sico_variables_m.F90

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!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!
!  Module :  s i c o _ v a r i a b l e s _ m
!
!> @file
!!
!! Declarations of global variables for SICOPOLIS.
!!
!! @section Copyright
!!
!! Copyright 2009-2018 Ralf Greve
!!
!! @section License
!!
!! This file is part of SICOPOLIS.
!!
!! SICOPOLIS is free software: you can redistribute it and/or modify
!! it under the terms of the GNU General Public License as published by
!! the Free Software Foundation, either version 3 of the License, or
!! (at your option) any later version.
!!
!! SICOPOLIS is distributed in the hope that it will be useful,
!! but WITHOUT ANY WARRANTY; without even the implied warranty of
!! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!! GNU General Public License for more details.
!!
!! You should have received a copy of the GNU General Public License
!! along with SICOPOLIS.  If not, see <http://www.gnu.org/licenses/>.
!<
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

!-------------------------------------------------------------------------------
!> Declarations of global variables for SICOPOLIS.
!<------------------------------------------------------------------------------
module sico_variables_m

use sico_types_m

implicit none
save

!-------- Field quantities --------

!> maske(j,i): Ice-land-ocean mask.
!>             0: grounded ice,
!>             1: ice-free land,
!>             2: ocean,
!>             3: floating ice
   integer(i2b), dimension(0:JMAX,0:IMAX) :: maske
!> maske_old(j,i): Old value of maske (at the previous time step)
   integer(i2b), dimension(0:JMAX,0:IMAX) :: maske_old
!> maske_neu(j,i): New value of maske computed during an integration step
   integer(i2b), dimension(0:JMAX,0:IMAX) :: maske_neu
!> mask_run(j,i): mask indicating melt type.
!>             2: visible (ocean, for later developments),
!>             1: visible (grounded ice),
!>            -1: hidden on land,
!>            -2: hidden in ocean
   integer(i2b), dimension(0:JMAX,0:IMAX) :: mask_run
!> n_cts(j,i): Mask for thermal conditions.
!>             -1: cold ice base,
!>              0: temperate ice base with cold ice above,
!>              1: temperate ice base with temperate ice layer above
!>                 (only for POLY)
   integer(i2b), dimension(0:JMAX,0:IMAX) :: n_cts
!> (.)_neu: New value of quantity (.) computed during an integration step
   integer(i2b), dimension(0:JMAX,0:IMAX) :: n_cts_neu
!> kc_cts(j,i): Position kc of the CTS (for COLD, ENTC, ENTM)
   integer(i2b), dimension(0:JMAX,0:IMAX) :: kc_cts
!> (.)_neu: New value of quantity (.) computed during an integration step
   integer(i2b), dimension(0:JMAX,0:IMAX) :: kc_cts_neu
!> flag_inner_point(j,i): Inner-point flag.
!>                              .true.: inner point,
!>                             .false.: margin point
   logical, dimension(0:JMAX,0:IMAX) :: flag_inner_point
!> flag_grounding_line_1(j,i): Grounding line flag.
!>                              .true.: grounding line point
!>                                      (grounded ice point with at least
!>                                      one floating ice neighbour),
!>                             .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_grounding_line_1
!> flag_grounding_line_2(j,i): Grounding line flag.
!>                              .true.: grounding line point
!>                                      (floating ice point with at least
!>                                      one grounded ice neighbour),
!>                             .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_grounding_line_2
!> flag_calving_front_1(j,i): Calving front flag.
!>                              .true.: calving front point
!>                                      (floating ice point with at least
!>                                      one ocean neighbour),
!>                             .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_calving_front_1
!> flag_calving_front_2(j,i): Calving front flag.
!>                              .true.: calving front point
!>                                      (ocean point with at least
!>                                      one floating ice neighbour),
!>                             .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_calving_front_2
!> flag_shelfy_stream_x(j,i): Shelfy stream flag in x-direction, at (i+1/2,j).
!>                             .true.: shelfy stream point
!>                            .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_shelfy_stream_x
!> flag_shelfy_stream_y(j,i): Shelfy stream flag in y-direction, at (i,j+1/2).
!>                             .true.: shelfy stream point
!>                            .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_shelfy_stream_y
!> flag_shelfy_stream(j,i):   Shelfy stream flag on the main grid.
!>                             .true.: grounded ice,
!>                                     and at least one neighbour on the
!>                                     staggered grid is a shelfy stream point
!>                            .false.: otherwise
   logical, dimension(0:JMAX,0:IMAX) :: flag_shelfy_stream
!> xi(i): Coordinate xi (= x) of grid point i
   real(dp), dimension(0:IMAX) :: xi
!> eta(j): Coordinate eta (= y) of grid point j
   real(dp), dimension(0:JMAX) :: eta
!> zeta_c(kc): Sigma coordinate zeta_c of grid point kc
   real(dp), dimension(0:KCMAX) :: zeta_c
!> zeta_t(kt): Sigma coordinate zeta_t of grid point kt
   real(dp), dimension(0:KTMAX) :: zeta_t
!> zeta_r(kr): Sigma coordinate zeta_r of grid point kr
   real(dp), dimension(0:KRMAX) :: zeta_r
!> aa: Exponential stretch parameter of the non-equidistant vertical grid
!>     in the upper (kc) ice domain
   real(dp) :: aa
!> flag_aa_nonzero: Flag for the exponential stretch parameter aa.
!>                    .true.: aa greater than zero (non-equidistant grid)
!>                   .false.: aa equal to zero (equidistant grid)
   logical :: flag_aa_nonzero
!> ea: Abbreviation for exp(aa)
   real(dp) :: ea
!> eaz_c(kc): Abbreviation for exp(aa*zeta(kc))
   real(dp), dimension(0:KCMAX) :: eaz_c
!> eaz_c_quotient(kc): Abbreviation for (eaz_c(kc)-1.0)/(ea-1.0)
   real(dp), dimension(0:KCMAX) :: eaz_c_quotient

!> lambda(j,i): Geographic longitude of grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: lambda
!> phi(j,i): Geographic latitude of grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: phi
!> area(j,i): Area of grid cell associated with grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: area
!> sq_g11_g(j,i): Square root of the coefficient g11 of the metric tensor
!>                on grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g11_g
!> sq_g22_g(j,i): Square root of the coefficient g22 of the metric tensor
!>                on grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g22_g
!> insq_g11_g(j,i): Inverse square root of g11 on grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: insq_g11_g
!> insq_g22_g(j,i): Inverse square root of g22 on grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: insq_g22_g
!> sq_g11_sgx(j,i): Square root of g11, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g11_sgx
!> sq_g11_sgy(j,i): Square root of g11, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g11_sgy
!> sq_g22_sgx(j,i): Square root of g22, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g22_sgx
!> sq_g22_sgy(j,i): Square root of g22, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: sq_g22_sgy
!> insq_g11_sgx(j,i): Inverse square root of g11, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: insq_g11_sgx
!> insq_g22_sgy(j,i): Inverse square root of g22, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: insq_g22_sgy
!> zs(j,i): Coordinate z of the surface topography
   real(dp), dimension(0:JMAX,0:IMAX) :: zs
!> zm(j,i): Coordinate z of the bottom of the upper (kc) ice domain
!>                               = top of the lower (kt) ice domain
!>          (position of the CTS for POLY,
!>          equal to zb for ISOT, COLD, ENTC, ENTM)
   real(dp), dimension(0:JMAX,0:IMAX) :: zm
!> zb(j,i): Coordinate z of the ice base
   real(dp), dimension(0:JMAX,0:IMAX) :: zb
!> zl(j,i): Coordinate z of the lithosphere surface
   real(dp), dimension(0:JMAX,0:IMAX) :: zl
!> zl0(j,i): zl for isostatically relaxed ice-free conditions
   real(dp), dimension(0:JMAX,0:IMAX) :: zl0
!> wss(j,i): Isostatic steady-state displacement of the lithosphere
   real(dp), dimension(0:JMAX,0:IMAX) :: wss
!> flex_rig_lith(j,i): Flexural rigidity of the lithosphere
   real(dp), dimension(0:JMAX,0:IMAX) :: flex_rig_lith
!> time_lag_asth(j,i): Time lag of the relaxing asthenosphere
   real(dp), dimension(0:JMAX,0:IMAX) :: time_lag_asth
!> H_c(j,i): Thickness of ice in the upper (kc) domain
!>           (thickness of the cold-ice layer for POLY,
!>           entire ice thickness for ISOT, COLD, ENTC, ENTM)
   real(dp), dimension(0:JMAX,0:IMAX) :: h_c
!> H_t(j,i): Thickness of ice in the lower (kt) domain
!>           (thickness of the temperate layer for POLY,
!>           redundant and thus set to zero for ISOT, COLD, ENTC, ENTM)
   real(dp), dimension(0:JMAX,0:IMAX) :: h_t
!> dzs_dxi(j,i): Derivative of zs by xi (at i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzs_dxi
!> dzm_dxi(j,i): Derivative of zm by xi (at i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzm_dxi
!> dzb_dxi(j,i): Derivative of zb by xi (at i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzb_dxi
!> dH_c_dxi(j,i): Derivative of H_c by xi (at i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_c_dxi
!> dH_t_dxi(j,i): Derivative of H_t by xi (at i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_t_dxi
!> dzs_deta(j,i): Derivative of zs by eta (at i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzs_deta
!> dzm_deta(j,i): Derivative of zm by eta (at i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzm_deta
!> dzb_deta(j,i): Derivative of zb by eta (at i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzb_deta
!> dH_c_deta(j,i): Derivative of H_c by eta (at i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_c_deta
!> dH_t_deta(j,i): Derivative of H_t by eta (at i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_t_deta
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzs_dxi_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzm_dxi_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzb_dxi_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_c_dxi_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_t_dxi_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzs_deta_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzm_deta_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzb_deta_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_c_deta_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_t_deta_g
!> dzs_dtau(j,i): Derivative of zs by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzs_dtau
!> dzm_dtau(j,i): Derivative of zm by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzm_dtau
!> dzb_dtau(j,i): Derivative of zb by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzb_dtau
!> dzl_dtau(j,i): Derivative of zl by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dzl_dtau
!> dH_c_dtau(j,i): Derivative of H_c by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_c_dtau
!> dH_t_dtau(j,i): Derivative of H_t by tau (time)
   real(dp), dimension(0:JMAX,0:IMAX) :: dh_t_dtau
!> p_weert(j,i): Weertman exponent for the basal shear stress
   integer(i4b), dimension(0:JMAX,0:IMAX) :: p_weert
!> q_weert(j,i): Weertman exponent for the basal pressure
   integer(i4b), dimension(0:JMAX,0:IMAX) :: q_weert
!> p_weert_inv(j,i): Inverse of p_weert
   real(dp), dimension(0:JMAX,0:IMAX) :: p_weert_inv
!> c_slide(j,i): Basal sliding coefficient
   real(dp), dimension(0:JMAX,0:IMAX) :: c_slide
!> d_help_b(j,i): Auxiliary quantity for the computation of vx_b and vy_b
   real(dp), dimension(0:JMAX,0:IMAX) :: d_help_b
!> c_drag(j,i): Auxiliary quantity for the computation of the basal drag
   real(dp), dimension(0:JMAX,0:IMAX) :: c_drag
!> p_b_w(j,i): Basal water pressure
   real(dp), dimension(0:JMAX,0:IMAX) :: p_b_w
!> vx_b(j,i): Velocity in x-direction at the ice base, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vx_b
!> vy_b(j,i): Velocity in y-direction at the ice base, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: vy_b
!> vx_m(j,i): Mean (depth-averaged) velocity in x-direction, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vx_m
!> vy_m(j,i): Mean (depth-averaged) velocity in y-direction, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: vy_m
!> ratio_sl_x(j,i): Ratio of basal to surface velocity (slip ratio)
!>                  in x-direction, at (i+1/2,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: ratio_sl_x
!> ratio_sl_y(j,i): Ratio of basal to surface velocity (slip ratio)
!>                  in y-direction, at (i,j+1/2)
   real(dp), dimension(0:JMAX,0:IMAX) :: ratio_sl_y
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vx_b_g
!> (.)_g(j,i): Staggered-grid quantity (.) interpolated to grid point (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vy_b_g
!> vz_b(j,i): Velocity in z-direction at the ice base, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vz_b
!> vz_m(j,i): Velocity in z-direction at the position z=zm (interface between
!>            the upper (kc) and the lower (kt) domain), at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vz_m
!> vx_s_g(j,i): Velocity in x-direction at the ice surface, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vx_s_g
!> vy_s_g(j,i): Velocity in x-direction at the ice surface, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vy_s_g
!> vz_s(j,i): Velocity in z-direction at the ice surface, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vz_s
!> flui_ave_sia(j,i): Depth-averaged fluidity of the SIA
   real(dp), dimension(0:JMAX,0:IMAX) :: flui_ave_sia
!> h_diff(j,i): Diffusivity of the SIA evolution equation of the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: h_diff
!> qx(j,i): Volume flux in x-direction (depth-integrated vx, at (i+1/2,j))
   real(dp), dimension(0:JMAX,0:IMAX) :: qx
!> qy(j,i): Volume flux in y-direction (depth-integrated vy, at (i,j+1/2))
   real(dp), dimension(0:JMAX,0:IMAX) :: qy
!> q_gl_g(j,i): Volume flux across the grounding line, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: q_gl_g
!> q_geo(j,i): Geothermal heat flux
   real(dp), dimension(0:JMAX,0:IMAX) :: q_geo
!> temp_b(j,i): Basal temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_b
!> temph_b(j,i): Basal temperature relative to the pressure melting point
   real(dp), dimension(0:JMAX,0:IMAX) :: temph_b
!> Q_bm(j,i): Basal melting rate
   real(dp), dimension(0:JMAX,0:IMAX) :: q_bm
!> Q_tld(j,i): Water drainage rate from the temperate layer
   real(dp), dimension(0:JMAX,0:IMAX) :: q_tld
!> Q_b_tot(j,i): Sum of Q_bm and Q_tld
   real(dp), dimension(0:JMAX,0:IMAX) :: q_b_tot
!> H_w(j,i): Thickness of the water column under the ice base
   real(dp), dimension(0:JMAX,0:IMAX) :: h_w
!> accum(j,i): Accumulation rate at the ice surface
!>             (includes liquid precipitation = rainfall!)
   real(dp), dimension(0:JMAX,0:IMAX) :: accum
!> snowfall(j,i): Snowfall rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: snowfall
!> rainfall(j,i): Rainfall rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: rainfall
!> ET(j,i): Temperature excess at the ice surface
!>          (positive degree days divided by time)
   real(dp), dimension(0:JMAX,0:IMAX) :: et
!> melt(j,i): Melting rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: melt
!> melt_star(j,i): Superimposed ice formation rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: melt_star
!> evap(j,i): Evaporation rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: evap
!> runoff(j,i): Runoff rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: runoff
!> as_perp(j,i): Accumulation-ablation function at the ice surface (SMB)
   real(dp), dimension(0:JMAX,0:IMAX) :: as_perp
!> as_perp_apl(j,i): Applied accumulation-ablation function (SMB)
   real(dp), dimension(0:JMAX,0:IMAX) :: as_perp_apl
!> smb_corr_prescribed(j,i): Prescribed SMB correction
   real(dp), dimension(0:JMAX,0:IMAX) :: smb_corr_prescribed
!> smb_corr(j,i): Diagnosed SMB correction
   real(dp), dimension(0:JMAX,0:IMAX) :: smb_corr
!> mb_source_apl(j,i): Applied mass balance source (SMB, BMB, calving)
   real(dp), dimension(0:JMAX,0:IMAX) :: mb_source_apl
!> runoff_top(j,i): Runoff rate (accounting at ice surface)
   real(dp), dimension(0:JMAX,0:IMAX) :: runoff_top
!> runoff_bot(j,i): Runoff rate (accounting at ice bottom)
   real(dp), dimension(0:JMAX,0:IMAX) :: runoff_bot
!> disc_top(j,i): Ice discharge rate (accounting at ice surface)
   real(dp), dimension(0:JMAX,0:IMAX) :: disc_top
!> calv_grounded(j,i): Calving rate of grounded ice
   real(dp), dimension(0:JMAX,0:IMAX) :: calv_grounded
!> temp_s(j,i): Ice surface temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_s
!> am_perp(j,i): Ice volume flux across the z=zm interface
   real(dp), dimension(0:JMAX,0:IMAX) :: am_perp
!> am_perp_st(j,i): Steady-state part of am_perp
!>                  (without contribution of dzm_dtau)
   real(dp), dimension(0:JMAX,0:IMAX) :: am_perp_st
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: zs_neu
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: zm_neu
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: zb_neu
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: zl_neu
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: h_c_neu
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:JMAX,0:IMAX) :: h_t_neu

!> zs_ref(j,i): Reference elevation for precip_present, temp_ma_present
!>              and temp_mj_present
   real(dp), dimension(0:JMAX,0:IMAX) :: zs_ref

!> accum_present(j,i): Present-day accumulation rate at the ice surface
!>                     (for EISMINT, ISMIP HEINO and the north and south
!>                     polar caps of Mars)
   real(dp), dimension(0:JMAX,0:IMAX) :: accum_present
!> precip_ma_present(j,i): Present-day mean annual precipitation rate
!>                         at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: precip_ma_present
!> precip_ma_lgm_anom(j,i): LGM anomaly (ratio LGM/present) of the mean annual
!>                          precipitation rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX) :: precip_ma_lgm_anom
!> temp_ma_present(j,i): Present-day mean annual surface temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_ma_present
!> temp_mj_present(j,i): Present-day mean summer (northern hemisphere: July,
!>                       southern hemisphere: January) surface temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_mj_present
!> temp_ma_lgm_anom(j,i): LGM anomaly (difference LGM - present) of the mean
!>                        annual surface temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_ma_lgm_anom
!> temp_mj_lgm_anom(j,i): LGM anomaly (difference LGM - present) of the mean
!>                        summer (northern hemisphere: July, southern
!>                        hemisphere: January) surface temperature
   real(dp), dimension(0:JMAX,0:IMAX) :: temp_mj_lgm_anom

!> dist_dxdy(jr,ir): Distance between grid points with delta_i=ir, delta_j=jr
   real(dp), dimension(-JMAX:JMAX,-IMAX:IMAX) :: dist_dxdy

!> precip_present(j,i,n): Present-day mean monthly precipitation rate
!>                        at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX,12) :: precip_present
!> precip_lgm_anom(j,i,n): LGM anomaly (ratio LGM/present) of the mean monthly
!>                         precipitation rate at the ice surface
   real(dp), dimension(0:JMAX,0:IMAX,12) :: precip_lgm_anom
!> gamma_precip_lgm_anom(j,i,n): negative natural logarithm of
!>                               precip_lgm_anom(j,i,n)
   real(dp), dimension(0:JMAX,0:IMAX,12) :: gamma_precip_lgm_anom
!> temp_mm_present(j,i,n): Present-day mean monthly surface temperature
   real(dp), dimension(0:JMAX,0:IMAX,12) :: temp_mm_present
!> temp_mm_lgm_anom(j,i,n): LGM anomaly (difference LGM - present) of the mean
!>                          monthly surface temperature
   real(dp), dimension(0:JMAX,0:IMAX,12) :: temp_mm_lgm_anom

!> d_help_c(kc,j,i): Auxiliary quantity for the computation of vx, vy und zs
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: d_help_c
!> vx_c(kc,j,i): Velocity in x-direction in the upper (kc) ice domain
!>               (at (i+1/2,j,kc))
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: vx_c
!> vy_c(kc,j,i): Velocity in y-direction in the upper (kc) ice domain
!>               (at (i,j+1/2,kc))
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: vy_c
!> vz_c(kc,j,i): Velocity in z-direction in the upper (kc) ice domain
!>               (at (i,j,kc+1/2))
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: vz_c
!> temp_c(kc,j,i): Temperature in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: temp_c
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: temp_c_neu
!> temp_c_m(kc,j,i): Melting temperature in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: temp_c_m
!> age_c(kc,j,i): Age in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: age_c
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: age_c_neu
!> txz_c(kc,j,i): Shear stress txz in the upper (kc) ice domain
!>                (at (i+1/2,j,kc))
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: txz_c
!> tyz_c(kc,j,i): Shear stress tyz in the upper (kc) ice domain
!>                (at (i,j+1/2,kc))
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: tyz_c
!> sigma_c(kc,j,i): Effective stress in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: sigma_c
!> enh_c(kc,j,i): Flow enhancement factor in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: enh_c

!> de_ssa(j,i): Effective strain rate of the SSA, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: de_ssa
!> vis_int_g(j,i): Depth-integrated viscosity of the SSA, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vis_int_g
!> vx_g(j,i): Velocity in x-direction of the SSA, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vx_g
!> vy_g(j,i): Velocity in y-direction of the SSA, at (i,j)
   real(dp), dimension(0:JMAX,0:IMAX) :: vy_g

!> d_help_t(kt,j,i): Auxiliary quantity for the computation of vx, vy und zs
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: d_help_t
!> vx_t(kt,j,i): Velocity in x-direction in the lower (kt) ice domain
!>               (at (i+1/2,j,kt))
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: vx_t
!> vy_t(kt,j,i): Velocity in y-direction in the lower (kt) ice domain
!>               (at (i,j+1/2,kt))
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: vy_t
!> vz_t(kt,j,i): Velocity in z-direction in the lower (kt) ice domain
!>               (at (i,j,kt+1/2))
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: vz_t
!> omega_t(kt,j,i): Water content in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: omega_t
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: omega_t_neu
!> temp_t_m(kt,j,i): Melting temperature in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: temp_t_m
!> age_t(kt,j,i): Age in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: age_t
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: age_t_neu
!> txz_t(kt,j,i): Shear stress txz in the lower (kt) ice domain
!>                (at (i+1/2,j,kt))
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: txz_t
!> tyz_t(kt,j,i): Shear stress tyz in the lower (kt) ice domain
!>                (at (i,j+1/2,kt))
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: tyz_t
!> sigma_t(kt,j,i): Effective stress in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: sigma_t
!> enh_t(kt,j,i): Flow enhancement factor in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: enh_t

!> temp_r(kr,j,i): Temperature in the bedrock
   real(dp), dimension(0:KRMAX,0:JMAX,0:IMAX) :: temp_r
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KRMAX,0:JMAX,0:IMAX) :: temp_r_neu

!> enth_c(kc,j,i): Enthalpy in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: enth_c
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: enth_c_neu
!> omega_c(kc,j,i): Water content in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: omega_c
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: omega_c_neu

!> enth_t(kt,j,i): Enthalpy in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: enth_t
!> (.)_neu: New value of quantity (.) computed during an integration step
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: enth_t_neu

!> dxx_c(kc,j,i): Strain rate dxx in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: dxx_c
!> dyy_c(kc,j,i): Strain rate dyy in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: dyy_c
!> dxy_c(kc,j,i): Strain rate dxy in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: dxy_c
!> dxz_c(kc,j,i): Strain rate dxz in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: dxz_c
!> dyz_c(kc,j,i): Strain rate dyz in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: dyz_c
!> de_c(kc,j,i): Full effective strain rate in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: de_c
!> lambda_shear_c(kc,j,i): Shear fraction in the upper (kc) ice domain
   real(dp), dimension(0:KCMAX,0:JMAX,0:IMAX) :: lambda_shear_c

!> dxx_t(kt,j,i): Strain rate dxx in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: dxx_t
!> dyy_t(kt,j,i): Strain rate dyy in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: dyy_t
!> dxy_t(kt,j,i): Strain rate dxy in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: dxy_t
!> dxz_t(kt,j,i): Strain rate dxz in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: dxz_t
!> dyz_t(kt,j,i): Strain rate dyz in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: dyz_t
!> de_t(kt,j,i): Full effective strain rate in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: de_t
!> lambda_shear_t(kt,j,i): Shear fraction in the lower (kt) ice domain
   real(dp), dimension(0:KTMAX,0:JMAX,0:IMAX) :: lambda_shear_t

!-------- Physical parameters --------

!> RHO: Density of ice
   real(dp) :: rho
!> RHO_W: Density of pure water
   real(dp) :: rho_w
!> RHO_SW: Density of sea water
   real(dp) :: rho_sw
!> L: Latent heat of ice
   real(dp) :: l
!> G: Acceleration due to gravity
   real(dp) :: g
!> NUE: Water diffusivity in ice
   real(dp) :: nue
!> BETA: Clausius-Clapeyron gradient of ice
   real(dp) :: beta
!> DELTA_TM_SW: Melting point depression of sea water due to its
!>              average salinity
   real(dp) :: delta_tm_sw
!> OMEGA_MAX: Threshold value for the water content of temperate ice
   real(dp) :: omega_max
!> H_R: Thickness of the modelled lithosphere layer
   real(dp) :: h_r
!> RHO_C_R: Density times specific heat of the lithosphere
   real(dp) :: rho_c_r
!> KAPPA_R: Heat conductivity of the lithosphere
   real(dp) :: kappa_r
!> RHO_A: Density of the asthenosphere
   real(dp) :: rho_a
!> R_T: Coefficient of the water-content dependence in the rate factor
!>      for temperate ice
   real(dp) :: r_t
!> R: Radius of the planet
   real(dp) :: r
!> A: Semi-major axis of the planet
   real(dp) :: a
!> B: Semi-minor axis of the planet
   real(dp) :: b
!> LAMBDA0: Reference longitude (central meridian) of the stereographic
!>          projection
   real(dp) :: lambda0
!> PHI0: Standard parallel of the stereographic projection
   real(dp) :: phi0
!> S_STAT_0: Standard deviation of the air termperature for the
!>           degree-day model
   real(dp) :: s_stat_0
!> BETA1_0: Degree-day factor for snow
   real(dp) :: beta1_0
!> BETA1_LT_0: Degree-day factor for snow at low summer temperatures
   real(dp) :: beta1_lt_0
!> BETA1_HT_0: Degree-day factor for snow at high summer temperatures
   real(dp) :: beta1_ht_0
!> BETA2_0: Degree-day factor for ice
   real(dp) :: beta2_0
!> BETA2_LT_0: Degree-day factor for ice at low summer temperatures
   real(dp) :: beta2_lt_0
!> BETA2_HT_0: Degree-day factor for ice at high summer temperatures
   real(dp) :: beta2_ht_0
!> PHI_SEP_0: Separation latitude for the computation of the degree-day
!>            factors beta1 and beta2: Equatorward of phi_sep, only the
!>            high-temperature values are used, whereas poleward of phi_sep,
!>            beta1 and beta2 are temperature-dependent
   real(dp) :: phi_sep_0
!> PMAX_0: Saturation factor for the formation of superimposed ice
   real(dp) :: pmax_0
!> MU_0: Firn-warming correction
   real(dp) :: mu_0

!> RF(n): Tabulated values for the rate factor of cold ice
   real(dp), dimension(-190:10) :: rf
!> KAPPA(n): Tabulated values for the heat conductivity of ice
   real(dp), dimension(-190:10) :: kappa
!> C(n): Tabulated values for the specific heat of ice
   real(dp), dimension(-190:10) :: c

!-------- Mathematical constants -------- 

!> pi: Constant pi
   real(dp), parameter :: pi = 3.141592653589793_dp
!> pi_inv: Inverse of pi
   real(dp), parameter :: pi_inv = 1.0_dp/pi
!> pi_180: pi divided by 180 (-> deg to rad)
   real(dp), parameter :: pi_180 = pi/180.0_dp
!> pi_180_inv: 180 divided by pi (-> rad to deg)
   real(dp), parameter :: pi_180_inv = 180.0_dp/pi

!> euler: Euler number
   real(dp), parameter :: euler = 2.718281828459045_dp

!> eps: Small number
   real(dp), parameter :: eps = 1.0e-05_dp
!> epsi: Very small number
   real(dp), parameter :: epsi = 1.0e-12_dp

#if (!defined(ALLOW_GRDCHK) && !defined(ALLOW_OPENAD)) /* Normal */

!> eps_sp: Small number to single-precision accuracy
   real(sp), parameter :: eps_sp = epsilon(1.0_sp)
!> eps_sp_dp: Small number to single-precision accuracy in double precision
   real(dp), parameter :: eps_sp_dp = eps_sp
!> eps_dp: Small number to double-precision accuracy
   real(dp), parameter :: eps_dp = epsilon(1.0_dp)

#else /* OpenAD */

   !myepsilon_sp was computed using the code below. 4 is the value of sp
   !real(4) :: y = 1.0
   !print *, EPSILON(y)
   real(sp), parameter :: myepsilon_sp  = 1.19209290e-07
   real(sp), parameter :: eps_sp = myepsilon_sp
   !myepsilon_sp_dp was computed using the code below. 4 is the value of sp,
   !8 is the value of dp
   !real(4) :: y = 1.0
   !print *, REAL(EPSILON(y),8)
   real(dp), parameter :: myepsilon_sp_dp  = 1.1920928955078125e-007
   real(dp), parameter :: eps_sp_dp = myepsilon_sp_dp
   !myepsilon_dp was computed using the code below. 8 is the value of dp
   !real(8) :: y = 1.0
   !print *, EPSILON(y)
   real(dp), parameter :: myepsilon_dp  = 2.2204460492503131e-016
   real(dp), parameter :: eps_dp = myepsilon_dp

#endif /* Normal vs. OpenAD */

!-------- Further quantities -------- 

!> year_zero: SICOPOLIS year zero in astronomical year numbering
!>            [ = signed year CE (AD) ]
   real(dp) :: year_zero

!> ch_domain_long: Long name of the computational domain
   character(len=64) :: ch_domain_long
!> ch_domain_short: Short name of the computational domain
   character(len=16) :: ch_domain_short

!> forcing_flag: Flag for the forcing type.
!>               1: forcing by a spatially constant surface temperature
!>                  anomaly (delta_ts),
!>               2: forcing by a glacial index (glac_index),
!>               3: forcing by time-dependent surface temperature
!>                  and precipitation data.
   integer(i2b) :: forcing_flag

!> n_core: Number of positions to be considered in the time-series file
!>         for deep boreholes
   integer(i4b) :: n_core
!> lambda_core(n): Geographical longitude of the prescribed borehole positions
   real(dp), dimension(:), allocatable :: lambda_core
!> phi_core(n): Geographical latitude of the prescribed borehole positions
   real(dp), dimension(:), allocatable :: phi_core
!> x_core(n): Coordinate xi (= x) of the prescribed borehole positions
   real(dp), dimension(:), allocatable :: x_core
!> y_core(n): Coordinate eta (= y) of the prescribed borehole positions
   real(dp), dimension(:), allocatable :: y_core
!> ch_core(n): Names of the prescribed borehole positions
   character(len=16), dimension(:), allocatable :: ch_core

!> grip_time_min: Minimum time of the data values for the
!>                surface temperature anomaly
   integer(i4b) :: grip_time_min
!> grip_time_stp: Time step of the data values for the
!>                surface temperature anomaly
   integer(i4b) :: grip_time_stp
!> grip_time_max: Maximum time of the data values for the
!>                surface temperature anomaly
   integer(i4b) :: grip_time_max
!> ndata_grip: Number of data values for the surface temperature anomaly
   integer(i4b) :: ndata_grip
!> griptemp(n): Data values for the surface temperature anomaly
   real(dp), dimension(:), allocatable :: griptemp

!> gi_time_min: Minimum time of the data values for the glacial index
   integer(i4b) :: gi_time_min
!> gi_time_stp: Time step of the data values for the glacial index
   integer(i4b) :: gi_time_stp
!> gi_time_max: Maximum time of the data values for the glacial index
   integer(i4b) :: gi_time_max
!> ndata_gi: Number of data values for the glacial index
   integer(i4b) :: ndata_gi
!> glacial_index(n): Data values for the glacial index
   real(dp), dimension(:), allocatable :: glacial_index

!> specmap_time_min: Minimum time of the data values for the sea level
   integer(i4b) :: specmap_time_min
!> specmap_time_stp: Time step of the data values for the sea level
   integer(i4b) :: specmap_time_stp
!> specmap_time_max: Maximum time of the data values for the sea level
   integer(i4b) :: specmap_time_max
!> ndata_specmap: Number of data values for the sea level
   integer(i4b) :: ndata_specmap
!> specmap_zsl(n): Data values for the sea level
   real(dp), dimension(:), allocatable :: specmap_zsl

!> time_target_topo_init: Initial time for target-topography adjustment
   real(dp) :: time_target_topo_init
!> time_target_topo_final: Final time for target-topography adjustment
   real(dp) :: time_target_topo_final
!> target_topo_tau: Relaxation time for target-topography adjustment
   real(dp) :: target_topo_tau
!> maske_target(j,i): Target topography (ice-land-ocean mask)
   integer(i2b), dimension(0:JMAX,0:IMAX) :: maske_target
!> zs_target(j,i): Target topography (ice surface)
   real(dp), dimension(0:JMAX,0:IMAX) :: zs_target
!> zb_target(j,i): Target topography (ice base)
   real(dp), dimension(0:JMAX,0:IMAX) :: zb_target
!> zl_target(j,i): Target topography (lithosphere surface)
   real(dp), dimension(0:JMAX,0:IMAX) :: zl_target
!> H_target(j,i): Target topography (ice thickness)
   real(dp), dimension(0:JMAX,0:IMAX) :: h_target

!> maske_maxextent(j,i): Maximum ice extent mask.
!>                       0: not allowed to glaciate,
!>                       1: allowed to glaciate.
   integer(i2b), dimension(0:JMAX,0:IMAX) :: maske_maxextent

!> ncid_ser: ID of the NetCDF time-series output file
   integer(i4b) :: ncid_ser
!> ncid_core: ID of the NetCDF time-series output file for the deep ice cores
   integer(i4b) :: ncid_core

!> ncid_temp_precip: ID of the NetCDF file containing the
!>                   surface-temperature and precipitation data
!>                   as functions of time
   integer(i4b) :: ncid_temp_precip
!> ndata_temp_precip: Number of surface-temperature
!>                    and precipitation data
   integer(i4b) :: ndata_temp_precip
!> temp_precip_time_min: Minimum time of the surface-temperature
!>                       and precipitation data
   real(dp) :: temp_precip_time_min
!> temp_precip_time_stp: Time step of the surface-temperature
!>                       and precipitation data
   real(dp) :: temp_precip_time_stp
!> temp_precip_time_max: Maximum time of the surface-temperature
!>                       and precipitation data
   real(dp) :: temp_precip_time_max
!> temp_precip_time(n): Times of the surface-temperature
!>                      and precipitation data
   real(dp), dimension(:), allocatable :: temp_precip_time

!> kei(n): Tabulated values of the kei function (Kelvin function of zero order)
   real(dp), dimension(-10000:10000) :: kei
!> n_data_kei: Number of tabulated values of the kei function
   integer(i4b):: n_data_kei
!> kei_r_max: Maximum value of the argument r of the tabulated kei function
   real(dp) :: kei_r_max
!> kei_r_incr: Increment of the argument r of the tabulated kei function
   real(dp) :: kei_r_incr

!> rcl1: Maximum length of record for input files
!>       (with factor 3 safety margin)
   integer(i4b), parameter :: rcl1 = 3*8*(imax+1)
!> rcl2: Maximum length of record for input mask files
!>       (with factor 3 safety margin)
   integer(i4b), parameter :: rcl2 = 3  *(imax+1)

!> no_value_pos_1: Positive no-value parameter
   real(dp), parameter :: no_value_pos_1 =  1.11e+11_dp
!> no_value_pos_2: Positive no-value parameter
   real(dp), parameter :: no_value_pos_2 =  9.999e+03_dp
!> no_value_neg_1: Negative no-value parameter
   real(dp), parameter :: no_value_neg_1 = -1.11e+11_dp
!> no_value_neg_2: Negative no-value parameter
   real(dp), parameter :: no_value_neg_2 = -9.999e+03_dp

!> errormsg: Error-message string
   character(len=256) :: errormsg
!> end_of_line: End-of-line string
   character, parameter :: end_of_line = char(10)

#if (defined(ALLOW_GRDCHK) || defined(ALLOW_OPENAD)) /* OpenAD */
!> fc: scalar cost function
   real(dp) :: fc
!> objf_test: cost function
   real(dp) :: objf_test
!> mult_test: cost function
   real(dp) :: mult_test
!> age_data: array of gridded ages to be used in adjoint mode
   real(dp), dimension(0:KTMAX+KCMAX+1,0:JMAX,0:IMAX) :: age_data
#endif /* OpenAD */

end module sico_variables_m
!