Trivia: methods for molar and isotopic mixing ratio and molecular isotope ratio; new test for Mv

PySDM/physics/constants_defaults.py

@@ -775,7 +775,6 @@ def compute_derived_values(c: dict):
     water molar mass is computed from molecular masses and VSMOW isotope abundances
     (and neglecting molecular binding energies)
     for discussion, see:
-    - caption of Table 2.1 in [Gat 2010](https://doi.org/10.1142/p027)
     - [IAPWS Guidelines](http://www.iapws.org/relguide/fundam.pdf)
     """
 
@@ -788,22 +787,41 @@ def compute_derived_values(c: dict):
     c["Mv"] = (
         (
             1
-            - 2 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_2H"])
-            - 2 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_3H"])
-            - 1 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_17O"])
-            - 1 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_18O"])
+            - 2
+            * Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+                isotopic_ratio=c["VSMOW_R_2H"]
+            )
+            - 2
+            * Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+                isotopic_ratio=c["VSMOW_R_3H"]
+            )
+            - Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+                isotopic_ratio=c["VSMOW_R_17O"]
+            )
+            - Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+                isotopic_ratio=c["VSMOW_R_18O"]
+            )
         )
         * c["M_1H2_16O"]
         + 2
-        * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_2H"])
+        * Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+            isotopic_ratio=c["VSMOW_R_2H"]
+        )
         * c["M_2H_1H_16O"]
         + 2
-        * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_3H"])
+        * Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+            isotopic_ratio=c["VSMOW_R_3H"]
+        )
         * c["M_3H_1H_16O"]
-        + 1 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_17O"]) * c["M_1H2_17O"]
-        + 1 * Trivia.mixing_ratio_to_specific_content(c["VSMOW_R_18O"]) * c["M_1H2_18O"]
+        + Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+            isotopic_ratio=c["VSMOW_R_17O"]
+        )
+        * c["M_1H2_17O"]
+        + Trivia.isotopic_fraction_assuming_single_heavy_isotope(
+            isotopic_ratio=c["VSMOW_R_18O"]
+        )
+        * c["M_1H2_18O"]
     )
-
     c["eps"] = c["Mv"] / c["Md"]
     c["Rd"] = c["R_str"] / c["Md"]
     c["Rv"] = c["R_str"] / c["Mv"]

PySDM/physics/trivia.py

@@ -134,6 +134,18 @@ def isotopic_enrichment_to_delta_SMOW(E, delta_0_SMOW):
     def mixing_ratio_to_specific_content(mixing_ratio):
         return mixing_ratio / (1 + mixing_ratio)
 
+    @staticmethod
+    def isotopic_fraction_assuming_single_heavy_isotope(*, isotopic_ratio):
+        """
+        isotopic ratio = n1/n2
+        isotopic fraction = n1/n_total = n1 / (n1 + n2) = isotopic_ratio / (1 + isotopic_ratio)
+        """
+        return isotopic_ratio / (1 + isotopic_ratio)
+
+    @staticmethod
+    def isotopic_ratio_assuming_single_heavy_isotope(isotopic_fraction):
+        return isotopic_fraction / (1 - isotopic_fraction)
+
     @staticmethod
     def dn_dlogr(r, dn_dr):
         return np.log(10) * r * dn_dr
@@ -172,12 +184,12 @@ def tau(Bo, dm_dt_over_m):
     @staticmethod
     def moles_heavy_atom(
         *,
+        atoms_per_heavy_molecule,
         mass_total,
         mass_other_heavy_isotopes,
         molar_mass_light_molecule,
         molar_mass_heavy_molecule,
-        molecular_isotope_ratio,
-        atoms_per_heavy_molecule,
+        molecular_isotopic_ratio,
     ):
         """
         Calculate moles of heavy atoms (e.g. deuterium, oxygen-17, oxygen-18)
@@ -187,8 +199,64 @@ def moles_heavy_atom(
         return (
             (mass_total - mass_other_heavy_isotopes)
             / (
-                molar_mass_light_molecule / molecular_isotope_ratio
+                molar_mass_light_molecule / molecular_isotopic_ratio
                 + molar_mass_heavy_molecule
             )
             / atoms_per_heavy_molecule
         )
+
+    @staticmethod
+    def molecular_isotopic_ratio(
+        *,
+        moles_heavy_molecule,
+        mass_total,
+        mass_other_heavy_isotopes,
+        molar_mass_light_molecule,
+        molar_mass_heavy_molecule,
+    ):
+        """
+        Molecular isotope ratio (e.g. moles of HDO to moles of H2O):
+            n_{heavy molecule}/n_{light molecule},
+        in contrast to (atomic) isotopic ratio as in VSMOW standard:
+            n_{heavy atom}/n_{light atom};
+
+        calculated with:
+            m_light = m_total - m_considered_heavy_isotope - m_other_heavy_isotopes
+        """
+        return (
+            moles_heavy_molecule
+            * molar_mass_light_molecule
+            / (
+                mass_total
+                - moles_heavy_molecule * molar_mass_heavy_molecule
+                - mass_other_heavy_isotopes
+            )
+        )
+
+    @staticmethod
+    def molality_in_dry_air(
+        isotopic_fraction, density_dry_air, total_vap_concentration
+    ):
+        """
+        n'/m_d [number of moles of isotopic molecules]/[mass of dry air] - molality in dry air
+        n/V - total vapour concentration
+
+        molality_in_dry_air = n'/m_d =
+            = n'/(rho_d * V) = isotopic_concentration / rho_d
+        isotopic_concentration = n'/V = n/V * n'/n =
+            = total vapour concentration * isotopic_fraction
+
+        where n'/n is an isotopic fraction (referred to as isotopic concentration (C)
+        on p. 10 in [Gat 2010](https://doi.org/10.1142/p027)
+        """
+        return total_vap_concentration * isotopic_fraction / density_dry_air
+
+    @staticmethod
+    def isotopic_fraction(
+        molality_in_dry_air, density_dry_air, total_vap_concentration
+    ):
+        """
+        isotopic_fraction: n'/n = n'/m_d / (n/V) * rho_d
+            = molality in dry air / total vapour concentration * dry air density
+        """
+        return molality_in_dry_air / total_vap_concentration * density_dry_air

docs/bibliography.json

@@ -382,7 +382,7 @@
   },
   "https://doi.org/10.1142/p027": {
     "usages": [
-      "PySDM/physics/constants_defaults.py"
+      "PySDM/physics/trivia.py"
     ],
     "label": "Gat 2010",
     "title": "Isotope Hydrology: A Study of the Water Cycle"
@@ -1004,5 +1004,12 @@
     ],
     "label": "Bartman 2020 (MSc thesis)",
     "title": "PySDM v1.0: Pythonic particle-based cloud microphysics package"
+  },
+    "https://web.archive.org/web/20220629123450/https://web.gps.caltech.edu/~als/research-articles/other_stuff/hayes-2004-3.pdf" : {
+    "usages": [
+      "tests/unit_tests/physics/test_constants.py"
+    ],
+    "label": "Hayes 2004",
+    "title": "An Introduction to Isotopic Calculations"
   }
 }

docs/generate_html.py

@@ -68,7 +68,7 @@ def check_urls(urls_from_json):
                 r"\b(https://digitallibrary\.un\.org/record/(?:[0-9])+)\b",
                 r"\b(http://mi\.mathnet\.ru/dan(?:[0-9])+)\b",
                 r"\b(https://archive\.org/details/(?:[0-9a-z_\.-])+)\b",
-                r"\b(https://web\.archive\.org/web/(?:[0-9])+/https://(?:[0-9a-zA-Z_\.-/])+)\b",
+                r"\b(https://web\.archive\.org/web/(?:[0-9])+/https://(?:[0-9a-zA-Z_\.\-/~%])+)\b",
                 r"\b(https://www\.ap\.uj\.edu\.pl/diplomas/attachments/file/download/(?:[0-9])+)\b",
             ):
                 urls = re.findall(pattern, text)