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MODULE icedyn_rhg_vp
!!======================================================================
!! *** MODULE icedyn_rhg_vp ***
!! Sea-Ice dynamics : Viscous-plastic rheology with LSR technique
!!======================================================================
!!
!! History : - ! 1997-20 (J. Zhang, M. Losch) Original code, implementation into mitGCM
!! 4.0 ! 2020-09 (M. Vancoppenolle) Adaptation to SI3
!!
!!----------------------------------------------------------------------
#if defined key_si3
!!----------------------------------------------------------------------
!! 'key_si3' SI3 sea-ice model
!!----------------------------------------------------------------------
!! ice_dyn_rhg_vp : computes ice velocities from VP rheolog with LSR solvery
!!----------------------------------------------------------------------
USE phycst ! Physical constants
USE dom_oce ! Ocean domain
USE sbc_oce , ONLY : ln_ice_embd, nn_fsbc, ssh_m
USE sbc_ice , ONLY : utau_ice, vtau_ice, snwice_mass, snwice_mass_b
USE ice ! sea-ice: ice variables
USE icevar ! ice_var_sshdyn
USE icedyn_rdgrft ! sea-ice: ice strength
USE bdy_oce , ONLY : ln_bdy
USE bdyice
#if defined key_agrif
USE agrif_ice_interp
#endif
!
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 prtctl ! Print control
USE netcdf ! NetCDF library for convergence test
IMPLICIT NONE
PRIVATE
PUBLIC ice_dyn_rhg_vp ! called by icedyn_rhg.F90
INTEGER :: nn_nvp ! total number of VP iterations (n_out_vp*n_inn_vp)
LOGICAL :: lp_zebra_vp =.TRUE. ! activate zebra (solve the linear system problem every odd j-band, then one every even one)
REAL(wp) :: zrelaxu_vp=0.95 ! U-relaxation factor (MV: can probably be merged with V-factor once ok)
REAL(wp) :: zrelaxv_vp=0.95 ! V-relaxation factor
REAL(wp) :: zuerr_max_vp=0.80 ! maximum velocity error, above which a forcing error is considered and solver is stopped
REAL(wp) :: zuerr_min_vp=1.e-06 ! minimum velocity error, beyond which convergence is assumed
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!! for convergence tests
INTEGER :: ncvgid ! netcdf file id
INTEGER :: nvarid_ures, nvarid_vres, nvarid_velres
INTEGER :: nvarid_uerr_max, nvarid_verr_max, nvarid_velerr_max
INTEGER :: nvarid_umad, nvarid_vmad, nvarid_velmad
INTEGER :: nvarid_umad_outer, nvarid_vmad_outer, nvarid_velmad_outer
INTEGER :: nvarid_mke
REAL(wp), DIMENSION(:,:), ALLOCATABLE :: fimask ! mask at F points for the ice
!! * Substitutions
# include "do_loop_substitute.h90"
!!----------------------------------------------------------------------
!! NEMO/ICE 4.0 , NEMO Consortium (2018)
!! $Id: icedyn_rhg_vp.F90 13279 2020-07-09 10:39:43Z clem $
!! Software governed by the CeCILL license (see ./LICENSE)
!!----------------------------------------------------------------------
CONTAINS
SUBROUTINE ice_dyn_rhg_vp( kt, pshear_i, pdivu_i, pdelta_i )
!!-------------------------------------------------------------------
!!
!! *** SUBROUTINE ice_dyn_rhg_vp ***
!! VP-LSR-C-grid
!!
!! ** Purpose : determines sea ice drift from wind stress, ice-ocean
!! stress and sea-surface slope. Internal forces assume viscous-plastic rheology (Hibler, 1979)
!!
!! ** Method
!!
!! The resolution algorithm follows from Zhang and Hibler (1997) and Losch (2010)
!! with elements from Lemieux and Tremblay (2008) and Lemieux and Tremblay (2009)
!!
!! The components of the momentum equations are arranged following the ideas of Zhang and Hibler (1997)
!!
!! f1(u) = g1(v)
!! f2(v) = g2(u)
!!
!! The right-hand side (RHS) is explicit
!! The left-hand side (LHS) is implicit
!! Coriolis is part of explicit terms, whereas ice-ocean drag is implicit
!!
!! Two iteration levels (outer and inner loops) are used to solve the equations
!!
!! The outer loop (OL, typically 10 iterations) is there to deal with the (strong) non-linearities in the equation
!!
!! The inner loop (IL, typically 1500 iterations) is there to solve the linear problem with a line-successive-relaxation algorithm
!!
!! The velocity used in the non-linear terms uses a "modified euler time step" (not sure its the correct term),
!!! with uk = ( uk-1 + uk-2 ) / 2.
!!
!! * Spatial discretization
!!
!! Assumes a C-grid
!!
!! The points in the C-grid look like this, my darling
!!
!! (ji,jj)
!! |
!! |
!! (ji-1,jj) | (ji,jj)
!! ---------
!! | |
!! | (ji,jj) |------(ji,jj)
!! | |
!! ---------
!! (ji-1,jj-1) (ji,jj-1)
!!
!! ** Inputs : - wind forcing (stress), oceanic currents
!! ice total volume (vt_i) per unit area
!! snow total volume (vt_s) per unit area
!!
!! ** Action :
!!
!! ** Steps :
!!
!! ** Notes :
!!
!! References : Zhang and Hibler, JGR 1997; Losch et al., OM 2010., Lemieux et al., 2008, 2009, ...
!!
!!
!!-------------------------------------------------------------------
!!
INTEGER , INTENT(in ) :: kt ! time step
REAL(wp), DIMENSION(:,:), INTENT( out) :: pshear_i , pdivu_i , pdelta_i !
!!
LOGICAL :: ll_u_iterate, ll_v_iterate ! continue iteration or not
!
INTEGER :: ji, ji2, jj, jj2, jn ! dummy loop indices
INTEGER :: i_out, i_inn, i_inn_tot !
INTEGER :: ji_min, jj_min !
INTEGER :: nn_zebra_vp ! number of zebra steps
!
REAL(wp) :: zrhoco ! rho0 * rn_cio
REAL(wp) :: ecc2, z1_ecc2 ! square of yield ellipse eccenticity
REAL(wp) :: zglob_area ! global ice area for diagnostics
REAL(wp) :: zkt ! isotropic tensile strength for landfast ice
REAL(wp) :: zm1, zm2, zm3, zmassU, zmassV ! ice/snow mass and volume
REAL(wp) :: zds2, zdt, zdt2, zdiv, zdiv2 ! temporary scalars
REAL(wp) :: zu_cV, zv_cU !
REAL(wp) :: zfac, zfac1, zfac2, zfac3
REAL(wp) :: zt12U, zt11U, zt22U, zt21U, zt122U, zt121U
REAL(wp) :: zt12V, zt11V, zt22V, zt21V, zt122V, zt121V
REAL(wp) :: zAA3, zw, ztau, zuerr_max, zverr_max
!
REAL(wp), DIMENSION(jpi,jpj) :: za_iU , za_iV ! ice fraction on U/V points
REAL(wp), DIMENSION(jpi,jpj) :: zmU_t, zmV_t ! Acceleration term contribution to RHS
REAL(wp), DIMENSION(jpi,jpj) :: zmassU_t, zmassV_t ! Mass per unit area divided by time step
!
REAL(wp), DIMENSION(jpi,jpj) :: zdelta ! Delta at T-points (now value)
REAL(wp), DIMENSION(jpi,jpj) :: zten_i, zshear ! Tension, shear
REAL(wp), DIMENSION(jpi,jpj) :: zvisc_t ! Bulk viscosity (P/delta*) at T points
REAL(wp), DIMENSION(jpi,jpj) :: zvisc_t_prev ! Bulk viscosity (next to last iterate) - for yield curve diag
REAL(wp), DIMENSION(jpi,jpj) :: zzt, zet ! Viscosity pre-factors at T points
REAL(wp), DIMENSION(jpi,jpj) :: zef ! Viscosity pre-factor at F point
!
REAL(wp), DIMENSION(jpi,jpj) :: zmt ! Mass per unit area at t-point
REAL(wp), DIMENSION(jpi,jpj) :: zmf ! Coriolis factor (m*f) at t-point
REAL(wp), DIMENSION(jpi,jpj) :: v_oceU, u_oceV, v_iceU, u_iceV ! ocean/ice u/v component on V/U points
REAL(wp), DIMENSION(jpi,jpj) :: zu_c, zv_c ! "current" ice velocity (m/s), average of previous two OL iterates
REAL(wp), DIMENSION(jpi,jpj) :: zu_b, zv_b ! velocity at previous sub-iterate
REAL(wp), DIMENSION(jpi,jpj) :: zuerr, zverr ! absolute U/Vvelocity difference between current and previous sub-iterates
REAL(wp), DIMENSION(jpi,jpj) :: zvel_res ! Residual of the linear system at last iteration
REAL(wp), DIMENSION(jpi,jpj) :: zvel_diff ! Absolute velocity difference @last outer iteration
!
REAL(wp), DIMENSION(jpi,jpj) :: zds ! shear
REAL(wp), DIMENSION(jpi,jpj) :: zsshdyn ! array used for the calculation of ice surface slope:
! ! ocean surface (ssh_m) if ice is not embedded
! ! ice bottom surface if ice is embedded
REAL(wp), DIMENSION(jpi,jpj) :: zCwU, zCwV ! ice-ocean drag pre-factor (rho*c*module(u))
REAL(wp), DIMENSION(jpi,jpj) :: zspgU, zspgV ! surface pressure gradient at U/V points
REAL(wp), DIMENSION(jpi,jpj) :: zCorU, zCorV ! Coriolis stress array
REAL(wp), DIMENSION(jpi,jpj) :: ztaux_ai, ztauy_ai ! ice-atm. stress at U-V points
REAL(wp), DIMENSION(jpi,jpj) :: ztaux_oi_rhsu, ztauy_oi_rhsv ! ice-ocean stress RHS contribution at U-V points
REAL(wp), DIMENSION(jpi,jpj) :: zs1_rhsu, zs2_rhsu, zs12_rhsu ! internal stress contributions to RHSU
REAL(wp), DIMENSION(jpi,jpj) :: zs1_rhsv, zs2_rhsv, zs12_rhsv ! internal stress contributions to RHSV
REAL(wp), DIMENSION(jpi,jpj) :: zf_rhsu, zf_rhsv ! U- and V- components of internal force RHS contributions
REAL(wp), DIMENSION(jpi,jpj) :: zrhsu, zrhsv ! U and V RHS
REAL(wp), DIMENSION(jpi,jpj) :: zAU, zBU, zCU, zDU, zEU ! Linear system coefficients, U equation
REAL(wp), DIMENSION(jpi,jpj) :: zAV, zBV, zCV, zDV, zEV ! Linear system coefficients, V equation
REAL(wp), DIMENSION(jpi,jpj) :: zFU, zFU_prime, zBU_prime ! Rearranged linear system coefficients, U equation
REAL(wp), DIMENSION(jpi,jpj) :: zFV, zFV_prime, zBV_prime ! Rearranged linear system coefficients, V equation
!!! REAL(wp), DIMENSION(jpi,jpj) :: ztaux_bi, ztauy_bi ! ice-OceanBottom stress at U-V points (landfast)
!!! REAL(wp), DIMENSION(jpi,jpj) :: ztaux_base, ztauy_base ! ice-bottom stress at U-V points (landfast)
!
REAL(wp), DIMENSION(jpi,jpj) :: zmsk, zmsk00
REAL(wp), DIMENSION(jpi,jpj) :: zmsk01x, zmsk01y ! mask for lots of ice (1), little ice (0)
REAL(wp), DIMENSION(jpi,jpj) :: zmsk00x, zmsk00y ! mask for ice presence (1), no ice (0)
!
REAL(wp), PARAMETER :: zepsi = 1.0e-20_wp ! tolerance parameter
REAL(wp), PARAMETER :: zmmin = 1._wp ! ice mass (kg/m2) below which ice velocity becomes very small
REAL(wp), PARAMETER :: zamin = 0.001_wp ! ice concentration below which ice velocity becomes very small
!! --- diags
REAL(wp) :: zsig1, zsig2, zsig12, z1_strength, zfac_x, zfac_y
REAL(wp), DIMENSION(jpi,jpj) :: zs1, zs2, zs12, zs12f ! stress tensor components
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zsig_I, zsig_II, zsig1_p, zsig2_p
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: ztaux_oi, ztauy_oi
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xmtrp_ice, zdiag_ymtrp_ice ! X/Y-component of ice mass transport (kg/s, SIMIP)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xmtrp_snw, zdiag_ymtrp_snw ! X/Y-component of snow mass transport (kg/s, SIMIP)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xatrp, zdiag_yatrp ! X/Y-component of area transport (m2/s, SIMIP)
!!----------------------------------------------------------------------------------------------------------------------
IF( kt == nit000 .AND. lwp ) WRITE(numout,*) '-- ice_dyn_rhg_vp: VP sea-ice rheology (LSR solver)'
IF( lwp ) WRITE(numout,*) '-- ice_dyn_rhg_vp: VP sea-ice rheology (LSR solver)'
!------------------------------------------------------------------------------!
!
! --- Initialization
!
!------------------------------------------------------------------------------!
IF ( lp_zebra_vp ) THEN; nn_zebra_vp = 2
ELSE; nn_zebra_vp = 1; ENDIF
!!clem
nn_zebra_vp=1
!!clem
nn_nvp = nn_vp_nout * nn_vp_ninn ! maximum number of iterations
IF( lwp ) WRITE(numout,*) ' lp_zebra_vp : ', lp_zebra_vp
IF( lwp ) WRITE(numout,*) ' nn_zebra_vp : ', nn_zebra_vp
IF( lwp ) WRITE(numout,*) ' nn_nvp : ', nn_nvp
! for diagnostics and convergence tests
DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
zmsk00(ji,jj) = MAX( 0._wp , SIGN( 1._wp , at_i(ji,jj) - epsi06 ) ) ! 1 if ice , 0 if no ice
zmsk (ji,jj) = MAX( 0._wp , SIGN( 1._wp , at_i(ji,jj) - epsi10 ) ) ! 1 if ice , 0 if no ice
END_2D
!---------------------------
! -- F-mask (code from EVP)
!---------------------------
IF( kt == nit000 ) THEN
! MartinV:
! In EVP routine, fimask is applied on shear at F-points, in order to enforce the lateral boundary condition (no-slip, ..., free-slip)
! I am not sure the same recipe applies here
! - ocean/land mask
ALLOCATE( fimask(jpi,jpj) )
IF( rn_ishlat == 0._wp ) THEN
DO_2D( 0, 0, 0, 0 )
fimask(ji,jj) = tmask(ji,jj,1) * tmask(ji+1,jj,1) * tmask(ji,jj+1,1) * tmask(ji+1,jj+1,1)
END_2D
ELSE
DO_2D( 0, 0, 0, 0 )
fimask(ji,jj) = tmask(ji,jj,1) * tmask(ji+1,jj,1) * tmask(ji,jj+1,1) * tmask(ji+1,jj+1,1)
! Lateral boundary conditions on velocity (modify fimask)
IF( fimask(ji,jj) == 0._wp ) THEN
fimask(ji,jj) = rn_ishlat * MIN( 1._wp , MAX( umask(ji,jj,1), umask(ji,jj+1,1), &
& vmask(ji,jj,1), vmask(ji+1,jj,1) ) )
ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_vp', fimask, 'F', 1._wp )
ENDIF
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! Initialise convergence checks
IF( nn_rhg_chkcvg /= 0 ) THEN
! ice area for global mean kinetic energy (m2)
zglob_area = glob_sum( 'ice_rhg_vp', at_i(:,:) * e1e2t(:,:) * tmask(:,:,1) )
ENDIF
! Landfast param from Lemieux(2016): add isotropic tensile strength (following Konig Beatty and Holland, 2010)
! MV: Not working yet...
IF( ln_landfast_L16 ) THEN ; zkt = rn_lf_tensile
ELSE ; zkt = 0._wp
ENDIF
!------------------------------------------------------------------------------!
!
! --- Time-independent quantities
!
!------------------------------------------------------------------------------!
zrhoco = rho0 * rn_cio
! ecc2: square of yield ellipse eccentricity
ecc2 = rn_ecc * rn_ecc
z1_ecc2 = 1._wp / ecc2
CALL ice_strength ! strength at T points
!----------------------------------------------------------------------------------------------------------
! -- Time-independent pre-factors for acceleration, ocean drag, coriolis, atmospheric drag, surface tilt
!----------------------------------------------------------------------------------------------------------
! Compute all terms & factors independent of velocities, or only depending on velocities at previous time step
! sea surface height
! embedded sea ice: compute representative ice top surface
! non-embedded sea ice: use ocean surface for slope calculation
zsshdyn(:,:) = ice_var_sshdyn( ssh_m, snwice_mass, snwice_mass_b)
DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
zmt(ji,jj) = rhos * vt_s(ji,jj) + rhoi * vt_i(ji,jj) ! Snow and ice mass at T-point
zmf(ji,jj) = zmt(ji,jj) * ff_t(ji,jj) ! Coriolis factor at T points (m*f)
END_2D
DO_2D( nn_hls, nn_hls-1, nn_hls-1, nn_hls )
! Ice fraction at U-V points
za_iU(ji,jj) = 0.5_wp * ( at_i(ji,jj) * e1e2t(ji,jj) + at_i(ji+1,jj) * e1e2t(ji+1,jj) ) * r1_e1e2u(ji,jj) * umask(ji,jj,1)
! Snow and ice mass at U-V points
zm1 = zmt(ji,jj)
zm2 = zmt(ji+1,jj)
zmassU = 0.5_wp * ( zm1 * e1e2t(ji,jj) + zm2 * e1e2t(ji+1,jj) ) * r1_e1e2u(ji,jj) * umask(ji,jj,1)
! Mass per unit area divided by time step
zmassU_t(ji,jj) = zmassU * r1_Dt_ice
! Acceleration term contribution to RHS (depends on velocity at previous time step)
zmU_t(ji,jj) = zmassU_t(ji,jj) * u_ice(ji,jj)
! Ocean currents at U-V points
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
v_oceU(ji,jj) = 0.25_wp * ( (v_oce(ji,jj) + v_oce(ji,jj-1)) + (v_oce(ji+1,jj) + v_oce(ji+1,jj-1)) ) * umask(ji,jj,1)
! Wind stress
ztaux_ai(ji,jj) = za_iU(ji,jj) * utau_ice(ji,jj)
! Force due to sea surface tilt(- m*g*GRAD(ssh))
zspgU(ji,jj) = - zmassU * grav * ( zsshdyn(ji+1,jj) - zsshdyn(ji,jj) ) * r1_e1u(ji,jj)
!!spgU(ji,jj) = - grav * ( zsshdyn(ji,jj) ) * r1_e1u(ji,jj)
! Mask for ice presence (1) or absence (0)
zmsk00x(ji,jj) = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zmassU ) ) ! 0 if no ice
! Mask for lots of ice (1) or little ice (0)
IF ( zmassU <= zmmin .AND. za_iU(ji,jj) <= zamin ) THEN ; zmsk01x(ji,jj) = 0._wp
ELSE ; zmsk01x(ji,jj) = 1._wp ; ENDIF
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END_2D
DO_2D( nn_hls-1, nn_hls, nn_hls, nn_hls-1 ) ! 2->jpj-1; 2->jpi-1
! Ice fraction at U-V points
za_iV(ji,jj) = 0.5_wp * ( at_i(ji,jj) * e1e2t(ji,jj) + at_i(ji,jj+1) * e1e2t(ji,jj+1) ) * r1_e1e2v(ji,jj) * vmask(ji,jj,1)
! Snow and ice mass at U-V points
zm1 = zmt(ji,jj)
zm3 = zmt(ji,jj+1)
zmassV = 0.5_wp * ( zm1 * e1e2t(ji,jj) + zm3 * e1e2t(ji,jj+1) ) * r1_e1e2v(ji,jj) * vmask(ji,jj,1)
! Mass per unit area divided by time step
zmassV_t(ji,jj) = zmassV * r1_Dt_ice
! Acceleration term contribution to RHS (depends on velocity at previous time step)
zmV_t(ji,jj) = zmassV_t(ji,jj) * v_ice(ji,jj)
! Ocean currents at U-V points
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
u_oceV(ji,jj) = 0.25_wp * ( (u_oce(ji,jj) + u_oce(ji-1,jj)) + (u_oce(ji,jj+1) + u_oce(ji-1,jj+1)) ) * vmask(ji,jj,1)
! Wind stress
ztauy_ai(ji,jj) = za_iV(ji,jj) * vtau_ice(ji,jj)
! Force due to sea surface tilt(- m*g*GRAD(ssh))
zspgV(ji,jj) = - zmassV * grav * ( zsshdyn(ji,jj+1) - zsshdyn(ji,jj) ) * r1_e2v(ji,jj)
! Mask for ice presence (1) or absence (0)
zmsk00y(ji,jj) = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zmassV ) ) ! 0 if no ice
! Mask for lots of ice (1) or little ice (0)
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IF ( zmassV <= zmmin .AND. za_iV(ji,jj) <= zamin ) THEN ; zmsk01y(ji,jj) = 0._wp
ELSE ; zmsk01y(ji,jj) = 1._wp ; ENDIF
END_2D
!------------------------------------------------------------------------------!
!
! --- Start outer loop
!
!------------------------------------------------------------------------------!
zu_c(:,:) = u_ice(:,:)
zv_c(:,:) = v_ice(:,:)
i_inn_tot = 0
DO i_out = 1, nn_vp_nout
! Velocities used in the non linear terms are the average of the past two iterates
! u_it = 0.5 * ( u_{it-1} + u_{it-2} )
! Also used in Hibler and Ackley (1983); Zhang and Hibler (1997); Lemieux and Tremblay (2009)
zu_c(:,:) = 0.5_wp * ( u_ice(:,:) + zu_c(:,:) )
zv_c(:,:) = 0.5_wp * ( v_ice(:,:) + zv_c(:,:) )
!------------------------------------------------------------------------------!
!
! --- Right-hand side (RHS) of the linear problem
!
!------------------------------------------------------------------------------!
! In the outer loop, one needs to update all RHS terms
! with explicit velocity dependencies (viscosities, coriolis, ocean stress)
! as a function of "current" velocities (uc, vc)
!------------------------------------------
! -- Strain rates, viscosities and P/Delta
!------------------------------------------
! --- divergence, tension & shear (Appendix B of Hunke & Dukowicz, 2002) --- !
DO_2D( nn_hls, nn_hls-1, nn_hls, nn_hls-1 ) ! 1->jpi-1
! loops start at 1 since there is no boundary condition (lbc_lnk) at i=1 and j=1 for F points
! shear at F points
zds(ji,jj) = ( ( zu_c(ji,jj+1) * r1_e1u(ji,jj+1) - zu_c(ji,jj) * r1_e1u(ji,jj) ) * e1f(ji,jj) * e1f(ji,jj) &
& + ( zv_c(ji+1,jj) * r1_e2v(ji+1,jj) - zv_c(ji,jj) * r1_e2v(ji,jj) ) * e2f(ji,jj) * e2f(ji,jj) &
& ) * r1_e1e2f(ji,jj) * fimask(ji,jj)
END_2D
! loop to jpi,jpj to avoid making a communication for zs1,zs2,zs12
! shear**2 at T points (doc eq. A16)
zds2 = ( zds(ji,jj ) * zds(ji,jj ) * e1e2f(ji,jj ) + zds(ji-1,jj ) * zds(ji-1,jj ) * e1e2f(ji-1,jj ) &
& + zds(ji,jj-1) * zds(ji,jj-1) * e1e2f(ji,jj-1) + zds(ji-1,jj-1) * zds(ji-1,jj-1) * e1e2f(ji-1,jj-1) &
& ) * 0.25_wp * r1_e1e2t(ji,jj)
! divergence at T points
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
zdiv = ( (e2u(ji,jj) * zu_c(ji,jj) - e2u(ji-1,jj) * zu_c(ji-1,jj)) &
& + (e1v(ji,jj) * zv_c(ji,jj) - e1v(ji,jj-1) * zv_c(ji,jj-1)) &
& ) * r1_e1e2t(ji,jj)
zdiv2 = zdiv * zdiv
! tension at T points
zdt = ( ( zu_c(ji,jj) * r1_e2u(ji,jj) - zu_c(ji-1,jj) * r1_e2u(ji-1,jj) ) * e2t(ji,jj) * e2t(ji,jj) &
& - ( zv_c(ji,jj) * r1_e1v(ji,jj) - zv_c(ji,jj-1) * r1_e1v(ji,jj-1) ) * e1t(ji,jj) * e1t(ji,jj) &
& ) * r1_e1e2t(ji,jj)
zdt2 = zdt * zdt
! delta at T points
zdelta(ji,jj) = SQRT( zdiv2 + ( zdt2 + zds2 ) * z1_ecc2 ) * zmsk(ji,jj)
! P/delta at T points
zvisc_t(ji,jj) = strength(ji,jj) / ( zdelta(ji,jj) + rn_creepl ) * zmsk(ji,jj)
zzt(ji,jj) = zvisc_t(ji,jj) * r1_e1e2t(ji,jj)
zet(ji,jj) = zzt(ji,jj) * z1_ecc2
CALL lbc_lnk( 'icedyn_rhg_vp', zdelta, 'T', 1.0_wp, zvisc_t, 'T', 1.0_wp, zzt, 'T', 1.0_wp, zet, 'T', 1.0_wp )
! Store bulk viscosity at last outer iteration for yield curve diagnostic
IF ( i_out == nn_vp_nout .AND. ( iom_use('sig1_pnorm') .OR. iom_use('sig2_pnorm') ) ) THEN
zvisc_t_prev(:,:) = zvisc_t(:,:)
ENDIF
DO_2D( nn_hls, nn_hls-1, nn_hls, nn_hls-1 )! 1-> jpj-1; 1->jpi-1
! P/delta* at F points
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
zvisc_f = 0.25_wp * ( (zvisc_t(ji,jj) + zvisc_t(ji+1,jj)) + (zvisc_t(ji,jj+1) + zvisc_t(ji+1,jj+1)) )
! Temporary zef factor at F-point
zef(ji,jj) = zvisc_f * r1_e1e2f(ji,jj) * z1_ecc2 * fimask(ji,jj) * 0.5_wp
END_2D
!---------------------------------------------------
! -- Ocean-ice drag and Coriolis RHS contributions
!---------------------------------------------------
DO_2D( nn_hls, nn_hls-1, nn_hls-1, nn_hls )
!--- ice u-velocity @V points, v-velocity @U points (for non-linear drag computation)
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
zv_cU = 0.25_wp * ( (zv_c(ji,jj) + zv_c(ji,jj-1)) + (zv_c(ji+1,jj) + zv_c(ji+1,jj-1)) ) * umask(ji,jj,1)
!--- non-linear drag coefficients (need to be updated at each outer loop, see Lemieux and Tremblay JGR09, p.3, beginning of Section 3)
zCwU(ji,jj) = za_iU(ji,jj) * zrhoco * SQRT( ( zu_c (ji,jj) - u_oce (ji,jj) ) * ( zu_c (ji,jj) - u_oce (ji,jj) ) &
& + ( zv_cU - v_oceU(ji,jj) ) * ( zv_cU - v_oceU(ji,jj) ) )
!--- Ocean-ice drag contributions to RHS
ztaux_oi_rhsu(ji,jj) = zCwU(ji,jj) * u_oce(ji,jj)
!--- U-component of Coriolis Force (energy conserving formulation)
zCorU(ji,jj) = 0.25_wp * r1_e1u(ji,jj) * &
& ( (zmf(ji ,jj) * ( e1v(ji ,jj) * zv_c(ji ,jj) + e1v(ji ,jj-1) * zv_c(ji ,jj-1) )) &
& + (zmf(ji+1,jj) * ( e1v(ji+1,jj) * zv_c(ji+1,jj) + e1v(ji+1,jj-1) * zv_c(ji+1,jj-1) )) )
END_2D
DO_2D( nn_hls-1, nn_hls, nn_hls, nn_hls-1 )
!--- ice u-velocity @V points, v-velocity @U points (for non-linear drag computation)
! (brackets added to fix the order of floating point operations for halo 1 - halo 2 compatibility)
zu_cV = 0.25_wp * ( (zu_c(ji,jj) + zu_c(ji-1,jj)) + (zu_c(ji,jj+1) + zu_c(ji-1,jj+1)) ) * vmask(ji,jj,1)
!--- non-linear drag coefficients (need to be updated at each outer loop, see Lemieux and Tremblay JGR09, p.3, beginning of Section 3)
zCwV(ji,jj) = za_iV(ji,jj) * zrhoco * SQRT( ( zv_c (ji,jj) - v_oce (ji,jj) ) * ( zv_c (ji,jj) - v_oce (ji,jj) ) &
& + ( zu_cV - u_oceV(ji,jj) ) * ( zu_cV - u_oceV(ji,jj) ) )
!--- Ocean-ice drag contributions to RHS
ztauy_oi_rhsv(ji,jj) = zCwV(ji,jj) * v_oce(ji,jj)
!--- V-component of Coriolis Force (energy conserving formulation)
zCorV(ji,jj) = - 0.25_wp * r1_e2v(ji,jj) * &
& ( (zmf(ji,jj ) * ( e2u(ji,jj ) * zu_c(ji,jj ) + e2u(ji-1,jj ) * zu_c(ji-1,jj ) )) &
& + (zmf(ji,jj+1) * ( e2u(ji,jj+1) * zu_c(ji,jj+1) + e2u(ji-1,jj+1) * zu_c(ji-1,jj+1) )) )
END_2D
!-------------------------------------
! -- Internal stress RHS contribution
!-------------------------------------
! --- Stress contributions at T-points
DO_2D( nn_hls, nn_hls, nn_hls-1, nn_hls ) ! 2 -> jpj; 2,jpi !!! CHECK !!!
! sig1 contribution to RHS of U-equation at T-points
zs1_rhsu(ji,jj) = zzt(ji,jj) * ( e1v(ji,jj) * zv_c(ji,jj) - e1v(ji,jj-1) * zv_c(ji,jj-1) ) &
& - zvisc_t(ji,jj) * zdelta(ji,jj)
! sig2 contribution to RHS of U-equation at T-points
zs2_rhsu(ji,jj) = - zet(ji,jj) * ( r1_e1v(ji,jj) * zv_c(ji,jj) - r1_e1v(ji,jj-1) * zv_c(ji,jj-1) ) * e1t(ji,jj) * e1t(ji,jj)
END_2D
DO_2D( nn_hls-1, nn_hls, nn_hls, nn_hls ) ! 2 -> jpj; 2,jpi !!! CHECK !!!
! sig1 contribution to RHS of V-equation at T-points
zs1_rhsv(ji,jj) = zzt(ji,jj) * ( e2u(ji,jj) * zu_c(ji,jj) - e2u(ji-1,jj) * zu_c(ji-1,jj) ) &
& - zvisc_t(ji,jj) * zdelta(ji,jj)
! sig2 contribution to RHS of V-equation at T-points
zs2_rhsv(ji,jj) = zet(ji,jj) * ( r1_e2u(ji,jj) * zu_c(ji,jj) - r1_e2u(ji-1,jj) * zu_c(ji-1,jj) ) * e2t(ji,jj) * e2t(ji,jj)
END_2D
! --- Stress contributions at F-points
! MV NOTE: I applied fimask on zds, by mimetism on EVP, but without deep understanding of what I was doing
! My guess is that this is the way to enforce boundary conditions on strain rate tensor
DO_2D( nn_hls, nn_hls-1, nn_hls, nn_hls-1 ) ! 1->jpi-1
! sig12 contribution to RHS of U equation at F-points
zs12_rhsu(ji,jj) = zef(ji,jj) * ( r1_e2v(ji+1,jj) * zv_c(ji+1,jj) + r1_e2v(ji,jj) * zv_c(ji,jj) ) * e2f(ji,jj) * e2f(ji,jj) * fimask(ji,jj)
! sig12 contribution to RHS of V equation at F-points
zs12_rhsv(ji,jj) = zef(ji,jj) * ( r1_e1u(ji,jj+1) * zu_c(ji,jj+1) + r1_e1u(ji,jj) * zu_c(ji,jj) ) * e1f(ji,jj) * e1f(ji,jj) * fimask(ji,jj)
END_2D
! --- Internal force contributions to RHS, taken as divergence of stresses (Appendix C of Hunke and Dukowicz, 2002)
! OPT: merge with next loop and use intermediate scalars for zf_rhsu
DO_2D( nn_hls, nn_hls-1, nn_hls-1, nn_hls-1 ) ! 2->jpj-1; 2->jpi-1
! --- U component of internal force contribution to RHS at U points
zf_rhsu(ji,jj) = 0.5_wp * r1_e1e2u(ji,jj) * &
( e2u(ji,jj) * ( zs1_rhsu(ji+1,jj) - zs1_rhsu(ji,jj) ) &
& + r1_e2u(ji,jj) * ( e2t(ji+1,jj) * e2t(ji+1,jj) * zs2_rhsu(ji+1,jj) - e2t(ji,jj) * e2t(ji,jj) * zs2_rhsu(ji,jj) ) &
& + 2._wp * r1_e1u(ji,jj) * ( e1f(ji,jj) * e1f(ji,jj) * zs12_rhsu(ji,jj) - e1f(ji,jj-1) * e1f(ji,jj-1) * zs12_rhsu(ji,jj-1) ) )
END_2D
DO_2D( nn_hls-1, nn_hls-1, nn_hls, nn_hls-1 ) ! 2->jpj-1; 2->jpi-1
! --- V component of internal force contribution to RHS at V points
zf_rhsv(ji,jj) = 0.5_wp * r1_e1e2v(ji,jj) * &
& ( e1v(ji,jj) * ( zs1_rhsv(ji,jj+1) - zs1_rhsv(ji,jj) ) &
& - r1_e1v(ji,jj) * ( e1t(ji,jj+1) * e1t(ji,jj+1) * zs2_rhsv(ji,jj+1) - e1t(ji,jj) * e1t(ji,jj) * zs2_rhsv(ji,jj) ) &
& + 2._wp * r1_e2v(ji,jj) * ( e2f(ji,jj) * e2f(ji,jj) * zs12_rhsv(ji,jj) - e2f(ji-1,jj) * e2f(ji-1,jj) * zs12_rhsv(ji-1,jj) ) )
END_2D
!---------------------------
! -- Sum RHS contributions
!---------------------------
! OPT: could use intermediate scalars to reduce memory access
DO_2D( nn_hls, nn_hls-1, nn_hls-1, nn_hls-1 ) ! 2->jpj-1; 2->jpi-1
zrhsu(ji,jj) = zmU_t(ji,jj) + ztaux_ai(ji,jj) + ztaux_oi_rhsu(ji,jj) + zspgU(ji,jj) + zCorU(ji,jj) + zf_rhsu(ji,jj)
END_2D
DO_2D( nn_hls-1, nn_hls-1, nn_hls, nn_hls-1 ) ! 2->jpj-1; 2->jpi-1
zrhsv(ji,jj) = zmV_t(ji,jj) + ztauy_ai(ji,jj) + ztauy_oi_rhsv(ji,jj) + zspgV(ji,jj) + zCorV(ji,jj) + zf_rhsv(ji,jj)
END_2D
!---------------------------------------------------------------------------------------!
!
! --- Linear system matrix
!
!---------------------------------------------------------------------------------------!
! Linear system matrix contains all implicit contributions
! 1) internal forces, 2) acceleration, 3) ice-ocean drag
! The linear system equation is written as follows
! AU * u_{i-1,j} + BU * u_{i,j} + CU * u_{i+1,j}
! = DU * u_{i,j-1} + EU * u_{i,j+1} + RHS (! my convention, not the same as ZH97 )
! MV Note 1: martin losch applies boundary condition to BU in mitGCM - check whether it is necessary here ?
! MV Note 2: "T" factor calculations can be optimized by putting things out of the loop
! only zzt and zet are iteration-dependent, other only depend on scale factors
DO_2D( nn_hls, nn_hls-1, nn_hls-1, nn_hls-1 )
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!-------------------------------------
! -- Internal forces LHS contribution
!-------------------------------------
!
! --- U-component
!
! "T" factors (intermediate results)
!
zfac = 0.5_wp * r1_e1e2u(ji,jj)
zfac1 = zfac * e2u(ji,jj)
zfac2 = zfac * r1_e2u(ji,jj)
zfac3 = 2._wp * zfac * r1_e1u(ji,jj)
zt11U = zfac1 * zzt(ji,jj)
zt12U = zfac1 * zzt(ji+1,jj)
zt21U = zfac2 * zet(ji,jj) * e2t(ji,jj) * e2t(ji,jj) * e2t(ji,jj) * e2t(ji,jj)
zt22U = zfac2 * zet(ji+1,jj) * e2t(ji+1,jj) * e2t(ji+1,jj) * e2t(ji+1,jj) * e2t(ji+1,jj)
zt121U = zfac3 * zef(ji,jj-1) * e1f(ji,jj-1) * e1f(ji,jj-1) * e1f(ji,jj-1) * e1f(ji,jj-1)
zt122U = zfac3 * zef(ji,jj) * e1f(ji,jj) * e1f(ji,jj) * e1f(ji,jj) * e1f(ji,jj)
!
! Linear system coefficients
!
zBU(ji,jj) = ( zt11U + zt12U ) * e2u(ji,jj) + ( zt21U + zt22U ) * r1_e2u(ji,jj) + ( zt121U + zt122U ) * r1_e1u(ji,jj)
zCU(ji,jj) = - zt12U * e2u(ji+1,jj) - zt22U * r1_e2u(ji+1,jj)
zDU(ji,jj) = zt121U * r1_e1u(ji,jj-1)
zEU(ji,jj) = zt122U * r1_e1u(ji,jj+1)
!-----------------------------------------------------
! -- Ocean-ice drag and acceleration LHS contribution
!-----------------------------------------------------
zBU(ji,jj) = zBU(ji,jj) + zCwU(ji,jj) + zmassU_t(ji,jj)
END_2D
DO_2D( nn_hls-1, nn_hls-1, nn_hls, nn_hls-1 )
!-------------------------------------
! -- Internal forces LHS contribution
!-------------------------------------
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!
! --- V-component
!
! "T" factors (intermediate results)
!
zfac = 0.5_wp * r1_e1e2v(ji,jj)
zfac1 = zfac * e1v(ji,jj)
zfac2 = zfac * r1_e1v(ji,jj)
zfac3 = 2._wp * zfac * r1_e2v(ji,jj)
zt11V = zfac1 * zzt(ji,jj)
zt12V = zfac1 * zzt(ji,jj+1)
zt21V = zfac2 * zet(ji,jj) * e1t(ji,jj) * e1t(ji,jj) * e1t(ji,jj) * e1t(ji,jj)
zt22V = zfac2 * zet(ji,jj+1) * e1t(ji,jj+1) * e1t(ji,jj+1) * e1t(ji,jj+1) * e1t(ji,jj+1)
zt121V = zfac3 * zef(ji-1,jj) * e2f(ji-1,jj) * e2f(ji-1,jj) * e2f(ji-1,jj) * e2f(ji-1,jj)
zt122V = zfac3 * zef(ji,jj) * e2f(ji,jj) * e2f(ji,jj) * e2f(ji,jj) * e2f(ji,jj)
!
! Linear system coefficients
!
zBV(ji,jj) = ( zt11V + zt12V ) * e1v(ji,jj) + ( zt21V + zt22V ) * r1_e1v(ji,jj) + ( zt122V + zt121V ) * r1_e2v(ji,jj)
zCV(ji,jj) = - zt12V * e1v(ji,jj+1) - zt22V * r1_e1v(ji,jj+1)
zDV(ji,jj) = zt121V * r1_e2v(ji-1,jj)
zEV(ji,jj) = zt122V * r1_e2v(ji+1,jj)
!-----------------------------------------------------
! -- Ocean-ice drag and acceleration LHS contribution
!-----------------------------------------------------
zBV(ji,jj) = zBV(ji,jj) + zCwV(ji,jj) + zmassV_t(ji,jj)
END_2D
DO_2D( nn_hls-1, nn_hls-1, nn_hls-1, nn_hls-1 )
! **U**
zfac = 0.5_wp * r1_e1e2u(ji,jj)
zfac1 = zfac * e2u(ji,jj)
zfac2 = zfac * r1_e2u(ji,jj)
zt11U = zfac1 * zzt(ji,jj)
zt21U = zfac2 * zet(ji,jj) * e2t(ji,jj) * e2t(ji,jj) * e2t(ji,jj) * e2t(ji,jj)
!
zAU(ji,jj) = - zt11U * e1u(ji-1,jj) - zt21U * r1_e1u(ji-1,jj) !!clem: because of this fuck we need to start at jpi=2
! **V**
zfac = 0.5_wp * r1_e1e2v(ji,jj)
zfac1 = zfac * e1v(ji,jj)
zfac2 = zfac * r1_e1v(ji,jj)
zt11V = zfac1 * zzt(ji,jj)
zt21V = zfac2 * zet(ji,jj) * e1t(ji,jj) * e1t(ji,jj) * e1t(ji,jj) * e1t(ji,jj)
!
zAV(ji,jj) = - zt11V * e1v(ji,jj-1) - zt21V * r1_e1v(ji,jj-1) !!clem: because of this fuck we need to start at jpj=2
END_2D
!! CALL lbc_lnk( 'icedyn_rhg_vp', zAU, 'U', -1._wp, zBU, 'U', -1._wp, zCU, 'U', -1._wp, zDU, 'U', -1._wp, zEU, 'U', -1._wp, &
!! & zAV, 'V', -1._wp, zBV, 'V', -1._wp, zCV, 'V', -1._wp, zDV, 'V', -1._wp, zEV, 'V', -1._wp )
!------------------------------------------------------------------------------!
!
! --- Inner loop: solve linear system, check convergence
!
!------------------------------------------------------------------------------!
! Inner loop solves the linear problem .. requires 1500 iterations
ll_u_iterate = .TRUE.
ll_v_iterate = .TRUE.
DO i_inn = 1, nn_vp_ninn ! inner loop iterations
!--- mitgcm computes initial value of residual here...
i_inn_tot = i_inn_tot + 1
! l_full_nf_update = i_inn_tot == nn_nvp ! false: disable full North fold update (performances) for iter = 1 to nn_nevp-1
zu_b(:,:) = u_ice(:,:) ! velocity at previous inner-iterate
zv_b(:,:) = v_ice(:,:)
IF ( ll_u_iterate .OR. ll_v_iterate ) THEN
! ---------------------------- !
IF ( ll_u_iterate ) THEN ! --- Solve for u-velocity --- !
! ---------------------------- !
! What follows could be subroutinized...
! Thomas Algorithm for tridiagonal solver
! A*u(i-1,j)+B*u(i,j)+C*u(i+1,j) = F
DO jn = 1, nn_zebra_vp ! "zebra" loop (! red-black-sor!!! )
! OPT: could be even better optimized with a true red-black SOR
IF ( jn == 1 ) THEN ; jj_min = ntsj-(nn_hls-1)
ELSE ; jj_min = ntsj-(nn_hls-1)+1
!! DO jj = jj_min, ntej+(nn_hls-1), nn_zebra_vp
!------------------------
! Independent term (zFU)
!------------------------
!! DO ji = ntsi-(nn_hls), ntei+(nn_hls-1)
DO ji = 1, jpi-1
! note: these are key lines linking information between processors
! u_ice/v_ice need to be lbc_linked
! sub-domain boundary condition substitution
! see Zhang and Hibler, 1997, Appendix B
zAA3 = 0._wp
!!$ IF ( ji == 2 ) zAA3 = zAA3 - zAU(ji,jj) * u_ice(ji-1,jj)
!!$ IF ( ji == jpi - 1 ) zAA3 = zAA3 - zCU(ji,jj) * u_ice(ji+1,jj)
! right hand side
zFU(ji,jj) = ( zrhsu(ji,jj) & ! right-hand side terms
& + zAA3 & ! boundary condition translation
& + zDU(ji,jj) * u_ice(ji,jj-1) & ! internal force, j-1
& + zEU(ji,jj) * u_ice(ji,jj+1) ) * umask(ji,jj,1) ! internal force, j+1
END DO
END DO
!!CALL lbc_lnk( 'icedyn_rhg_vp', zFU, 'U', -1._wp )
!---------------
! Forward sweep
!---------------
DO jj = jj_min, jpj - 1, nn_zebra_vp
!! DO jj = jj_min, ntej+(nn_hls-1), nn_zebra_vp
!!$ zBU_prime(2,jj) = zBU(2,jj)
!!$ zFU_prime(2,jj) = zFU(2,jj)
DO ji = 2, jpi-1
!! DO ji = ntsi-(nn_hls-1), ntei+(nn_hls-1)
zfac = SIGN( 1._wp , zBU(ji-1,jj) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBU(ji-1,jj) ) - epsi20 ) )
zw = zfac * zAU(ji,jj) / MAX ( ABS( zBU(ji-1,jj) ) , epsi20 )
zBU_prime(ji,jj) = zBU(ji,jj) - zw * zCU(ji-1,jj)
zFU_prime(ji,jj) = zFU(ji,jj) - zw * zFU(ji-1,jj)
END DO
END DO
!-----------------------------
! Backward sweep & relaxation
!-----------------------------
DO jj = jj_min, jpj - 1, nn_zebra_vp
!!DO jj = jj_min, ntej+(nn_hls-1), nn_zebra_vp
!!$ zfac = SIGN( 1._wp , zBU_prime(jpi-1,jj) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBU_prime(jpi-1,jj) ) - epsi20 ) )
!!$ u_ice(jpi-1,jj) = zfac * zFU_prime(jpi-1,jj) / MAX( ABS ( zBU_prime(jpi-1,jj) ) , epsi20 ) &
!!$ & * umask(jpi-1,jj,1)
!!clem => should be backward but then no repro!!!
!!DO ji = jpi - 1 , 2, -1 ! all other rows ! ---> original backward loop
!!DO ji = ntei+(nn_hls-1), ntsi-(nn_hls-1), -1
DO ji = 2, jpi - 1 ! all other rows !
zfac = SIGN( 1._wp , zBU_prime(ji,jj) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBU_prime(ji,jj) ) - epsi20 ) )
u_ice(ji,jj) = zfac * ( zFU_prime(ji,jj) - zCU(ji,jj) * u_ice(ji+1,jj) ) * umask(ji,jj,1) &
& / MAX ( ABS ( zBU_prime(ji,jj) ) , epsi20 )
END DO
!--- Relaxation and masking (for low-ice/no-ice cases)
DO ji = 2, jpi - 1
!!DO ji = ntsi-(nn_hls-1), ntei+(nn_hls-1)
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u_ice(ji,jj) = zu_b(ji,jj) + zrelaxu_vp * ( u_ice(ji,jj) - zu_b(ji,jj) ) ! relaxation
u_ice(ji,jj) = zmsk00x(ji,jj) & ! masking
& * ( zmsk01x(ji,jj) * u_ice(ji,jj) &
& + ( 1._wp - zmsk01x(ji,jj) ) * u_oce(ji,jj) * 0.01_wp ) * umask(ji,jj,1)
END DO
END DO ! jj
END DO ! zebra loop
ENDIF ! ll_u_iterate
! ! ---------------------------- !
IF ( ll_v_iterate ) THEN ! --- Solve for V-velocity --- !
! ! ---------------------------- !
! MV OPT: what follows could be subroutinized...
! Thomas Algorithm for tridiagonal solver
! A*v(i,j-1)+B*v(i,j)+C*v(i,j+1) = F
! It is intentional to have a ji then jj loop for V-velocity
!!! ZH97 explain it is critical for convergence speed
DO jn = 1, nn_zebra_vp ! "zebra" loop
IF ( jn == 1 ) THEN ; ji_min = 2
ELSE ; ji_min = 3
ENDIF
DO ji = ji_min, jpi - 1, nn_zebra_vp
!------------------------
! Independent term (zFV)
!------------------------
! subdomain boundary condition substitution (check it is correctly applied !!!)
! see Zhang and Hibler, 1997, Appendix B
zAA3 = 0._wp
!!$ IF ( jj == 2 ) zAA3 = zAA3 - zAV(ji,jj) * v_ice(ji,jj-1)
!!$ IF ( jj == jpj - 1 ) zAA3 = zAA3 - zCV(ji,jj) * v_ice(ji,jj+1)
! right hand side
zFV(ji,jj) = ( zrhsv(ji,jj) & ! right-hand side terms
& + zAA3 & ! boundary condition translation
& + zDV(ji,jj) * v_ice(ji-1,jj) & ! internal force, j-1
& + zEV(ji,jj) * v_ice(ji+1,jj) ) * vmask(ji,jj,1) ! internal force, j+1
END DO
END DO
!---------------
! Forward sweep
!---------------
DO ji = ji_min, jpi - 1, nn_zebra_vp
!!$ zBV_prime(ji,2) = zBV(ji,2)
!!$ zFV_prime(ji,2) = zFV(ji,2)
zfac = SIGN( 1._wp , zBV(ji,jj-1) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBV(ji,jj-1) ) - epsi20 ) )
zw = zfac * zAV(ji,jj) / MAX ( ABS( zBV(ji,jj-1) ) , epsi20 )
zBV_prime(ji,jj) = zBV(ji,jj) - zw * zCV(ji,jj-1)
zFV_prime(ji,jj) = zFV(ji,jj) - zw * zFV(ji,jj-1)
END DO
END DO
!-----------------------------
! Backward sweep & relaxation
!-----------------------------
DO ji = ji_min, jpi - 1, nn_zebra_vp
! --- Backward sweep
!!$ ! last row
!!$ zfac = SIGN( 1._wp , zBV_prime(ji,jpj-1) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBV_prime(ji,jpj-1) ) - epsi20 ) )
!!$ v_ice(ji,jpj-1) = zfac * zFV_prime(ji,jpj-1) / MAX ( ABS(zBV_prime(ji,jpj-1) ) , epsi20 ) &
!!$ & * vmask(ji,jpj-1,1) ! last row
!!clem => should be backward but then no repro!!!
!!DO jj = jpj-1, 2, -1 ! original back loop
DO jj = 2, jpj-1
zfac = SIGN( 1._wp , zBV_prime(ji,jj) ) * MAX( 0._wp , SIGN( 1._wp , ABS( zBV_prime(ji,jj) ) - epsi20 ) )
v_ice(ji,jj) = zfac * ( zFV_prime(ji,jj) - zCV(ji,jj) * v_ice(ji,jj+1) ) * vmask(ji,jj,1) &
& / MAX ( ABS( zBV_prime(ji,jj) ) , epsi20 )
END DO
! --- Relaxation & masking
DO jj = 2, jpj - 1
v_ice(ji,jj) = zv_b(ji,jj) + zrelaxv_vp * ( v_ice(ji,jj) - zv_b(ji,jj) ) ! relaxation
v_ice(ji,jj) = zmsk00y(ji,jj) & ! masking
& * ( zmsk01y(ji,jj) * v_ice(ji,jj) &
& + ( 1._wp - zmsk01y(ji,jj) ) * v_oce(ji,jj) * 0.01_wp ) * vmask(ji,jj,1)
END DO ! jj
END DO ! ji
END DO ! zebra loop
ENDIF ! ll_v_iterate
CALL lbc_lnk( 'icedyn_rhg_vp', u_ice, 'U', -1._wp, v_ice, 'V', -1._wp )
! I suspect the communication should go into the zebra loop if we want reproducibility
!--------------------------------------------------------------------------------------
! -- Check convergence based on maximum velocity difference, continue or stop the loop
!--------------------------------------------------------------------------------------
!------
! on U
!------
! MV OPT: if the number of iterations to convergence is really variable, and keep the convergence check
! then we must optimize the use of the mpp_max, which is prohibitive
zuerr_max = 0._wp
IF ( ll_u_iterate .AND. MOD ( i_inn, nn_vp_chkcvg ) == 0 ) THEN
! - Maximum U-velocity difference
zuerr(:,:) = 0._wp
zuerr(ji,jj) = ABS ( ( u_ice(ji,jj) - zu_b(ji,jj) ) ) * umask(ji,jj,1)
END_2D
zuerr_max = MAXVAL( zuerr )
CALL mpp_max( 'icedyn_rhg_vp', zuerr_max ) ! max over the global domain - damned!
! - Stop if max error is too large ("safeguard against bad forcing" of original Zhang routine)
IF ( i_inn > 1 .AND. zuerr_max > zuerr_max_vp ) THEN
IF ( lwp ) WRITE(numout,*) " VP rheology error was too large : ", zuerr_max, " in outer U-iteration ", i_out, " after ", i_inn, " iterations, we stopped "
ll_u_iterate = .FALSE.
ENDIF
! - Stop if error small enough
IF ( zuerr_max < zuerr_min_vp ) THEN
IF ( lwp ) WRITE(numout,*) " VP rheology nicely done in outer U-iteration ", i_out, " after ", i_inn, " iterations, finished! "
ll_u_iterate = .FALSE.
ENDIF
ENDIF ! ll_u_iterate
!------
! on V
!------
zverr_max = 0._wp
IF ( ll_v_iterate .AND. MOD ( i_inn, nn_vp_chkcvg ) == 0 ) THEN
! - Maximum V-velocity difference
zverr(:,:) = 0._wp
zverr(ji,jj) = ABS ( ( v_ice(ji,jj) - zv_b(ji,jj) ) ) * vmask(ji,jj,1)
END_2D
zverr_max = MAXVAL( zverr )
CALL mpp_max( 'icedyn_rhg_vp', zverr_max ) ! max over the global domain - damned!