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MODULE icedyn_rdgrft
   !!======================================================================
   !!                       ***  MODULE icedyn_rdgrft ***
   !!    sea-ice : Mechanical impact on ice thickness distribution
   !!======================================================================
   !! History :       !  2006-02  (M. Vancoppenolle) Original code
   !!            4.0  !  2018     (many people)      SI3 [aka Sea Ice cube]
   !!----------------------------------------------------------------------
#if defined key_si3
   !!----------------------------------------------------------------------
   !!   'key_si3'                                       SI3 sea-ice model
   !!----------------------------------------------------------------------
   !!   ice_dyn_rdgrft       : ridging/rafting of sea ice
   !!   ice_dyn_rdgrft_init  : initialization of ridging/rafting of sea ice
   !!   ice_strength         : ice strength calculation
   !!----------------------------------------------------------------------
   USE dom_oce        ! ocean domain
   USE phycst         ! physical constants (ocean directory)
   USE sbc_oce , ONLY : sss_m, sst_m   ! surface boundary condition: ocean fields
   USE ice1D          ! sea-ice: thermodynamics
   USE ice            ! sea-ice: variables
   USE icetab         ! sea-ice: 1D <==> 2D transformation
   USE icevar         ! sea-ice: operations
   USE icectl         ! sea-ice: control prints
   !
   USE in_out_manager ! I/O manager
   USE iom            ! I/O manager library
   USE lib_mpp        ! MPP library
   USE lib_fortran    ! fortran utilities (glob_sum + no signed zero)
   USE lbclnk         ! lateral boundary conditions (or mpp links)
   USE timing         ! Timing

   IMPLICIT NONE
   PRIVATE

   PUBLIC   ice_dyn_rdgrft        ! called by icestp
   PUBLIC   ice_dyn_rdgrft_init   ! called by icedyn
   PUBLIC   ice_strength          ! called by icedyn_rhg_evp

   INTEGER ::              nice_str   ! choice of the type of strength
   !                                        ! associated indices:
   INTEGER, PARAMETER ::   np_strh79     = 1   ! Hibler 79
   INTEGER, PARAMETER ::   np_strr75     = 2   ! Rothrock 75
   INTEGER, PARAMETER ::   np_strcst     = 3   ! Constant value

   ! Variables shared among ridging subroutines
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:)     ::   closing_net     ! net rate at which area is removed    (1/s)
      !                                                               ! (ridging ice area - area of new ridges) / dt
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:)     ::   opning          ! rate of opening due to divergence/shear
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:)     ::   closing_gross   ! rate at which area removed, not counting area of new ridges
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   apartf          ! participation function; fraction of ridging/closing associated w/ category n
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   hrmin           ! minimum ridge thickness
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   hrmax           ! maximum ridge thickness
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   hrexp           ! e-folding ridge thickness
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   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   hraft           ! thickness of rafted ice
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   hi_hrdg         ! thickness of ridging ice / mean ridge thickness
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   aridge          ! participating ice ridging
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:)   ::   araft           ! participating ice rafting
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:,:) ::   ze_i_2d
   REAL(wp), ALLOCATABLE, SAVE, DIMENSION(:,:,:) ::   ze_s_2d
   !
   REAL(wp), PARAMETER ::   hrdg_hi_min = 1.1_wp    ! min ridge thickness multiplier: min(hrdg/hi)
   REAL(wp), PARAMETER ::   hi_hrft     = 0.5_wp    ! rafting multiplier: (hi/hraft)
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   !
   ! ** namelist (namdyn_rdgrft) **
   LOGICAL  ::   ln_str_R75       ! ice strength parameterization: Rothrock 75
   REAL(wp) ::   rn_pe_rdg        !    coef accounting for frictional dissipation
   LOGICAL  ::   ln_str_CST       ! ice strength parameterization: Constant
   REAL(wp) ::   rn_str           !    constant value of ice strength
   LOGICAL  ::   ln_str_smooth    ! ice strength spatial smoothing
   LOGICAL  ::   ln_distf_lin     ! redistribution of ridged ice: linear (Hibler, 1980)
   LOGICAL  ::   ln_distf_exp     ! redistribution of ridged ice: exponential (Lipscomb et al., 2007)
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   REAL(wp) ::   rn_murdg         !    gives e-folding scale of ridged ice (m^.5)
   REAL(wp) ::   rn_csrdg         ! fraction of shearing energy contributing to ridging
   LOGICAL  ::   ln_partf_lin     ! participation function: linear (Thorndike et al., 1975)
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   REAL(wp) ::   rn_gstar         !    fractional area of young ice contributing to ridging
   LOGICAL  ::   ln_partf_exp     ! participation function: exponential (Lipscomb et al., 2007)
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   REAL(wp) ::   rn_astar         !    equivalent of G* for an exponential participation function
   LOGICAL  ::   ln_ridging       ! ridging of ice or not
   REAL(wp) ::   rn_hstar         !    thickness that determines the maximal thickness of ridged ice
   REAL(wp) ::   rn_porordg       !    initial porosity of ridges (0.3 regular value)
   REAL(wp) ::   rn_fsnwrdg       !    fractional snow loss to the ocean during ridging
   REAL(wp) ::   rn_fpndrdg       !    fractional pond loss to the ocean during ridging
   LOGICAL  ::   ln_rafting       ! rafting of ice or not
   REAL(wp) ::   rn_hraft         !    threshold thickness (m) for rafting / ridging
   REAL(wp) ::   rn_craft         !    coefficient for smoothness of the hyperbolic tangent in rafting
   REAL(wp) ::   rn_fsnwrft       !    fractional snow loss to the ocean during rafting
   REAL(wp) ::   rn_fpndrft       !    fractional pond loss to the ocean during rafting
   !
   !! * Substitutions
#  include "do_loop_substitute.h90"
   !!----------------------------------------------------------------------
   !! NEMO/ICE 4.0 , NEMO Consortium (2018)
   !! $Id: icedyn_rdgrft.F90 15549 2021-11-28 20:00:36Z clem $
   !! Software governed by the CeCILL licence     (./LICENSE)
   !!----------------------------------------------------------------------
CONTAINS

   INTEGER FUNCTION ice_dyn_rdgrft_alloc()
      !!-------------------------------------------------------------------
      !!                ***  ROUTINE ice_dyn_rdgrft_alloc ***
      !!-------------------------------------------------------------------
      ALLOCATE( closing_net(jpij) , opning(jpij)    , closing_gross(jpij),                   &
         &      apartf(jpij,0:jpl), hrmin (jpij,jpl), hraft(jpij,jpl)    , aridge(jpij,jpl), &
         &      hrmax (jpij,jpl)  , hrexp (jpij,jpl), hi_hrdg(jpij,jpl)  , araft(jpij,jpl) , &
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         &      ze_i_2d(jpij,nlay_i,jpl), ze_s_2d(jpij,nlay_s,jpl), STAT=ice_dyn_rdgrft_alloc )

      CALL mpp_sum ( 'icedyn_rdgrft', ice_dyn_rdgrft_alloc )
      IF( ice_dyn_rdgrft_alloc /= 0 )   CALL ctl_stop( 'STOP',  'ice_dyn_rdgrft_alloc: failed to allocate arrays'  )
      !
   END FUNCTION ice_dyn_rdgrft_alloc


   SUBROUTINE ice_dyn_rdgrft( kt )
      !!-------------------------------------------------------------------
      !!                ***  ROUTINE ice_dyn_rdgrft ***
      !!
      !! ** Purpose :   computes the mechanical redistribution of ice thickness
      !!
      !! ** Method  :   Steps :
      !!       0) Identify grid cells with ice
      !!       1) Calculate closing rate, divergence and opening
      !!       2) Identify grid cells with ridging
      !!       3) Start ridging iterations
      !!          - prep = ridged and rafted ice + closing_gross
      !!          - shift = move ice from one category to another
      !!
      !! ** Details
      !!    step1: The net rate of closing is due to convergence and shear, based on Flato and Hibler (1995).
      !!           The energy dissipation rate is equal to the net closing rate times the ice strength.
      !!
      !!    step3: The gross closing rate is equal to the first two terms (open
      !!           water closing and thin ice ridging) without the third term
      !!           (thick, newly ridged ice).
      !!
      !! References :   Flato, G. M., and W. D. Hibler III, 1995, JGR, 100, 18,611-18,626.
      !!                Hibler, W. D. III, 1980, MWR, 108, 1943-1973, 1980.
      !!                Rothrock, D. A., 1975: JGR, 80, 4514-4519.
      !!                Thorndike et al., 1975, JGR, 80, 4501-4513.
      !!                Bitz et al., JGR, 2001
      !!                Amundrud and Melling, JGR 2005
      !!                Babko et al., JGR 2002
      !!
      !!     This routine is based on CICE code and authors William H. Lipscomb,
      !!     and Elizabeth C. Hunke, LANL are gratefully acknowledged
      !!-------------------------------------------------------------------
      INTEGER, INTENT(in) ::   kt     ! number of iteration
      !!
      INTEGER  ::   ji, jj, jk, jl             ! dummy loop index
      INTEGER  ::   iter, iterate_ridging      ! local integer
      INTEGER  ::   ipti                       ! local integer
      REAL(wp) ::   zfac                       ! local scalar
      INTEGER , DIMENSION(jpij) ::   iptidx        ! compute ridge/raft or not
      REAL(wp), DIMENSION(jpij) ::   zdivu, zdelt  ! 1D divu_i & delta_i
      REAL(wp), DIMENSION(jpij) ::   zconv         ! 1D rdg_conv (if EAP rheology)
      !
      INTEGER, PARAMETER ::   jp_itermax = 20
      !!-------------------------------------------------------------------
      ! controls
      IF( ln_timing    )   CALL timing_start('icedyn_rdgrft')                                                             ! timing
      IF( ln_icediachk )   CALL ice_cons_hsm(0, 'icedyn_rdgrft', rdiag_v, rdiag_s, rdiag_t, rdiag_fv, rdiag_fs, rdiag_ft) ! conservation
      IF( ln_icediachk )   CALL ice_cons2D  (0, 'icedyn_rdgrft',  diag_v,  diag_s,  diag_t,  diag_fv,  diag_fs,  diag_ft) ! conservation

      IF( kt == nit000 ) THEN
         IF(lwp) WRITE(numout,*)
         IF(lwp) WRITE(numout,*)'ice_dyn_rdgrft: ice ridging and rafting'
         IF(lwp) WRITE(numout,*)'~~~~~~~~~~~~~~'
      ENDIF

      !--------------------------------
      ! 0) Identify grid cells with ice
      !--------------------------------
      at_i(:,:) = SUM( a_i, dim=3 )
      !
      npti = 0   ;   nptidx(:) = 0
      ipti = 0   ;   iptidx(:) = 0
      DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
         IF ( at_i(ji,jj) > epsi10 ) THEN
            npti           = npti + 1
            nptidx( npti ) = (jj - 1) * jpi + ji
         ENDIF
      END_2D

      !--------------------------------------------------------
      ! 1) Dynamical inputs (closing rate, divergence, opening)
      !--------------------------------------------------------
      IF( npti > 0 ) THEN

         ! just needed here
         CALL tab_2d_1d( npti, nptidx(1:npti), zdelt   (1:npti)      , delta_i )
         CALL tab_2d_1d( npti, nptidx(1:npti), zconv   (1:npti)      , rdg_conv )
         ! needed here and in the iteration loop
         CALL tab_2d_1d( npti, nptidx(1:npti), zdivu   (1:npti)      , divu_i) ! zdivu is used as a work array here (no change in divu_i)
         CALL tab_3d_2d( npti, nptidx(1:npti), a_i_2d  (1:npti,1:jpl), a_i   )
         CALL tab_3d_2d( npti, nptidx(1:npti), v_i_2d  (1:npti,1:jpl), v_i   )
         CALL tab_2d_1d( npti, nptidx(1:npti), ato_i_1d(1:npti)      , ato_i )

         DO ji = 1, npti
            ! closing_net = rate at which open water area is removed + ice area removed by ridging
            !                                                        - ice area added in new ridges
            IF( ln_rhg_EVP .OR. ln_rhg_VP ) &
               &               closing_net(ji) = rn_csrdg * 0.5_wp * ( zdelt(ji) - ABS( zdivu(ji) ) ) - MIN( zdivu(ji), 0._wp )
            IF( ln_rhg_EAP )   closing_net(ji) = zconv(ji)
            !
            IF( zdivu(ji) < 0._wp )   closing_net(ji) = MAX( closing_net(ji), -zdivu(ji) )   ! make sure the closing rate is large enough
            !                                                                                ! to give asum = 1.0 after ridging
            ! Opening rate (non-negative) that will give asum = 1.0 after ridging.
            opning(ji) = closing_net(ji) + zdivu(ji)
         END DO
         !
         !------------------------------------
         ! 2) Identify grid cells with ridging
         !------------------------------------
         CALL rdgrft_prep( a_i_2d, v_i_2d, ato_i_1d, closing_net )

         DO ji = 1, npti
            IF( SUM( apartf(ji,1:jpl) ) > 0._wp .AND. closing_gross(ji) > 0._wp ) THEN
               ipti = ipti + 1
               iptidx     (ipti)   = nptidx     (ji)
               ! adjust to new indices
               a_i_2d     (ipti,:) = a_i_2d     (ji,:)
               v_i_2d     (ipti,:) = v_i_2d     (ji,:)
               ato_i_1d   (ipti)   = ato_i_1d   (ji)
               closing_net(ipti)   = closing_net(ji)
               zdivu      (ipti)   = zdivu      (ji)
               opning     (ipti)   = opning     (ji)
            ENDIF
         END DO

      ENDIF

      ! grid cells with ridging
      nptidx(:) = iptidx(:)
      npti      = ipti

      !-----------------
      ! 3) Start ridging
      !-----------------
      IF( npti > 0 ) THEN

         CALL ice_dyn_1d2d( 1 )            ! --- Move to 1D arrays --- !

         iter            = 1
         iterate_ridging = 1
         !                                                        !----------------------!
         DO WHILE( iterate_ridging > 0 .AND. iter < jp_itermax )  !  ridging iterations  !
            !                                                     !----------------------!
            ! Calculate participation function (apartf)
            !       and transfer      function
            !       and closing_gross (+correction on opening)
            CALL rdgrft_prep( a_i_2d, v_i_2d, ato_i_1d, closing_net )

            ! Redistribute area, volume, and energy between categories
            CALL rdgrft_shift

            ! Do we keep on iterating?
            !-------------------------
            ! Check whether a_i + ato_i = 0
            ! If not, because the closing and opening rates were reduced above, ridge again with new rates
            iterate_ridging = 0
            DO ji = 1, npti
               zfac = 1._wp - ( ato_i_1d(ji) + SUM( a_i_2d(ji,:) ) )
               IF( ABS( zfac ) < epsi10 ) THEN
                  closing_net(ji) = 0._wp
                  opning     (ji) = 0._wp
                  ato_i_1d   (ji) = MAX( 0._wp, 1._wp - SUM( a_i_2d(ji,:) ) )
               ELSE
                  iterate_ridging  = 1
                  zdivu      (ji) = zfac * r1_Dt_ice
                  closing_net(ji) = MAX( 0._wp, -zdivu(ji) )
                  opning     (ji) = MAX( 0._wp,  zdivu(ji) )
               ENDIF
            END DO
            !
            iter = iter + 1
            IF( iter  >  jp_itermax )    CALL ctl_stop( 'STOP',  'icedyn_rdgrft: non-converging ridging scheme'  )
            !
         END DO

         CALL ice_dyn_1d2d( 2 )            ! --- Move to 2D arrays --- !

      ENDIF

      CALL ice_var_agg( 1 )

      ! controls
      IF( sn_cfctl%l_prtctl )   CALL ice_prt3D('icedyn_rdgrft')                                                           ! prints
      IF( ln_icectl    )   CALL ice_prt     (kt, iiceprt, jiceprt,-1, ' - ice dyn rdgrft - ')                             ! prints
      IF( ln_icediachk )   CALL ice_cons_hsm(1, 'icedyn_rdgrft', rdiag_v, rdiag_s, rdiag_t, rdiag_fv, rdiag_fs, rdiag_ft) ! conservation
      IF( ln_icediachk )   CALL ice_cons2D  (1, 'icedyn_rdgrft',  diag_v,  diag_s,  diag_t,  diag_fv,  diag_fs,  diag_ft) ! conservation
      IF( ln_timing    )   CALL timing_stop ('icedyn_rdgrft')                                                             ! timing
      !
   END SUBROUTINE ice_dyn_rdgrft


   SUBROUTINE rdgrft_prep( pa_i, pv_i, pato_i, pclosing_net )
      !!-------------------------------------------------------------------
      !!                ***  ROUTINE rdgrft_prep ***
      !!
      !! ** Purpose :   preparation for ridging calculations
      !!
      !! ** Method  :   Compute the thickness distribution of the ice and open water
      !!                participating in ridging and of the resulting ridges.
      !!-------------------------------------------------------------------
      REAL(wp), DIMENSION(:,:), INTENT(in) ::   pa_i, pv_i
      REAL(wp), DIMENSION(:)  , INTENT(in) ::   pato_i
      REAL(wp), DIMENSION(:)  , INTENT(in), OPTIONAL ::   pclosing_net
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      !!
      INTEGER  ::   ji, jl                     ! dummy loop indices
      REAL(wp) ::   z1_gstar, z1_astar, zhmean, zfac   ! local scalar
      REAL(wp), DIMENSION(jpij)        ::   zasum, z1_asum, zaksum   ! sum of a_i+ato_i and reverse
      REAL(wp), DIMENSION(jpij,jpl)    ::   zhi                      ! ice thickness
      REAL(wp), DIMENSION(jpij,-1:jpl) ::   zGsum                    ! zGsum(n) = sum of areas in categories 0 to n
      !--------------------------------------------------------------------

      z1_gstar = 1._wp / rn_gstar
      z1_astar = 1._wp / rn_astar

      !                       ! Ice thickness needed for rafting
      ! In single precision there were floating point invalids due a sqrt of zhi which happens to have negative values
      ! To solve that an extra check about the value of pv_i was added.
      ! Although adding this condition is safe, the double definition (one for single other for double) has been kept to preserve the results of the sette test.
#if defined key_single

      WHERE( pa_i(1:npti,:) > epsi10 .and. pv_i(1:npti,:) > epsi10 )   ;   zhi(1:npti,:) = pv_i(1:npti,:) / pa_i(1:npti,:)
#else
      WHERE( pa_i(1:npti,:) > epsi10 )   ;   zhi(1:npti,:) = pv_i(1:npti,:) / pa_i(1:npti,:)
#endif
      ELSEWHERE                          ;   zhi(1:npti,:) = 0._wp
      END WHERE

      ! 1) Participation function (apartf): a(h) = b(h).g(h)
      !-----------------------------------------------------------------
      ! Compute the participation function = total area lost due to ridging/closing
      ! This is analogous to
      !   a(h) = b(h)g(h) as defined in Thorndike et al. (1975).
      !   assuming b(h) = (2/Gstar) * (1 - G(h)/Gstar).
      !
      ! apartf = integrating b(h)g(h) between the category boundaries
      ! apartf is always >= 0 and SUM(apartf(0:jpl))=1
      !-----------------------------------------------------------------
      !
      ! Compute total area of ice plus open water.
      ! This is in general not equal to one because of divergence during transport
      zasum(1:npti) = pato_i(1:npti) + SUM( pa_i(1:npti,:), dim=2 )
      !
      WHERE( zasum(1:npti) > epsi10 )   ;   z1_asum(1:npti) = 1._wp / zasum(1:npti)
      ELSEWHERE                         ;   z1_asum(1:npti) = 0._wp
      END WHERE
      !
      ! Compute cumulative thickness distribution function
      ! Compute the cumulative thickness distribution function zGsum,
      ! where zGsum(n) is the fractional area in categories 0 to n.
      ! initial value (in h = 0) = open water area
      zGsum(1:npti,-1) = 0._wp
      zGsum(1:npti,0 ) = pato_i(1:npti) * z1_asum(1:npti)
      DO jl = 1, jpl
         zGsum(1:npti,jl) = ( pato_i(1:npti) + SUM( pa_i(1:npti,1:jl), dim=2 ) ) * z1_asum(1:npti)  ! sum(1:jl) is correct (and not jpl)
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      END DO
      !
      IF( ln_partf_lin ) THEN          !--- Linear formulation (Thorndike et al., 1975)
         DO jl = 0, jpl
            DO ji = 1, npti
               IF    ( zGsum(ji,jl)   < rn_gstar ) THEN
                  apartf(ji,jl) = z1_gstar * ( zGsum(ji,jl) - zGsum(ji,jl-1) ) * &
                     &                       ( 2._wp - ( zGsum(ji,jl-1) + zGsum(ji,jl) ) * z1_gstar )
               ELSEIF( zGsum(ji,jl-1) < rn_gstar ) THEN
                  apartf(ji,jl) = z1_gstar * ( rn_gstar     - zGsum(ji,jl-1) ) *  &
                     &                       ( 2._wp - ( zGsum(ji,jl-1) + rn_gstar     ) * z1_gstar )
               ELSE
                  apartf(ji,jl) = 0._wp
               ENDIF
            END DO
         END DO
         !
      ELSEIF( ln_partf_exp ) THEN      !--- Exponential, more stable formulation (Lipscomb et al., 2007)
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         !
         zfac = 1._wp / ( 1._wp - EXP(-z1_astar) )
         DO jl = -1, jpl
            DO ji = 1, npti
               zGsum(ji,jl) = EXP( -zGsum(ji,jl) * z1_astar ) * zfac
            END DO
         END DO
         DO jl = 0, jpl
            DO ji = 1, npti
               apartf(ji,jl) = zGsum(ji,jl-1) - zGsum(ji,jl)
            END DO
         END DO
         !
      ENDIF

      !                                !--- Ridging and rafting participation concentrations
      IF( ln_rafting .AND. ln_ridging ) THEN             !- ridging & rafting
         DO jl = 1, jpl
            DO ji = 1, npti
               aridge(ji,jl) = ( 1._wp + TANH ( rn_craft * ( zhi(ji,jl) - rn_hraft ) ) ) * 0.5_wp * apartf(ji,jl)
               araft (ji,jl) = apartf(ji,jl) - aridge(ji,jl)
            END DO
         END DO
      ELSEIF( ln_ridging .AND. .NOT. ln_rafting ) THEN   !- ridging alone
         DO jl = 1, jpl
            DO ji = 1, npti
               aridge(ji,jl) = apartf(ji,jl)
               araft (ji,jl) = 0._wp
            END DO
         END DO
      ELSEIF( ln_rafting .AND. .NOT. ln_ridging ) THEN   !- rafting alone
         DO jl = 1, jpl
            DO ji = 1, npti
               aridge(ji,jl) = 0._wp
               araft (ji,jl) = apartf(ji,jl)
            END DO
         END DO
      ELSE                                               !- no ridging & no rafting
         DO jl = 1, jpl
            DO ji = 1, npti
               aridge(ji,jl) = 0._wp
               araft (ji,jl) = 0._wp
            END DO
         END DO
      ENDIF

      ! 2) Transfer function
      !-----------------------------------------------------------------
      ! If assuming ridged ice is uniformly distributed between hrmin and  
      ! hrmax (ln_distf_lin):
      !
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      ! Compute max and min ridged ice thickness for each ridging category.
      !
      ! This parameterization is a modified version of Hibler (1980).
      ! The mean ridging thickness, zhmean, is proportional to hi^(0.5)
      !  and for very thick ridging ice must be >= hrdg_hi_min*hi
      !
      ! The minimum ridging thickness, hrmin, is equal to 2*hi
      !  (i.e., rafting) and for very thick ridging ice is
      !  constrained by hrmin <= (zhmean + hi)/2.
      !
      ! The maximum ridging thickness, hrmax, is determined by zhmean and hrmin.
      !
      ! These modifications have the effect of reducing the ice strength
      ! (relative to the Hibler formulation) when very thick ice is ridging.
      !
      !-----------------------------------------------------------------
      ! If assuming ridged ice ITD is a negative exponential 
      ! (ln_distf_exp) and following CICE implementation: 
      ! 
      !  g(h) ~ exp[-(h-hrmin)/hrexp], h >= hrmin 
      ! 
      ! where hrmin is the minimum thickness of ridging ice and 
      ! hrexp is the e-folding thickness.
      ! 
      ! Here, assume as above that hrmin = min(2*hi, hi+maxraft).
      ! That is, the minimum ridge thickness results from rafting,
      !  unless the ice is thicker than maxraft.
      !
      ! Also, assume that hrexp = mu_rdg*sqrt(hi).
      ! The parameter mu_rdg is tuned to give e-folding scales mostly
      !  in the range 2-4 m as observed by upward-looking sonar.
      !
      ! Values of mu_rdg in the right column give ice strengths
      !  roughly equal to values of Hstar in the left column
      !  (within ~10 kN/m for typical ITDs):
      !
      !   Hstar     mu_rdg
      !
      !     25        3.0
      !     50        4.0
      !     75        5.0
      !    100        6.0
      !
      ! zaksum = net area removed/ total area participating
      ! where total area participating = area of ice that ridges
      !         net area removed = total area participating - area of new ridges
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      !-----------------------------------------------------------------
      zfac = 1._wp / hi_hrft
      zaksum(1:npti) = apartf(1:npti,0)
      !
      DO jl = 1, jpl
         DO ji = 1, npti
            IF ( apartf(ji,jl) > 0._wp ) THEN
               zhmean         = MAX( SQRT( rn_hstar * zhi(ji,jl) ), zhi(ji,jl) * hrdg_hi_min )
               hrmin  (ji,jl) = MIN( 2._wp * zhi(ji,jl), 0.5_wp * ( zhmean + zhi(ji,jl) ) )
               hraft  (ji,jl) = zhi(ji,jl) * zfac
               !
               IF( ln_distf_lin ) THEN
                  hrmax  (ji,jl) = 2._wp * zhmean - hrmin(ji,jl)
                  hi_hrdg(ji,jl) = zhi(ji,jl) / MAX( zhmean, epsi20 )
               ELSEIF( ln_distf_exp ) THEN
                  hrexp  (ji,jl) = rn_murdg * SQRT( zhi(ji,jl) )
                  hi_hrdg(ji,jl) = zhi(ji,jl) / ( hrmin(ji,jl) + hrexp(ji,jl) )
                  !!clem: set a mini for zhi??
               ENDIF
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               !
               ! Normalization factor : zaksum, ensures mass conservation
               zaksum(ji) = zaksum(ji) + aridge(ji,jl) * ( 1._wp - hi_hrdg(ji,jl) )    &
                  &                    + araft (ji,jl) * ( 1._wp - hi_hrft )
            ELSE
               hrmin  (ji,jl) = 0._wp
               hrmax  (ji,jl) = 0._wp
               hrexp  (ji,jl) = 0._wp
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               hraft  (ji,jl) = 0._wp
               hi_hrdg(ji,jl) = 1._wp
               !!clem zaksum(ji,jl) = 0._wp
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            ENDIF
         END DO
      END DO
      !
      IF( PRESENT( pclosing_net ) ) THEN
         !
         ! 3) closing_gross
         !-----------------
         ! Based on the ITD of ridging and ridged ice, convert the net closing rate to a gross closing rate.
         ! NOTE: 0 < aksum <= 1
         WHERE( zaksum(1:npti) > epsi10 )   ;   closing_gross(1:npti) = pclosing_net(1:npti) / zaksum(1:npti)
         ELSEWHERE                          ;   closing_gross(1:npti) = 0._wp
         END WHERE
         ! correction to closing rate if excessive ice removal
         !----------------------------------------------------
         ! Reduce the closing rate if more than 100% of any ice category would be removed
         ! Reduce the opening rate in proportion
         DO jl = 1, jpl
            DO ji = 1, npti
               zfac = apartf(ji,jl) * closing_gross(ji) * rDt_ice
               IF( zfac > pa_i(ji,jl) .AND. apartf(ji,jl) /= 0._wp ) THEN
                  closing_gross(ji) = pa_i(ji,jl) / apartf(ji,jl) * r1_Dt_ice
               ENDIF
            END DO
         END DO
         ! 4) correction to opening if excessive open water removal
         !---------------------------------------------------------
         ! Reduce the closing rate if more than 100% of the open water would be removed
         ! Reduce the opening rate in proportion
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         DO ji = 1, npti
            zfac = pato_i(ji) + ( opning(ji) - apartf(ji,0) * closing_gross(ji) ) * rDt_ice
            IF( zfac < 0._wp ) THEN           ! would lead to negative ato_i
               opning(ji) = apartf(ji,0) * closing_gross(ji) - pato_i(ji) * r1_Dt_ice
            ELSEIF( zfac > zasum(ji) ) THEN   ! would lead to ato_i > asum
               opning(ji) = apartf(ji,0) * closing_gross(ji) + ( zasum(ji) - pato_i(ji) ) * r1_Dt_ice
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            ENDIF
         END DO
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      !
   END SUBROUTINE rdgrft_prep


   SUBROUTINE rdgrft_shift
      !!-------------------------------------------------------------------
      !!                ***  ROUTINE rdgrft_shift ***
      !!
      !! ** Purpose :   shift ridging ice among thickness categories of ice thickness
      !!
      !! ** Method  :   Remove area, volume, and energy from each ridging category
      !!                and add to thicker ice categories.
      !!-------------------------------------------------------------------
      !
      INTEGER  ::   ji, jj, jl, jl1, jl2, jk   ! dummy loop indices
      REAL(wp) ::   hL, hR, farea              ! left and right limits of integration and new area going to jl2
      REAL(wp) ::   expL, expR                 ! exponentials involving hL, hR
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      REAL(wp) ::   vsw                        ! vol of water trapped into ridges
      REAL(wp) ::   afrdg, afrft               ! fraction of category area ridged/rafted
      REAL(wp)                  ::   airdg1, oirdg1, aprdg1, virdg1, sirdg1
      REAL(wp)                  ::   airft1, oirft1, aprft1
      REAL(wp), DIMENSION(jpij) ::   airdg2, oirdg2, aprdg2, virdg2, sirdg2, vsrdg, vprdg, vlrdg  ! area etc of new ridges
      REAL(wp), DIMENSION(jpij) ::   airft2, oirft2, aprft2, virft , sirft , vsrft, vprft, vlrft  ! area etc of rafted ice
      !
      REAL(wp), DIMENSION(jpij) ::   ersw             ! enth of water trapped into ridges
      REAL(wp), DIMENSION(jpij) ::   zswitch, fvol    ! new ridge volume going to jl2
      REAL(wp), DIMENSION(jpij) ::   z1_ai            ! 1 / a
      REAL(wp), DIMENSION(jpij) ::   zvti             ! sum(v_i)
      !
      REAL(wp), DIMENSION(jpij,nlay_s) ::   esrft     ! snow energy of rafting ice
      REAL(wp), DIMENSION(jpij,nlay_i) ::   eirft     ! ice  energy of rafting ice
      REAL(wp), DIMENSION(jpij,nlay_s) ::   esrdg     ! enth*volume of new ridges
      REAL(wp), DIMENSION(jpij,nlay_i) ::   eirdg     ! enth*volume of new ridges
      !
      INTEGER , DIMENSION(jpij) ::   itest_rdg, itest_rft   ! test for conservation
      LOGICAL , DIMENSION(jpij) ::   ll_shift         ! logical for doing calculation or not
      !!-------------------------------------------------------------------
      !
      zvti(1:npti) = SUM( v_i_2d(1:npti,:), dim=2 )   ! total ice volume
      !
      ! 1) Change in open water area due to closing and opening
      !--------------------------------------------------------
      DO ji = 1, npti
         ato_i_1d(ji) = MAX( 0._wp, ato_i_1d(ji) + ( opning(ji) - apartf(ji,0) * closing_gross(ji) ) * rDt_ice )
      END DO

      ! 2) compute categories in which ice is removed (jl1)
      !----------------------------------------------------
      DO jl1 = 1, jpl

         IF( nn_icesal /= 2 )  THEN
            CALL tab_2d_1d( npti, nptidx(1:npti), s_i_1d(1:npti), s_i(:,:,jl1) )
         ENDIF

         DO ji = 1, npti

            ! set logical to true when ridging
            IF( apartf(ji,jl1) > 0._wp .AND. closing_gross(ji) > 0._wp ) THEN   ;   ll_shift(ji) = .TRUE.
            ELSE                                                                ;   ll_shift(ji) = .FALSE.
            ENDIF
            
            IF( ll_shift(ji) ) THEN   ! only if ice is ridging

               IF( a_i_2d(ji,jl1) > epsi10 ) THEN   ;   z1_ai(ji) = 1._wp / a_i_2d(ji,jl1)
               ELSE                                 ;   z1_ai(ji) = 0._wp
               ENDIF

               ! area of ridging / rafting ice (airdg1) and of new ridge (airdg2)
               airdg1 = aridge(ji,jl1) * closing_gross(ji) * rDt_ice
               airft1 = araft (ji,jl1) * closing_gross(ji) * rDt_ice

               airdg2(ji) = airdg1 * hi_hrdg(ji,jl1)
               airft2(ji) = airft1 * hi_hrft

               ! ridging /rafting fractions
               afrdg = airdg1 * z1_ai(ji)
               afrft = airft1 * z1_ai(ji)

               ! volume and enthalpy (J/m2, >0) of seawater trapped into ridges
               IF    ( zvti(ji) <= 10. ) THEN ; vsw = v_i_2d(ji,jl1) * afrdg * rn_porordg                                           ! v <= 10m then porosity = rn_porordg
               ELSEIF( zvti(ji) >= 20. ) THEN ; vsw = 0._wp                                                                         ! v >= 20m then porosity = 0
               ELSE                           ; vsw = v_i_2d(ji,jl1) * afrdg * rn_porordg * MAX( 0._wp, 2._wp - 0.1_wp * zvti(ji) ) ! v > 10m and v < 20m then porosity = linear transition to 0
               ENDIF
               ersw(ji) = -rhoi * vsw * rcp * sst_1d(ji)   ! clem: if sst>0, then ersw <0 (is that possible?)

               ! volume etc of ridging / rafting ice and new ridges (vi, vs, sm, oi, es, ei)
               virdg1     = v_i_2d (ji,jl1)   * afrdg
               virdg2(ji) = v_i_2d (ji,jl1)   * afrdg + vsw
               vsrdg(ji)  = v_s_2d (ji,jl1)   * afrdg
               sirdg1     = sv_i_2d(ji,jl1)   * afrdg
               sirdg2(ji) = sv_i_2d(ji,jl1)   * afrdg + vsw * sss_1d(ji)
               oirdg1     = oa_i_2d(ji,jl1)   * afrdg
               oirdg2(ji) = oa_i_2d(ji,jl1)   * afrdg * hi_hrdg(ji,jl1)

               virft(ji)  = v_i_2d (ji,jl1)   * afrft
               vsrft(ji)  = v_s_2d (ji,jl1)   * afrft
               sirft(ji)  = sv_i_2d(ji,jl1)   * afrft
               oirft1     = oa_i_2d(ji,jl1)   * afrft
               oirft2(ji) = oa_i_2d(ji,jl1)   * afrft * hi_hrft

               IF ( ln_pnd_LEV .OR. ln_pnd_TOPO ) THEN
                  aprdg1     = a_ip_2d(ji,jl1) * afrdg
                  aprdg2(ji) = a_ip_2d(ji,jl1) * afrdg * hi_hrdg(ji,jl1)
                  vprdg (ji) = v_ip_2d(ji,jl1) * afrdg
                  aprft1     = a_ip_2d(ji,jl1) * afrft
                  aprft2(ji) = a_ip_2d(ji,jl1) * afrft * hi_hrft
                  vprft (ji) = v_ip_2d(ji,jl1) * afrft
                  IF ( ln_pnd_lids ) THEN
                     vlrdg (ji) = v_il_2d(ji,jl1) * afrdg
                     vlrft (ji) = v_il_2d(ji,jl1) * afrft
                  ENDIF
               ENDIF

               ! Ice-ocean exchanges associated with ice porosity
               wfx_dyn_1d(ji) = wfx_dyn_1d(ji) - vsw * rhoi * r1_Dt_ice   ! increase in ice volume due to seawater frozen in voids
               sfx_dyn_1d(ji) = sfx_dyn_1d(ji) - vsw * sss_1d(ji) * rhoi * r1_Dt_ice
               hfx_dyn_1d(ji) = hfx_dyn_1d(ji) + ersw(ji) * r1_Dt_ice          ! > 0 [W.m-2]

               ! Put the snow lost by ridging into the ocean
               !  Note that esrdg > 0; the ocean must cool to melt snow. If the ocean temp = Tf already, new ice must grow.
               wfx_snw_dyn_1d(ji) = wfx_snw_dyn_1d(ji) + ( rhos * vsrdg(ji) * ( 1._wp - rn_fsnwrdg )   &   ! fresh water source for ocean
                  &                                      + rhos * vsrft(ji) * ( 1._wp - rn_fsnwrft ) ) * r1_Dt_ice

               ! virtual salt flux to keep salinity constant
               IF( nn_icesal /= 2 )  THEN
                  sirdg2(ji)     = sirdg2(ji)     - vsw * ( sss_1d(ji) - s_i_1d(ji) )       ! ridge salinity = s_i
                  sfx_bri_1d(ji) = sfx_bri_1d(ji) + sss_1d(ji) * vsw * rhoi * r1_Dt_ice  &  ! put back sss_m into the ocean
                     &                            - s_i_1d(ji) * vsw * rhoi * r1_Dt_ice     ! and get  s_i  from the ocean
               ENDIF

               ! Remove area, volume of new ridge to each category jl1
               !------------------------------------------------------
               a_i_2d (ji,jl1) = a_i_2d (ji,jl1) - airdg1    - airft1
               v_i_2d (ji,jl1) = v_i_2d (ji,jl1) - virdg1    - virft(ji)
               v_s_2d (ji,jl1) = v_s_2d (ji,jl1) - vsrdg(ji) - vsrft(ji)
               sv_i_2d(ji,jl1) = sv_i_2d(ji,jl1) - sirdg1    - sirft(ji)
               oa_i_2d(ji,jl1) = oa_i_2d(ji,jl1) - oirdg1    - oirft1
               IF ( ln_pnd_LEV .OR. ln_pnd_TOPO ) THEN
                  a_ip_2d(ji,jl1) = a_ip_2d(ji,jl1) - aprdg1    - aprft1
                  v_ip_2d(ji,jl1) = v_ip_2d(ji,jl1) - vprdg(ji) - vprft(ji)
                  IF ( ln_pnd_lids ) THEN
                     v_il_2d(ji,jl1) = v_il_2d(ji,jl1) - vlrdg(ji) - vlrft(ji)
                  ENDIF
               ENDIF
            ENDIF

         END DO ! ji

         ! special loop for e_s because of layers jk
         DO jk = 1, nlay_s
            DO ji = 1, npti
               IF( ll_shift(ji) ) THEN
                  ! Compute ridging /rafting fractions
                  afrdg = aridge(ji,jl1) * closing_gross(ji) * rDt_ice * z1_ai(ji)
                  afrft = araft (ji,jl1) * closing_gross(ji) * rDt_ice * z1_ai(ji)
                  ! Compute ridging /rafting ice and new ridges for es
                  esrdg(ji,jk) = ze_s_2d (ji,jk,jl1) * afrdg
                  esrft(ji,jk) = ze_s_2d (ji,jk,jl1) * afrft
                  ! Put the snow lost by ridging into the ocean
                  hfx_dyn_1d(ji) = hfx_dyn_1d(ji) + ( - esrdg(ji,jk) * ( 1._wp - rn_fsnwrdg )   &                 ! heat sink for ocean (<0, W.m-2)
                     &                                - esrft(ji,jk) * ( 1._wp - rn_fsnwrft ) ) * r1_Dt_ice
                  !
                  ! Remove energy of new ridge to each category jl1
                  !-------------------------------------------------
                  ze_s_2d(ji,jk,jl1) = ze_s_2d(ji,jk,jl1) * ( 1._wp - afrdg - afrft )
               ENDIF
            END DO
         END DO

         ! special loop for e_i because of layers jk
         DO jk = 1, nlay_i
            DO ji = 1, npti
               IF( ll_shift(ji) ) THEN
                  ! Compute ridging /rafting fractions
                  afrdg = aridge(ji,jl1) * closing_gross(ji) * rDt_ice * z1_ai(ji)
                  afrft = araft (ji,jl1) * closing_gross(ji) * rDt_ice * z1_ai(ji)
                  ! Compute ridging ice and new ridges for ei
                  eirdg(ji,jk) = ze_i_2d (ji,jk,jl1) * afrdg + ersw(ji) * r1_nlay_i
                  eirft(ji,jk) = ze_i_2d (ji,jk,jl1) * afrft
                  !
                  ! Remove energy of new ridge to each category jl1
                  !-------------------------------------------------
                  ze_i_2d(ji,jk,jl1) = ze_i_2d(ji,jk,jl1) * ( 1._wp - afrdg - afrft )
               ENDIF
            END DO
         END DO

         ! 3) compute categories in which ice is added (jl2)
         !--------------------------------------------------
         itest_rdg(1:npti) = 0
         itest_rft(1:npti) = 0
         DO jl2  = 1, jpl
            !
            DO ji = 1, npti

               IF( ll_shift(ji) ) THEN

                  ! Compute the fraction of ridged ice area and volume going to thickness category jl2
                  IF( ln_distf_lin ) THEN ! Hibler (1980) linear formulation
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                     !
                     IF( hrmin(ji,jl1) <= hi_max(jl2) .AND. hrmax(ji,jl1) > hi_max(jl2-1) ) THEN
                        hL = MAX( hrmin(ji,jl1), hi_max(jl2-1) )
                        hR = MIN( hrmax(ji,jl1), hi_max(jl2)   )
                        farea    = ( hR      - hL      ) / ( hrmax(ji,jl1)                 - hrmin(ji,jl1)                 )
                        fvol(ji) = ( hR * hR - hL * hL ) / ( hrmax(ji,jl1) * hrmax(ji,jl1) - hrmin(ji,jl1) * hrmin(ji,jl1) )
                        !
                        itest_rdg(ji) = 1   ! test for conservation
                     ELSE
                        farea    = 0._wp
                        fvol(ji) = 0._wp
                     ENDIF
                     !
                  ELSEIF( ln_distf_exp ) THEN ! Lipscomb et al. (2007) exponential formulation                      
                     !
                     IF( jl2 < jpl ) THEN
                        !
                        IF( hrmin(ji,jl1) <= hi_max(jl2) ) THEN
                           hL       = MAX( hrmin(ji,jl1), hi_max(jl2-1) )
                           hR       = hi_max(jl2)
                           expL     = EXP( -( hL - hrmin(ji,jl1) ) / hrexp(ji,jl1) )
                           expR     = EXP( -( hR - hrmin(ji,jl1) ) / hrexp(ji,jl1) )
                           farea    = expL - expR
                           fvol(ji) = ( ( hL + hrexp(ji,jl1) ) * expL  &
                              - ( hR + hrexp(ji,jl1) ) * expR ) / ( hrmin(ji,jl1) + hrexp(ji,jl1) )
                        ELSE
                           farea    = 0._wp
                           fvol(ji) = 0._wp
                        END IF
                        !                 
                     ELSE             ! jl2 = jpl
                        !
                        hL       = MAX( hrmin(ji,jl1), hi_max(jl2-1) )
                        expL     = EXP(-( hL - hrmin(ji,jl1) ) / hrexp(ji,jl1) )
                        farea    = expL
                        fvol(ji) = ( hL + hrexp(ji,jl1) ) * expL / ( hrmin(ji,jl1) + hrexp(ji,jl1) )
                        !
                     END IF            ! jl2 < jpl
                     ! 
                     itest_rdg(ji) = 1   ! test for conservation => clem: I am not sure about that
                     !
                  END IF             ! ridge redistribution
                     
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                  ! Compute the fraction of rafted ice area and volume going to thickness category jl2
                  IF( hraft(ji,jl1) <= hi_max(jl2) .AND. hraft(ji,jl1) >  hi_max(jl2-1) ) THEN
                     zswitch(ji) = 1._wp
                     !
                     itest_rft(ji) = 1   ! test for conservation
                  ELSE
                     zswitch(ji) = 0._wp
                  ENDIF
                  !
                  ! Patch to ensure perfect conservation if ice thickness goes mad
                  ! Sometimes thickness is larger than hi_max(jpl) because of advection scheme (for very small areas)
                  ! Then ice volume is removed from one category but the ridging/rafting scheme
                  ! does not know where to move it, leading to a conservation issue.
                  IF( itest_rdg(ji) == 0 .AND. jl2 == jpl ) THEN   ;   farea = 1._wp   ;   fvol(ji) = 1._wp   ;   ENDIF
                  IF( itest_rft(ji) == 0 .AND. jl2 == jpl )      zswitch(ji) = 1._wp
                  !
                  ! Add area, volume of new ridge to category jl2
                  !----------------------------------------------
                  a_i_2d (ji,jl2) = a_i_2d (ji,jl2) + ( airdg2(ji) * farea    + airft2(ji) * zswitch(ji) )
                  oa_i_2d(ji,jl2) = oa_i_2d(ji,jl2) + ( oirdg2(ji) * farea    + oirft2(ji) * zswitch(ji) )
                  v_i_2d (ji,jl2) = v_i_2d (ji,jl2) + ( virdg2(ji) * fvol(ji) + virft (ji) * zswitch(ji) )
                  sv_i_2d(ji,jl2) = sv_i_2d(ji,jl2) + ( sirdg2(ji) * fvol(ji) + sirft (ji) * zswitch(ji) )
                  v_s_2d (ji,jl2) = v_s_2d (ji,jl2) + ( vsrdg (ji) * rn_fsnwrdg * fvol(ji)  +  &
                     &                                  vsrft (ji) * rn_fsnwrft * zswitch(ji) )
                  IF ( ln_pnd_LEV .OR. ln_pnd_TOPO ) THEN
                     v_ip_2d (ji,jl2) = v_ip_2d(ji,jl2) + (   vprdg (ji) * rn_fpndrdg * fvol   (ji)   &
                        &                                   + vprft (ji) * rn_fpndrft * zswitch(ji)   )
                     a_ip_2d (ji,jl2) = a_ip_2d(ji,jl2) + (   aprdg2(ji) * rn_fpndrdg * farea         &
                        &                                   + aprft2(ji) * rn_fpndrft * zswitch(ji)   )
                     IF ( ln_pnd_lids ) THEN
                        v_il_2d (ji,jl2) = v_il_2d(ji,jl2) + (   vlrdg(ji) * rn_fpndrdg * fvol   (ji) &
                           &                                   + vlrft(ji) * rn_fpndrft * zswitch(ji) )
                     ENDIF
                  ENDIF

               ENDIF

            END DO
            ! Add snow energy of new ridge to category jl2
            !---------------------------------------------
            DO jk = 1, nlay_s
               DO ji = 1, npti
                  IF( ll_shift(ji) )   &
                     &   ze_s_2d(ji,jk,jl2) = ze_s_2d(ji,jk,jl2) + ( esrdg(ji,jk) * rn_fsnwrdg * fvol(ji)  +  &
                     &                                               esrft(ji,jk) * rn_fsnwrft * zswitch(ji) )
               END DO
            END DO
            ! Add ice energy of new ridge to category jl2
            !--------------------------------------------
            DO jk = 1, nlay_i
               DO ji = 1, npti
                  IF( ll_shift(ji) )   &
                     &   ze_i_2d(ji,jk,jl2) = ze_i_2d(ji,jk,jl2) + eirdg(ji,jk) * fvol(ji) + eirft(ji,jk) * zswitch(ji)
               END DO
            END DO
            !
         END DO ! jl2
         !
      END DO ! jl1
      !
      ! roundoff errors
      !----------------
      ! In case ridging/rafting lead to very small negative values (sometimes it happens)
      CALL ice_var_roundoff( a_i_2d, v_i_2d, v_s_2d, sv_i_2d, oa_i_2d, a_ip_2d, v_ip_2d, v_il_2d, ze_s_2d, ze_i_2d )
      !
   END SUBROUTINE rdgrft_shift


   SUBROUTINE ice_strength
      !!----------------------------------------------------------------------
      !!                ***  ROUTINE ice_strength ***
      !!
      !! ** Purpose :   computes ice strength used in dynamics routines of ice thickness
      !!
      !! ** Method  :   Compute the strength of the ice pack, defined as the energy (J m-2) 
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      !!              dissipated per unit area removed from the ice pack under compression,
      !!              and assumed proportional to the change in potential energy caused
      !!              by ridging. Note that ice strength using Hibler's formulation must be
      !!              smoothed.
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      !!----------------------------------------------------------------------
      INTEGER             ::   ji, jj, jl  ! dummy loop indices
      REAL(wp)            ::   z1_3        ! local scalars
      REAL(wp), DIMENSION(jpi,jpj) ::   zmsk, zworka           ! temporary array used here
      !!
      LOGICAL             ::   ln_str_R75
      REAL(wp)            ::   zhi, zcp
      REAL(wp)            ::   h2rdg                     ! mean value of h^2 for new ridge
      REAL(wp), PARAMETER ::   zmax_strength = 200.e3_wp ! Richter-Menge and Elder (1998) estimate maximum in Beaufort Sea in wintertime of the order 150 kN/m
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      REAL(wp), DIMENSION(jpij) ::   zstrength, zaksum   ! strength in 1D      
      !!----------------------------------------------------------------------
      ! prepare the mask
      at_i(:,:) = SUM( a_i, dim=3 )
      DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
         zmsk(ji,jj) = MAX( 0._wp , SIGN( 1._wp , at_i(ji,jj) - epsi10  ) ) ! 1 if ice    , 0 if no ice
      END_2D
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      !
      SELECT CASE( nice_str )          !--- Set which ice strength is chosen

      CASE ( np_strr75 )           !== Rothrock(1975)'s method ==!

         ! this should be defined once for all at the 1st time step
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         zcp = 0.5_wp * grav * (rho0-rhoi) * rhoi * r1_rho0   ! proport const for PE
         !
         strength(:,:) = 0._wp
         !
         ! Identify grid cells with ice
         npti = 0   ;   nptidx(:) = 0
         DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
            IF ( at_i(ji,jj) > epsi10 ) THEN
               npti           = npti + 1
               nptidx( npti ) = (jj - 1) * jpi + ji
            ENDIF
         END_2D

         IF( npti > 0 ) THEN
            CALL tab_3d_2d( npti, nptidx(1:npti), a_i_2d  (1:npti,1:jpl), a_i   )
            CALL tab_3d_2d( npti, nptidx(1:npti), v_i_2d  (1:npti,1:jpl), v_i   )
            CALL tab_2d_1d( npti, nptidx(1:npti), ato_i_1d(1:npti)      , ato_i )
            CALL tab_2d_1d( npti, nptidx(1:npti), zstrength(1:npti)     , strength )

            CALL rdgrft_prep( a_i_2d, v_i_2d, ato_i_1d )
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            !
            zaksum(1:npti) = apartf(1:npti,0) !clem: aksum should be defined in the header => local to module
            DO jl = 1, jpl
               DO ji = 1, npti
                  IF ( apartf(ji,jl) > 0._wp ) THEN
                     zaksum(ji) = zaksum(ji) + aridge(ji,jl) * ( 1._wp - hi_hrdg(ji,jl) )    &
                        &                    + araft (ji,jl) * ( 1._wp - hi_hrft )
                  ENDIF
               END DO
            END DO
            !
            z1_3 = 1._wp / 3._wp
            DO jl = 1, jpl
               DO ji = 1, npti
                  !
                  IF( apartf(ji,jl) > 0._wp ) THEN
                     !
                     IF( ln_distf_lin ) THEN       ! Uniform redistribution of ridged ice               
                        h2rdg = z1_3 * ( hrmax(ji,jl) * hrmax(ji,jl) +     & ! (a**3-b**3)/(a-b) = a*a+ab+b*b
                           &             hrmin(ji,jl) * hrmin(ji,jl) +     &
                           &             hrmax(ji,jl) * hrmin(ji,jl) )
                        !
                     ELSEIF( ln_distf_exp ) THEN   ! Exponential redistribution of ridged ice
                        h2rdg =          hrmin(ji,jl) * hrmin(ji,jl)   &
                           &   + 2._wp * hrmin(ji,jl) * hrexp(ji,jl)   &
                           &   + 2._wp * hrexp(ji,jl) * hrexp(ji,jl)
                     END IF
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                     !
                     IF( a_i_2d(ji,jl) > epsi10 ) THEN   ;   zhi = v_i_2d(ji,jl) / a_i_2d(ji,jl)
                     ELSE                                ;   zhi = 0._wp
                     ENDIF
!!$                     zstrength(ji) = zstrength(ji) -         apartf(ji,jl) * zhi * zhi                  ! PE loss from deforming ice
!!$                     zstrength(ji) = zstrength(ji) + 2._wp * araft (ji,jl) * zhi * zhi                  ! PE gain from rafting ice
!!$                     zstrength(ji) = zstrength(ji) +         aridge(ji,jl) * hi_hrdg(ji,jl) * z1_3 *  & ! PE gain from ridging ice
!!$                        &                                   ( hrmax(ji,jl) * hrmax(ji,jl) +           & ! (a**3-b**3)/(a-b) = a*a+ab+b*b
!!$                        &                                     hrmin(ji,jl) * hrmin(ji,jl) +           &
!!$                        &                                     hrmax(ji,jl) * hrmin(ji,jl) )
                     zstrength(ji) = zstrength(ji) - apartf(ji,jl) * zhi * zhi                  ! PE loss
                     zstrength(ji) = zstrength(ji) + 2._wp * araft(ji,jl) * zhi * zhi           ! PE gain (rafting)
                     zstrength(ji) = zstrength(ji) + aridge(ji,jl) * h2rdg *  hi_hrdg(ji,jl)    ! PE gain (ridging)
                     
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                  ENDIF
                  !
               END DO
            END DO
            !
            zstrength(1:npti) = rn_pe_rdg * zcp * zstrength(1:npti) / zaksum(1:npti)
            !
            ! Enforce a maximum for strength
            ! Richter-Menge and Elder (1998) estimate maximum in Beaufort Sea in wintertime of the order 150 kN/m
            WHERE( zstrength(1:npti) > zmax_strength ) ; zstrength(1:npti) = zmax_strength
            ENDWHERE
            !
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            CALL tab_1d_2d( npti, nptidx(1:npti), zstrength(1:npti), strength )
            !
         ENDIF
         !
      CASE ( np_strh79 )           !== Hibler(1979)'s method ==!
         strength(:,:) = rn_pstar * SUM( v_i(:,:,:), dim=3 ) * EXP( -rn_crhg * ( 1._wp - at_i(:,:) ) ) * zmsk(:,:)
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         !
      CASE ( np_strcst )           !== Constant strength ==!
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         !
      END SELECT
      !
      IF( ln_str_smooth ) THEN         !--- Spatial smoothing
         DO_2D( 0, 0, 0, 0 )
            IF ( SUM( a_i(ji,jj,:) ) > 0._wp ) THEN
               zworka(ji,jj) = ( 4._wp * strength(ji,jj)              &
                  &                    + strength(ji-1,jj) * tmask(ji-1,jj,1) + strength(ji+1,jj) * tmask(ji+1,jj,1) &
                  &                    + strength(ji,jj-1) * tmask(ji,jj-1,1) + strength(ji,jj+1) * tmask(ji,jj+1,1) &
                  &            ) / ( 4._wp + tmask(ji-1,jj,1) + tmask(ji+1,jj,1) + tmask(ji,jj-1,1) + tmask(ji,jj+1,1) )
            ELSE
               zworka(ji,jj) = 0._wp
            ENDIF
         END_2D

         DO_2D( 0, 0, 0, 0 )
            strength(ji,jj) = zworka(ji,jj) * zmsk(ji,jj)
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         END_2D
         CALL lbc_lnk( 'icedyn_rdgrft', strength, 'T', 1.0_wp )
         !
      ENDIF
      !
   END SUBROUTINE ice_strength


   SUBROUTINE ice_dyn_1d2d( kn )
      !!-----------------------------------------------------------------------
      !!                   ***  ROUTINE ice_dyn_1d2d ***
      !!
      !! ** Purpose :   move arrays between 1d <=> 2d forms and
      !!                            between 2d <=> 3d forms