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fractal_meanfield_mod.F90
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fractal_meanfield_mod.F90
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!! This module (fractal_meanfield_mod.F90) contains the main routines
!! necessary to calculate the solution of the mean field approximation
!! for a dry fractal particle composed of identical spherical monomers.
!! This is used to generate optical properties for these paticles in CARMA.
!!
!! See Botet et al. 1997 "Mean-field approximation of Mie
!! scattering by fractal aggregates of identical spheres."
!! Applied Optics 36(33) 8791-8797
!!
!! Original code from P. Rannou and R. Botet.
!! Translated to F90 and ported into CARMA by E. Wolf
!!
!! master: fractal_meanfield calling: cmie,ludcmpc,lubksbc,dqagi
!!
!! calculating the monomer Mie scattering
!! - SUBROUTINE cmie() calling: intmie()
!! - SUBROUTINE intmie() calling: intmie()
!!
!! calculating the matrix elements
!! - FUNCTION funa() calling: dqag,fpl
!! - FUNCTION fpl() calling: calcpoly
!! - FUNCTION calcpoly()
!! - FUNCTION funb_n()
!! - FUNCTION funs_n() calling: dq2agi,xfreal_n,xfimag_n
!! - FUNCTION xfreal_n() calling: besseljy,phi
!! - FUNCTION xfimag_n() calling: besseljy,phi
!! - FUNCTION BESSELJY()
!!
!! Routines to calculate the scattered wave
!! of monomer:
!! - FUNCTION fpi() calling: calcpoly()
!! - FUNCTION ftau() calling: calcpoly()
!! of agglomerate/cluster:
!! - FUNCTION fp1()
!!
!! Routines related to the probability distribution:
!! - FUNCTION anorm() calling: dqdagi,fdval
!! - FUNCTION fdval()
!! - FUNCTION phi() calling: fdval
!! - FUNCTION fco() calling: fdval
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! version March 2013
module fractal_meanfield_mod
use carma_precision_mod
use carma_enums_mod
use carma_constants_mod
use carma_types_mod
use carma_mod
use adgaquad_types_mod
use adgaquad_mod
implicit none
private
public :: fractal_meanfield
! Private module varibles: Moved from COMMON blocks
integer, parameter :: nmi=40
integer, parameter :: n2m = 2*nmi
contains
!!
!! Generate optical properties for CARMA fractal particles.
!!
!! See Botet et al. 1997 "Mean-field approximation of Mie
!! scattering by fractal aggregates of identical spheres."
!! Applied Optics 36(33) 8791-8797
!!
!! @author P.Rannou, R.Botet, Eric Wolf
!! @version March 2013
subroutine fractal_meanfield(carma, xl_in, xk_in, xn_in, nb_in, alpha_in, &
df_in, rmon,xv, ang, Qext, Qsca, gfac, rc)
! some of these may be included in carma object
type(carma_type), intent(in) :: carma !! the carma object
real(kind=f),intent(in) :: xl_in !! Wavelength [microns]
real(kind=f),intent(in) :: xk_in !! imaginary index of refraction
real(kind=f),intent(in) :: xn_in !! real index of refraction
real(kind=f),intent(in) :: nb_in !! number of monomers
real(kind=f),intent(in) :: alpha_in !! Packing coefficient
real(kind=f),intent(in) :: df_in !! Fractal dimension
real(kind=f),intent(in) :: rmon !! monomer size [microns]
real(kind=f),intent(in) :: xv !! set to 1
real(kind=f),intent(in) :: ang !! angle set to zero
real(kind=f),intent(out) :: Qext !! EFFICIENCY FACTOR FOR EXTINCTION
real(kind=f),intent(out) :: Qsca !! EFFICIENCY FACTOR FOR SCATTERING
real(kind=f),intent(out) :: gfac !! asymmetry factor
integer,intent(inout) :: rc !! return code, negative indicates failure
! Local declarations
integer, parameter :: nth = 10001
integer, parameter :: maxsub = 50000
integer, parameter :: lenw = 200000
integer :: nstop ! index with the last mie-coefficient
integer :: n1stop
integer :: pp,tt,mm
real(kind=f) :: krg,rg ! Particle structure
real(kind=f) :: sigmas,sigmae,nc1
real(kind=f) :: sigext ! extinction cross section
real(kind=f) :: sigsca ! scattering cross section
real(kind=f) :: sigabs ! absorption cross section
real(kind=f) :: totg ! asymmetry parameter
real(kind=f) :: sigext2,sigext3
real(kind=f) :: rems ! radius of equivalent mass sphere
real(kind=f) :: gems ! geometric cross-section of equivalent mass sphere
real(kind=f) :: dthetar,angler,weight
real(kind=f) :: sumsca
real(kind=f) :: lbd,beta ! optical characteristics
real(kind=f) :: sstest(0:nf)
real(kind=f) :: setest(0:nf)
real(kind=f) :: xl(39) ! place holder for wavelength
real(kind=f) :: xn(39) ! place holder for real index of refraction
real(kind=f) :: xk(39) ! place holder for imaginary index of refraction
real(kind=f) :: val
real(kind=f) :: funca(nmi,nmi,0:n2m) ! for storage of funa(nu,n;p)
complex(kind=f) :: res ! for storage of funs_n
complex(kind=f) :: funcs(0:n2m)
real(kind=f) :: s11(0:nth-1)
real(kind=f) :: s11_n(0:nth-1)
real(kind=f) :: xint(0:nth-1)
real(kind=f) :: wom
real(kind=f) :: pol(0:nth-1)
complex(kind=f) :: s01,s02
complex(kind=f) :: s1(0:nth-1)
complex(kind=f) :: s2(0:nth-1)
complex(kind=f) :: ajt
complex(kind=f) :: an(nf)
complex(kind=f) :: bn(nf)
complex(kind=f) :: ni,i,id,onec,zeroc
complex(kind=f) :: d1(nmi)
complex(kind=f) :: d2(nmi)
complex(kind=f) :: Ap1(nmi,nmi)
complex(kind=f) :: Bp1(nmi,nmi)
complex(kind=f) :: cvec(n2m)
complex(kind=f) :: EpABC(n2m,n2m)
integer :: luindx(n2m) ! For LU decomposition
real(kind=f) :: dlu
integer :: ifail
integer :: iwork(maxsub)
integer :: neval,nsubin
real(kind=f) :: work(lenw)
! Previously these were implicitly defined
real(kind=f) :: angle, pi, rn, ri
real(kind=f) :: deltas, deltae, xfact
real(kind=f) :: bound, errrel, p1, dp1
real(kind=f) :: errabs, total
integer :: n2stop, n3stop, ntheta, ii, kk, nn, jj, iy, ir, q, interv
real(kind=f) :: a0, c0, a1, c1, a2, c2
real(kind=f) :: qabs
! Previously these were globals, which wouldn't be thread safe.
type(adgaquad_vars_type) :: fx_vars
integer :: info
external ZGESV ! lapack routine to solve complex matrix eqn.
errabs=0.0_f
! Set the return code to default to okay.
rc = RC_OK
! *** Set from input arguments
fx_vars%nb = nb_in
fx_vars%df = df_in
fx_vars%alpha = alpha_in
xl(1) = xl_in
xk(1) = xk_in
xn(1) = xn_in
! *** Complex constants 1, 1, identity(1,1), zero(0,0) :
i = cmplx(0._f,1._f,kind=f)
onec = cmplx(1._f,0._f,kind=f)
id = cmplx(1._f,1._f,kind=f)
zeroc = cmplx(0._f,0._f,kind=f)
! Other initializations
funca(:,:,:) = 0.0_f
fx_vars%a = rmon *1.e-2_f ! a = r_monomer in m
beta=ang*(3.1415926_f / 180._f) ! =0 when ang=0
Ap1(:,:) = zeroc
Bp1(:,:) = zeroc
sstest(:) = 0.0_f
setest(:) = 0.0_f
! ****************************************************************
! *** Definition and calculation of factorials 0 - nf
! *** (nf set in adgaqaud_types_mod.F90)
! *** and storage [ real*8 fact() (double prec.) ]
! ****************************************************************
fx_vars%fact(0)=1._f ! factorials fact(n)=n!
do ii=1,nf
fx_vars%fact(ii) = fx_vars%fact(ii-1)*ii*1._f
end do
pi=4._f*atan(1._f) ! 3.1415926535
fx_vars%coeff=anorm(carma,fx_vars,rc)
if (rc < 0) return
! ****************************************************************
! anorm() integrated INT_0^inf[ x**(df-1.)*exp(-x**df/2._f) dx ]
! and occupied
! anorm := 4 pi * INT_0^inf[ x**(df-1.)*exp(-x**df/2._f) dx ]
! == geometric scalingfactor Eq.(10) in [Botet et al, 1995]
! c := 0.5
! ****************************************************************
kk=1
ni=xn(kk)*1._f+i*xk(kk)*xv*1._f ! ni := complex index of refraction of monomer
! (xv := 1 ; input parameter in file "calpha")
lbd=xl(kk)*1.e-6_f ! lbd := wavelength in m
! (in matrix medium / material !)
fx_vars%k=2._f*pi/lbd ! k := abs.val. of wavevector in m^-1
! (in matrix medium / material !)
! *** ******************************************************************
! *** Calculation of Mie coefficients for monomer scattering
! *** up to a maximum order of nf=50
! *** ******************************************************************
do ii=1,nf
an(ii) = zeroc
bn(ii) = zeroc
end do
rn=xn(kk) ! Re(relative_n_complex,monomer)
ri=xk(kk)*xv ! Im(relative_n_complex,monomer)
! xv should be set to 1 (sse above)
! a = monomer sphere radius
! lbd = wavelength in matrix medium
! Call Mie routine
call cmie(lbd,rn,ri,fx_vars%a,an,bn,nstop)
do ii=1,nf
if (an(ii).ne.0._f) nstop=ii
end do
! nstop is now the index with the last mie-coefficient
! (highest index i) an(i) not equal zero.
! since all the an were set to zero before calling
! cmie(), nstop is the termination index used in cmie()
! or in intmie(). Usually, a termination index
! nstop = INT( 2 + x + 4 x^(1/3) ) is used; in intmie(),
! however, a value of
! nstop := MAX( INT(...), |m*x| )+15 is used !???
sigmas=0._f
sigmae=0._f
do nn=1,nstop
nc1=abs(an(nn))**2._f+abs(bn(nn))**2._f
nc1=nc1*(2._f*nn+1._f)
sigmas=sigmas+nc1*(2._f*3.14159265_f)/(fx_vars%k**2._f)
nc1=real(an(nn)+bn(nn))
nc1=nc1*(2._f*nn+1._f)
sigmae=sigmae+nc1*(2._f*3.14159265_f)/(fx_vars%k**2._f)
sstest(nn)=sigmas
setest(nn)=sigmae
deltas=abs(sstest(nn-1)-sstest(nn))/sstest(nn)
deltae=abs(setest(nn-1)-setest(nn))/setest(nn)
if(deltas.gt.1.e-6_f) n2stop=nn
if(deltae.gt.1.e-6_f) n3stop=nn
end do
n1stop=n2stop
if (n3stop.gt.n2stop) n1stop=n3stop
! The order of the set of linear equations is chosen
! as the number of mie coefficients where the sum yielding
! the monomer ext./scatt. cross sections do not change more
! than 1.D-3 compared to the values with one summand less.
rg=fx_vars%alpha*fx_vars%nb**(1._f/fx_vars%df)*fx_vars%a ! rg := radius of gyration
krg=fx_vars%k*rg
ntheta=7800 !180-int(krg**.5*28*log10(dxk(kk)))
if (ntheta*0.5_f .eq. (ntheta/2)*1._f) ntheta=ntheta+1
! *** ******************************************************************
! *** FIRST PART: Solution of self consistent mean field equation,
! *** i.e. the set of linear equations (SLE) defining the
! *** mean field coefficients d^1_1,n and d^2_1,n
! *** according to Eq.(12) of (Botet 1997)
! *** To do so,
! *** - matrix elements A^1,nu_1,n and B^1,nu_1,n
! *** are calculated with Eq.(13), using Eqns.(14)-(16)
! *** - the set of lin. Eqns. is solved yielding the d's
! *** ******************************************************************
! *** Eq.(12) of Botet et al 1997 defines a matrix eqn. of order 2N :
! *** (since N=n1stop, 2N = 2 * n1stop = order of SLE)
! ***
! *** EpABC * dvec = cvec with
! ***
! *** dvec and cvec being the 2N-vectors
! ***
! *** ( d^(1)_1,1 ) ( a_1 )
! *** ( d^(1)_1,2 ) ( a_2 )
! *** ( ... ) ( ... )
! *** ( d^(1)_1,n ) ( a_n )
! *** dvec := ( d^(2)_1,1 ) and cvec := ( b_1 ) and further
! *** ( d^(2)_1,2 ) ( b_2 )
! *** ( ... ) ( ... )
! *** ( d^(2)_1,n ) ( b_n )
! ***
! ***
! *** EpABC := 1 + AB * C where AB, 1, C are the 2N*2N - matrices
! ***
! *** ( a_1 0 0 ... ... 0 )
! *** ( 0 a_2 0 ... ... 0 )
! *** AB := ( 0 0 ... 0 0 0 ... 0 ) 1 := 2N*2N unity matrix
! *** ( 0 0 ... a_n 0 0 ... 0 )
! *** ( 0 0 ... 0 b_1 0 ... 0 )
! *** ( ... ... ... 0 )
! *** ( ... ... 0 b_n-1 0 )
! *** ( 0 0 ... ... 0 0 b_n)
! ***
! *** and ( A B )
! *** C := ( ) where A and B are the two N*N matrices
! *** ( B A ) given by Eq.(13),
! *** including the factor (N_monomers - 1):
! ***
! *** A_n,nu := (N_m-1) * A_(1,n)^(1,nu) and
! *** B_n,nu := (N_m-1) * B_(1,n)^(1,nu)
! ***
! *** (A_(1,n... and B_(1,n... according to Eq.(13) of Botet 1997)
! *** ******************************************************************
n2stop = 2 * n1stop ! n2stop = order of SLE
! *** ******************************************************************
! *** Error handling moved from xfreal_n, xfimag_n. Calculations fail
! *** in integration package when n2stop > 48. n2stop is related to the
! *** number of complex mie scattering coefficients used in teh calculation
! *** which is in turn related to the size parameter of monomers.
! *** If nstop>48 end calculation here instead of continuing.
if (n2stop.gt.48) then
if (carma%f_do_print) then
write(carma%f_LUNOPRT, *) "fractal_meanfield_mod::n2stop greater &
&than 48. Size parameter (2*pi*rmon/lambda): ", &
2._f*3.14159265_f*fx_vars%a/lbd, "Monomer Size parameter &
&must be less than ~17."
end if
rc = RC_ERROR
return
endif
! *** ******************************************************************
do ii=1,n1stop
cvec(ii) = an(ii) ! right hand side vector
cvec(n1stop+ii) = bn(ii)
end do
do pp=0,n2stop !variable p
res = funs_n(carma,fx_vars,pp,rc) ! Eq.(16) S_p(k R_g)
if (rc < 0) return
funcs(pp) = res
end do
! Calculate terms A and B
! *** loops over indices nu,n,p :
do ii=1,n1stop !variable n
do jj=1,n1stop !variable nu
mm = IABS(ii-jj)
tt = ii+jj
! *** ******************************************************************
! calculation of A_(1,n)^(1,nu) according to Eq.(13)
! *** ******************************************************************
do pp=mm,tt
funca(jj,ii,pp) = funa(carma,fx_vars,jj,ii,pp,rc) ! Eq.(14) a(nu,n;p)a
if (rc < 0) return
Ap1(ii,jj) = Ap1(ii,jj) + &
( onec * (ii*(ii+1)+jj*(jj+1)-pp*(pp+1)) ) &
* funca(jj,ii,pp) * funcs(pp)
end do ! loop over pp (variable p)
! scaling factors of eq.(13), factor (N_mon-1) from eq.(12)
Ap1(ii,jj) = Ap1(ii,jj) * (2._f*jj+1._f)/(jj*(jj*1._f+1._f))
Ap1(ii,jj) = Ap1(ii,jj) * (fx_vars%nb-1._f) / (ii*(ii*1._f+1._f))
! *** ******************************************************************
! calculation of B_(1,n)^(1,nu) according to Eq.(13)
! *** ******************************************************************
do pp=mm,tt
Bp1(ii,jj) = Bp1(ii,jj) + funb_n(jj,ii,pp,funca) * funcs(pp)
end do ! loop over pp (variable p)
! scaling factors of eq.(13), factor (N_mon-1) from eq.(12)
Bp1(ii,jj) = Bp1(ii,jj) * (2._f*jj+1._f)/(jj*(jj*1._f+1._f))
Bp1(ii,jj) = Bp1(ii,jj) * (fx_vars%nb-1._f) * 2._f/(ii*(ii*1._f+1._f))
end do ! loop over jj=1,n1stop (variable nu)
end do ! loop over ii=1,n1stop (variable n)
! *** ******************************************************************
! End of Calculation of terms A and B
! *** ******************************************************************
! *** Setup and solution of matrix equation of order 2N ( = n2stop )
! *** constituted by eq.(12)
! *** ******************************************************************
! *** matrix product (AB * C) (definitions see above)
do ii=1,n1stop
do jj=1,n1stop
EpABC(ii,jj) = an(ii) * Ap1(ii,jj)
EpABC(ii,jj+n1stop) = an(ii) * Bp1(ii,jj)
EpABC(ii+n1stop,jj) = bn(ii) * Bp1(ii,jj)
EpABC(ii+n1stop,jj+n1stop) = bn(ii) * Ap1(ii,jj)
end do
end do
! *** ******************************************************************
! *** add 2N*2N unity matrix
do ii=1,n1stop
EpABC(ii,ii) = EpABC(ii,ii) + onec
EpABC(ii+n1stop,ii+n1stop) = EpABC(ii+n1stop,ii+n1stop) + onec
end do
! ======================================================================
! *** solve matrix equation using lapack routine
call ZGESV(n2stop, 1, EpABC, n2m, luindx, cvec, n2m, info)
if (info/=0) then
rc = info
return
end if
do ii=1,n1stop
d1(ii) = cvec(ii)
d2(ii) = cvec(ii+n1stop)
end do
! *** ******************************************************************
! *** SECOND PART: Recomposition of the total wave scattered by
! *** the entire agglomerate/cluster by adding the
! *** waves scattered by each monomer taking into
! *** account the respective phase of the single waves.
! *** ******************************************************************
! *** ******************************************************************
! ----------------------------------------------------------------------
! 1) Calculate the amplitude functions |S1^j(th)| et |S2^j(th)|
! of one monomer of the agglomerate/cluster:
! ( see e.g. Bohren, Huffman (1983) p.112, Eq.(4.74) with the
! substitutions a_n -> d^1_1,n and b_n -> d^2_1,n
! or Rannou (1999) Eq.(1)-(6) )
! ----------------------------------------------------------------------
! *** ******************************************************************
do iy=0,ntheta-1,1 ! loop over angles
angle=iy*180._f/(ntheta-1)
s1(iy)=0._f
s2(iy)=0._f
wom=cos(angle*3.1415926353_f/180._f)
do ir=1,n1stop ! loop over Mie - indices
xfact=2._f*(2._f*ir+1._f)/(ir*1._f*(ir*1._f+1._f))
ajt=d1(ir)*fpi(ir,wom,fx_vars)+d2(ir)*tau(ir,wom,fx_vars)
s1(iy)=s1(iy)+xfact*ajt
ajt=d1(ir)*tau(ir,wom,fx_vars)+d2(ir)*fpi(ir,wom,fx_vars)
s2(iy)=s2(iy)+xfact*ajt
end do
s11(iy)=abs(s1(iy))**2._f+abs(s2(iy))**2._f
pol(iy)=abs(s1(iy))**2._f-abs(s2(iy))**2._f
pol(iy)=pol(iy)/(abs(s1(iy))**2_f+abs(s2(iy))**2._f)
! *** S_11(theta) = 1/2 * ( |S_1|^2 + |S_2|^2 )
! *** above, s1(theta) = 2 * S_1(theta)
! *** =>S_11(theta) = 1/2 * ( |1/2*s1|^2 + |1/2*s2|^2 )
! = 1/8 * ( |s1|^2 + |s2|^2 )
s11_n(iy)=.125_f*(abs(s1(iy))**2._f+abs(s2(iy))**2._f)
end do
s01=s1(0)
s02=s2(0)
! *** Extinction cross section sigext( d^1_1,n , d^2_1,n ) ***
sigext=0._f
do ir=1,n1stop ! loop (sum) over Mie-indices
sigext=sigext+(2._f*ir+1._f)*REAL(d1(ir)+d2(ir))
end do
sigext = fx_vars%nb * 2._f*pi/fx_vars%k**2._f * sigext ! Eq.(27)
! *** Alternatively (in a test, all values agreed with rel.acc. 1e-6),
! *** Extinction cross section sigext( S(0 deg) ) (optical theorem) ***
! *** (see e.g. Bohren, Huffman (1983), Eq. (4.76))
! *** S(0)=S_1(0)=S_2(0); sigma_ext = 4 pi / k^2 * Re(S(0))
! *** above, s1(theta) = 2 * S_1(theta) (factor 2 in 'xfact')
! sigext2 = nb * 4._f*pi/k**2._f * 0.5_f*REAL(s01)
! sigext3 = nb * 4._f*pi/k**2._f * 0.5_f*REAL(s02)
! *** ******************************************************************
! *** ******************************************************************
! ----------------------------------------------------------------------
! 2) Calculate the phase integral in Eq.(26) with P(r) already
! substituted ( compare Eq.(10) and (Botet 1995) ) :
! INT(0;infinity)[ sin(2XuZ) u^(d-2) f_co(u) du ]
! taking into account the different phases of the single
! scattered waves.
! ----------------------------------------------------------------------
! *** ******************************************************************
do q=0,ntheta-1,1
angle=q*180._f/(ntheta-1)
if (angle .eq. 0._f) angle=0.001_f
if (angle .eq. 180._f) angle=179.999_f
fx_vars%zed=sin(angle*3.1415928353_f/180._f/2._f)
bound=0._f
interv=1
errrel=1e-5_f
p1=0._f
dp1=0._f
!======================================================================
!--- Version using the QUADPACK - routine :
!----------------------------------------------------------------------
ifail = 0
CALL dqagi(fp1,fx_vars,bound,interv,errabs,errrel,p1,dp1,neval,ifail,maxsub,lenw,nsubin,iwork,work)
if(ifail.ne.0) then
if (carma%f_do_print) write(carma%f_LUNOPRT, *) "fractal_meanfield_mod::ifail=",ifail," returned by dqagi()!"
rc = RC_ERROR
return
endif
!======================================================================
p1=2._f*pi * (fx_vars%nb-1._f) / (fx_vars%coeff*fx_vars%zed*krg)*p1 + 1._f
xint(q)=p1
end do
! *** now, xint(theta) contains the square bracket terms in
! *** Botet (1997) Eq.(26) or Rannou (1999) Eq.(1)
! *** ******************************************************************
! ----------------------------------------------------------------------
! 3) Calculation of the phase function, calculation of the optical
! properties (asymmetrie factor g, scatt. cross section sigma_s)
! by angular integration: INT_0^180[ ... d_theta ]
! ----------------------------------------------------------------------
! *** ******************************************************************
total=0._f
totg=0._f
do q=1,ntheta-2,2
angle=(q-1)*180._f/(ntheta-1) ! angle in deg
a0=fx_vars%nb*xint(q-1)*s11(q-1)*sin(angle*3.1415926353_f/180._f)
c0=cos(angle*3.1415926353_f/180._f)
angle=q*180._f/(ntheta-1)
a1=fx_vars%nb*xint(q)*s11(q)*sin(angle*3.1415926353_f/180._f)
c1=cos(angle*3.1415926353_f/180._f)
angle=(q+1)*180._f/(ntheta-1)
a2=fx_vars%nb*xint(q+1)*s11(q+1)*sin(angle*3.1415926353_f/180._f)
c2=cos(angle*3.1415926353_f/180._f)
total=total+2._f/6._f*3.1415926353_f/(ntheta-1)*(a0+4._f*a1+a2)
totg=totg+2._f/6._f*3.1415926353_f/(ntheta-1)*(a0*c0+4._f*a1*c1+a2*c2)
end do
totg=totg/total
! *** ******************************************************************
! *** angular integration of I(theta) according to
! *** Botet (1997) Eq.(26) or Rannou (1999) Eq.(1)
! *** I(theta) = N 2pi/k^2 * S(theta) * [ phase integral ]
! *** with
! *** S(theta) = s11_n(i)
! *** [ phase i. ] = xint(i)
! *** Perfom integration using the following rule:
! *** Integral_0^pi[ I(theta) sin(theta) d_theta ]
! ***
! *** = Sum_q=1^ntheta-1{ Integral_th_(i-1)^th_i [
! ***
! *** 1/2(I(th_(i-1))+I(th_i)) * sin(th) d_th ] }
! ***
! *** = sin(delta_theta/2) * Sum_q=1^ntheta-1{
! ***
! *** ( I(th_(i-1)) + I(th_i) ) * sin(th_middle) }
! ***
! *** ******************************************************************
!dthetad = 180._f / (ntheta-1) ! angular interval in deg
dthetar = pi / (ntheta-1) ! angular interval in rad
sumsca = 0._f
do q=1,ntheta-1,1
angler = (DBLE(q)-.5_f)*dthetar ! middle of interval in rad
weight = SIN(angler) ! integration weight
val = s11_n(q-1)*xint(q-1) + s11_n(q)*xint(q)
sumsca = sumsca + val*weight
end do
sumsca = sin(.5_f*dthetar) * sumsca ! interval width factor
! *** Scattering cross section
sigsca = 2._f * pi / fx_vars%k**2._f * DBLE(fx_vars%nb) * sumsca
! Warning! sigabs is well computed using this approximation
sigabs=fx_vars%nb*(sigmae-sigmas)
! sigext=sigabs+sigsca is better than the mean-field value
! previously defined. This is used hereafter. (P.Rannou)
! *** Radius of equivalent mass sphere
rems = fx_vars%a * fx_vars%nb**(1._f/3._f)
! *** reference area in definition of efficiencies is the geometrical
! *** cross section of equivalent mass sphere
gems = pi * rems**2._f
! *** Extinction and scattering efficiencies:
qsca = sigsca / gems
qabs = sigabs / gems
qext = qabs + qsca
gfac = totg
end subroutine fractal_meanfield
!!
!! Mie-scattering routine calling interface
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
subroutine cmie(lambda,xn,xk,rad,an,bn,nstop)
! Arguments
real(kind=f), intent(in) :: lambda !! wavelength (microns)
real(kind=f), intent(in) :: xn !! real index of refraction
real(kind=f), intent(in) :: xk !! imaginary index of refraction
real(kind=f), intent(in) :: rad !! monomer radius (meters)
complex(kind=f), intent(out) :: an(50) !! Mie wave coefficient an
complex(kind=f), intent(out) :: bn(50) !! Mie wave coefficient bn
integer, intent(out) :: nstop !! index of last mie-coefficent
! Local declarations
integer, parameter :: nang = 451 ! number of angles
complex(kind=f) :: refrel ! complex index of refraction
real(kind=f) :: theta(10000)
real(kind=f) :: x,dang
refrel=cmplx(xn,xk,kind=f)
x=2._f*3.14159265_f*rad/lambda ! size parameter of monomer
dang=1.570796327_f/real(nang-1,kind=f)
call intmie(x,refrel,nang,an,bn,nstop)
return
end subroutine cmie
!!
!! Mie scattering calculations
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
SUBROUTINE intmie(x,refrel,nang,an,bn,nstop)
! Arguments
real(kind=f), intent(in) :: x !! size parameter of monomer
complex(kind=f), intent(in) :: refrel !! complex index of refraction
integer, intent(in) :: nang !! number of angles
complex(kind=f), intent(out) :: an(nf) !! Mie wave coefficient an
complex(kind=f), intent(out) :: bn(nf) !! Mie wave coefficient an
integer, intent(out) :: nstop !! index of last mie-coefficent
! Local declarations
real(kind=f) :: amu(10000),pi(10000)
real(kind=f) :: pi0(10000),pi1(10000)
complex(kind=f) :: d(3000),y,xi,xi0,xi1
complex(kind=f) :: s1(2000),s2(2000)
real(kind=f) psi0,psi1,psi,dn,dx
integer :: nmx,nn,n,j
real(kind=f) :: rn, xstop, dang, ymod, chi0, chi1, apsi0, apsi1, fn, chi, apsi
dx=x
y=x*refrel
xstop=x+4._f*x**.3333_f+2._f
nstop=xstop
ymod=abs(y)
nmx=dmax1(xstop,ymod)+15
dang=1.570796327_f/real(nang-1,kind=f)
! Initializations
pi0(:) = 0._f
pi1(:) = 0._f
s1(:) = cmplx(0._f,0._f,kind=f)
s2(:) = cmplx(0._f,0._f,kind=f)
amu(:) = 0.0_f
pi(:) = 0.0_f
d(:) = cmplx(0._f,0._f,kind=f)
nn=nmx-1
do n=1,nn
rn=nmx-n+1
d(nmx-n)=(rn/y)-(1._f/(d(nmx-n+1)+rn/y))
end do
do j=1,nang
pi0(j)=0._f ! Legendre functions
pi1(j)=1._f
end do
nn=2*nang-1
do j=1,nn
s1(j)=cmplx(0._f,0._f,kind=f)
s2(j)=cmplx(0._f,0._f,kind=f)
end do
psi0=cos(dx) ! Initialize Bessel functions
psi1=sin(dx)
chi0=-sin(x)
chi1=cos(x)
apsi0=psi0
apsi1=psi1
xi0=cmplx(apsi0,-chi0,kind=f)
xi1=cmplx(apsi1,-chi1,kind=f)
n=1
! ************* iterate over index n *************
200 dn=n
rn=n
fn=(2._f*rn+1._f)/(rn*(rn+1._f))
psi=(2._f*dn-1._f)*psi1/dx-psi0 ! calculate Bessel functions
chi=(2._f*rn-1._f)*chi1/x-chi0
apsi=psi
xi=cmplx(apsi,-chi,kind=f)
an(n)=(d(n)/refrel+rn/x)*apsi-apsi1
an(n)=an(n)/((d(n)/refrel+rn/x)*xi-xi1)
bn(n)=(refrel*d(n)+rn/x)*apsi-apsi1
bn(n)=bn(n)/((refrel*d(n)+rn/x)*xi-xi1)
psi0=psi1
psi1=psi
apsi1=psi1
chi0=chi1
chi1=chi
xi1=cmplx(apsi1,-chi1,kind=f)
n=n+1
rn=n
do 999 j=1,nang
pi1(j)=((2._f*rn-1._f)/(rn-1._f))*amu(j)*pi(j)
pi1(j)=pi1(j)-rn*pi0(j)/(rn-1._f)
999 pi0(j)=pi(j)
if (n-1-nstop) 200,300,300
300 continue
return
END SUBROUTINE intmie
!!
!!
!! CALLS: FUNCTION dqag/dqdag/DADAPT_() Integration
!! FUNCTION fpl() Integrand
!!
!! Integral in eq. 14, Botet et al. 1997
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
function funa(carma,fx_vars,nu,n,p,rc)
type(carma_type), intent(in) :: carma !! the carma object
type(adgaquad_vars_type), intent(inout) :: fx_vars !! varaibles for functions being integrated
integer, intent(in) :: n !! indices
integer, intent(in) :: nu !! indices
integer, intent(in) :: p !! indices
integer, intent(inout) :: rc !! return code
real(kind=f) :: funa !!
! Local declarations
integer, parameter :: maxsub=1000
real(kind=f) :: r,xa,xb,era,erl
integer :: interv
integer :: ifail
integer, parameter :: lenw=4000 ! .ge. 4*maxsub
integer :: iwork(maxsub),neval,nsubin ! nsubin=last
real(kind=f) :: work(lenw)
real(kind=f) :: bound, rres, rerr
! Set return code assuming success.
rc = RC_OK
! Initializations
funa=0._f
fx_vars%u1=n
fx_vars%u2=1
fx_vars%u3=nu
fx_vars%u4=1
fx_vars%u5=p
fx_vars%u6=0
xa=-1._f
xb=1._f
bound=0._f
interv=1
era=0._f
erl=1.e-4_f
rres=0._f
rerr=0._f
!======================================================================
!--- Version using the QUADPACK - routine :
!----------------------------------------------------------------------
ifail = 0
call dqag(fpl,fx_vars,xa,xb,era,erl,3,rres,rerr,neval,ifail,maxsub,lenw,nsubin,iwork,work)
if (ifail.ne.0) then
if (carma%f_do_print) then
write(carma%f_LUNOPRT, *) "funa::ifail=",ifail, &
" returned by dqag() during call of funa(",nu,",",n,",",p,")"
end if
rc = RC_ERROR
return
endif
rres=rres-2._f ! ceci est un artifice pour eviter que
! la routine se plante quand la fonction
! est paire (res=0.;err=1.d-3 impossible
! a atteindre!! j'ai fpl'=fpl+1....d'ou
! int(fpl)=int(fpl')-2. integr de -1 a 1!
r = (2._f*p+1._f)/2._f
funa = r * rres
return
END FUNCTION funa
!!
!! CALLS: FUNCTION calcpoly() Legendre-Functions
!!
!! Used in funa. Integrand of eq. 14, Botet et al. 1997
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
FUNCTION fpl(x, fx_vars)
! Arguments
real(kind=f),intent(in) :: x !!
type(adgaquad_vars_type), intent(inout) :: fx_vars !! varaibles for functions being integrated
! Local declarations
real(kind=f) :: fpl
integer :: m,n,mu,nu,p,pmu
real(kind=f) :: c1,c2,c3
c1 = calcpoly(fx_vars%u1,fx_vars%u2,x,fx_vars)
c2 = calcpoly(fx_vars%u3,fx_vars%u4,x,fx_vars)
c3 = calcpoly(fx_vars%u5,fx_vars%u6,x,fx_vars)
fpl=c1*c2*c3+1._f !this is a trick!
return
END FUNCTION fpl
!!
!! Calculate Legendre Polynomials, used in eq. 14 Botet et al. 1997
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
FUNCTION calcpoly(l,m,x,fx_vars)
! Arguments
integer, intent(in) :: l !! indices
integer, intent(in) :: m !! indices
real(kind=f), intent(in) :: x !! return result
type(adgaquad_vars_type), intent(in) :: fx_vars !! variables for functions being integrated
! Local declarations
real(kind=f) :: calcpoly
integer ::lbl
real(kind=f) :: pll, pmm, somx2, pmmp1
integer :: i, ll
real(kind=f) :: fact1
integer :: mstar
mstar=m
lbl=0
calcpoly=0._f
if (mstar.lt.0)then
mstar=-m
lbl=1
endif
if (mstar.gt.l) then
pll=0._f
calcpoly=0._f
return ! si m>l, Pl,m=0 !
endif
pmm=1._f
if(mstar.gt.0) then
somx2=sqrt((1._f-x)*(1._f+x))
fact1=1._f
do i=1,mstar
pmm=+pmm*fact1*somx2 !cghmt - en + !!
fact1=fact1+2._f
end do
endif
if(l.eq.mstar) then
calcpoly=pmm
else
pmmp1=x*(2*mstar+1)*pmm
if(l.eq.mstar+1) then
calcpoly=pmmp1
else
do ll=mstar+2,l
pll=(x*(2*ll-1)*pmmp1-(ll+mstar-1)*pmm)/(ll-mstar)
pmm=pmmp1
pmmp1=pll
end do
calcpoly=pll
endif
endif
if (lbl.eq.1) then
calcpoly=(-1)**mstar*(fx_vars%fact(l-mstar)/fx_vars%fact(l+mstar))*calcpoly
mstar=-m !restitution du parametre m!!!!!
endif
return
END FUNCTION calcpoly
!!
!! replaces funb(nu,n,p) in original code,
!! saving n*n re-calculations of funa(nu,n,p).
!!
!! Calculates eq. 15, Botet et al. 1997
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version March 2013
FUNCTION funb_n(nu,n,p,funca)
! Arguments
integer, intent(in) :: nu !! indices
integer, intent(in) :: n !! indices
integer, intent(in) :: p !! indices
real(kind=f), intent(in) :: funca(nmi,nmi,0:n2m) !! return result
! Local Declarations
real(kind=f) :: funb_n
integer :: i, l, j
real(kind=f) :: var
funb_n = 0._f
i = int((p*1._f-1._f-abs(n*1._f-nu*1._f))*1._f/2._f)
!print*,nu,n,p,i
do l=0,i
j = p-2*l-1
! omit j = -1 (when nu=n and p=l=i=0)
IF (j .GE. 0) THEN
var = funca(nu,n,j) ! in main, a(nu,n,p) was stored in
! funca(nu,n;p)
funb_n = funb_n + var
ENDIF
end do
funb_n = (2._f*p+1._f) * funb_n
return
END FUNCTION funb_n
!! Replaces funs(pp,k) in original code
!!
!! CALLS:
!! FUNCTION dqagi/dq2agi/DADAPT_() Integration
!! FUNCTION xfreal_n() Integrand
!! FUNCTION xfimag_n() Integrand
!!
!! Calculates eq. 16 , Botet et al. 1997
!!
!! @author P. Rannou, R. Botet, Eric Wolf
!! @version Mar 2013
function funs_n(carma,fx_vars,pp,rc)
! Arguments
type(carma_type), intent(in) :: carma !! the carma object
type(adgaquad_vars_type), intent(inout) :: fx_vars !! varaibles for functions being integrated
integer, intent(in) :: pp !! indices