Index: branches/UKMO/dev_r5107_hadgem3_mct/NEMOGCM/NEMO/OPA_SRC/TRA/zpshde.F90
===================================================================
--- branches/UKMO/dev_r5107_hadgem3_mct/NEMOGCM/NEMO/OPA_SRC/TRA/zpshde.F90	(revision 5279)
+++ branches/UKMO/dev_r5107_hadgem3_mct/NEMOGCM/NEMO/OPA_SRC/TRA/zpshde.F90	(revision 5568)
@@ -8,4 +8,5 @@
    !!             -   !  2004-03  (C. Ethe)  adapted for passive tracers
    !!            3.3  !  2010-05  (C. Ethe, G. Madec)  merge TRC-TRA 
+   !!            3.6  !  2014-11  (P. Mathiot) Add zps_hde_isf (needed to open a cavity)
    !!======================================================================
    
@@ -27,5 +28,6 @@
    PRIVATE
 
-   PUBLIC   zps_hde    ! routine called by step.F90
+   PUBLIC   zps_hde     ! routine called by step.F90
+   PUBLIC   zps_hde_isf ! routine called by step.F90
 
    !! * Substitutions
@@ -40,6 +42,161 @@
 
    SUBROUTINE zps_hde( kt, kjpt, pta, pgtu, pgtv,   &
+      &                          prd, pgru, pgrv    )
+      !!----------------------------------------------------------------------
+      !!                     ***  ROUTINE zps_hde  ***
+      !!                    
+      !! ** Purpose :   Compute the horizontal derivative of T, S and rho
+      !!      at u- and v-points with a linear interpolation for z-coordinate
+      !!      with partial steps.
+      !!
+      !! ** Method  :   In z-coord with partial steps, scale factors on last 
+      !!      levels are different for each grid point, so that T, S and rd 
+      !!      points are not at the same depth as in z-coord. To have horizontal
+      !!      gradients again, we interpolate T and S at the good depth : 
+      !!      Linear interpolation of T, S   
+      !!         Computation of di(tb) and dj(tb) by vertical interpolation:
+      !!          di(t) = t~ - t(i,j,k) or t(i+1,j,k) - t~
+      !!          dj(t) = t~ - t(i,j,k) or t(i,j+1,k) - t~
+      !!         This formulation computes the two cases:
+      !!                 CASE 1                   CASE 2  
+      !!         k-1  ___ ___________   k-1   ___ ___________
+      !!                    Ti  T~                  T~  Ti+1
+      !!                  _____                        _____
+      !!         k        |   |Ti+1     k           Ti |   |
+      !!                  |   |____                ____|   |
+      !!              ___ |   |   |           ___  |   |   |
+      !!                  
+      !!      case 1->   e3w(i+1) >= e3w(i) ( and e3w(j+1) >= e3w(j) ) then
+      !!          t~ = t(i+1,j  ,k) + (e3w(i+1) - e3w(i)) * dk(Ti+1)/e3w(i+1)
+      !!        ( t~ = t(i  ,j+1,k) + (e3w(j+1) - e3w(j)) * dk(Tj+1)/e3w(j+1)  )
+      !!          or
+      !!      case 2->   e3w(i+1) <= e3w(i) ( and e3w(j+1) <= e3w(j) ) then
+      !!          t~ = t(i,j,k) + (e3w(i) - e3w(i+1)) * dk(Ti)/e3w(i )
+      !!        ( t~ = t(i,j,k) + (e3w(j) - e3w(j+1)) * dk(Tj)/e3w(j ) )
+      !!          Idem for di(s) and dj(s)          
+      !!
+      !!      For rho, we call eos which will compute rd~(t~,s~) at the right
+      !!      depth zh from interpolated T and S for the different formulations
+      !!      of the equation of state (eos).
+      !!      Gradient formulation for rho :
+      !!          di(rho) = rd~ - rd(i,j,k)   or   rd(i+1,j,k) - rd~
+      !!
+      !! ** Action  : compute for top interfaces
+      !!              - pgtu, pgtv: horizontal gradient of tracer at u- & v-points
+      !!              - pgru, pgrv: horizontal gradient of rho (if present) at u- & v-points
+      !!----------------------------------------------------------------------
+      INTEGER                              , INTENT(in   )           ::  kt          ! ocean time-step index
+      INTEGER                              , INTENT(in   )           ::  kjpt        ! number of tracers
+      REAL(wp), DIMENSION(jpi,jpj,jpk,kjpt), INTENT(in   )           ::  pta         ! 4D tracers fields
+      REAL(wp), DIMENSION(jpi,jpj,    kjpt), INTENT(  out)           ::  pgtu, pgtv  ! hor. grad. of ptra at u- & v-pts 
+      REAL(wp), DIMENSION(jpi,jpj,jpk     ), INTENT(in   ), OPTIONAL ::  prd         ! 3D density anomaly fields
+      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pgru, pgrv  ! hor. grad of prd at u- & v-pts (bottom)
+      !
+      INTEGER  ::   ji, jj, jn      ! Dummy loop indices
+      INTEGER  ::   iku, ikv, ikum1, ikvm1   ! partial step level (ocean bottom level) at u- and v-points
+      REAL(wp) ::  ze3wu, ze3wv, zmaxu, zmaxv  ! temporary scalars
+      REAL(wp), DIMENSION(jpi,jpj)      ::  zri, zrj, zhi, zhj   ! NB: 3rd dim=1 to use eos
+      REAL(wp), DIMENSION(jpi,jpj,kjpt) ::  zti, ztj             ! 
+      !!----------------------------------------------------------------------
+      !
+      IF( nn_timing == 1 )  CALL timing_start( 'zps_hde')
+      !
+      pgtu(:,:,:)=0.0_wp ; pgtv(:,:,:)=0.0_wp ;
+      zti (:,:,:)=0.0_wp ; ztj (:,:,:)=0.0_wp ;
+      zhi (:,:  )=0.0_wp ; zhj (:,:  )=0.0_wp ;
+      !
+      DO jn = 1, kjpt      !==   Interpolation of tracers at the last ocean level   ==!
+         !
+         DO jj = 1, jpjm1
+            DO ji = 1, jpim1
+               iku = mbku(ji,jj)   ;   ikum1 = MAX( iku - 1 , 1 )    ! last and before last ocean level at u- & v-points
+               ikv = mbkv(ji,jj)   ;   ikvm1 = MAX( ikv - 1 , 1 )    ! if level first is a p-step, ik.m1=1
+               ze3wu = fse3w(ji+1,jj  ,iku) - fse3w(ji,jj,iku)
+               ze3wv = fse3w(ji  ,jj+1,ikv) - fse3w(ji,jj,ikv)
+               !
+               ! i- direction
+               IF( ze3wu >= 0._wp ) THEN      ! case 1
+                  zmaxu =  ze3wu / fse3w(ji+1,jj,iku)
+                  ! interpolated values of tracers
+                  zti (ji,jj,jn) = pta(ji+1,jj,iku,jn) + zmaxu * ( pta(ji+1,jj,ikum1,jn) - pta(ji+1,jj,iku,jn) )
+                  ! gradient of  tracers
+                  pgtu(ji,jj,jn) = umask(ji,jj,1) * ( zti(ji,jj,jn) - pta(ji,jj,iku,jn) )
+               ELSE                           ! case 2
+                  zmaxu = -ze3wu / fse3w(ji,jj,iku)
+                  ! interpolated values of tracers
+                  zti (ji,jj,jn) = pta(ji,jj,iku,jn) + zmaxu * ( pta(ji,jj,ikum1,jn) - pta(ji,jj,iku,jn) )
+                  ! gradient of tracers
+                  pgtu(ji,jj,jn) = umask(ji,jj,1) * ( pta(ji+1,jj,iku,jn) - zti(ji,jj,jn) )
+               ENDIF
+               !
+               ! j- direction
+               IF( ze3wv >= 0._wp ) THEN      ! case 1
+                  zmaxv =  ze3wv / fse3w(ji,jj+1,ikv)
+                  ! interpolated values of tracers
+                  ztj (ji,jj,jn) = pta(ji,jj+1,ikv,jn) + zmaxv * ( pta(ji,jj+1,ikvm1,jn) - pta(ji,jj+1,ikv,jn) )
+                  ! gradient of tracers
+                  pgtv(ji,jj,jn) = vmask(ji,jj,1) * ( ztj(ji,jj,jn) - pta(ji,jj,ikv,jn) )
+               ELSE                           ! case 2
+                  zmaxv =  -ze3wv / fse3w(ji,jj,ikv)
+                  ! interpolated values of tracers
+                  ztj (ji,jj,jn) = pta(ji,jj,ikv,jn) + zmaxv * ( pta(ji,jj,ikvm1,jn) - pta(ji,jj,ikv,jn) )
+                  ! gradient of tracers
+                  pgtv(ji,jj,jn) = vmask(ji,jj,1) * ( pta(ji,jj+1,ikv,jn) - ztj(ji,jj,jn) )
+               ENDIF
+            END DO
+         END DO
+         CALL lbc_lnk( pgtu(:,:,jn), 'U', -1. )   ;   CALL lbc_lnk( pgtv(:,:,jn), 'V', -1. )   ! Lateral boundary cond.
+         !
+      END DO
+
+      ! horizontal derivative of density anomalies (rd)
+      IF( PRESENT( prd ) ) THEN         ! depth of the partial step level
+         pgru(:,:)=0.0_wp   ; pgrv(:,:)=0.0_wp ; 
+         DO jj = 1, jpjm1
+            DO ji = 1, jpim1
+               iku = mbku(ji,jj)
+               ikv = mbkv(ji,jj)
+               ze3wu  = fse3w(ji+1,jj  ,iku) - fse3w(ji,jj,iku)
+               ze3wv  = fse3w(ji  ,jj+1,ikv) - fse3w(ji,jj,ikv)
+               IF( ze3wu >= 0._wp ) THEN   ;   zhi(ji,jj) = fsdept(ji  ,jj,iku)     ! i-direction: case 1
+               ELSE                        ;   zhi(ji,jj) = fsdept(ji+1,jj,iku)     ! -     -      case 2
+               ENDIF
+               IF( ze3wv >= 0._wp ) THEN   ;   zhj(ji,jj) = fsdept(ji,jj  ,ikv)     ! j-direction: case 1
+               ELSE                        ;   zhj(ji,jj) = fsdept(ji,jj+1,ikv)     ! -     -      case 2
+               ENDIF
+            END DO
+         END DO
+
+         ! Compute interpolated rd from zti, ztj for the 2 cases at the depth of the partial
+         ! step and store it in  zri, zrj for each  case
+         CALL eos( zti, zhi, zri )  
+         CALL eos( ztj, zhj, zrj )
+
+         ! Gradient of density at the last level 
+         DO jj = 1, jpjm1
+            DO ji = 1, jpim1
+               iku = mbku(ji,jj)
+               ikv = mbkv(ji,jj)
+               ze3wu  = fse3w(ji+1,jj  ,iku) - fse3w(ji,jj,iku)
+               ze3wv  = fse3w(ji  ,jj+1,ikv) - fse3w(ji,jj,ikv)
+               IF( ze3wu >= 0._wp ) THEN   ;   pgru(ji,jj) = umask(ji,jj,1) * ( zri(ji  ,jj    ) - prd(ji,jj,iku) )   ! i: 1
+               ELSE                        ;   pgru(ji,jj) = umask(ji,jj,1) * ( prd(ji+1,jj,iku) - zri(ji,jj    ) )   ! i: 2
+               ENDIF
+               IF( ze3wv >= 0._wp ) THEN   ;   pgrv(ji,jj) = vmask(ji,jj,1) * ( zrj(ji,jj      ) - prd(ji,jj,ikv) )   ! j: 1
+               ELSE                        ;   pgrv(ji,jj) = vmask(ji,jj,1) * ( prd(ji,jj+1,ikv) - zrj(ji,jj    ) )   ! j: 2
+               ENDIF
+            END DO
+         END DO
+         CALL lbc_lnk( pgru , 'U', -1. )   ;   CALL lbc_lnk( pgrv , 'V', -1. )   ! Lateral boundary conditions
+         !
+      END IF
+      !
+      IF( nn_timing == 1 )  CALL timing_stop( 'zps_hde')
+      !
+   END SUBROUTINE zps_hde
+   !
+   SUBROUTINE zps_hde_isf( kt, kjpt, pta, pgtu, pgtv,   &
       &                          prd, pgru, pgrv, pmru, pmrv, pgzu, pgzv, pge3ru, pge3rv,  &
-      &                   sgtu, sgtv, sgru, sgrv, smru, smrv, sgzu, sgzv, sge3ru, sge3rv )
+      &                   pgtui, pgtvi, pgrui, pgrvi, pmrui, pmrvi, pgzui, pgzvi, pge3rui, pge3rvi )
       !!----------------------------------------------------------------------
       !!                     ***  ROUTINE zps_hde  ***
@@ -82,9 +239,9 @@
       !!
       !! ** Action  : compute for top and bottom interfaces
-      !!              - pgtu, pgtv, sgtu, sgtv: horizontal gradient of tracer at u- & v-points
-      !!              - pgru, pgrv, sgru, sgtv: horizontal gradient of rho (if present) at u- & v-points
-      !!              - pmru, pmrv, smru, smrv: horizontal sum of rho at u- & v- point (used in dynhpg with vvl)
-      !!              - pgzu, pgzv, sgzu, sgzv: horizontal gradient of z at u- and v- point (used in dynhpg with vvl)
-      !!              - pge3ru, pge3rv, sge3ru, sge3rv: horizontal gradient of rho weighted by local e3w at u- & v-points 
+      !!              - pgtu, pgtv, pgtui, pgtvi: horizontal gradient of tracer at u- & v-points
+      !!              - pgru, pgrv, pgrui, pgtvi: horizontal gradient of rho (if present) at u- & v-points
+      !!              - pmru, pmrv, pmrui, pmrvi: horizontal sum of rho at u- & v- point (used in dynhpg with vvl)
+      !!              - pgzu, pgzv, pgzui, pgzvi: horizontal gradient of z at u- and v- point (used in dynhpg with vvl)
+      !!              - pge3ru, pge3rv, pge3rui, pge3rvi: horizontal gradient of rho weighted by local e3w at u- & v-points 
       !!----------------------------------------------------------------------
       INTEGER                              , INTENT(in   )           ::  kt          ! ocean time-step index
@@ -92,5 +249,5 @@
       REAL(wp), DIMENSION(jpi,jpj,jpk,kjpt), INTENT(in   )           ::  pta         ! 4D tracers fields
       REAL(wp), DIMENSION(jpi,jpj,    kjpt), INTENT(  out)           ::  pgtu, pgtv  ! hor. grad. of ptra at u- & v-pts 
-      REAL(wp), DIMENSION(jpi,jpj,    kjpt), INTENT(  out)           ::  sgtu, sgtv  ! hor. grad. of stra at u- & v-pts (ISF)
+      REAL(wp), DIMENSION(jpi,jpj,    kjpt), INTENT(  out)           ::  pgtui, pgtvi  ! hor. grad. of stra at u- & v-pts (ISF)
       REAL(wp), DIMENSION(jpi,jpj,jpk     ), INTENT(in   ), OPTIONAL ::  prd         ! 3D density anomaly fields
       REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pgru, pgrv      ! hor. grad of prd at u- & v-pts (bottom)
@@ -98,8 +255,8 @@
       REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pgzu, pgzv      ! hor. grad of z   at u- & v-pts (bottom)
       REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pge3ru, pge3rv  ! hor. grad of prd weighted by local e3w at u- & v-pts (bottom)
-      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  sgru, sgrv      ! hor. grad of prd at u- & v-pts (top)
-      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  smru, smrv      ! hor. sum  of prd at u- & v-pts (top)
-      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  sgzu, sgzv      ! hor. grad of z   at u- & v-pts (top)
-      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  sge3ru, sge3rv  ! hor. grad of prd weighted by local e3w at u- & v-pts (top)
+      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pgrui, pgrvi      ! hor. grad of prd at u- & v-pts (top)
+      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pmrui, pmrvi      ! hor. sum  of prd at u- & v-pts (top)
+      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pgzui, pgzvi      ! hor. grad of z   at u- & v-pts (top)
+      REAL(wp), DIMENSION(jpi,jpj         ), INTENT(  out), OPTIONAL ::  pge3rui, pge3rvi  ! hor. grad of prd weighted by local e3w at u- & v-pts (top)
       !
       INTEGER  ::   ji, jj, jn      ! Dummy loop indices
@@ -110,8 +267,8 @@
       !!----------------------------------------------------------------------
       !
-      IF( nn_timing == 1 )  CALL timing_start( 'zps_hde')
+      IF( nn_timing == 1 )  CALL timing_start( 'zps_hde_isf')
       !
       pgtu(:,:,:)=0.0_wp ; pgtv(:,:,:)=0.0_wp ;
-      sgtu(:,:,:)=0.0_wp ; sgtv(:,:,:)=0.0_wp ;
+      pgtui(:,:,:)=0.0_wp ; pgtvi(:,:,:)=0.0_wp ;
       zti (:,:,:)=0.0_wp ; ztj (:,:,:)=0.0_wp ;
       zhi (:,:  )=0.0_wp ; zhj (:,:  )=0.0_wp ;
@@ -256,5 +413,5 @@
                   zti(ji,jj,jn) = pta(ji+1,jj,iku,jn) + zmaxu * ( pta(ji+1,jj,iku+1,jn) - pta(ji+1,jj,iku,jn) )
                   ! gradient of tracers
-                  sgtu(ji,jj,jn) = umask(ji,jj,iku) * ( zti(ji,jj,jn) - pta(ji,jj,iku,jn) )
+                  pgtui(ji,jj,jn) = umask(ji,jj,iku) * ( zti(ji,jj,jn) - pta(ji,jj,iku,jn) )
                ELSE                           ! case 2
                   zmaxu = - ze3wu / fse3w(ji,jj,iku+1)
@@ -262,5 +419,5 @@
                   zti(ji,jj,jn) = pta(ji,jj,iku,jn) + zmaxu * ( pta(ji,jj,iku+1,jn) - pta(ji,jj,iku,jn) )
                   ! gradient of  tracers
-                  sgtu(ji,jj,jn) = umask(ji,jj,iku) * ( pta(ji+1,jj,iku,jn) - zti(ji,jj,jn) )
+                  pgtui(ji,jj,jn) = umask(ji,jj,iku) * ( pta(ji+1,jj,iku,jn) - zti(ji,jj,jn) )
                ENDIF
                !
@@ -271,5 +428,5 @@
                   ztj(ji,jj,jn) = pta(ji,jj+1,ikv,jn) + zmaxv * ( pta(ji,jj+1,ikv+1,jn) - pta(ji,jj+1,ikv,jn) )
                   ! gradient of tracers
-                  sgtv(ji,jj,jn) = vmask(ji,jj,ikv) * ( ztj(ji,jj,jn) - pta(ji,jj,ikv,jn) )
+                  pgtvi(ji,jj,jn) = vmask(ji,jj,ikv) * ( ztj(ji,jj,jn) - pta(ji,jj,ikv,jn) )
                ELSE                           ! case 2
                   zmaxv =  - ze3wv / fse3w(ji,jj,ikv+1)
@@ -277,9 +434,9 @@
                   ztj(ji,jj,jn) = pta(ji,jj,ikv,jn) + zmaxv * ( pta(ji,jj,ikv+1,jn) - pta(ji,jj,ikv,jn) )
                   ! gradient of tracers
-                  sgtv(ji,jj,jn) = vmask(ji,jj,ikv) * ( pta(ji,jj+1,ikv,jn) - ztj(ji,jj,jn) )
+                  pgtvi(ji,jj,jn) = vmask(ji,jj,ikv) * ( pta(ji,jj+1,ikv,jn) - ztj(ji,jj,jn) )
                ENDIF
             END DO!!
          END DO!!
-         CALL lbc_lnk( sgtu(:,:,jn), 'U', -1. )   ;   CALL lbc_lnk( sgtv(:,:,jn), 'V', -1. )   ! Lateral boundary cond.
+         CALL lbc_lnk( pgtui(:,:,jn), 'U', -1. )   ;   CALL lbc_lnk( pgtvi(:,:,jn), 'V', -1. )   ! Lateral boundary cond.
          !
       END DO
@@ -287,8 +444,8 @@
       ! horizontal derivative of density anomalies (rd)
       IF( PRESENT( prd ) ) THEN         ! depth of the partial step level
-         sgru(:,:)  =0.0_wp ; sgrv(:,:)  =0.0_wp ;
-         sgzu(:,:)  =0.0_wp ; sgzv(:,:)  =0.0_wp ;
-         smru(:,:)  =0.0_wp ; smru(:,:)  =0.0_wp ;
-         sge3ru(:,:)=0.0_wp ; sge3rv(:,:)=0.0_wp ;
+         pgrui(:,:)  =0.0_wp ; pgrvi(:,:)  =0.0_wp ;
+         pgzui(:,:)  =0.0_wp ; pgzvi(:,:)  =0.0_wp ;
+         pmrui(:,:)  =0.0_wp ; pmrui(:,:)  =0.0_wp ;
+         pge3rui(:,:)=0.0_wp ; pge3rvi(:,:)=0.0_wp ;
 
          DO jj = 1, jpjm1
@@ -321,31 +478,31 @@
                ze3wv  = (gdepw_0(ji,jj+1,ikv+1) - gdept_0(ji,jj+1,ikv)) - (gdepw_0(ji,jj,ikv+1) - gdept_0(ji,jj,ikv))
                IF( ze3wu >= 0._wp ) THEN
-                 sgzu  (ji,jj) = (fsde3w(ji+1,jj,iku) + ze3wu) - fsde3w(ji,jj,iku)
-                 sgru  (ji,jj) = umask(ji,jj,iku)   * ( zri(ji,jj) - prd(ji,jj,iku) )          ! i: 1
-                 smru  (ji,jj) = umask(ji,jj,iku)   * ( zri(ji,jj) + prd(ji,jj,iku) )          ! i: 1 
-                 sge3ru(ji,jj) = umask(ji,jj,iku+1)                                                                  &
+                 pgzui  (ji,jj) = (fsde3w(ji+1,jj,iku) + ze3wu) - fsde3w(ji,jj,iku)
+                 pgrui  (ji,jj) = umask(ji,jj,iku)   * ( zri(ji,jj) - prd(ji,jj,iku) )          ! i: 1
+                 pmrui  (ji,jj) = umask(ji,jj,iku)   * ( zri(ji,jj) + prd(ji,jj,iku) )          ! i: 1 
+                 pge3rui(ji,jj) = umask(ji,jj,iku+1)                                                                  &
                                 * ( (fse3w(ji+1,jj,iku+1) - ze3wu) * (zri(ji,jj    ) + prd(ji+1,jj,iku+1) + 2._wp)   &
                                    - fse3w(ji  ,jj,iku+1)          * (prd(ji,jj,iku) + prd(ji  ,jj,iku+1) + 2._wp)   ) ! i: 1
                ELSE
-                 sgzu  (ji,jj) = fsde3w(ji+1,jj,iku) - (fsde3w(ji,jj,iku) - ze3wu)
-                 sgru  (ji,jj) = umask(ji,jj,iku)   * ( prd(ji+1,jj,iku) - zri(ji,jj) )      ! i: 2
-                 smru  (ji,jj) = umask(ji,jj,iku)   * ( prd(ji+1,jj,iku) + zri(ji,jj) )      ! i: 2
-                 sge3ru(ji,jj) = umask(ji,jj,iku+1)                                                                   &
+                 pgzui  (ji,jj) = fsde3w(ji+1,jj,iku) - (fsde3w(ji,jj,iku) - ze3wu)
+                 pgrui  (ji,jj) = umask(ji,jj,iku)   * ( prd(ji+1,jj,iku) - zri(ji,jj) )      ! i: 2
+                 pmrui  (ji,jj) = umask(ji,jj,iku)   * ( prd(ji+1,jj,iku) + zri(ji,jj) )      ! i: 2
+                 pge3rui(ji,jj) = umask(ji,jj,iku+1)                                                                   &
                                 * (  fse3w(ji+1,jj,iku+1)          * (prd(ji+1,jj,iku) + prd(ji+1,jj,iku+1) + 2._wp)  &
                                    -(fse3w(ji  ,jj,iku+1) + ze3wu) * (zri(ji,jj      ) + prd(ji  ,jj,iku+1) + 2._wp)  )     ! i: 2
                ENDIF
                IF( ze3wv >= 0._wp ) THEN
-                 sgzv  (ji,jj) = (fsde3w(ji,jj+1,ikv) + ze3wv) - fsde3w(ji,jj,ikv) 
-                 sgrv  (ji,jj) = vmask(ji,jj,ikv)   * ( zrj(ji,jj  ) - prd(ji,jj,ikv) )        ! j: 1
-                 smrv  (ji,jj) = vmask(ji,jj,ikv)   * ( zrj(ji,jj  ) + prd(ji,jj,ikv) )        ! j: 1
-                 sge3rv(ji,jj) = vmask(ji,jj,ikv+1)                                                                  & 
+                 pgzvi  (ji,jj) = (fsde3w(ji,jj+1,ikv) + ze3wv) - fsde3w(ji,jj,ikv) 
+                 pgrvi  (ji,jj) = vmask(ji,jj,ikv)   * ( zrj(ji,jj  ) - prd(ji,jj,ikv) )        ! j: 1
+                 pmrvi  (ji,jj) = vmask(ji,jj,ikv)   * ( zrj(ji,jj  ) + prd(ji,jj,ikv) )        ! j: 1
+                 pge3rvi(ji,jj) = vmask(ji,jj,ikv+1)                                                                  & 
                                 * ( (fse3w(ji,jj+1,ikv+1) - ze3wv) * ( zrj(ji,jj    ) + prd(ji,jj+1,ikv+1) + 2._wp)  &
                                    - fse3w(ji,jj  ,ikv+1)          * ( prd(ji,jj,ikv) + prd(ji,jj  ,ikv+1) + 2._wp)  ) ! j: 1
                                   ! + 2 due to the formulation in density and not in anomalie in hpg sco
                ELSE
-                 sgzv  (ji,jj) = fsde3w(ji,jj+1,ikv) - (fsde3w(ji,jj,ikv) - ze3wv)
-                 sgrv  (ji,jj) = vmask(ji,jj,ikv)   * ( prd(ji,jj+1,ikv) - zrj(ji,jj) )     ! j: 2
-                 smrv  (ji,jj) = vmask(ji,jj,ikv)   * ( prd(ji,jj+1,ikv) + zrj(ji,jj) )     ! j: 2
-                 sge3rv(ji,jj) = vmask(ji,jj,ikv+1)                                                                   &
+                 pgzvi  (ji,jj) = fsde3w(ji,jj+1,ikv) - (fsde3w(ji,jj,ikv) - ze3wv)
+                 pgrvi  (ji,jj) = vmask(ji,jj,ikv)   * ( prd(ji,jj+1,ikv) - zrj(ji,jj) )     ! j: 2
+                 pmrvi  (ji,jj) = vmask(ji,jj,ikv)   * ( prd(ji,jj+1,ikv) + zrj(ji,jj) )     ! j: 2
+                 pge3rvi(ji,jj) = vmask(ji,jj,ikv+1)                                                                   &
                                 * (  fse3w(ji,jj+1,ikv+1)          * ( prd(ji,jj+1,ikv) + prd(ji,jj+1,ikv+1) + 2._wp) &
                                    -(fse3w(ji,jj  ,ikv+1) + ze3wv) * ( zrj(ji,jj      ) + prd(ji,jj  ,ikv+1) + 2._wp) )  ! j: 2
@@ -353,15 +510,14 @@
             END DO
          END DO
-         CALL lbc_lnk( sgru   , 'U', -1. )   ;   CALL lbc_lnk( sgrv   , 'V', -1. )   ! Lateral boundary conditions
-         CALL lbc_lnk( smru   , 'U',  1. )   ;   CALL lbc_lnk( smrv   , 'V',  1. )   ! Lateral boundary conditions
-         CALL lbc_lnk( sgzu   , 'U', -1. )   ;   CALL lbc_lnk( sgzv   , 'V', -1. )   ! Lateral boundary conditions
-         CALL lbc_lnk( sge3ru , 'U', -1. )   ;   CALL lbc_lnk( sge3rv , 'V', -1. )   ! Lateral boundary conditions
+         CALL lbc_lnk( pgrui   , 'U', -1. )   ;   CALL lbc_lnk( pgrvi   , 'V', -1. )   ! Lateral boundary conditions
+         CALL lbc_lnk( pmrui   , 'U',  1. )   ;   CALL lbc_lnk( pmrvi   , 'V',  1. )   ! Lateral boundary conditions
+         CALL lbc_lnk( pgzui   , 'U', -1. )   ;   CALL lbc_lnk( pgzvi   , 'V', -1. )   ! Lateral boundary conditions
+         CALL lbc_lnk( pge3rui , 'U', -1. )   ;   CALL lbc_lnk( pge3rvi , 'V', -1. )   ! Lateral boundary conditions
          !
       END IF  
       !
-      IF( nn_timing == 1 )  CALL timing_stop( 'zps_hde')
-      !
-   END SUBROUTINE zps_hde
-
+      IF( nn_timing == 1 )  CALL timing_stop( 'zps_hde_isf')
+      !
+   END SUBROUTINE zps_hde_isf
    !!======================================================================
 END MODULE zpshde
