vcoord_gen.f90

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subroutine vcoord_gen(levs,lupp,pbot,psig,ppre,pupp,ptop,&
                   dpbot,dpsig,dppre,dpupp,dptop,pmin,ak,bk)
  implicit none
  integer,intent(in):: levs,lupp
  real,intent(in):: pbot,psig,ppre,pupp,ptop
  real,intent(in):: dpbot,dpsig,dppre,dpupp,dptop
  real,intent(out):: pmin,ak(0:levs),bk(0:levs)
  integer,parameter:: lo=100,li=10  ! outer and inner N-R iterations
  real pdif        ! thickness from pbot to pupp
  real delb        ! delta sig at bot
  real delt        ! delta sig at top
  real sig1        ! crossing parameter 1
  real sig2        ! crossing parameter 2
  real c1          ! proportionality constant
  real c2          ! integration constant
  real sig         ! sig variable
  real dsig        ! delta sig variable
  real delbio0     ! initial guess at delta sig at bot
  real deltio0     ! initial guess at delta sig at top
  real delbio      ! guess at delta sig at bot
  real deltio      ! guess at delta sig at top
  real c1sig1      ! d(c1)/d(sig1)
  real c1sig2      ! d(c1)/d(sig2)
  real p(2)        ! rhs in N-R iteration
  real fjac(2,2)   ! lhs in N-R iteration
  integer indx(2)  ! permutations in N-R iteration
  real ppred       ! pressure at T1-T2 boundary
  real spre        ! sig at P-T1 boundary
  real spred       ! sig at T1-T2 boundary
  real rkpre       ! level at P-T1 boundary
  real rkpred      ! level at T1-T2 boundary
  real pkpre       ! dp/dk at P-T1 boundary
  real pkkpre      ! d2p/dk2 at P-T1 boundary
  real psigd       ! pressure at T2-T3 boundary
  real ssig        ! sig at T3-S boundary
  real ssigd       ! sig at T2-T3 boundary
  real rksig       ! level at T3-S boundary
  real rksigd      ! level at T2-T3 boundary
  real pksig       ! dp/dk at T3-S boundary
  real pkksig      ! d2p/dk2 at T3-S boundary
  real p2sig       ! pressure at T3-S boundary for pmin surface pressure
  real p2sigd      ! pressure at T2-T3 boundary for pmin surface pressure
  real p2pred      ! pressure at T1-T2 boundary for pmin surface pressure
  real x2(9)       ! rhs in linear solver
  real a2(9,9)     ! lhs in linear solver
  integer indx2(9) ! permutations in linear solver
  real pkupp       ! dp/dk at U-P boundary
  real pkkupp      ! d2p/dk2 at U-P boundary
  real x3(3)       ! rhs in linear solver
  real a3(3,3)     ! lhs in linear solver
  integer indx3(3) ! permutations in linear solver
  real p1          ! pressure variable for pbot surface pressure
  real p2          ! pressure variable for pmin surface pressure
  real d           ! determinant permutation 
  integer io,ii,k
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  STEP 1.
  pdif=pbot-pupp
  delb=dpbot/pdif
  delt=dpupp/pdif
  sig1=1+delb
  sig2=-delt
  c1=log(-sig2*(1-sig1)/sig1/(sig2-1))/lupp
  c2=log((sig2-1)/(1-sig1))
  sig=1
  dsig=(sig2-sig)*(sig-sig1)*c1/(sig1-sig2)
  delbio0=-dsig
  sig=0
  dsig=(sig2-sig)*(sig-sig1)*c1/(sig1-sig2)
  deltio0=-dsig
!  Newton-Raphson iterations
  do io=1,lo
    delbio=delbio0+(delb-delbio0)*io/lo
    deltio=deltio0+(delt-deltio0)*io/lo
    do ii=1,li
      c1sig1=-1/(sig1*(1-sig1)*lupp)
      c1sig2=-1/(sig2*(sig2-1)*lupp)
      sig=1
      dsig=(sig2-sig)*(sig-sig1)*c1/(sig1-sig2)
      p(1)=-delbio-dsig
      fjac(1,1)=((-c1*(sig+sig2)+(sig-sig1)*c1sig1*(sig1+sig2)) &
                *(sig2-sig)/(sig1+sig2)**2)
      fjac(1,2)=((c1*(sig+sig1)+(sig2-sig)*c1sig2*(sig1+sig2)) &
                *(sig-sig1)/(sig1+sig2)**2)
      sig=0
      dsig=(sig2-sig)*(sig-sig1)*c1/(sig1-sig2)
      p(2)=-deltio-dsig
      fjac(2,1)=((-c1*(sig+sig2)+(sig-sig1)*c1sig1*(sig1+sig2)) &
                *(sig2-sig)/(sig1+sig2)**2)
      fjac(2,2)=((c1*(sig+sig1)+(sig2-sig)*c1sig2*(sig1+sig2)) &
                *(sig-sig1)/(sig1+sig2)**2)
      call ludcmp(fjac,2,2,indx,d)
      call lubksb(fjac,2,2,indx,p)
      sig1=sig1+p(1)
      sig2=sig2+p(2)
      c1=log(-sig2*(1-sig1)/sig1/(sig2-1))/lupp
      c2=log((sig2-1)/(1-sig1))
    enddo
  enddo
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  STEP 2.
!  Compute minimum surface pressure 
  ppred=ppre+dppre
  spre=(ppre-pupp)/pdif
  spred=(ppred-pupp)/pdif
  rkpre=(log((spre-sig2)/(sig1-spre))-c2)/c1
  rkpred=(log((spred-sig2)/(sig1-spred))-c2)/c1
  pkpre=pdif*c1*(sig2-spre)*(spre-sig1)/(sig1-sig2)
  pkkpre=pkpre*c1*(sig2+sig1-2*spre)/(sig1-sig2)
  psigd=psig-dpsig
  ssig=(psig-pupp)/pdif
  ssigd=(psigd-pupp)/pdif
  rksig=(log((ssig-sig2)/(sig1-ssig))-c2)/c1
  rksigd=(log((ssigd-sig2)/(sig1-ssigd))-c2)/c1
  pksig=pdif*c1*(sig2-ssig)*(ssig-sig1)/(sig1-sig2)
  pkksig=pksig*c1*(sig2+sig1-2*ssig)/(sig1-sig2)
  x2=0
  a2=0
  x2(1)=pkpre
  a2(1,1)=1
  a2(1,2)=rkpre
  a2(1,3)=rkpre**2
  x2(2)=pkkpre
  a2(2,2)=1
  a2(2,3)=2*rkpre
  a2(3,1)=1
  a2(3,2)=rkpred
  a2(3,3)=rkpred**2
  a2(3,4)=-1
  a2(3,5)=-rkpred
  a2(4,2)=1
  a2(4,3)=2*rkpred
  a2(4,5)=-1
  a2(5,4)=-1
  a2(5,5)=-rksigd
  a2(5,6)=1
  a2(5,7)=rksigd
  a2(5,8)=rksigd**2
  a2(6,5)=-1
  a2(6,7)=1
  a2(6,8)=2*rksigd
  a2(7,6)=1
  a2(7,7)=rksig
  a2(7,8)=rksig**2
  a2(7,9)=-pksig/pbot
  a2(8,7)=1
  a2(8,8)=2*rksig
  a2(8,9)=-pkksig/pbot
  x2(9)=ppre
  a2(9,1)=(rkpre-rkpred)
  a2(9,2)=(rkpre**2-rkpred**2)/2
  a2(9,3)=(rkpre**3-rkpred**3)/3
  a2(9,4)=(rkpred-rksigd)
  a2(9,5)=(rkpred**2-rksigd**2)/2
  a2(9,6)=(rksigd-rksig)
  a2(9,7)=(rksigd**2-rksig**2)/2
  a2(9,8)=(rksigd**3-rksig**3)/3
  a2(9,9)=psig/pbot
  call ludcmp(a2,9,9,indx2,d)
  call lubksb(a2,9,9,indx2,x2)
  pmin=x2(9)
  p2sig=psig/pbot*pmin
  p2sigd=p2sig &
        +x2(6)*(rksigd-rksig) &
        +x2(7)*(rksigd**2-rksig**2)/2 &
        +x2(8)*(rksigd**3-rksig**3)/3
  p2pred=p2sigd &
        +x2(4)*(rkpred-rksigd) &
        +x2(5)*(rkpred**2-rksigd**2)/2
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  STEP 3.
  if(lupp.lt.levs) then
    pkupp=pdif*c1*(sig2-0)*(0-sig1)/(sig1-sig2)
    pkkupp=pkupp*c1*(sig2+sig1-2*0)/(sig1-sig2)
    x3=0
    a3=0
    x3(1)=pkupp
    a3(1,1)=pupp
    x3(2)=pkkupp*pupp-pkupp**2
    a3(2,2)=pupp**2
    x3(3)=log((ptop+dptop)/pupp)
    a3(3,1)=levs-1-lupp
    a3(3,2)=(levs-1-lupp)**2/2
    a3(3,3)=(levs-1-lupp)**3/3
    call ludcmp(a3,3,3,indx3,d)
    call lubksb(a3,3,3,indx3,x3)
  endif
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  STEP 4.
!  Compute hybrid interface values
  ak(0)=0
  bk(0)=1
  do k=1,levs-1
    if(k.ge.lupp) then
      p1=pupp*exp(x3(1)*(k-lupp)+x3(2)*(k-lupp)**2/2+x3(3)*(k-lupp)**3/3)
    else
      p1=(sig1*exp(c1*k+c2)+sig2)/(exp(c1*k+c2)+1)*pdif+pupp
    endif
    if(k.ge.rkpre) then
      p2=p1
    elseif(k.ge.rkpred) then
      p2=p2pred+x2(1)*(k-rkpred) &
               +x2(2)*(k**2-rkpred**2)/2 &
               +x2(3)*(k**3-rkpred**3)/3
    elseif(k.ge.rksigd) then
      p2=p2sigd+x2(4)*(k-rksigd) &
               +x2(5)*(k**2-rksigd**2)/2
    elseif(k.ge.rksig) then
      p2=p2sig+x2(6)*(k-rksig) &
              +x2(7)*(k**2-rksig**2)/2 &
              +x2(8)*(k**3-rksig**3)/3
    else
      p2=p1/pbot*pmin
    endif
    ak(k)=(p2*pbot-p1*pmin)/(pbot-pmin)
    bk(k)=(p1-p2)/(pbot-pmin)
  enddo
  ak(levs)=ptop
  bk(levs)=0
end subroutine