                                    ],
                                    "op": "*"
                                  }
                                ],
                                "op": "/"
                              }
                            ],
                            "op": "+"
                          },
                          "ΔH_v",
                          "ρ_w"
                        ],
                        "op": "*"
                      },
                      {
                        "args": [
                          4,
                          "T",
                          "k_a_prime"
                        ],
                        "op": "*"
                      }
                    ],
                    "op": "/"
                  },
                  {
                    "args": [
                      {
                        "args": [
                          "T",
                          "R",
                          "ρ_w"
                        ],
                        "op": "*"
                      },
                      {
                        "args": [
                          4,
                          "p_sat",
                          "M_w",
                          "D_v_prime"
                        ],
                        "op": "*"
                      }
                    ],
                    "op": "/"
                  }
                ],
                "op": "+"
              },
              "D_p"
            ],
            "op": "*"
          }
        ],
        "op": "/"
      },
      "lhs": "dDp_dt"
    }
  ],
  "tests": [
    {
      "id": "baseline_supersat_293K",
      "description": "Seinfeld & Pandis (2006) base case: T=293.15 K, p_a=101325 Pa, S_v=1.01, D_p=1 μm, pure water (n_s=0, d_u=0). Checks all 9 observed variables against Tsit5 ground truth at three times. Primary behavioral gate: reproduces upstream 'DropletGrowth physical behavior' testitem in cloud_physics_test.jl (dDp_dt > 0, D_v ≈ 2.4e-5 m^2/s, k_a ≈ 0.025 J/(m·s·K)).\n\nTolerance rationale (test-level rel=1e-4, applied uniformly across this and all other tests in this file): reference values from upstream Aerosol.jl Tsit5 ground truth at the test conditions; 1e-4 absorbs Tsit5↔Python LSODA truncation over tspan=[0,1] s plus float64 roundoff in the kinetic-correction expressions for D_v' (Eq. 17.61) and k_a' (Eq. 17.71) and the Köhler composite S_eq (Eq. 17.27) inlined into Eq. 17.70. Per-assertion `tolerance` blocks duplicate the test-level value (no numerical reason to differ); the redundancy is a serialization artifact that could be removed in a follow-up cleanup.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "D_v",
          "time": 0.0,
          "expected": 2.4225978261e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "D_v",
          "time": 1.0,
          "expected": 2.4225978261e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "D_v_prime",
          "time": 0.5,
          "expected": 1.821237252e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a",
          "time": 0.0,
          "expected": 0.02520365,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a",
          "time": 0.5,
          "expected": 0.02520365,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a_prime",
          "time": 0.5,
          "expected": 0.018533519773,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ρ_air",
          "time": 0.0,
          "expected": 1.2043845867,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "p_sat",
          "time": 0.5,
          "expected": 2333.4406231,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ΔH_v",
          "time": 0.5,
          "expected": 2451959.6638,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "S_eq",
          "time": 0.5,
          "expected": 1.0021612223,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.0,
          "expected": 2.8822501313e-06,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 1.0,
          "expected": 2.8822501313e-06,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "subsat_293K_evaporation",
      "description": "Same thermodynamic state as baseline, but S_v=0.99 (subsaturated). Mirrors upstream 'DropletGrowth physical behavior' subsat branch: growth rate must flip sign. dDp_dt < 0 proves the driving force (S_v − S_eq) is handled with correct sign.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied to the evaporation branch. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 0.99,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "S_eq",
          "time": 0.5,
          "expected": 1.0021612223,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.5,
          "expected": -4.4715752844e-06,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "small_Dp_grows_faster",
      "description": "D_p=5e-7 m with same S_v=1.01: Eq. 17.70 has a 1/D_p prefactor and Kelvin S_eq is larger at smaller D_p (Eq. 17.27). Net effect at this D_p range: |dDp_dt| is larger. Mirrors upstream 'smaller droplet should grow faster' assertion.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied to a smaller-D_p variant of the baseline. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 5e-07,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "D_v_prime",
          "time": 0.0,
          "expected": 1.4590562514e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a_prime",
          "time": 0.0,
          "expected": 0.014655065772,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "S_eq",
          "time": 0.0,
          "expected": 1.0043271155,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.0,
          "expected": 3.3112759979e-06,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "cold_cloud_263K",
      "description": "Cold cloud at T=263.15 K. D_v scales as (T/273)^1.94 so it drops ~20%, p_sat (Magnus) drops by an order of magnitude, and ΔH_v rises slightly. Net growth rate is much slower than the 293 K baseline. Guards against T-dependence regressions in Eq. 17.61, 17.71, the Magnus formula, and Eq. 17.3.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied to a cold-cloud regime. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 263.15,
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "D_v",
          "time": 0.5,
          "expected": 1.9648141009e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a",
          "time": 0.5,
          "expected": 0.02307365,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ρ_air",
          "time": 0.5,
          "expected": 1.3416885487,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "p_sat",
          "time": 0.5,
          "expected": 286.77296062,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ΔH_v",
          "time": 0.5,
          "expected": 2524697.6096,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "S_eq",
          "time": 0.5,
          "expected": 1.0024079055,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.5,
          "expected": 7.923430706e-07,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "large_Dp_10um",
      "description": "D_p=10 μm (drizzle drop): curvature is weak so S_eq is close to 1 and the kinetic corrections to D_v and k_a are minor. Tests the large-D_p limit where the Maxwellian diffusion picture should dominate; |dDp_dt| is ~6× smaller than the 1 μm baseline at the same S_v.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied to the drizzle-drop regime. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 1e-05,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "D_v_prime",
          "time": 0.0,
          "expected": 2.345162114e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a_prime",
          "time": 0.0,
          "expected": 0.024328092902,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "S_eq",
          "time": 0.0,
          "expected": 1.0002159123,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.0,
          "expected": 4.6957270508e-07,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "solute_lowers_Seq",
      "description": "Add 1e-15 mol solute (n_s) into a 1 μm droplet. Eq. 17.27's Raoult term pulls S_eq below unity so the droplet grows even at only 0.1% supersaturation (S_v=1.001). Confirms the composite Köhler expression embedded in Eq. 17.70 handles the n_s term with correct sign.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied with the Köhler n_s term active. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 1.001,
        "D_p": 1e-06,
        "n_s": 1e-15,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "S_eq",
          "time": 0.0,
          "expected": 0.9681669338,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.0,
          "expected": 1.2072431836e-05,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "low_pressure_500hPa",
      "description": "Upper-tropospheric conditions: p_a=500 hPa, T=273.15 K. D_v nearly doubles via the p_ref/p_a factor in Eq. 17.61, and air density falls. Used to guard the p_a dependence of Eq. 17.61 and of ρ_air (ideal-gas law), plus the T-dependence of k_a and ΔH_v.\n\nTolerance rationale (rel=1e-4): see baseline_supersat_293K — same Tsit5↔LSODA truncation profile applied to upper-tropospheric pressure/temperature. Per-assertion blocks are redundant with test-level.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 273.15,
        "p_a": 50000.0,
        "S_v": 1.01,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "tolerance": {
        "rel": 0.0001
      },
      "assertions": [
        {
          "variable": "D_v",
          "time": 0.5,
          "expected": 4.2804740205e-05,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "k_a",
          "time": 0.5,
          "expected": 0.02378365,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ρ_air",
          "time": 0.5,
          "expected": 0.63783342539,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "p_sat",
          "time": 0.5,
          "expected": 610.94,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "ΔH_v",
          "time": 0.5,
          "expected": 2500000.0,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "S_eq",
          "time": 0.5,
          "expected": 1.0023196501,
          "tolerance": {
            "rel": 0.0001
          }
        },
        {
          "variable": "dDp_dt",
          "time": 0.5,
          "expected": 1.5566265402e-06,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    }
  ],
  "reference": {
    "citation": "Seinfeld, J. H. and Pandis, S. N.: Atmospheric Chemistry and Physics — From Air Pollution to Climate Change, 2nd Edition, John Wiley & Sons, Chapter 17 (Cloud Physics), 2006.",
    "notes": "Single-droplet diffusional growth. Equations 17.61, 17.62, 17.70, 17.71, 17.72 (air diffusivity + kinetic correction, thermal conductivity + kinetic correction, simplified droplet-growth equation). Köhler (Eq. 17.27) composite term for S_eq uses the solute moles n_s and an insoluble-core diameter d_u; pass n_s = 0, d_u = 0 for pure-water droplets. The `vol_eps` term (1e-60 m^3) is a numerical regularizer that prevents division by zero when D_p^3 == d_u^3 — at any physical droplet size it contributes O(1e-40) to a quantity of order D_p^3 and is negligible. `one_val`, `π_val`, `T_unit`, `vol_eps` are numerical-stability / dimensionless-constant parameters that the upstream `@constants` block declared with explicit units (1 or m^3) — they should not be treated as tunable. Upstream repo: https://github.com/EarthSciML/Aerosol.jl; pinned commit c8c640bb636b22f2425460e9ed5cba50793bfee1.",
    "url": "https://www.wiley.com/en-us/Atmospheric+Chemistry+and+Physics%3A+From+Air+Pollution+to+Climate+Change%2C+2nd+Edition-p-9780471720188"
  },
  "analyses": [
    {
      "id": "growth_rate_vs_Dp",
      "description": "Droplet-diameter growth rate dD_p/dt as a function of droplet diameter at fixed T, p_a, S_v, and n_s. Reproduces the canonical shape of Seinfeld & Pandis Fig. 17.6–17.7 (Köhler curve): at very small D_p the Kelvin term dominates and growth is suppressed / negative; above the critical diameter the droplet grows freely with the ~1/D_p Maxwell scaling.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameters": {
        "T": 293.15,
        "p_a": 101325.0,
        "S_v": 1.003,
        "n_s": 1e-16,
        "d_u": 0.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "D_p",
            "values": [
              1e-08,
              2e-08,
              5e-08,
              1e-07,
              2e-07,
              5e-07,
              1e-06,
              2e-06,
              5e-06,
              1e-05
            ]
          }
        ]
      },
      "plots": [
        {
          "id": "dDp_dt_vs_Dp_log",
          "type": "line",
          "description": "dD_p/dt (m/s) vs D_p (m) at S_v=1.003 with a small solute mass — resembles the activation-curve shape in Seinfeld & Pandis Fig. 17.6.",
          "x": {
            "variable": "D_p",
            "label": "Droplet diameter D_p (m)"
          },
          "y": {
            "variable": "dDp_dt",
            "label": "Growth rate dD_p/dt (m/s)"
          }
        }
      ]
    },
    {
      "id": "Dv_vs_T",
      "description": "Water-vapor diffusivity D_v as a function of temperature at 1 atm. Reproduces Eq. 17.61: D_v = 0.211e-4 × (p_ref/p_a) × (T/273)^1.94. The ~(T/273)^1.94 scaling is a textbook result (Seinfeld & Pandis §17.4.1) and should cross 2.4e-5 m²/s at ~293 K.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameters": {
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "T",
            "values": [
              233.15,
              243.15,
              253.15,
              263.15,
              273.15,
              283.15,
              293.15,
              303.15,
              313.15
            ]
          }
        ]
      },
      "plots": [
        {
          "id": "Dv_vs_T_curve",
          "type": "line",
          "description": "Water-vapor diffusivity D_v vs T (Eq. 17.61).",
          "x": {
            "variable": "T",
            "label": "Temperature (K)"
          },
          "y": {
            "variable": "D_v",
            "label": "Diffusivity D_v (m²/s)"
          }
        },
        {
          "id": "ka_vs_T_curve",
          "type": "line",
          "description": "Air thermal conductivity k_a vs T (Eq. 17.71).",
          "x": {
            "variable": "T",
            "label": "Temperature (K)"
          },
          "y": {
            "variable": "k_a",
            "label": "Thermal conductivity k_a (J/(m·s·K))"
          }
        }
      ]
    },
    {
      "id": "psat_vs_T_magnus",
      "description": "Saturation vapor pressure p_sat as a function of T via the Magnus formula. Reproduces the water-vapor pressure curve in Seinfeld & Pandis Table 17.2 / Fig. 17.1: rises from a few hundred Pa at 0 °C to several thousand Pa at 30 °C.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameters": {
        "p_a": 101325.0,
        "S_v": 1.01,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "T",
            "values": [
              253.15,
              263.15,
              273.15,
              283.15,
              293.15,
              303.15,
              313.15,
              323.15
            ]
          }
        ]
      },
      "plots": [
        {
          "id": "psat_vs_T",
          "type": "line",
          "description": "Magnus saturation vapor pressure vs T (S&P Table 17.2).",
          "x": {
            "variable": "T",
            "label": "Temperature (K)"
          },
          "y": {
            "variable": "p_sat",
            "label": "Saturation pressure p_sat (Pa)"
          }
        }
      ]
    },
    {
      "id": "growth_rate_vs_Sv",
      "description": "Droplet growth rate dD_p/dt as a function of environmental saturation ratio S_v. The curve crosses zero at S_v = S_eq (Kelvin-limited equilibrium) and grows linearly in (S_v − S_eq). Useful for inspecting model sensitivity to ambient humidity, as in Seinfeld & Pandis §17.4.2.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameters": {
        "T": 293.15,
        "p_a": 101325.0,
        "D_p": 1e-06,
        "n_s": 0.0,
        "d_u": 0.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "S_v",
            "values": [
              0.99,
              0.995,
              1.0,
              1.002,
              1.005,
              1.01,
              1.02,
              1.05
            ]
          }
        ]
      },
      "plots": [
        {
          "id": "dDp_dt_vs_Sv",
          "type": "line",
          "description": "Growth rate vs ambient saturation ratio S_v.",
          "x": {
            "variable": "S_v",
            "label": "Environmental saturation ratio S_v"
          },
          "y": {
            "variable": "dDp_dt",
            "label": "Growth rate dD_p/dt (m/s)"
          }
        }
      ]
    }
  ]
}