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
!! 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) :: zvisc_f !
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
ENDIF
CALL lbc_lnk( 'icedyn_rhg_vp', fimask, 'F', 1._wp )
ENDIF
! 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 the North Pole reproducibility
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
! Note the use of 0.5*(2-umask) in order to unmask the stress along coastlines
! and the use of MAX(tmask(i,j),tmask(i+1,j) is to mask tau over ice shelves
ztaux_ai(ji,jj) = za_iU(ji,jj) * 0.5_wp * ( utau_ice(ji,jj) + utau_ice(ji+1,jj) ) * &
& ( 2. - umask(ji,jj,1) ) * MAX( tmask(ji,jj,1), tmask(ji+1,jj,1) )
! 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
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 the North Pole reproducibility
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
! Note the use of 0.5*(2-umask) in order to unmask the stress along coastlines
! and the use of MAX(tmask(i,j),tmask(i+1,j) is to mask tau over ice shelves
ztauy_ai(ji,jj) = za_iV(ji,jj) * 0.5_wp * ( vtau_ice(ji,jj) + vtau_ice(ji,jj+1) ) * &
& ( 2. - vmask(ji,jj,1) ) * MAX( tmask(ji,jj,1), tmask(ji,jj+1,1) )
! 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)
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
DO_2D( 0, 0, 0, 0 )
! loop to jpi,jpj to avoid making a communication for zs1,zs2,zs12
! shear**2 at T points (doc eq. A16)
! (brackets added to fix the order of floating point operations for the North Pole reproducibility
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 the North Pole reproducibility
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)
! Temporary zzt and zet factors at T-points
zzt(ji,jj) = zvisc_t(ji,jj) * r1_e1e2t(ji,jj)
zet(ji,jj) = zzt(ji,jj) * z1_ecc2
END_2D
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 the North Pole reproducibility
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 the North Pole reproducibility
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 the North Pole reproducibility
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 )
!-------------------------------------
! -- 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
!-------------------------------------
!
! --- 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
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
ENDIF
DO jj = jj_min, jpj - 1, nn_zebra_vp
!! 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
! --- Backward sweep
! last row
!!$ 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)
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)
!------------------------
DO jj = 1, jpj - 1
! 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)
DO jj = 2, jpj - 1
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
! other rows
!!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
DO_2D( 0, 0, 0, 0 )
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
DO_2D( 0, 0, 0, 0 )
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!
! - Stop if error is too large ("safeguard against bad forcing" of original Zhang routine)
IF ( i_inn > 1 .AND. zverr_max > zuerr_max_vp ) THEN
IF ( lwp ) WRITE(numout,*) " VP rheology error was too large : ", zverr_max, " in outer V-iteration ", i_out, " after ", i_inn, " iterations, we stopped "
ll_v_iterate = .FALSE.
ENDIF
! - Stop if error small enough
IF ( zverr_max < zuerr_min_vp ) THEN
IF ( lwp ) WRITE(numout,*) " VP rheology nicely done in outer V-iteration ", i_out, " after ", i_inn, " iterations, finished! "
ll_v_iterate = .FALSE.
ENDIF
ENDIF ! ll_v_iterate
ENDIF ! --- end ll_u_iterate or ll_v_iterate
!---------------------------------------------------------------------------------------
!
! --- Calculate extra convergence diagnostics and save them
!
!---------------------------------------------------------------------------------------
IF( nn_rhg_chkcvg/=0 .AND. MOD ( i_inn - 1, nn_vp_chkcvg ) == 0 ) THEN
CALL rhg_cvg_vp( kt, i_out, i_inn, i_inn_tot, nn_vp_nout, nn_vp_ninn, nn_nvp, &
& u_ice, v_ice, zu_b, zv_b, zu_c, zv_c, &
& zmt, za_iU, za_iV, zuerr_max, zverr_max, zglob_area, &
& zrhsu, zAU, zBU, zCU, zDU, zEU, zFU, &
& zrhsv, zAV, zBV, zCV, zDV, zEV, zFV, &
zvel_res, zvel_diff )
ENDIF
END DO ! i_inn, end of inner loop
END DO ! End of outer loop (i_out) =============================================================================================
IF( nn_rhg_chkcvg/=0 ) THEN
IF( iom_use('velo_res') ) CALL iom_put( 'velo_res', zvel_res ) ! linear system residual @last inner&outer iteration
IF( iom_use('velo_ero') ) CALL iom_put( 'velo_ero', zvel_diff ) ! abs velocity difference @last outer iteration
IF( iom_use('uice_eri') ) CALL iom_put( 'uice_eri', zuerr ) ! abs velocity difference @last inner iteration
IF( iom_use('vice_eri') ) CALL iom_put( 'vice_eri', zverr ) ! abs velocity difference @last inner iteration
ENDIF ! nn_rhg_chkcvg
!------------------------------------------------------------------------------!
!
! --- Recompute delta, shear and div (inputs for mechanical redistribution)
!
!------------------------------------------------------------------------------!
!
DO_2D( nn_hls, nn_hls-1, nn_hls, nn_hls-1 ) ! 1->jpj-1; 1->jpi-1
! shear at F points
zds(ji,jj) = ( ( u_ice(ji,jj+1) * r1_e1u(ji,jj+1) - u_ice(ji,jj) * r1_e1u(ji,jj) ) * e1f(ji,jj) * e1f(ji,jj) &
& + ( v_ice(ji+1,jj) * r1_e2v(ji+1,jj) - v_ice(ji,jj) * r1_e2v(ji,jj) ) * e2f(ji,jj) * e2f(ji,jj) &
& ) * r1_e1e2f(ji,jj) * fimask(ji,jj)
END_2D
DO_2D( 0, 0, 0, 0 ) ! 2->jpj-1; 2->jpi-1
! shear**2 at T points (doc eq. A16)
! (brackets added to fix the order of floating point operations for the North Pole reproducibility
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)
! tension**2 at T points
zdt = ( ( u_ice(ji,jj) * r1_e2u(ji,jj) - u_ice(ji-1,jj) * r1_e2u(ji-1,jj) ) * e2t(ji,jj) * e2t(ji,jj) &
& - ( v_ice(ji,jj) * r1_e1v(ji,jj) - v_ice(ji,jj-1) * r1_e1v(ji,jj-1) ) * e1t(ji,jj) * e1t(ji,jj) &
& ) * r1_e1e2t(ji,jj)
zdt2 = zdt * zdt
zten_i(ji,jj) = zdt * zmsk(ji,jj)
! maximum shear rate at T points (includes tension, output only)
pshear_i(ji,jj) = SQRT( zdt2 + zds2 ) * zmsk(ji,jj)
! shear at T-points
zshear(ji,jj) = SQRT( zds2 ) * zmsk(ji,jj)
! divergence at T points
! (brackets added to fix the order of floating point operations for the North Pole reproducibility
pdivu_i(ji,jj) = ( ( e2u(ji,jj) * u_ice(ji,jj) - e2u(ji-1,jj) * u_ice(ji-1,jj) ) &
& + ( e1v(ji,jj) * v_ice(ji,jj) - e1v(ji,jj-1) * v_ice(ji,jj-1) ) &
& ) * r1_e1e2t(ji,jj) * zmsk(ji,jj)
! delta at T points
zfac = SQRT( pdivu_i(ji,jj) * pdivu_i(ji,jj) + ( zdt2 + zds2 ) * z1_ecc2 ) * zmsk(ji,jj) ! delta
zdelta(ji,jj) = zfac
! delta* at T points
rswitch = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zfac ) ) ! 0 if delta=0
pdelta_i(ji,jj) = zfac + rn_creepl ! * rswitch
END_2D
CALL lbc_lnk( 'icedyn_rhg_vp', pshear_i, 'T', 1._wp, pdivu_i, 'T', 1._wp, pdelta_i, 'T', 1._wp, &
& zdelta , 'T', 1._wp, zten_i , 'T', 1._wp, zshear , 'T', 1._wp )
! --- Sea ice stresses at T-points --- !
IF ( iom_use('normstr') .OR. iom_use('sheastr') .OR. &
& iom_use('intstrx') .OR. iom_use('intstry') .OR. &
& iom_use('sig1_pnorm') .OR. iom_use('sig2_pnorm') ) THEN
! sigma1, sigma2, sigma12 are some recombination of the stresses (HD MWR002, Bouillon et al., OM2013)
! not to be confused with stress tensor components, stress invariants, or stress principal components
DO_2D( nn_hls, nn_hls, nn_hls, nn_hls ) ! 2->jpj-1; 2->jpi-1
zvisc_t(ji,jj) = strength(ji,jj) / pdelta_i(ji,jj) ! update viscosity
zfac = zvisc_t(ji,jj)
zs1(ji,jj) = zfac * ( pdivu_i(ji,jj) - zdelta(ji,jj) )
zs2(ji,jj) = zfac * z1_ecc2 * zten_i(ji,jj)
zs12(ji,jj) = zfac * z1_ecc2 * zshear(ji,jj) * 0.5_wp
END_2D
!!$ CALL lbc_lnk( 'icedyn_rhg_vp', zs1, 'T', 1., zs2, 'T', 1., zs12, 'T', 1. )
ENDIF
! --- Shear (s12) at F-points --- !
IF ( iom_use('intstrx') .OR. iom_use('intstry') ) THEN
DO_2D( nn_hls, nn_hls-1, nn_hls, nn_hls-1 ) ! 1->jpj-1; 1->jpi-1
! P/delta* at F points
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) )
! s12 at F-points
zs12f(ji,jj) = zvisc_f * z1_ecc2 * zds(ji,jj)
END_2D
CALL lbc_lnk( 'icedyn_rhg_vp', zs12f, 'F', 1. )
ENDIF
!------------------------------------------------------------------------------!
!
! --- Diagnostics
!
!------------------------------------------------------------------------------!
!
! --- Ice-ocean, ice-atm. & ice-ocean bottom (landfast) stresses --- !
IF( iom_use('utau_oi') .OR. iom_use('vtau_oi') .OR. iom_use('utau_ai') .OR. iom_use('vtau_ai') .OR. &
& iom_use('utau_bi') .OR. iom_use('vtau_bi') ) THEN
ALLOCATE( ztaux_oi(jpi,jpj) , ztauy_oi(jpi,jpj) )
!--- Recalculate oceanic stress at last inner iteration
DO_2D( nn_hls-1, nn_hls-1, nn_hls-1, nn_hls-1 ) ! 2->jpj-1; 2->jpi-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 the North Pole reproducibility
zu_cV = 0.25_wp * ( ( u_ice(ji,jj) + u_ice(ji-1,jj) ) + ( u_ice(ji,jj+1) + u_ice(ji-1,jj+1) ) ) * vmask(ji,jj,1)
zv_cU = 0.25_wp * ( ( v_ice(ji,jj) + v_ice(ji,jj-1) ) + ( v_ice(ji+1,jj) + v_ice(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( ( u_ice(ji,jj) - u_oce (ji,jj) ) * ( u_ice(ji,jj) - u_oce (ji,jj) ) &
& + ( zv_cU - v_oceU(ji,jj) ) * ( zv_cU - v_oceU(ji,jj) ) )
zCwV(ji,jj) = za_iV(ji,jj) * zrhoco * SQRT( ( v_ice(ji,jj) - v_oce (ji,jj) ) * ( v_ice(ji,jj) - v_oce (ji,jj) ) &
& + ( zu_cV - u_oceV(ji,jj) ) * ( zu_cV - u_oceV(ji,jj) ) )
!--- Ocean-ice stress
ztaux_oi(ji,jj) = zCwU(ji,jj) * ( u_oce(ji,jj) - u_ice(ji,jj) )
ztauy_oi(ji,jj) = zCwV(ji,jj) * ( v_oce(ji,jj) - v_ice(ji,jj) )
END_2D
!
CALL lbc_lnk( 'icedyn_rhg_vp', ztaux_oi, 'U', -1., ztauy_oi, 'V', -1., ztaux_ai, 'U', -1., ztauy_ai, 'V', -1. ) !, &
! & ztaux_bi, 'U', -1., ztauy_bi, 'V', -1. )
!
CALL iom_put( 'utau_oi' , ztaux_oi * zmsk00 )
CALL iom_put( 'vtau_oi' , ztauy_oi * zmsk00 )
CALL iom_put( 'utau_ai' , ztaux_ai * zmsk00 )
CALL iom_put( 'vtau_ai' , ztauy_ai * zmsk00 )
! CALL iom_put( 'utau_bi' , ztaux_bi * zmsk00 )
! CALL iom_put( 'vtau_bi' , ztauy_bi * zmsk00 )
DEALLOCATE( ztaux_oi , ztauy_oi )
ENDIF
! --- Divergence, shear and strength --- !
IF( iom_use('icediv') ) CALL iom_put( 'icediv' , pdivu_i * zmsk00 ) ! divergence
IF( iom_use('iceshe') ) CALL iom_put( 'iceshe' , pshear_i * zmsk00 ) ! maximum shear rate
IF( iom_use('icedlt') ) CALL iom_put( 'icedlt' , zdelta * zmsk00 ) ! delta
IF( iom_use('icestr') ) CALL iom_put( 'icestr' , strength * zmsk00 ) ! strength
! --- Stress tensor invariants (SIMIP diags) --- !
IF( iom_use('normstr') .OR. iom_use('sheastr') ) THEN
!
ALLOCATE( zsig_I(jpi,jpj) , zsig_II(jpi,jpj) )
!
! Stress invariants (sigma_I, sigma_II, Coon 1974, Feltham 2008)
DO_2D( nn_hls, nn_hls, nn_hls, nn_hls ) ! 2->jpj-1; 2->jpi-1
zsig_I(ji,jj) = 0.5_wp * zs1(ji,jj)
zsig_II(ji,jj) = 0.5_wp * SQRT ( zs2(ji,jj) * zs2(ji,jj) + 4. * zs12(ji,jj) * zs12(ji,jj) )
END_2D
!!$ CALL lbc_lnk( 'icedyn_rhg_vp', zsig_I, 'T', 1., zsig_II, 'T', 1.)
IF( iom_use('normstr') ) CALL iom_put( 'normstr' , zsig_I(:,:) * zmsk00(:,:) ) ! Normal stress
IF( iom_use('sheastr') ) CALL iom_put( 'sheastr' , zsig_II(:,:) * zmsk00(:,:) ) ! Maximum shear stress
DEALLOCATE ( zsig_I, zsig_II )
ENDIF
! --- Normalized stress tensor principal components --- !
! These are used to plot the normalized yield curve (Lemieux & Dupont, GMD 2020)
! To plot the yield curve and evaluate physical convergence, they have two recommendations
! Recommendation 1 : Use ice strength, not replacement pressure
! Recommendation 2 : Need to use deformations at PREVIOUS iterate for viscosities (see p. 1765)
! R2 means we need to recompute stresses
IF( iom_use('sig1_pnorm') .OR. iom_use('sig2_pnorm') ) THEN
!
ALLOCATE( zsig1_p(jpi,jpj) , zsig2_p(jpi,jpj) , zsig_I(jpi,jpj) , zsig_II(jpi,jpj) )
!
DO_2D( 0, 0, 0, 0 ) ! clem: check bounds
! Ice stresses computed with **viscosities** (P/delta) at **previous** iterates
! and **deformations** at current iterates
! following Lemieux & Dupont (2020)
zfac = zvisc_t_prev(ji,jj)
zsig1 = zfac * ( pdivu_i(ji,jj) - zdelta(ji,jj) )
zsig2 = zfac * z1_ecc2 * zten_i(ji,jj)
zsig12 = zfac * z1_ecc2 * zshear(ji,jj) * 0.5_wp
! Stress invariants (sigma_I, sigma_II, Coon 1974, Feltham 2008), T-point
zsig_I(ji,jj) = 0.5_wp * zsig1 ! normal stress
zsig_II(ji,jj) = 0.5_wp * SQRT ( zsig2 * zsig2 + 4. *zsig12 * zsig12 ) ! max shear stress
! Normalized principal stresses (used to display the ellipse)
z1_strength = 1._wp / MAX ( 1._wp , strength(ji,jj) )
zsig1_p(ji,jj) = ( zsig_I(ji,jj) + zsig_II(ji,jj) ) * z1_strength
zsig2_p(ji,jj) = ( zsig_I(ji,jj) - zsig_II(ji,jj) ) * z1_strength
END_2D
!
! CALL lbc_lnk( 'icedyn_rhg_vp', zsig1_p, 'T', 1., zsig2_p, 'T', 1.)
!
CALL iom_put( 'sig1_pnorm' , zsig1_p )
CALL iom_put( 'sig2_pnorm' , zsig2_p )
DEALLOCATE( zsig1_p , zsig2_p , zsig_I , zsig_II )
ENDIF
! --- SIMIP, terms of tendency for momentum equation --- !
IF( iom_use('dssh_dx') .OR. iom_use('dssh_dy') .OR. &
& iom_use('corstrx') .OR. iom_use('corstry') ) THEN
! --- Recalculate Coriolis stress at last inner iteration
DO_2D( 0, 0, 0, 0 ) ! clem: check bounds
! --- U-component
zCorU(ji,jj) = 0.25_wp * r1_e1u(ji,jj) * &
& ( zmf(ji ,jj) * ( e1v(ji ,jj) * v_ice(ji ,jj) + e1v(ji ,jj-1) * v_ice(ji ,jj-1) ) &
& + zmf(ji+1,jj) * ( e1v(ji+1,jj) * v_ice(ji+1,jj) + e1v(ji+1,jj-1) * v_ice(ji+1,jj-1) ) )
zCorV(ji,jj) = - 0.25_wp * r1_e2v(ji,jj) * &
& ( zmf(ji,jj ) * ( e2u(ji,jj ) * u_ice(ji,jj ) + e2u(ji-1,jj ) * u_ice(ji-1,jj ) ) &
& + zmf(ji,jj+1) * ( e2u(ji,jj+1) * u_ice(ji,jj+1) + e2u(ji-1,jj+1) * u_ice(ji-1,jj+1) ) )
END_2D
!
CALL lbc_lnk( 'icedyn_rhg_vp', zspgU, 'U', -1., zspgV, 'V', -1., &
& zCorU, 'U', -1., zCorV, 'V', -1. )
!
CALL iom_put( 'dssh_dx' , zspgU * zmsk00 ) ! Sea-surface tilt term in force balance (x)
CALL iom_put( 'dssh_dy' , zspgV * zmsk00 ) ! Sea-surface tilt term in force balance (y)
CALL iom_put( 'corstrx' , zCorU * zmsk00 ) ! Coriolis force term in force balance (x)
CALL iom_put( 'corstry' , zCorV * zmsk00 ) ! Coriolis force term in force balance (y)
ENDIF
IF ( iom_use('intstrx') .OR. iom_use('intstry') ) THEN
! Recalculate internal forces (divergence of stress tensor) at last inner iteration
DO_2D( 0, 0, 0, 0 ) ! clem: check bounds
zfU(ji,jj) = 0.5_wp * ( ( zs1(ji+1,jj) - zs1(ji,jj) ) * e2u(ji,jj) &
& + ( zs2(ji+1,jj) * e2t(ji+1,jj) * e2t(ji+1,jj) - zs2(ji,jj) * e2t(ji,jj) * e2t(ji,jj) &
& ) * r1_e2u(ji,jj) &
& + ( zs12f(ji,jj) * e1f(ji,jj) * e1f(ji,jj) - zs12f(ji,jj-1) * e1f(ji,jj-1) * e1f(ji,jj-1) &
& ) * 2._wp * r1_e1u(ji,jj) &
& ) * r1_e1e2u(ji,jj)
zfV(ji,jj) = 0.5_wp * ( ( zs1(ji,jj+1) - zs1(ji,jj) ) * e1v(ji,jj) &
& - ( zs2(ji,jj+1) * e1t(ji,jj+1) * e1t(ji,jj+1) - zs2(ji,jj) * e1t(ji,jj) * e1t(ji,jj) &
& ) * r1_e1v(ji,jj) &
& + ( zs12f(ji,jj) * e2f(ji,jj) * e2f(ji,jj) - zs12f(ji-1,jj) * e2f(ji-1,jj) * e2f(ji-1,jj) &
& ) * 2._wp * r1_e2v(ji,jj) &
& ) * r1_e1e2v(ji,jj)
END_2D
CALL lbc_lnk( 'icedyn_rhg_vp', zfU, 'U', -1., zfV, 'V', -1. )
CALL iom_put( 'intstrx' , zfU * zmsk00 ) ! Internal force term in force balance (x)
CALL iom_put( 'intstry' , zfV * zmsk00 ) ! Internal force term in force balance (y)
ENDIF
! --- Ice & snow mass and ice area transports
IF( iom_use('xmtrpice') .OR. iom_use('ymtrpice') .OR. &
& iom_use('xmtrpsnw') .OR. iom_use('ymtrpsnw') .OR. iom_use('xatrp') .OR. iom_use('yatrp') ) THEN
!
ALLOCATE( zdiag_xmtrp_ice(jpi,jpj) , zdiag_ymtrp_ice(jpi,jpj) , &
& zdiag_xmtrp_snw(jpi,jpj) , zdiag_ymtrp_snw(jpi,jpj) , zdiag_xatrp(jpi,jpj) , zdiag_yatrp(jpi,jpj) )
!
DO_2D( 0, 0, 0, 0 ) ! clem: check bounds
zfac_x = 0.5 * u_ice(ji,jj) * e2u(ji,jj) * zmsk00(ji,jj)
zfac_y = 0.5 * v_ice(ji,jj) * e1v(ji,jj) * zmsk00(ji,jj)
zdiag_xmtrp_ice(ji,jj) = rhoi * zfac_x * ( vt_i(ji+1,jj) + vt_i(ji,jj) ) ! ice mass transport, X-component
zdiag_ymtrp_ice(ji,jj) = rhoi * zfac_y * ( vt_i(ji,jj+1) + vt_i(ji,jj) ) ! '' Y- ''
zdiag_xmtrp_snw(ji,jj) = rhos * zfac_x * ( vt_s(ji+1,jj) + vt_s(ji,jj) ) ! snow mass transport, X-component
zdiag_ymtrp_snw(ji,jj) = rhos * zfac_y * ( vt_s(ji,jj+1) + vt_s(ji,jj) ) ! '' Y- ''
zdiag_xatrp(ji,jj) = zfac_x * ( at_i(ji+1,jj) + at_i(ji,jj) ) ! area transport, X-component
zdiag_yatrp(ji,jj) = zfac_y * ( at_i(ji,jj+1) + at_i(ji,jj) ) ! '' Y- ''
END_2D
CALL lbc_lnk( 'icedyn_rhg_vp', zdiag_xmtrp_ice, 'U', -1., zdiag_ymtrp_ice, 'V', -1., &
& zdiag_xmtrp_snw, 'U', -1., zdiag_ymtrp_snw, 'V', -1., &
& zdiag_xatrp , 'U', -1., zdiag_yatrp , 'V', -1. )
CALL iom_put( 'xmtrpice' , zdiag_xmtrp_ice ) ! X-component of sea-ice mass transport (kg/s)
CALL iom_put( 'ymtrpice' , zdiag_ymtrp_ice ) ! Y-component of sea-ice mass transport
CALL iom_put( 'xmtrpsnw' , zdiag_xmtrp_snw ) ! X-component of snow mass transport (kg/s)
CALL iom_put( 'ymtrpsnw' , zdiag_ymtrp_snw ) ! Y-component of snow mass transport
CALL iom_put( 'xatrp' , zdiag_xatrp ) ! X-component of ice area transport
CALL iom_put( 'yatrp' , zdiag_yatrp ) ! Y-component of ice area transport
DEALLOCATE( zdiag_xmtrp_ice , zdiag_ymtrp_ice , &
& zdiag_xmtrp_snw , zdiag_ymtrp_snw , zdiag_xatrp , zdiag_yatrp )
ENDIF
END SUBROUTINE ice_dyn_rhg_vp
SUBROUTINE rhg_cvg_vp( kt, kitout, kitinn, kitinntot, kitoutmax, kitinnmax, kitinntotmax , &
& pu, pv, pub, pvb, pub_outer, pvb_outer , &
& pmt, pat_iu, pat_iv, puerr_max, pverr_max, pglob_area , &
& prhsu, pAU, pBU, pCU, pDU, pEU, pFU , &
& prhsv, pAV, pBV, pCV, pDV, pEV, pFV , &
& pvel_res, pvel_diff )
!!
!!----------------------------------------------------------------------
!! *** ROUTINE rhg_cvg_vp ***
!!
!! ** Purpose : check convergence of VP ice rheology
!!
!! ** Method : create a file ice_cvg.nc containing a few convergence diagnostics
!! This routine is called every sub-iteration, so it is cpu expensive
!!
!! Calculates / stores
!! - maximum absolute U-V difference (uice_cvg, u_dif, v_dif, m/s)
!! - residuals in U, V and UV-mean taken as square-root of area-weighted mean square residual (u_res, v_res, vel_res, N/m2)
!! - mean kinetic energy (mke_ice, J/m2)
!!
!! ** Note : for the first sub-iteration, uice_cvg is set to 0 (too large otherwise)
!!
!!----------------------------------------------------------------------
!!
INTEGER , INTENT(in) :: kt, kitout, kitinn, kitinntot ! ocean model iterate, outer, inner and total n-iterations
INTEGER , INTENT(in) :: kitoutmax, kitinnmax ! max number of outer & inner iterations
INTEGER , INTENT(in) :: kitinntotmax ! max number of total sub-iterations
REAL(wp), DIMENSION(:,:), INTENT(in) :: pu, pv, pub, pvb ! now & sub-iter-before velocities
REAL(wp), DIMENSION(:,:), INTENT(in) :: pub_outer, pvb_outer ! velocities @before outer iterations
REAL(wp), DIMENSION(:,:), INTENT(in) :: pmt, pat_iu, pat_iv ! mass at T-point, ice concentration at U&V
REAL(wp), INTENT(in) :: puerr_max, pverr_max ! absolute mean velocity difference
REAL(wp), INTENT(in) :: pglob_area ! global ice area
REAL(wp), DIMENSION(:,:), INTENT(in) :: prhsu, pAU, pBU, pCU, pDU, pEU, pFU ! linear system coefficients
REAL(wp), DIMENSION(:,:), INTENT(in) :: prhsv, pAV, pBV, pCV, pDV, pEV, pFV
REAL(wp), DIMENSION(:,:), INTENT(inout) :: pvel_res ! velocity residual @last inner iteration
REAL(wp), DIMENSION(:,:), INTENT(inout) :: pvel_diff ! velocity difference @last outer iteration
!!
INTEGER :: idtime, istatus, ix_dim, iy_dim
INTEGER :: ji, jj ! dummy loop indices
INTEGER :: it_inn_file, it_out_file
REAL(wp) :: zu_res_mean, zv_res_mean, zvel_res_mean ! mean residuals of the linear system
REAL(wp) :: zu_mad, zv_mad, zvel_mad ! mean absolute deviation, sub-iterates
REAL(wp) :: zu_mad_outer, zv_mad_outer, zvel_mad_outer ! mean absolute deviation, outer-iterates
REAL(wp) :: zvel_err_max, zmke, zu, zv ! local scalars
REAL(wp) :: z1_pglob_area ! inverse global ice area
REAL(wp), DIMENSION(jpi,jpj) :: zu_res, zv_res, zvel2 ! local arrays
REAL(wp), DIMENSION(jpi,jpj) :: zu_diff, zv_diff ! local arrays
CHARACTER(len=20) :: clname
!!----------------------------------------------------------------------
IF( lwp ) THEN
WRITE(numout,*)
WRITE(numout,*) 'rhg_cvg_vp : ice rheology convergence control'
WRITE(numout,*) '~~~~~~~~~~~'
WRITE(numout,*) ' kt = : ', kt
WRITE(numout,*) ' kitout = : ', kitout
WRITE(numout,*) ' kitinn = : ', kitinn
WRITE(numout,*) ' kitinntot = : ', kitinntot
WRITE(numout,*) ' kitoutmax (nn_vp_nout) = ', kitoutmax
WRITE(numout,*) ' kitinnmax (nn_vp_ninn) = ', kitinnmax
WRITE(numout,*) ' kitinntotmax (nn_nvp) = ', kitinntotmax
WRITE(numout,*)
ENDIF
z1_pglob_area = 1._wp / pglob_area ! inverse global ice area
! create file
IF( kt == nit000 .AND. kitinntot == 1 ) THEN
!
IF( lwm ) THEN
clname = 'ice_cvg.nc'
IF( .NOT. Agrif_Root() ) clname = TRIM(Agrif_CFixed())//"_"//TRIM(clname)
istatus = NF90_CREATE( TRIM(clname), NF90_CLOBBER, ncvgid )
istatus = NF90_DEF_DIM( ncvgid, 'time' , NF90_UNLIMITED, idtime )
istatus = NF90_DEF_DIM( ncvgid, 'x' , jpi, ix_dim )
istatus = NF90_DEF_DIM( ncvgid, 'y' , jpj, iy_dim )
istatus = NF90_DEF_VAR( ncvgid, 'u_res' , NF90_DOUBLE , (/ idtime /), nvarid_ures )
istatus = NF90_DEF_VAR( ncvgid, 'v_res' , NF90_DOUBLE , (/ idtime /), nvarid_vres )
istatus = NF90_DEF_VAR( ncvgid, 'vel_res' , NF90_DOUBLE , (/ idtime /), nvarid_velres )
istatus = NF90_DEF_VAR( ncvgid, 'uerr_max_sub' , NF90_DOUBLE , (/ idtime /), nvarid_uerr_max )
istatus = NF90_DEF_VAR( ncvgid, 'verr_max_sub' , NF90_DOUBLE , (/ idtime /), nvarid_verr_max )
istatus = NF90_DEF_VAR( ncvgid, 'velerr_max_sub', NF90_DOUBLE , (/ idtime /), nvarid_velerr_max )
istatus = NF90_DEF_VAR( ncvgid, 'umad_sub' , NF90_DOUBLE , (/ idtime /), nvarid_umad )
istatus = NF90_DEF_VAR( ncvgid, 'vmad_sub' , NF90_DOUBLE , (/ idtime /), nvarid_vmad )
istatus = NF90_DEF_VAR( ncvgid, 'velmad_sub' , NF90_DOUBLE , (/ idtime /), nvarid_velmad )
istatus = NF90_DEF_VAR( ncvgid, 'umad_outer' , NF90_DOUBLE , (/ idtime /), nvarid_umad_outer )
istatus = NF90_DEF_VAR( ncvgid, 'vmad_outer' , NF90_DOUBLE , (/ idtime /), nvarid_vmad_outer )
istatus = NF90_DEF_VAR( ncvgid, 'velmad_outer' , NF90_DOUBLE , (/ idtime /), nvarid_velmad_outer )
istatus = NF90_DEF_VAR( ncvgid, 'mke_ice', NF90_DOUBLE , (/ idtime /), nvarid_mke )
istatus = NF90_ENDDEF(ncvgid)
ENDIF
!
ENDIF
!------------------------------------------------------------
!
! Max absolute velocity difference with previous sub-iterate
! ( zvel_err_max )
!
!------------------------------------------------------------
!
! This comes from the criterion used to stop the iterative procedure
zvel_err_max = 0.5_wp * ( puerr_max + pverr_max ) ! average of U- and V- maximum error over the whole domain
!----------------------------------------------
!
! Mean-absolute-deviation (sub-iterates)
! ( zu_mad, zv_mad, zvel_mad)
!
!----------------------------------------------
!
! U
DO_2D( 0, 0, 0, 0 ) !clem check bounds
zu_diff(ji,jj) = ABS ( ( pu(ji,jj) - pub(ji,jj) ) ) * e1e2u(ji,jj) * pat_iu(ji,jj) * umask(ji,jj,1) * z1_pglob_area
zv_diff(ji,jj) = ABS ( ( pv(ji,jj) - pvb(ji,jj) ) ) * e1e2v(ji,jj) * pat_iv(ji,jj) * vmask(ji,jj,1) * z1_pglob_area
END_2D
! global sum & U-V average
zu_mad = glob_sum( 'icedyn_rhg_vp : ', zu_diff )
zv_mad = glob_sum( 'icedyn_rhg_vp : ', zv_diff )
zvel_mad = 0.5_wp * ( zu_mad + zv_mad )
!-----------------------------------------------
!
! Mean-absolute-deviation (outer-iterates)
! ( zu_mad_outer, zv_mad_outer, zvel_mad_outer)
!
!-----------------------------------------------
!
IF ( kitinn == kitinnmax ) THEN ! only work at the end of outer iterates
DO_2D( 0, 0, 0, 0 ) !clem check bounds
zu_diff(ji,jj) = ABS ( ( pu(ji,jj) - pub_outer(ji,jj) ) ) * e1e2u(ji,jj) * pat_iu(ji,jj) * umask(ji,jj,1) * &
& z1_pglob_area
zv_diff(ji,jj) = ABS ( ( pv(ji,jj) - pvb_outer(ji,jj) ) ) * e1e2v(ji,jj) * pat_iv(ji,jj) * vmask(ji,jj,1) * &
& z1_pglob_area
END_2D
! Global ice-concentration, grid-cell-area weighted mean
zu_mad_outer = glob_sum( 'icedyn_rhg_vp : ', zu_diff )
zv_mad_outer = glob_sum( 'icedyn_rhg_vp : ', zv_diff )
! Average of both U & V
zvel_mad_outer = 0.5_wp * ( zu_mad_outer + zv_mad_outer )
ENDIF
! --- Spatially-resolved absolute difference to send back to main routine
! (last iteration only, T-point)
IF ( kitinntot == kitinntotmax) THEN
DO_2D( 0, 0, 0, 0 ) !clem check bounds
zu_diff(ji,jj) = ( ABS ( ( pu(ji-1,jj) - pub_outer(ji-1,jj) ) ) * umask(ji-1,jj,1) &
& + ABS ( ( pu(ji,jj ) - pub_outer(ji,jj) ) ) * umask(ji,jj,1) ) &
& / ( umask(ji-1,jj,1) + umask(ji,jj,1) )
zv_diff(ji,jj) = ( ABS ( ( pv(ji,jj-1) - pvb_outer(ji,jj-1) ) ) * vmask(ji,jj-1,1) &
& + ABS ( ( pv(ji,jj ) - pvb_outer(ji,jj) ) ) * vmask(ji,jj,1) &
& / ( vmask(ji,jj-1,1) + vmask(ji,jj,1) ) )
pvel_diff(ji,jj) = 0.5_wp * ( zu_diff(ji,jj) + zv_diff(ji,jj) )
END_2D
CALL lbc_lnk( 'icedyn_rhg_cvg_vp', pvel_diff, 'T', 1._wp )
ELSE
pvel_diff(:,:) = 0._wp
ENDIF
!---------------------------------------
!
! --- Mean residual & kinetic energy
!
!---------------------------------------
IF ( kitinntot == 1 ) THEN
zu_res_mean = 0._wp
zv_res_mean = 0._wp
zvel_res_mean = 0._wp
zmke = 0._wp
ELSE
! * Mean residual (N/m2)
! Here we take the residual of the linear system (N/m2),
! We define it as in mitgcm: global area-weighted mean of square-root residual
! Local residual r = Ax - B expresses to which extent the momentum balance is verified
! i.e., how close we are to a solution
DO_2D( 0, 0, 0, 0 ) !clem check bounds
! (brackets added to fix the order of floating point operations for the North Pole reproducibility
zu_res(ji,jj) = prhsu(ji,jj) + ( pDU(ji,jj) * pu(ji ,jj-1) + pEU(ji,jj) * pu(ji ,jj+1) ) &
& - pBU(ji,jj) * pu(ji,jj) - ( pAU(ji,jj) * pu(ji-1,jj ) + pCU(ji,jj) * pu(ji+1,jj ) )
zv_res(ji,jj) = prhsv(ji,jj) + ( pDV(ji,jj) * pv(ji-1,jj ) + pEV(ji,jj) * pv(ji+1,jj ) ) &
& - pBV(ji,jj) * pv(ji,jj) - ( pAV(ji,jj) * pv(ji ,jj-1) + pCV(ji,jj) * pv(ji ,jj+1) )
! zu_res(ji,jj) = pFU(ji,jj) - pAU(ji,jj) * pu(ji-1,jj) - pBU(ji,jj) * pu(ji,jj) - pCU(ji,jj) * pu(ji+1,jj)
! zv_res(ji,jj) = pFV(ji,jj) - pAV(ji,jj) * pv(ji,jj-1) - pBV(ji,jj) * pv(ji,jj) - pCV(ji,jj) * pv(ji,jj+1)
zu_res(ji,jj) = SQRT( zu_res(ji,jj) * zu_res(ji,jj) ) * umask(ji,jj,1) * pat_iu(ji,jj) * e1e2u(ji,jj) * z1_pglob_area
zv_res(ji,jj) = SQRT( zv_res(ji,jj) * zv_res(ji,jj) ) * vmask(ji,jj,1) * pat_iv(ji,jj) * e1e2v(ji,jj) * z1_pglob_area
END_2D
! Global ice-concentration, grid-cell-area weighted mean
zu_res_mean = glob_sum( 'ice_rhg_vp', zu_res(:,:) )
zv_res_mean = glob_sum( 'ice_rhg_vp', zv_res(:,:) )
zvel_res_mean = 0.5_wp * ( zu_res_mean + zv_res_mean )
! --- Global mean kinetic energy per unit area (J/m2)
zvel2(:,:) = 0._wp
DO_2D( 0, 0, 0, 0 ) !clem check bounds
zu = 0.5_wp * ( pu(ji-1,jj) + pu(ji,jj) ) ! u-vel at T-point
zv = 0.5_wp * ( pv(ji,jj-1) + pv(ji,jj) )
zvel2(ji,jj) = zu*zu + zv*zv ! square of ice velocity at T-point
END_2D
zmke = 0.5_wp * glob_sum( 'ice_rhg_vp', pmt(:,:) * e1e2t(:,:) * zvel2(:,:) ) / pglob_area
ENDIF ! kitinntot
!--- Spatially-resolved residual at last iteration to send back to main routine (last iteration only)
!--- Calculation @T-point
IF ( kitinntot == kitinntotmax) THEN
DO_2D( nn_hls-1, nn_hls-1, nn_hls-1, nn_hls-1 )
! (brackets added to fix the order of floating point operations for the North Pole reproducibility
zu_res(ji,jj) = prhsu(ji,jj) + ( pDU(ji,jj) * pu(ji ,jj-1) + pEU(ji,jj) * pu(ji ,jj+1) ) &
& - pBU(ji,jj) * pu(ji,jj) - ( pAU(ji,jj) * pu(ji-1,jj ) + pCU(ji,jj) * pu(ji+1,jj ) )
zv_res(ji,jj) = prhsv(ji,jj) + ( pDV(ji,jj) * pv(ji-1,jj ) + pEV(ji,jj) * pv(ji+1,jj ) ) &
& - pBV(ji,jj) * pv(ji,jj) - ( pAV(ji,jj) * pv(ji ,jj-1) + pCV(ji,jj) * pv(ji ,jj+1) )
zu_res(ji,jj) = SQRT( zu_res(ji,jj) * zu_res(ji,jj) ) * umask(ji,jj,1)
zv_res(ji,jj) = SQRT( zv_res(ji,jj) * zv_res(ji,jj) ) * vmask(ji,jj,1)
END_2D
IF( nn_hls == 1 ) CALL lbc_lnk( 'icedyn_rhg_cvg_vp', zu_res, 'U', 1., zv_res , 'V', 1. )
DO_2D( 0, 0, 0, 0 ) !clem check bounds
pvel_res(ji,jj) = 0.25_wp * ( ( zu_res(ji-1,jj) + zu_res(ji,jj) ) + ( zv_res(ji,jj-1) + zv_res(ji,jj) ) )
END_2D
CALL lbc_lnk( 'icedyn_rhg_cvg_vp', pvel_res, 'T', 1. )
ELSE
pvel_res(:,:) = 0._wp
ENDIF
! ! ==================== !
it_inn_file = ( kt - nit000 ) * kitinntotmax + kitinntot ! time step in the file
it_out_file = ( kt - nit000 ) * kitoutmax + kitout
! write variables
IF( lwm ) THEN
istatus = NF90_PUT_VAR( ncvgid, nvarid_ures , (/zu_res_mean/), (/it_inn_file/), (/1/) ) ! Residuals of the linear system, area weighted mean
istatus = NF90_PUT_VAR( ncvgid, nvarid_vres , (/zv_res_mean/), (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_velres, (/zvel_res_mean/), (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_uerr_max , (/puerr_max/), (/it_inn_file/), (/1/) ) ! Max velocit_inn_filey error, sub-it_inn_fileerates
istatus = NF90_PUT_VAR( ncvgid, nvarid_verr_max , (/pverr_max/), (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_velerr_max, (/zvel_err_max/), (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_umad , (/zu_mad/) , (/it_inn_file/), (/1/) ) ! velocit_inn_filey MAD, area/sic-weighted, sub-it_inn_fileerates
istatus = NF90_PUT_VAR( ncvgid, nvarid_vmad , (/zv_mad/) , (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_velmad , (/zvel_mad/), (/it_inn_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_mke, (/zmke/), (/kitinntot/), (/1/) ) ! mean kinetic energy
IF ( kitinn == kitinnmax ) THEN ! only print outer mad at the end of inner loop
istatus = NF90_PUT_VAR( ncvgid, nvarid_umad_outer , (/zu_mad_outer/) , (/it_out_file/), (/1/) ) ! velocity MAD, area/sic-weighted, outer-iterates
istatus = NF90_PUT_VAR( ncvgid, nvarid_vmad_outer , (/zv_mad_outer/) , (/it_out_file/), (/1/) ) !
istatus = NF90_PUT_VAR( ncvgid, nvarid_velmad_outer , (/zvel_mad_outer/), (/it_out_file/), (/1/) ) !
ENDIF
IF( kt == nitend - nn_fsbc + 1 .AND. kitinntot == kitinntotmax ) istatus = NF90_CLOSE( ncvgid )
ENDIF
END SUBROUTINE rhg_cvg_vp
#else
!!----------------------------------------------------------------------
!! Default option Empty module NO SI3 sea-ice model
!!----------------------------------------------------------------------
#endif
!!==============================================================================
END MODULE icedyn_rhg_vp