| 1 | MODULE limthd_zdf |
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| 2 | #if defined key_ice_lim |
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| 3 | !!====================================================================== |
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| 4 | !! *** MODULE limthd_zdf *** |
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| 5 | !! thermodynamic growth and decay of the ice |
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| 6 | !!====================================================================== |
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| 7 | |
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| 8 | !!---------------------------------------------------------------------- |
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| 9 | !! lim_thd_zdf : vertical accr./abl. and lateral ablation of sea ice |
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| 10 | !! * Modules used |
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| 11 | USE par_oce ! ocean parameters |
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| 12 | USE phycst ! ??? |
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| 13 | USE ice_oce ! ice variables |
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| 14 | USE thd_ice |
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| 15 | USE iceini |
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| 16 | USE limistate |
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| 17 | USE in_out_manager |
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| 18 | |
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| 19 | IMPLICIT NONE |
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| 20 | PRIVATE |
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| 21 | |
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| 22 | !! * Routine accessibility |
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| 23 | PUBLIC lim_thd_zdf ! called by lim_thd |
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| 24 | |
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| 25 | !! * Module variables |
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| 26 | REAL(wp) :: & ! constant values |
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| 27 | epsi20 = 1e-20 , & |
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| 28 | epsi13 = 1e-13 , & |
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| 29 | zzero = 0.0 , & |
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| 30 | zone = 1.0 |
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| 31 | !!---------------------------------------------------------------------- |
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| 32 | !! LIM 2.0 , UCL-LODYC-IPSL (2003) |
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| 33 | !!---------------------------------------------------------------------- |
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| 34 | CONTAINS |
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| 35 | |
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| 36 | SUBROUTINE lim_thd_zdf( kideb , kiut ) |
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| 37 | !!------------------------------------------------------------------ |
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| 38 | !! *** ROUTINE lim_thd_zdf *** |
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| 39 | !! |
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| 40 | !! ** Purpose : This routine determines the time evolution of snow |
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| 41 | !! and sea-ice thicknesses, concentration and heat content |
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| 42 | !! due to the vertical and lateral thermodynamic accretion- |
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| 43 | !! ablation processes. One only treats the case of lat. abl. |
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| 44 | !! For lateral accretion, see routine lim_lat_accr |
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| 45 | !! |
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| 46 | !! ** Method : The representation of vertical growth and decay of |
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| 47 | !! the sea-ice model is based upon the diffusion of heat |
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| 48 | !! through the external and internal boundaries of a |
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| 49 | !! three-layer system (two layers of ice and one layer and |
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| 50 | !! one layer of snow, if present, on top of the ice). |
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| 51 | !! |
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| 52 | !! ** Action : - Calculation of some intermediates variables |
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| 53 | !! - Calculation of surface temperature |
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| 54 | !! - Calculation of available heat for surface ablation |
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| 55 | !! - Calculation of the changes in internal temperature |
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| 56 | !! of the three-layer system, due to vertical diffusion |
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| 57 | !! processes |
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| 58 | !! - Performs surface ablation and bottom accretion-ablation |
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| 59 | !! - Performs snow-ice formation |
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| 60 | !! - Performs lateral ablation |
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| 61 | !! |
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| 62 | !! References : |
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| 63 | !! Fichefet T. and M. Maqueda 1997, J. Geophys. Res., 102(C6), 12609-12646 |
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| 64 | !! Fichefet T. and M. Maqueda 1999, Clim. Dyn, 15(4), 251-268 |
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| 65 | !! |
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| 66 | !! History : |
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| 67 | !! original : 01-04 (LIM) |
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| 68 | !! addition : 02-08 (C. Ethe, G. Madec) |
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| 69 | !!------------------------------------------------------------------ |
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| 70 | !! * Arguments |
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| 71 | INTEGER , INTENT (in) :: & |
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| 72 | kideb , & ! Start point on which the the computation is applied |
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| 73 | kiut ! End point on which the the computation is applied |
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| 74 | |
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| 75 | !! * Local variables |
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| 76 | INTEGER :: ji ! dummy loop indices |
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| 77 | |
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| 78 | REAL(wp) , DIMENSION(jpij,2) :: & |
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| 79 | zqcmlt ! energy due to surface( /1 ) and bottom melting( /2 ) |
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| 80 | |
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| 81 | REAL(wp), DIMENSION(jpij) :: & |
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| 82 | ztsmlt & ! snow/ice surface melting temperature |
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| 83 | ,ztbif & ! int. temp. at the mid-point of the 1st layer of the snow/ice sys. |
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| 84 | ,zksn & ! effective conductivity of snow |
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| 85 | ,zkic & ! effective conductivity of ice |
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| 86 | ,zksndh & ! thermal cond. at the mid-point of the 1st layer of the snow/ice sys. |
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| 87 | , zfcsu & ! conductive heat flux at the surface of the snow/ice system |
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| 88 | , zfcsudt & ! = zfcsu * dt |
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| 89 | , zi0 & ! frac. of the net SW rad. which is not absorbed at the surface |
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| 90 | , z1mi0 & ! fraction of the net SW radiation absorbed at the surface |
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| 91 | , zqmax & ! maximum energy stored in brine pockets |
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| 92 | , zrcpdt & ! h_su*rho_su*cp_su/dt(h_su being the thick. of surf. layer) |
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| 93 | , zts_old & ! previous surface temperature |
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| 94 | , zidsn , z1midsn , zidsnic ! tempory variables |
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| 95 | |
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| 96 | REAL(wp), DIMENSION(jpij) :: & |
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| 97 | zfnet & ! net heat flux at the top surface( incl. conductive heat flux) |
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| 98 | , zsprecip & ! snow accumulation |
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| 99 | , zhsnw_old & ! previous snow thickness |
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| 100 | , zdhictop & ! change in ice thickness due to top surf ablation/accretion |
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| 101 | , zdhicbot & ! change in ice thickness due to bottom surf abl/acc |
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| 102 | , zqsup & ! energy transmitted to ocean (coming from top surf abl/acc) |
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| 103 | , zqocea & ! energy transmitted to ocean (coming from bottom sur abl/acc) |
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| 104 | , zfrl_old & ! previous sea/ice fraction |
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| 105 | , zfrld_1d & ! new sea/ice fraction |
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| 106 | , zep ! internal temperature of the 2nd layer of the snow/ice system |
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| 107 | |
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| 108 | REAL(wp), DIMENSION(3) :: & |
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| 109 | zplediag & ! principle diagonal, subdiag. and supdiag. of the |
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| 110 | , zsubdiag & ! tri-diagonal matrix coming from the computation |
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| 111 | , zsupdiag & ! of the temperatures inside the snow-ice system |
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| 112 | , zsmbr ! second member |
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| 113 | |
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| 114 | REAL(wp) :: & |
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| 115 | zhsu & ! thickness of surface layer |
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| 116 | , zhe & ! effective thickness for compu. of equ. thermal conductivity |
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| 117 | , zheshth & ! = zhe / thth |
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| 118 | , zghe & ! correction factor of the thermal conductivity |
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| 119 | , zumsb & ! parameter for numerical method to solve heat-diffusion eq. |
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| 120 | , zkhsn & ! conductivity at the snow layer |
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| 121 | , zkhic & ! conductivity at the ice layers |
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| 122 | , zkint & ! equivalent conductivity at the snow-ice interface |
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| 123 | , zkhsnint & ! = zkint*dt / (hsn*rhosn*cpsn) |
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| 124 | , zkhicint & ! = 2*zkint*dt / (hic*rhoic*cpic) |
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| 125 | , zpiv1 , zpiv2 & ! tempory scalars used to solve the tri-diagonal system |
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| 126 | , zb2 , zd2 , zb3 , zd3 & |
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| 127 | , ztint ! equivalent temperature at the snow-ice interface |
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| 128 | |
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| 129 | REAL(wp) :: & |
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| 130 | zexp & ! exponential function of the ice thickness |
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| 131 | , zfsab & ! part of solar radiation stored in brine pockets |
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| 132 | , zfts & ! value of energy balance function when the temp. equal surf. temp. |
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| 133 | , zdfts & ! value of derivative of ztfs when the temp. equal surf. temp. |
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| 134 | , zdts & ! surface temperature increment |
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| 135 | , zqsnw_mlt & ! energy needed to melt snow |
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| 136 | , zdhsmlt & ! change in snow thickness due to melt |
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| 137 | , zhsn & ! snow thickness (previous+accumulation-melt) |
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| 138 | , zqsn_mlt_rem & ! remaining heat coming from snow melting |
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| 139 | , zqice_top_mlt & ! energy used to melt ice at top surface |
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| 140 | , zdhssub & ! change in snow thick. due to sublimation or evaporation |
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| 141 | , zdhisub & ! change in ice thick. due to sublimation or evaporation |
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| 142 | , zdhsn & ! snow ice thickness increment |
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| 143 | , zdtsn & ! snow internal temp. increment |
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| 144 | , zdtic & ! ice internal temp. increment |
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| 145 | , zqnes ! conductive energy due to ice melting in the first ice layer |
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| 146 | |
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| 147 | REAL(wp) :: & |
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| 148 | ztbot & ! temperature at the bottom surface |
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| 149 | , zfcbot & ! conductive heat flux at bottom surface |
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| 150 | , zqice_bot & ! energy used for bottom melting/growing |
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| 151 | , zqice_bot_mlt & ! energy used for bottom melting |
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| 152 | , zqstbif_bot & ! part of energy stored in brine pockets used for bottom melting |
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| 153 | , zqstbif_old & ! tempory var. for zqstbif_bot |
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| 154 | , zdhicmlt & ! change in ice thickness due to bottom melting |
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| 155 | , zdhicm & ! change in ice thickness var. |
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| 156 | , zdhsnm & ! change in snow thickness var. |
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| 157 | , zhsnfi & ! snow thickness var. |
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| 158 | , zc1, zpc1, zc2, zpc2, zp1, zp2 & ! tempory variables |
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| 159 | , ztb2, ztb3 |
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| 160 | |
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| 161 | REAL(wp) :: & |
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| 162 | zdrmh & ! change in snow/ice thick. after snow-ice formation |
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| 163 | , zhicnew & ! new ice thickness |
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| 164 | , zhsnnew & ! new snow thickness |
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| 165 | , zquot , ztneq & ! tempory temp. variables |
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| 166 | , zqice, zqicetot & ! total heat inside the snow/ice system |
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| 167 | , zdfrl & ! change in ice concentration |
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| 168 | , zdvsnvol & ! change in snow volume |
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| 169 | , zdrfrl1, zdrfrl2 & ! tempory scalars |
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| 170 | , zihsn, zidhb, zihic, zihe, zihq, ziexp, ziqf, zihnf, zibmlt, ziqr, zihgnew, zind |
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| 171 | !!---------------------------------------------------------------------- |
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| 172 | |
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| 173 | !----------------------------------------------------------------------- |
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| 174 | ! 1. Boundaries conditions for snow/ice system internal temperature |
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| 175 | ! - If tbif_1d(ji,1) > rt0_snow, tbif_1d(ji,1) = rt0_snow |
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| 176 | ! - If tbif_1d(ji,2/3) > rt0_ice, tbif_1d(ji,2/3) = rt0_ice |
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| 177 | ! Computation of energies due to surface and bottom melting |
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| 178 | !----------------------------------------------------------------------- |
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| 179 | |
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| 180 | |
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| 181 | DO ji = kideb , kiut |
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| 182 | zihsn = MAX( zzero , SIGN( zone , hsndif - h_snow_1d(ji) ) ) |
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| 183 | zihic = MAX( zzero , SIGN( zone , hicdif - h_ice_1d(ji) ) ) |
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| 184 | !--computation of energy due to surface melting |
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| 185 | zqcmlt(ji,1) = ( MAX ( zzero , & |
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| 186 | & rcpsn * h_snow_1d(ji) * ( tbif_1d(ji,1) - rt0_snow ) ) ) * ( 1.0 - zihsn ) |
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| 187 | !--computation of energy due to bottom melting |
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| 188 | zqcmlt(ji,2) = ( MAX( zzero , & |
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| 189 | & rcpic * ( tbif_1d(ji,2) - rt0_ice ) * ( h_ice_1d(ji) / 2. ) ) & |
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| 190 | & + MAX( zzero , & |
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| 191 | & rcpic * ( tbif_1d(ji,3) - rt0_ice ) * ( h_ice_1d(ji) / 2. ) ) & |
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| 192 | & ) * ( 1.0 - zihic ) |
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| 193 | !--limitation of snow/ice system internal temperature |
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| 194 | tbif_1d(ji,1) = MIN( rt0_snow, tbif_1d(ji,1) ) |
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| 195 | tbif_1d(ji,2) = MIN( rt0_ice , tbif_1d(ji,2) ) |
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| 196 | tbif_1d(ji,3) = MIN( rt0_ice , tbif_1d(ji,3) ) |
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| 197 | END DO |
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| 198 | |
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| 199 | |
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| 200 | !------------------------------------------- |
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| 201 | ! 2. Calculate some intermediate variables. |
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| 202 | !------------------------------------------- |
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| 203 | |
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| 204 | ! initialisation of the thickness of surface layer |
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| 205 | zhsu = hnzst |
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| 206 | |
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| 207 | DO ji = kideb , kiut |
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| 208 | zind = MAX( zzero , SIGN( zone , zhsu - h_snow_1d(ji) ) ) |
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| 209 | zihsn = MAX( zzero , SIGN( zone , hsndif - h_snow_1d(ji) ) ) |
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| 210 | zihsn = MAX( zihsn , zind ) |
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| 211 | zihic = MAX( zzero , sign( zone , hicdif - h_ice_1d(ji) ) ) |
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| 212 | ! 2.1. Computation of surface melting temperature |
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| 213 | !---------------------------------------------------- |
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| 214 | zind = MAX( zzero , SIGN( zone , -h_snow_1d(ji) ) ) |
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| 215 | ztsmlt(ji) = ( 1.0 - zind ) * rt0_snow + zind * rt0_ice |
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| 216 | ! |
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| 217 | ! 2.2. Effective conductivity of snow and ice |
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| 218 | !----------------------------------------------- |
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| 219 | |
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| 220 | !---computation of the correction factor on the thermal conductivity |
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| 221 | !-- (Morales Maqueda, 1995 ; Fichefet and Morales Maqueda, 1997) |
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| 222 | zhe = ( rcdsn / ( rcdsn + rcdic ) ) * h_ice_1d(ji) & |
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| 223 | & + ( rcdic / ( rcdsn + rcdic ) ) * h_snow_1d(ji) |
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| 224 | zihe = MAX( zzero , SIGN( zone , 2.0 * zhe - thth ) ) |
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| 225 | zheshth = zhe / thth |
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| 226 | zghe = ( 1.0 - zihe ) * zheshth * ( 2.0 - zheshth ) & |
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| 227 | & + zihe * 0.5 * ( 1.5 + LOG( 2.0 * zheshth ) ) |
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| 228 | #if defined key_lim_cp3 |
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| 229 | zghe = 1.0 |
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| 230 | #endif |
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| 231 | |
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| 232 | !---effective conductivities |
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| 233 | zksn(ji) = zghe * rcdsn |
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| 234 | zkic(ji) = zghe * rcdic |
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| 235 | |
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| 236 | ! |
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| 237 | ! 2.3. Computation of the conductive heat flux from the snow/ice |
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| 238 | ! system interior toward the top surface |
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| 239 | !------------------------------------------------------------------ |
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| 240 | |
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| 241 | !---Thermal conductivity at the mid-point of the first snow/ice system layer |
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| 242 | zksndh(ji) = ( ( 1.0 - zihsn ) * 2.0 * zksn(ji) + zihsn * 4.0 * zkic(ji) ) & |
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| 243 | & / ( ( 1.0 - zihsn ) * h_snow_1d(ji) & |
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| 244 | & + zihsn * ( ( 1.0 + 3.0 * zihic ) * h_ice_1d(ji) & |
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| 245 | & + 4.0 * zkic(ji)/zksn(ji) * h_snow_1d(ji) ) ) |
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| 246 | |
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| 247 | !---internal temperature at the mid-point of the first snow/ice system layer |
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| 248 | ztbif(ji) = ( 1.0 - zihsn ) * tbif_1d(ji,1) & |
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| 249 | & + zihsn * ( ( 1.0 - zihic ) * tbif_1d(ji,2) & |
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| 250 | & + zihic * tfu_1d(ji) ) |
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| 251 | !---conductive heat flux |
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| 252 | zfcsu(ji) = zksndh(ji) * ( ztbif(ji) - sist_1d(ji) ) |
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| 253 | |
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| 254 | END DO |
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| 255 | |
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| 256 | !-------------------------------------------------------------------- |
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| 257 | ! 3. Calculate : |
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| 258 | ! - fstbif_1d, part of solar radiation absorbing inside the ice |
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| 259 | ! assuming an exponential absorption (Grenfell and Maykut, 1977) |
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| 260 | ! - zqmax, maximum energy stored in brine pockets |
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| 261 | ! - qstbif_1d, total energy stored in brine pockets (updating) |
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| 262 | !------------------------------------------------------------------- |
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| 263 | |
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| 264 | |
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| 265 | DO ji = kideb , kiut |
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| 266 | zihsn = MAX( zzero , SIGN (zone , -h_snow_1d(ji) ) ) |
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| 267 | zihic = MAX( zzero , 1.0 - ( h_ice_1d(ji) / zhsu ) ) |
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| 268 | zind = MAX( zzero , SIGN (zone , hicdif - h_ice_1d(ji) ) ) |
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| 269 | !--Computation of the fraction of the net shortwave radiation which |
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| 270 | !--penetrates inside the ice cover ( See Forcat) |
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| 271 | zi0(ji) = zihsn * ( fr1_i0_1d(ji) + zihic * fr2_i0_1d(ji) ) |
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| 272 | zexp = MIN( zone , EXP( -1.5 * ( h_ice_1d(ji) - zhsu ) ) ) |
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| 273 | fstbif_1d(ji) = zi0(ji) * qsr_ice_1d(ji) * zexp |
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| 274 | !--Computation of maximum energy stored in brine pockets zqmax and update |
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| 275 | !--the total energy stored in brine pockets, if less than zqmax |
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| 276 | zqmax(ji) = MAX( zzero , 0.5 * xlic * ( h_ice_1d(ji) - hicmin ) ) |
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| 277 | zfsab = zi0(ji) * qsr_ice_1d(ji) * ( 1.0 - zexp ) |
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| 278 | zihq = ( 1.0 - zind ) * MAX(zzero, SIGN( zone , qstbif_1d(ji) - zqmax(ji) ) ) & |
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| 279 | & + zind * zone |
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| 280 | qstbif_1d(ji) = ( qstbif_1d(ji) + ( 1.0 - zihq ) * zfsab * rdt_ice ) * swiqst |
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| 281 | !--fraction of shortwave radiation absorbed at surface |
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| 282 | ziexp = zihq * zexp + ( 1.0 - zihq ) * ( swiqst + ( 1.0 - swiqst ) * zexp ) |
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| 283 | z1mi0(ji) = 1.0 - zi0(ji) * ziexp |
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| 284 | END DO |
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| 285 | |
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| 286 | |
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| 287 | !-------------------------------------------------------------------------------- |
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| 288 | ! 4. Computation of the surface temperature : determined by considering the |
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| 289 | ! budget of a thin layer of thick. zhsu at the top surface (H. Grenier, 1995) |
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| 290 | ! and based on a surface energy balance : |
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| 291 | ! hsu * rcp * dT/dt = Fsr + Fnsr(T) + Fcs(T), |
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| 292 | ! where - Fsr is the net absorbed solar radiation, |
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| 293 | ! - Fnsr is the total non solar radiation (incoming and outgoing long-wave, |
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| 294 | ! sensible and latent heat fluxes) |
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| 295 | ! - Fcs the conductive heat flux at the top of surface |
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| 296 | !------------------------------------------------------------------------------ |
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| 297 | |
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| 298 | ! 4.1. Computation of intermediate values |
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| 299 | !--------------------------------------------- |
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| 300 | DO ji = kideb, kiut |
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| 301 | zrcpdt(ji) = ( rcpsn * MIN( h_snow_1d(ji) , zhsu ) & |
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| 302 | & + rcpic * MAX( zhsu - h_snow_1d(ji) , zzero ) ) / rdt_ice |
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| 303 | zts_old(ji) = sist_1d(ji) |
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| 304 | END DO |
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| 305 | |
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| 306 | ! 4.2. Computation of surface temperature by expanding the eq. of energy balance |
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| 307 | ! with Ts = Tp + DT. One obtain , F(Tp) + DT * DF(Tp) = 0 |
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| 308 | ! where - F(Tp) = Fsr + Fnsr(Tp) + Fcs(Tp) |
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| 309 | ! - DF(Tp)= (dFnsr(Tp)/dT) + (dFcs(Tp)/dT) - hsu*rcp/dt |
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| 310 | !--------------------------------------------------------------------------------- |
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| 311 | |
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| 312 | DO ji = kideb, kiut |
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| 313 | !---computation of the derivative of energy balance function |
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| 314 | #if defined key_coupled |
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| 315 | # if defined key_lim_cp2 |
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| 316 | zdfts = zksndh(ji) & ! contribution of the conductive heat flux |
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| 317 | & + zrcpdt(ji) & ! contribution of hsu * rcp / dt |
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| 318 | & - dqns_ice_1d(ji) ! contribution of the total non solar radiation |
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| 319 | # else |
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| 320 | zdfts = zksndh(ji) & ! contribution of the conductive heat flux |
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| 321 | & + zrcpdt(ji) ! contribution of hsu * rcp / dt |
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| 322 | # endif |
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| 323 | |
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| 324 | #else |
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| 325 | zdfts = zksndh(ji) & ! contribution of the conductive heat flux |
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| 326 | & + zrcpdt(ji) & ! contribution of hsu * rcp / dt |
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| 327 | & - dqns_ice_1d (ji) ! contribution of the total non solar radiation |
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| 328 | #endif |
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| 329 | !---computation of the energy balance function |
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| 330 | zfts = - z1mi0 (ji) * qsr_ice_1d(ji) & ! net absorbed solar radiation |
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| 331 | & - qnsr_ice_1d(ji) & ! total non solar radiation |
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| 332 | & - zfcsu (ji) ! conductive heat flux from the surface |
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| 333 | !---computation of surface temperature increment |
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| 334 | zdts = -zfts / zdfts |
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| 335 | #if defined key_lim_cp3 |
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| 336 | zdts = zdts / 3.0 |
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| 337 | #endif |
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| 338 | !---computation of the new surface temperature |
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| 339 | sist_1d(ji) = sist_1d(ji) + zdts |
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| 340 | |
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| 341 | END DO |
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| 342 | |
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| 343 | !---------------------------------------------------------------------------- |
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| 344 | ! 5. Boundary condition at the top surface |
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| 345 | !-- IF Tsb < Tmelt, Fnet = Fcs (the net heat flux equal the conductive heat flux) |
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| 346 | ! Otherwise Tsb = Tmelt and Qnet(Tmelt) > 0 |
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| 347 | ! Fnet(Tmelt) is therefore the net surface flux needed for melting |
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| 348 | !---------------------------------------------------------------------------- |
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| 349 | |
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| 350 | |
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| 351 | ! 5.1. Limitation of surface temperature and update total non solar fluxes, |
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| 352 | ! latent heat flux and conductive flux at the top surface |
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| 353 | !---------------------------------------------------------------------- |
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| 354 | |
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| 355 | DO ji = kideb, kiut |
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| 356 | sist_1d(ji) = MIN( ztsmlt(ji) , sist_1d(ji) ) |
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| 357 | #if defined key_coupled |
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| 358 | qnsr_ice_1d(ji) = qnsr_ice_1d(ji) - ( emic * stefan * sist_1d(ji) * sist_1d(ji) * sist_1d(ji) * sist_1d(ji) ) |
|---|
| 359 | #else |
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| 360 | qnsr_ice_1d(ji) = qnsr_ice_1d(ji) + dqns_ice_1d(ji) * ( sist_1d(ji) - zts_old(ji) ) |
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| 361 | qla_ice_1d (ji) = qla_ice_1d (ji) + dqla_ice_1d(ji) * ( sist_1d(ji) - zts_old(ji) ) |
|---|
| 362 | #endif |
|---|
| 363 | zfcsu(ji) = zksndh(ji) * ( ztbif(ji) - sist_1d(ji) ) |
|---|
| 364 | END DO |
|---|
| 365 | |
|---|
| 366 | |
|---|
| 367 | |
|---|
| 368 | ! 5.2. Calculate available heat for surface ablation. |
|---|
| 369 | !--------------------------------------------------------------------- |
|---|
| 370 | |
|---|
| 371 | DO ji = kideb, kiut |
|---|
| 372 | zfnet(ji) = qnsr_ice_1d(ji) + z1mi0(ji) * qsr_ice_1d(ji) + zfcsu(ji) |
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| 373 | zfnet(ji) = MAX( zzero , zfnet(ji) ) |
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| 374 | zfnet(ji) = zfnet(ji) * MAX( zzero , SIGN( zone , sist_1d(ji) - ztsmlt(ji) ) ) |
|---|
| 375 | END DO |
|---|
| 376 | |
|---|
| 377 | !------------------------------------------------------------------------- |
|---|
| 378 | ! 6. Calculate changes in internal temperature due to vertical diffusion |
|---|
| 379 | ! processes. The evolution of this temperature is governed by the one- |
|---|
| 380 | ! dimensionnal heat-diffusion equation. |
|---|
| 381 | ! Given the temperature tbif(1/2/3), at time m we solve a set |
|---|
| 382 | ! of finite difference equations to obtain new tempe. Each tempe is coupled |
|---|
| 383 | ! to the temp. immediatly above and below by heat conduction terms. Thus |
|---|
| 384 | ! we have a set of equations of the form A * T = B, where A is a tridiagonal |
|---|
| 385 | ! matrix, T a vector whose components are the unknown new temp. |
|---|
| 386 | !------------------------------------------------------------------------- |
|---|
| 387 | |
|---|
| 388 | !--parameter for the numerical methode use to solve the heat-diffusion equation |
|---|
| 389 | !- implicit, explicit or Crank-Nicholson |
|---|
| 390 | zumsb = 1.0 - sbeta |
|---|
| 391 | DO ji = kideb, kiut |
|---|
| 392 | zidsn(ji) = MAX ( zzero, SIGN( zone, hsndif - h_snow_1d(ji) ) ) |
|---|
| 393 | z1midsn(ji) = 1.0 - zidsn(ji) |
|---|
| 394 | zihic = MAX ( zzero, SIGN( zone, hicdif - h_ice_1d(ji) ) ) |
|---|
| 395 | zidsnic(ji) = zidsn(ji) * zihic |
|---|
| 396 | zfcsudt(ji) = zfcsu(ji) * rdt_ice |
|---|
| 397 | END DO |
|---|
| 398 | |
|---|
| 399 | DO ji = kideb, kiut |
|---|
| 400 | |
|---|
| 401 | ! 6.1 Calculate intermediate variables. |
|---|
| 402 | !---------------------------------------- |
|---|
| 403 | |
|---|
| 404 | !--conductivity at the snow surface |
|---|
| 405 | zkhsn = 2.0 * zksn(ji) * rdt_ice / rcpsn |
|---|
| 406 | !--conductivity at the ice surface |
|---|
| 407 | zkhic = 4.0 * zkic(ji) * rdt_ice / MAX( h_ice_1d(ji) * h_ice_1d(ji) * rcpic , epsi20 ) |
|---|
| 408 | !--conductivity at the snow/ice interface |
|---|
| 409 | zkint = 4.0 * zksn(ji) * zkic(ji) & |
|---|
| 410 | & / ( zksn(ji) * h_ice_1d(ji) + 2.0 * zkic(ji) * h_snow_1d(ji) * z1midsn(ji)) |
|---|
| 411 | zkhsnint = zkint * rdt_ice / rcpsn |
|---|
| 412 | zkhicint = zkint * 2.0 * rdt_ice / MAX( h_ice_1d(ji) * rcpic , epsi20 ) |
|---|
| 413 | |
|---|
| 414 | ! 6.2. Fulfill the linear system matrix. |
|---|
| 415 | !----------------------------------------- |
|---|
| 416 | !$$$ zplediag(1) = 1 + sbeta * z1midsn(ji) * ( zkhsn + zkhsnint ) |
|---|
| 417 | zplediag(1) = zidsn(ji) + z1midsn(ji) * h_snow_1d(ji) & |
|---|
| 418 | & + sbeta * z1midsn(ji) * zkhsnint |
|---|
| 419 | zplediag(2) = 1 + sbeta * ( z1midsn(ji) * zkhicint + zkhic ) |
|---|
| 420 | zplediag(3) = 1 + 3.0 * sbeta * zkhic |
|---|
| 421 | |
|---|
| 422 | zsubdiag(1) = 0.0 |
|---|
| 423 | zsubdiag(2) = -1.0 * z1midsn(ji) * sbeta * zkhicint |
|---|
| 424 | zsubdiag(3) = -1.0 * sbeta * zkhic |
|---|
| 425 | |
|---|
| 426 | zsupdiag(1) = -1.0 * z1midsn(ji) * sbeta * zkhsnint |
|---|
| 427 | zsupdiag(2) = zsubdiag(3) |
|---|
| 428 | zsupdiag(3) = 0.0 |
|---|
| 429 | |
|---|
| 430 | ! 6.3. Fulfill the idependent term vector. |
|---|
| 431 | !------------------------------------------- |
|---|
| 432 | |
|---|
| 433 | !$$$ zsmbr(1) = zidsn(ji) * sist_1d(ji) + z1midsn(ji) * & |
|---|
| 434 | !$$$ & ( tbif_1d(ji,1) + zkhsn * sist_1d(ji) |
|---|
| 435 | !$$$ & - zumsb * ( zkhsn * tbif_1d(ji,1) |
|---|
| 436 | !$$$ & + zkhsnint * ( tbif_1d(ji,1) - tbif_1d(ji,2) ) ) ) |
|---|
| 437 | zsmbr(1) = zidsn(ji) * sist_1d(ji) + z1midsn(ji) * & |
|---|
| 438 | & ( h_snow_1d(ji) * tbif_1d(ji,1) - ( zfcsudt(ji) / rcpsn ) & |
|---|
| 439 | & - zumsb * zkhsnint * ( tbif_1d(ji,1) - tbif_1d(ji,2) ) ) |
|---|
| 440 | |
|---|
| 441 | zsmbr(2) = tbif_1d(ji,2) & |
|---|
| 442 | & - zidsn(ji) * ( 1.0 - zidsnic(ji) ) & |
|---|
| 443 | & * ( zfcsudt(ji) / MAX( h_ice_1d(ji) * rcpic , epsi20 ) ) & |
|---|
| 444 | & + zumsb * ( zkhicint * ( tbif_1d(ji,1) - tbif_1d(ji,2) ) & |
|---|
| 445 | & - zkhic * ( tbif_1d(ji,2) - tbif_1d(ji,3) ) ) |
|---|
| 446 | |
|---|
| 447 | zsmbr(3) = tbif_1d(ji,3) & |
|---|
| 448 | & + zkhic * ( 2.0 * tfu_1d(ji) & |
|---|
| 449 | & + zumsb * ( tbif_1d(ji,2) - 3.0 * tbif_1d(ji,3) ) ) |
|---|
| 450 | |
|---|
| 451 | ! 6.4. Solve the system (Gauss elimination method). |
|---|
| 452 | !---------------------------------------------------- |
|---|
| 453 | |
|---|
| 454 | zpiv1 = zsubdiag(2) / zplediag(1) |
|---|
| 455 | zb2 = zplediag(2) - zpiv1 * zsupdiag(1) |
|---|
| 456 | zd2 = zsmbr(2) - zpiv1 * zsmbr(1) |
|---|
| 457 | |
|---|
| 458 | zpiv2 = zsubdiag(3) / zb2 |
|---|
| 459 | zb3 = zplediag(3) - zpiv2 * zsupdiag(2) |
|---|
| 460 | zd3 = zsmbr(3) - zpiv2 * zd2 |
|---|
| 461 | |
|---|
| 462 | tbif_1d(ji,3) = zd3 / zb3 |
|---|
| 463 | tbif_1d(ji,2) = ( zd2 - zsupdiag(2) * tbif_1d(ji,3) ) / zb2 |
|---|
| 464 | tbif_1d(ji,1) = ( zsmbr(1) - zsupdiag(1) * tbif_1d(ji,2) ) / zplediag(1) |
|---|
| 465 | |
|---|
| 466 | !--- taking into account the particular case of zidsnic(ji) = 1 |
|---|
| 467 | ztint = ( zkic(ji) * h_snow_1d(ji) * tfu_1d (ji) & |
|---|
| 468 | & + zksn(ji) * h_ice_1d(ji) * sist_1d(ji) ) & |
|---|
| 469 | & / ( zkic(ji) * h_snow_1d(ji) + zksn(ji) * h_ice_1d(ji) ) |
|---|
| 470 | |
|---|
| 471 | tbif_1d(ji,1) = ( 1.0 - zidsnic(ji) ) * tbif_1d(ji,1) & |
|---|
| 472 | & + zidsnic(ji) * ( ztint + sist_1d(ji) ) / 2.0 |
|---|
| 473 | tbif_1d(ji,2) = ( 1.0 - zidsnic(ji) ) * tbif_1d(ji,2) & |
|---|
| 474 | & + zidsnic(ji) * ( 3.0 * ztint + tfu_1d(ji) ) / 4.0 |
|---|
| 475 | tbif_1d(ji,3) = ( 1.0 - zidsnic(ji) ) * tbif_1d(ji,3) & |
|---|
| 476 | & + zidsnic(ji) * ( ztint + 3.0 * tfu_1d(ji) ) / 4.0 |
|---|
| 477 | END DO |
|---|
| 478 | |
|---|
| 479 | !---------------------------------------------------------------------- |
|---|
| 480 | ! 9. Take into account surface ablation and bottom accretion-ablation.| |
|---|
| 481 | !---------------------------------------------------------------------- |
|---|
| 482 | |
|---|
| 483 | !---Snow accumulation in one thermodynamic time step |
|---|
| 484 | zsprecip(kideb:kiut) = sprecip_1d(kideb:kiut) * rdt_ice / rhosn |
|---|
| 485 | |
|---|
| 486 | |
|---|
| 487 | DO ji = kideb, kiut |
|---|
| 488 | |
|---|
| 489 | ! 9.1. Surface ablation and update of snow thickness and qstbif_1d |
|---|
| 490 | !-------------------------------------------------------------------- |
|---|
| 491 | |
|---|
| 492 | !-------------------------------------------------------------------------- |
|---|
| 493 | !-- Melting snow processes : |
|---|
| 494 | !-- Melt at the upper surface is computed from the difference between |
|---|
| 495 | !-- the net heat flux (including the conductive heat flux) at the upper |
|---|
| 496 | !-- surface and the pre-existing energy due to surface melting |
|---|
| 497 | !------------------------------------------------------------------------------ |
|---|
| 498 | |
|---|
| 499 | !-- store the snow thickness |
|---|
| 500 | zhsnw_old(ji) = h_snow_1d(ji) |
|---|
| 501 | !--computation of the energy needed to melt snow |
|---|
| 502 | zqsnw_mlt = zfnet(ji) * rdt_ice - zqcmlt(ji,1) |
|---|
| 503 | !--change in snow thickness due to melt |
|---|
| 504 | zdhsmlt = - zqsnw_mlt / xlsn |
|---|
| 505 | |
|---|
| 506 | !-- compute new snow thickness, taking into account the part of snow accumulation |
|---|
| 507 | ! (as snow precipitation) and the part of snow lost due to melt |
|---|
| 508 | zhsn = h_snow_1d(ji) + zsprecip(ji) + zdhsmlt |
|---|
| 509 | h_snow_1d(ji) = MAX( zzero , zhsn ) |
|---|
| 510 | !-- compute the volume of snow lost after surface melting and the associated mass |
|---|
| 511 | dvsbq_1d(ji) = ( 1.0 - frld_1d(ji) ) * ( h_snow_1d(ji) - zhsnw_old(ji) - zsprecip(ji) ) |
|---|
| 512 | dvsbq_1d(ji) = MIN( zzero , dvsbq_1d(ji) ) |
|---|
| 513 | rdmsnif_1d(ji) = rhosn * dvsbq_1d(ji) |
|---|
| 514 | !-- If the snow is completely melted the remaining heat is used to melt ice |
|---|
| 515 | zqsn_mlt_rem = MAX( zzero , -zhsn ) * xlsn |
|---|
| 516 | zqice_top_mlt = zqsn_mlt_rem |
|---|
| 517 | zqstbif_old = qstbif_1d(ji) |
|---|
| 518 | |
|---|
| 519 | !-------------------------------------------------------------------------- |
|---|
| 520 | !-- Melting ice processes at the top surface : |
|---|
| 521 | !-- The energy used to melt ice, zqice_top_mlt, is taken from the energy |
|---|
| 522 | !-- stored in brine pockets qstbif_1d and the remaining energy coming |
|---|
| 523 | !-- from the melting snow process zqsn_mlt_rem. |
|---|
| 524 | !-- If qstbif_1d > zqsn_mlt_rem then, one uses only a zqsn_mlt_rem part |
|---|
| 525 | !-- of qstbif_1d to melt ice, |
|---|
| 526 | !-- zqice_top_mlt = zqice_top_mlt + zqsn_mlt_rem |
|---|
| 527 | !-- qstbif_1d = qstbif_1d - zqsn_mlt_rem |
|---|
| 528 | !-- Otherwise one uses all qstbif_1d to melt ice |
|---|
| 529 | !-- zqice_top_mlt = zqice_top_mlt + qstbif_1d |
|---|
| 530 | !-- qstbif_1d = 0 |
|---|
| 531 | !------------------------------------------------------ |
|---|
| 532 | |
|---|
| 533 | ziqf = MAX ( zzero , SIGN( zone , qstbif_1d(ji) - zqsn_mlt_rem ) ) |
|---|
| 534 | zqice_top_mlt = ziqf * ( zqice_top_mlt + zqsn_mlt_rem ) & |
|---|
| 535 | & + ( 1.0 - ziqf ) * ( zqice_top_mlt + qstbif_1d(ji) ) |
|---|
| 536 | |
|---|
| 537 | qstbif_1d(ji) = ziqf * ( qstbif_1d(ji) - zqsn_mlt_rem ) & |
|---|
| 538 | & + ( 1.0 - ziqf ) * ( qstbif_1d(ji) - qstbif_1d(ji) ) |
|---|
| 539 | |
|---|
| 540 | !-- The contribution of the energy stored in brine pockets qstbif_1d to melt |
|---|
| 541 | !-- ice is taking into account only when qstbif_1d is less than zqmax. |
|---|
| 542 | !-- Otherwise, only the remaining energy coming from the melting snow |
|---|
| 543 | !-- process is used |
|---|
| 544 | zihq = MAX ( zzero , SIGN( zone , qstbif_1d(ji) - zqmax(ji) ) ) |
|---|
| 545 | |
|---|
| 546 | zqice_top_mlt = zihq * zqice_top_mlt & |
|---|
| 547 | & + ( 1.0 - zihq ) * zqsn_mlt_rem |
|---|
| 548 | |
|---|
| 549 | qstbif_1d(ji) = zihq * qstbif_1d(ji) & |
|---|
| 550 | & + ( 1.0 - zihq ) * zqstbif_old |
|---|
| 551 | |
|---|
| 552 | !--change in ice thickness due to melt at the top surface |
|---|
| 553 | zdhictop(ji) = -zqice_top_mlt / xlic |
|---|
| 554 | !--compute the volume formed after surface melting |
|---|
| 555 | dvsbq_1d(ji) = zdhictop(ji) * ( 1.0 - frld_1d(ji) ) |
|---|
| 556 | |
|---|
| 557 | !------------------------------------------------------------------------- |
|---|
| 558 | !-- A small variation at the surface also occurs because of sublimation |
|---|
| 559 | !-- associated with the latent flux. If qla_ice_1d is negative, snow condensates at |
|---|
| 560 | ! the surface. Otherwise, snow evaporates |
|---|
| 561 | !----------------------------------------------------------------------- |
|---|
| 562 | !----change in snow and ice thicknesses due to sublimation or evaporation |
|---|
| 563 | zdhssub = parsub * ( qla_ice_1d(ji) / ( rhosn * xsn ) ) * rdt_ice |
|---|
| 564 | zhsn = h_snow_1d(ji) - zdhssub |
|---|
| 565 | zdhisub = MAX( zzero , -zhsn ) * rhosn/rhoic |
|---|
| 566 | zdhictop(ji) = zdhictop(ji) - zdhisub |
|---|
| 567 | h_snow_1d(ji) = MAX( zzero , zhsn ) |
|---|
| 568 | !------------------------------------------------- |
|---|
| 569 | !-- Update Internal temperature and qstbif_1d. |
|---|
| 570 | !------------------------------------------- |
|---|
| 571 | zihsn = MAX( zzero , SIGN( zone, -h_snow_1d(ji) ) ) |
|---|
| 572 | tbif_1d(ji,1) = ( 1.0 - zihsn ) * tbif_1d(ji,1) + zihsn * tfu_1d(ji) |
|---|
| 573 | !--change in snow internal temperature if snow has increased |
|---|
| 574 | zihnf = MAX( zzero , SIGN( zone , h_snow_1d(ji) - zhsnw_old(ji) ) ) |
|---|
| 575 | zdhsn = 1.0 - zhsnw_old(ji) / MAX( h_snow_1d(ji) , epsi20 ) |
|---|
| 576 | zdtsn = zdhsn * ( sist_1d(ji) - tbif_1d(ji,1) ) |
|---|
| 577 | tbif_1d(ji,1) = tbif_1d(ji,1) + z1midsn(ji) * zihnf * zdtsn |
|---|
| 578 | !--energy created due to ice melting in the first ice layer |
|---|
| 579 | zqnes = ( rt0_ice - tbif_1d(ji,2) ) * rcpic * ( h_ice_1d(ji) / 2. ) |
|---|
| 580 | !--change in first ice layer internal temperature |
|---|
| 581 | ziqr = MAX( zzero , SIGN( zone , qstbif_1d(ji) - zqnes ) ) |
|---|
| 582 | zdtic = qstbif_1d(ji) / ( rcpic * ( h_ice_1d(ji) / 2. ) ) |
|---|
| 583 | tbif_1d(ji,2) = ziqr * rt0_ice + ( 1 - ziqr ) * ( tbif_1d(ji,2) + zdtic ) |
|---|
| 584 | !--update qstbif_1d |
|---|
| 585 | qstbif_1d(ji) = ziqr * ( qstbif_1d(ji) - zqnes ) * swiqst |
|---|
| 586 | |
|---|
| 587 | |
|---|
| 588 | !-- 9.2. Calculate bottom accretion-ablation and update qstbif_1d. |
|---|
| 589 | ! Growth and melting at bottom ice surface are governed by |
|---|
| 590 | ! -xlic * Dh = (Fcb - Fbot ) * Dt |
|---|
| 591 | ! where Fbot is the net downward heat flux from ice to the ocean |
|---|
| 592 | ! and Fcb is the conductive heat flux at the bottom surface |
|---|
| 593 | !--------------------------------------------------------------------------- |
|---|
| 594 | ztbot = ( 1.0 - zidsnic(ji) ) * tbif_1d(ji,3) + zidsnic(ji) * sist_1d(ji) |
|---|
| 595 | !---computes conductive heat flux at bottom surface |
|---|
| 596 | zfcbot = 4.0 * zkic(ji) * ( tfu_1d(ji) - ztbot ) & |
|---|
| 597 | & / ( h_ice_1d(ji) + zidsnic(ji) * ( 3. * h_ice_1d(ji) & |
|---|
| 598 | & + 4.0 * zkic(ji)/zksn(ji) * h_snow_1d(ji) ) ) |
|---|
| 599 | !---computation of net energy needed for bottom melting/growing |
|---|
| 600 | zqice_bot = ( zfcbot - ( fbif_1d(ji) + qlbbq_1d(ji) ) ) * rdt_ice |
|---|
| 601 | zqstbif_bot = qstbif_1d(ji) |
|---|
| 602 | !---switch to know if bottom surface melts ( = 1 ) or grows ( = 0 )occurs |
|---|
| 603 | zibmlt = MAX( zzero , SIGN( zone , -zqice_bot ) ) |
|---|
| 604 | !--particular case of melting (in the same way as the top surface) |
|---|
| 605 | zqice_bot_mlt = zqice_bot |
|---|
| 606 | zqstbif_old = zqstbif_bot |
|---|
| 607 | |
|---|
| 608 | ziqf = MAX ( zzero , SIGN( zone , qstbif_1d(ji) + zqice_bot_mlt ) ) |
|---|
| 609 | zqice_bot_mlt = ziqf * ( zqice_bot_mlt + zqice_bot_mlt ) & |
|---|
| 610 | & + ( 1.0 - ziqf ) * ( zqice_bot_mlt + qstbif_1d(ji) ) |
|---|
| 611 | qstbif_1d(ji) = ziqf * ( qstbif_1d(ji) + zqice_bot_mlt ) & |
|---|
| 612 | & + ( 1.0 - ziqf ) * ( qstbif_1d(ji) - qstbif_1d(ji) ) |
|---|
| 613 | !-- The contribution of the energy stored in brine pockets qstbif_1d to melt |
|---|
| 614 | !-- ice is taking into account only when qstbif_1d is less than zqmax. |
|---|
| 615 | zihq = MAX ( zzero , SIGN( zone , qstbif_1d(ji) - zqmax(ji) ) ) |
|---|
| 616 | zqice_bot_mlt = zihq * zqice_bot_mlt & |
|---|
| 617 | & + ( 1.0 - zihq ) * zqice_bot |
|---|
| 618 | qstbif_1d(ji) = zihq * qstbif_1d(ji) & |
|---|
| 619 | & + ( 1.0 - zihq ) * zqstbif_old |
|---|
| 620 | |
|---|
| 621 | !---treatment of the case of melting/growing |
|---|
| 622 | zqice_bot = zibmlt * ( zqice_bot_mlt - zqcmlt(ji,2) ) & |
|---|
| 623 | & + ( 1.0 - zibmlt ) * ( zqice_bot - zqcmlt(ji,2) ) |
|---|
| 624 | qstbif_1d(ji) = zibmlt * qstbif_1d(ji) & |
|---|
| 625 | & + ( 1.0 - zibmlt ) * zqstbif_bot |
|---|
| 626 | |
|---|
| 627 | !--computes change in ice thickness due to melt or growth |
|---|
| 628 | zdhicbot(ji) = zqice_bot / xlic |
|---|
| 629 | !--limitation of bottom melting if so : hmelt maximum melting at bottom |
|---|
| 630 | zdhicmlt = MAX( hmelt , zdhicbot(ji) ) |
|---|
| 631 | !--energy after bottom melting/growing |
|---|
| 632 | zqsup(ji) = ( 1.0 - frld_1d(ji) ) * xlic * ( zdhicmlt - zdhicbot(ji) ) |
|---|
| 633 | !-- compute the new thickness and the newly formed volume after bottom melting/growing |
|---|
| 634 | zdhicbot(ji) = zdhicmlt |
|---|
| 635 | dvbbq_1d(ji) = ( 1.0 - frld_1d(ji) ) * zdhicbot(ji) |
|---|
| 636 | |
|---|
| 637 | |
|---|
| 638 | ! 9.3. Updating ice thickness after top surface ablation |
|---|
| 639 | ! and bottom surface accretion/ablation |
|---|
| 640 | !--------------------------------------------------------------- |
|---|
| 641 | zhicnew = h_ice_1d(ji) + zdhictop(ji) + zdhicbot(ji) |
|---|
| 642 | |
|---|
| 643 | ! |
|---|
| 644 | ! 9.4. Case of total ablation (ice is gone but snow may be left) |
|---|
| 645 | !------------------------------------------------------------------- |
|---|
| 646 | zhsn = h_snow_1d(ji) |
|---|
| 647 | zihgnew = 1.0 - MAX( zzero , SIGN( zone , -zhicnew ) ) |
|---|
| 648 | zihsn = MAX( zzero , SIGN( zone , -zhsn ) ) |
|---|
| 649 | !---convert |
|---|
| 650 | zdhicm = ( 1.0 - zihgnew ) * ( zhicnew - qstbif_1d(ji) / xlic ) |
|---|
| 651 | zdhsnm = ( 1.0 - zihsn ) * zdhicm * rhoic / rhosn |
|---|
| 652 | !---updating new ice thickness and computing the newly formed ice mass |
|---|
| 653 | zhicnew = zihgnew * zhicnew |
|---|
| 654 | rdmicif_1d(ji) = rdmicif_1d(ji) + ( 1.0 - frld_1d(ji) ) * ( zhicnew - h_ice_1d(ji) ) * rhoic |
|---|
| 655 | !---updating new snow thickness and computing the newly formed snow mass |
|---|
| 656 | zhsnfi = zhsn + zdhsnm |
|---|
| 657 | h_snow_1d(ji) = MAX( zzero , zhsnfi ) |
|---|
| 658 | rdmsnif_1d(ji) = rdmsnif_1d(ji) + ( 1.0 - frld_1d(ji) ) * ( h_snow_1d(ji) - zhsn ) * rhosn |
|---|
| 659 | !--remaining energy in case of total ablation |
|---|
| 660 | zqocea(ji) = - ( zihsn * xlic * zdhicm + xlsn * ( zhsnfi - h_snow_1d(ji) ) ) * ( 1.0 - frld_1d(ji) ) |
|---|
| 661 | qstbif_1d(ji) = zihgnew * qstbif_1d(ji) |
|---|
| 662 | |
|---|
| 663 | ! |
|---|
| 664 | ! 9.5. Update internal temperature and ice thickness. |
|---|
| 665 | !------------------------------------------------------- |
|---|
| 666 | ! |
|---|
| 667 | sist_1d(ji) = zihgnew * sist_1d(ji) + ( 1.0 - zihgnew ) * tfu_1d(ji) |
|---|
| 668 | zidhb = MAX( zzero , SIGN( zone , - zdhicbot(ji) ) ) |
|---|
| 669 | zc1 = - zhicnew * 0.5 |
|---|
| 670 | zpc1 = MIN( 0.5 * zone , - h_ice_1d(ji) * 0.5 - zdhictop(ji) ) |
|---|
| 671 | zc2 = - zhicnew |
|---|
| 672 | zpc2 = zidhb * zc2 + ( 1.0 - zidhb ) * ( - h_ice_1d(ji) - zdhictop(ji) ) |
|---|
| 673 | zp1 = MAX( zpc1 , zc1 ) |
|---|
| 674 | zp2 = MAX( zpc2 , zc1 ) |
|---|
| 675 | zep(ji) = tbif_1d(ji,2) |
|---|
| 676 | ztb2 = 2.0 * ( - zp1 * tbif_1d(ji,2) & |
|---|
| 677 | & + ( zp1 - zp2 ) * tbif_1d(ji,3) & |
|---|
| 678 | & + ( zp2 - zc1 ) * tfu_1d(ji) ) / MAX( zhicnew , epsi20 ) |
|---|
| 679 | tbif_1d(ji,2) = zihgnew * ztb2 + ( 1.0 - zihgnew ) * tfu_1d(ji) |
|---|
| 680 | !--- |
|---|
| 681 | zp1 = MIN( zpc1 , zc1 ) |
|---|
| 682 | zp2 = MIN( zpc2 , zc1 ) |
|---|
| 683 | zp1 = MAX( zc2 , zp1 ) |
|---|
| 684 | ztb3 = 2.0 * ( ( 1.0 - zidhb ) * ( ( zc1 - zp2 ) * tbif_1d(ji,3) & |
|---|
| 685 | & + ( zp2 - zc2 ) * tfu_1d(ji) ) & |
|---|
| 686 | & + zidhb * ( ( zc1 - zp1 ) * zep(ji) & |
|---|
| 687 | & + ( zp1 - zc2 ) * tbif_1d(ji,3)) ) / MAX( zhicnew , epsi20 ) |
|---|
| 688 | tbif_1d(ji,3) = zihgnew * ztb3 + ( 1.0 - zihgnew ) * tfu_1d(ji) |
|---|
| 689 | h_ice_1d(ji) = zhicnew |
|---|
| 690 | END DO |
|---|
| 691 | |
|---|
| 692 | |
|---|
| 693 | !---------------------------------------------------------------------------- |
|---|
| 694 | ! 10. Surface accretion. |
|---|
| 695 | ! The change of ice thickness after snow/ice formation is such that |
|---|
| 696 | ! the interface between snow and ice is located at the same height |
|---|
| 697 | ! as the ocean surface. It is given by (Fichefet and Morales Maqueda 1999) |
|---|
| 698 | ! D(h_ice) = (- D(hsn)/alph) = [rhosn*hsn - (rau0 - rhoic)*hic] |
|---|
| 699 | ! / [alph*rhosn+rau0 - rhoic] |
|---|
| 700 | !---------------------------------------------------------------------------- |
|---|
| 701 | ! |
|---|
| 702 | DO ji = kideb , kiut |
|---|
| 703 | |
|---|
| 704 | !-- Computation of the change of ice thickness after snow-ice formation |
|---|
| 705 | zdrmh = ( rhosn * h_snow_1d(ji) + ( rhoic - rau0 ) * h_ice_1d(ji) ) & |
|---|
| 706 | & / ( alphs * rhosn + rau0 - rhoic ) |
|---|
| 707 | zdrmh = MAX( zzero , zdrmh ) |
|---|
| 708 | |
|---|
| 709 | !--New ice and snow thicknesses Fichefet and Morales Maqueda (1999) |
|---|
| 710 | zhicnew = MAX( h_ice_1d(ji) , h_ice_1d(ji) + zdrmh ) |
|---|
| 711 | zhsnnew = MIN( h_snow_1d(ji) , h_snow_1d(ji) - alphs * zdrmh ) |
|---|
| 712 | !---Compute new ice temperatures. snow temperature remains unchanged |
|---|
| 713 | ! Lepparanta (1983): |
|---|
| 714 | zihic = 1.0 - MAX( zzero , SIGN( zone , -zhicnew ) ) |
|---|
| 715 | zquot = ( 1.0 - zihic ) & |
|---|
| 716 | & + zihic * MIN( zone , h_ice_1d(ji) / MAX( zhicnew , epsi20 ) ) |
|---|
| 717 | ztneq = alphs * cnscg * tbif_1d(ji,1) & |
|---|
| 718 | & + ( 1.0 - alphs * ( rhosn/rhoic ) ) * tfu_1d(ji) |
|---|
| 719 | zep(ji) = tbif_1d(ji,2) |
|---|
| 720 | tbif_1d(ji,2) = ztneq - zquot * zquot * ( ztneq - tbif_1d(ji,2) ) |
|---|
| 721 | tbif_1d(ji,3) = 2.0 * ztneq & |
|---|
| 722 | & + zquot * ( tbif_1d(ji,3) + zep(ji) - 2.0 * ztneq ) - tbif_1d(ji,2) |
|---|
| 723 | |
|---|
| 724 | !--- Lepparanta (1983) (latent heat released during white ice formation |
|---|
| 725 | ! goes to the ocean -for lateral ablation-) |
|---|
| 726 | qldif_1d(ji) = qldif_1d(ji) + zdrmh * ( 1.0 - alphs * ( rhosn/rhoic ) ) * xlic * ( 1.0 - frld_1d(ji) ) |
|---|
| 727 | !-- Changes in ice volume and ice mass Lepparanta (1983): |
|---|
| 728 | dvnbq_1d(ji) = ( 1.0 - frld_1d(ji) ) * ( zhicnew - h_ice_1d(ji) ) |
|---|
| 729 | dmgwi_1d(ji) = dmgwi_1d(ji) + ( 1.0 -frld_1d(ji) ) * ( h_snow_1d(ji) - zhsnnew ) * rhosn |
|---|
| 730 | ! case of natural freshwater flux |
|---|
| 731 | #if defined key_lim_fdd |
|---|
| 732 | rdmicif_1d(ji) = rdmicif_1d(ji) + ( 1.0 - frld_1d(ji) ) * ( zhicnew - h_ice_1d(ji) ) * rhoic |
|---|
| 733 | rdmsnif_1d(ji) = rdmsnif_1d(ji) + ( 1.0 - frld_1d(ji) ) * ( zhsnnew - h_snow_1d(ji) ) * rhosn |
|---|
| 734 | |
|---|
| 735 | #else |
|---|
| 736 | rdmicif_1d(ji) = rdmicif_1d(ji) + ( 1.0 - frld_1d(ji) ) * ( ( zhicnew - h_ice_1d(ji) ) * rhoic & |
|---|
| 737 | & + ( zhsnnew - h_snow_1d(ji) ) * rhosn ) |
|---|
| 738 | #endif |
|---|
| 739 | |
|---|
| 740 | !--- Actualize new snow and ice thickness. |
|---|
| 741 | |
|---|
| 742 | h_snow_1d(ji) = zhsnnew |
|---|
| 743 | h_ice_1d(ji) = zhicnew |
|---|
| 744 | |
|---|
| 745 | END DO |
|---|
| 746 | |
|---|
| 747 | !---------------------------------------------------- |
|---|
| 748 | ! 11. Lateral ablation (Changes in sea/ice fraction) |
|---|
| 749 | !---------------------------------------------------- |
|---|
| 750 | DO ji = kideb , kiut |
|---|
| 751 | zfrl_old(ji) = frld_1d(ji) |
|---|
| 752 | zihic = 1.0 - MAX( zzero , SIGN( zone , -h_ice_1d(ji) ) ) |
|---|
| 753 | zihsn = 1.0 - MAX( zzero , SIGN( zone , -h_snow_1d(ji) ) ) |
|---|
| 754 | !--In the case of total ablation (all the ice ice has melted) frld = 1 |
|---|
| 755 | frld_1d(ji) = ( 1.0 - zihic ) + zihic * zfrl_old(ji) |
|---|
| 756 | !--Part of solar radiation absorbing inside the ice and going |
|---|
| 757 | !--through the ocean |
|---|
| 758 | fscbq_1d(ji) = ( 1.0 - zfrl_old(ji) ) * ( 1.0 - thcm_1d(ji) ) * fstbif_1d(ji) |
|---|
| 759 | !--Total remaining energy after bottom melting/growing |
|---|
| 760 | qfvbq_1d(ji) = zqsup(ji) + ( 1.0 - zihic ) * zqocea(ji) |
|---|
| 761 | !--Updating of total heat from the ocean |
|---|
| 762 | qldif_1d(ji) = qldif_1d(ji) + qfvbq_1d(ji) + ( 1.0 - zihic ) * fscbq_1d(ji) * rdt_ice |
|---|
| 763 | !--Computation of total heat inside the snow/ice system |
|---|
| 764 | zqice = h_snow_1d(ji) * xlsn + h_ice_1d(ji) * xlic |
|---|
| 765 | zqicetot = ( 1.0 - frld_1d(ji) ) * zqice |
|---|
| 766 | !--The concentration of ice is reduced (frld increases) if the heat |
|---|
| 767 | !--exchange between ice and ocean is positive |
|---|
| 768 | ziqf = MAX( zzero , SIGN( zone , zqicetot - qldif_1d(ji) ) ) |
|---|
| 769 | zdfrl = qldif_1d(ji) / MAX( epsi20 , zqice ) |
|---|
| 770 | frld_1d(ji) = ( 1.0 - ziqf ) & |
|---|
| 771 | & + ziqf * ( frld_1d(ji) + MAX( zzero , zdfrl ) ) |
|---|
| 772 | fltbif_1d(ji) = ( ( 1.0 - zfrl_old(ji) ) * qstbif_1d(ji) - zqicetot ) / rdt_ice |
|---|
| 773 | !-- Opening of leads: Hakkinen & Mellor, 1992. |
|---|
| 774 | zdfrl = - ( zdhictop(ji) + zdhicbot(ji) ) * hakspl * ( 1.0 - zfrl_old(ji) ) & |
|---|
| 775 | & / MAX( epsi13 , h_ice_1d(ji) + h_snow_1d(ji) * rhosn/rhoic ) |
|---|
| 776 | zfrld_1d(ji) = frld_1d(ji) + MAX( zzero , zdfrl ) |
|---|
| 777 | !--Limitation of sea-ice fraction <= 1 |
|---|
| 778 | zfrld_1d(ji) = ziqf * MIN( 0.99 * zone , zfrld_1d(ji) ) + ( 1 - ziqf ) |
|---|
| 779 | !---Update surface and internal temperature and snow/ice thicknesses. |
|---|
| 780 | sist_1d(ji) = sist_1d(ji) + ( 1.0 - ziqf ) * ( tfu_1d(ji) - sist_1d(ji) ) |
|---|
| 781 | tbif_1d(ji,1) = tbif_1d(ji,1) + ( 1.0 - ziqf ) * ( tfu_1d(ji) - tbif_1d(ji,1) ) |
|---|
| 782 | tbif_1d(ji,2) = tbif_1d(ji,2) + ( 1.0 - ziqf ) * ( tfu_1d(ji) - tbif_1d(ji,2) ) |
|---|
| 783 | tbif_1d(ji,3) = tbif_1d(ji,3) + ( 1.0 - ziqf ) * ( tfu_1d(ji) - tbif_1d(ji,3) ) |
|---|
| 784 | !--variation of ice volume and ice mass |
|---|
| 785 | dvlbq_1d(ji) = zihic * ( zfrl_old(ji) - frld_1d(ji) ) * h_ice_1d(ji) |
|---|
| 786 | rdmicif_1d(ji) = rdmicif_1d(ji) + dvlbq_1d(ji) * rhoic |
|---|
| 787 | !--variation of snow volume and snow mass |
|---|
| 788 | zdvsnvol = zihsn * ( zfrl_old(ji) - frld_1d(ji) ) * h_snow_1d(ji) |
|---|
| 789 | rdmsnif_1d(ji) = rdmsnif_1d(ji) + zdvsnvol * rhosn |
|---|
| 790 | h_snow_1d(ji) = ziqf * h_snow_1d(ji) |
|---|
| 791 | |
|---|
| 792 | zdrfrl1 = ziqf * ( 1.0 - frld_1d(ji) ) / MAX( epsi20 , 1.0 - zfrld_1d(ji) ) |
|---|
| 793 | zdrfrl2 = ziqf * ( 1.0 - zfrl_old(ji) ) / MAX( epsi20 , 1.0 - zfrld_1d(ji) ) |
|---|
| 794 | |
|---|
| 795 | h_snow_1d (ji) = zdrfrl1 * h_snow_1d(ji) |
|---|
| 796 | h_ice_1d (ji) = zdrfrl1 * h_ice_1d(ji) |
|---|
| 797 | qstbif_1d(ji) = zdrfrl2 * qstbif_1d(ji) |
|---|
| 798 | frld_1d(ji) = zfrld_1d(ji) |
|---|
| 799 | |
|---|
| 800 | END DO |
|---|
| 801 | |
|---|
| 802 | END SUBROUTINE lim_thd_zdf |
|---|
| 803 | #else |
|---|
| 804 | !!====================================================================== |
|---|
| 805 | !! *** MODULE limthd_zdf *** |
|---|
| 806 | !! no sea ice model |
|---|
| 807 | !!====================================================================== |
|---|
| 808 | CONTAINS |
|---|
| 809 | SUBROUTINE lim_thd_zdf ! Empty routine |
|---|
| 810 | END SUBROUTINE lim_thd_zdf |
|---|
| 811 | #endif |
|---|
| 812 | END MODULE limthd_zdf |
|---|
