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derivatives.f90
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MODULE derivatives
USE parameters
USE fft
USE mpi
IMPLICIT NONE
! integer :: kx,ky,kh2
! integer :: ikx,iky
! integer :: izh0,izh1,izh2,izh3
CONTAINS
SUBROUTINE convol(nuk,nvk,nwk,nbk,nur,nvr,nwr,nbr,uk,vk,wk,bk,ur,vr,wr,br)
! this subroutine computes 1/rho d/dxj( rho uj ui) in the divergence form on a staggered grid.
double complex, dimension(iktx,ikty,n3h2) :: uk,vk,wk,bk
double complex, dimension(iktx,ikty,n3h1) :: nuk,nvk,nwk,nbk
double precision, dimension(n1d,n2d,n3h2) :: ur,vr,wr,br
double precision, dimension(n1d,n2d,n3h1) :: nur,nvr,nwr,nbr
double complex, dimension(iktx,ikty,n3h2) :: umem,vmem,wmem,bmem
!Suboptimal but simple.
double complex, dimension(iktx,ikty,n3h1) :: Auk,Buk,Cuk,Avk,Bvk,Cvk,Awk,Bwk,Cwk,Abk,Bbk,Cbk
double precision, dimension(n1d,n2d,n3h1) :: Aur,Bur,Cur,Avr,Bvr,Cvr,Awr,Bwr,Cwr,Abr,Bbr,Cbr
equivalence(Aur,Auk)
equivalence(Avr,Avk)
equivalence(Awr,Awk)
equivalence(Abr,Abk)
equivalence(Bur,Buk)
equivalence(Bvr,Bvk)
equivalence(Bwr,Bwk)
equivalence(Bbr,Bbk)
equivalence(Cur,Cuk)
equivalence(Cvr,Cvk)
equivalence(Cwr,Cwk)
equivalence(Cbr,Cbk)
!DO A COPY OF VELOCITY, FASTER THAN TAKE A TRANSFORM BACK
umem=uk
vmem=vk
wmem=wk
bmem=bk
call fft_c2r(uk,ur,n3h2)
call fft_c2r(vk,vr,n3h2)
call fft_c2r(wk,wr,n3h2)
call fft_c2r(bk,br,n3h2)
!Assign values to A, B and C for each of the three components of the nlt
!nuk = ikx Auk + iky Buk + 1/2dz Cuk (Same for nvk and nwk but u->v or ->w)
!With A,B and C defined as:
!Aur_n = u_n^2
!Bur_n = v_n*u_n
!Cur_n = w_n*u_n+1 + w_n*u_n - u_n*w_n-1 - u_n-1*w_n-1
!(same as for u)
!Avr_n = u_n*v_n
!Bvr_n = v_n^2
!Cvr_n = w_n*v_n+1 + w_n*v_n - w_n-1*v_n - w_n-1*v_n-1
!Awr_n = 1/2 ( u_n+1*w_n + u_n*w_n)
!Bwr_n = 1/2 ( v_n+1*w_n + v_n*w_n)
!Cwr_n = 1/2 ( w_n+1^2 - w_n-1^2 ) + w_n * (w_n+1 - w_n-1)
!Atr_n = 1/2 ( u_n+1*t_n + u_n*t_n)
!Btr_n = 1/2 ( v_n+1*t_n + v_n*t_n)
!Ctr_n = 1/2 [ w_n+1*t_n+1 + w_n+1*t_n + w_n*t_n+1 - w_n*t_n-1 - w_n-1*t_n - w_n-1*b_n-1 ]
!DAVID'S CHEAT (NOT USED IN THE CURRENT VERSION): modify the point below the top and the one above the bottom for the unstaggered field w
!David's cheat make the w-nonlinear term conservative in the momentum equation (triple integral of n_w = 0) but NOT in the energy equation
!(triple integral of w*n_w is NOT = 0). It is unlikely that this tweak makes energy conservation better, and it causes a O(dz) error at
!the modified grid points.
!See journal Feb 19th 2016 or code documentation. March 25h 2016: reverted to the original, non-David-cheated algorithm.
do izh1=1,n3h1
izh2=izh1+1
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
Aur(ix,iy,izh1) = ur(ix,iy,izh2)*ur(ix,iy,izh2)
Bur(ix,iy,izh1) = vr(ix,iy,izh2)*ur(ix,iy,izh2)
Cur(ix,iy,izh1) = rho_u(izh2)*wr(ix,iy,izh2)*ur(ix,iy,izh2+1) + rho_u(izh2)*wr(ix,iy,izh2)*ur(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*ur(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*ur(ix,iy,izh2-1)
Avr(ix,iy,izh1) = ur(ix,iy,izh2)*vr(ix,iy,izh2)
Bvr(ix,iy,izh1) = vr(ix,iy,izh2)*vr(ix,iy,izh2)
Cvr(ix,iy,izh1) = rho_u(izh2)*wr(ix,iy,izh2)*vr(ix,iy,izh2+1) + rho_u(izh2)*wr(ix,iy,izh2)*vr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*vr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*vr(ix,iy,izh2-1)
Awr(ix,iy,izh1) = 0.5 * ( ur(ix,iy,izh2+1)*wr(ix,iy,izh2) + ur(ix,iy,izh2)*wr(ix,iy,izh2) )
Bwr(ix,iy,izh1) = 0.5 * ( vr(ix,iy,izh2+1)*wr(ix,iy,izh2) + vr(ix,iy,izh2)*wr(ix,iy,izh2) )
Cwr(ix,iy,izh1) = 0.5 * ( rho_u(izh2+1)*wr(ix,iy,izh2+1)*wr(ix,iy,izh2+1) + rho_u(izh2+1)*wr(ix,iy,izh2+1)*wr(ix,iy,izh2) + rho_u(izh2)*wr(ix,iy,izh2)*wr(ix,iy,izh2+1) - rho_u(izh2)*wr(ix,iy,izh2)*wr(ix,iy,izh2-1) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*wr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*wr(ix,iy,izh2-1) )
Abr(ix,iy,izh1) = ur(ix,iy,izh2)*br(ix,iy,izh2)
Bbr(ix,iy,izh1) = vr(ix,iy,izh2)*br(ix,iy,izh2)
Cbr(ix,iy,izh1) = rho_u(izh2)*wr(ix,iy,izh2)*br(ix,iy,izh2+1) + rho_u(izh2)*wr(ix,iy,izh2)*br(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*br(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*br(ix,iy,izh2-1)
!Unstaggered buoyancy tk
! Atr(ix,iy,izh1) = 0.5 * ( ur(ix,iy,izh2+1)*tr(ix,iy,izh2) + ur(ix,iy,izh2)*tr(ix,iy,izh2) )
! Btr(ix,iy,izh1) = 0.5 * ( vr(ix,iy,izh2+1)*tr(ix,iy,izh2) + vr(ix,iy,izh2)*tr(ix,iy,izh2) )
! Ctr(ix,iy,izh1) = 0.5 * ( rho_u(izh2+1)*wr(ix,iy,izh2+1)*tr(ix,iy,izh2+1) + rho_u(izh2+1)*wr(ix,iy,izh2+1)*tr(ix,iy,izh2) + rho_u(izh2)*wr(ix,iy,izh2)*tr(ix,iy,izh2+1) - rho_u(izh2)*wr(ix,iy,izh2)*tr(ix,iy,izh2-1) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*tr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*tr(ix,iy,izh2-1) )
else
Aur(ix,iy,izh1) = 0.D0
Bur(ix,iy,izh1) = 0.D0
Cur(ix,iy,izh1) = 0.D0
Avr(ix,iy,izh1) = 0.D0
Bvr(ix,iy,izh1) = 0.D0
Cvr(ix,iy,izh1) = 0.D0
Awr(ix,iy,izh1) = 0.D0
Bwr(ix,iy,izh1) = 0.D0
Cwr(ix,iy,izh1) = 0.D0
Abr(ix,iy,izh1) = 0.D0
Bbr(ix,iy,izh1) = 0.D0
Cbr(ix,iy,izh1) = 0.D0
end if
end do
end do
end do
!Boundary conditions! X-marked regions were assigned with insignificant values. We give the correct values here
if(mype==0) then
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
!Aur, Avr, Bur and Bur are fine already.
Cur(ix,iy,izbot1) = rho_u(izbot2)*wr(ix,iy,izbot2) * ( ur(ix,iy,izbot2+1) + ur(ix,iy,izbot2) )
Cvr(ix,iy,izbot1) = rho_u(izbot2)*wr(ix,iy,izbot2) * ( vr(ix,iy,izbot2+1) + vr(ix,iy,izbot2) )
Cwr(ix,iy,izbot1) = 0.5 * (rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2+1) + rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2) + rho_u(izbot2)*wr(ix,iy,izbot2)*wr(ix,iy,izbot2+1) ) !Original, non-Davidcheat
! Cwr(ix,iy,izbot1) = 0.5 * (rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2+1) + rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2) + rho_u(izbot2)*wr(ix,iy,izbot2)*wr(ix,iy,izbot2+1) + rho_u(izbot2)*wr(ix,iy,izbot2)*wr(ix,iy,izbot2) ) !David's cheat
Cbr(ix,iy,izbot1) = rho_u(izbot2)*wr(ix,iy,izbot2) * ( br(ix,iy,izbot2+1) + br(ix,iy,izbot2) )
! Ctr(ix,iy,izbot1) = 0.5 * (wr(ix,iy,izbot2+1)*tr(ix,iy,izbot2+1) + wr(ix,iy,izbot2+1)*tr(ix,iy,izbot2) + wr(ix,iy,izbot2)*tr(ix,iy,izbot2+1) + wr(ix,iy,izbot2)*tr(ix,iy,izbot2) )
else
Cur(ix,iy,izbot1) = 0.D0
Cvr(ix,iy,izbot1) = 0.D0
Cwr(ix,iy,izbot1) = 0.D0
Cbr(ix,iy,izbot1) = 0.D0
end if
end do
end do
else if(mype==(npe-1)) then
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
!Aur, Avr, Awr, Bur, Bvr and Bwr are fine already.
Cur(ix,iy,iztop1) = - rho_u(iztop2-1)*wr(ix,iy,iztop2-1) * ( ur(ix,iy,iztop2) + ur(ix,iy,iztop2-1) )
Cvr(ix,iy,iztop1) = - rho_u(iztop2-1)*wr(ix,iy,iztop2-1) * ( vr(ix,iy,iztop2) + vr(ix,iy,iztop2-1) )
Cbr(ix,iy,iztop1) = - rho_u(iztop2-1)*wr(ix,iy,iztop2-1) * ( br(ix,iy,iztop2) + br(ix,iy,iztop2-1) )
Awr(ix,iy,iztop1) = 0.D0
Bwr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1) = 0.D0
! Atr(ix,iy,iztop1) = 0.D0
! Btr(ix,iy,iztop1) = 0.D0
! Ctr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1-1) = - 0.5* (rho_u(iztop2-1)*wr(ix,iy,iztop2-1)*wr(ix,iy,iztop2-2) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-1) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-2) ) !Original
! Cwr(ix,iy,iztop1-1) = - 0.5* (rho_u(iztop2-1)*wr(ix,iy,iztop2-1)*wr(ix,iy,iztop2-1) + rho_u(iztop2-1)*wr(ix,iy,iztop2-1)*wr(ix,iy,iztop2-2) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-1) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-2) ) !David's cheat
! Ctr(ix,iy,iztop1-1) = - 0.5* (wr(ix,iy,iztop2-1)*tr(ix,iy,iztop2-1) + wr(ix,iy,iztop2-1)*tr(ix,iy,iztop2-2) + wr(ix,iy,iztop2-2)*tr(ix,iy,iztop2-1) + wr(ix,iy,iztop2-2)*tr(ix,iy,iztop2-2) )
else
Cur(ix,iy,iztop1) = 0.D0
Cvr(ix,iy,iztop1) = 0.D0
Cbr(ix,iy,iztop1) = 0.D0
Awr(ix,iy,iztop1) = 0.D0
Bwr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1) = 0.D0
! Atr(ix,iy,iztop1) = 0.D0
! Btr(ix,iy,iztop1) = 0.D0
! Ctr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1-1) = 0.D0
! Ctr(ix,iy,iztop1-1) = 0.D0
end if
end do
end do
end if
!Move to h-space
call fft_r2c(Aur,Auk,n3h1)
call fft_r2c(Avr,Avk,n3h1)
call fft_r2c(Awr,Awk,n3h1)
call fft_r2c(Abr,Abk,n3h1)
call fft_r2c(Bur,Buk,n3h1)
call fft_r2c(Bvr,Bvk,n3h1)
call fft_r2c(Bwr,Bwk,n3h1)
call fft_r2c(Bbr,Bbk,n3h1)
call fft_r2c(Cur,Cuk,n3h1)
call fft_r2c(Cvr,Cvk,n3h1)
call fft_r2c(Cwr,Cwk,n3h1)
call fft_r2c(Cbr,Cbk,n3h1)
!nuk = ikx Auk + iky Buk + 1/2dz Cuk (Same for nvk, nwk and ntk but u->v, ->w or -> t)
do izh1=1,n3h1
izh2=izh1+1
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
nuk(ikx,iky,izh1) = i*kx*Auk(ikx,iky,izh1) + i*ky*Buk(ikx,iky,izh1) + Cuk(ikx,iky,izh1)/(2*dz*rho_s(izh2))
nvk(ikx,iky,izh1) = i*kx*Avk(ikx,iky,izh1) + i*ky*Bvk(ikx,iky,izh1) + Cvk(ikx,iky,izh1)/(2*dz*rho_s(izh2))
nwk(ikx,iky,izh1) = i*kx*Awk(ikx,iky,izh1) + i*ky*Bwk(ikx,iky,izh1) + Cwk(ikx,iky,izh1)/(2*dz*rho_u(izh2))
nbk(ikx,iky,izh1) = i*kx*Abk(ikx,iky,izh1) + i*ky*Bbk(ikx,iky,izh1) + Cbk(ikx,iky,izh1)/(2*dz*rho_s(izh2))
! ntk(ikx,iky,izh1) = i*kx*Atk(ikx,iky,izh1) + i*ky*Btk(ikx,iky,izh1) + Ctk(ikx,iky,izh1)/(2*dz*rho_u(izh2))
else
nuk(ikx,iky,izh1) = (0.D0,0.D0)
nvk(ikx,iky,izh1) = (0.D0,0.D0)
nwk(ikx,iky,izh1) = (0.D0,0.D0)
nbk(ikx,iky,izh1) = (0.D0,0.D0)
endif
enddo
enddo
enddo
!Copy back velocity to avoid inverse FFTs!
uk=umem
vk=vmem
wk=wmem
bk=bmem
END SUBROUTINE convol
!The original version!
SUBROUTINE convol2(nuk,nvk,nwk,ntk,nur,nvr,nwr,ntr,uk,vk,wk,tk,ur,vr,wr,tr)
! this subroutine computes ((u.grad)u)i = d/dxj(uj ui) in the divergence form on a staggered grid.
double complex, dimension(iktx,ikty,n3h2) :: uk,vk,wk,tk
double complex, dimension(iktx,ikty,n3h1) :: nuk,nvk,nwk,ntk
double precision, dimension(n1d,n2d,n3h2) :: ur,vr,wr,tr
double precision, dimension(n1d,n2d,n3h1) :: nur,nvr,nwr,ntr
double complex, dimension(iktx,ikty,n3h2) :: umem,vmem,wmem,tmem
!Suboptimal but simple.
double complex, dimension(iktx,ikty,n3h1) :: Auk,Buk,Cuk,Avk,Bvk,Cvk,Awk,Bwk,Cwk,Atk,Btk,Ctk
double precision, dimension(n1d,n2d,n3h1) :: Aur,Bur,Cur,Avr,Bvr,Cvr,Awr,Bwr,Cwr,Atr,Btr,Ctr
equivalence(Aur,Auk)
equivalence(Avr,Avk)
equivalence(Awr,Awk)
equivalence(Atr,Atk)
equivalence(Bur,Buk)
equivalence(Bvr,Bvk)
equivalence(Bwr,Bwk)
equivalence(Btr,Btk)
equivalence(Cur,Cuk)
equivalence(Cvr,Cvk)
equivalence(Cwr,Cwk)
equivalence(Ctr,Ctk)
!DO A COPY OF VELOCITY, FASTER THAN TAKE A TRANSFORM BACK
umem=uk
vmem=vk
wmem=wk
tmem=tk
call fft_c2r(uk,ur,n3h2)
call fft_c2r(vk,vr,n3h2)
call fft_c2r(wk,wr,n3h2)
call fft_c2r(tk,tr,n3h2)
!Assign values to A, B and C for each of the three components of the nlt
!nuk = ikx Auk + iky Buk + 1/2dz Cuk (Same for nvk and nwk but u->v or ->w)
!With A,B and C defined as:
!Aur_n = u_n^2
!Bur_n = v_n*u_n
!Cur_n = w_n*u_n+1 + w_n*u_n - u_n*w_n-1 - u_n-1*w_n-1
!(same as for u)
!Avr_n = u_n*v_n
!Bvr_n = v_n^2
!Cvr_n = w_n*v_n+1 + w_n*v_n - w_n-1*v_n - w_n-1*v_n-1
!Awr_n = 1/2 ( u_n+1*w_n + u_n*w_n)
!Bwr_n = 1/2 ( v_n+1*w_n + v_n*w_n)
!Cwr_n = 1/2 ( w_n+1^2 - w_n-1^2 ) + w_n * (w_n+1 - w_n-1)
!Atr_n = 1/2 ( u_n+1*t_n + u_n*t_n)
!Btr_n = 1/2 ( v_n+1*t_n + v_n*t_n)
!Ctr_n = 1/2 [ w_n+1*t_n+1 + w_n+1*t_n + w_n*t_n+1 - w_n*t_n-1 - w_n-1*t_n - w_n-1*b_n-1 ]
!*****************************************************************************!
!Attention: I am starting to doubt this is the best way to do it. Two comments:
! 1. Using the analytic rho_0 sometimes is not NECESSARLY better than using rho_0 at the points dictated by FDs
! 2. For the interpolation of fields (ex. wwz and wbz terms) there are other possibilities that would turn out simpler.
! For instance d\dz(rho w b)| idz could be just [rho w b|(i+1)dz - rho w b|(i-1)]/2dz
! and I am not sure if this is better or worse than what we use now with complicated cross terms...
! When code will be re-written, I want to make sure I got the best discretization possible.
do izh1=1,n3h1
izh2=izh1+1
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
Aur(ix,iy,izh1) = ur(ix,iy,izh2)*ur(ix,iy,izh2)
Bur(ix,iy,izh1) = vr(ix,iy,izh2)*ur(ix,iy,izh2)
Cur(ix,iy,izh1) = rho_u(izh2)*wr(ix,iy,izh2)*ur(ix,iy,izh2+1) + rho_u(izh2)*wr(ix,iy,izh2)*ur(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*ur(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*ur(ix,iy,izh2-1)
Avr(ix,iy,izh1) = ur(ix,iy,izh2)*vr(ix,iy,izh2)
Bvr(ix,iy,izh1) = vr(ix,iy,izh2)*vr(ix,iy,izh2)
Cvr(ix,iy,izh1) = rho_u(izh2)*wr(ix,iy,izh2)*vr(ix,iy,izh2+1) + rho_u(izh2)*wr(ix,iy,izh2)*vr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*vr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*vr(ix,iy,izh2-1)
Awr(ix,iy,izh1) = 0.5 * ( ur(ix,iy,izh2+1)*wr(ix,iy,izh2) + ur(ix,iy,izh2)*wr(ix,iy,izh2) )
Bwr(ix,iy,izh1) = 0.5 * ( vr(ix,iy,izh2+1)*wr(ix,iy,izh2) + vr(ix,iy,izh2)*wr(ix,iy,izh2) )
Cwr(ix,iy,izh1) = 0.5 * ( rho_u(izh2+1)*wr(ix,iy,izh2+1)*wr(ix,iy,izh2+1) + rho_u(izh2+1)*wr(ix,iy,izh2+1)*wr(ix,iy,izh2) + rho_u(izh2)*wr(ix,iy,izh2)*wr(ix,iy,izh2+1) - rho_u(izh2)*wr(ix,iy,izh2)*wr(ix,iy,izh2-1) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*wr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*wr(ix,iy,izh2-1) )
Atr(ix,iy,izh1) = 0.5 * ( ur(ix,iy,izh2+1)*tr(ix,iy,izh2) + ur(ix,iy,izh2)*tr(ix,iy,izh2) )
Btr(ix,iy,izh1) = 0.5 * ( vr(ix,iy,izh2+1)*tr(ix,iy,izh2) + vr(ix,iy,izh2)*tr(ix,iy,izh2) )
Ctr(ix,iy,izh1) = 0.5 * ( rho_u(izh2+1)*wr(ix,iy,izh2+1)*tr(ix,iy,izh2+1) + rho_u(izh2+1)*wr(ix,iy,izh2+1)*tr(ix,iy,izh2) + rho_u(izh2)*wr(ix,iy,izh2)*tr(ix,iy,izh2+1) - rho_u(izh2)*wr(ix,iy,izh2)*tr(ix,iy,izh2-1) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*tr(ix,iy,izh2) - rho_u(izh2-1)*wr(ix,iy,izh2-1)*tr(ix,iy,izh2-1) )
else
Aur(ix,iy,izh1) = 0.D0
Bur(ix,iy,izh1) = 0.D0
Cur(ix,iy,izh1) = 0.D0
Avr(ix,iy,izh1) = 0.D0
Bvr(ix,iy,izh1) = 0.D0
Cvr(ix,iy,izh1) = 0.D0
Awr(ix,iy,izh1) = 0.D0
Bwr(ix,iy,izh1) = 0.D0
Cwr(ix,iy,izh1) = 0.D0
Atr(ix,iy,izh1) = 0.D0
Btr(ix,iy,izh1) = 0.D0
Ctr(ix,iy,izh1) = 0.D0
end if
end do
end do
end do
!Boundary conditions! X-marked regions were assigned with insignificant values. We give the correct values here
if(mype==0) then
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
!Aur, Avr, Bur and Bur are fine already.
Cur(ix,iy,izbot1) = rho_u(izbot2)*wr(ix,iy,izbot2) * ( ur(ix,iy,izbot2+1) + ur(ix,iy,izbot2) )
Cvr(ix,iy,izbot1) = rho_u(izbot2)*wr(ix,iy,izbot2) * ( vr(ix,iy,izbot2+1) + vr(ix,iy,izbot2) )
Cwr(ix,iy,izbot1) = 0.5 * (rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2+1) + rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*wr(ix,iy,izbot2) + rho_u(izbot2)*wr(ix,iy,izbot2)*wr(ix,iy,izbot2+1) )
Ctr(ix,iy,izbot1) = 0.5 * (rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*tr(ix,iy,izbot2+1) + rho_u(izbot2+1)*wr(ix,iy,izbot2+1)*tr(ix,iy,izbot2) + rho_u(izbot2)*wr(ix,iy,izbot2)*tr(ix,iy,izbot2+1) ) !1/4dz (W2*B2 + W2*B1 + W1*B2)
else
Cur(ix,iy,izbot1) = 0.D0
Cvr(ix,iy,izbot1) = 0.D0
Cwr(ix,iy,izbot1) = 0.D0
Ctr(ix,iy,izbot1) = 0.D0
end if
end do
end do
else if(mype==(npe-1)) then
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
!Aur, Avr, Awr, Bur, Bvr and Bwr are fine already.
Cur(ix,iy,iztop1) = - rho_u(iztop2-1)*wr(ix,iy,iztop2-1) * ( ur(ix,iy,iztop2) + ur(ix,iy,iztop2-1) )
Cvr(ix,iy,iztop1) = - rho_u(iztop2-1)*wr(ix,iy,iztop2-1) * ( vr(ix,iy,iztop2) + vr(ix,iy,iztop2-1) )
Awr(ix,iy,iztop1) = 0.D0
Bwr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1) = 0.D0
Atr(ix,iy,iztop1) = 0.D0
Btr(ix,iy,iztop1) = 0.D0
Ctr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1-1) = - 0.5* (rho_u(iztop2-1)*wr(ix,iy,iztop2-1)*wr(ix,iy,iztop2-2) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-1) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*wr(ix,iy,iztop2-2) ) !1/4dz (WN-1*WN-2 + WN-2*WN-1 + WN-2*WN-2)
Ctr(ix,iy,iztop1-1) = - 0.5* (rho_u(iztop2-1)*wr(ix,iy,iztop2-1)*tr(ix,iy,iztop2-2) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*tr(ix,iy,iztop2-1) + rho_u(iztop2-2)*wr(ix,iy,iztop2-2)*tr(ix,iy,iztop2-2) ) !1/4dz (WN-1*BN-2 + WN-2*BN-1 + WN-2*BN-2)
else
Cur(ix,iy,iztop1) = 0.D0
Cvr(ix,iy,iztop1) = 0.D0
Awr(ix,iy,iztop1) = 0.D0
Bwr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1) = 0.D0
Atr(ix,iy,iztop1) = 0.D0
Btr(ix,iy,iztop1) = 0.D0
Ctr(ix,iy,iztop1) = 0.D0
Cwr(ix,iy,iztop1-1) = 0.D0
Ctr(ix,iy,iztop1-1) = 0.D0
end if
end do
end do
end if
!Move to h-space
call fft_r2c(Aur,Auk,n3h1)
call fft_r2c(Avr,Avk,n3h1)
call fft_r2c(Awr,Awk,n3h1)
call fft_r2c(Atr,Atk,n3h1)
call fft_r2c(Bur,Buk,n3h1)
call fft_r2c(Bvr,Bvk,n3h1)
call fft_r2c(Bwr,Bwk,n3h1)
call fft_r2c(Btr,Btk,n3h1)
call fft_r2c(Cur,Cuk,n3h1)
call fft_r2c(Cvr,Cvk,n3h1)
call fft_r2c(Cwr,Cwk,n3h1)
call fft_r2c(Ctr,Ctk,n3h1)
!nuk = ikx Auk + iky Buk + 1/2dz Cuk (Same for nvk, nwk and ntk but u->v, ->w or -> t)
do izh1=1,n3h1
izh2=izh1+1
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
nuk(ikx,iky,izh1) = i*kx*Auk(ikx,iky,izh1) + i*ky*Buk(ikx,iky,izh1) + Cuk(ikx,iky,izh1)/(2*dz*rho_s(izh2))
nvk(ikx,iky,izh1) = i*kx*Avk(ikx,iky,izh1) + i*ky*Bvk(ikx,iky,izh1) + Cvk(ikx,iky,izh1)/(2*dz*rho_s(izh2))
nwk(ikx,iky,izh1) = i*kx*Awk(ikx,iky,izh1) + i*ky*Bwk(ikx,iky,izh1) + Cwk(ikx,iky,izh1)/(2*dz*rho_u(izh2))
ntk(ikx,iky,izh1) = i*kx*Atk(ikx,iky,izh1) + i*ky*Btk(ikx,iky,izh1) + Ctk(ikx,iky,izh1)/(2*dz*rho_u(izh2))
else
nuk(ikx,iky,izh1) = (0.D0,0.D0)
nvk(ikx,iky,izh1) = (0.D0,0.D0)
nwk(ikx,iky,izh1) = (0.D0,0.D0)
ntk(ikx,iky,izh1) = (0.D0,0.D0)
endif
enddo
enddo
enddo
!Copy back velocity to avoid inverse FFTs!
uk=umem
vk=vmem
wk=wmem
tk=tmem
END SUBROUTINE convol2
SUBROUTINE convol_q(nqk,nqr,uk,vk,qk,ur,vr,qr)
! this subroutine computes ((u.grad)q) = d/dxj(uj q) in the divergence form on a staggered grid.
! Notice that this routine outputs the r-space velocity fields.
double complex, dimension(iktx,ikty,n3h2) :: uk,vk
double complex, dimension(iktx,ikty,n3h1) :: qk
double complex, dimension(iktx,ikty,n3h0) :: nqk
double complex, dimension(iktx,ikty,n3h1) :: qmem
double precision, dimension(n1d,n2d,n3h2) :: ur,vr
double precision, dimension(n1d,n2d,n3h1) :: qr
double precision, dimension(n1d,n2d,n3h0) :: nqr
!Suboptimal but simple.
double complex, dimension(iktx,ikty,n3h0) :: Aqk,Bqk
double precision, dimension(n1d,n2d,n3h0) :: Aqr,Bqr
equivalence(Aqr,Aqk)
equivalence(Bqr,Bqk)
!I think we don't need to keep a copy of u in qg, right?
call fft_c2r(uk,ur,n3h2)
call fft_c2r(vk,vr,n3h2)
qmem = qk
call fft_c2r(qk,qr,n3h1)
!Assign values to A and B
!nqk = ikx Aqk + iky Bqk
!With A,B and C defined as:
!Aqr_n = u_n*q_n
!Bqr_n = v_n*q_n
Aqr=0.
Bqr=0.
do izh0=1,n3h0
izh1=izh0+1
izh2=izh0+2
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
Aqr(ix,iy,izh0) = ur(ix,iy,izh2)*qr(ix,iy,izh1)
Bqr(ix,iy,izh0) = vr(ix,iy,izh2)*qr(ix,iy,izh1)
end if
end do
end do
end do
!Move to h-space
call fft_r2c(Aqr,Aqk,n3h0)
call fft_r2c(Bqr,Bqk,n3h0)
!nqk = ikx Aqk + iky Bqk
do izh0=1,n3h0
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
nqk(ikx,iky,izh0) = i*kx*Aqk(ikx,iky,izh0) + i*ky*Bqk(ikx,iky,izh0)
else
nqk(ikx,iky,izh0) = (0.D0,0.D0)
endif
enddo
enddo
enddo
qk = qmem
END SUBROUTINE convol_q
SUBROUTINE compute_streamfunction(uk,vk,psik) !Computes the QG streamfunction (rotational part of the horizontal flow)
double complex, dimension(iktx,ikty,n3h2), intent(in) :: uk,vk
double complex, dimension(iktx,ikty,n3h1), intent(out) :: psik
!Compute psi
do izh1=1,n3h1
izh2=izh1+1
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2 = kx*kx+ky*ky
psik(ikx,iky,izh1) = i * ( ky*uk(ikx,iky,izh2) - kx*vk(ikx,iky,izh2) )/kh2
if(kh2==0) psik(ikx,iky,izh1) = (0.D0,0.D0)
end do
end do
end do
end SUBROUTINE compute_streamfunction
SUBROUTINE compute_velo(uk,vk,wk,bk,psik) !Computes the QG velocity and buoyancy fields from the streamfunction, assuming psi_z=0 at the top/bottom. bk is staggered.
double complex, dimension(iktx,ikty,n3h2), intent(out) :: uk,vk,wk,bk
double complex, dimension(iktx,ikty,n3h1), intent(in) :: psik
!Compute velocity/buoyancy fields
do izh1=1,n3h1
izh2=izh1+1
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
uk(ikx,iky,izh2) = - i*ky*psik(ikx,iky,izh1)
vk(ikx,iky,izh2) = i*kx*psik(ikx,iky,izh1)
wk(ikx,iky,izh2) = (0.D0,0.D0)
bk(ikx,iky,izh2) = ( psik(ikx,iky,izh1+1) - psik(ikx,iky,izh1-1) )/(r_1s(izh2)*2.*dz) !1/r_1 d psi/dz
else
uk(ikx,iky,izh2) = (0.D0,0.D0)
vk(ikx,iky,izh2) = (0.D0,0.D0)
wk(ikx,iky,izh2) = (0.D0,0.D0)
bk(ikx,iky,izh2) = (0.D0,0.D0)
endif
enddo
enddo
enddo
!Explicit treatment of boundary conditions
if(mype==0) then
do iky=1,ikty
do ikx=1,iktx
bk(ikx,iky,izbot2) = ( psik(ikx,iky,izbot1+1) - psik(ikx,iky,izbot1) )/(r_1s(izbot2)*2.*dz) !1/r_1 d psi/dz
enddo
enddo
elseif(mype==(npe-1)) then
do iky=1,ikty
do ikx=1,iktx
bk(ikx,iky,iztop2) = ( psik(ikx,iky,iztop1) - psik(ikx,iky,iztop1-1) )/(r_1s(iztop2)*2.*dz) !1/r_1 d psi/dz
enddo
enddo
end if
end SUBROUTINE compute_velo
SUBROUTINE omega_verification2(wak,lhs)
double complex, dimension(iktx,ikty,n3h1) :: wak
double complex, dimension(iktx,ikty,n3h0) :: lhs
do izh0=1,n3h0
izh1=izh0+1
izh2=izh0+2
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,izh0) = (wak(ikx,iky,izh1+1) -2.*wak(ikx,iky,izh1) +wak(ikx,iky,izh1-1))/(dz*dz) + r_3u(izh2)*(wak(ikx,iky,izh1+1) -wak(ikx,iky,izh1-1))/(2.*dz) + (r_5u(izh2)-kh2/a_ell_u(izh2))*wak(ikx,iky,izh1)
else
lhs(ikx,iky,izh0) = (0.D0,0.D0)
endif
enddo
enddo
enddo
if(mype==0) then
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,1) = (wak(ikx,iky,izbot1+1) -2.*wak(ikx,iky,izbot1) )/(dz*dz) + r_3u(izbot2)*wak(ikx,iky,izbot1+1)/(2.*dz) + (r_5u(izbot2)-kh2/a_ell_u(izbot2))*wak(ikx,iky,izbot1)
else
lhs(ikx,iky,1) = (0.D0,0.D0)
endif
enddo
enddo
end if
if(mype==(npe-1)) then
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,n3h0-1) = (wak(ikx,iky,iztop1-2) -2.*wak(ikx,iky,iztop1-1))/(dz*dz) - r_3u(iztop2-1)*wak(ikx,iky,iztop1-2)/(2.*dz) + (r_5u(iztop2-1)-kh2/a_ell_u(iztop2-1))*wak(ikx,iky,iztop1-1)
lhs(ikx,iky,n3h0) = (0.D0,0.D0)
else
lhs(ikx,iky,n3h0-1) = (0.D0,0.D0)
lhs(ikx,iky,n3h0) = (0.D0,0.D0)
endif
enddo
enddo
end if
end SUBROUTINE omega_verification2
SUBROUTINE omega_verification(wak,lhs)
double complex, dimension(iktx,ikty,n3h1) :: wak
double complex, dimension(iktx,ikty,n3h0) :: lhs
do izh0=1,n3h0
izh1=izh0+1
izh2=izh0+2
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,izh0) = (rho_u(izh2+1)/rho_s(izh2+1))*wak(ikx,iky,izh1+1) - (rho_u(izh2)*(rho_s(izh2+1)+rho_s(izh2))/(rho_s(izh2+1)*rho_s(izh2)) + r_1(izh2)*r_2(izh2)*dz*dz*kh2/big_F)*wak(ikx,iky,izh1) + (rho_u(izh2-1)/rho_s(izh2))*wak(ikx,iky,izh1-1)
else
lhs(ikx,iky,izh0) = (0.D0,0.D0)
endif
enddo
enddo
enddo
if(mype==0) then
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,1) = (rho_u(izbot2+1)/rho_s(izbot2+1))*wak(ikx,iky,izbot1+1) - (rho_u(izbot2)*(rho_s(izbot2+1)+rho_s(izbot2))/(rho_s(izbot2+1)*rho_s(izbot2)) + r_1(izbot2)*r_2(izbot2)*dz*dz*kh2/big_F)*wak(ikx,iky,izbot1)
else
lhs(ikx,iky,1) = (0.D0,0.D0)
endif
enddo
enddo
end if
if(mype==(npe-1)) then
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
kh2=kx*kx+ky*ky
if (L(ikx,iky).eq.1) then
lhs(ikx,iky,n3h0-1) = - (rho_u(iztop2-1)*(rho_s(iztop2)+rho_s(iztop2-1))/(rho_s(iztop2)*rho_s(iztop2-1)) + r_1(iztop2-1)*r_2(iztop2-1)*dz*dz*kh2/big_F)*wak(ikx,iky,iztop1-1) + (rho_u(iztop2-2)/rho_s(iztop2-1))*wak(ikx,iky,iztop1-2)
lhs(ikx,iky,n3h0) = (0.D0,0.D0)
else
lhs(ikx,iky,n3h0-1) = (0.D0,0.D0)
lhs(ikx,iky,n3h0) = (0.D0,0.D0)
endif
enddo
enddo
end if
lhs=lhs/(dz*dz)
end SUBROUTINE omega_verification
SUBROUTINE omega_eqn_rhs(rhs,rhsr,psik)
! this subroutine computes the right hand side of the omega equation = 2 J(psi_z,nabla^2 psi)
! Notice that this routine outputs the r-space velocity fields.
double complex, dimension(iktx,ikty,n3h1) :: psik
double complex, dimension(iktx,ikty,n3h0) :: rhs
double precision, dimension(n1d,n2d,n3h0) :: rhsr
!Suboptimal but simple.
double complex, dimension(iktx,ikty,n3h0) :: bxk,byk,xxk,xyk
double precision, dimension(n1d,n2d,n3h0) :: bxr,byr,xxr,xyr
equivalence(bxr,bxk)
equivalence(byr,byk)
equivalence(xxr,xxk)
equivalence(xyr,xyk)
!First compute psik_z and store (temporarily) in rhs
rhs = (0.D0,0.D0)
do izh0=1,n3h0
izh1=izh0+1
do iky=1,ikty
do ikx=1,iktx
if (L(ikx,iky).eq.1) then
rhs(ikx,iky,izh0) = ( psik(ikx,iky,izh1+1) - psik(ikx,iky,izh1) )/dz
else
rhs(ikx,iky,izh0) = (0.D0,0.D0)
endif
enddo
enddo
enddo
if(mype==(npe-1)) then
do iky=1,ikty
do ikx=1,iktx
rhs(ikx,iky,n3h0) = (0.D0,0.D0)
enddo
enddo
end if
!Now compute psi_z and lap(psi)'s horizontal derivatives...
bxk = (0.D0,0.D0)
byk = (0.D0,0.D0)
xxk = (0.D0,0.D0)
xyk = (0.D0,0.D0)
do izh0=1,n3h0
izh1=izh0+1
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
bxk(ikx,iky,izh0) = i*kx*rhs(ikx,iky,izh0)
byk(ikx,iky,izh0) = i*ky*rhs(ikx,iky,izh0)
xxk(ikx,iky,izh0) = -i*kx*(kx*kx+ky*ky)*0.5*( psik(ikx,iky,izh1+1) + psik(ikx,iky,izh1) ) !Need to interpolate so everything is unstaggered.
xyk(ikx,iky,izh0) = -i*ky*(kx*kx+ky*ky)*0.5*( psik(ikx,iky,izh1+1) + psik(ikx,iky,izh1) )
else
bxk(ikx,iky,izh0) = (0.D0,0.D0)
byk(ikx,iky,izh0) = (0.D0,0.D0)
xxk(ikx,iky,izh0) = (0.D0,0.D0)
xyk(ikx,iky,izh0) = (0.D0,0.D0)
endif
enddo
enddo
enddo
if(mype==(npe-1)) then
do iky=1,ikty
ky = kya(iky)
do ikx=1,iktx
kx = kxa(ikx)
if (L(ikx,iky).eq.1) then
bxk(ikx,iky,n3h0) = (0.D0,0.D0)
byk(ikx,iky,n3h0) = (0.D0,0.D0)
xxk(ikx,iky,n3h0) = -i*kx*(kx*kx+ky*ky)*psik(ikx,iky,iztop1)
xyk(ikx,iky,n3h0) = -i*ky*(kx*kx+ky*ky)*psik(ikx,iky,iztop1)
else
bxk(ikx,iky,n3h0) = (0.D0,0.D0)
byk(ikx,iky,n3h0) = (0.D0,0.D0)
xxk(ikx,iky,n3h0) = (0.D0,0.D0)
xyk(ikx,iky,n3h0) = (0.D0,0.D0)
endif
enddo
enddo
end if
call fft_c2r(bxk,bxr,n3h0)
call fft_c2r(byk,byr,n3h0)
call fft_c2r(xxk,xxr,n3h0)
call fft_c2r(xyk,xyr,n3h0)
rhsr = 0.D0
!Now compute the r-space RHS.
do izh0=1,n3h0
do ix=1,n1d
do iy=1,n2d
if(ix<=n1) then
rhsr(ix,iy,izh0) = 2.D0*( bxr(ix,iy,izh0)*xyr(ix,iy,izh0) - byr(ix,iy,izh0)*xxr(ix,iy,izh0) )
else
rhsr(ix,iy,izh0) = 0.D0
end if
end do
end do
end do
call fft_r2c(rhsr,rhs,n3h0)
END SUBROUTINE omega_eqn_rhs
SUBROUTINE compute_w(psik,psi_old,uk,vk,wak,ur,vr)
!This subroutine computes the first order vertical velocity w_1 = -1/(r1r2) (Fr/Ro)^2 D/Dt psi_z
! this subroutine computes ((u.grad)psiz) = d/dxj(uj psiz) in the divergence form on a unstaggered grid.
double complex, dimension(iktx,ikty,n3h2) :: uk,vk,wak
double complex, dimension(iktx,ikty,n3h0) :: psizk
double complex, dimension(iktx,ikty,n3h1) :: psik
double complex, dimension(iktx,ikty,n3h1) :: psi_old
double complex, dimension(iktx,ikty,n3h0) :: psiz_old
double precision, dimension(n1d,n2d,n3h2) :: ur,vr
double precision, dimension(n1d,n2d,n3h0) :: psizr
double complex, dimension(iktx,ikty,n3h2) :: umem,vmem
!Suboptimal but simple.
double complex, dimension(iktx,ikty,n3h0) :: Aqk,Bqk
double precision, dimension(n1d,n2d,n3h0) :: Aqr,Bqr
equivalence(Aqr,Aqk)
equivalence(Bqr,Bqk)
equivalence(psizr,psizk)
!First compute psi_z at current and past time steps:
do izh0=1,n3h0
izh1=izh0+1
do iky=1,ikty
do ikx=1,iktx
if (L(ikx,iky).eq.1) then
psizk(ikx,iky,izh0) = ( psik(ikx,iky,izh1+1) - psik(ikx,iky,izh1) )/dz
psiz_old(ikx,iky,izh0) = ( psi_old(ikx,iky,izh1+1) - psi_old(ikx,iky,izh1) )/dz
else
psizk(ikx,iky,izh0) = (0.,0.)
psiz_old(ikx,iky,izh0) = (0.,0.)
endif
enddo
enddo
enddo
if(mype==(npe-1)) then !Correction at the top... psi_z = 0 right at the top and bot...
do iky=1,ikty
do ikx=1,iktx
psizk(ikx,iky,n3h0) = (0.,0.)
psiz_old(ikx,iky,n3h0) = (0.,0.)
enddo
enddo
end if
!Compute the Eularian time derivative of psi_z using Euler \partial_t psi_z = [ psi_z(n+1) - psi_z(n) ] / dt
!Overwrite the psiz_old
psiz_old = ( psizk - psiz_old )/delt