source: palm/trunk/SOURCE/basic_constants_and_equations_mod.f90 @ 4880

!> @file basic_constants_and_equations_mod.f90
!--------------------------------------------------------------------------------------------------!
! This file is part of the PALM model system.
!
! PALM 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.
!
! PALM 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 PALM. If not, see
! <http://www.gnu.org/licenses/>.
!
! Copyright 1997-2021 Leibniz Universitaet Hannover
!--------------------------------------------------------------------------------------------------!
!
! Current revisions:
! -----------------
!
!
! Former revisions:
! -----------------
! $Id: basic_constants_and_equations_mod.f90 4828 2021-01-05 11:21:41Z suehring $
! Implement snow and graupel (bulk microphysics)
! 
! 4509 2020-04-26 15:57:55Z raasch
! file re-formatted to follow the PALM coding standard
!
! 4502 2020-04-17 16:14:16Z schwenkel
! Implementation of ice microphysics
!
! 4400 2020-02-10 20:32:41Z suehring
! Move routine to transform coordinates from netcdf_interface_mod to
! basic_constants_and_equations_mod
!
! 4360 2020-01-07 11:25:50Z suehring
! Corrected "Former revisions" section
!
! 4088 2019-07-11 13:57:56Z Giersch
! Comment of barometric formula improved, function for ideal gas law revised
!
! 4084 2019-07-10 17:09:11Z knoop
! Changed precomputed fractions to be variable based
!
! 4055 2019-06-27 09:47:29Z suehring
! Added rgas_univ (universal gas constant) (E.C. Chan)
!
!
! 3655 2019-01-07 16:51:22Z knoop
! OpenACC port for SPEC
! 3361 2018-10-16 20:39:37Z knoop
! New module (introduced with modularization of bulk cloud physics model)
!
!
!
!
! Description:
! ------------
!> This module contains all basic (physical) constants and functions for the calculation of
!> diagnostic quantities.
!-                    -----------------------------------------------------------------------------!
 MODULE basic_constants_and_equations_mod


    USE kinds

    IMPLICIT NONE


    REAL(wp), PARAMETER ::  c_p = 1005.0_wp                           !< heat capacity of dry air (J kg-1 K-1)
    REAL(wp), PARAMETER ::  c_w = 4185.0_wp                           !< heat capacity of water at 0°C (J kg-1 K-1)
    REAL(wp), PARAMETER ::  degc_to_k = 273.15_wp                     !< temperature (in K) of 0 deg C (K)
    REAL(wp), PARAMETER ::  g = 9.81_wp                               !< gravitational acceleration (m s-2)
    REAL(wp), PARAMETER ::  kappa = 0.4_wp                            !< von Karman constant
    REAL(wp), PARAMETER ::  l_m = 0.33E+06_wp                         !< latent heat of water melting (J kg-1)
    REAL(wp), PARAMETER ::  l_v = 2.5E+06_wp                          !< latent heat of water vaporization (J kg-1)
    REAL(wp), PARAMETER ::  l_s = l_m + l_v                           !< latent heat of water sublimation (J kg-1)
    REAL(wp), PARAMETER ::  molecular_weight_of_nacl = 0.05844_wp     !< mol. m. NaCl (kg mol-1)
    REAL(wp), PARAMETER ::  molecular_weight_of_c3h4o4 = 0.10406_wp   !< mol. m. malonic acid (kg mol-1)
    REAL(wp), PARAMETER ::  molecular_weight_of_nh4no3 = 0.08004_wp   !< mol. m. ammonium sulfate (kg mol-1)
    REAL(wp), PARAMETER ::  molecular_weight_of_water = 0.01801528_wp !< mol. m. H2O (kg mol-1)
    REAL(wp), PARAMETER ::  pi = 3.141592654_wp                       !< PI
    !$ACC DECLARE COPYIN(pi)
    REAL(wp), PARAMETER ::  rgas_univ = 8.31446261815324_wp           !< universal gas constant (J K-1 mol-1)
    REAL(wp), PARAMETER ::  rho_i = 916.7_wp                          !> density of pure ice (kg m-3)
    REAL(wp), PARAMETER ::  rho_l = 1.0E3_wp                          !< density of water (kg m-3)
    REAL(wp), PARAMETER ::  rho_nacl = 2165.0_wp                      !< density of NaCl (kg m-3)
    REAL(wp), PARAMETER ::  rho_c3h4o4 = 1600.0_wp                    !< density of malonic acid (kg m-3)
    REAL(wp), PARAMETER ::  rho_nh4no3 = 1720.0_wp                    !< density of ammonium sulfate (kg m-3)
    REAL(wp), PARAMETER ::  r_d = 287.0_wp                            !< sp. gas const. dry air (J kg-1 K-1)
    REAL(wp), PARAMETER ::  r_v = 461.51_wp                           !< sp. gas const. water vapor (J kg-1 K-1)
    REAL(wp), PARAMETER ::  sigma_sb = 5.67037E-08_wp                 !< Stefan-Boltzmann constant
    REAL(wp), PARAMETER ::  solar_constant = 1368.0_wp                !< solar constant at top of atmosphere
    REAL(wp), PARAMETER ::  vanthoff_nacl = 2.0_wp                    !< van't Hoff factor for NaCl
    REAL(wp), PARAMETER ::  vanthoff_c3h4o4 = 1.37_wp                 !< van't Hoff factor for malonic acid
    REAL(wp), PARAMETER ::  vanthoff_nh4no3 = 2.31_wp                 !< van't Hoff factor for ammonium sulfate

    REAL(wp), PARAMETER ::  p_0 = 100000.0_wp                         !< standard pressure reference state

    REAL(wp), PARAMETER ::  cp_d_rd = c_p / r_d   !< precomputed c_p / r_d
    REAL(wp), PARAMETER ::  g_d_cp  = g   / c_p   !< precomputed g / c_p
    REAL(wp), PARAMETER ::  lv_d_cp = l_v / c_p   !< precomputed l_v / c_p
    REAL(wp), PARAMETER ::  ls_d_cp = l_s / c_p   !< precomputed l_s / c_p
    REAL(wp), PARAMETER ::  lv_d_rd = l_v / r_d   !< precomputed l_v / r_d
    REAL(wp), PARAMETER ::  rd_d_rv = r_d / r_v   !< precomputed r_d / r_v
    REAL(wp), PARAMETER ::  rd_d_cp = r_d / c_p   !< precomputed r_d / c_p

    REAL(wp) ::  molecular_weight_of_solute = molecular_weight_of_nacl  !< mol. m. NaCl (kg mol-1)
    REAL(wp) ::  rho_s = rho_nacl                                       !< density of NaCl (kg m-3)
    REAL(wp) ::  vanthoff = vanthoff_nacl                               !< van't Hoff factor for NaCl

    SAVE

    PRIVATE magnus_0d,                                                                             &
            magnus_1d,                                                                             &
            magnus_tl_0d,                                                                          &
            magnus_tl_1d,                                                                          &
            magnus_0d_ice,                                                                         &
            magnus_1d_ice,                                                                         &
            ideal_gas_law_rho_0d,                                                                  &
            ideal_gas_law_rho_1d,                                                                  &
            ideal_gas_law_rho_pt_0d,                                                               &
            ideal_gas_law_rho_pt_1d,                                                               &
            exner_function_0d,                                                                     &
            exner_function_1d,                                                                     &
            exner_function_invers_0d,                                                              &
            exner_function_invers_1d,                                                              &
            barometric_formula_0d,                                                                 &
            barometric_formula_1d


    INTERFACE convert_utm_to_geographic
       MODULE PROCEDURE convert_utm_to_geographic
    END INTERFACE convert_utm_to_geographic

    INTERFACE magnus
       MODULE PROCEDURE magnus_0d
       MODULE PROCEDURE magnus_1d
    END INTERFACE magnus

    INTERFACE magnus_tl
       MODULE PROCEDURE magnus_tl_0d
       MODULE PROCEDURE magnus_tl_1d
    END INTERFACE magnus_tl

    INTERFACE magnus_ice
       MODULE PROCEDURE magnus_0d_ice
       MODULE PROCEDURE magnus_1d_ice
    END INTERFACE magnus_ice

    INTERFACE ideal_gas_law_rho
       MODULE PROCEDURE ideal_gas_law_rho_0d
       MODULE PROCEDURE ideal_gas_law_rho_1d
    END INTERFACE ideal_gas_law_rho

    INTERFACE ideal_gas_law_rho_pt
       MODULE PROCEDURE ideal_gas_law_rho_pt_0d
       MODULE PROCEDURE ideal_gas_law_rho_pt_1d
    END INTERFACE ideal_gas_law_rho_pt

    INTERFACE exner_function
       MODULE PROCEDURE exner_function_0d
       MODULE PROCEDURE exner_function_1d
    END INTERFACE exner_function

    INTERFACE exner_function_invers
       MODULE PROCEDURE exner_function_invers_0d
       MODULE PROCEDURE exner_function_invers_1d
    END INTERFACE exner_function_invers

    INTERFACE barometric_formula
       MODULE PROCEDURE barometric_formula_0d
       MODULE PROCEDURE barometric_formula_1d
    END INTERFACE barometric_formula
!
!-- Public routines
    PUBLIC convert_utm_to_geographic

 CONTAINS


!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Convert UTM coordinates into geographic latitude and longitude. Conversion is based on the work
!> of KrÃŒger (1912) DOI: 10.2312/GFZ.b103-krueger28 and Karney (2013) DOI: 10.1007/s00190-012-0578-z
!> Based on a JavaScript of the geodesy function library written by chrisveness
!> https://github.com/chrisveness/geodesy
!--------------------------------------------------------------------------------------------------!
 SUBROUTINE convert_utm_to_geographic( crs, eutm, nutm, lon, lat )

    INTEGER(iwp) ::  j   !< loop index

    REAL(wp), INTENT(in)  ::  eutm !< easting (UTM)
    REAL(wp), INTENT(out) ::  lat  !< geographic latitude in degree
    REAL(wp), INTENT(out) ::  lon  !< geographic longitude in degree
    REAL(wp), INTENT(in)  ::  nutm !< northing (UTM)

    REAL(wp) ::  a           !< 2*pi*a is the circumference of a meridian
    REAL(wp) ::  cos_eta_s   !< cos(eta_s)
    REAL(wp) ::  delta_i     !<
    REAL(wp) ::  delta_tau_i !<
    REAL(wp) ::  e           !< eccentricity
    REAL(wp) ::  eta         !<
    REAL(wp) ::  eta_s       !<
    REAL(wp) ::  n           !< 3rd flattening
    REAL(wp) ::  n2          !< n^2
    REAL(wp) ::  n3          !< n^3
    REAL(wp) ::  n4          !< n^4
    REAL(wp) ::  n5          !< n^5
    REAL(wp) ::  n6          !< n^6
    REAL(wp) ::  nu          !<
    REAL(wp) ::  nu_s        !<
    REAL(wp) ::  sin_eta_s   !< sin(eta_s)
    REAL(wp) ::  sinh_nu_s   !< sinush(nu_s)
    REAL(wp) ::  tau_i       !<
    REAL(wp) ::  tau_i_s     !<
    REAL(wp) ::  tau_s       !<
    REAL(wp) ::  x           !< adjusted easting
    REAL(wp) ::  y           !< adjusted northing

    REAL(wp), DIMENSION(6) ::  beta !< 6th order KrÃŒger expressions

    REAL(wp), DIMENSION(8), INTENT(in) ::  crs !< coordinate reference system, consists of
                                               !< (/semi_major_axis,
                                               !<   inverse_flattening,
                                               !<   longitude_of_prime_meridian,
                                               !<   longitude_of_central_meridian,
                                               !<   scale_factor_at_central_meridian,
                                               !<   latitude_of_projection_origin,
                                               !<   false_easting,
                                               !<   false_northing /)

    x = eutm - crs(7)  ! remove false easting
    y = nutm - crs(8)  ! remove false northing
!
!-- From Karney 2011 Eq 15-22, 36:
    e = SQRT( 1.0_wp / crs(2) * ( 2.0_wp - 1.0_wp / crs(2) ) )
    n = 1.0_wp / crs(2) / ( 2.0_wp - 1.0_wp / crs(2) )
    n2 = n * n
    n3 = n * n2
    n4 = n * n3
    n5 = n * n4
    n6 = n * n5

    a = crs(1) / ( 1.0_wp + n ) * ( 1.0_wp + 0.25_wp * n2 + 0.015625_wp * n4 + 3.90625E-3_wp * n6 )

    nu  = x / ( crs(5) * a )
    eta = y / ( crs(5) * a )

!-- According to KrÃŒger (1912), eq. 26*
    beta(1) =          0.5_wp                  * n                                                 &
              -        2.0_wp /         3.0_wp * n2                                                &
              +       37.0_wp /        96.0_wp * n3                                                &
              -        1.0_wp /       360.0_wp * n4                                                &
              -       81.0_wp /       512.0_wp * n5                                                &
              +    96199.0_wp /    604800.0_wp * n6

    beta(2) =          1.0_wp /        48.0_wp * n2                                                &
              +        1.0_wp /        15.0_wp * n3                                                &
              -      437.0_wp /      1440.0_wp * n4                                                &
              +       46.0_wp /       105.0_wp * n5                                                &
              -  1118711.0_wp /   3870720.0_wp * n6

    beta(3) =         17.0_wp /       480.0_wp * n3                                                &
              -       37.0_wp /       840.0_wp * n4                                                &
              -      209.0_wp /      4480.0_wp * n5                                                &
              +     5569.0_wp /     90720.0_wp * n6

    beta(4) =       4397.0_wp /    161280.0_wp * n4                                                &
              -       11.0_wp /       504.0_wp * n5                                                &
              -   830251.0_wp /   7257600.0_wp * n6

    beta(5) =       4583.0_wp /    161280.0_wp * n5                                                &
              -   108847.0_wp /   3991680.0_wp * n6

    beta(6) =   20648693.0_wp / 638668800.0_wp * n6

    eta_s = eta
    nu_s  = nu
    DO  j = 1, 6
      eta_s = eta_s - beta(j) * SIN(2.0_wp * j * eta) * COSH(2.0_wp * j * nu)
      nu_s  = nu_s  - beta(j) * COS(2.0_wp * j * eta) * SINH(2.0_wp * j * nu)
    ENDDO

    sinh_nu_s = SINH( nu_s )
    sin_eta_s = SIN( eta_s )
    cos_eta_s = COS( eta_s )

    tau_s = sin_eta_s / SQRT( sinh_nu_s**2 + cos_eta_s**2 )

    tau_i = tau_s
    delta_tau_i = 1.0_wp

    DO WHILE ( ABS( delta_tau_i ) > 1.0E-12_wp )

      delta_i = SINH( e * ATANH( e * tau_i / SQRT( 1.0_wp + tau_i**2 ) ) )

      tau_i_s = tau_i   * SQRT( 1.0_wp + delta_i**2 ) - delta_i * SQRT( 1.0_wp + tau_i**2 )

      delta_tau_i = ( tau_s - tau_i_s ) / SQRT( 1.0_wp + tau_i_s**2 )                              &
                    * ( 1.0_wp + ( 1.0_wp - e**2 ) * tau_i**2 )                                    &
                    / ( ( 1.0_wp - e**2 ) * SQRT( 1.0_wp + tau_i**2 ) )

      tau_i = tau_i + delta_tau_i

    ENDDO

    lat = ATAN( tau_i ) / pi * 180.0_wp
    lon = ATAN2( sinh_nu_s, cos_eta_s ) / pi * 180.0_wp + crs(4)

 END SUBROUTINE convert_utm_to_geographic

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992).
!> The magnus formula is needed to calculate the saturation vapor pressure.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_0d( t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  t  !< temperature (K)

    REAL(wp) ::  magnus_0d

!
!-- Saturation vapor pressure for a specific temperature:
    magnus_0d =  611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / ( t - 29.65_wp  ) )

 END FUNCTION magnus_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992).
!> The magnus formula is needed to calculate the saturation vapor pressure.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_1d( t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  t  !< temperature (K)

    REAL(wp), DIMENSION(size(t)) ::  magnus_1d

!
!-- Saturation vapor pressure for a specific temperature:
    magnus_1d =  611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / ( t - 29.65_wp  ) )

 END FUNCTION magnus_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992) using the (ice-) liquid water
!> potential temperature.
!> The magnus formula is needed to calculate the saturation vapor pressure over a plane liquid water
!> surface.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_tl_0d( t_l )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  t_l  !< liquid water temperature (K)

    REAL(wp) ::  magnus_tl_0d

!
!-- Saturation vapor pressure for a specific temperature:
    magnus_tl_0d =  610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / ( t_l - 35.86_wp  ) )

 END FUNCTION magnus_tl_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992) using the (ice-) liquid water
!> potential temperature.
!> The magnus formula is needed to calculate the saturation vapor pressure over a plane liquid water
!> surface.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_tl_1d( t_l )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  t_l  !< liquid water temperature (K)

    REAL(wp), DIMENSION(size(t_l)) ::  magnus_tl_1d
!
!-- Saturation vapor pressure for a specific temperature:
    magnus_tl_1d =  610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / ( t_l - 35.86_wp  ) )

 END FUNCTION magnus_tl_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992).
!> The magnus formula is needed to calculate the saturation vapor pressure over a plane ice surface.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_0d_ice( t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  t  !< temperature (K)

    REAL(wp) ::  magnus_0d_ice

!
!-- Saturation vapor pressure for a specific temperature:
    !magnus_0d_ice =  611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / ( t - 0.53_wp  ) )
    magnus_0d_ice =  610.78_wp * EXP( 21.875_wp * ( t - degc_to_k ) / ( t - 7.66_wp  ) )


 END FUNCTION magnus_0d_ice

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> This function computes the magnus formula (Press et al., 1992).
!> The magnus formula is needed to calculate the saturation vapor pressure over a plane ice surface.
!--------------------------------------------------------------------------------------------------!
 FUNCTION magnus_1d_ice( t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  t  !< temperature (K)

    REAL(wp), DIMENSION(size(t)) ::  magnus_1d_ice

!
!-- Saturation vapor pressure for a specific temperature:
    !magnus_1d_ice =  611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / ( t - 0.53_wp  ) )
    magnus_1d_ice =  610.78_wp * EXP( 21.875_wp * ( t - degc_to_k ) / ( t - 7.66_wp  ) )


 END FUNCTION magnus_1d_ice

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the ideal gas law for scalar arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION ideal_gas_law_rho_0d( p, t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  p  !< pressure (Pa)
    REAL(wp), INTENT(IN) ::  t  !< temperature (K)

    REAL(wp) ::  ideal_gas_law_rho_0d

!
!-- Compute density according to ideal gas law:
    ideal_gas_law_rho_0d = p / (r_d * t)

 END FUNCTION ideal_gas_law_rho_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the ideal gas law for 1-D array arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION ideal_gas_law_rho_1d( p, t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  p  !< pressure (Pa)
    REAL(wp), INTENT(IN), DIMENSION(:) ::  t  !< temperature (K)

    REAL(wp), DIMENSION(size(p)) ::  ideal_gas_law_rho_1d

!
!-- Compute density according to ideal gas law:
    ideal_gas_law_rho_1d = p / (r_d * t)

 END FUNCTION ideal_gas_law_rho_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the ideal gas law for scalar arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION ideal_gas_law_rho_pt_0d( p, t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  p  !< pressure (Pa)
    REAL(wp), INTENT(IN) ::  t  !< temperature (K)

    REAL(wp) ::  ideal_gas_law_rho_pt_0d

!
!-- Compute density according to ideal gas law:
    ideal_gas_law_rho_pt_0d = p / (r_d * exner_function(p) * t)

 END FUNCTION ideal_gas_law_rho_pt_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the ideal gas law for 1-D array arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION ideal_gas_law_rho_pt_1d( p, t )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  p  !< pressure (Pa)
    REAL(wp), INTENT(IN), DIMENSION(:) ::  t  !< temperature (K)

    REAL(wp), DIMENSION(size(p)) ::  ideal_gas_law_rho_pt_1d

!
!-- Compute density according to ideal gas law:
    ideal_gas_law_rho_pt_1d = p / (r_d * exner_function(p) * t)

 END FUNCTION ideal_gas_law_rho_pt_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the exner function for scalar arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION exner_function_0d( p )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  p    !< pressure (Pa)

    REAL(wp) ::  exner_function_0d

!
!-- Compute exner function:
    exner_function_0d = ( p / p_0 )**( rd_d_cp )

 END FUNCTION exner_function_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the exner function for 1-D array arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION exner_function_1d( p )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  p  !< pressure (Pa)

    REAL(wp), DIMENSION(size(p)) ::  exner_function_1d

!
!-- Compute exner function:
    exner_function_1d = ( p / p_0 )**( rd_d_cp )

 END FUNCTION exner_function_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the exner function for scalar arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION exner_function_invers_0d( p )

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  p    !< pressure (Pa)

    REAL(wp) ::  exner_function_invers_0d

!
!-- Compute exner function:
    exner_function_invers_0d = ( p_0 / p )**( rd_d_cp )

 END FUNCTION exner_function_invers_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the exner function for 1-D array arguments.
!--------------------------------------------------------------------------------------------------!
 FUNCTION exner_function_invers_1d( p )

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  p  !< pressure (Pa)

    REAL(wp), DIMENSION(size(p)) ::  exner_function_invers_1d

!
!-- Compute exner function:
    exner_function_invers_1d = ( p_0 / p )**( rd_d_cp )

 END FUNCTION exner_function_invers_1d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the barometric formula for scalar arguments. The calculation is based on the assumption
!> of a polytropic atmosphere and neutral stratification, where the temperature lapse rate is g/cp.
!--------------------------------------------------------------------------------------------------!
 FUNCTION barometric_formula_0d( z, t_0, p_0)

    IMPLICIT NONE

    REAL(wp), INTENT(IN) ::  z    !< height (m)
    REAL(wp), INTENT(IN) ::  t_0  !< temperature reference state (K)
    REAL(wp), INTENT(IN) ::  p_0  !< surface pressure (Pa)

    REAL(wp) ::  barometric_formula_0d

!
!-- Compute barometric formula:
    barometric_formula_0d =  p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )

 END FUNCTION barometric_formula_0d

!--------------------------------------------------------------------------------------------------!
! Description:
! ------------
!> Compute the barometric formula for 1-D array arguments. The calculation is based on the
!> assumption of a polytropic atmosphere and neutral stratification, where the temperature lapse
!> rate is g/cp.
!--------------------------------------------------------------------------------------------------!
 FUNCTION barometric_formula_1d( z, t_0, p_0)

    IMPLICIT NONE

    REAL(wp), INTENT(IN), DIMENSION(:) ::  z  !< height (m)
    REAL(wp), INTENT(IN) ::  t_0              !< temperature reference state (K)
    REAL(wp), INTENT(IN) ::  p_0              !< surface pressure (Pa)

    REAL(wp), DIMENSION(size(z)) ::  barometric_formula_1d

!
!-- Compute barometric formula:
    barometric_formula_1d =  p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )

 END FUNCTION barometric_formula_1d

 END MODULE basic_constants_and_equations_mod