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Nc_Ncn_eqns.F90
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Nc_Ncn_eqns.F90
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!---------------------------------------------------------------------------
! $Id$
!===============================================================================
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( 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, <Nc>, is calculated from
! the PDF parameters involving Ncn. The in-cloud mean of cloud droplet
! concentration is calculated from <Nc> 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
real( kind = core_rknd ), 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 ) :: &
Nc_in_cloud ! Mean cloud droplet concentration (in-cloud) [num/kg]
! Local Variable
real( kind = core_rknd ) :: &
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
if ( cloud_frac > cloud_frac_min ) then
! There is cloud found at this grid level. Calculate Nc_in_cloud.
Ncm = Ncnm_to_Ncm( 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
else ! cloud_frac <= cloud_frac_min
! This level is entirely clear. Set Nc_in_cloud to <Ncn>.
! Since <Ncn> = mu_Ncn_1 = mu_Ncn_2, use mu_Ncn_1 here.
Nc_in_cloud = mu_Ncn_1
endif
return
end function Ncnm_to_Nc_in_cloud
!=============================================================================
function Nc_in_cloud_to_Ncnm( 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, <Ncn>, is
! calculated from the in-cloud mean of cloud droplet concentration, <Nc>,
! 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, <Nc>, is calculated from
! Nc_in_cloud and cloud fraction. The value of <Ncn> is calculated from
! <Nc> 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
real( kind = core_rknd ), 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 [-]
real( kind = core_rknd ), intent(in) :: &
Nc_in_cloud, & ! Mean cloud droplet conc. (in-cloud) [num/kg]
cloud_frac_1, & ! Cloud fraction (1st PDF component) [-]
cloud_frac_2, & ! Cloud fraction (2nd PDF component) [-]
const_Ncnp2_on_Ncnm2, & ! Prescribed ratio of <Ncn'^2> to <Ncn>^2 [-]
const_corr_chi_Ncn ! Prescribed correlation of chi and Ncn [-]
! Return Variable
real( kind = core_rknd ) :: &
Ncnm ! Mean simplified cloud nuclei concentration (overall) [num/kg]
! Local Variable
real( kind = core_rknd ) :: &
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
if ( cloud_frac > cloud_frac_min &
.and. abs(const_corr_chi_Ncn * const_Ncnp2_on_Ncnm2) > eps ) then
! There is cloud found at this grid level. Additionally, Ncn varies.
! Calculate Nc_in_cloud.
Ncm = Nc_in_cloud * cloud_frac
Ncnm = Ncm_to_Ncnm( 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 )
else ! 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, <Ncn>, which is based on
! Nc_in_cloud, here, must be set to something. Set <Ncn> to Nc_in_cloud.
Ncnm = Nc_in_cloud
endif
return
end function Nc_in_cloud_to_Ncnm
!=============================================================================
function Ncnm_to_Ncm( 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, <Nc>, 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, <Nc>, is found by
! integrating over the PDF of chi and Ncn, such that:
!
! <Nc> = INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P(chi,Ncn) dNcn dchi;
!
! which can also be written as:
!
! <Nc> = 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:
!
! <Nc> = 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
real( kind = core_rknd ), 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 ) :: &
Ncm ! Mean cloud droplet concentration (overall) [num/kg]
! Calculate mean cloud droplet concentration (overall), <Nc>.
Ncm &
= mixt_frac &
* bivar_NL_chi_Ncn_mean( 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( 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( 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, <Ncn>, is
! calculated from the overall mean of cloud droplet concentration, <Nc>, 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, <Nc>, is found by
! integrating over the PDF of chi and Ncn, such that:
!
! <Nc> = INT(-inf:inf) INT(0:inf) Ncn * H(chi) * P(chi,Ncn) dNcn dchi;
!
! which can also be written as:
!
! <Nc> = 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:
!
! <Nc> = 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 <Ncn>
! =================
!
! 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 = <Ncn>;
! mu_Ncn_1_n = mu_Ncn_2_n = mu_Ncn_i_n;
! sigma_Ncn_1 = sigma_Ncn_2 = sigma_Ncn_i = sqrt( <Ncn'^2> );
! 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( <Ncn'^2> / <Ncn>^2 ).
!
! Substituting all of this into the equation for <Nc>, the equation for <Nc>
! becomes:
!
! <Nc> = <Ncn>
! * SUM(i=1,n) mixt_frac_i
! ---
! | (1/2) * erfc( - ( 1 / sqrt(2) )
! | * ( ( mu_chi_i / sigma_chi_i )
! | + rho_chi_Ncn * sqrt(<Ncn'^2>/<Ncn>^2) ) );
! | where sigma_chi_i > 0 and <Ncn'^2> > 0;
! |
! * | (1/2) * erfc( - ( mu_chi_i / ( sqrt(2) * sigma_chi_i ) ) );
! | where sigma_chi_i > 0 and <Ncn'^2> = 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 <Ncn>, the value of <Ncn'^2>/<Ncn>^2 is set to a
! constant value, const_Ncn. The value of this constant does not depend on
! <Ncn>. Likewise, the value of rho_chi_Ncn does not depend on <Ncn>.
! Solving for <Ncn>, the equation becomes:
!
! <Ncn>
! = <Nc> / ( 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 (<Nc>) and the denominator have a value of 0, and <Ncn> 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
real( kind = core_rknd ), 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 [-]
real( kind = core_rknd ), intent(in) :: &
Ncm, & ! Mean cloud droplet conc. (overall) [num/kg]
const_Ncnp2_on_Ncnm2, & ! Prescribed ratio of <Ncn'^2> to <Ncn>^2 [-]
const_corr_chi_Ncn, & ! Prescribed correlation of chi and Ncn [-]
Ncnm_val_denom_0 ! Ncnm value -- denominator in eqn. is 0 [num/kg]
! Return Variable
real( kind = core_rknd ) :: &
Ncnm ! Mean simplified cloud nuclei concentration (overall) [num/kg]
! Local Variable
real( kind = core_rknd ) :: &
denominator_term ! Denominator in the equation for <Ncn> [-]
denominator_term &
= mixt_frac &
* bivar_Ncnm_eqn_comp( mu_chi_1, sigma_chi_1, &
const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn ) &
+ ( one - mixt_frac ) &
* bivar_Ncnm_eqn_comp( mu_chi_2, sigma_chi_2, &
const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn )
if ( denominator_term > zero ) then
Ncnm = Ncm / denominator_term
else ! 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
endif ! denominator_term > 0
return
end function Ncm_to_Ncnm
!=============================================================================
function bivar_NL_chi_Ncn_mean( 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
real( kind = core_rknd ), 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 ) :: &
bivar_NL_chi_Ncn_mean
if ( sigma_chi_i <=chi_tol .and. sigma_Ncn_i <= Ncn_tol ) then
! The ith PDF component variances of both chi and Ncn are 0.
if ( mu_chi_i > zero ) then
bivar_NL_chi_Ncn_mean = mu_Ncn_i
else ! mu_chi_i <= 0
bivar_NL_chi_Ncn_mean = zero
endif
elseif ( sigma_chi_i <= chi_tol ) then
! The ith PDF component variance of chi is 0.
if ( mu_chi_i > zero ) then
bivar_NL_chi_Ncn_mean = mu_Ncn_i
else ! mu_chi_i <= 0
bivar_NL_chi_Ncn_mean = zero
endif
elseif ( sigma_Ncn_i <= Ncn_tol ) then
! 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 ) ) )
else
! 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 ) )
endif
return
end function bivar_NL_chi_Ncn_mean
!=============================================================================
function bivar_Ncnm_eqn_comp( mu_chi_i, sigma_chi_i, &
const_Ncnp2_on_Ncnm2, const_corr_chi_Ncn )
! Description:
! When <Ncn> is found based on the value of <Nc>, the following equation is
! used:
!
! <Ncn>
! = <Nc> / ( 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 = <Ncn'^2> / <Ncn>^2. It is a constant,
! prescribed parameter. Likewise, rho_chi_Ncn is a parameter that is not
! based on the value of <Ncn>.
!
! When the denominator term is 0, there is only clear air. Both the
! numerator (<Nc>) and the denominator have a value of 0, and <Ncn> 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
real( kind = core_rknd ), 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 <Ncn'^2> to <Ncn>^2 [-]
const_corr_chi_Ncn ! Prescribed correlation of chi and Ncn [-]
! Return Variable
real( kind = core_rknd ) :: &
bivar_Ncnm_eqn_comp
if ( sigma_chi_i <= chi_tol ) then
! 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.
if ( mu_chi_i > zero ) then
bivar_Ncnm_eqn_comp = one
else ! mu_chi_i <= 0
bivar_Ncnm_eqn_comp = zero
endif
else
! 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 ) ) )
endif
return
end function bivar_Ncnm_eqn_comp
!===============================================================================
end module Nc_Ncn_eqns