MODULE icethd_zdf !!====================================================================== !! *** MODULE icethd_zdf *** !! heat diffusion in sea ice !! computation of surface and inner T !!====================================================================== !! History : LIM ! 02-2003 (M. Vancoppenolle) original 1D code !! ! 06-2005 (M. Vancoppenolle) 3d version !! ! 11-2006 (X Fettweis) Vectorization by Xavier !! ! 04-2007 (M. Vancoppenolle) Energy conservation !! 4.0 ! 2011-02 (G. Madec) dynamical allocation !! - ! 2012-05 (C. Rousset) add penetration solar flux !!---------------------------------------------------------------------- #if defined key_lim3 !!---------------------------------------------------------------------- !! 'key_lim3' LIM3 sea-ice model !!---------------------------------------------------------------------- USE par_oce ! ocean parameters USE phycst ! physical constants (ocean directory) USE ice ! sea-ice: variables USE ice1D ! sea-ice: thermodynamics ! USE in_out_manager ! I/O manager USE lib_mpp ! MPP library USE lib_fortran ! Fortran utilities (allows no signed zero when 'key_nosignedzero' defined) IMPLICIT NONE PRIVATE PUBLIC ice_thd_zdf ! called by ice_thd PUBLIC ice_thd_zdf_init ! called by ice_stp !!** namelist (namthd_zdf) ** LOGICAL :: ln_zdf_Beer ! Heat diffusion follows a Beer Law LOGICAL :: ln_cndi_U64 ! thermal conductivity: Untersteiner (1964) LOGICAL :: ln_cndi_P07 ! thermal conductivity: Pringle et al (2007) REAL(wp) :: rn_cnd_s ! thermal conductivity of the snow [W/m/K] REAL(wp) :: rn_kappa_i ! coef. for the extinction of radiation Grenfell et al. (2006) [1/m] LOGICAL :: ln_dqns_i ! change non-solar surface flux with changing surface temperature (T) or not (F) !!---------------------------------------------------------------------- !! NEMO/ICE 4.0 , NEMO Consortium (2017) !! $Id: icethd_zdf.F90 8420 2017-08-08 12:18:46Z clem $ !! Software governed by the CeCILL licence (NEMOGCM/NEMO_CeCILL.txt) !!---------------------------------------------------------------------- CONTAINS SUBROUTINE ice_thd_zdf !!------------------------------------------------------------------ !! *** ROUTINE ice_thd_zdf *** !! ** Purpose : !! This routine determines the time evolution of snow and sea-ice !! temperature profiles. !! ** Method : !! This is done by solving the heat equation diffusion with !! a Neumann boundary condition at the surface and a Dirichlet one !! at the bottom. Solar radiation is partially absorbed into the ice. !! The specific heat and thermal conductivities depend on ice salinity !! and temperature to take into account brine pocket melting. The !! numerical !! scheme is an iterative Crank-Nicolson on a non-uniform multilayer grid !! in the ice and snow system. !! !! The successive steps of this routine are !! 1. Thermal conductivity at the interfaces of the ice layers !! 2. Internal absorbed radiation !! 3. Scale factors due to non-uniform grid !! 4. Kappa factors !! Then iterative procedure begins !! 5. specific heat in the ice !! 6. eta factors !! 7. surface flux computation !! 8. tridiagonal system terms !! 9. solving the tridiagonal system with Gauss elimination !! Iterative procedure ends according to a criterion on evolution !! of temperature !! !! ** Inputs / Ouputs : (global commons) !! surface temperature : t_su_1d !! ice/snow temperatures : t_i_1d, t_s_1d !! ice salinities : s_i_1d !! number of layers in the ice/snow: nlay_i, nlay_s !! total ice/snow thickness : ht_i_1d, ht_s_1d !!------------------------------------------------------------------ INTEGER :: ji, jk ! spatial loop index INTEGER :: numeq ! current reference number of equation INTEGER :: minnumeqmin, maxnumeqmax INTEGER :: iconv ! number of iterations in iterative procedure INTEGER :: iconv_max = 50 ! max number of iterations in iterative procedure INTEGER, DIMENSION(jpij) :: numeqmin ! reference number of top equation INTEGER, DIMENSION(jpij) :: numeqmax ! reference number of bottom equation REAL(wp) :: zg1s = 2._wp ! for the tridiagonal system REAL(wp) :: zg1 = 2._wp ! REAL(wp) :: zgamma = 18009._wp ! for specific heat REAL(wp) :: zbeta = 0.117_wp ! for thermal conductivity (could be 0.13) REAL(wp) :: zraext_s = 10._wp ! extinction coefficient of radiation in the snow REAL(wp) :: zkimin = 0.10_wp ! minimum ice thermal conductivity REAL(wp) :: ztsu_err = 1.e-5_wp ! range around which t_su is considered at 0C REAL(wp) :: zdti_bnd = 1.e-4_wp ! maximal authorized error on temperature REAL(wp) :: ztmelt_i ! ice melting temperature REAL(wp) :: z1_hsu REAL(wp) :: zdti_max ! current maximal error on temperature REAL(wp) :: zcpi ! Ice specific heat REAL(wp) :: zhfx_err, zdq ! diag errors on heat REAL(wp) :: zfac ! dummy factor REAL(wp), DIMENSION(jpij) :: isnow ! switch for presence (1) or absence (0) of snow REAL(wp), DIMENSION(jpij) :: ztsub ! surface temperature at previous iteration REAL(wp), DIMENSION(jpij) :: zh_i, z1_h_i ! ice layer thickness REAL(wp), DIMENSION(jpij) :: zh_s, z1_h_s ! snow layer thickness REAL(wp), DIMENSION(jpij) :: zfsw ! solar radiation absorbed at the surface REAL(wp), DIMENSION(jpij) :: zqns_ice_b ! solar radiation absorbed at the surface REAL(wp), DIMENSION(jpij) :: zf ! surface flux function REAL(wp), DIMENSION(jpij) :: zdqns_ice_b ! derivative of the surface flux function REAL(wp), DIMENSION(jpij) :: zftrice ! solar radiation transmitted through the ice REAL(wp), DIMENSION(jpij,nlay_i) :: ztiold ! Old temperature in the ice REAL(wp), DIMENSION(jpij,nlay_s) :: ztsold ! Old temperature in the snow REAL(wp), DIMENSION(jpij,nlay_i) :: ztib ! Temporary temperature in the ice to check the convergence REAL(wp), DIMENSION(jpij,nlay_s) :: ztsb ! Temporary temperature in the snow to check the convergence REAL(wp), DIMENSION(jpij,0:nlay_i) :: ztcond_i ! Ice thermal conductivity REAL(wp), DIMENSION(jpij,0:nlay_i) :: zradtr_i ! Radiation transmitted through the ice REAL(wp), DIMENSION(jpij,0:nlay_i) :: zradab_i ! Radiation absorbed in the ice REAL(wp), DIMENSION(jpij,0:nlay_i) :: zkappa_i ! Kappa factor in the ice REAL(wp), DIMENSION(jpij,0:nlay_i) :: zeta_i ! Eta factor in the ice REAL(wp), DIMENSION(jpij,0:nlay_s) :: zradtr_s ! Radiation transmited through the snow REAL(wp), DIMENSION(jpij,0:nlay_s) :: zradab_s ! Radiation absorbed in the snow REAL(wp), DIMENSION(jpij,0:nlay_s) :: zkappa_s ! Kappa factor in the snow REAL(wp), DIMENSION(jpij,0:nlay_s) :: zeta_s ! Eta factor in the snow REAL(wp), DIMENSION(jpij,nlay_i+3) :: zindterm ! 'Ind'ependent term REAL(wp), DIMENSION(jpij,nlay_i+3) :: zindtbis ! Temporary 'ind'ependent term REAL(wp), DIMENSION(jpij,nlay_i+3) :: zdiagbis ! Temporary 'dia'gonal term REAL(wp), DIMENSION(jpij,nlay_i+3,3) :: ztrid ! Tridiagonal system terms REAL(wp), DIMENSION(jpij) :: zq_ini ! diag errors on heat REAL(wp), DIMENSION(jpij) :: zghe ! G(he), th. conduct enhancement factor, mono-cat ! Mono-category REAL(wp) :: zepsilon ! determines thres. above which computation of G(h) is done REAL(wp) :: zhe ! dummy factor REAL(wp) :: zcnd_i ! mean sea ice thermal conductivity !!------------------------------------------------------------------ ! --- diag error on heat diffusion - PART 1 --- ! DO ji = 1, nidx zq_ini(ji) = ( SUM( e_i_1d(ji,1:nlay_i) ) * ht_i_1d(ji) * r1_nlay_i + & & SUM( e_s_1d(ji,1:nlay_s) ) * ht_s_1d(ji) * r1_nlay_s ) END DO !------------------------------------------------------------------------------! ! 1) Initialization ! !------------------------------------------------------------------------------! DO ji = 1, nidx isnow(ji)= 1._wp - MAX( 0._wp , SIGN(1._wp, - ht_s_1d(ji) ) ) ! is there snow or not ! layer thickness zh_i(ji) = ht_i_1d(ji) * r1_nlay_i zh_s(ji) = ht_s_1d(ji) * r1_nlay_s END DO ! WHERE( zh_i(1:nidx) >= epsi10 ) ; z1_h_i(1:nidx) = 1._wp / zh_i(1:nidx) ELSEWHERE ; z1_h_i(1:nidx) = 0._wp END WHERE WHERE( zh_s(1:nidx) >= epsi10 ) ; z1_h_s(1:nidx) = 1._wp / zh_s(1:nidx) ELSEWHERE ; z1_h_s(1:nidx) = 0._wp END WHERE ! ! temperatures ztsub (1:nidx) = t_su_1d(1:nidx) ! temperature at the previous iteration ztsold (1:nidx,:) = t_s_1d(1:nidx,:) ! Old snow temperature ztiold (1:nidx,:) = t_i_1d(1:nidx,:) ! Old ice temperature t_su_1d(1:nidx) = MIN( t_su_1d(1:nidx), rt0 - ztsu_err ) ! necessary ! !------------------------------------------------------------------------------| ! 2) Radiation | !------------------------------------------------------------------------------| ! z1_hsu = 1._wp / 0.1_wp ! threshold for the computation of i0 DO ji = 1, nidx !------------------- ! Computation of i0 !------------------- ! i0 describes the fraction of solar radiation which does not contribute ! to the surface energy budget but rather penetrates inside the ice. ! We assume that no radiation is transmitted through the snow ! If there is no no snow ! zfsw = (1-i0).qsr_ice is absorbed at the surface ! zftrice = io.qsr_ice is below the surface ! ftr_ice = io.qsr_ice.exp(-k(h_i)) transmitted below the ice ! fr1_i0_1d = i0 for a thin ice cover, fr1_i0_2d = i0 for a thick ice cover zfac = MAX( 0._wp , 1._wp - ( ht_i_1d(ji) * z1_hsu ) ) i0(ji) = ( 1._wp - isnow(ji) ) * ( fr1_i0_1d(ji) + zfac * fr2_i0_1d(ji) ) !------------------------------------------------------- ! Solar radiation absorbed / transmitted at the surface ! Derivative of the non solar flux !------------------------------------------------------- zfsw (ji) = qsr_ice_1d(ji) * ( 1 - i0(ji) ) ! Shortwave radiation absorbed at surface zftrice(ji) = qsr_ice_1d(ji) * i0(ji) ! Solar radiation transmitted below the surface layer zdqns_ice_b(ji) = dqns_ice_1d(ji) ! derivative of incoming nonsolar flux zqns_ice_b (ji) = qns_ice_1d(ji) ! store previous qns_ice_1d value END DO !--------------------------------------------------------- ! Transmission - absorption of solar radiation in the ice !--------------------------------------------------------- zradtr_s(1:nidx,0) = zftrice(1:nidx) DO jk = 1, nlay_s DO ji = 1, nidx ! ! radiation transmitted below the layer-th snow layer zradtr_s(ji,jk) = zradtr_s(ji,0) * EXP( - zraext_s * zh_s(ji) * REAL(jk) ) ! ! radiation absorbed by the layer-th snow layer zradab_s(ji,jk) = zradtr_s(ji,jk-1) - zradtr_s(ji,jk) END DO END DO zradtr_i(1:nidx,0) = zradtr_s(1:nidx,nlay_s) * isnow(1:nidx) + zftrice(1:nidx) * ( 1._wp - isnow(1:nidx) ) DO jk = 1, nlay_i DO ji = 1, nidx ! ! radiation transmitted below the layer-th ice layer zradtr_i(ji,jk) = zradtr_i(ji,0) * EXP( - rn_kappa_i * zh_i(ji) * REAL(jk) ) ! ! radiation absorbed by the layer-th ice layer zradab_i(ji,jk) = zradtr_i(ji,jk-1) - zradtr_i(ji,jk) END DO END DO ftr_ice_1d(1:nidx) = zradtr_i(1:nidx,nlay_i) ! record radiation transmitted below the ice !------------------------------------------------------------------------------| ! 3) Iterative procedure begins | !------------------------------------------------------------------------------| ! iconv = 0 ! number of iterations zdti_max = 1000._wp ! maximal value of error on all points DO WHILE ( zdti_max > zdti_bnd .AND. iconv < iconv_max ) ! iconv = iconv + 1 ! ztib(1:nidx,:) = t_i_1d(1:nidx,:) ztsb(1:nidx,:) = t_s_1d(1:nidx,:) ! !------------------------------------------------------------------------------| ! 4) Sea ice thermal conductivity | !------------------------------------------------------------------------------| ! IF( ln_cndi_U64 ) THEN !-- Untersteiner (1964) formula: k = k0 + beta.S/T ! DO ji = 1, nidx ztcond_i(ji,0) = rcdic + zbeta * s_i_1d(ji,1) / MIN( -epsi10, t_i_1d(ji,1) - rt0 ) ztcond_i(ji,nlay_i) = rcdic + zbeta * s_i_1d(ji,nlay_i) / MIN( -epsi10, t_bo_1d(ji) - rt0 ) END DO DO jk = 1, nlay_i-1 DO ji = 1, nidx ztcond_i(ji,jk) = rcdic + zbeta * 0.5_wp * ( s_i_1d(ji,jk) + s_i_1d(ji,jk+1) ) / & & MIN( -epsi10, 0.5_wp * (t_i_1d(ji,jk) + t_i_1d(ji,jk+1)) - rt0 ) END DO END DO ! ELSEIF( ln_cndi_P07 ) THEN !-- Pringle et al formula: k = k0 + beta1.S/T - beta2.T ! DO ji = 1, nidx ztcond_i(ji,0) = rcdic + 0.09_wp * s_i_1d(ji,1) / MIN( -epsi10, t_i_1d(ji,1) - rt0 ) & & - 0.011_wp * ( t_i_1d(ji,1) - rt0 ) ztcond_i(ji,nlay_i) = rcdic + 0.09_wp * s_i_1d(ji,nlay_i) / MIN( -epsi10, t_bo_1d(ji) - rt0 ) & & - 0.011_wp * ( t_bo_1d(ji) - rt0 ) END DO DO jk = 1, nlay_i-1 DO ji = 1, nidx ztcond_i(ji,jk) = rcdic + 0.09_wp * 0.5_wp * ( s_i_1d(ji,jk) + s_i_1d(ji,jk+1) ) / & & MIN( -epsi10, 0.5_wp * (t_i_1d(ji,jk) + t_i_1d(ji,jk+1)) - rt0 ) & & - 0.011_wp * ( 0.5_wp * (t_i_1d(ji,jk) + t_i_1d(ji,jk+1)) - rt0 ) END DO END DO ! ENDIF ztcond_i(1:nidx,:) = MAX( zkimin, ztcond_i(1:nidx,:) ) ! !------------------------------------------------------------------------------| ! 5) G(he) - enhancement of thermal conductivity in mono-category case | !------------------------------------------------------------------------------| ! ! Computation of effective thermal conductivity G(h) ! Used in mono-category case only to simulate an ITD implicitly ! Fichefet and Morales Maqueda, JGR 1997 zghe(1:nidx) = 1._wp SELECT CASE ( nn_monocat ) CASE ( 1 , 3 ) zepsilon = 0.1_wp DO ji = 1, nidx ! Mean sea ice thermal conductivity zcnd_i = SUM( ztcond_i(ji,:) ) / REAL( nlay_i+1, wp ) ! Effective thickness he (zhe) zhe = ( rn_cnd_s * ht_i_1d(ji) + zcnd_i * ht_s_1d(ji) ) / ( rn_cnd_s + zcnd_i ) ! G(he) IF( zhe >= zepsilon * 0.5_wp * EXP(1._wp) ) THEN zghe(ji) = MIN( 2._wp, 0.5_wp * ( 1._wp + LOG( 2._wp * zhe / zepsilon ) ) ) ENDIF END DO END SELECT ! !------------------------------------------------------------------------------| ! 6) kappa factors | !------------------------------------------------------------------------------| ! !--- Snow DO jk = 0, nlay_s-1 DO ji = 1, nidx zkappa_s(ji,jk) = zghe(ji) * rn_cnd_s * z1_h_s(ji) END DO END DO DO ji = 1, nidx ! Snow-ice interface zfac = 0.5_wp * ( ztcond_i(ji,0) * zh_s(ji) + rn_cnd_s * zh_i(ji) ) IF( zfac > epsi10 ) THEN zkappa_s(ji,nlay_s) = zghe(ji) * rn_cnd_s * ztcond_i(ji,0) / zfac ELSE zkappa_s(ji,nlay_s) = 0._wp ENDIF END DO !--- Ice DO jk = 0, nlay_i DO ji = 1, nidx zkappa_i(ji,jk) = zghe(ji) * ztcond_i(ji,jk) * z1_h_i(ji) END DO END DO DO ji = 1, nidx ! Snow-ice interface zkappa_i(ji,0) = zkappa_s(ji,nlay_s) * isnow(ji) + zkappa_i(ji,0) * ( 1._wp - isnow(ji) ) END DO ! !------------------------------------------------------------------------------| ! 7) Sea ice specific heat, eta factors | !------------------------------------------------------------------------------| ! DO jk = 1, nlay_i DO ji = 1, nidx zcpi = cpic + zgamma * s_i_1d(ji,jk) / MAX( ( t_i_1d(ji,jk) - rt0 ) * ( ztiold(ji,jk) - rt0 ), epsi10 ) zeta_i(ji,jk) = rdt_ice * r1_rhoic * z1_h_i(ji) / MAX( epsi10, zcpi ) END DO END DO DO jk = 1, nlay_s DO ji = 1, nidx zeta_s(ji,jk) = rdt_ice * r1_rhosn * r1_cpic * z1_h_s(ji) END DO END DO ! !------------------------------------------------------------------------------| ! 8) surface flux computation | !------------------------------------------------------------------------------| ! IF ( ln_dqns_i ) THEN DO ji = 1, nidx ! update of the non solar flux according to the update in T_su qns_ice_1d(ji) = qns_ice_1d(ji) + dqns_ice_1d(ji) * ( t_su_1d(ji) - ztsub(ji) ) END DO ENDIF DO ji = 1, nidx zf(ji) = zfsw(ji) + qns_ice_1d(ji) ! incoming = net absorbed solar radiation + non solar total flux (LWup, LWdw, SH, LH) END DO ! !------------------------------------------------------------------------------| ! 9) tridiagonal system terms | !------------------------------------------------------------------------------| ! !!layer denotes the number of the layer in the snow or in the ice !!numeq denotes the reference number of the equation in the tridiagonal !!system, terms of tridiagonal system are indexed as following : !!1 is subdiagonal term, 2 is diagonal and 3 is superdiagonal one !!ice interior terms (top equation has the same form as the others) DO numeq=1,nlay_i+3 DO ji = 1, nidx ztrid(ji,numeq,1) = 0. ztrid(ji,numeq,2) = 0. ztrid(ji,numeq,3) = 0. zindterm(ji,numeq)= 0. zindtbis(ji,numeq)= 0. zdiagbis(ji,numeq)= 0. ENDDO ENDDO DO numeq = nlay_s + 2, nlay_s + nlay_i DO ji = 1, nidx jk = numeq - nlay_s - 1 ztrid(ji,numeq,1) = - zeta_i(ji,jk) * zkappa_i(ji,jk-1) ztrid(ji,numeq,2) = 1.0 + zeta_i(ji,jk) * ( zkappa_i(ji,jk-1) + zkappa_i(ji,jk) ) ztrid(ji,numeq,3) = - zeta_i(ji,jk) * zkappa_i(ji,jk) zindterm(ji,numeq) = ztiold(ji,jk) + zeta_i(ji,jk) * zradab_i(ji,jk) END DO ENDDO numeq = nlay_s + nlay_i + 1 DO ji = 1, nidx !!ice bottom term ztrid(ji,numeq,1) = - zeta_i(ji,nlay_i)*zkappa_i(ji,nlay_i-1) ztrid(ji,numeq,2) = 1.0 + zeta_i(ji,nlay_i) * ( zkappa_i(ji,nlay_i) * zg1 + zkappa_i(ji,nlay_i-1) ) ztrid(ji,numeq,3) = 0.0 zindterm(ji,numeq) = ztiold(ji,nlay_i) + zeta_i(ji,nlay_i) * & & ( zradab_i(ji,nlay_i) + zkappa_i(ji,nlay_i) * zg1 * t_bo_1d(ji) ) ENDDO DO ji = 1, nidx IF ( ht_s_1d(ji) > 0.0 ) THEN ! !------------------------------------------------------------------------------| ! snow-covered cells | !------------------------------------------------------------------------------| ! !!snow interior terms (bottom equation has the same form as the others) DO numeq = 3, nlay_s + 1 jk = numeq - 1 ztrid(ji,numeq,1) = - zeta_s(ji,jk) * zkappa_s(ji,jk-1) ztrid(ji,numeq,2) = 1.0 + zeta_s(ji,jk) * ( zkappa_s(ji,jk-1) + zkappa_s(ji,jk) ) ztrid(ji,numeq,3) = - zeta_s(ji,jk)*zkappa_s(ji,jk) zindterm(ji,numeq) = ztsold(ji,jk) + zeta_s(ji,jk) * zradab_s(ji,jk) END DO !!case of only one layer in the ice (ice equation is altered) IF ( nlay_i == 1 ) THEN ztrid(ji,nlay_s+2,3) = 0.0 zindterm(ji,nlay_s+2) = zindterm(ji,nlay_s+2) + zkappa_i(ji,1) * t_bo_1d(ji) ENDIF IF ( t_su_1d(ji) < rt0 ) THEN !------------------------------------------------------------------------------| ! case 1 : no surface melting - snow present | !------------------------------------------------------------------------------| numeqmin(ji) = 1 numeqmax(ji) = nlay_i + nlay_s + 1 !!surface equation ztrid(ji,1,1) = 0.0 ztrid(ji,1,2) = zdqns_ice_b(ji) - zg1s * zkappa_s(ji,0) ztrid(ji,1,3) = zg1s * zkappa_s(ji,0) zindterm(ji,1) = zdqns_ice_b(ji) * t_su_1d(ji) - zf(ji) !!first layer of snow equation ztrid(ji,2,1) = - zkappa_s(ji,0) * zg1s * zeta_s(ji,1) ztrid(ji,2,2) = 1.0 + zeta_s(ji,1) * ( zkappa_s(ji,1) + zkappa_s(ji,0) * zg1s ) ztrid(ji,2,3) = - zeta_s(ji,1)* zkappa_s(ji,1) zindterm(ji,2) = ztsold(ji,1) + zeta_s(ji,1) * zradab_s(ji,1) ELSE ! !------------------------------------------------------------------------------| ! case 2 : surface is melting - snow present | !------------------------------------------------------------------------------| ! numeqmin(ji) = 2 numeqmax(ji) = nlay_i + nlay_s + 1 !!first layer of snow equation ztrid(ji,2,1) = 0.0 ztrid(ji,2,2) = 1.0 + zeta_s(ji,1) * ( zkappa_s(ji,1) + zkappa_s(ji,0) * zg1s ) ztrid(ji,2,3) = - zeta_s(ji,1)*zkappa_s(ji,1) zindterm(ji,2) = ztsold(ji,1) + zeta_s(ji,1) * & & ( zradab_s(ji,1) + zkappa_s(ji,0) * zg1s * t_su_1d(ji) ) ENDIF ELSE ! !------------------------------------------------------------------------------| ! cells without snow | !------------------------------------------------------------------------------| ! IF ( t_su_1d(ji) < rt0 ) THEN ! !------------------------------------------------------------------------------| ! case 3 : no surface melting - no snow | !------------------------------------------------------------------------------| ! numeqmin(ji) = nlay_s + 1 numeqmax(ji) = nlay_i + nlay_s + 1 !!surface equation ztrid(ji,numeqmin(ji),1) = 0.0 ztrid(ji,numeqmin(ji),2) = zdqns_ice_b(ji) - zkappa_i(ji,0)*zg1 ztrid(ji,numeqmin(ji),3) = zkappa_i(ji,0)*zg1 zindterm(ji,numeqmin(ji)) = zdqns_ice_b(ji)*t_su_1d(ji) - zf(ji) !!first layer of ice equation ztrid(ji,numeqmin(ji)+1,1) = - zkappa_i(ji,0) * zg1 * zeta_i(ji,1) ztrid(ji,numeqmin(ji)+1,2) = 1.0 + zeta_i(ji,1) * ( zkappa_i(ji,1) + zkappa_i(ji,0) * zg1 ) ztrid(ji,numeqmin(ji)+1,3) = - zeta_i(ji,1) * zkappa_i(ji,1) zindterm(ji,numeqmin(ji)+1)= ztiold(ji,1) + zeta_i(ji,1) * zradab_i(ji,1) !!case of only one layer in the ice (surface & ice equations are altered) IF ( nlay_i == 1 ) THEN ztrid(ji,numeqmin(ji),1) = 0.0 ztrid(ji,numeqmin(ji),2) = zdqns_ice_b(ji) - zkappa_i(ji,0) * 2.0 ztrid(ji,numeqmin(ji),3) = zkappa_i(ji,0) * 2.0 ztrid(ji,numeqmin(ji)+1,1) = -zkappa_i(ji,0) * 2.0 * zeta_i(ji,1) ztrid(ji,numeqmin(ji)+1,2) = 1.0 + zeta_i(ji,1) * ( zkappa_i(ji,0) * 2.0 + zkappa_i(ji,1) ) ztrid(ji,numeqmin(ji)+1,3) = 0.0 zindterm(ji,numeqmin(ji)+1) = ztiold(ji,1) + zeta_i(ji,1) * & & ( zradab_i(ji,1) + zkappa_i(ji,1) * t_bo_1d(ji) ) ENDIF ELSE ! !------------------------------------------------------------------------------| ! case 4 : surface is melting - no snow | !------------------------------------------------------------------------------| ! numeqmin(ji) = nlay_s + 2 numeqmax(ji) = nlay_i + nlay_s + 1 !!first layer of ice equation ztrid(ji,numeqmin(ji),1) = 0.0 ztrid(ji,numeqmin(ji),2) = 1.0 + zeta_i(ji,1) * ( zkappa_i(ji,1) + zkappa_i(ji,0) * zg1 ) ztrid(ji,numeqmin(ji),3) = - zeta_i(ji,1) * zkappa_i(ji,1) zindterm(ji,numeqmin(ji)) = ztiold(ji,1) + zeta_i(ji,1) * & & ( zradab_i(ji,1) + zkappa_i(ji,0) * zg1 * t_su_1d(ji) ) !!case of only one layer in the ice (surface & ice equations are altered) IF ( nlay_i == 1 ) THEN ztrid(ji,numeqmin(ji),1) = 0.0 ztrid(ji,numeqmin(ji),2) = 1.0 + zeta_i(ji,1) * ( zkappa_i(ji,0) * 2.0 + zkappa_i(ji,1) ) ztrid(ji,numeqmin(ji),3) = 0.0 zindterm(ji,numeqmin(ji)) = ztiold(ji,1) + zeta_i(ji,1) * ( zradab_i(ji,1) + zkappa_i(ji,1) * t_bo_1d(ji) ) & & + t_su_1d(ji) * zeta_i(ji,1) * zkappa_i(ji,0) * 2.0 ENDIF ENDIF ENDIF END DO ! !------------------------------------------------------------------------------| ! 10) tridiagonal system solving | !------------------------------------------------------------------------------| ! ! Solve the tridiagonal system with Gauss elimination method. ! Thomas algorithm, from Computational fluid Dynamics, J.D. ANDERSON, ! McGraw-Hill 1984. maxnumeqmax = 0 minnumeqmin = nlay_i+5 DO ji = 1, nidx zindtbis(ji,numeqmin(ji)) = zindterm(ji,numeqmin(ji)) zdiagbis(ji,numeqmin(ji)) = ztrid(ji,numeqmin(ji),2) minnumeqmin = MIN(numeqmin(ji),minnumeqmin) maxnumeqmax = MAX(numeqmax(ji),maxnumeqmax) END DO DO jk = minnumeqmin+1, maxnumeqmax DO ji = 1, nidx numeq = min(max(numeqmin(ji)+1,jk),numeqmax(ji)) zdiagbis(ji,numeq) = ztrid(ji,numeq,2) - ztrid(ji,numeq,1) * ztrid(ji,numeq-1,3) / zdiagbis(ji,numeq-1) zindtbis(ji,numeq) = zindterm(ji,numeq) - ztrid(ji,numeq,1) * zindtbis(ji,numeq-1) / zdiagbis(ji,numeq-1) END DO END DO DO ji = 1, nidx ! ice temperatures t_i_1d(ji,nlay_i) = zindtbis(ji,numeqmax(ji)) / zdiagbis(ji,numeqmax(ji)) END DO DO numeq = nlay_i + nlay_s, nlay_s + 2, -1 DO ji = 1, nidx jk = numeq - nlay_s - 1 t_i_1d(ji,jk) = ( zindtbis(ji,numeq) - ztrid(ji,numeq,3) * t_i_1d(ji,jk+1) ) / zdiagbis(ji,numeq) END DO END DO DO ji = 1, nidx ! snow temperatures IF( ht_s_1d(ji) > 0._wp ) THEN t_s_1d(ji,nlay_s) = ( zindtbis(ji,nlay_s+1) - ztrid(ji,nlay_s+1,3) * t_i_1d(ji,1) ) & & / zdiagbis(ji,nlay_s+1) ENDIF ! surface temperature ztsub(ji) = t_su_1d(ji) IF( t_su_1d(ji) < rt0 ) THEN t_su_1d(ji) = ( zindtbis(ji,numeqmin(ji)) - ztrid(ji,numeqmin(ji),3) * & & ( isnow(ji) * t_s_1d(ji,1) + ( 1._wp - isnow(ji) ) * t_i_1d(ji,1) ) ) / zdiagbis(ji,numeqmin(ji)) ENDIF END DO ! !-------------------------------------------------------------------------- ! 11) Has the scheme converged ?, end of the iterative procedure | !-------------------------------------------------------------------------- ! ! check that nowhere it has started to melt ! zdti_max is a measure of error, it has to be under zdti_bnd zdti_max = 0._wp DO ji = 1, nidx t_su_1d(ji) = MAX( MIN( t_su_1d(ji) , rt0 ) , rt0 - 100._wp ) zdti_max = MAX( zdti_max, ABS( t_su_1d(ji) - ztsub(ji) ) ) END DO DO jk = 1, nlay_s DO ji = 1, nidx t_s_1d(ji,jk) = MAX( MIN( t_s_1d(ji,jk), rt0 ), rt0 - 100._wp ) zdti_max = MAX( zdti_max, ABS( t_s_1d(ji,jk) - ztsb(ji,jk) ) ) END DO END DO DO jk = 1, nlay_i DO ji = 1, nidx ztmelt_i = -tmut * s_i_1d(ji,jk) + rt0 t_i_1d(ji,jk) = MAX( MIN( t_i_1d(ji,jk), ztmelt_i ), rt0 - 100._wp ) zdti_max = MAX( zdti_max, ABS( t_i_1d(ji,jk) - ztib(ji,jk) ) ) END DO END DO ! Compute spatial maximum over all errors ! note that this could be optimized substantially by iterating only the non-converging points IF( lk_mpp ) CALL mpp_max( zdti_max, kcom=ncomm_ice ) END DO ! End of the do while iterative procedure IF( ln_icectl .AND. lwp ) THEN WRITE(numout,*) ' zdti_max : ', zdti_max WRITE(numout,*) ' iconv : ', iconv ENDIF ! !-------------------------------------------------------------------------! ! 12) Fluxes at the interfaces ! !-------------------------------------------------------------------------! DO ji = 1, nidx ! ! surface ice conduction flux fc_su(ji) = - isnow(ji) * zkappa_s(ji,0) * zg1s * (t_s_1d(ji,1) - t_su_1d(ji)) & & - ( 1._wp - isnow(ji) ) * zkappa_i(ji,0) * zg1 * (t_i_1d(ji,1) - t_su_1d(ji)) ! ! bottom ice conduction flux fc_bo_i(ji) = - zkappa_i(ji,nlay_i) * ( zg1*(t_bo_1d(ji) - t_i_1d(ji,nlay_i)) ) END DO ! --- computes sea ice energy of melting compulsory for icethd_dh --- ! CALL ice_thd_enmelt ! --- diagnose the change in non-solar flux due to surface temperature change --- ! IF ( ln_dqns_i ) THEN DO ji = 1, nidx hfx_err_dif_1d(ji) = hfx_err_dif_1d(ji) - ( qns_ice_1d(ji) - zqns_ice_b(ji) ) * a_i_1d(ji) END DO END IF ! --- diag conservation imbalance on heat diffusion - PART 2 --- ! ! hfx_dif = Heat flux used to warm/cool ice in W.m-2 ! zhfx_err = correction on the diagnosed heat flux due to non-convergence of the algorithm used to solve heat equation DO ji = 1, nidx zdq = - zq_ini(ji) + ( SUM( e_i_1d(ji,1:nlay_i) ) * ht_i_1d(ji) * r1_nlay_i + & & SUM( e_s_1d(ji,1:nlay_s) ) * ht_s_1d(ji) * r1_nlay_s ) IF( t_su_1d(ji) < rt0 ) THEN ! case T_su < 0degC zhfx_err = ( qns_ice_1d(ji) + qsr_ice_1d(ji) - zradtr_i(ji,nlay_i) - fc_bo_i(ji) + zdq * r1_rdtice ) * a_i_1d(ji) ELSE ! case T_su = 0degC zhfx_err = ( fc_su(ji) + i0(ji) * qsr_ice_1d(ji) - zradtr_i(ji,nlay_i) - fc_bo_i(ji) + zdq * r1_rdtice ) * a_i_1d(ji) ENDIF hfx_dif_1d(ji) = hfx_dif_1d(ji) - zdq * r1_rdtice * a_i_1d(ji) ! total heat that is sent to the ocean (i.e. not used in the heat diffusion equation) hfx_err_dif_1d(ji) = hfx_err_dif_1d(ji) + zhfx_err END DO ! --- Diagnostics SIMIP --- ! DO ji = 1, nidx !--- Conduction fluxes (positive downwards) diag_fc_bo_1d(ji) = diag_fc_bo_1d(ji) + fc_bo_i(ji) * a_i_1d(ji) / at_i_1d(ji) diag_fc_su_1d(ji) = diag_fc_su_1d(ji) + fc_su(ji) * a_i_1d(ji) / at_i_1d(ji) !--- Snow-ice interfacial temperature (diagnostic SIMIP) zfac = rn_cnd_s * zh_i(ji) + ztcond_i(ji,1) * zh_s(ji) IF( zh_s(ji) >= 1.e-3 .AND. zfac > epsi10 ) THEN t_si_1d(ji) = ( rn_cnd_s * zh_i(ji) * t_s_1d(ji,1) + & & ztcond_i(ji,1) * zh_s(ji) * t_i_1d(ji,1) ) / zfac ELSE t_si_1d(ji) = t_su_1d(ji) ENDIF END DO ! END SUBROUTINE ice_thd_zdf SUBROUTINE ice_thd_enmelt !!----------------------------------------------------------------------- !! *** ROUTINE ice_thd_enmelt *** !! !! ** Purpose : Computes sea ice energy of melting q_i (J.m-3) from temperature !! !! ** Method : Formula (Bitz and Lipscomb, 1999) !!------------------------------------------------------------------- INTEGER :: ji, jk ! dummy loop indices REAL(wp) :: ztmelts ! local scalar !!------------------------------------------------------------------- ! DO jk = 1, nlay_i ! Sea ice energy of melting DO ji = 1, nidx ztmelts = - tmut * s_i_1d(ji,jk) t_i_1d(ji,jk) = MIN( t_i_1d(ji,jk), ztmelts + rt0 ) ! Force t_i_1d to be lower than melting point ! (sometimes dif scheme produces abnormally high temperatures) e_i_1d(ji,jk) = rhoic * ( cpic * ( ztmelts - ( t_i_1d(ji,jk) - rt0 ) ) & & + lfus * ( 1._wp - ztmelts / ( t_i_1d(ji,jk) - rt0 ) ) & & - rcp * ztmelts ) END DO END DO DO jk = 1, nlay_s ! Snow energy of melting DO ji = 1, nidx e_s_1d(ji,jk) = rhosn * ( cpic * ( rt0 - t_s_1d(ji,jk) ) + lfus ) END DO END DO ! END SUBROUTINE ice_thd_enmelt SUBROUTINE ice_thd_zdf_init !!----------------------------------------------------------------------- !! *** ROUTINE ice_thd_zdf_init *** !! !! ** Purpose : Physical constants and parameters associated with !! ice thermodynamics !! !! ** Method : Read the namthd_zdf namelist and check the parameters !! called at the first timestep (nit000) !! !! ** input : Namelist namthd_zdf !!------------------------------------------------------------------- INTEGER :: ios ! Local integer output status for namelist read !! NAMELIST/namthd_zdf/ ln_zdf_Beer, ln_cndi_U64, ln_cndi_P07, rn_cnd_s, rn_kappa_i, ln_dqns_i !!------------------------------------------------------------------- ! REWIND( numnam_ice_ref ) ! Namelist namthd_zdf in reference namelist : Ice thermodynamics READ ( numnam_ice_ref, namthd_zdf, IOSTAT = ios, ERR = 901) 901 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namthd_zdf in reference namelist', lwp ) REWIND( numnam_ice_cfg ) ! Namelist namthd_zdf in configuration namelist : Ice thermodynamics READ ( numnam_ice_cfg, namthd_zdf, IOSTAT = ios, ERR = 902 ) 902 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namthd_zdf in configuration namelist', lwp ) IF(lwm) WRITE ( numoni, namthd_zdf ) ! ! IF(lwp) THEN ! control print WRITE(numout,*) 'ice_thd_zdf_init: Ice vertical heat diffusion' WRITE(numout,*) '~~~~~~~~~~~~~~~~' WRITE(numout,*) ' Namelist namthd_zdf:' WRITE(numout,*) ' Diffusion follows a Beer Law ln_zdf_Beer = ', ln_zdf_Beer WRITE(numout,*) ' thermal conductivity in the ice (Untersteiner 1964) ln_cndi_U64 = ', ln_cndi_U64 WRITE(numout,*) ' thermal conductivity in the ice (Pringle et al 2007) ln_cndi_P07 = ', ln_cndi_P07 WRITE(numout,*) ' thermal conductivity in the snow rn_cnd_s = ', rn_cnd_s WRITE(numout,*) ' extinction radiation parameter in sea ice rn_kappa_i = ', rn_kappa_i WRITE(numout,*) ' change the surface non-solar flux with Tsu or not ln_dqns_i = ', ln_dqns_i ENDIF ! IF ( ( ln_cndi_U64 .AND. ln_cndi_P07 ) .OR. ( .NOT.ln_cndi_U64 .AND. .NOT.ln_cndi_P07 ) ) THEN CALL ctl_stop( 'ice_thd_zdf_init: choose one and only one formulation for thermal conductivity (ln_cndi_U64 or ln_cndi_P07)' ) ENDIF ! END SUBROUTINE ice_thd_zdf_init #else !!---------------------------------------------------------------------- !! Dummy Module No ESIM sea-ice model !!---------------------------------------------------------------------- #endif !!====================================================================== END MODULE icethd_zdf