!> @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 ! . ! ! Copyright 1997-2021 Leibniz Universitaet Hannover !--------------------------------------------------------------------------------------------------! ! ! Current revisions: ! ----------------- ! ! ! Former revisions: ! ----------------- ! $Id: basic_constants_and_equations_mod.f90 4828 2021-01-05 11:21:41Z gronemeier $ ! 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 Krüger (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 Krüger 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 Krüger (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