!--------------------------------------------------------------------------- ! $Id$ !=============================================================================== module Nc_Ncn_eqns ! Description: ! Equations are provided to perform calculations back-and-forth between Nc and ! Ncn, where Nc is cloud droplet concentration and Ncn is simplified cloud ! nuclei concentration. The equation that relates the two is: ! ! Nc = Ncn * H(chi); ! ! where chi is extended liquid water mixing ratio, which is equal to cloud ! water mixing ratio, rc, when both are positive. However, chi is negative in ! subsaturated air. ! ! Equation are provided relating mean cloud droplet concentration (overall), ! Ncm, and/or mean cloud droplet concentration (in-cloud), Nc_in_cloud, to ! mean simplified cloud nuclei concentration, Ncnm. ! Notes: ! ! Meaning of Nc flag combinations: ! ! l_const_Nc_in_cloud: ! When this flag is enabled, cloud droplet concentration (in-cloud) is ! constant (spatially) at a grid level (it is constant over the subgrid ! domain, but could vary over time depending on the value of l_predict_Nc). ! The value of in-cloud Nc does not vary at a grid level. This also means ! that Ncn is constant across the entire grid level. When this flag is turned ! off, both in-cloud Nc and Ncn vary at a grid level. ! ! l_predict_Nc: ! When this flag is enabled, Nc_in_cloud (or alternatively Ncm) is predicted. ! It is advanced every time step by a predictive equation, and can change ! at every time step at a grid level. When this flag is turned off, ! Nc_in_cloud does not change at a grid level over the course of a model run. ! ! 1) l_predict_Nc turned on and l_const_Nc_in_cloud turned on: ! The value of Nc_in_cloud (mean in-cloud Nc) is predicted and can change ! at every timestep at a grid level. However, the value of in-cloud Nc is ! constant (spatially) at a grid level (no subgrid variability). ! ! 2) l_predict_Nc turned on and l_const_Nc_in_cloud turned off: ! The value of Nc_in_cloud (mean in-cloud Nc) is predicted and can change ! at every timestep at a grid level. The value of in-cloud Nc also varies ! (spatially) at a grid level (subgrid variability around mean ! in-cloud Nc). ! ! 3) l_predict_Nc turned off and l_const_Nc_in_cloud turned on: ! The value of Nc_in_cloud (mean in-cloud Nc) is constant over time at a ! grid level. It retains its initial value. Additionally, the value of ! in-cloud Nc is constant (spatially) at a grid level (no subgrid ! variability). This configuration is used most often in idealized cases. ! ! 4) l_predict_Nc turned off and l_const_Nc_in_cloud turned off: ! The value of Nc_in_cloud (mean in-cloud Nc) is constant over time at a ! grid level. It retains its initial value. However, the value of ! in-cloud Nc varies (spatially) at a grid level (subgrid variability ! around mean in-cloud Nc). ! ! ! ! Nc_in_cloud/Nc - Ncn flow chart of CLUBB code: ! ! (Please update when warranted). ! ! ! Ncm/Nc-in-cloud Ncnm/Ncn PDF params. ! ---> ! | | Start of CLUBB main time step loop ! | | ! | | advance_clubb_core ! | | ! | | ! | |\ ! | | \ ! | | (intent in)-------setup_pdf_parameters-------->calc. Ncnm (local) ! | | | ! | | \ / ! | | mu_Ncn_i, sigma_Ncn_i, ! | | corr_xNcn_i ! | | | ! | | \ / ! | | PDF param. arrays: ! | | mu_x_i_n, sigma_x_i_n, ! | | corr_array_i_n ! | | (intent out) ! | | | ! | | | ! | | | ! | | | ! | | | ! | | | ! | |--(intent in)---calc_microphys_scheme_tendcies----(intent in) ! | | | ! | | | ! | | call a microphysics scheme ! | | | ! | | Local micro. scheme-----------Latin Hypercube-----------Upscaled KK ! | | | | | ! | | Ncm/Nc-in-cloud: Populate sample points Use PDF params. ! | | used to find micro. using PDF params (Ncn). of Ncn ! | | tendencies. At every sample point: (mu_Ncn_i, etc.) ! | | | Nc = Ncn * H(chi). to find micro. ! | | | Use sample-point Nc to tendencies. ! | | | find micro. tendencies | ! | | | when calling micro. scheme. | ! | | | | | ! | | hydromet_mc/-----------------hydromet_mc/-------------hydromet_mc ! | | Ncm_mc (intent out) | Ncm_mc (intent out) (intent out) ! | | | ! | | | ! | | | ! | | | ! | | | ! | | | ! | | | ! | | (intent in) ! | | | ! | |--(intent inout)----advance_microphys ! | | ! | | ! | | advance microphysics variables (hydromet, Nc_in_cloud/Ncm) one timestep ! | | ! | | l_predict_Nc = true: ! | | Nc_in_cloud/Ncm necessary for starting ! | | value of Nc_in_cloud/Ncm when advancing ! | | one timestep using predictive equation. ! | | ! | | ! | | End of CLUBB main time step loop ! <--- ! References: !------------------------------------------------------------------------- implicit none private ! default scope public :: Ncnm_to_Nc_in_cloud, & Nc_in_cloud_to_Ncnm, & Ncnm_to_Ncm, & Ncm_to_Ncnm private :: bivar_NL_chi_Ncn_mean, & bivar_Ncnm_eqn_comp contains !============================================================================= function Ncnm_to_Nc_in_cloud( nz, mu_chi_1, mu_chi_2, mu_Ncn_1, mu_Ncn_2, & sigma_chi_1, sigma_chi_2, sigma_Ncn_1, & sigma_Ncn_2, sigma_Ncn_1_n, sigma_Ncn_2_n, & corr_chi_Ncn_1_n, corr_chi_Ncn_2_n, mixt_frac, & cloud_frac_1, cloud_frac_2 ) & result( Nc_in_cloud ) ! Description: ! The in-cloud mean of cloud droplet concentration is calculated from the ! PDF parameters involving simplified cloud nuclei concentration, Ncn, and ! cloud fraction. At any point, cloud droplet concentration, Nc, is given ! by: ! ! Nc = Ncn * H(chi); ! ! where extended liquid water mixing ratio, chi, is equal to cloud water ! ratio, rc, when positive. When the atmosphere is saturated at this point, ! cloud water is found, and Nc = Ncn. Otherwise, only clear air is found, ! and Nc = 0. ! ! The overall mean of cloud droplet concentration, , is calculated from ! the PDF parameters involving Ncn. The in-cloud mean of cloud droplet ! concentration is calculated from and cloud fraction. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & one, & ! Constant(s) cloud_frac_min use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_1, & ! Mean of chi (old s) (1st PDF component) [kg/kg] mu_chi_2, & ! Mean of chi (old s) (2nd PDF component) [kg/kg] mu_Ncn_1, & ! Mean of Ncn (1st PDF component) [num/kg] mu_Ncn_2, & ! Mean of Ncn (2nd PDF component) [num/kg] sigma_chi_1, & ! Standard deviation of chi (1st PDF comp.) [kg/kg] sigma_chi_2, & ! Standard deviation of chi (2nd PDF comp.) [kg/kg] sigma_Ncn_1, & ! Standard deviation of Ncn (1st PDF comp.) [num/kg] sigma_Ncn_2, & ! Standard deviation of Ncn (2nd PDF comp.) [num/kg] sigma_Ncn_1_n, & ! Standard deviation of ln Ncn (1st PDF component) [-] sigma_Ncn_2_n, & ! Standard deviation of ln Ncn (2nd PDF component) [-] corr_chi_Ncn_1_n, & ! Correlation of chi and ln Ncn (1st PDF comp.) [-] corr_chi_Ncn_2_n, & ! Correlation of chi and ln Ncn (2nd PDF comp.) [-] mixt_frac, & ! Mixture fraction [-] cloud_frac_1, & ! Cloud fraction (1st PDF component) [-] cloud_frac_2 ! Cloud fraction (2nd PDF component) [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & Nc_in_cloud ! Mean cloud droplet concentration (in-cloud) [num/kg] ! Local Variable real( kind = core_rknd ), dimension(nz) :: & Ncm, & ! Mean cloud droplet concentration (overall) [num/kg] cloud_frac ! Cloud fraction [-] ! Calculate overall cloud fraction as calculated by the PDF. ! The variable cloud_frac is not used here because it is altered by factors ! such as the trapezoidal rule calculation. ! Cloud fraction can be recalculated here from cloud_frac_1 and cloud_frac_2 ! as long neither of these variables are altered by any factor. They can ! only be calculated from PDF. cloud_frac = mixt_frac * cloud_frac_1 + ( one - mixt_frac ) * cloud_frac_2 where ( cloud_frac > cloud_frac_min ) ! There is cloud found at this grid level. Calculate Nc_in_cloud. Ncm = Ncnm_to_Ncm( nz, mu_chi_1, mu_chi_2, mu_Ncn_1, mu_Ncn_2, & sigma_chi_1, sigma_chi_2, sigma_Ncn_1, & sigma_Ncn_2, sigma_Ncn_1_n, sigma_Ncn_2_n, & corr_chi_Ncn_1_n, corr_chi_Ncn_2_n, mixt_frac ) Nc_in_cloud = Ncm / cloud_frac elsewhere ! cloud_frac <= cloud_frac_min ! This level is entirely clear. Set Nc_in_cloud to . ! Since = mu_Ncn_1 = mu_Ncn_2, use mu_Ncn_1 here. Nc_in_cloud = mu_Ncn_1 endwhere return end function Ncnm_to_Nc_in_cloud !============================================================================= function Nc_in_cloud_to_Ncnm( nz, mu_chi_1, mu_chi_2, sigma_chi_1, & sigma_chi_2, mixt_frac, Nc_in_cloud, & cloud_frac_1, cloud_frac_2, & const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn ) & result( Ncnm ) ! Description: ! The overall mean of simplified cloud nuclei concentration, , is ! calculated from the in-cloud mean of cloud droplet concentration, , ! cloud fraction, and some of the PDF parameters. ! ! At any point, cloud droplet concentration, Nc, is given by: ! ! Nc = Ncn * H(chi); ! ! where extended liquid water mixing ratio, chi, is equal to cloud water ! ratio, rc, when positive. When the atmosphere is saturated at this point, ! cloud water is found, and Nc = Ncn. Otherwise, only clear air is found, ! and Nc = 0. ! ! The overall mean of cloud droplet concentration, , is calculated from ! Nc_in_cloud and cloud fraction. The value of is calculated from ! and PDF parameters. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & one, & ! Constant(s) cloud_frac_min, & eps use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_1, & ! Mean of chi (old s) (1st PDF component) [kg/kg] mu_chi_2, & ! Mean of chi (old s) (2nd PDF component) [kg/kg] sigma_chi_1, & ! Standard deviation of chi (1st PDF component) [kg/kg] sigma_chi_2, & ! Standard deviation of chi (2nd PDF component) [kg/kg] mixt_frac, & ! Mixture fraction [-] Nc_in_cloud, & ! Mean cloud droplet concentration (in-cloud) [num/kg] cloud_frac_1, & ! Cloud fraction (1st PDF component) [-] cloud_frac_2 ! Cloud fraction (2nd PDF component) [-] real( kind = core_rknd ), intent(in) :: & const_Ncnp2_on_Ncnm2, & ! Prescribed ratio of to ^2 [-] const_corr_chi_Ncn ! Prescribed correlation of chi and Ncn [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & Ncnm ! Mean simplified cloud nuclei concentration (overall) [num/kg] ! Local Variable real( kind = core_rknd ), dimension(nz) :: & Ncm, & ! Mean cloud droplet concentration (overall) [num/kg] cloud_frac ! Cloud fraction [-] ! Calculate overall cloud fraction as calculated by the PDF. ! The variable cloud_frac is not used here because it is altered by factors ! such as the trapezoidal rule calculation. ! Cloud fraction can be recalculated here from cloud_frac_1 and cloud_frac_2 ! as long neither of these variables are altered by any factor. They can ! only be calculated from the PDF. cloud_frac = mixt_frac * cloud_frac_1 + ( one - mixt_frac ) * cloud_frac_2 Ncm = Nc_in_cloud * cloud_frac where ( cloud_frac > cloud_frac_min & .and. abs( const_corr_chi_Ncn * const_Ncnp2_on_Ncnm2 ) > eps ) ! There is cloud found at this grid level. Additionally, Ncn varies. ! Calculate . Ncnm = Ncm_to_Ncnm( nz, mu_chi_1, mu_chi_2, sigma_chi_1, sigma_chi_2, & mixt_frac, Ncm, const_Ncnp2_on_Ncnm2, & const_corr_chi_Ncn, Nc_in_cloud ) elsewhere ! cloud_frac <= cloud_frac_min .or. const_Ncnp2_on_Ncnm2 = 0 ! When Ncn is constant a a grid level, it is equal to Nc_in_cloud. ! Additionally, when a level is entirely clear, , which is based on ! Nc_in_cloud, here, must be set to something. Set to Nc_in_cloud. Ncnm = Nc_in_cloud endwhere return end function Nc_in_cloud_to_Ncnm !============================================================================= function Ncnm_to_Ncm( nz, mu_chi_1, mu_chi_2, mu_Ncn_1, mu_Ncn_2, & sigma_chi_1, sigma_chi_2, sigma_Ncn_1, & sigma_Ncn_2, sigma_Ncn_1_n, sigma_Ncn_2_n, & corr_chi_Ncn_1_n, corr_chi_Ncn_2_n, mixt_frac ) & result( Ncm ) ! Description: ! The overall mean of cloud droplet concentration, , is calculated from ! the PDF parameters involving the simplified cloud nuclei concentration, ! Ncn. At any point, cloud droplet concentration, Nc, is given by: ! ! Nc = Ncn * H(chi); ! ! where extended liquid water mixing ratio, chi, is equal to cloud water ! ratio, rc, when positive. When the atmosphere is saturated at this point, ! cloud water is found, and Nc = Ncn. Otherwise, only clear air is found, ! and Nc = 0. ! ! The overall mean of cloud droplet concentration, , is found by ! integrating over the PDF of chi and Ncn, such that: ! ! = INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P(chi,Ncn) dNcn dchi; ! ! which can also be written as: ! ! = SUM(i=1,n) mixt_frac_i ! * INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P_i(chi,Ncn) dNcn dchi; ! ! where n is the number of multivariate joint PDF components, mixt_frac_i is ! the weight of the ith PDF component, and P_i is the functional form of the ! multivariate joint PDF in the ith PDF component. ! ! This equation is rewritten as: ! ! = SUM(i=1,n) mixt_frac_i ! * INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi. ! ! When both chi and Ncn vary in the ith PDF component, the integral is ! evaluated and the result is: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * exp{ mu_Ncn_i_n + (1/2) * sigma_Ncn_i_n^2 } ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ); ! ! which can be reduced to: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * mu_Ncn_i ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ). ! ! When chi is constant, but Ncn varies, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! ! When chi varies, but Ncn is constant, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = mu_Ncn_i * (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ). ! ! When both chi and Ncn are constant in the ith PDF component, the integral ! is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & one ! Constant(s) use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_1, & ! Mean of chi (old s) (1st PDF component) [kg/kg] mu_chi_2, & ! Mean of chi (old s) (2nd PDF component) [kg/kg] mu_Ncn_1, & ! Mean of Ncn (1st PDF component) [num/kg] mu_Ncn_2, & ! Mean of Ncn (2nd PDF component) [num/kg] sigma_chi_1, & ! Standard deviation of chi (1st PDF comp.) [kg/kg] sigma_chi_2, & ! Standard deviation of chi (2nd PDF comp.) [kg/kg] sigma_Ncn_1, & ! Standard deviation of Ncn (1st PDF comp.) [num/kg] sigma_Ncn_2, & ! Standard deviation of Ncn (2nd PDF comp.) [num/kg] sigma_Ncn_1_n, & ! Standard deviation of ln Ncn (1st PDF component) [-] sigma_Ncn_2_n, & ! Standard deviation of ln Ncn (2nd PDF component) [-] corr_chi_Ncn_1_n, & ! Correlation of chi and ln Ncn (1st PDF comp.) [-] corr_chi_Ncn_2_n, & ! Correlation of chi and ln Ncn (2nd PDF comp.) [-] mixt_frac ! Mixture fraction [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & Ncm ! Mean cloud droplet concentration (overall) [num/kg] ! Calculate mean cloud droplet concentration (overall), . Ncm & = mixt_frac & * bivar_NL_chi_Ncn_mean( nz, mu_chi_1, mu_Ncn_1, sigma_chi_1, & sigma_Ncn_1, sigma_Ncn_1_n, corr_chi_Ncn_1_n ) & + ( one - mixt_frac ) & * bivar_NL_chi_Ncn_mean( nz, mu_chi_2, mu_Ncn_2, sigma_chi_2, & sigma_Ncn_2, sigma_Ncn_2_n, corr_chi_Ncn_2_n ) return end function Ncnm_to_Ncm !============================================================================= function Ncm_to_Ncnm( nz, mu_chi_1, mu_chi_2, sigma_chi_1, sigma_chi_2, & mixt_frac, Ncm, const_Ncnp2_on_Ncnm2, & const_corr_chi_Ncn, Ncnm_val_denom_0 ) & result( Ncnm ) ! Description: ! The overall mean of simplified cloud nuclei concentration, , is ! calculated from the overall mean of cloud droplet concentration, , and ! some of the PDF parameters. ! ! At any point, cloud droplet concentration, Nc, is given by: ! ! Nc = Ncn * H(chi); ! ! where extended liquid water mixing ratio, chi, is equal to cloud water ! ratio, rc, when positive. When the atmosphere is saturated at this point, ! cloud water is found, and Nc = Ncn. Otherwise, only clear air is found, ! and Nc = 0. ! ! The overall mean of cloud droplet concentration, , is found by ! integrating over the PDF of chi and Ncn, such that: ! ! = INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P(chi,Ncn) dNcn dchi; ! ! which can also be written as: ! ! = SUM(i=1,n) mixt_frac_i ! * INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P_i(chi,Ncn) dNcn dchi; ! ! where n is the number of multivariate joint PDF components, mixt_frac_i is ! the weight of the ith PDF component, and P_i is the functional form of the ! multivariate joint PDF in the ith PDF component. ! ! This equation is rewritten as: ! ! = SUM(i=1,n) mixt_frac_i ! * INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi. ! ! When both chi and Ncn vary in the ith PDF component, the integral is ! evaluated and the result is: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * exp{ mu_Ncn_i_n + (1/2) * sigma_Ncn_i_n^2 } ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ); ! ! which can be reduced to: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * mu_Ncn_i ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ). ! ! When chi is constant, but Ncn varies, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! ! When chi varies, but Ncn is constant, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = mu_Ncn_i * (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ). ! ! When both chi and Ncn are constant in the ith PDF component, the integral ! is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! ! ! Solving for ! ================= ! ! The individual marginal for simplified cloud nuclei concentration, Ncn, is ! a single lognormal distribution over the entire horizontal domain. In ! order to accomplish this in a two-component PDF structure, the PDF ! parameters involving Ncn are set equal between the two components. This ! results in: ! ! mu_Ncn_1 = mu_Ncn_2 = mu_Ncn_i = ; ! mu_Ncn_1_n = mu_Ncn_2_n = mu_Ncn_i_n; ! sigma_Ncn_1 = sigma_Ncn_2 = sigma_Ncn_i = sqrt( ); ! sigma_Ncn_1_n = sigma_Ncn_2_n = sigma_Ncn_i_n; ! rho_chi_Ncn_1 = rho_chi_Ncn_2 = rho_chi_Ncn_i = rho_chi_Ncn; and ! rho_chi_Ncn_1_n = rho_chi_Ncn_2_n = rho_chi_Ncn_i_n. ! ! Additionally, the equation for sigma_Ncn_i_n is: ! ! sigma_Ncn_i_n = sqrt( ln( 1 + ( sigma_Ncn_i^2 / mu_Ncn_i^2 ) ) ); ! ! and the equation for rho_chi_Ncn_i_n is: ! ! rho_chi_Ncn_i_n ! = rho_chi_Ncn_i * sqrt( exp{ sigma_Ncn_i_n^2 } - 1 ) / sigma_Ncn_i_n. ! ! The product of rho_chi_Ncn_i_n and sigma_Ncn_i_n is: ! ! rho_chi_Ncn_i_n * sigma_Ncn_i_n ! = rho_chi_Ncn_i * sqrt( exp{ sigma_Ncn_i_n^2 } - 1 ). ! ! After substituting for sigma_Ncn_i_n^2, the equation for the product of ! rho_chi_Ncn_i_n and sigma_Ncn_i_n is: ! ! rho_chi_Ncn_i_n * sigma_Ncn_i_n ! = rho_chi_Ncn_i * sqrt( sigma_Ncn_i^2 / mu_Ncn_i^2 ); ! ! which can be rewritten as: ! ! rho_chi_Ncn_i_n * sigma_Ncn_i_n ! = rho_chi_Ncn * sqrt( / ^2 ). ! ! Substituting all of this into the equation for , the equation for ! becomes: ! ! = ! * SUM(i=1,n) mixt_frac_i ! --- ! | (1/2) * erfc( - ( 1 / sqrt(2) ) ! | * ( ( mu_chi_i / sigma_chi_i ) ! | + rho_chi_Ncn * sqrt(/^2) ) ); ! | where sigma_chi_i > 0 and > 0; ! | ! * | (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ); ! | where sigma_chi_i > 0 and = 0; ! | ! | 1; where sigma_chi_i = 0 and mu_chi_i > 0; ! | ! | 0; where sigma_chi_i = 0 and mu_chi_i <= 0. ! --- ! ! In order to isolate , the value of /^2 is set to a ! constant value, const_Ncn. The value of this constant does not depend on ! . Likewise, the value of rho_chi_Ncn does not depend on . ! Solving for , the equation becomes: ! ! ! = / ( SUM(i=1,n) mixt_frac_i ! --- ! | (1/2) * erfc( - ( 1 / sqrt(2) ) ! | * ( ( mu_chi_i / sigma_chi_i ) ! | + rho_chi_Ncn * sqrt( const_Ncn ) ) ); ! | where sigma_chi_i > 0 and const_Ncn > 0; ! | ! * | (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ); ! | where sigma_chi_i > 0 and const_Ncn = 0; ! | ! | 1; where sigma_chi_i = 0 and mu_chi_i > 0; ! | ! | 0; where sigma_chi_i = 0 and mu_chi_i <= 0 ). ! --- ! ! When the denominator term is 0, there is only clear air. Both the ! numerator () and the denominator have a value of 0, and is set ! to an appropriate value. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & one, & ! Constant(s) zero use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_1, & ! Mean of chi (old s) (1st PDF component) [kg/kg] mu_chi_2, & ! Mean of chi (old s) (2nd PDF component) [kg/kg] sigma_chi_1, & ! Standard deviation of chi (1st PDF comp.) [kg/kg] sigma_chi_2, & ! Standard deviation of chi (2nd PDF comp.) [kg/kg] mixt_frac, & ! Mixture fraction [-] Ncm, & ! Mean cloud droplet conc. (overall) [num/kg] Ncnm_val_denom_0 ! Ncnm value -- denominator in eqn. is 0 [num/kg] real( kind = core_rknd ), intent(in) :: & const_Ncnp2_on_Ncnm2, & ! Prescribed ratio of to ^2 [-] const_corr_chi_Ncn ! Prescribed correlation of chi and Ncn [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & Ncnm ! Mean simplified cloud nuclei concentration (overall) [num/kg] ! Local Variable real( kind = core_rknd ), dimension(nz) :: & denominator_term ! Denominator in the equation for [-] denominator_term & = mixt_frac & * bivar_Ncnm_eqn_comp( nz, mu_chi_1, sigma_chi_1, & const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn ) & + ( one - mixt_frac ) & * bivar_Ncnm_eqn_comp( nz, mu_chi_2, sigma_chi_2, & const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn ) where ( denominator_term > zero ) Ncnm = Ncm / denominator_term elsewhere ! denominator_term = 0 ! When the denominator is 0, it is usually because there is only clear ! air. In that scenario, Ncm should also be 0. Set Ncnm to a value that ! is usual or typical Ncnm = Ncnm_val_denom_0 endwhere ! denominator_term > 0 return end function Ncm_to_Ncnm !============================================================================= function bivar_NL_chi_Ncn_mean( nz, mu_chi_i, mu_Ncn_i, sigma_chi_i, & sigma_Ncn_i, sigma_Ncn_i_n, corr_chi_Ncn_i_n ) ! Description: ! The double integral over Ncn * H(chi) multiplied by the ! bivariate normal-lognormal joint PDF of chi and Ncn is evaluated. The ! integral is given by: ! ! INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P_i(chi,Ncn) dNcn dchi; ! ! which reduces to: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi; ! ! where the individual marginal distribution of chi is normal in the ith PDF ! component and the individual marginal distribution of Ncn is lognormal in ! the ith PDF component. ! ! When both chi and Ncn vary in the ith PDF component, the integral is ! evaluated and the result is: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * exp{ mu_Ncn_i_n + (1/2) * sigma_Ncn_i_n^2 } ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ); ! ! which can be reduced to: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = (1/2) * mu_Ncn_i ! * erfc( - ( 1 / sqrt(2) ) * ( ( mu_chi_i / sigma_chi_i ) ! + rho_chi_Ncn_i_n * sigma_Ncn_i_n ) ). ! ! When chi is constant, but Ncn varies, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! ! When chi varies, but Ncn is constant, in the ith PDF component, the ! integral is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi ! = mu_Ncn_i * (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ). ! ! When both chi and Ncn are constant in the ith PDF component, the integral ! is evaluated and results in: ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = mu_Ncn_i; ! ! when mu_chi_i > 0; and ! ! INT(0:inf) INT(0:inf) Ncn * P_i(chi,Ncn) dNcn dchi = 0; ! ! when mu_chi_i <= 0. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & sqrt_2, & ! Constant(s) one, & one_half, & zero, & chi_tol, & Ncn_tol use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_i, & ! Mean of chi (old s) (ith PDF component) [kg/kg] mu_Ncn_i, & ! Mean of Ncn (ith PDF component) [num/kg] sigma_chi_i, & ! Standard deviation of chi (ith PDF comp.) [kg/kg] sigma_Ncn_i, & ! Standard deviation of Ncn (ith PDF comp.) [num/kg] sigma_Ncn_i_n, & ! Standard deviation of ln Ncn (ith PDF component) [-] corr_chi_Ncn_i_n ! Correlation of chi and ln Ncn (ith PDF comp.) [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & bivar_NL_chi_Ncn_mean where ( sigma_chi_i <= chi_tol .and. sigma_Ncn_i <= Ncn_tol ) ! The ith PDF component variances of both chi and Ncn are 0. where ( mu_chi_i > zero ) bivar_NL_chi_Ncn_mean = mu_Ncn_i elsewhere ! mu_chi_i <= 0 bivar_NL_chi_Ncn_mean = zero endwhere elsewhere ( sigma_chi_i <= chi_tol ) ! The ith PDF component variance of chi is 0. where ( mu_chi_i > zero ) bivar_NL_chi_Ncn_mean = mu_Ncn_i elsewhere ! mu_chi_i <= 0 bivar_NL_chi_Ncn_mean = zero endwhere elsewhere ( sigma_Ncn_i <= Ncn_tol ) ! The ith PDF component variance of Ncn is 0. bivar_NL_chi_Ncn_mean & = mu_Ncn_i * one_half * erfc( - ( mu_chi_i / ( sqrt_2 * sigma_chi_i ) ) ) elsewhere ! Both chi and Ncn vary in the ith PDF component. bivar_NL_chi_Ncn_mean & = one_half * mu_Ncn_i & * erfc( - ( one / sqrt_2 ) & * ( ( mu_chi_i / sigma_chi_i ) & + corr_chi_Ncn_i_n * sigma_Ncn_i_n ) ) endwhere return end function bivar_NL_chi_Ncn_mean !============================================================================= function bivar_Ncnm_eqn_comp( nz, mu_chi_i, sigma_chi_i, & const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn ) ! Description: ! When is found based on the value of , the following equation is ! used: ! ! ! = / ( SUM(i=1,n) mixt_frac_i ! --- ! | (1/2) * erfc( - ( 1 / sqrt(2) ) ! | * ( ( mu_chi_i / sigma_chi_i ) ! | + rho_chi_Ncn * sqrt( const_Ncn ) ) ); ! | where sigma_chi_i > 0 and const_Ncn > 0; ! | ! * | (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) ); ! | where sigma_chi_i > 0 and const_Ncn = 0; ! | ! | 1; where sigma_chi_i = 0 and mu_chi_i > 0; ! | ! | 0; where sigma_chi_i = 0 and mu_chi_i <= 0 ). ! --- ! ! In the above equation, const_Ncn = / ^2. It is a constant, ! prescribed parameter. Likewise, rho_chi_Ncn is a parameter that is not ! based on the value of . ! ! When the denominator term is 0, there is only clear air. Both the ! numerator () and the denominator have a value of 0, and is set ! to an appropriate value. ! ! The contribution of the ith PDF component to the denominator term in the ! equation is calculated here. ! References: !----------------------------------------------------------------------- use constants_clubb, only: & sqrt_2, & ! Constant(s) one, & one_half, & zero, & chi_tol use clubb_precision, only: & core_rknd ! Variable(s) implicit none ! Input Variables integer, intent(in) :: & nz ! Number of vertical levels real( kind = core_rknd ), dimension(nz), intent(in) :: & mu_chi_i, & ! Mean of chi (old s) (ith PDF component) [kg/kg] sigma_chi_i ! Standard deviation of chi (ith PDF component) [kg/kg] real( kind = core_rknd ), intent(in) :: & const_Ncnp2_on_Ncnm2, & ! Prescribed ratio of to ^2 [-] const_corr_chi_Ncn ! Prescribed correlation of chi and Ncn [-] ! Return Variable real( kind = core_rknd ), dimension(nz) :: & bivar_Ncnm_eqn_comp where ( sigma_chi_i <= chi_tol ) ! The ith PDF component variances of chi is 0. The value of the ith PDF ! component variance of Ncn does not matter in this scenario. where ( mu_chi_i > zero ) bivar_Ncnm_eqn_comp = one elsewhere ! mu_chi_i <= 0 bivar_Ncnm_eqn_comp = zero endwhere elsewhere ! Both chi and Ncn vary in the ith PDF component. bivar_Ncnm_eqn_comp & = one_half * erfc( - ( one / sqrt_2 ) * ( ( mu_chi_i / sigma_chi_i ) & + const_corr_chi_Ncn * sqrt( const_Ncnp2_on_Ncnm2 ) ) ) endwhere return end function bivar_Ncnm_eqn_comp !=============================================================================== end module Nc_Ncn_eqns