[5841] | 1 | !> \file mocsy_phsolvers.f90 |
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| 2 | !! \BRIEF |
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| 3 | !> Module with routines needed to solve pH-total alkalinity equation (Munhoven, 2013, GMD) |
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| 4 | MODULE mocsy_phsolvers |
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| 5 | ! Module of fastest solvers from Munhoven (2013, Geosci. Model Dev., 6, 1367-1388) |
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| 6 | ! ! Taken from SolveSAPHE (mod_phsolvers.F90) & adapted very slightly for use with mocsy |
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| 7 | ! ! SolveSaphe is distributed under the GNU Lesser General Public License |
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| 8 | ! ! mocsy is distributed under the MIT License |
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| 9 | ! |
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| 10 | ! Modifications J. C. Orr, LSCE/IPSL, CEA-CNRS-UVSQ, France, 11 Sep 2014: |
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| 11 | ! 1) kept only the 3 fastest solvers (atgen, atsec, atfast) and routines which they call |
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| 12 | ! 2) reduced vertical white space: deleted many blank lines & comment lines that served as divisions |
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| 13 | ! 3) converted name from .F90 to .f90, deleting a few optional preprocesse if statements |
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| 14 | ! 4) read in mocsy computed equilibrium constants (as arguments) instead of USE MOD_CHEMCONST |
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| 15 | ! 5) converted routine names from upper case to lower case |
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| 16 | ! 6) commented out arguments and equations for NH4 and H2S acid systems |
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| 17 | |
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| 18 | USE mocsy_singledouble |
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| 19 | IMPLICIT NONE |
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| 20 | |
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| 21 | ! General parameters |
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| 22 | REAL(KIND=wp), PARAMETER :: pp_rdel_ah_target = 1.E-8_wp |
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| 23 | REAL(KIND=wp), PARAMETER :: pp_ln10 = 2.302585092994045684018_wp |
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| 24 | |
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| 25 | ! Maximum number of iterations for each method |
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| 26 | INTEGER, PARAMETER :: jp_maxniter_atgen = 50 |
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| 27 | INTEGER, PARAMETER :: jp_maxniter_atsec = 50 |
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| 28 | INTEGER, PARAMETER :: jp_maxniter_atfast = 50 |
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| 29 | |
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| 30 | ! Bookkeeping variables for each method |
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| 31 | ! - SOLVE_AT_GENERAL |
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| 32 | INTEGER :: niter_atgen = jp_maxniter_atgen |
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| 33 | |
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| 34 | ! - SOLVE_AT_GENERAL_SEC |
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| 35 | INTEGER :: niter_atsec = jp_maxniter_atsec |
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| 36 | |
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| 37 | ! - SOLVE_AT_FAST (variant of SOLVE_AT_GENERAL w/o bracketing |
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| 38 | INTEGER :: niter_atfast = jp_maxniter_atfast |
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| 39 | |
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| 40 | ! Keep the following functions private to avoid conflicts with |
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| 41 | ! other modules that provide similar ones. |
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| 42 | !PRIVATE AHINI_FOR_AT |
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| 43 | |
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| 44 | CONTAINS |
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| 45 | !=============================================================================== |
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| 46 | SUBROUTINE anw_infsup(p_dictot, p_bortot, & |
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| 47 | p_po4tot, p_siltot, & |
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| 48 | p_so4tot, p_flutot, & |
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| 49 | p_alknw_inf, p_alknw_sup) |
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| 50 | |
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| 51 | ! Subroutine returns the lower and upper bounds of "non-water-selfionization" |
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| 52 | ! contributions to total alkalinity (the infimum and the supremum), i.e |
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| 53 | ! inf(TA - [OH-] + [H+]) and sup(TA - [OH-] + [H+]) |
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| 54 | |
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| 55 | USE mocsy_singledouble |
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| 56 | IMPLICIT NONE |
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| 57 | |
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| 58 | ! Argument variables |
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| 59 | REAL(KIND=wp), INTENT(IN) :: p_dictot |
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| 60 | REAL(KIND=wp), INTENT(IN) :: p_bortot |
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| 61 | REAL(KIND=wp), INTENT(IN) :: p_po4tot |
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| 62 | REAL(KIND=wp), INTENT(IN) :: p_siltot |
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| 63 | !REAL(KIND=wp), INTENT(IN) :: p_nh4tot |
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| 64 | !REAL(KIND=wp), INTENT(IN) :: p_h2stot |
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| 65 | REAL(KIND=wp), INTENT(IN) :: p_so4tot |
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| 66 | REAL(KIND=wp), INTENT(IN) :: p_flutot |
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| 67 | REAL(KIND=wp), INTENT(OUT) :: p_alknw_inf |
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| 68 | REAL(KIND=wp), INTENT(OUT) :: p_alknw_sup |
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| 69 | |
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| 70 | p_alknw_inf = -p_po4tot - p_so4tot - p_flutot |
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| 71 | p_alknw_sup = p_dictot + p_dictot + p_bortot & |
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| 72 | + p_po4tot + p_po4tot + p_siltot !& |
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| 73 | ! + p_nh4tot + p_h2stot |
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| 74 | |
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| 75 | RETURN |
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| 76 | END SUBROUTINE anw_infsup |
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| 77 | |
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| 78 | !=============================================================================== |
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| 79 | |
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| 80 | FUNCTION equation_at(p_alktot, p_h, p_dictot, p_bortot, & |
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| 81 | p_po4tot, p_siltot, & |
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| 82 | p_so4tot, p_flutot, & |
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| 83 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
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| 84 | p_deriveqn) |
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| 85 | |
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| 86 | USE mocsy_singledouble |
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| 87 | IMPLICIT NONE |
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| 88 | REAL(KIND=wp) :: equation_at |
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| 89 | |
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| 90 | ! Argument variables |
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| 91 | REAL(KIND=wp), INTENT(IN) :: p_alktot |
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| 92 | REAL(KIND=wp), INTENT(IN) :: p_h |
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| 93 | REAL(KIND=wp), INTENT(IN) :: p_dictot |
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| 94 | REAL(KIND=wp), INTENT(IN) :: p_bortot |
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| 95 | REAL(KIND=wp), INTENT(IN) :: p_po4tot |
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| 96 | REAL(KIND=wp), INTENT(IN) :: p_siltot |
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| 97 | !REAL(KIND=wp), INTENT(IN) :: p_nh4tot |
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| 98 | !REAL(KIND=wp), INTENT(IN) :: p_h2stot |
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| 99 | REAL(KIND=wp), INTENT(IN) :: p_so4tot |
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| 100 | REAL(KIND=wp), INTENT(IN) :: p_flutot |
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| 101 | REAL(KIND=wp), INTENT(IN) :: K0, K1, K2, Kb, Kw, Ks, Kf |
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| 102 | REAL(KIND=wp), INTENT(IN) :: K1p, K2p, K3p, Ksi |
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| 103 | REAL(KIND=wp), INTENT(OUT), OPTIONAL :: p_deriveqn |
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| 104 | |
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| 105 | ! Local variables |
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| 106 | !----------------- |
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| 107 | REAL(KIND=wp) :: znumer_dic, zdnumer_dic, zdenom_dic, zalk_dic, zdalk_dic |
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| 108 | REAL(KIND=wp) :: znumer_bor, zdnumer_bor, zdenom_bor, zalk_bor, zdalk_bor |
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| 109 | REAL(KIND=wp) :: znumer_po4, zdnumer_po4, zdenom_po4, zalk_po4, zdalk_po4 |
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| 110 | REAL(KIND=wp) :: znumer_sil, zdnumer_sil, zdenom_sil, zalk_sil, zdalk_sil |
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| 111 | REAL(KIND=wp) :: znumer_nh4, zdnumer_nh4, zdenom_nh4, zalk_nh4, zdalk_nh4 |
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| 112 | REAL(KIND=wp) :: znumer_h2s, zdnumer_h2s, zdenom_h2s, zalk_h2s, zdalk_h2s |
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| 113 | REAL(KIND=wp) :: znumer_so4, zdnumer_so4, zdenom_so4, zalk_so4, zdalk_so4 |
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| 114 | REAL(KIND=wp) :: znumer_flu, zdnumer_flu, zdenom_flu, zalk_flu, zdalk_flu |
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| 115 | REAL(KIND=wp) :: zalk_wat, zdalk_wat |
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| 116 | REAL(KIND=wp) :: aphscale |
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| 117 | |
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| 118 | ! TOTAL H+ scale: conversion factor for Htot = aphscale * Hfree |
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| 119 | aphscale = 1._wp + p_so4tot/Ks |
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| 120 | |
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| 121 | ! H2CO3 - HCO3 - CO3 : n=2, m=0 |
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| 122 | znumer_dic = 2._wp*K1*K2 + p_h* K1 |
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| 123 | zdenom_dic = K1*K2 + p_h*( K1 + p_h) |
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| 124 | zalk_dic = p_dictot * (znumer_dic/zdenom_dic) |
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| 125 | |
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| 126 | ! B(OH)3 - B(OH)4 : n=1, m=0 |
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| 127 | znumer_bor = Kb |
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| 128 | zdenom_bor = Kb + p_h |
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| 129 | zalk_bor = p_bortot * (znumer_bor/zdenom_bor) |
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| 130 | |
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| 131 | ! H3PO4 - H2PO4 - HPO4 - PO4 : n=3, m=1 |
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| 132 | znumer_po4 = 3._wp*K1p*K2p*K3p + p_h*(2._wp*K1p*K2p + p_h* K1p) |
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| 133 | zdenom_po4 = K1p*K2p*K3p + p_h*( K1p*K2p + p_h*(K1p + p_h)) |
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| 134 | zalk_po4 = p_po4tot * (znumer_po4/zdenom_po4 - 1._wp) ! Zero level of H3PO4 = 1 |
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| 135 | |
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| 136 | ! H4SiO4 - H3SiO4 : n=1, m=0 |
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| 137 | znumer_sil = Ksi |
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| 138 | zdenom_sil = Ksi + p_h |
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| 139 | zalk_sil = p_siltot * (znumer_sil/zdenom_sil) |
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| 140 | |
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| 141 | ! NH4 - NH3 : n=1, m=0 |
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| 142 | !znumer_nh4 = api1_nh4 |
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| 143 | !zdenom_nh4 = api1_nh4 + p_h |
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| 144 | !zalk_nh4 = p_nh4tot * (znumer_nh4/zdenom_nh4) |
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| 145 | ! Note: api1_nh4 = Knh4 |
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| 146 | |
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| 147 | ! H2S - HS : n=1, m=0 |
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| 148 | !znumer_h2s = api1_h2s |
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| 149 | !zdenom_h2s = api1_h2s + p_h |
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| 150 | !zalk_h2s = p_h2stot * (znumer_h2s/zdenom_h2s) |
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| 151 | ! Note: api1_h2s = Kh2s |
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| 152 | |
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| 153 | ! HSO4 - SO4 : n=1, m=1 |
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| 154 | znumer_so4 = Ks |
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| 155 | zdenom_so4 = Ks + p_h |
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| 156 | zalk_so4 = p_so4tot * (znumer_so4/zdenom_so4 - 1._wp) |
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| 157 | |
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| 158 | ! HF - F : n=1, m=1 |
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| 159 | znumer_flu = Kf |
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| 160 | zdenom_flu = Kf + p_h |
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| 161 | zalk_flu = p_flutot * (znumer_flu/zdenom_flu - 1._wp) |
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| 162 | |
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| 163 | ! H2O - OH |
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| 164 | zalk_wat = Kw/p_h - p_h/aphscale |
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| 165 | |
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| 166 | equation_at = zalk_dic + zalk_bor + zalk_po4 + zalk_sil & |
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| 167 | + zalk_so4 + zalk_flu & |
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| 168 | + zalk_wat - p_alktot |
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| 169 | |
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| 170 | IF(PRESENT(p_deriveqn)) THEN |
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| 171 | ! H2CO3 - HCO3 - CO3 : n=2 |
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| 172 | zdnumer_dic = K1*K1*K2 + p_h*(4._wp*K1*K2 & |
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| 173 | + p_h* K1 ) |
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| 174 | zdalk_dic = -p_dictot*(zdnumer_dic/zdenom_dic**2) |
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| 175 | |
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| 176 | ! B(OH)3 - B(OH)4 : n=1 |
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| 177 | zdnumer_bor = Kb |
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| 178 | zdalk_bor = -p_bortot*(zdnumer_bor/zdenom_bor**2) |
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| 179 | |
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| 180 | ! H3PO4 - H2PO4 - HPO4 - PO4 : n=3 |
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| 181 | zdnumer_po4 = K1p*K2p*K1p*K2p*K3p + p_h*(4._wp*K1p*K1p*K2p*K3p & |
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| 182 | + p_h*(9._wp*K1p*K2p*K3p + K1p*K1p*K2p & |
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| 183 | + p_h*(4._wp*K1p*K2p & |
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| 184 | + p_h* K1p))) |
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| 185 | zdalk_po4 = -p_po4tot * (zdnumer_po4/zdenom_po4**2) |
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| 186 | |
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| 187 | ! H4SiO4 - H3SiO4 : n=1 |
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| 188 | zdnumer_sil = Ksi |
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| 189 | zdalk_sil = -p_siltot * (zdnumer_sil/zdenom_sil**2) |
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| 190 | |
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| 191 | ! ! NH4 - NH3 : n=1 |
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| 192 | ! zdnumer_nh4 = Knh4 |
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| 193 | ! zdalk_nh4 = -p_nh4tot * (zdnumer_nh4/zdenom_nh4**2) |
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| 194 | |
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| 195 | ! ! H2S - HS : n=1 |
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| 196 | ! zdnumer_h2s = api1_h2s |
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| 197 | ! zdalk_h2s = -p_h2stot * (zdnumer_h2s/zdenom_h2s**2) |
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| 198 | |
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| 199 | ! HSO4 - SO4 : n=1 |
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| 200 | zdnumer_so4 = Ks |
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| 201 | zdalk_so4 = -p_so4tot * (zdnumer_so4/zdenom_so4**2) |
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| 202 | |
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| 203 | ! HF - F : n=1 |
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| 204 | zdnumer_flu = Kf |
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| 205 | zdalk_flu = -p_flutot * (zdnumer_flu/zdenom_flu**2) |
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| 206 | |
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| 207 | ! p_deriveqn = zdalk_dic + zdalk_bor + zdalk_po4 + zdalk_sil & |
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| 208 | ! + zdalk_nh4 + zdalk_h2s + zdalk_so4 + zdalk_flu & |
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| 209 | ! - Kw/p_h**2 - 1._wp/aphscale |
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| 210 | p_deriveqn = zdalk_dic + zdalk_bor + zdalk_po4 + zdalk_sil & |
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| 211 | + zdalk_so4 + zdalk_flu & |
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| 212 | - Kw/p_h**2 - 1._wp/aphscale |
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| 213 | ENDIF |
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| 214 | RETURN |
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| 215 | END FUNCTION equation_at |
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| 216 | |
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| 217 | !=============================================================================== |
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| 218 | |
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| 219 | SUBROUTINE ahini_for_at(p_alkcb, p_dictot, p_bortot, K1, K2, Kb, p_hini) |
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| 220 | |
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| 221 | ! Subroutine returns the root for the 2nd order approximation of the |
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| 222 | ! DIC -- B_T -- A_CB equation for [H+] (reformulated as a cubic polynomial) |
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| 223 | ! around the local minimum, if it exists. |
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| 224 | |
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| 225 | ! Returns * 1E-03_wp if p_alkcb <= 0 |
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| 226 | ! * 1E-10_wp if p_alkcb >= 2*p_dictot + p_bortot |
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| 227 | ! * 1E-07_wp if 0 < p_alkcb < 2*p_dictot + p_bortot |
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| 228 | ! and the 2nd order approximation does not have a solution |
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| 229 | |
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| 230 | !USE MOD_CHEMCONST, ONLY : api1_dic, api2_dic, api1_bor |
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| 231 | |
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| 232 | USE mocsy_singledouble |
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| 233 | IMPLICIT NONE |
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| 234 | |
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| 235 | ! Argument variables |
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| 236 | !-------------------- |
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| 237 | REAL(KIND=wp), INTENT(IN) :: p_alkcb, p_dictot, p_bortot |
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| 238 | REAL(KIND=wp), INTENT(IN) :: K1, K2, Kb |
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| 239 | REAL(KIND=wp), INTENT(OUT) :: p_hini |
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| 240 | |
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| 241 | ! Local variables |
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| 242 | !----------------- |
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| 243 | REAL(KIND=wp) :: zca, zba |
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| 244 | REAL(KIND=wp) :: zd, zsqrtd, zhmin |
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| 245 | REAL(KIND=wp) :: za2, za1, za0 |
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| 246 | |
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| 247 | IF (p_alkcb <= 0._wp) THEN |
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| 248 | p_hini = 1.e-3_wp |
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| 249 | ELSEIF (p_alkcb >= (2._wp*p_dictot + p_bortot)) THEN |
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| 250 | p_hini = 1.e-10_wp |
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| 251 | ELSE |
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| 252 | zca = p_dictot/p_alkcb |
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| 253 | zba = p_bortot/p_alkcb |
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| 254 | |
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| 255 | ! Coefficients of the cubic polynomial |
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| 256 | za2 = Kb*(1._wp - zba) + K1*(1._wp-zca) |
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| 257 | za1 = K1*Kb*(1._wp - zba - zca) + K1*K2*(1._wp - (zca+zca)) |
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| 258 | za0 = K1*K2*Kb*(1._wp - zba - (zca+zca)) |
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| 259 | ! Taylor expansion around the minimum |
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| 260 | zd = za2*za2 - 3._wp*za1 ! Discriminant of the quadratic equation |
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| 261 | ! for the minimum close to the root |
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| 262 | |
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| 263 | IF(zd > 0._wp) THEN ! If the discriminant is positive |
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| 264 | zsqrtd = SQRT(zd) |
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| 265 | IF(za2 < 0) THEN |
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| 266 | zhmin = (-za2 + zsqrtd)/3._wp |
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| 267 | ELSE |
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| 268 | zhmin = -za1/(za2 + zsqrtd) |
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| 269 | ENDIF |
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| 270 | p_hini = zhmin + SQRT(-(za0 + zhmin*(za1 + zhmin*(za2 + zhmin)))/zsqrtd) |
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| 271 | ELSE |
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| 272 | p_hini = 1.e-7_wp |
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| 273 | ENDIF |
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| 274 | |
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| 275 | ENDIF |
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| 276 | RETURN |
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| 277 | END SUBROUTINE ahini_for_at |
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| 278 | |
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| 279 | !=============================================================================== |
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| 280 | |
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| 281 | FUNCTION solve_at_general(p_alktot, p_dictot, p_bortot, & |
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| 282 | p_po4tot, p_siltot, & |
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| 283 | p_so4tot, p_flutot, & |
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| 284 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
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| 285 | p_hini, p_val) |
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| 286 | |
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| 287 | ! Universal pH solver that converges from any given initial value, |
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| 288 | ! determines upper an lower bounds for the solution if required |
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| 289 | |
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| 290 | USE mocsy_singledouble |
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| 291 | IMPLICIT NONE |
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| 292 | REAL(KIND=wp) :: SOLVE_AT_GENERAL |
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| 293 | |
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| 294 | ! Argument variables |
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| 295 | !-------------------- |
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| 296 | REAL(KIND=wp), INTENT(IN) :: p_alktot |
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| 297 | REAL(KIND=wp), INTENT(IN) :: p_dictot |
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| 298 | REAL(KIND=wp), INTENT(IN) :: p_bortot |
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| 299 | REAL(KIND=wp), INTENT(IN) :: p_po4tot |
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| 300 | REAL(KIND=wp), INTENT(IN) :: p_siltot |
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| 301 | !REAL(KIND=wp), INTENT(IN) :: p_nh4tot |
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| 302 | !REAL(KIND=wp), INTENT(IN) :: p_h2stot |
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| 303 | REAL(KIND=wp), INTENT(IN) :: p_so4tot |
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| 304 | REAL(KIND=wp), INTENT(IN) :: p_flutot |
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| 305 | REAL(KIND=wp), INTENT(IN) :: K0, K1, K2, Kb, Kw, Ks, Kf |
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| 306 | REAL(KIND=wp), INTENT(IN) :: K1p, K2p, K3p, Ksi |
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| 307 | REAL(KIND=wp), INTENT(IN), OPTIONAL :: p_hini |
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| 308 | REAL(KIND=wp), INTENT(OUT), OPTIONAL :: p_val |
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| 309 | |
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| 310 | ! Local variables |
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| 311 | !----------------- |
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| 312 | REAL(KIND=wp) :: zh_ini, zh, zh_prev, zh_lnfactor |
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| 313 | REAL(KIND=wp) :: zalknw_inf, zalknw_sup |
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| 314 | REAL(KIND=wp) :: zh_min, zh_max |
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| 315 | REAL(KIND=wp) :: zdelta, zh_delta |
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| 316 | REAL(KIND=wp) :: zeqn, zdeqndh, zeqn_absmin |
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| 317 | REAL(KIND=wp) :: aphscale |
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| 318 | LOGICAL :: l_exitnow |
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| 319 | REAL(KIND=wp), PARAMETER :: pz_exp_threshold = 1.0_wp |
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| 320 | |
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| 321 | ! TOTAL H+ scale: conversion factor for Htot = aphscale * Hfree |
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| 322 | aphscale = 1._wp + p_so4tot/Ks |
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| 323 | |
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| 324 | IF(PRESENT(p_hini)) THEN |
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| 325 | zh_ini = p_hini |
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| 326 | ELSE |
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| 327 | CALL ahini_for_at(p_alktot, p_dictot, p_bortot, K1, K2, Kb, zh_ini) |
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| 328 | ENDIF |
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| 329 | |
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| 330 | CALL anw_infsup(p_dictot, p_bortot, & |
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| 331 | p_po4tot, p_siltot, & |
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| 332 | p_so4tot, p_flutot, & |
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| 333 | zalknw_inf, zalknw_sup) |
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| 334 | |
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| 335 | zdelta = (p_alktot-zalknw_inf)**2 + 4._wp*Kw/aphscale |
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| 336 | |
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| 337 | IF(p_alktot >= zalknw_inf) THEN |
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| 338 | zh_min = 2._wp*Kw /( p_alktot-zalknw_inf + SQRT(zdelta) ) |
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| 339 | ELSE |
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| 340 | zh_min = aphscale*(-(p_alktot-zalknw_inf) + SQRT(zdelta) ) / 2._wp |
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| 341 | ENDIF |
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| 342 | |
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| 343 | zdelta = (p_alktot-zalknw_sup)**2 + 4._wp*Kw/aphscale |
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| 344 | |
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| 345 | IF(p_alktot <= zalknw_sup) THEN |
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| 346 | zh_max = aphscale*(-(p_alktot-zalknw_sup) + SQRT(zdelta) ) / 2._wp |
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| 347 | ELSE |
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| 348 | zh_max = 2._wp*Kw /( p_alktot-zalknw_sup + SQRT(zdelta) ) |
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| 349 | ENDIF |
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| 350 | |
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| 351 | zh = MAX(MIN(zh_max, zh_ini), zh_min) |
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| 352 | niter_atgen = 0 ! Reset counters of iterations |
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| 353 | zeqn_absmin = HUGE(1._wp) |
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| 354 | |
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| 355 | DO |
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| 356 | IF(niter_atgen >= jp_maxniter_atgen) THEN |
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| 357 | zh = -1._wp |
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| 358 | EXIT |
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| 359 | ENDIF |
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| 360 | |
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| 361 | zh_prev = zh |
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| 362 | zeqn = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 363 | p_po4tot, p_siltot, & |
---|
| 364 | p_so4tot, p_flutot, & |
---|
| 365 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
---|
| 366 | P_DERIVEQN = zdeqndh) |
---|
| 367 | |
---|
| 368 | ! Adapt bracketing interval |
---|
| 369 | IF(zeqn > 0._wp) THEN |
---|
| 370 | zh_min = zh_prev |
---|
| 371 | ELSEIF(zeqn < 0._wp) THEN |
---|
| 372 | zh_max = zh_prev |
---|
| 373 | ELSE |
---|
| 374 | ! zh is the root; unlikely but, one never knows |
---|
| 375 | EXIT |
---|
| 376 | ENDIF |
---|
| 377 | |
---|
| 378 | ! Now determine the next iterate zh |
---|
| 379 | niter_atgen = niter_atgen + 1 |
---|
| 380 | |
---|
| 381 | IF(ABS(zeqn) >= 0.5_wp*zeqn_absmin) THEN |
---|
| 382 | ! if the function evaluation at the current point is |
---|
| 383 | ! not decreasing faster than with a bisection step (at least linearly) |
---|
| 384 | ! in absolute value take one bisection step on [ph_min, ph_max] |
---|
| 385 | ! ph_new = (ph_min + ph_max)/2d0 |
---|
| 386 | ! |
---|
| 387 | ! In terms of [H]_new: |
---|
| 388 | ! [H]_new = 10**(-ph_new) |
---|
| 389 | ! = 10**(-(ph_min + ph_max)/2d0) |
---|
| 390 | ! = SQRT(10**(-(ph_min + phmax))) |
---|
| 391 | ! = SQRT(zh_max * zh_min) |
---|
| 392 | zh = SQRT(zh_max * zh_min) |
---|
| 393 | zh_lnfactor = (zh - zh_prev)/zh_prev ! Required to test convergence below |
---|
| 394 | ELSE |
---|
| 395 | ! dzeqn/dpH = dzeqn/d[H] * d[H]/dpH |
---|
| 396 | ! = -zdeqndh * LOG(10) * [H] |
---|
| 397 | ! \Delta pH = -zeqn/(zdeqndh*d[H]/dpH) = zeqn/(zdeqndh*[H]*LOG(10)) |
---|
| 398 | ! |
---|
| 399 | ! pH_new = pH_old + \deltapH |
---|
| 400 | ! |
---|
| 401 | ! [H]_new = 10**(-pH_new) |
---|
| 402 | ! = 10**(-pH_old - \Delta pH) |
---|
| 403 | ! = [H]_old * 10**(-zeqn/(zdeqndh*[H]_old*LOG(10))) |
---|
| 404 | ! = [H]_old * EXP(-LOG(10)*zeqn/(zdeqndh*[H]_old*LOG(10))) |
---|
| 405 | ! = [H]_old * EXP(-zeqn/(zdeqndh*[H]_old)) |
---|
| 406 | |
---|
| 407 | zh_lnfactor = -zeqn/(zdeqndh*zh_prev) |
---|
| 408 | |
---|
| 409 | IF(ABS(zh_lnfactor) > pz_exp_threshold) THEN |
---|
| 410 | zh = zh_prev*EXP(zh_lnfactor) |
---|
| 411 | ELSE |
---|
| 412 | zh_delta = zh_lnfactor*zh_prev |
---|
| 413 | zh = zh_prev + zh_delta |
---|
| 414 | ENDIF |
---|
| 415 | |
---|
| 416 | IF( zh < zh_min ) THEN |
---|
| 417 | ! if [H]_new < [H]_min |
---|
| 418 | ! i.e., if ph_new > ph_max then |
---|
| 419 | ! take one bisection step on [ph_prev, ph_max] |
---|
| 420 | ! ph_new = (ph_prev + ph_max)/2d0 |
---|
| 421 | ! In terms of [H]_new: |
---|
| 422 | ! [H]_new = 10**(-ph_new) |
---|
| 423 | ! = 10**(-(ph_prev + ph_max)/2d0) |
---|
| 424 | ! = SQRT(10**(-(ph_prev + phmax))) |
---|
| 425 | ! = SQRT([H]_old*10**(-ph_max)) |
---|
| 426 | ! = SQRT([H]_old * zh_min) |
---|
| 427 | zh = SQRT(zh_prev * zh_min) |
---|
| 428 | zh_lnfactor = (zh - zh_prev)/zh_prev ! Required to test convergence below |
---|
| 429 | ENDIF |
---|
| 430 | |
---|
| 431 | IF( zh > zh_max ) THEN |
---|
| 432 | ! if [H]_new > [H]_max |
---|
| 433 | ! i.e., if ph_new < ph_min, then |
---|
| 434 | ! take one bisection step on [ph_min, ph_prev] |
---|
| 435 | ! ph_new = (ph_prev + ph_min)/2d0 |
---|
| 436 | ! In terms of [H]_new: |
---|
| 437 | ! [H]_new = 10**(-ph_new) |
---|
| 438 | ! = 10**(-(ph_prev + ph_min)/2d0) |
---|
| 439 | ! = SQRT(10**(-(ph_prev + ph_min))) |
---|
| 440 | ! = SQRT([H]_old*10**(-ph_min)) |
---|
| 441 | ! = SQRT([H]_old * zhmax) |
---|
| 442 | zh = SQRT(zh_prev * zh_max) |
---|
| 443 | zh_lnfactor = (zh - zh_prev)/zh_prev ! Required to test convergence below |
---|
| 444 | ENDIF |
---|
| 445 | ENDIF |
---|
| 446 | |
---|
| 447 | zeqn_absmin = MIN( ABS(zeqn), zeqn_absmin) |
---|
| 448 | |
---|
| 449 | ! Stop iterations once |\delta{[H]}/[H]| < rdel |
---|
| 450 | ! <=> |(zh - zh_prev)/zh_prev| = |EXP(-zeqn/(zdeqndh*zh_prev)) -1| < rdel |
---|
| 451 | ! |EXP(-zeqn/(zdeqndh*zh_prev)) -1| ~ |zeqn/(zdeqndh*zh_prev)| |
---|
| 452 | |
---|
| 453 | ! Alternatively: |
---|
| 454 | ! |\Delta pH| = |zeqn/(zdeqndh*zh_prev*LOG(10))| |
---|
| 455 | ! ~ 1/LOG(10) * |\Delta [H]|/[H] |
---|
| 456 | ! < 1/LOG(10) * rdel |
---|
| 457 | |
---|
| 458 | ! Hence |zeqn/(zdeqndh*zh)| < rdel |
---|
| 459 | |
---|
| 460 | ! rdel <-- pp_rdel_ah_target |
---|
| 461 | |
---|
| 462 | l_exitnow = (ABS(zh_lnfactor) < pp_rdel_ah_target) |
---|
| 463 | |
---|
| 464 | IF(l_exitnow) EXIT |
---|
| 465 | ENDDO |
---|
| 466 | |
---|
| 467 | solve_at_general = zh |
---|
| 468 | |
---|
| 469 | IF(PRESENT(p_val)) THEN |
---|
| 470 | IF(zh > 0._wp) THEN |
---|
| 471 | p_val = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 472 | p_po4tot, p_siltot, & |
---|
| 473 | p_so4tot, p_flutot, & |
---|
| 474 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 475 | ELSE |
---|
| 476 | p_val = HUGE(1._wp) |
---|
| 477 | ENDIF |
---|
| 478 | ENDIF |
---|
| 479 | RETURN |
---|
| 480 | END FUNCTION solve_at_general |
---|
| 481 | |
---|
| 482 | !=============================================================================== |
---|
| 483 | |
---|
| 484 | FUNCTION solve_at_general_sec(p_alktot, p_dictot, p_bortot, & |
---|
| 485 | p_po4tot, p_siltot, & |
---|
| 486 | p_so4tot, p_flutot, & |
---|
| 487 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
---|
| 488 | p_hini, p_val) |
---|
| 489 | |
---|
| 490 | ! Universal pH solver that converges from any given initial value, |
---|
| 491 | ! determines upper an lower bounds for the solution if required |
---|
| 492 | |
---|
| 493 | !USE MOD_CHEMCONST, ONLY: api1_wat, aphscale |
---|
| 494 | USE mocsy_singledouble |
---|
| 495 | IMPLICIT NONE |
---|
| 496 | REAL(KIND=wp) :: SOLVE_AT_GENERAL_SEC |
---|
| 497 | |
---|
| 498 | ! Argument variables |
---|
| 499 | REAL(KIND=wp), INTENT(IN) :: p_alktot |
---|
| 500 | REAL(KIND=wp), INTENT(IN) :: p_dictot |
---|
| 501 | REAL(KIND=wp), INTENT(IN) :: p_bortot |
---|
| 502 | REAL(KIND=wp), INTENT(IN) :: p_po4tot |
---|
| 503 | REAL(KIND=wp), INTENT(IN) :: p_siltot |
---|
| 504 | !REAL(KIND=wp), INTENT(IN) :: p_nh4tot |
---|
| 505 | !REAL(KIND=wp), INTENT(IN) :: p_h2stot |
---|
| 506 | REAL(KIND=wp), INTENT(IN) :: p_so4tot |
---|
| 507 | REAL(KIND=wp), INTENT(IN) :: p_flutot |
---|
| 508 | REAL(KIND=wp), INTENT(IN) :: K0, K1, K2, Kb, Kw, Ks, Kf |
---|
| 509 | REAL(KIND=wp), INTENT(IN) :: K1p, K2p, K3p, Ksi |
---|
| 510 | REAL(KIND=wp), INTENT(IN), OPTIONAL :: p_hini |
---|
| 511 | REAL(KIND=wp), INTENT(OUT), OPTIONAL :: p_val |
---|
| 512 | |
---|
| 513 | ! Local variables |
---|
| 514 | REAL(KIND=wp) :: zh_ini, zh, zh_1, zh_2, zh_delta |
---|
| 515 | REAL(KIND=wp) :: zalknw_inf, zalknw_sup |
---|
| 516 | REAL(KIND=wp) :: zh_min, zh_max |
---|
| 517 | REAL(KIND=wp) :: zeqn, zeqn_1, zeqn_2, zeqn_absmin |
---|
| 518 | REAL(KIND=wp) :: zdelta |
---|
| 519 | REAL(KIND=wp) :: aphscale |
---|
| 520 | LOGICAL :: l_exitnow |
---|
| 521 | |
---|
| 522 | ! TOTAL H+ scale: conversion factor for Htot = aphscale * Hfree |
---|
| 523 | aphscale = 1._wp + p_so4tot/Ks |
---|
| 524 | |
---|
| 525 | IF(PRESENT(p_hini)) THEN |
---|
| 526 | zh_ini = p_hini |
---|
| 527 | ELSE |
---|
| 528 | CALL ahini_for_at(p_alktot, p_dictot, p_bortot, K1, K2, Kb, zh_ini) |
---|
| 529 | ENDIF |
---|
| 530 | |
---|
| 531 | CALL anw_infsup(p_dictot, p_bortot, & |
---|
| 532 | p_po4tot, p_siltot, & |
---|
| 533 | p_so4tot, p_flutot, & |
---|
| 534 | zalknw_inf, zalknw_sup) |
---|
| 535 | |
---|
| 536 | zdelta = (p_alktot-zalknw_inf)**2 + 4._wp*Kw/aphscale |
---|
| 537 | |
---|
| 538 | IF(p_alktot >= zalknw_inf) THEN |
---|
| 539 | zh_min = 2._wp*Kw /( p_alktot-zalknw_inf + SQRT(zdelta) ) |
---|
| 540 | ELSE |
---|
| 541 | zh_min = aphscale*(-(p_alktot-zalknw_inf) + SQRT(zdelta) ) / 2._wp |
---|
| 542 | ENDIF |
---|
| 543 | |
---|
| 544 | zdelta = (p_alktot-zalknw_sup)**2 + 4._wp*Kw/aphscale |
---|
| 545 | |
---|
| 546 | IF(p_alktot <= zalknw_sup) THEN |
---|
| 547 | zh_max = aphscale*(-(p_alktot-zalknw_sup) + SQRT(zdelta) ) / 2._wp |
---|
| 548 | ELSE |
---|
| 549 | zh_max = 2._wp*Kw /( p_alktot-zalknw_sup + SQRT(zdelta) ) |
---|
| 550 | ENDIF |
---|
| 551 | |
---|
| 552 | zh = MAX(MIN(zh_max, zh_ini), zh_min) |
---|
| 553 | niter_atsec = 0 ! Reset counters of iterations |
---|
| 554 | |
---|
| 555 | ! Prepare the secant iterations: two initial (zh, zeqn) pairs are required |
---|
| 556 | ! We have the starting value, that needs to be completed by the evaluation |
---|
| 557 | ! of the equation value it produces. |
---|
| 558 | |
---|
| 559 | ! Complete the initial value with its equation evaluation |
---|
| 560 | ! (will take the role of the $n-2$ iterate at the first secant evaluation) |
---|
| 561 | |
---|
| 562 | niter_atsec = 0 ! zh_2 is the initial value; |
---|
| 563 | |
---|
| 564 | zh_2 = zh |
---|
| 565 | zeqn_2 = equation_at(p_alktot, zh_2, p_dictot, p_bortot, & |
---|
| 566 | p_po4tot, p_siltot, & |
---|
| 567 | p_so4tot, p_flutot, & |
---|
| 568 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 569 | |
---|
| 570 | zeqn_absmin = ABS(zeqn_2) |
---|
| 571 | |
---|
| 572 | ! Adapt bracketing interval and heuristically set zh_1 |
---|
| 573 | IF(zeqn_2 < 0._wp) THEN |
---|
| 574 | ! If zeqn_2 < 0, then we adjust zh_max: |
---|
| 575 | ! we can be sure that zh_min < zh_2 < zh_max. |
---|
| 576 | zh_max = zh_2 |
---|
| 577 | ! for zh_1, try 25% (0.1 pH units) below the current zh_max, |
---|
| 578 | ! but stay above SQRT(zh_min*zh_max), which would be equivalent |
---|
| 579 | ! to a bisection step on [pH@zh_min, pH@zh_max] |
---|
| 580 | zh_1 = MAX(zh_max/1.25_wp, SQRT(zh_min*zh_max)) |
---|
| 581 | ELSEIF(zeqn_2 > 0._wp) THEN |
---|
| 582 | ! If zeqn_2 < 0, then we adjust zh_min: |
---|
| 583 | ! we can be sure that zh_min < zh_2 < zh_max. |
---|
| 584 | zh_min = zh_2 |
---|
| 585 | ! for zh_1, try 25% (0.1 pH units) above the current zh_min, |
---|
| 586 | ! but stay below SQRT(zh_min*zh_max) which would be equivalent |
---|
| 587 | ! to a bisection step on [pH@zh_min, pH@zh_max] |
---|
| 588 | zh_1 = MIN(zh_min*1.25_wp, SQRT(zh_min*zh_max)) |
---|
| 589 | ELSE ! we have got the root; unlikely, but one never knows |
---|
| 590 | solve_at_general_sec = zh_2 |
---|
| 591 | IF(PRESENT(p_val)) p_val = zeqn_2 |
---|
| 592 | RETURN |
---|
| 593 | ENDIF |
---|
| 594 | |
---|
| 595 | ! We now have the first pair completed (zh_2, zeqn_2). |
---|
| 596 | ! Define the second one (zh_1, zeqn_1), which is also the first iterate. |
---|
| 597 | ! zh_1 has already been set above |
---|
| 598 | niter_atsec = 1 ! Update counter of iterations |
---|
| 599 | |
---|
| 600 | zeqn_1 = equation_at(p_alktot, zh_1, p_dictot, p_bortot, & |
---|
| 601 | p_po4tot, p_siltot, & |
---|
| 602 | p_so4tot, p_flutot, & |
---|
| 603 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 604 | |
---|
| 605 | ! Adapt bracketing interval: we know that zh_1 <= zh <= zh_max (if zeqn_1 > 0) |
---|
| 606 | ! or zh_min <= zh <= zh_1 (if zeqn_1 < 0), so this can always be done |
---|
| 607 | IF(zeqn_1 > 0._wp) THEN |
---|
| 608 | zh_min = zh_1 |
---|
| 609 | ELSEIF(zeqn_1 < 0._wp) THEN |
---|
| 610 | zh_max = zh_1 |
---|
| 611 | ELSE ! zh_1 is the root |
---|
| 612 | solve_at_general_sec = zh_1 |
---|
| 613 | IF(PRESENT(p_val)) p_val = zeqn_1 |
---|
| 614 | ENDIF |
---|
| 615 | |
---|
| 616 | IF(ABS(zeqn_1) > zeqn_absmin) THEN ! Swap zh_2 and zh_1 if ABS(zeqn_2) < ABS(zeqn_1) |
---|
| 617 | ! so that zh_2 and zh_1 lead to decreasing equation |
---|
| 618 | ! values (in absolute value) |
---|
| 619 | zh = zh_1 |
---|
| 620 | zeqn = zeqn_1 |
---|
| 621 | zh_1 = zh_2 |
---|
| 622 | zeqn_1 = zeqn_2 |
---|
| 623 | zh_2 = zh |
---|
| 624 | zeqn_2 = zeqn |
---|
| 625 | ELSE |
---|
| 626 | zeqn_absmin = ABS(zeqn_1) |
---|
| 627 | ENDIF |
---|
| 628 | |
---|
| 629 | ! Pre-calculate the first secant iterate (this is the second iterate) |
---|
| 630 | niter_atsec = 2 |
---|
| 631 | |
---|
| 632 | zh_delta = -zeqn_1/((zeqn_2-zeqn_1)/(zh_2 - zh_1)) |
---|
| 633 | zh = zh_1 + zh_delta |
---|
| 634 | |
---|
| 635 | ! Make sure that zh_min < zh < zh_max (if not, |
---|
| 636 | ! bisect around zh_1 which is the best estimate) |
---|
| 637 | IF (zh > zh_max) THEN ! this can only happen if zh_2 < zh_1 |
---|
| 638 | ! and zeqn_2 > zeqn_1 > 0 |
---|
| 639 | zh = SQRT(zh_1*zh_max) |
---|
| 640 | ENDIF |
---|
| 641 | |
---|
| 642 | IF (zh < zh_min) THEN ! this can only happen if zh_2 > zh_1 |
---|
| 643 | ! and zeqn_2 < zeqn_1 < 0 |
---|
| 644 | zh = SQRT(zh_1*zh_min) |
---|
| 645 | ENDIF |
---|
| 646 | |
---|
| 647 | DO |
---|
| 648 | IF(niter_atsec >= jp_maxniter_atsec) THEN |
---|
| 649 | zh = -1._wp |
---|
| 650 | EXIT |
---|
| 651 | ENDIF |
---|
| 652 | |
---|
| 653 | zeqn = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 654 | p_po4tot, p_siltot, & |
---|
| 655 | p_so4tot, p_flutot, & |
---|
| 656 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 657 | |
---|
| 658 | ! Adapt bracketing interval: since initially, zh_min <= zh <= zh_max |
---|
| 659 | ! we are sure that zh will improve either bracket, depending on the sign |
---|
| 660 | ! of zeqn |
---|
| 661 | IF(zeqn > 0._wp) THEN |
---|
| 662 | zh_min = zh |
---|
| 663 | ELSEIF(zeqn < 0._wp) THEN |
---|
| 664 | zh_max = zh |
---|
| 665 | ELSE |
---|
| 666 | ! zh is the root |
---|
| 667 | EXIT |
---|
| 668 | ENDIF |
---|
| 669 | |
---|
| 670 | ! start calculation of next iterate |
---|
| 671 | niter_atsec = niter_atsec + 1 |
---|
| 672 | |
---|
| 673 | zh_2 = zh_1 |
---|
| 674 | zeqn_2 = zeqn_1 |
---|
| 675 | zh_1 = zh |
---|
| 676 | zeqn_1 = zeqn |
---|
| 677 | |
---|
| 678 | IF(ABS(zeqn) >= 0.5_wp*zeqn_absmin) THEN |
---|
| 679 | ! if the function evaluation at the current point |
---|
| 680 | ! is not decreasing faster in absolute value than |
---|
| 681 | ! we may expect for a bisection step, then take |
---|
| 682 | ! one bisection step on [ph_min, ph_max] |
---|
| 683 | ! ph_new = (ph_min + ph_max)/2d0 |
---|
| 684 | ! In terms of [H]_new: |
---|
| 685 | ! [H]_new = 10**(-ph_new) |
---|
| 686 | ! = 10**(-(ph_min + ph_max)/2d0) |
---|
| 687 | ! = SQRT(10**(-(ph_min + phmax))) |
---|
| 688 | ! = SQRT(zh_max * zh_min) |
---|
| 689 | zh = SQRT(zh_max * zh_min) |
---|
| 690 | zh_delta = zh - zh_1 |
---|
| 691 | ELSE |
---|
| 692 | ! \Delta H = -zeqn_1*(h_2 - h_1)/(zeqn_2 - zeqn_1) |
---|
| 693 | ! H_new = H_1 + \Delta H |
---|
| 694 | zh_delta = -zeqn_1/((zeqn_2-zeqn_1)/(zh_2 - zh_1)) |
---|
| 695 | zh = zh_1 + zh_delta |
---|
| 696 | |
---|
| 697 | IF( zh < zh_min ) THEN |
---|
| 698 | ! if [H]_new < [H]_min |
---|
| 699 | ! i.e., if ph_new > ph_max then |
---|
| 700 | ! take one bisection step on [ph_prev, ph_max] |
---|
| 701 | ! ph_new = (ph_prev + ph_max)/2d0 |
---|
| 702 | ! In terms of [H]_new: |
---|
| 703 | ! [H]_new = 10**(-ph_new) |
---|
| 704 | ! = 10**(-(ph_prev + ph_max)/2d0) |
---|
| 705 | ! = SQRT(10**(-(ph_prev + phmax))) |
---|
| 706 | ! = SQRT([H]_old*10**(-ph_max)) |
---|
| 707 | ! = SQRT([H]_old * zh_min) |
---|
| 708 | zh = SQRT(zh_1 * zh_min) |
---|
| 709 | zh_delta = zh - zh_1 |
---|
| 710 | ENDIF |
---|
| 711 | |
---|
| 712 | IF( zh > zh_max ) THEN |
---|
| 713 | ! if [H]_new > [H]_max |
---|
| 714 | ! i.e., if ph_new < ph_min, then |
---|
| 715 | ! take one bisection step on [ph_min, ph_prev] |
---|
| 716 | ! ph_new = (ph_prev + ph_min)/2d0 |
---|
| 717 | ! In terms of [H]_new: |
---|
| 718 | ! [H]_new = 10**(-ph_new) |
---|
| 719 | ! = 10**(-(ph_prev + ph_min)/2d0) |
---|
| 720 | ! = SQRT(10**(-(ph_prev + ph_min))) |
---|
| 721 | ! = SQRT([H]_old*10**(-ph_min)) |
---|
| 722 | ! = SQRT([H]_old * zhmax) |
---|
| 723 | zh = SQRT(zh_1 * zh_max) |
---|
| 724 | zh_delta = zh - zh_1 |
---|
| 725 | ENDIF |
---|
| 726 | ENDIF |
---|
| 727 | |
---|
| 728 | zeqn_absmin = MIN(ABS(zeqn), zeqn_absmin) |
---|
| 729 | |
---|
| 730 | ! Stop iterations once |([H]-[H_1])/[H_1]| < rdel |
---|
| 731 | l_exitnow = (ABS(zh_delta) < pp_rdel_ah_target*zh_1) |
---|
| 732 | |
---|
| 733 | IF(l_exitnow) EXIT |
---|
| 734 | ENDDO |
---|
| 735 | |
---|
| 736 | SOLVE_AT_GENERAL_SEC = zh |
---|
| 737 | |
---|
| 738 | IF(PRESENT(p_val)) THEN |
---|
| 739 | IF(zh > 0._wp) THEN |
---|
| 740 | p_val = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 741 | p_po4tot, p_siltot, & |
---|
| 742 | p_so4tot, p_flutot, & |
---|
| 743 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 744 | ELSE |
---|
| 745 | p_val = HUGE(1._wp) |
---|
| 746 | ENDIF |
---|
| 747 | ENDIF |
---|
| 748 | |
---|
| 749 | RETURN |
---|
| 750 | END FUNCTION SOLVE_AT_GENERAL_SEC |
---|
| 751 | |
---|
| 752 | !=============================================================================== |
---|
| 753 | |
---|
| 754 | FUNCTION SOLVE_AT_FAST(p_alktot, p_dictot, p_bortot, & |
---|
| 755 | p_po4tot, p_siltot, & |
---|
| 756 | p_so4tot, p_flutot, & |
---|
| 757 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
---|
| 758 | p_hini, p_val) |
---|
| 759 | |
---|
| 760 | ! Fast version of SOLVE_AT_GENERAL, without any bounds checking. |
---|
| 761 | |
---|
| 762 | USE mocsy_singledouble |
---|
| 763 | IMPLICIT NONE |
---|
| 764 | REAL(KIND=wp) :: SOLVE_AT_FAST |
---|
| 765 | |
---|
| 766 | ! Argument variables |
---|
| 767 | REAL(KIND=wp), INTENT(IN) :: p_alktot |
---|
| 768 | REAL(KIND=wp), INTENT(IN) :: p_dictot |
---|
| 769 | REAL(KIND=wp), INTENT(IN) :: p_bortot |
---|
| 770 | REAL(KIND=wp), INTENT(IN) :: p_po4tot |
---|
| 771 | REAL(KIND=wp), INTENT(IN) :: p_siltot |
---|
| 772 | !REAL(KIND=wp), INTENT(IN) :: p_nh4tot |
---|
| 773 | !REAL(KIND=wp), INTENT(IN) :: p_h2stot |
---|
| 774 | REAL(KIND=wp), INTENT(IN) :: p_so4tot |
---|
| 775 | REAL(KIND=wp), INTENT(IN) :: p_flutot |
---|
| 776 | REAL(KIND=wp), INTENT(IN) :: K0, K1, K2, Kb, Kw, Ks, Kf |
---|
| 777 | REAL(KIND=wp), INTENT(IN) :: K1p, K2p, K3p, Ksi |
---|
| 778 | REAL(KIND=wp), INTENT(IN), OPTIONAL :: p_hini |
---|
| 779 | REAL(KIND=wp), INTENT(OUT), OPTIONAL :: p_val |
---|
| 780 | |
---|
| 781 | ! Local variables |
---|
| 782 | REAL(KIND=wp) :: zh_ini, zh, zh_prev, zh_lnfactor |
---|
| 783 | REAL(KIND=wp) :: zhdelta |
---|
| 784 | REAL(KIND=wp) :: zeqn, zdeqndh |
---|
| 785 | !REAL(KIND=wp) :: aphscale |
---|
| 786 | LOGICAL :: l_exitnow |
---|
| 787 | REAL(KIND=wp), PARAMETER :: pz_exp_threshold = 1.0_wp |
---|
| 788 | |
---|
| 789 | ! TOTAL H+ scale: conversion factor for Htot = aphscale * Hfree |
---|
| 790 | !aphscale = 1._wp + p_so4tot/Ks |
---|
| 791 | |
---|
| 792 | IF(PRESENT(p_hini)) THEN |
---|
| 793 | zh_ini = p_hini |
---|
| 794 | ELSE |
---|
| 795 | CALL AHINI_FOR_AT(p_alktot, p_dictot, p_bortot, K1, K2, Kb, zh_ini) |
---|
| 796 | ENDIF |
---|
| 797 | zh = zh_ini |
---|
| 798 | |
---|
| 799 | niter_atfast = 0 ! Reset counters of iterations |
---|
| 800 | DO |
---|
| 801 | niter_atfast = niter_atfast + 1 |
---|
| 802 | IF(niter_atfast > jp_maxniter_atfast) THEN |
---|
| 803 | zh = -1._wp |
---|
| 804 | EXIT |
---|
| 805 | ENDIF |
---|
| 806 | |
---|
| 807 | zh_prev = zh |
---|
| 808 | |
---|
| 809 | zeqn = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 810 | p_po4tot, p_siltot, & |
---|
| 811 | p_so4tot, p_flutot, & |
---|
| 812 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi, & |
---|
| 813 | P_DERIVEQN = zdeqndh) |
---|
| 814 | |
---|
| 815 | IF(zeqn == 0._wp) EXIT ! zh is the root |
---|
| 816 | |
---|
| 817 | zh_lnfactor = -zeqn/(zdeqndh*zh_prev) |
---|
| 818 | IF(ABS(zh_lnfactor) > pz_exp_threshold) THEN |
---|
| 819 | zh = zh_prev*EXP(zh_lnfactor) |
---|
| 820 | ELSE |
---|
| 821 | zhdelta = zh_lnfactor*zh_prev |
---|
| 822 | zh = zh_prev + zhdelta |
---|
| 823 | ENDIF |
---|
| 824 | |
---|
| 825 | ! Stop iterations once |\delta{[H]}/[H]| < rdel |
---|
| 826 | ! <=> |(zh - zh_prev)/zh_prev| = |EXP(-zeqn/(zdeqndh*zh_prev)) -1| < rdel |
---|
| 827 | ! |EXP(-zeqn/(zdeqndh*zh_prev)) -1| ~ |zeqn/(zdeqndh*zh_prev)| |
---|
| 828 | |
---|
| 829 | ! Alternatively: |
---|
| 830 | ! |\Delta pH| = |zeqn/(zdeqndh*zh_prev*LOG(10))| |
---|
| 831 | ! ~ 1/LOG(10) * |\Delta [H]|/[H] |
---|
| 832 | ! < 1/LOG(10) * rdel |
---|
| 833 | |
---|
| 834 | ! Hence |zeqn/(zdeqndh*zh)| < rdel |
---|
| 835 | |
---|
| 836 | ! rdel <- pp_rdel_ah_target |
---|
| 837 | |
---|
| 838 | l_exitnow = (ABS(zh_lnfactor) < pp_rdel_ah_target) |
---|
| 839 | |
---|
| 840 | IF(l_exitnow) EXIT |
---|
| 841 | ENDDO |
---|
| 842 | |
---|
| 843 | SOLVE_AT_FAST = zh |
---|
| 844 | |
---|
| 845 | IF(PRESENT(p_val)) THEN |
---|
| 846 | IF(zh > 0._wp) THEN |
---|
| 847 | p_val = equation_at(p_alktot, zh, p_dictot, p_bortot, & |
---|
| 848 | p_po4tot, p_siltot, & |
---|
| 849 | p_so4tot, p_flutot, & |
---|
| 850 | K0, K1, K2, Kb, Kw, Ks, Kf, K1p, K2p, K3p, Ksi) |
---|
| 851 | ELSE |
---|
| 852 | p_val = HUGE(1._wp) |
---|
| 853 | ENDIF |
---|
| 854 | ENDIF |
---|
| 855 | |
---|
| 856 | RETURN |
---|
| 857 | END FUNCTION solve_at_fast |
---|
| 858 | !=============================================================================== |
---|
| 859 | |
---|
| 860 | END MODULE mocsy_phsolvers |
---|