SnowCompaction

{
  "reference": {
    "notes": "source_ref: UrbanCanopy.jl@9200edd4:src/hydrology.jl:174\nUpstream `SnowCompaction` is an MTK `@component` where T_i, w_ice, w_liq, Δz, P_s, f_ice_n, and f_ice_n1 are `@parameters` (driven by a parent layer/timestepper). For standalone serialization (and to make the component well-posed in isolation), they are declared as parameters here — matching the migration tracker's `(M, equations)` classification. The seven algebraic equations are unchanged from upstream. Δz is renamed to `dz` and η to `eta` for ASCII parameter names.\n\nTolerance rationale (model-level rel=1e-6): observed-only algebraic system (no ODE integration); 1e-6 covers float64 roundoff in the chained exp/ifelse/max closed-form expressions across the relevant snow regimes (T_i 230–273 K, snow densities 50–500 kg/m³, overburden pressures 0–10⁴ kg/m²)."
  },
  "tolerance": {
    "rel": 1e-06
  },
  "variables": {
    "T_i": {
      "type": "parameter",
      "units": "K",
      "default": 263.15,
      "description": "Snow layer temperature"
    },
    "w_ice": {
      "type": "parameter",
      "units": "kg/m^2",
      "default": 50.0,
      "description": "Mass of ice in layer"
    },
    "w_liq": {
      "type": "parameter",
      "units": "kg/m^2",
      "default": 1.0,
      "description": "Mass of liquid water in layer"
    },
    "dz": {
      "type": "parameter",
      "units": "m",
      "default": 0.5,
      "description": "Snow layer thickness (Δz in upstream Julia source)"
    },
    "P_s": {
      "type": "parameter",
      "units": "kg/m^2",
      "default": 100.0,
      "description": "Snow load pressure on this layer (Eq. 5.30)"
    },
    "f_ice_n": {
      "type": "parameter",
      "units": "1",
      "default": 0.9,
      "description": "Ice fraction at time n (dimensionless)"
    },
    "f_ice_n1": {
      "type": "parameter",
      "units": "1",
      "default": 0.9,
      "description": "Ice fraction at time n+1 (dimensionless)"
    },
    "T_f": {
      "type": "parameter",
      "units": "K",
      "default": 273.15,
      "description": "Freezing temperature of water (Oleson et al. 2010 Table 1.4)"
    },
    "c_3": {
      "type": "parameter",
      "units": "s^-1",
      "default": 2.777e-06,
      "description": "Fractional compaction rate at T_f (Eq. 5.26)"
    },
    "c_4": {
      "type": "parameter",
      "units": "K^-1",
      "default": 0.04,
      "description": "Temperature coefficient for metamorphism (Eq. 5.26)"
    },
    "eta_0": {
      "type": "parameter",
      "units": "kg*s/m^2",
      "default": 900000.0,
      "description": "Reference viscosity (Eq. 5.29)"
    },
    "c_5": {
      "type": "parameter",
      "units": "K^-1",
      "default": 0.08,
      "description": "Viscosity temperature coefficient (Eq. 5.29)"
    },
    "c_6": {
      "type": "parameter",
      "units": "m^3/kg",
      "default": 0.023,
      "description": "Viscosity density coefficient (Eq. 5.29)"
    },
    "hundred_kgpm3": {
      "type": "parameter",
      "units": "kg/m^3",
      "default": 100.0,
      "description": "Reference density 100 kg/m^3 (Eq. 5.27 threshold)"
    },
    "thresh_liq_density": {
      "type": "parameter",
      "units": "kg/m^3",
      "default": 0.01,
      "description": "Liquid water density threshold (Eq. 5.27)"
    },
    "coeff_0046": {
      "type": "parameter",
      "units": "m^3/kg",
      "default": 0.046,
      "description": "Density coefficient (Eq. 5.27)"
    },
    "one_per_s": {
      "type": "parameter",
      "units": "s^-1",
      "default": 1.0,
      "description": "Unit rate for nondimensionalization (Eq. 5.31)"
    },
    "c_1": {
      "type": "unknown",
      "units": "1",
      "description": "Density-dependent coefficient (Eq. 5.27, dimensionless)"
    },
    "c_2": {
      "type": "unknown",
      "units": "1",
      "description": "Liquid-dependent coefficient (Eq. 5.27, dimensionless)"
    },
    "C_R1": {
      "type": "unknown",
      "units": "s^-1",
      "description": "Destructive metamorphism compaction rate (Eq. 5.26)"
    },
    "eta": {
      "type": "unknown",
      "units": "kg*s/m^2",
      "description": "Viscosity coefficient (Eq. 5.29; η in upstream Julia source)"
    },
    "C_R2": {
      "type": "unknown",
      "units": "s^-1",
      "description": "Overburden compaction rate (Eq. 5.28)"
    },
    "C_R3": {
      "type": "unknown",
      "units": "s^-1",
      "description": "Melt compaction rate (Eq. 5.31)"
    },
    "C_R": {
      "type": "unknown",
      "units": "s^-1",
      "description": "Total fractional compaction rate (Eq. 5.24)"
    }
  },
  "equations": [
    {
      "lhs": "c_1",
      "rhs": {
        "op": "ifelse",
        "args": [
          {
            "op": ">",
            "args": [
              {
                "op": "/",
                "args": [
                  "w_ice",
                  "dz"
                ]
              },
              "hundred_kgpm3"
            ]
          },
          {
            "op": "exp",
            "args": [
              {
                "op": "*",
                "args": [
                  -1,
                  "coeff_0046",
                  {
                    "op": "-",
                    "args": [
                      {
                        "op": "/",
                        "args": [
                          "w_ice",
                          "dz"
                        ]
                      },
                      "hundred_kgpm3"
                    ]
                  }
                ]
              }
            ]
          },
          1.0
        ]
      }
    },
    {
      "lhs": "c_2",
      "rhs": {
        "op": "ifelse",
        "args": [
          {
            "op": ">",
            "args": [
              {
                "op": "/",
                "args": [
                  "w_liq",
                  "dz"
                ]
              },
              "thresh_liq_density"
            ]
          },
          2.0,
          1.0
        ]
      }
    },
    {
      "lhs": "C_R1",
      "rhs": {
        "op": "*",
        "args": [
          -1,
          "c_3",
          "c_1",
          "c_2",
          {
            "op": "exp",
            "args": [
              {
                "op": "*",
                "args": [
                  -1,
                  "c_4",
                  {
                    "op": "-",
                    "args": [
                      "T_f",
                      "T_i"
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "eta",
      "rhs": {
        "op": "*",
        "args": [
          "eta_0",
          {
            "op": "exp",
            "args": [
              {
                "op": "+",
                "args": [
                  {
                    "op": "*",
                    "args": [
                      "c_5",
                      {
                        "op": "-",
                        "args": [
                          "T_f",
                          "T_i"
                        ]
                      }
                    ]
                  },
                  {
                    "op": "*",
                    "args": [
                      "c_6",
                      {
                        "op": "/",
                        "args": [
                          "w_ice",
                          "dz"
                        ]
                      }
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "C_R2",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              -1,
              "P_s"
            ]
          },
          "eta"
        ]
      }
    },
    {
      "lhs": "C_R3",
      "rhs": {
        "op": "*",
        "args": [
          -1,
          "one_per_s",
          {
            "op": "max",
            "args": [
              0.0,
              {
                "op": "/",
                "args": [
                  {
                    "op": "-",
                    "args": [
                      "f_ice_n",
                      "f_ice_n1"
                    ]
                  },
                  {
                    "op": "max",
                    "args": [
                      "f_ice_n",
                      1e-10
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "C_R",
      "rhs": {
        "op": "+",
        "args": [
          "C_R1",
          "C_R2",
          "C_R3"
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_state",
      "description": "Default parameter values: cold (T_i=263.15 K), boundary-density snow (w_ice/dz = 100 kg/m³ — branch FALSE → c_1 = 1.0), wet (w_liq/dz = 2.0 kg/m³ > 0.01 → c_2 = 2.0), moderate overburden, no melt (f_ice_n = f_ice_n1 = 0.9 → C_R3 = 0).",
      "parameter_overrides": {
        "T_i": 263.15,
        "w_ice": 50.0,
        "w_liq": 1.0,
        "dz": 0.5,
        "P_s": 100.0,
        "f_ice_n": 0.9,
        "f_ice_n1": 0.9
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_1",
          "time": 0.0,
          "expected": 1.0
        },
        {
          "variable": "c_2",
          "time": 0.0,
          "expected": 2.0
        },
        {
          "variable": "C_R1",
          "time": 0.0,
          "expected": -3.7229575356819407e-06
        },
        {
          "variable": "eta",
          "time": 0.0,
          "expected": 19978156.15329746
        },
        {
          "variable": "C_R2",
          "time": 0.0,
          "expected": -5.005466932617536e-06
        },
        {
          "variable": "C_R3",
          "time": 0.0,
          "expected": 0.0
        },
        {
          "variable": "C_R",
          "time": 0.0,
          "expected": -8.728424468299477e-06
        }
      ]
    },
    {
      "id": "dense_snow_metamorphism",
      "description": "Higher-density snow (w_ice/dz = 200 kg/m³ > 100 → c_1 < 1 branch), dry layer (w_liq/dz < threshold → c_2 = 1), no overburden, no melt. Isolates the destructive-metamorphism path with the density-suppression factor.",
      "parameter_overrides": {
        "T_i": 263.15,
        "w_ice": 200.0,
        "w_liq": 0.001,
        "dz": 1.0,
        "P_s": 0.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_1",
          "time": 0.0,
          "expected": 0.010051835744633586
        },
        {
          "variable": "c_2",
          "time": 0.0,
          "expected": 1.0
        },
        {
          "variable": "C_R1",
          "time": 0.0,
          "expected": -1.871127881646035e-08
        },
        {
          "variable": "C_R2",
          "time": 0.0,
          "expected": 0.0
        },
        {
          "variable": "C_R3",
          "time": 0.0,
          "expected": 0.0
        }
      ]
    },
    {
      "id": "warm_low_density",
      "description": "Near-freezing layer (T_i = 272 K), low density (w_ice/dz = 80 kg/m³ < 100 → c_1 = 1), wet layer (c_2 = 2). Maximizes destructive metamorphism rate (T_i close to T_f).",
      "parameter_overrides": {
        "T_i": 272.0,
        "w_ice": 40.0,
        "w_liq": 5.0,
        "dz": 0.5,
        "P_s": 0.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_1",
          "time": 0.0,
          "expected": 1.0
        },
        {
          "variable": "c_2",
          "time": 0.0,
          "expected": 2.0
        },
        {
          "variable": "C_R1",
          "time": 0.0,
          "expected": -5.304303058007234e-06
        }
      ]
    },
    {
      "id": "overburden_dominated",
      "description": "Heavy overburden pressure (P_s = 5000 kg/m²) with cold dense snow. Eta is high (cold + dense), but P_s/eta still dominates the total rate.",
      "parameter_overrides": {
        "T_i": 250.0,
        "w_ice": 150.0,
        "w_liq": 0.0,
        "dz": 0.5,
        "P_s": 5000.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "eta",
          "time": 0.0,
          "expected": 5690989902.689874
        },
        {
          "variable": "C_R2",
          "time": 0.0,
          "expected": -8.785817732055239e-07
        },
        {
          "variable": "C_R",
          "time": 0.0,
          "expected": -8.786929233306654e-07
        }
      ]
    },
    {
      "id": "melt_compaction",
      "description": "Active melt: f_ice drops from 0.9 to 0.7 in one step → C_R3 = -1.0 * (0.2/0.9). C_R1 and C_R2 are smaller-magnitude contributions; melt dominates.",
      "parameter_overrides": {
        "T_i": 272.0,
        "w_ice": 50.0,
        "w_liq": 5.0,
        "dz": 0.5,
        "P_s": 100.0,
        "f_ice_n": 0.9,
        "f_ice_n1": 0.7
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "C_R3",
          "time": 0.0,
          "expected": -0.2222222222222223
        },
        {
          "variable": "C_R",
          "time": 0.0,
          "expected": -0.2222376872594633
        }
      ]
    },
    {
      "id": "no_melt_negative_diff",
      "description": "f_ice_n1 > f_ice_n (refreezing): the max(0, ...) clamps C_R3 to zero. Verifies the no-negative-melt-rate guard from Eq. 5.31.",
      "parameter_overrides": {
        "T_i": 263.15,
        "w_ice": 50.0,
        "w_liq": 1.0,
        "dz": 0.5,
        "P_s": 0.0,
        "f_ice_n": 0.7,
        "f_ice_n1": 0.9
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "C_R3",
          "time": 0.0,
          "expected": 0.0
        }
      ]
    },
    {
      "id": "very_cold",
      "description": "Very cold layer (T_i = 230 K, 43 K below T_f). Both metamorphism and viscosity-driven overburden compaction are strongly suppressed (exp(-0.04*43) ≈ 0.179; exp(0.08*43) ≈ 31).",
      "parameter_overrides": {
        "T_i": 230.0,
        "w_ice": 60.0,
        "w_liq": 0.001,
        "dz": 0.5,
        "P_s": 200.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "C_R1",
          "time": 0.0,
          "expected": -1.9698478402571534e-07
        },
        {
          "variable": "eta",
          "time": 0.0,
          "expected": 448827884.87023425
        },
        {
          "variable": "C_R2",
          "time": 0.0,
          "expected": -4.4560511220871287e-07
        }
      ]
    },
    {
      "id": "all_three_processes_active",
      "description": "All three compaction mechanisms simultaneously active: dense wet snow (c_1<1, c_2=2), nontrivial overburden, and active melt. Verifies C_R = C_R1 + C_R2 + C_R3 sum (Eq. 5.24).",
      "parameter_overrides": {
        "T_i": 270.0,
        "w_ice": 200.0,
        "w_liq": 10.0,
        "dz": 1.0,
        "P_s": 1000.0,
        "f_ice_n": 0.85,
        "f_ice_n1": 0.8
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_1",
          "time": 0.0,
          "expected": 0.010051835744633586
        },
        {
          "variable": "c_2",
          "time": 0.0,
          "expected": 2.0
        },
        {
          "variable": "C_R1",
          "time": 0.0,
          "expected": -4.921870173644911e-08
        },
        {
          "variable": "eta",
          "time": 0.0,
          "expected": 115196513.64921543
        },
        {
          "variable": "C_R2",
          "time": 0.0,
          "expected": -8.680818267166463e-06
        },
        {
          "variable": "C_R3",
          "time": 0.0,
          "expected": -0.05882352941176463
        },
        {
          "variable": "C_R",
          "time": 0.0,
          "expected": -0.058832259448733534
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "metamorphism_vs_temperature",
      "description": "Destructive-metamorphism compaction rate C_R1 as a function of snow layer temperature T_i, sweeping the cold-pack to near-freezing range (240–273 K). Holds the c_1 = 1 (w_ice/Δz ≤ 100 kg/m³) and c_2 = 2 (wet) branches of Eq. 5.27 by using the default w_ice/dz = 100 kg/m³ and w_liq/dz = 2 kg/m³. Overburden and melt contributions are zeroed (P_s = 0, f_ice_n = f_ice_n1 = 0.5) so the C_R1 curve stands alone. Surfaces the canonical Anderson-1976 exp(−c_4 (T_f − T_i)) temperature dependence of Eq. 5.26 — C_R1 grows by ≈ exp(c_4 × 33) ≈ 3.7× across the 240→273 K sweep.",
      "initial_state": {},
      "parameters": {
        "w_ice": 50.0,
        "w_liq": 1.0,
        "dz": 0.5,
        "P_s": 0.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "T_i",
            "range": {
              "start": 240.0,
              "stop": 273.0,
              "count": 34,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "C_R1_curve",
          "type": "line",
          "description": "C_R1 vs T_i: Anderson-1976 exponential ramp (Eq. 5.26) from cold-snow floor toward the near-freezing maximum.",
          "x": {
            "variable": "T_i",
            "label": "Snow layer temperature T_i (K)"
          },
          "y": {
            "variable": "C_R1",
            "label": "Destructive metamorphism rate C_R1 (s^-1)"
          }
        }
      ]
    },
    {
      "id": "viscosity_temperature_sweep",
      "description": "Snow viscosity coefficient η (Eq. 5.29) and the resulting overburden compaction rate C_R2 (Eq. 5.28) as functions of layer temperature, sweeping 240–273 K at a fixed mid-pack snow density (w_ice/Δz = 200 kg/m³) and fixed overburden P_s = 1000 kg/m². η decreases by ≈ exp(c_5 × 33) ≈ 14× as T_i warms, so |C_R2| = P_s / η grows by the same factor. Surfaces the strong temperature suppression of viscosity-controlled compaction in cold snowpacks. Melt is zeroed (f_ice_n = f_ice_n1 = 0.5) so C_R2 stands alone aside from a weak C_R1 contribution.",
      "initial_state": {},
      "parameters": {
        "w_ice": 100.0,
        "w_liq": 0.0,
        "dz": 0.5,
        "P_s": 1000.0,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "T_i",
            "range": {
              "start": 240.0,
              "stop": 273.0,
              "count": 34,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "eta_curve",
          "type": "line",
          "description": "η vs T_i: viscosity drops near-exponentially as the layer warms (Eq. 5.29) at fixed density 200 kg/m³.",
          "x": {
            "variable": "T_i",
            "label": "Snow layer temperature T_i (K)"
          },
          "y": {
            "variable": "eta",
            "label": "Viscosity coefficient η (kg·s/m^2)"
          }
        },
        {
          "id": "C_R2_curve",
          "type": "line",
          "description": "C_R2 vs T_i at P_s = 1000 kg/m²: −P_s/η warms toward larger magnitude as η falls.",
          "x": {
            "variable": "T_i",
            "label": "Snow layer temperature T_i (K)"
          },
          "y": {
            "variable": "C_R2",
            "label": "Overburden compaction rate C_R2 (s^-1)"
          }
        }
      ]
    },
    {
      "id": "overburden_load_sweep",
      "description": "Overburden compaction rate C_R2 as a function of snow load pressure P_s at a fixed cold (T_i = 250 K) mid-density (w_ice/Δz = 200 kg/m³) layer. With temperature and density fixed, η is constant, so C_R2 = −P_s/η is a unit-slope line through the origin (Eq. 5.28). Sweeps 0–5000 kg/m² — the upper end is representative of a deep snowpack column overburden. Surfaces the linear P_s dependence and the magnitude of overburden compaction relative to metamorphism. Melt zeroed (f_ice_n = f_ice_n1 = 0.5).",
      "initial_state": {},
      "parameters": {
        "T_i": 250.0,
        "w_ice": 100.0,
        "w_liq": 0.0,
        "dz": 0.5,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "P_s",
            "range": {
              "start": 0.0,
              "stop": 5000.0,
              "count": 21,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "C_R2_line",
          "type": "line",
          "description": "C_R2 vs P_s at fixed T_i = 250 K, ρ_snow = 200 kg/m³: linear −P_s/η behavior (Eq. 5.28).",
          "x": {
            "variable": "P_s",
            "label": "Snow load pressure P_s (kg/m^2)"
          },
          "y": {
            "variable": "C_R2",
            "label": "Overburden compaction rate C_R2 (s^-1)"
          }
        }
      ]
    },
    {
      "id": "melt_compaction_ramp",
      "description": "Melt-induced compaction rate C_R3 as a function of the post-melt ice fraction f_ice_n1, holding the pre-melt fraction f_ice_n = 0.9 fixed and sweeping f_ice_n1 across the full physical range [0, 1] in 21 points. For f_ice_n1 < f_ice_n, C_R3 = −(f_ice_n − f_ice_n1)/f_ice_n is a unit-slope ramp from −1.0 (complete ice loss in one step) to 0 (no melt). For f_ice_n1 ≥ f_ice_n (refreezing), the max(0, ·) clamp of Eq. 5.31 pins C_R3 = 0 — the no-negative-melt-rate guard. The two regimes meet at f_ice_n1 = 0.9 with a hinge in the curve. Metamorphism and overburden terms are zeroed at the canonical-cold defaults so C_R3 stands alone.",
      "initial_state": {},
      "parameters": {
        "T_i": 263.15,
        "w_ice": 50.0,
        "w_liq": 1.0,
        "dz": 0.5,
        "P_s": 0.0,
        "f_ice_n": 0.9
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "f_ice_n1",
            "range": {
              "start": 0.0,
              "stop": 1.0,
              "count": 21,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "C_R3_curve",
          "type": "line",
          "description": "C_R3 vs f_ice_n1 (with f_ice_n = 0.9): linear ramp from −1.0 to 0 over [0, 0.9], then clamped to 0 for f_ice_n1 ≥ 0.9 (Eq. 5.31 max(0, ·) guard).",
          "x": {
            "variable": "f_ice_n1",
            "label": "Post-step ice fraction f_ice_n1 (—)"
          },
          "y": {
            "variable": "C_R3",
            "label": "Melt compaction rate C_R3 (s^-1)"
          }
        }
      ]
    },
    {
      "id": "total_compaction_T_Ps_heatmap",
      "description": "Total fractional compaction rate C_R = C_R1 + C_R2 + C_R3 (Eq. 5.24) on the (T_i, P_s) plane at no-melt (f_ice_n = f_ice_n1 = 0.5) and fixed mid-pack density (w_ice/Δz = 200 kg/m³). 13×13 Cartesian sweep of T_i ∈ [240, 273] K against P_s ∈ [0, 5000] kg/m². Surfaces the joint behavior of the two non-melt mechanisms: at P_s = 0 the surface reduces to the C_R1(T_i) temperature curve; along the temperature-fixed axis the slope vs P_s is −1/η, with the slope steepening rapidly toward T_f as η drops. The result is a strongly bowed surface where the largest compaction magnitudes concentrate in the warm, heavily-loaded corner — the operational regime where overburden compaction dominates the snowpack mass-budget.",
      "initial_state": {},
      "parameters": {
        "w_ice": 100.0,
        "w_liq": 0.0,
        "dz": 0.5,
        "f_ice_n": 0.5,
        "f_ice_n1": 0.5
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "T_i",
            "range": {
              "start": 240.0,
              "stop": 273.0,
              "count": 13,
              "scale": "linear"
            }
          },
          {
            "parameter": "P_s",
            "range": {
              "start": 0.0,
              "stop": 5000.0,
              "count": 13,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "C_R_surface",
          "type": "heatmap",
          "description": "C_R(T_i, P_s) at fixed ρ_snow = 200 kg/m³, no melt: combined metamorphism + overburden surface; strongest compaction in the warm/heavy-load corner.",
          "x": {
            "variable": "T_i",
            "label": "Snow layer temperature T_i (K)"
          },
          "y": {
            "variable": "P_s",
            "label": "Snow load pressure P_s (kg/m^2)"
          },
          "value": {
            "variable": "C_R",
            "reduce": "final"
          }
        }
      ]
    }
  ]
}