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MODULE icedyn_rhg_eap
!!======================================================================
!! *** MODULE icedyn_rhg_eap ***
!! Sea-Ice dynamics : rheology Elasto-Viscous-Plastic
!!======================================================================
!! History : - ! 2007-03 (M.A. Morales Maqueda, S. Bouillon) Original code
!! 3.0 ! 2008-03 (M. Vancoppenolle) adaptation to new model
!! - ! 2008-11 (M. Vancoppenolle, S. Bouillon, Y. Aksenov) add surface tilt in ice rheolohy
!! 3.3 ! 2009-05 (G.Garric) addition of the evp case
!! 3.4 ! 2011-01 (A. Porter) dynamical allocation
!! 3.5 ! 2012-08 (R. Benshila) AGRIF
!! 3.6 ! 2016-06 (C. Rousset) Rewriting + landfast ice + mEVP (Bouillon 2013)
!! 3.7 ! 2017 (C. Rousset) add aEVP (Kimmritz 2016-2017)
!! 4.0 ! 2018 (many people) SI3 [aka Sea Ice cube]
!! ! 2019 (S. Rynders, Y. Aksenov, C. Rousset) change into eap rheology from
!! CICE code (Tsamados, Heorton)
!!----------------------------------------------------------------------
#if defined key_si3
!!----------------------------------------------------------------------
!! 'key_si3' SI3 sea-ice model
!!----------------------------------------------------------------------
!! ice_dyn_rhg_eap : computes ice velocities from EVP rheology
!! rhg_eap_rst : read/write EVP fields in ice restart
!!----------------------------------------------------------------------
USE phycst ! Physical constant
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_eap ! called by icedyn_rhg.F90
PUBLIC rhg_eap_rst ! called by icedyn_rhg.F90
REAL(wp), PARAMETER :: pphi = 3.141592653589793_wp/12._wp ! diamond shaped floe smaller angle (default phi = 30 deg)
! look-up table for calculating structure tensor
INTEGER, PARAMETER :: nx_yield = 41
INTEGER, PARAMETER :: ny_yield = 41
INTEGER, PARAMETER :: na_yield = 21
REAL(wp), DIMENSION(nx_yield, ny_yield, na_yield) :: s11r, s12r, s22r, s11s, s12s, s22s
REAL(wp), DIMENSION(:,:), ALLOCATABLE :: fimask ! mask at F points for the ice
!! for convergence tests
INTEGER :: ncvgid ! netcdf file id
INTEGER :: nvarid ! netcdf variable id
!! * Substitutions
# include "do_loop_substitute.h90"
# include "domzgr_substitute.h90"
!!----------------------------------------------------------------------
!! NEMO/ICE 4.0 , NEMO Consortium (2018)
!! $Id: icedyn_rhg_eap.F90 11536 2019-09-11 13:54:18Z smasson $
!! Software governed by the CeCILL license (see ./LICENSE)
!!----------------------------------------------------------------------
CONTAINS
SUBROUTINE ice_dyn_rhg_eap( kt, Kmm, pstress1_i, pstress2_i, pstress12_i, pshear_i, pdivu_i, pdelta_i, paniso_11, paniso_12, prdg_conv )
!!-------------------------------------------------------------------
!! *** SUBROUTINE ice_dyn_rhg_eap ***
!! EAP-C-grid
!!
!! ** purpose : determines sea ice drift from wind stress, ice-ocean
!! stress and sea-surface slope. Ice-ice interaction is described by
!! a non-linear elasto-anisotropic-plastic (EAP) law including shear
!! strength and a bulk rheology .
!!
!! The points in the C-grid look like this, dear reader
!!
!! (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 : - compute u_ice, v_ice : the components of the
!! sea-ice velocity vector
!! - compute delta_i, shear_i, divu_i, which are inputs
!! of the ice thickness distribution
!!
!! ** Steps : 0) compute mask at F point
!! 1) Compute ice snow mass, ice strength
!! 2) Compute wind, oceanic stresses, mass terms and
!! coriolis terms of the momentum equation
!! 3) Solve the momentum equation (iterative procedure)
!! 4) Recompute delta, shear and divergence
!! (which are inputs of the ITD) & store stress
!! for the next time step
!! 5) Diagnostics including charge ellipse
!!
!! ** Notes : There is the possibility to use aEVP from the nice work of Kimmritz et al. (2016 & 2017)
!! by setting up ln_aEVP=T (i.e. changing alpha and beta parameters).
!! This is an upgraded version of mEVP from Bouillon et al. 2013
!! (i.e. more stable and better convergence)
!!
!! References : Hunke and Dukowicz, JPO97
!! Bouillon et al., Ocean Modelling 2009
!! Bouillon et al., Ocean Modelling 2013
!! Kimmritz et al., Ocean Modelling 2016 & 2017
!!-------------------------------------------------------------------
INTEGER , INTENT(in ) :: kt ! time step
INTEGER , INTENT(in ) :: Kmm ! ocean time level index
REAL(wp), DIMENSION(:,:), INTENT(inout) :: pstress1_i, pstress2_i, pstress12_i !
REAL(wp), DIMENSION(:,:), INTENT( out) :: pshear_i , pdivu_i , pdelta_i !
REAL(wp), DIMENSION(:,:), INTENT(inout) :: paniso_11 , paniso_12 ! structure tensor components
REAL(wp), DIMENSION(:,:), INTENT(inout) :: prdg_conv ! for ridging
!!
INTEGER :: ji, jj ! dummy loop indices
INTEGER :: jter ! local integers
!
REAL(wp) :: zrhoco ! rau0 * rn_cio
REAL(wp) :: zdtevp, z1_dtevp ! time step for subcycling
REAL(wp) :: ecc2, z1_ecc2 ! square of yield ellipse eccenticity
REAL(wp) :: zalph1, z1_alph1, zalph2, z1_alph2 ! alpha coef from Bouillon 2009 or Kimmritz 2017
REAl(wp) :: zbetau, zbetav
REAL(wp) :: zm1, zm2, zm3, zmassU, zmassV, zvU, zvV ! ice/snow mass and volume
REAL(wp) :: zds2, zdt, zdt2, zdiv, zdiv2, zdsT ! temporary scalars
REAL(wp) :: zTauO, zTauB, zRHS, zvel ! temporary scalars
REAL(wp) :: zkt ! isotropic tensile strength for landfast ice
REAL(wp) :: zvCr ! critical ice volume above which ice is landfast
!
REAL(wp) :: zintb, zintn ! dummy argument
REAL(wp) :: zfac_x, zfac_y
REAL(wp) :: zshear, zdum1, zdum2
REAL(wp) :: zstressptmp, zstressmtmp, zstress12tmpF ! anisotropic stress tensor components
REAL(wp) :: zalphar, zalphas ! for mechanical redistribution
REAL(wp) :: zmresult11, zmresult12, z1dtevpkth, zp5kth, z1_dtevp_A ! for structure tensor evolution
!
REAL(wp), DIMENSION(jpi,jpj) :: zstress12tmp ! anisotropic stress tensor component for regridding
REAL(wp), DIMENSION(jpi,jpj) :: zyield11, zyield22, zyield12 ! yield surface tensor for history
REAL(wp), DIMENSION(jpi,jpj) :: zdelta, zp_delt ! delta and P/delta at T points
REAL(wp), DIMENSION(jpi,jpj) :: zten_i ! tension
REAL(wp), DIMENSION(jpi,jpj) :: zbeta ! beta coef from Kimmritz 2017
!
REAL(wp), DIMENSION(jpi,jpj) :: zdt_m ! (dt / ice-snow_mass) on T points
REAL(wp), DIMENSION(jpi,jpj) :: zaU , zaV ! ice fraction on U/V points
REAL(wp), DIMENSION(jpi,jpj) :: zmU_t, zmV_t ! (ice-snow_mass / dt) on U/V points
REAL(wp), DIMENSION(jpi,jpj) :: zmf ! coriolis parameter at T points
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) :: zds ! shear
REAL(wp), DIMENSION(jpi,jpj) :: zs1, zs2, zs12 ! stress tensor components
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) :: zfU , zfV ! internal stresses
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, ztauy_oi ! ice-ocean stress at U-V points
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) :: zmsk00, zmsk15
REAL(wp), DIMENSION(jpi,jpj) :: zmsk01x, zmsk01y ! dummy arrays
REAL(wp), DIMENSION(jpi,jpj) :: zmsk00x, zmsk00y ! mask for ice presence
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
!! --- check convergence
REAL(wp), DIMENSION(jpi,jpj) :: zu_ice, zv_ice
!! --- diags
REAL(wp) :: zsig1, zsig2, zsig12, zfac, z1_strength
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zsig_I, zsig_II, zsig1_p, zsig2_p
!! --- SIMIP diags
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xmtrp_ice ! X-component of ice mass transport (kg/s)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_ymtrp_ice ! Y-component of ice mass transport (kg/s)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xmtrp_snw ! X-component of snow mass transport (kg/s)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_ymtrp_snw ! Y-component of snow mass transport (kg/s)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_xatrp ! X-component of area transport (m2/s)
REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zdiag_yatrp ! Y-component of area transport (m2/s)
!!-------------------------------------------------------------------
IF( kt == nit000 .AND. lwp ) WRITE(numout,*) '-- ice_dyn_rhg_eap: EAP sea-ice rheology'
!
! for diagnostics and convergence tests
DO_2D( 1, 1, 1, 1 )
zmsk00(ji,jj) = MAX( 0._wp , SIGN( 1._wp , at_i(ji,jj) - epsi06 ) ) ! 1 if ice , 0 if no ice
END_2D
IF( nn_rhg_chkcvg > 0 ) THEN
DO_2D( 1, 1, 1, 1 )
zmsk15(ji,jj) = MAX( 0._wp , SIGN( 1._wp , at_i(ji,jj) - 0.15_wp ) ) ! 1 if 15% ice, 0 if less
END_2D
ENDIF
!
!------------------------------------------------------------------------------!
! 0) mask at F points for the ice
!------------------------------------------------------------------------------!
IF( kt == nit000 ) THEN
! 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_eap', fimask, 'F', 1.0_wp )
ENDIF
!------------------------------------------------------------------------------!
! 1) define some variables and initialize arrays
!------------------------------------------------------------------------------!
zrhoco = rho0 * rn_cio
! ecc2: square of yield ellipse eccenticrity
ecc2 = rn_ecc * rn_ecc
z1_ecc2 = 1._wp / ecc2
! alpha parameters (Bouillon 2009)
IF( .NOT. ln_aEVP ) THEN
zdtevp = rDt_ice / REAL( nn_nevp )
zalph1 = 2._wp * rn_relast * REAL( nn_nevp )
zalph2 = zalph1 * z1_ecc2
z1_alph1 = 1._wp / ( zalph1 + 1._wp )
z1_alph2 = 1._wp / ( zalph2 + 1._wp )
ELSE
zdtevp = rdt_ice
! zalpha parameters set later on adaptatively
ENDIF
z1_dtevp = 1._wp / zdtevp
! Initialise stress tensor
zs1 (:,:) = pstress1_i (:,:)
zs2 (:,:) = pstress2_i (:,:)
zs12(:,:) = pstress12_i(:,:)
! constants for structure tensor
z1_dtevp_A = z1_dtevp/10.0_wp
z1dtevpkth = 1._wp / (z1_dtevp_A + 0.00002_wp)
zp5kth = 0.5_wp * 0.00002_wp
! Ice strength
CALL ice_strength
! landfast param from Lemieux(2016): add isotropic tensile strength (following Konig Beatty and Holland, 2010)
IF( ln_landfast_L16 ) THEN ; zkt = rn_lf_tensile
ELSE ; zkt = 0._wp
ENDIF
!
!------------------------------------------------------------------------------!
! 2) Wind / ocean stress, mass terms, coriolis terms
!------------------------------------------------------------------------------!
! 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( 0, 0, 0, 0 )
! ice fraction at U-V points
zaU(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)
zaV(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)
! Ice/snow mass at U-V points
zm1 = ( rhos * vt_s(ji ,jj ) + rhoi * vt_i(ji ,jj ) )
zm2 = ( rhos * vt_s(ji+1,jj ) + rhoi * vt_i(ji+1,jj ) )
zm3 = ( rhos * vt_s(ji ,jj+1) + rhoi * vt_i(ji ,jj+1) )
zmassU = 0.5_wp * ( zm1 * e1e2t(ji,jj) + zm2 * e1e2t(ji+1,jj) ) * r1_e1e2u(ji,jj) * umask(ji,jj,1)
zmassV = 0.5_wp * ( zm1 * e1e2t(ji,jj) + zm3 * e1e2t(ji,jj+1) ) * r1_e1e2v(ji,jj) * vmask(ji,jj,1)
! Ocean currents at U-V points
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)
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)
! Coriolis at T points (m*f)
zmf(ji,jj) = zm1 * ff_t(ji,jj)
! dt/m at T points (for alpha and beta coefficients)
zdt_m(ji,jj) = zdtevp / MAX( zm1, zmmin )
! m/dt
zmU_t(ji,jj) = zmassU * z1_dtevp
zmV_t(ji,jj) = zmassV * z1_dtevp
! Drag ice-atm.
ztaux_ai(ji,jj) = zaU(ji,jj) * utau_ice(ji,jj)
ztauy_ai(ji,jj) = zaV(ji,jj) * vtau_ice(ji,jj)
! Surface pressure gradient (- m*g*GRAD(ssh)) at U-V points
zspgU(ji,jj) = - zmassU * grav * ( zsshdyn(ji+1,jj) - zsshdyn(ji,jj) ) * r1_e1u(ji,jj)
zspgV(ji,jj) = - zmassV * grav * ( zsshdyn(ji,jj+1) - zsshdyn(ji,jj) ) * r1_e2v(ji,jj)
! masks
zmsk00x(ji,jj) = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zmassU ) ) ! 0 if no ice
zmsk00y(ji,jj) = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zmassV ) ) ! 0 if no ice
! switches
IF( zmassU <= zmmin .AND. zaU(ji,jj) <= zamin ) THEN ; zmsk01x(ji,jj) = 0._wp
ELSE ; zmsk01x(ji,jj) = 1._wp ; ENDIF
IF( zmassV <= zmmin .AND. zaV(ji,jj) <= zamin ) THEN ; zmsk01y(ji,jj) = 0._wp
ELSE ; zmsk01y(ji,jj) = 1._wp ; ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', zmf, 'T', 1.0_wp, zdt_m, 'T', 1.0_wp )
!
! !== Landfast ice parameterization ==!
!
IF( ln_landfast_L16 ) THEN !-- Lemieux 2016
DO_2D( 0, 0, 0, 0 )
! ice thickness at U-V points
zvU = 0.5_wp * ( vt_i(ji,jj) * e1e2t(ji,jj) + vt_i(ji+1,jj) * e1e2t(ji+1,jj) ) * r1_e1e2u(ji,jj) * umask(ji,jj,1)
zvV = 0.5_wp * ( vt_i(ji,jj) * e1e2t(ji,jj) + vt_i(ji,jj+1) * e1e2t(ji,jj+1) ) * r1_e1e2v(ji,jj) * vmask(ji,jj,1)
! ice-bottom stress at U points
zvCr = zaU(ji,jj) * rn_lf_depfra * hu(ji,jj,Kmm) * ( 1._wp - icb_mask(ji,jj) ) ! if grounded icebergs are read: ocean depth = 0
ztaux_base(ji,jj) = - rn_lf_bfr * MAX( 0._wp, zvU - zvCr ) * EXP( -rn_crhg * ( 1._wp - zaU(ji,jj) ) )
! ice-bottom stress at V points
zvCr = zaV(ji,jj) * rn_lf_depfra * hv(ji,jj,Kmm) * ( 1._wp - icb_mask(ji,jj) ) ! if grounded icebergs are read: ocean depth = 0
ztauy_base(ji,jj) = - rn_lf_bfr * MAX( 0._wp, zvV - zvCr ) * EXP( -rn_crhg * ( 1._wp - zaV(ji,jj) ) )
! ice_bottom stress at T points
zvCr = at_i(ji,jj) * rn_lf_depfra * ht(ji,jj) * ( 1._wp - icb_mask(ji,jj) ) ! if grounded icebergs are read: ocean depth = 0
tau_icebfr(ji,jj) = - rn_lf_bfr * MAX( 0._wp, vt_i(ji,jj) - zvCr ) * EXP( -rn_crhg * ( 1._wp - at_i(ji,jj) ) )
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', tau_icebfr(:,:), 'T', 1.0_wp )
!
ELSE !-- no landfast
DO_2D( 0, 0, 0, 0 )
ztaux_base(ji,jj) = 0._wp
ztauy_base(ji,jj) = 0._wp
END_2D
ENDIF
!------------------------------------------------------------------------------!
! 3) Solution of the momentum equation, iterative procedure
!------------------------------------------------------------------------------!
!
! ! ==================== !
DO jter = 1 , nn_nevp ! loop over jter !
! ! ==================== !
l_full_nf_update = jter == nn_nevp ! false: disable full North fold update (performances) for iter = 1 to nn_nevp-1
!
! convergence test
IF( nn_rhg_chkcvg == 1 .OR. nn_rhg_chkcvg == 2 ) THEN
DO_2D( nn_hls, nn_hls, nn_hls, nn_hls )
zu_ice(ji,jj) = u_ice(ji,jj) * umask(ji,jj,1) ! velocity at previous time step
zv_ice(ji,jj) = v_ice(ji,jj) * vmask(ji,jj,1)
END_2D
ENDIF
! --- divergence, tension & shear (Appendix B of Hunke & Dukowicz, 2002) --- !
DO_2D( 1, 0, 1, 0 )
! 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 )
! 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
zdiv = ( 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)
zdiv2 = zdiv * zdiv
! tension 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
! delta at T points
zdelta(ji,jj) = SQRT( zdiv2 + ( zdt2 + zds2 ) * z1_ecc2 )
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', zdelta, 'T', 1.0_wp )
! P/delta at T points
DO_2D( 1, 1, 1, 1 )
zp_delt(ji,jj) = strength(ji,jj) / ( zdelta(ji,jj) + rn_creepl )
END_2D
DO_2D( 0, 1, 0, 1 ) ! loop ends at jpi,jpj so that no lbc_lnk are needed for zs1 and zs2
! shear at T points
zdsT = ( zds(ji,jj ) * e1e2f(ji,jj ) + zds(ji-1,jj ) * e1e2f(ji-1,jj ) &
& + zds(ji,jj-1) * e1e2f(ji,jj-1) + zds(ji-1,jj-1) * e1e2f(ji-1,jj-1) &
& ) * 0.25_wp * r1_e1e2t(ji,jj)
! divergence at T points (duplication to avoid communications)
zdiv = ( 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)
! tension at T points (duplication to avoid communications)
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)
! --- anisotropic stress calculation --- !
CALL update_stress_rdg (jter, nn_nevp, zdiv, zdt, zdsT, paniso_11(ji,jj), paniso_12(ji,jj), &
zstressptmp, zstressmtmp, zstress12tmp(ji,jj), strength(ji,jj), zalphar, zalphas)
! structure tensor update
CALL calc_ffrac(zstressptmp, zstressmtmp, zstress12tmp(ji,jj), paniso_11(ji,jj), paniso_12(ji,jj), zmresult11, zmresult12)
paniso_11(ji,jj) = (paniso_11(ji,jj) + 0.5*2.e-5*zdtevp + zmresult11*zdtevp) / (1. + 2.e-5*zdtevp) ! implicit
paniso_12(ji,jj) = (paniso_12(ji,jj) + zmresult12*zdtevp) / (1. + 2.e-5*zdtevp) ! implicit
IF (jter == nn_nevp) THEN
zyield11(ji,jj) = 0.5_wp * (zstressptmp + zstressmtmp)
zyield22(ji,jj) = 0.5_wp * (zstressptmp - zstressmtmp)
zyield12(ji,jj) = zstress12tmp(ji,jj)
prdg_conv(ji,jj) = -min( zalphar, 0._wp)
ENDIF
! alpha for aEVP
! gamma = 0.5*P/(delta+creepl) * (c*pi)**2/Area * dt/m
! alpha = beta = sqrt(4*gamma)
IF( ln_aEVP ) THEN
zalph1 = MAX( 50._wp, rpi * SQRT( 0.5_wp * zp_delt(ji,jj) * r1_e1e2t(ji,jj) * zdt_m(ji,jj) ) )
z1_alph1 = 1._wp / ( zalph1 + 1._wp )
zalph2 = zalph1
z1_alph2 = z1_alph1
! explicit:
! z1_alph1 = 1._wp / zalph1
! z1_alph2 = 1._wp / zalph1
! zalph1 = zalph1 - 1._wp
! zalph2 = zalph1
ENDIF
! stress at T points (zkt/=0 if landfast)
zs1(ji,jj) = ( zs1(ji,jj) * zalph1 + zstressptmp ) * z1_alph1
zs2(ji,jj) = ( zs2(ji,jj) * zalph1 + zstressmtmp ) * z1_alph1
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', zstress12tmp, 'T', 1.0_wp , paniso_11, 'T', 1.0_wp , paniso_12, 'T', 1.0_wp)
! Save beta at T-points for further computations
IF( ln_aEVP ) THEN
DO_2D( 1, 1, 1, 1 )
zbeta(ji,jj) = MAX( 50._wp, rpi * SQRT( 0.5_wp * zp_delt(ji,jj) * r1_e1e2t(ji,jj) * zdt_m(ji,jj) ) )
END_2D
ENDIF
DO_2D( 1, 0, 1, 0 )
! stress12tmp at F points
zstress12tmpF = ( zstress12tmp(ji,jj+1) * e1e2t(ji,jj+1) + zstress12tmp(ji+1,jj+1) * e1e2t(ji+1,jj+1) &
& + zstress12tmp(ji,jj ) * e1e2t(ji,jj ) + zstress12tmp(ji+1,jj ) * e1e2t(ji+1,jj ) &
& ) * 0.25_wp * r1_e1e2f(ji,jj)
! alpha for aEVP
IF( ln_aEVP ) THEN
zalph2 = MAX( zbeta(ji,jj), zbeta(ji+1,jj), zbeta(ji,jj+1), zbeta(ji+1,jj+1) )
z1_alph2 = 1._wp / ( zalph2 + 1._wp )
! explicit:
! z1_alph2 = 1._wp / zalph2
! zalph2 = zalph2 - 1._wp
ENDIF
! stress at F points (zkt/=0 if landfast)
zs12(ji,jj) = ( zs12(ji,jj) * zalph1 + zstress12tmpF ) * z1_alph1
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', zs1, 'T', 1.0_wp, zs2, 'T', 1.0_wp, zs12, 'F', 1.0_wp )
! --- Ice internal stresses (Appendix C of Hunke and Dukowicz, 2002) --- !
DO_2D( 0, 0, 0, 0 )
! !--- U points
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) &
& + ( zs12(ji,jj) * e1f(ji,jj) * e1f(ji,jj) - zs12(ji,jj-1) * e1f(ji,jj-1) * e1f(ji,jj-1) &
& ) * 2._wp * r1_e1u(ji,jj) &
& ) * r1_e1e2u(ji,jj)
!
! !--- V points
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) &
& + ( zs12(ji,jj) * e2f(ji,jj) * e2f(ji,jj) - zs12(ji-1,jj) * e2f(ji-1,jj) * e2f(ji-1,jj) &
& ) * 2._wp * r1_e2v(ji,jj) &
& ) * r1_e1e2v(ji,jj)
!
! !--- ice currents at U-V point
v_iceU(ji,jj) = 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)
u_iceV(ji,jj) = 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)
!
END_2D
!
! --- Computation of ice velocity --- !
! Bouillon et al. 2013 (eq 47-48) => unstable unless alpha, beta vary as in Kimmritz 2016 & 2017
! Bouillon et al. 2009 (eq 34-35) => stable
IF( MOD(jter,2) == 0 ) THEN ! even iterations
!
DO_2D( 0, 0, 0, 0 )
! !--- tau_io/(v_oce - v_ice)
zTauO = zaV(ji,jj) * zrhoco * SQRT( ( v_ice (ji,jj) - v_oce (ji,jj) ) * ( v_ice (ji,jj) - v_oce (ji,jj) ) &
& + ( u_iceV(ji,jj) - u_oceV(ji,jj) ) * ( u_iceV(ji,jj) - u_oceV(ji,jj) ) )
! !--- Ocean-to-Ice stress
ztauy_oi(ji,jj) = zTauO * ( v_oce(ji,jj) - v_ice(ji,jj) )
!
! !--- tau_bottom/v_ice
zvel = 5.e-05_wp + SQRT( v_ice(ji,jj) * v_ice(ji,jj) + u_iceV(ji,jj) * u_iceV(ji,jj) )
zTauB = ztauy_base(ji,jj) / zvel
! !--- OceanBottom-to-Ice stress
ztauy_bi(ji,jj) = zTauB * v_ice(ji,jj)
!
! !--- Coriolis at V-points (energy conserving formulation)
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) ) )
!
! !--- Sum of external forces (explicit solution) = F + tau_ia + Coriolis + spg + tau_io
zRHS = zfV(ji,jj) + ztauy_ai(ji,jj) + zCorV(ji,jj) + zspgV(ji,jj) + ztauy_oi(ji,jj)
!
! !--- landfast switch => 0 = static friction : TauB > RHS & sign(TauB) /= sign(RHS)
! 1 = sliding friction : TauB < RHS
rswitch = 1._wp - MIN( 1._wp, ABS( SIGN( 1._wp, zRHS + ztauy_base(ji,jj) ) - SIGN( 1._wp, zRHS ) ) )
!
IF( ln_aEVP ) THEN !--- ice velocity using aEVP (Kimmritz et al 2016 & 2017)
zbetav = MAX( zbeta(ji,jj), zbeta(ji,jj+1) )
v_ice(ji,jj) = ( ( rswitch * ( zmV_t(ji,jj) * ( zbetav * v_ice(ji,jj) + v_ice_b(ji,jj) ) & ! previous velocity
& + zRHS + zTauO * v_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmV_t(ji,jj) * ( zbetav + 1._wp ) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * ( v_ice_b(ji,jj) &
& + v_ice (ji,jj) * MAX( 0._wp, zbetav - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) / ( zbetav + 1._wp ) &
& ) * zmsk01y(ji,jj) + v_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01y(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00y(ji,jj)
ELSE !--- ice velocity using EVP implicit formulation (cf Madec doc & Bouillon 2009)
v_ice(ji,jj) = ( ( rswitch * ( zmV_t(ji,jj) * v_ice(ji,jj) & ! previous velocity
& + zRHS + zTauO * v_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmV_t(ji,jj) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * v_ice(ji,jj) * MAX( 0._wp, 1._wp - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) * zmsk01y(ji,jj) + v_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01y(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00y(ji,jj)
ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', v_ice, 'V', -1.0_wp )
!
#if defined key_agrif
!! CALL agrif_interp_ice( 'V', jter, nn_nevp )
CALL agrif_interp_ice( 'V' )
#endif
IF( ln_bdy ) CALL bdy_ice_dyn( 'V' )
!
DO_2D( 0, 0, 0, 0 )
! !--- tau_io/(u_oce - u_ice)
zTauO = zaU(ji,jj) * zrhoco * SQRT( ( u_ice (ji,jj) - u_oce (ji,jj) ) * ( u_ice (ji,jj) - u_oce (ji,jj) ) &
& + ( v_iceU(ji,jj) - v_oceU(ji,jj) ) * ( v_iceU(ji,jj) - v_oceU(ji,jj) ) )
! !--- Ocean-to-Ice stress
ztaux_oi(ji,jj) = zTauO * ( u_oce(ji,jj) - u_ice(ji,jj) )
!
! !--- tau_bottom/u_ice
zvel = 5.e-05_wp + SQRT( v_iceU(ji,jj) * v_iceU(ji,jj) + u_ice(ji,jj) * u_ice(ji,jj) )
zTauB = ztaux_base(ji,jj) / zvel
! !--- OceanBottom-to-Ice stress
ztaux_bi(ji,jj) = zTauB * u_ice(ji,jj)
!
! !--- Coriolis at U-points (energy conserving formulation)
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) ) )
!
! !--- Sum of external forces (explicit solution) = F + tau_ia + Coriolis + spg + tau_io
zRHS = zfU(ji,jj) + ztaux_ai(ji,jj) + zCorU(ji,jj) + zspgU(ji,jj) + ztaux_oi(ji,jj)
!
! !--- landfast switch => 0 = static friction : TauB > RHS & sign(TauB) /= sign(RHS)
! 1 = sliding friction : TauB < RHS
rswitch = 1._wp - MIN( 1._wp, ABS( SIGN( 1._wp, zRHS + ztaux_base(ji,jj) ) - SIGN( 1._wp, zRHS ) ) )
!
IF( ln_aEVP ) THEN !--- ice velocity using aEVP (Kimmritz et al 2016 & 2017)
zbetau = MAX( zbeta(ji,jj), zbeta(ji+1,jj) )
u_ice(ji,jj) = ( ( rswitch * ( zmU_t(ji,jj) * ( zbetau * u_ice(ji,jj) + u_ice_b(ji,jj) ) & ! previous velocity
& + zRHS + zTauO * u_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmU_t(ji,jj) * ( zbetau + 1._wp ) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * ( u_ice_b(ji,jj) &
& + u_ice (ji,jj) * MAX( 0._wp, zbetau - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) / ( zbetau + 1._wp ) &
& ) * zmsk01x(ji,jj) + u_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01x(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00x(ji,jj)
ELSE !--- ice velocity using EVP implicit formulation (cf Madec doc & Bouillon 2009)
u_ice(ji,jj) = ( ( rswitch * ( zmU_t(ji,jj) * u_ice(ji,jj) & ! previous velocity
& + zRHS + zTauO * u_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmU_t(ji,jj) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * u_ice(ji,jj) * MAX( 0._wp, 1._wp - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) * zmsk01x(ji,jj) + u_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01x(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00x(ji,jj)
ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', u_ice, 'U', -1.0_wp )
!
#if defined key_agrif
!! CALL agrif_interp_ice( 'U', jter, nn_nevp )
CALL agrif_interp_ice( 'U' )
#endif
IF( ln_bdy ) CALL bdy_ice_dyn( 'U' )
!
ELSE ! odd iterations
!
DO_2D( 0, 0, 0, 0 )
! !--- tau_io/(u_oce - u_ice)
zTauO = zaU(ji,jj) * zrhoco * SQRT( ( u_ice (ji,jj) - u_oce (ji,jj) ) * ( u_ice (ji,jj) - u_oce (ji,jj) ) &
& + ( v_iceU(ji,jj) - v_oceU(ji,jj) ) * ( v_iceU(ji,jj) - v_oceU(ji,jj) ) )
! !--- Ocean-to-Ice stress
ztaux_oi(ji,jj) = zTauO * ( u_oce(ji,jj) - u_ice(ji,jj) )
!
! !--- tau_bottom/u_ice
zvel = 5.e-05_wp + SQRT( v_iceU(ji,jj) * v_iceU(ji,jj) + u_ice(ji,jj) * u_ice(ji,jj) )
zTauB = ztaux_base(ji,jj) / zvel
! !--- OceanBottom-to-Ice stress
ztaux_bi(ji,jj) = zTauB * u_ice(ji,jj)
!
! !--- Coriolis at U-points (energy conserving formulation)
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) ) )
!
! !--- Sum of external forces (explicit solution) = F + tau_ia + Coriolis + spg + tau_io
zRHS = zfU(ji,jj) + ztaux_ai(ji,jj) + zCorU(ji,jj) + zspgU(ji,jj) + ztaux_oi(ji,jj)
!
! !--- landfast switch => 0 = static friction : TauB > RHS & sign(TauB) /= sign(RHS)
! 1 = sliding friction : TauB < RHS
rswitch = 1._wp - MIN( 1._wp, ABS( SIGN( 1._wp, zRHS + ztaux_base(ji,jj) ) - SIGN( 1._wp, zRHS ) ) )
!
IF( ln_aEVP ) THEN !--- ice velocity using aEVP (Kimmritz et al 2016 & 2017)
zbetau = MAX( zbeta(ji,jj), zbeta(ji+1,jj) )
u_ice(ji,jj) = ( ( rswitch * ( zmU_t(ji,jj) * ( zbetau * u_ice(ji,jj) + u_ice_b(ji,jj) ) & ! previous velocity
& + zRHS + zTauO * u_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmU_t(ji,jj) * ( zbetau + 1._wp ) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * ( u_ice_b(ji,jj) &
& + u_ice (ji,jj) * MAX( 0._wp, zbetau - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) / ( zbetau + 1._wp ) &
& ) * zmsk01x(ji,jj) + u_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01x(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00x(ji,jj)
ELSE !--- ice velocity using EVP implicit formulation (cf Madec doc & Bouillon 2009)
u_ice(ji,jj) = ( ( rswitch * ( zmU_t(ji,jj) * u_ice(ji,jj) & ! previous velocity
& + zRHS + zTauO * u_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmU_t(ji,jj) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * u_ice(ji,jj) * MAX( 0._wp, 1._wp - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) * zmsk01x(ji,jj) + u_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01x(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00x(ji,jj)
ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', u_ice, 'U', -1.0_wp )
!
#if defined key_agrif
!! CALL agrif_interp_ice( 'U', jter, nn_nevp )
CALL agrif_interp_ice( 'U' )
#endif
IF( ln_bdy ) CALL bdy_ice_dyn( 'U' )
!
DO_2D( 0, 0, 0, 0 )
! !--- tau_io/(v_oce - v_ice)
zTauO = zaV(ji,jj) * zrhoco * SQRT( ( v_ice (ji,jj) - v_oce (ji,jj) ) * ( v_ice (ji,jj) - v_oce (ji,jj) ) &
& + ( u_iceV(ji,jj) - u_oceV(ji,jj) ) * ( u_iceV(ji,jj) - u_oceV(ji,jj) ) )
! !--- Ocean-to-Ice stress
ztauy_oi(ji,jj) = zTauO * ( v_oce(ji,jj) - v_ice(ji,jj) )
!
! !--- tau_bottom/v_ice
zvel = 5.e-05_wp + SQRT( v_ice(ji,jj) * v_ice(ji,jj) + u_iceV(ji,jj) * u_iceV(ji,jj) )
zTauB = ztauy_base(ji,jj) / zvel
! !--- OceanBottom-to-Ice stress
ztauy_bi(ji,jj) = zTauB * v_ice(ji,jj)
!
! !--- Coriolis at v-points (energy conserving formulation)
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) ) )
!
! !--- Sum of external forces (explicit solution) = F + tau_ia + Coriolis + spg + tau_io
zRHS = zfV(ji,jj) + ztauy_ai(ji,jj) + zCorV(ji,jj) + zspgV(ji,jj) + ztauy_oi(ji,jj)
!
! !--- landfast switch => 0 = static friction : TauB > RHS & sign(TauB) /= sign(RHS)
! 1 = sliding friction : TauB < RHS
rswitch = 1._wp - MIN( 1._wp, ABS( SIGN( 1._wp, zRHS + ztauy_base(ji,jj) ) - SIGN( 1._wp, zRHS ) ) )
!
IF( ln_aEVP ) THEN !--- ice velocity using aEVP (Kimmritz et al 2016 & 2017)
zbetav = MAX( zbeta(ji,jj), zbeta(ji,jj+1) )
v_ice(ji,jj) = ( ( rswitch * ( zmV_t(ji,jj) * ( zbetav * v_ice(ji,jj) + v_ice_b(ji,jj) ) & ! previous velocity
& + zRHS + zTauO * v_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmV_t(ji,jj) * ( zbetav + 1._wp ) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * ( v_ice_b(ji,jj) &
& + v_ice (ji,jj) * MAX( 0._wp, zbetav - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) / ( zbetav + 1._wp ) &
& ) * zmsk01y(ji,jj) + v_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01y(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00y(ji,jj)
ELSE !--- ice velocity using EVP implicit formulation (cf Madec doc & Bouillon 2009)
v_ice(ji,jj) = ( ( rswitch * ( zmV_t(ji,jj) * v_ice(ji,jj) & ! previous velocity
& + zRHS + zTauO * v_ice(ji,jj) & ! F + tau_ia + Coriolis + spg + tau_io(only ocean part)
& ) / MAX( zepsi, zmV_t(ji,jj) + zTauO - zTauB ) & ! m/dt + tau_io(only ice part) + landfast
& + ( 1._wp - rswitch ) * v_ice(ji,jj) * MAX( 0._wp, 1._wp - zdtevp * rn_lf_relax ) & ! static friction => slow decrease to v=0
& ) * zmsk01y(ji,jj) + v_oce(ji,jj) * 0.01_wp * ( 1._wp - zmsk01y(ji,jj) ) & ! v_ice = v_oce/100 if mass < zmmin & conc < zamin
& ) * zmsk00y(ji,jj)
ENDIF
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', v_ice, 'V', -1.0_wp )
!
#if defined key_agrif
!! CALL agrif_interp_ice( 'V', jter, nn_nevp )
CALL agrif_interp_ice( 'V' )
#endif
IF( ln_bdy ) CALL bdy_ice_dyn( 'V' )
!
ENDIF
! convergence test
IF( nn_rhg_chkcvg == 2 ) CALL rhg_cvg_eap( kt, jter, nn_nevp, u_ice, v_ice, zu_ice, zv_ice, zmsk15 )
!
! ! ==================== !
END DO ! end loop over jter !
! ! ==================== !
IF( ln_aEVP ) CALL iom_put( 'beta_evp' , zbeta )
!
CALL lbc_lnk( 'icedyn_rhg_eap', prdg_conv, 'T', 1.0_wp ) ! only need this in ridging module after rheology completed
!
!------------------------------------------------------------------------------!
! 4) Recompute delta, shear and div (inputs for mechanical redistribution)
!------------------------------------------------------------------------------!
DO_2D( 1, 0, 1, 0 )
! 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 )
! 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
! 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)
! shear at T points
pshear_i(ji,jj) = SQRT( zdt2 + zds2 )
! divergence at T points
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)
! delta at T points
zfac = SQRT( pdivu_i(ji,jj) * pdivu_i(ji,jj) + ( zdt2 + zds2 ) * z1_ecc2 ) ! delta
rswitch = 1._wp - MAX( 0._wp, SIGN( 1._wp, -zfac ) ) ! 0 if delta=0
pdelta_i(ji,jj) = zfac + rn_creepl * rswitch ! delta+creepl
END_2D
CALL lbc_lnk( 'icedyn_rhg_eap', pshear_i, 'T', 1.0_wp, pdivu_i, 'T', 1.0_wp, pdelta_i, 'T', 1.0_wp, &
& zten_i, 'T', 1.0_wp, zs1 , 'T', 1.0_wp, zs2 , 'T', 1.0_wp, &
& zs12, 'F', 1.0_wp )
! --- Store the stress tensor for the next time step --- !
pstress1_i (:,:) = zs1 (:,:)
pstress2_i (:,:) = zs2 (:,:)
pstress12_i(:,:) = zs12(:,:)
!
!------------------------------------------------------------------------------!
! 5) diagnostics
!------------------------------------------------------------------------------!
! --- ice-ocean, ice-atm. & ice-oceanbottom(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
!
CALL lbc_lnk( 'icedyn_rhg_eap', ztaux_oi, 'U', -1.0_wp, ztauy_oi, 'V', -1.0_wp, ztaux_ai, 'U', -1.0_wp, &
& ztauy_ai, 'V', -1.0_wp, ztaux_bi, 'U', -1.0_wp, ztauy_bi, 'V', -1.0_wp )
!
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 )
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 ) ! shear
IF( iom_use('icedlt') ) CALL iom_put( 'icedlt' , pdelta_i * 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) )
!
DO_2D( 1, 1, 1, 1 )
! Ice stresses
! sigma1, sigma2, sigma12 are some useful recombination of the stresses (Hunke and Dukowicz MWR 2002, Bouillon et al., OM2013)
! These are NOT stress tensor components, neither stress invariants, neither stress principal components
! I know, this can be confusing...
zfac = strength(ji,jj) / ( pdelta_i(ji,jj) + rn_creepl )
zsig1 = zfac * ( pdivu_i(ji,jj) - pdelta_i(ji,jj) )
zsig2 = zfac * z1_ecc2 * zten_i(ji,jj)
zsig12 = zfac * z1_ecc2 * pshear_i(ji,jj)
! Stress invariants (sigma_I, sigma_II, Coon 1974, Feltham 2008)
zsig_I (ji,jj) = zsig1 * 0.5_wp ! 1st stress invariant, aka average normal stress, aka negative pressure
zsig_II(ji,jj) = SQRT ( zsig2 * zsig2 * 0.25_wp + zsig12 * zsig12 ) ! 2nd '' '' , aka maximum shear stress
END_2D
!
! Stress tensor invariants (normal and shear stress N/m) - SIMIP diags - definitions following Coon (1974) and Feltham (2008)
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 --- !
! This are used to plot the normalized yield curve, see Lemieux & Dupont, 2020
! Recommendation 1 : we use ice strength, not replacement pressure
! Recommendation 2 : need to use deformations at PREVIOUS iterate for viscosities
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( 1, 1, 1, 1 )
! Ice stresses computed with **viscosities** (delta, p/delta) at **previous** iterates
! and **deformations** at current iterates
! following Lemieux & Dupont (2020)
zfac = zp_delt(ji,jj)
zsig1 = zfac * ( pdivu_i(ji,jj) - ( zdelta(ji,jj) + rn_creepl ) )
zsig2 = zfac * z1_ecc2 * zten_i(ji,jj)
zsig12 = zfac * z1_ecc2 * pshear_i(ji,jj)
! Stress invariants (sigma_I, sigma_II, Coon 1974, Feltham 2008), T-point
zsig_I(ji,jj) = zsig1 * 0.5_wp ! 1st stress invariant, aka average normal stress, aka negative pressure
zsig_II(ji,jj) = SQRT ( zsig2 * zsig2 * 0.25_wp + zsig12 * zsig12 ) ! 2nd '' '' , aka maximum 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 iom_put( 'sig1_pnorm' , zsig1_p )
CALL iom_put( 'sig2_pnorm' , zsig2_p )
DEALLOCATE( zsig1_p , zsig2_p , zsig_I, zsig_II )
ENDIF
! --- yieldcurve --- !
IF( iom_use('yield11') .OR. iom_use('yield12') .OR. iom_use('yield22')) THEN
CALL lbc_lnk( 'icedyn_rhg_eap', zyield11, 'T', 1.0_wp, zyield22, 'T', 1.0_wp, zyield12, 'T', 1.0_wp )
CALL iom_put( 'yield11', zyield11 * zmsk00 )
CALL iom_put( 'yield22', zyield22 * zmsk00 )
CALL iom_put( 'yield12', zyield12 * zmsk00 )
ENDIF
! --- anisotropy tensor --- !
IF( iom_use('aniso') ) THEN
CALL lbc_lnk( 'icedyn_rhg_eap', paniso_11, 'T', 1.0_wp )
CALL iom_put( 'aniso' , paniso_11 * zmsk00 )
ENDIF
! --- SIMIP --- !
IF( iom_use('dssh_dx') .OR. iom_use('dssh_dy') .OR. &
& iom_use('corstrx') .OR. iom_use('corstry') .OR. iom_use('intstrx') .OR. iom_use('intstry') ) THEN
!
CALL lbc_lnk( 'icedyn_rhg_eap', zspgU, 'U', -1.0_wp, zspgV, 'V', -1.0_wp, &
& zCorU, 'U', -1.0_wp, zCorV, 'V', -1.0_wp, &
& zfU, 'U', -1.0_wp, zfV, 'V', -1.0_wp )
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)
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
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 )
! 2D ice mass, snow mass, area transport arrays (X, Y)
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_eap', zdiag_xmtrp_ice, 'U', -1.0_wp, zdiag_ymtrp_ice, 'V', -1.0_wp, &
& zdiag_xmtrp_snw, 'U', -1.0_wp, zdiag_ymtrp_snw, 'V', -1.0_wp, &
& zdiag_xatrp , 'U', -1.0_wp, zdiag_yatrp , 'V', -1.0_wp )
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
!
! --- convergence tests --- !
IF( nn_rhg_chkcvg == 1 .OR. nn_rhg_chkcvg == 2 ) THEN
IF( iom_use('uice_cvg') ) THEN
IF( ln_aEVP ) THEN ! output: beta * ( u(t=nn_nevp) - u(t=nn_nevp-1) )
CALL iom_put( 'uice_cvg', MAX( ABS( u_ice(:,:) - zu_ice(:,:) ) * zbeta(:,:) * umask(:,:,1) , &
& ABS( v_ice(:,:) - zv_ice(:,:) ) * zbeta(:,:) * vmask(:,:,1) ) * zmsk15(:,:) )
ELSE ! output: nn_nevp * ( u(t=nn_nevp) - u(t=nn_nevp-1) )
CALL iom_put( 'uice_cvg', REAL( nn_nevp ) * MAX( ABS( u_ice(:,:) - zu_ice(:,:) ) * umask(:,:,1) , &
& ABS( v_ice(:,:) - zv_ice(:,:) ) * vmask(:,:,1) ) * zmsk15(:,:) )
ENDIF
ENDIF
ENDIF
!
END SUBROUTINE ice_dyn_rhg_eap
SUBROUTINE rhg_cvg_eap( kt, kiter, kitermax, pu, pv, pub, pvb, pmsk15 )
!!----------------------------------------------------------------------
!! *** ROUTINE rhg_cvg_eap ***
!!
!! ** Purpose : check convergence of oce rheology
!!
!! ** Method : create a file ice_cvg.nc containing the convergence of ice velocity
!! during the sub timestepping of rheology so as:
!! uice_cvg = MAX( u(t+1) - u(t) , v(t+1) - v(t) )
!! This routine is called every sub-iteration, so it is cpu expensive
!!
!! ** Note : for the first sub-iteration, uice_cvg is set to 0 (too large otherwise)
!!----------------------------------------------------------------------
INTEGER , INTENT(in) :: kt, kiter, kitermax ! ocean time-step index
REAL(wp), DIMENSION(:,:), INTENT(in) :: pu, pv, pub, pvb ! now and before velocities
REAL(wp), DIMENSION(:,:), INTENT(in) :: pmsk15
!!
INTEGER :: it, idtime, istatus
INTEGER :: ji, jj ! dummy loop indices
REAL(wp) :: zresm ! local real
CHARACTER(len=20) :: clname
!!----------------------------------------------------------------------
! create file