1 | MODULE limthd_dif |
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2 | #if defined key_lim3 |
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3 | !!---------------------------------------------------------------------- |
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4 | !! 'key_lim3' LIM3 sea-ice model |
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5 | !!---------------------------------------------------------------------- |
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6 | !!====================================================================== |
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7 | !! *** MODULE limthd_dif *** |
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8 | !! heat diffusion in sea ice |
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9 | !! computation of surface and inner T |
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10 | !!====================================================================== |
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11 | |
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12 | !!---------------------------------------------------------------------- |
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13 | !! * Modules used |
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14 | USE par_oce ! ocean parameters |
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15 | USE phycst ! physical constants (ocean directory) |
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16 | USE thd_ice |
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17 | USE in_out_manager |
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18 | USE ice |
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19 | USE par_ice |
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20 | USE lib_mpp |
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21 | |
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22 | IMPLICIT NONE |
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23 | PRIVATE |
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24 | |
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25 | !! * Routine accessibility |
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26 | PUBLIC lim_thd_dif ! called by lim_thd |
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27 | |
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28 | !! * Module variables |
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29 | REAL(wp) :: & ! constant values |
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30 | epsi20 = 1e-20 , & |
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31 | epsi13 = 1e-13 , & |
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32 | zzero = 0.e0 , & |
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33 | zone = 1.e0 |
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34 | |
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35 | !!---------------------------------------------------------------------- |
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36 | !! LIM 3.0, UCL-LOCEAN-IPSL (2005) |
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37 | !! $Id$ |
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38 | !! Software governed by the CeCILL licence (modipsl/doc/NEMO_CeCILL.txt) |
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39 | !!---------------------------------------------------------------------- |
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40 | |
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41 | CONTAINS |
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42 | |
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43 | SUBROUTINE lim_thd_dif( kideb , kiut , jl ) |
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44 | !!------------------------------------------------------------------ |
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45 | !! *** ROUTINE lim_thd_dif *** |
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46 | !! ** Purpose : |
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47 | !! This routine determines the time evolution of snow and sea-ice |
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48 | !! temperature profiles. |
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49 | !! ** Method : |
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50 | !! This is done by solving the heat equation diffusion with |
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51 | !! a Neumann boundary condition at the surface and a Dirichlet one |
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52 | !! at the bottom. Solar radiation is partially absorbed into the ice. |
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53 | !! The specific heat and thermal conductivities depend on ice salinity |
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54 | !! and temperature to take into account brine pocket melting. The |
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55 | !! numerical |
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56 | !! scheme is an iterative Crank-Nicolson on a non-uniform multilayer grid |
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57 | !! in the ice and snow system. |
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58 | !! |
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59 | !! The successive steps of this routine are |
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60 | !! 1. Thermal conductivity at the interfaces of the ice layers |
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61 | !! 2. Internal absorbed radiation |
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62 | !! 3. Scale factors due to non-uniform grid |
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63 | !! 4. Kappa factors |
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64 | !! Then iterative procedure begins |
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65 | !! 5. specific heat in the ice |
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66 | !! 6. eta factors |
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67 | !! 7. surface flux computation |
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68 | !! 8. tridiagonal system terms |
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69 | !! 9. solving the tridiagonal system with Gauss elimination |
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70 | !! Iterative procedure ends according to a criterion on evolution |
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71 | !! of temperature |
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72 | !! |
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73 | !! ** Arguments : |
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74 | !! kideb , kiut : Starting and ending points on which the |
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75 | !! the computation is applied |
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76 | !! |
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77 | !! ** Inputs / Ouputs : (global commons) |
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78 | !! surface temperature : t_su_b |
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79 | !! ice/snow temperatures : t_i_b, t_s_b |
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80 | !! ice salinities : s_i_b |
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81 | !! number of layers in the ice/snow: nlay_i, nlay_s |
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82 | !! profile of the ice/snow layers : z_i, z_s |
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83 | !! total ice/snow thickness : ht_i_b, ht_s_b |
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84 | !! |
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85 | !! ** External : |
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86 | !! |
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87 | !! ** References : |
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88 | !! |
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89 | !! ** History : |
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90 | !! (02-2003) Martin Vancoppenolle, Louvain-la-Neuve, Belgium |
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91 | !! (06-2005) Martin Vancoppenolle, 3d version |
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92 | !! (11-2006) Vectorized by Xavier Fettweis (UCL-ASTR) |
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93 | !! (04-2007) Energy conservation tested by M. Vancoppenolle |
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94 | !! |
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95 | !!------------------------------------------------------------------ |
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96 | !! * Arguments |
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97 | |
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98 | INTEGER , INTENT (in) :: & |
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99 | kideb , & ! Start point on which the the computation is applied |
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100 | kiut , & ! End point on which the the computation is applied |
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101 | jl ! Category number |
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102 | |
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103 | !! * Local variables |
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104 | INTEGER :: ji, & ! spatial loop index |
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105 | zji, zjj, & ! temporary dummy loop index |
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106 | numeq, & ! current reference number of equation |
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107 | layer, & ! vertical dummy loop index |
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108 | nconv, & ! number of iterations in iterative procedure |
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109 | minnumeqmin, & ! |
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110 | maxnumeqmax |
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111 | |
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112 | INTEGER , DIMENSION(kiut) :: & |
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113 | numeqmin, & ! reference number of top equation |
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114 | numeqmax, & ! reference number of bottom equation |
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115 | isnow ! switch for presence (1) or absence (0) of snow |
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116 | |
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117 | !! * New local variables |
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118 | REAL(wp) , DIMENSION(kiut,0:nlay_i) :: & |
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119 | ztcond_i, & !Ice thermal conductivity |
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120 | zradtr_i, & !Radiation transmitted through the ice |
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121 | zradab_i, & !Radiation absorbed in the ice |
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122 | zkappa_i !Kappa factor in the ice |
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123 | |
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124 | REAL(wp) , DIMENSION(kiut,0:nlay_s) :: & |
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125 | zradtr_s, & !Radiation transmited through the snow |
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126 | zradab_s, & !Radiation absorbed in the snow |
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127 | zkappa_s !Kappa factor in the snow |
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128 | |
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129 | REAL(wp) , DIMENSION(kiut,0:nlay_i) :: & |
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130 | ztiold, & !Old temperature in the ice |
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131 | zeta_i, & !Eta factor in the ice |
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132 | ztitemp, & !Temporary temperature in the ice to check the convergence |
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133 | zspeche_i, & !Ice specific heat |
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134 | z_i !Vertical cotes of the layers in the ice |
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135 | |
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136 | REAL(wp) , DIMENSION(kiut,0:nlay_s) :: & |
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137 | zeta_s, & !Eta factor in the snow |
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138 | ztstemp, & !Temporary temperature in the snow to check the convergence |
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139 | ztsold, & !Temporary temperature in the snow |
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140 | z_s !Vertical cotes of the layers in the snow |
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141 | |
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142 | REAL(wp) , DIMENSION(kiut,jkmax+2) :: & |
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143 | zindterm, & ! Independent term |
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144 | zindtbis, & ! temporary independent term |
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145 | zdiagbis |
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146 | |
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147 | REAL(wp) , DIMENSION(kiut,jkmax+2,3) :: & |
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148 | ztrid ! tridiagonal system terms |
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149 | |
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150 | REAL(wp), DIMENSION(kiut) :: & |
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151 | ztfs , & ! ice melting point |
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152 | ztsuold , & ! old surface temperature (before the iterative |
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153 | ! procedure ) |
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154 | ztsuoldit, & ! surface temperature at previous iteration |
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155 | zh_i , & !ice layer thickness |
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156 | zh_s , & !snow layer thickness |
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157 | zfsw , & !solar radiation absorbed at the surface |
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158 | zf , & ! surface flux function |
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159 | dzf ! derivative of the surface flux function |
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160 | |
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161 | REAL(wp) :: & ! constant values |
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162 | zeps = 1.0e-10, & ! |
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163 | zg1s = 2.0, & !: for the tridiagonal system |
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164 | zg1 = 2.0, & |
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165 | zgamma = 18009.0, & !: for specific heat |
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166 | zbeta = 0.117, & !: for thermal conductivity (could be 0.13) |
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167 | zraext_s = 1.0e08, & !: extinction coefficient of radiation in the snow |
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168 | zkimin = 0.10 , & !: minimum ice thermal conductivity |
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169 | zht_smin = 1.0e-4 !: minimum snow depth |
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170 | |
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171 | REAL(wp) :: & ! local variables |
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172 | ztmelt_i, & ! ice melting temperature |
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173 | zerritmax ! current maximal error on temperature |
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174 | |
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175 | REAL(wp), DIMENSION(kiut) :: & |
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176 | zerrit, & ! current error on temperature |
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177 | zdifcase, & ! case of the equation resolution (1->4) |
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178 | zftrice, & ! solar radiation transmitted through the ice |
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179 | zihic, zhsu |
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180 | |
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181 | ! |
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182 | !------------------------------------------------------------------------------! |
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183 | ! 1) Initialization ! |
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184 | !------------------------------------------------------------------------------! |
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185 | ! |
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186 | DO ji = kideb , kiut |
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187 | ! is there snow or not |
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188 | isnow(ji)= INT ( 1.0 - MAX( 0.0 , SIGN (1.0, - ht_s_b(ji) ) ) ) |
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189 | ! surface temperature of fusion |
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190 | ztfs(ji) = isnow(ji) * rtt + (1.0-isnow(ji)) * rtt |
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191 | ! layer thickness |
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192 | zh_i(ji) = ht_i_b(ji) / nlay_i |
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193 | zh_s(ji) = ht_s_b(ji) / nlay_s |
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194 | END DO |
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195 | |
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196 | !-------------------- |
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197 | ! Ice / snow layers |
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198 | !-------------------- |
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199 | |
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200 | z_s(:,0) = 0.0 ! vert. coord. of the up. lim. of the 1st snow layer |
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201 | z_i(:,0) = 0.0 ! vert. coord. of the up. lim. of the 1st ice layer |
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202 | |
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203 | DO layer = 1, nlay_s |
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204 | DO ji = kideb , kiut |
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205 | ! vert. coord of the up. lim. of the layer-th snow layer |
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206 | z_s(ji,layer) = z_s(ji,layer-1) + ht_s_b(ji) / nlay_s |
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207 | END DO |
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208 | END DO |
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209 | |
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210 | DO layer = 1, nlay_i |
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211 | DO ji = kideb , kiut |
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212 | ! vert. coord of the up. lim. of the layer-th ice layer |
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213 | z_i(ji,layer) = z_i(ji,layer-1) + ht_i_b(ji) / nlay_i |
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214 | END DO |
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215 | END DO |
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216 | ! |
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217 | !------------------------------------------------------------------------------| |
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218 | ! 2) Radiations | |
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219 | !------------------------------------------------------------------------------| |
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220 | ! |
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221 | !------------------- |
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222 | ! Computation of i0 |
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223 | !------------------- |
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224 | ! i0 describes the fraction of solar radiation which does not contribute |
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225 | ! to the surface energy budget but rather penetrates inside the ice. |
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226 | ! We assume that no radiation is transmitted through the snow |
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227 | ! If there is no no snow |
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228 | ! zfsw = (1-i0).qsr_ice is absorbed at the surface |
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229 | ! zftrice = io.qsr_ice is below the surface |
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230 | ! fstbif = io.qsr_ice.exp(-k(h_i)) transmitted below the ice |
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231 | |
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232 | DO ji = kideb , kiut |
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233 | ! switches |
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234 | isnow(ji) = INT ( 1.0 - MAX ( 0.0 , SIGN ( 1.0 , - ht_s_b(ji) ) ) ) |
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235 | ! hs > 0, isnow = 1 |
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236 | zhsu(ji) = hnzst !threshold for the computation of i0 |
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237 | zihic(ji) = MAX( zzero , 1.0 - ( ht_i_b(ji) / zhsu(ji) ) ) |
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238 | |
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239 | i0(ji) = ( 1.0 - isnow(ji) ) * & |
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240 | ( fr1_i0_1d(ji) + zihic(ji) * fr2_i0_1d(ji) ) |
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241 | !fr1_i0_1d = i0 for a thin ice surface |
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242 | !fr1_i0_2d = i0 for a thick ice surface |
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243 | ! a function of the cloud cover |
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244 | ! |
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245 | !i0(ji) = (1.0-FLOAT(isnow(ji)))*3.0/(100*ht_s_b(ji)+10.0) |
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246 | !formula used in Cice |
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247 | END DO |
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248 | |
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249 | !------------------------------------------------------- |
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250 | ! Solar radiation absorbed / transmitted at the surface |
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251 | ! Derivative of the non solar flux |
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252 | !------------------------------------------------------- |
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253 | DO ji = kideb , kiut |
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254 | |
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255 | ! Shortwave radiation absorbed at surface |
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256 | zfsw(ji) = qsr_ice_1d(ji) * ( 1 - i0(ji) ) |
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257 | |
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258 | ! Solar radiation transmitted below the surface layer |
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259 | zftrice(ji)= qsr_ice_1d(ji) * i0(ji) |
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260 | |
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261 | ! derivative of incoming nonsolar flux |
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262 | dzf(ji) = dqns_ice_1d(ji) |
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263 | |
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264 | END DO |
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265 | |
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266 | !--------------------------------------------------------- |
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267 | ! Transmission - absorption of solar radiation in the ice |
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268 | !--------------------------------------------------------- |
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269 | |
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270 | DO ji = kideb , kiut |
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271 | ! Initialization |
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272 | zradtr_s(ji,0) = zftrice(ji) ! radiation penetrating through snow |
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273 | END DO |
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274 | |
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275 | ! Radiation through snow |
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276 | DO layer = 1, nlay_s |
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277 | DO ji = kideb , kiut |
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278 | ! radiation transmitted below the layer-th snow layer |
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279 | zradtr_s(ji,layer) = zradtr_s(ji,0) * EXP ( - zraext_s * ( MAX ( 0.0 , & |
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280 | z_s(ji,layer) ) ) ) |
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281 | ! radiation absorbed by the layer-th snow layer |
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282 | zradab_s(ji,layer) = zradtr_s(ji,layer-1) - zradtr_s(ji,layer) |
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283 | END DO |
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284 | END DO |
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285 | |
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286 | ! Radiation through ice |
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287 | DO ji = kideb , kiut |
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288 | zradtr_i(ji,0) = zradtr_s(ji,nlay_s) * isnow(ji) + & |
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289 | zftrice(ji) * ( 1 - isnow(ji) ) |
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290 | END DO |
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291 | |
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292 | DO layer = 1, nlay_i |
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293 | DO ji = kideb , kiut |
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294 | ! radiation transmitted below the layer-th ice layer |
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295 | zradtr_i(ji,layer) = zradtr_i(ji,0) * EXP ( - kappa_i * ( MAX ( 0.0 , & |
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296 | z_i(ji,layer) ) ) ) |
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297 | ! radiation absorbed by the layer-th ice layer |
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298 | zradab_i(ji,layer) = zradtr_i(ji,layer-1) - zradtr_i(ji,layer) |
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299 | END DO |
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300 | END DO |
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301 | |
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302 | ! Radiation transmitted below the ice |
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303 | DO ji = kideb , kiut |
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304 | fstbif_1d(ji) = fstbif_1d(ji) + & |
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305 | zradtr_i(ji,nlay_i) * a_i_b(ji) / at_i_b(ji) |
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306 | END DO |
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307 | |
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308 | ! +++++ |
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309 | ! just to check energy conservation |
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310 | DO ji = kideb , kiut |
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311 | zji = MOD( npb(ji) - 1, jpi ) + 1 |
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312 | zjj = ( npb(ji) - 1 ) / jpi + 1 |
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313 | fstroc(zji,zjj,jl) = & |
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314 | zradtr_i(ji,nlay_i) |
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315 | END DO |
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316 | ! +++++ |
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317 | |
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318 | DO layer = 1, nlay_i |
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319 | DO ji = kideb , kiut |
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320 | radab(ji,layer) = zradab_i(ji,layer) |
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321 | END DO |
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322 | END DO |
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323 | |
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324 | |
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325 | ! |
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326 | !------------------------------------------------------------------------------| |
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327 | ! 3) Iterative procedure begins | |
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328 | !------------------------------------------------------------------------------| |
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329 | ! |
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330 | ! Old surface temperature |
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331 | DO ji = kideb, kiut |
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332 | ztsuold(ji) = t_su_b(ji) ! temperature at the beg of iter pr. |
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333 | ztsuoldit(ji) = t_su_b(ji) ! temperature at the previous iter |
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334 | t_su_b(ji) = MIN(t_su_b(ji),ztfs(ji)-0.00001) !necessary |
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335 | zerrit(ji) = 1000.0 ! initial value of error |
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336 | END DO |
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337 | !RB Min global ?? |
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338 | |
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339 | ! Old snow temperature |
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340 | DO layer = 1, nlay_s |
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341 | DO ji = kideb , kiut |
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342 | ztsold(ji,layer) = t_s_b(ji,layer) |
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343 | END DO |
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344 | END DO |
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345 | |
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346 | ! Old ice temperature |
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347 | DO layer = 1, nlay_i |
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348 | DO ji = kideb , kiut |
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349 | ztiold(ji,layer) = t_i_b(ji,layer) |
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350 | END DO |
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351 | END DO |
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352 | |
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353 | nconv = 0 ! number of iterations |
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354 | zerritmax = 1000.0 ! maximal value of error on all points |
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355 | |
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356 | DO WHILE ((zerritmax > maxer_i_thd).AND.(nconv < nconv_i_thd)) |
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357 | |
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358 | nconv = nconv+1 |
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359 | |
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360 | ! |
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361 | !------------------------------------------------------------------------------| |
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362 | ! 4) Sea ice thermal conductivity | |
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363 | !------------------------------------------------------------------------------| |
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364 | ! |
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365 | IF ( thcon_i_swi .EQ. 0 ) THEN |
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366 | ! Untersteiner (1964) formula |
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367 | DO ji = kideb , kiut |
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368 | ztcond_i(ji,0) = rcdic + zbeta*s_i_b(ji,1) / & |
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369 | MIN(-zeps,t_i_b(ji,1)-rtt) |
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370 | ztcond_i(ji,0) = MAX(ztcond_i(ji,0),zkimin) |
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371 | END DO |
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372 | ENDIF |
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373 | |
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374 | IF ( thcon_i_swi .EQ. 1 ) THEN |
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375 | ! Pringle et al formula included, |
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376 | ! 2.11 + 0.09 S/T - 0.011.T |
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377 | DO ji = kideb , kiut |
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378 | ztcond_i(ji,0) = rcdic + 0.09*s_i_b(ji,1) / & |
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379 | MIN(-zeps,t_i_b(ji,1)-rtt) - & |
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380 | 0.011* ( t_i_b(ji,1) - rtt ) |
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381 | ztcond_i(ji,0) = MAX(ztcond_i(ji,0),zkimin) |
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382 | END DO |
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383 | ENDIF |
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384 | |
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385 | IF ( thcon_i_swi .EQ. 0 ) THEN ! Untersteiner |
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386 | DO layer = 1, nlay_i-1 |
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387 | DO ji = kideb , kiut |
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388 | ztcond_i(ji,layer) = rcdic + zbeta*( s_i_b(ji,layer) & |
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389 | + s_i_b(ji,layer+1) ) / MIN(-zeps, & |
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390 | t_i_b(ji,layer)+t_i_b(ji,layer+1)-2.0*rtt) |
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391 | ztcond_i(ji,layer) = MAX(ztcond_i(ji,layer),zkimin) |
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392 | END DO |
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393 | END DO |
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394 | ENDIF |
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395 | |
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396 | IF ( thcon_i_swi .EQ. 1 ) THEN ! Pringle |
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397 | DO layer = 1, nlay_i-1 |
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398 | DO ji = kideb , kiut |
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399 | ztcond_i(ji,layer) = rcdic + 0.09*( s_i_b(ji,layer) & |
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400 | + s_i_b(ji,layer+1) ) / MIN(-zeps, & |
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401 | t_i_b(ji,layer)+t_i_b(ji,layer+1)-2.0*rtt) - & |
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402 | 0.011* ( t_i_b(ji,layer) + t_i_b(ji,layer+1) - 2.0*rtt ) |
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403 | ztcond_i(ji,layer) = MAX(ztcond_i(ji,layer),zkimin) |
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404 | END DO |
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405 | END DO |
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406 | ENDIF |
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407 | |
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408 | IF ( thcon_i_swi .EQ. 0 ) THEN ! Untersteiner |
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409 | DO ji = kideb , kiut |
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410 | ztcond_i(ji,nlay_i) = rcdic + zbeta*s_i_b(ji,nlay_i) / & |
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411 | MIN(-zeps,t_bo_b(ji)-rtt) |
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412 | ztcond_i(ji,nlay_i) = MAX(ztcond_i(ji,nlay_i),zkimin) |
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413 | END DO |
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414 | ENDIF |
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415 | |
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416 | IF ( thcon_i_swi .EQ. 1 ) THEN ! Pringle |
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417 | DO ji = kideb , kiut |
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418 | ztcond_i(ji,nlay_i) = rcdic + 0.09*s_i_b(ji,nlay_i) / & |
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419 | MIN(-zeps,t_bo_b(ji)-rtt) - & |
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420 | 0.011* ( t_bo_b(ji) - rtt ) |
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421 | ztcond_i(ji,nlay_i) = MAX(ztcond_i(ji,nlay_i),zkimin) |
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422 | END DO |
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423 | ENDIF |
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424 | ! |
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425 | !------------------------------------------------------------------------------| |
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426 | ! 5) kappa factors | |
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427 | !------------------------------------------------------------------------------| |
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428 | ! |
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429 | DO ji = kideb, kiut |
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430 | |
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431 | !-- Snow kappa factors |
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432 | zkappa_s(ji,0) = rcdsn / MAX(zeps,zh_s(ji)) |
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433 | zkappa_s(ji,nlay_s) = rcdsn / MAX(zeps,zh_s(ji)) |
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434 | END DO |
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435 | |
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436 | DO layer = 1, nlay_s-1 |
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437 | DO ji = kideb , kiut |
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438 | zkappa_s(ji,layer) = 2.0 * rcdsn / & |
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439 | MAX(zeps,2.0*zh_s(ji)) |
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440 | END DO |
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441 | END DO |
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442 | |
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443 | DO layer = 1, nlay_i-1 |
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444 | DO ji = kideb , kiut |
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445 | !-- Ice kappa factors |
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446 | zkappa_i(ji,layer) = 2.0*ztcond_i(ji,layer)/ & |
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447 | MAX(zeps,2.0*zh_i(ji)) |
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448 | END DO |
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449 | END DO |
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450 | |
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451 | DO ji = kideb , kiut |
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452 | zkappa_i(ji,0) = ztcond_i(ji,0)/MAX(zeps,zh_i(ji)) |
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453 | zkappa_i(ji,nlay_i) = ztcond_i(ji,nlay_i) / MAX(zeps,zh_i(ji)) |
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454 | !-- Interface |
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455 | zkappa_s(ji,nlay_s) = 2.0*rcdsn*ztcond_i(ji,0)/MAX(zeps, & |
---|
456 | (ztcond_i(ji,0)*zh_s(ji) + rcdsn*zh_i(ji))) |
---|
457 | zkappa_i(ji,0) = zkappa_s(ji,nlay_s)*isnow(ji) & |
---|
458 | + zkappa_i(ji,0)*(1.0-isnow(ji)) |
---|
459 | END DO |
---|
460 | ! |
---|
461 | !------------------------------------------------------------------------------| |
---|
462 | ! 6) Sea ice specific heat, eta factors | |
---|
463 | !------------------------------------------------------------------------------| |
---|
464 | ! |
---|
465 | DO layer = 1, nlay_i |
---|
466 | DO ji = kideb , kiut |
---|
467 | ztitemp(ji,layer) = t_i_b(ji,layer) |
---|
468 | zspeche_i(ji,layer) = cpic + zgamma*s_i_b(ji,layer)/ & |
---|
469 | MAX((t_i_b(ji,layer)-rtt)*(ztiold(ji,layer)-rtt),zeps) |
---|
470 | zeta_i(ji,layer) = rdt_ice / MAX(rhoic*zspeche_i(ji,layer)*zh_i(ji), & |
---|
471 | zeps) |
---|
472 | END DO |
---|
473 | END DO |
---|
474 | |
---|
475 | DO layer = 1, nlay_s |
---|
476 | DO ji = kideb , kiut |
---|
477 | ztstemp(ji,layer) = t_s_b(ji,layer) |
---|
478 | zeta_s(ji,layer) = rdt_ice / MAX(rhosn*cpic*zh_s(ji),zeps) |
---|
479 | END DO |
---|
480 | END DO |
---|
481 | ! |
---|
482 | !------------------------------------------------------------------------------| |
---|
483 | ! 7) surface flux computation | |
---|
484 | !------------------------------------------------------------------------------| |
---|
485 | ! |
---|
486 | DO ji = kideb , kiut |
---|
487 | |
---|
488 | ! update of the non solar flux according to the update in T_su |
---|
489 | qnsr_ice_1d(ji) = qnsr_ice_1d(ji) + dqns_ice_1d(ji) * & |
---|
490 | ( t_su_b(ji) - ztsuoldit(ji) ) |
---|
491 | |
---|
492 | ! update incoming flux |
---|
493 | zf(ji) = zfsw(ji) & ! net absorbed solar radiation |
---|
494 | + qnsr_ice_1d(ji) ! non solar total flux |
---|
495 | ! (LWup, LWdw, SH, LH) |
---|
496 | |
---|
497 | END DO |
---|
498 | |
---|
499 | ! |
---|
500 | !------------------------------------------------------------------------------| |
---|
501 | ! 8) tridiagonal system terms | |
---|
502 | !------------------------------------------------------------------------------| |
---|
503 | ! |
---|
504 | !!layer denotes the number of the layer in the snow or in the ice |
---|
505 | !!numeq denotes the reference number of the equation in the tridiagonal |
---|
506 | !!system, terms of tridiagonal system are indexed as following : |
---|
507 | !!1 is subdiagonal term, 2 is diagonal and 3 is superdiagonal one |
---|
508 | |
---|
509 | !!ice interior terms (top equation has the same form as the others) |
---|
510 | |
---|
511 | DO numeq=1,jkmax+2 |
---|
512 | DO ji = kideb , kiut |
---|
513 | ztrid(ji,numeq,1) = 0. |
---|
514 | ztrid(ji,numeq,2) = 0. |
---|
515 | ztrid(ji,numeq,3) = 0. |
---|
516 | zindterm(ji,numeq)= 0. |
---|
517 | zindtbis(ji,numeq)= 0. |
---|
518 | zdiagbis(ji,numeq)= 0. |
---|
519 | ENDDO |
---|
520 | ENDDO |
---|
521 | |
---|
522 | DO numeq = nlay_s + 2, nlay_s + nlay_i |
---|
523 | DO ji = kideb , kiut |
---|
524 | layer = numeq - nlay_s - 1 |
---|
525 | ztrid(ji,numeq,1) = - zeta_i(ji,layer)*zkappa_i(ji,layer-1) |
---|
526 | ztrid(ji,numeq,2) = 1.0 + zeta_i(ji,layer)*(zkappa_i(ji,layer-1) + & |
---|
527 | zkappa_i(ji,layer)) |
---|
528 | ztrid(ji,numeq,3) = - zeta_i(ji,layer)*zkappa_i(ji,layer) |
---|
529 | zindterm(ji,numeq) = ztiold(ji,layer) + zeta_i(ji,layer)* & |
---|
530 | zradab_i(ji,layer) |
---|
531 | END DO |
---|
532 | ENDDO |
---|
533 | |
---|
534 | numeq = nlay_s + nlay_i + 1 |
---|
535 | DO ji = kideb , kiut |
---|
536 | !!ice bottom term |
---|
537 | ztrid(ji,numeq,1) = - zeta_i(ji,nlay_i)*zkappa_i(ji,nlay_i-1) |
---|
538 | ztrid(ji,numeq,2) = 1.0 + zeta_i(ji,nlay_i)*( zkappa_i(ji,nlay_i)*zg1 & |
---|
539 | + zkappa_i(ji,nlay_i-1) ) |
---|
540 | ztrid(ji,numeq,3) = 0.0 |
---|
541 | zindterm(ji,numeq) = ztiold(ji,nlay_i) + zeta_i(ji,nlay_i)* & |
---|
542 | ( zradab_i(ji,nlay_i) + zkappa_i(ji,nlay_i)*zg1 & |
---|
543 | * t_bo_b(ji) ) |
---|
544 | ENDDO |
---|
545 | |
---|
546 | |
---|
547 | DO ji = kideb , kiut |
---|
548 | IF ( ht_s_b(ji).gt.0.0 ) THEN |
---|
549 | ! |
---|
550 | !------------------------------------------------------------------------------| |
---|
551 | ! snow-covered cells | |
---|
552 | !------------------------------------------------------------------------------| |
---|
553 | ! |
---|
554 | !!snow interior terms (bottom equation has the same form as the others) |
---|
555 | DO numeq = 3, nlay_s + 1 |
---|
556 | layer = numeq - 1 |
---|
557 | ztrid(ji,numeq,1) = - zeta_s(ji,layer)*zkappa_s(ji,layer-1) |
---|
558 | ztrid(ji,numeq,2) = 1.0 + zeta_s(ji,layer)*( zkappa_s(ji,layer-1) + & |
---|
559 | zkappa_s(ji,layer) ) |
---|
560 | ztrid(ji,numeq,3) = - zeta_s(ji,layer)*zkappa_s(ji,layer) |
---|
561 | zindterm(ji,numeq) = ztsold(ji,layer) + zeta_s(ji,layer)* & |
---|
562 | zradab_s(ji,layer) |
---|
563 | END DO |
---|
564 | |
---|
565 | !!case of only one layer in the ice (ice equation is altered) |
---|
566 | IF ( nlay_i.eq.1 ) THEN |
---|
567 | ztrid(ji,nlay_s+2,3) = 0.0 |
---|
568 | zindterm(ji,nlay_s+2) = zindterm(ji,nlay_s+2) + zkappa_i(ji,1)* & |
---|
569 | t_bo_b(ji) |
---|
570 | ENDIF |
---|
571 | |
---|
572 | IF ( t_su_b(ji) .LT. rtt ) THEN |
---|
573 | |
---|
574 | !------------------------------------------------------------------------------| |
---|
575 | ! case 1 : no surface melting - snow present | |
---|
576 | !------------------------------------------------------------------------------| |
---|
577 | zdifcase(ji) = 1.0 |
---|
578 | numeqmin(ji) = 1 |
---|
579 | numeqmax(ji) = nlay_i + nlay_s + 1 |
---|
580 | |
---|
581 | !!surface equation |
---|
582 | ztrid(ji,1,1) = 0.0 |
---|
583 | ztrid(ji,1,2) = dzf(ji) - zg1s*zkappa_s(ji,0) |
---|
584 | ztrid(ji,1,3) = zg1s*zkappa_s(ji,0) |
---|
585 | zindterm(ji,1) = dzf(ji)*t_su_b(ji) - zf(ji) |
---|
586 | |
---|
587 | !!first layer of snow equation |
---|
588 | ztrid(ji,2,1) = - zkappa_s(ji,0)*zg1s*zeta_s(ji,1) |
---|
589 | ztrid(ji,2,2) = 1.0 + zeta_s(ji,1)*(zkappa_s(ji,1) + zkappa_s(ji,0)*zg1s) |
---|
590 | ztrid(ji,2,3) = - zeta_s(ji,1)* zkappa_s(ji,1) |
---|
591 | zindterm(ji,2) = ztsold(ji,1) + zeta_s(ji,1)*zradab_s(ji,1) |
---|
592 | |
---|
593 | ELSE |
---|
594 | ! |
---|
595 | !------------------------------------------------------------------------------| |
---|
596 | ! case 2 : surface is melting - snow present | |
---|
597 | !------------------------------------------------------------------------------| |
---|
598 | ! |
---|
599 | zdifcase(ji) = 2.0 |
---|
600 | numeqmin(ji) = 2 |
---|
601 | numeqmax(ji) = nlay_i + nlay_s + 1 |
---|
602 | |
---|
603 | !!first layer of snow equation |
---|
604 | ztrid(ji,2,1) = 0.0 |
---|
605 | ztrid(ji,2,2) = 1.0 + zeta_s(ji,1) * ( zkappa_s(ji,1) + & |
---|
606 | zkappa_s(ji,0) * zg1s ) |
---|
607 | ztrid(ji,2,3) = - zeta_s(ji,1)*zkappa_s(ji,1) |
---|
608 | zindterm(ji,2) = ztsold(ji,1) + zeta_s(ji,1) * & |
---|
609 | ( zradab_s(ji,1) + & |
---|
610 | zkappa_s(ji,0) * zg1s * t_su_b(ji) ) |
---|
611 | ENDIF |
---|
612 | ELSE |
---|
613 | ! |
---|
614 | !------------------------------------------------------------------------------| |
---|
615 | ! cells without snow | |
---|
616 | !------------------------------------------------------------------------------| |
---|
617 | ! |
---|
618 | IF (t_su_b(ji) .LT. rtt) THEN |
---|
619 | ! |
---|
620 | !------------------------------------------------------------------------------| |
---|
621 | ! case 3 : no surface melting - no snow | |
---|
622 | !------------------------------------------------------------------------------| |
---|
623 | ! |
---|
624 | zdifcase(ji) = 3.0 |
---|
625 | numeqmin(ji) = nlay_s + 1 |
---|
626 | numeqmax(ji) = nlay_i + nlay_s + 1 |
---|
627 | |
---|
628 | !!surface equation |
---|
629 | ztrid(ji,numeqmin(ji),1) = 0.0 |
---|
630 | ztrid(ji,numeqmin(ji),2) = dzf(ji) - zkappa_i(ji,0)*zg1 |
---|
631 | ztrid(ji,numeqmin(ji),3) = zkappa_i(ji,0)*zg1 |
---|
632 | zindterm(ji,numeqmin(ji)) = dzf(ji)*t_su_b(ji) - zf(ji) |
---|
633 | |
---|
634 | !!first layer of ice equation |
---|
635 | ztrid(ji,numeqmin(ji)+1,1) = - zkappa_i(ji,0) * zg1 * zeta_i(ji,1) |
---|
636 | ztrid(ji,numeqmin(ji)+1,2) = 1.0 + zeta_i(ji,1) * ( zkappa_i(ji,1) & |
---|
637 | + zkappa_i(ji,0) * zg1 ) |
---|
638 | ztrid(ji,numeqmin(ji)+1,3) = - zeta_i(ji,1)*zkappa_i(ji,1) |
---|
639 | zindterm(ji,numeqmin(ji)+1)= ztiold(ji,1) + zeta_i(ji,1)*zradab_i(ji,1) |
---|
640 | |
---|
641 | !!case of only one layer in the ice (surface & ice equations are altered) |
---|
642 | |
---|
643 | IF (nlay_i.eq.1) THEN |
---|
644 | ztrid(ji,numeqmin(ji),1) = 0.0 |
---|
645 | ztrid(ji,numeqmin(ji),2) = dzf(ji) - zkappa_i(ji,0)*2.0 |
---|
646 | ztrid(ji,numeqmin(ji),3) = zkappa_i(ji,0)*2.0 |
---|
647 | ztrid(ji,numeqmin(ji)+1,1) = -zkappa_i(ji,0)*2.0*zeta_i(ji,1) |
---|
648 | ztrid(ji,numeqmin(ji)+1,2) = 1.0 + zeta_i(ji,1)*(zkappa_i(ji,0)*2.0 + & |
---|
649 | zkappa_i(ji,1)) |
---|
650 | ztrid(ji,numeqmin(ji)+1,3) = 0.0 |
---|
651 | |
---|
652 | zindterm(ji,numeqmin(ji)+1) = ztiold(ji,1) + zeta_i(ji,1)* & |
---|
653 | ( zradab_i(ji,1) + zkappa_i(ji,1)*t_bo_b(ji) ) |
---|
654 | ENDIF |
---|
655 | |
---|
656 | ELSE |
---|
657 | |
---|
658 | ! |
---|
659 | !------------------------------------------------------------------------------| |
---|
660 | ! case 4 : surface is melting - no snow | |
---|
661 | !------------------------------------------------------------------------------| |
---|
662 | ! |
---|
663 | zdifcase(ji) = 4.0 |
---|
664 | numeqmin(ji) = nlay_s + 2 |
---|
665 | numeqmax(ji) = nlay_i + nlay_s + 1 |
---|
666 | |
---|
667 | !!first layer of ice equation |
---|
668 | ztrid(ji,numeqmin(ji),1) = 0.0 |
---|
669 | ztrid(ji,numeqmin(ji),2) = 1.0 + zeta_i(ji,1)*(zkappa_i(ji,1) + zkappa_i(ji,0)* & |
---|
670 | zg1) |
---|
671 | ztrid(ji,numeqmin(ji),3) = - zeta_i(ji,1) * zkappa_i(ji,1) |
---|
672 | zindterm(ji,numeqmin(ji)) = ztiold(ji,1) + zeta_i(ji,1)*( zradab_i(ji,1) + & |
---|
673 | zkappa_i(ji,0) * zg1 * t_su_b(ji) ) |
---|
674 | |
---|
675 | !!case of only one layer in the ice (surface & ice equations are altered) |
---|
676 | IF (nlay_i.eq.1) THEN |
---|
677 | ztrid(ji,numeqmin(ji),1) = 0.0 |
---|
678 | ztrid(ji,numeqmin(ji),2) = 1.0 + zeta_i(ji,1)*(zkappa_i(ji,0)*2.0 + & |
---|
679 | zkappa_i(ji,1)) |
---|
680 | ztrid(ji,numeqmin(ji),3) = 0.0 |
---|
681 | zindterm(ji,numeqmin(ji)) = ztiold(ji,1) + zeta_i(ji,1)* & |
---|
682 | (zradab_i(ji,1) + zkappa_i(ji,1)*t_bo_b(ji)) & |
---|
683 | + t_su_b(ji)*zeta_i(ji,1)*zkappa_i(ji,0)*2.0 |
---|
684 | ENDIF |
---|
685 | |
---|
686 | ENDIF |
---|
687 | ENDIF |
---|
688 | |
---|
689 | END DO |
---|
690 | |
---|
691 | ! |
---|
692 | !------------------------------------------------------------------------------| |
---|
693 | ! 9) tridiagonal system solving | |
---|
694 | !------------------------------------------------------------------------------| |
---|
695 | ! |
---|
696 | |
---|
697 | ! Solve the tridiagonal system with Gauss elimination method. |
---|
698 | ! Thomas algorithm, from Computational fluid Dynamics, J.D. ANDERSON, |
---|
699 | ! McGraw-Hill 1984. |
---|
700 | |
---|
701 | maxnumeqmax = 0 |
---|
702 | minnumeqmin = jkmax+4 |
---|
703 | |
---|
704 | DO ji = kideb , kiut |
---|
705 | zindtbis(ji,numeqmin(ji)) = zindterm(ji,numeqmin(ji)) |
---|
706 | zdiagbis(ji,numeqmin(ji)) = ztrid(ji,numeqmin(ji),2) |
---|
707 | minnumeqmin = MIN(numeqmin(ji),minnumeqmin) |
---|
708 | maxnumeqmax = MAX(numeqmax(ji),maxnumeqmax) |
---|
709 | END DO |
---|
710 | |
---|
711 | DO layer = minnumeqmin+1, maxnumeqmax |
---|
712 | DO ji = kideb , kiut |
---|
713 | numeq = min(max(numeqmin(ji)+1,layer),numeqmax(ji)) |
---|
714 | zdiagbis(ji,numeq) = ztrid(ji,numeq,2) - ztrid(ji,numeq,1)* & |
---|
715 | ztrid(ji,numeq-1,3)/zdiagbis(ji,numeq-1) |
---|
716 | zindtbis(ji,numeq) = zindterm(ji,numeq) - ztrid(ji,numeq,1)* & |
---|
717 | zindtbis(ji,numeq-1)/zdiagbis(ji,numeq-1) |
---|
718 | END DO |
---|
719 | END DO |
---|
720 | |
---|
721 | DO ji = kideb , kiut |
---|
722 | ! ice temperatures |
---|
723 | t_i_b(ji,nlay_i) = zindtbis(ji,numeqmax(ji))/zdiagbis(ji,numeqmax(ji)) |
---|
724 | END DO |
---|
725 | |
---|
726 | DO numeq = nlay_i + nlay_s + 1, nlay_s + 2, -1 |
---|
727 | DO ji = kideb , kiut |
---|
728 | layer = numeq - nlay_s - 1 |
---|
729 | t_i_b(ji,layer) = (zindtbis(ji,numeq) - ztrid(ji,numeq,3)* & |
---|
730 | t_i_b(ji,layer+1))/zdiagbis(ji,numeq) |
---|
731 | END DO |
---|
732 | END DO |
---|
733 | |
---|
734 | DO ji = kideb , kiut |
---|
735 | ! snow temperatures |
---|
736 | IF (ht_s_b(ji).GT.0) & |
---|
737 | t_s_b(ji,nlay_s) = (zindtbis(ji,nlay_s+1) - ztrid(ji,nlay_s+1,3) & |
---|
738 | * t_i_b(ji,1))/zdiagbis(ji,nlay_s+1) & |
---|
739 | * MAX(0.0,SIGN(1.0,ht_s_b(ji)-zeps)) |
---|
740 | |
---|
741 | ! surface temperature |
---|
742 | isnow(ji) = INT(1.0-max(0.0,sign(1.0,-ht_s_b(ji)))) |
---|
743 | ztsuoldit(ji) = t_su_b(ji) |
---|
744 | IF (t_su_b(ji) .LT. ztfs(ji)) & |
---|
745 | t_su_b(ji) = ( zindtbis(ji,numeqmin(ji)) - ztrid(ji,numeqmin(ji),3)* & |
---|
746 | ( isnow(ji)*t_s_b(ji,1) + & |
---|
747 | (1.0-isnow(ji))*t_i_b(ji,1) ) ) / & |
---|
748 | zdiagbis(ji,numeqmin(ji)) |
---|
749 | END DO |
---|
750 | ! |
---|
751 | !-------------------------------------------------------------------------- |
---|
752 | ! 10) Has the scheme converged ?, end of the iterative procedure | |
---|
753 | !-------------------------------------------------------------------------- |
---|
754 | ! |
---|
755 | ! check that nowhere it has started to melt |
---|
756 | ! zerrit(ji) is a measure of error, it has to be under maxer_i_thd |
---|
757 | DO ji = kideb , kiut |
---|
758 | t_su_b(ji) = MAX(MIN(t_su_b(ji),ztfs(ji)),190.0) |
---|
759 | zerrit(ji) = ABS(t_su_b(ji)-ztsuoldit(ji)) |
---|
760 | END DO |
---|
761 | |
---|
762 | DO layer = 1, nlay_s |
---|
763 | DO ji = kideb , kiut |
---|
764 | zji = MOD( npb(ji) - 1, jpi ) + 1 |
---|
765 | zjj = ( npb(ji) - 1 ) / jpi + 1 |
---|
766 | t_s_b(ji,layer) = MAX(MIN(t_s_b(ji,layer),rtt),190.0) |
---|
767 | zerrit(ji) = MAX(zerrit(ji),ABS(t_s_b(ji,layer) & |
---|
768 | - ztstemp(ji,layer))) |
---|
769 | END DO |
---|
770 | END DO |
---|
771 | |
---|
772 | DO layer = 1, nlay_i |
---|
773 | DO ji = kideb , kiut |
---|
774 | ztmelt_i = -tmut*s_i_b(ji,layer) +rtt |
---|
775 | t_i_b(ji,layer) = MAX(MIN(t_i_b(ji,layer),ztmelt_i),190.0) |
---|
776 | zerrit(ji) = MAX(zerrit(ji),ABS(t_i_b(ji,layer) - ztitemp(ji,layer))) |
---|
777 | END DO |
---|
778 | END DO |
---|
779 | |
---|
780 | ! Compute spatial maximum over all errors |
---|
781 | ! note that this could be optimized substantially by iterating only |
---|
782 | ! the non-converging points |
---|
783 | zerritmax = 0.0 |
---|
784 | DO ji = kideb , kiut |
---|
785 | zerritmax = MAX(zerritmax,zerrit(ji)) |
---|
786 | END DO |
---|
787 | IF( lk_mpp ) CALL mpp_max(zerritmax, kcom=ncomm_ice) |
---|
788 | |
---|
789 | END DO ! End of the do while iterative procedure |
---|
790 | |
---|
791 | IF( ln_nicep ) THEN |
---|
792 | WRITE(numout,*) ' zerritmax : ', zerritmax |
---|
793 | WRITE(numout,*) ' nconv : ', nconv |
---|
794 | ENDIF |
---|
795 | |
---|
796 | ! |
---|
797 | !-------------------------------------------------------------------------- |
---|
798 | ! 11) Fluxes at the interfaces | |
---|
799 | !-------------------------------------------------------------------------- |
---|
800 | ! |
---|
801 | DO ji = kideb, kiut |
---|
802 | ! update of latent heat fluxes |
---|
803 | qla_ice_1d (ji) = qla_ice_1d (ji) + & |
---|
804 | dqla_ice_1d(ji) * ( t_su_b(ji) - ztsuold(ji) ) |
---|
805 | |
---|
806 | ! surface ice conduction flux |
---|
807 | isnow(ji) = int(1.0-max(0.0,sign(1.0,-ht_s_b(ji)))) |
---|
808 | fc_su(ji) = - isnow(ji)*zkappa_s(ji,0)*zg1s*(t_s_b(ji,1) - & |
---|
809 | t_su_b(ji)) & |
---|
810 | - (1.0-isnow(ji))*zkappa_i(ji,0)*zg1* & |
---|
811 | (t_i_b(ji,1) - t_su_b(ji)) |
---|
812 | |
---|
813 | ! bottom ice conduction flux |
---|
814 | fc_bo_i(ji) = - zkappa_i(ji,nlay_i)* & |
---|
815 | ( zg1*(t_bo_b(ji) - t_i_b(ji,nlay_i)) ) |
---|
816 | |
---|
817 | END DO |
---|
818 | |
---|
819 | !-------------------------! |
---|
820 | ! Heat conservation ! |
---|
821 | !-------------------------! |
---|
822 | IF ( con_i ) THEN |
---|
823 | |
---|
824 | DO ji = kideb, kiut |
---|
825 | ! Upper snow value |
---|
826 | fc_s(ji,0) = - isnow(ji)* & |
---|
827 | zkappa_s(ji,0) * zg1s * ( t_s_b(ji,1) - & |
---|
828 | t_su_b(ji) ) |
---|
829 | ! Bott. snow value |
---|
830 | fc_s(ji,1) = - isnow(ji)* & |
---|
831 | zkappa_s(ji,1) * ( t_i_b(ji,1) - & |
---|
832 | t_s_b(ji,1) ) |
---|
833 | END DO |
---|
834 | |
---|
835 | ! Upper ice layer |
---|
836 | DO ji = kideb, kiut |
---|
837 | fc_i(ji,0) = - isnow(ji) * & ! interface flux if there is snow |
---|
838 | ( zkappa_i(ji,0) * ( t_i_b(ji,1) - t_s_b(ji,nlay_s ) ) ) & |
---|
839 | - ( 1.0 - isnow(ji) ) * ( zkappa_i(ji,0) * & |
---|
840 | zg1 * ( t_i_b(ji,1) - t_su_b(ji) ) ) ! upper flux if not |
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841 | END DO |
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842 | |
---|
843 | ! Internal ice layers |
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844 | DO layer = 1, nlay_i - 1 |
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845 | DO ji = kideb, kiut |
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846 | fc_i(ji,layer) = - zkappa_i(ji,layer) * ( t_i_b(ji,layer+1) - & |
---|
847 | t_i_b(ji,layer) ) |
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848 | zji = MOD( npb(ji) - 1, jpi ) + 1 |
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849 | zjj = ( npb(ji) - 1 ) / jpi + 1 |
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850 | END DO |
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851 | END DO |
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852 | |
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853 | ! Bottom ice layers |
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854 | DO ji = kideb, kiut |
---|
855 | fc_i(ji,nlay_i) = - zkappa_i(ji,nlay_i)* & |
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856 | ( zg1*(t_bo_b(ji) - t_i_b(ji,nlay_i)) ) |
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857 | END DO |
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858 | |
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859 | ENDIF |
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860 | |
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861 | END SUBROUTINE lim_thd_dif |
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862 | |
---|
863 | #else |
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864 | !!====================================================================== |
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865 | !! *** MODULE limthd_dif *** |
---|
866 | !! no sea ice model |
---|
867 | !!====================================================================== |
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868 | CONTAINS |
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869 | SUBROUTINE lim_thd_dif ! Empty routine |
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870 | END SUBROUTINE lim_thd_dif |
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871 | #endif |
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872 | END MODULE limthd_dif |
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