MODULE tl_biology_mod ! !git $Id$ !svn $Id: tl_npzd_Powell.h 1151 2023-02-09 03:08:53Z arango $ !================================================== Hernan G. Arango === ! Copyright (c) 2002-2023 The ROMS/TOMS Group Andrew M. Moore ! ! Licensed under a MIT/X style license ! ! See License_ROMS.md ! !======================================================================= ! ! ! Nutrient-Phytoplankton-Zooplankton-Detritus Model. ! ! ! ! This routine computes the biological sources and sinks and adds ! ! then the global biological fields. ! ! ! ! Reference: ! ! ! ! Powell, T.M., C.V.W. Lewis, E. Curchitser, D. Haidvogel, ! ! Q. Hermann and E. Dobbins, 2006: Results from a three- ! ! dimensional, nested biological-physical model of the ! ! California Current System: Comparisons with Statistics ! ! from Satellite Imagery, J. Geophys. Res. ! ! ! !======================================================================= ! implicit none ! PRIVATE PUBLIC :: tl_biology ! CONTAINS ! !*********************************************************************** SUBROUTINE tl_biology (ng,tile) !*********************************************************************** ! USE mod_param USE mod_forces USE mod_grid USE mod_ncparam USE mod_ocean USE mod_stepping ! ! Imported variable declarations. ! integer, intent(in) :: ng, tile ! ! Local variable declarations. ! character (len=*), parameter :: MyFile = & & __FILE__ ! #include "tile.h" ! ! Set header file name. ! #ifdef DISTRIBUTE IF (Lbiofile(iTLM)) THEN #else IF (Lbiofile(iTLM).and.(tile.eq.0)) THEN #endif Lbiofile(iTLM)=.FALSE. BIONAME(iTLM)=MyFile END IF ! #ifdef PROFILE CALL wclock_on (ng, iTLM, 15, __LINE__, MyFile) #endif CALL tl_npzd_powell_tile (ng, tile, & & LBi, UBi, LBj, UBj, N(ng), NT(ng), & & IminS, ImaxS, JminS, JmaxS, & & nstp(ng), nnew(ng), & #ifdef MASKING & GRID(ng) % rmask, & #endif & GRID(ng) % Hz, & & GRID(ng) % tl_Hz, & & GRID(ng) % z_r, & & GRID(ng) % tl_z_r, & & GRID(ng) % z_w, & & GRID(ng) % tl_z_w, & & FORCES(ng) % srflx, & & FORCES(ng) % tl_srflx, & & OCEAN(ng) % t, & & OCEAN(ng) % tl_t) #ifdef PROFILE CALL wclock_off (ng, iTLM, 15, __LINE__, MyFile) #endif ! RETURN END SUBROUTINE tl_biology ! !----------------------------------------------------------------------- SUBROUTINE tl_npzd_powell_tile (ng, tile, & & LBi, UBi, LBj, UBj, UBk, UBt, & & IminS, ImaxS, JminS, JmaxS, & & nstp, nnew, & #ifdef MASKING & rmask, & #endif & Hz, tl_Hz, & & z_r, tl_z_r, & & z_w, tl_z_w, & & srflx, tl_srflx, & & t, tl_t) !----------------------------------------------------------------------- ! USE mod_param USE mod_biology USE mod_ncparam USE mod_scalars ! ! Imported variable declarations. ! integer, intent(in) :: ng, tile integer, intent(in) :: LBi, UBi, LBj, UBj, UBk, UBt integer, intent(in) :: IminS, ImaxS, JminS, JmaxS integer, intent(in) :: nstp, nnew #ifdef ASSUMED_SHAPE # ifdef MASKING real(r8), intent(in) :: rmask(LBi:,LBj:) # endif real(r8), intent(in) :: Hz(LBi:,LBj:,:) real(r8), intent(in) :: z_r(LBi:,LBj:,:) real(r8), intent(in) :: z_w(LBi:,LBj:,0:) real(r8), intent(in) :: srflx(LBi:,LBj:) real(r8), intent(in) :: t(LBi:,LBj:,:,:,:) real(r8), intent(in) :: tl_Hz(LBi:,LBj:,:) real(r8), intent(in) :: tl_z_r(LBi:,LBj:,:) real(r8), intent(in) :: tl_z_w(LBi:,LBj:,0:) real(r8), intent(in) :: tl_srflx(LBi:,LBj:) real(r8), intent(inout) :: tl_t(LBi:,LBj:,:,:,:) #else # ifdef MASKING real(r8), intent(in) :: rmask(LBi:UBi,LBj:UBj) # endif real(r8), intent(in) :: Hz(LBi:UBi,LBj:UBj,UBk) real(r8), intent(in) :: z_r(LBi:UBi,LBj:UBj,UBk) real(r8), intent(in) :: z_w(LBi:UBi,LBj:UBj,0:UBk) real(r8), intent(in) :: srflx(LBi:UBi,LBj:UBj) real(r8), intent(in) :: t(LBi:UBi,LBj:UBj,UBk,3,UBt) real(r8), intent(in) :: tl_Hz(LBi:UBi,LBj:UBj,UBk) real(r8), intent(in) :: tl_z_r(LBi:UBi,LBj:UBj,UBk) real(r8), intent(in) :: tl_z_w(LBi:UBi,LBj:UBj,0:UBk) real(r8), intent(in) :: tl_srflx(LBi:UBi,LBj:UBj) real(r8), intent(inout) :: tl_t(LBi:UBi,LBj:UBj,UBk,3,UBt) #endif ! ! Local variable declarations. ! integer, parameter :: Nsink = 2 integer :: Iter, i, ibio, isink, itime, itrc, iTrcMax, j, k, ks integer :: Iteradj integer, dimension(Nsink) :: idsink real(r8), parameter :: MinVal = 1.0e-6_r8 real(r8) :: Att, ExpAtt, Itop, PAR real(r8) :: tl_Att, tl_ExpAtt, tl_Itop, tl_PAR real(r8) :: cff, cff1, cff2, cff3, cff4, dtdays real(r8) :: tl_cff, tl_cff1, tl_cff4 real(r8) :: cffL, cffR, cu, dltL, dltR real(r8) :: tl_cffL, tl_cffR, tl_cu, tl_dltL, tl_dltR real(r8), dimension(Nsink) :: Wbio real(r8), dimension(Nsink) :: tl_Wbio integer, dimension(IminS:ImaxS,N(ng)) :: ksource real(r8), dimension(IminS:ImaxS) :: PARsur real(r8), dimension(IminS:ImaxS) :: tl_PARsur real(r8), dimension(NT(ng),2) :: BioTrc real(r8), dimension(NT(ng),2) :: BioTrc1 real(r8), dimension(NT(ng),2) :: tl_BioTrc real(r8), dimension(IminS:ImaxS,N(ng),NT(ng)) :: Bio real(r8), dimension(IminS:ImaxS,N(ng),NT(ng)) :: Bio1 real(r8), dimension(IminS:ImaxS,N(ng),NT(ng)) :: Bio_old real(r8), dimension(IminS:ImaxS,N(ng),NT(ng)) :: tl_Bio real(r8), dimension(IminS:ImaxS,N(ng),NT(ng)) :: tl_Bio_old real(r8), dimension(IminS:ImaxS,0:N(ng)) :: FC real(r8), dimension(IminS:ImaxS,0:N(ng)) :: tl_FC real(r8), dimension(IminS:ImaxS,N(ng)) :: Hz_inv real(r8), dimension(IminS:ImaxS,N(ng)) :: Hz_inv2 real(r8), dimension(IminS:ImaxS,N(ng)) :: Hz_inv3 real(r8), dimension(IminS:ImaxS,N(ng)) :: Light real(r8), dimension(IminS:ImaxS,N(ng)) :: WL real(r8), dimension(IminS:ImaxS,N(ng)) :: WR real(r8), dimension(IminS:ImaxS,N(ng)) :: bL real(r8), dimension(IminS:ImaxS,N(ng)) :: bL1 real(r8), dimension(IminS:ImaxS,N(ng)) :: bR real(r8), dimension(IminS:ImaxS,N(ng)) :: bR1 real(r8), dimension(IminS:ImaxS,N(ng)) :: qc real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_Hz_inv real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_Hz_inv2 real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_Hz_inv3 real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_Light real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_WL real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_WR real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_bL real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_bR real(r8), dimension(IminS:ImaxS,N(ng)) :: tl_qc #include "set_bounds.h" ! !----------------------------------------------------------------------- ! Add biological Source/Sink terms. !----------------------------------------------------------------------- ! ! Avoid computing source/sink terms if no biological iterations. ! IF (BioIter(ng).le.0) RETURN ! ! Set time-stepping size (days) according to the number of iterations. ! dtdays=dt(ng)*sec2day/REAL(BioIter(ng),r8) ! ! Set vertical sinking indentification vector. ! idsink(1)=iPhyt ! Phytoplankton idsink(2)=iSdet ! Small detritus ! ! Set vertical sinking velocity vector in the same order as the ! identification vector, IDSINK. ! Wbio(1)=wPhy(ng) ! Phytoplankton tl_Wbio(1)=tl_wPhy(ng) ! Phytoplankton Wbio(2)=wDet(ng) ! Small detritus tl_Wbio(2)=tl_wDet(ng) ! Small detritus ! J_LOOP : DO j=Jstr,Jend ! ! Compute inverse thickness to avoid repeated divisions. ! DO k=1,N(ng) DO i=Istr,Iend Hz_inv(i,k)=1.0_r8/Hz(i,j,k) tl_Hz_inv(i,k)=-Hz_inv(i,k)*Hz_inv(i,k)*tl_Hz(i,j,k) END DO END DO DO k=1,N(ng)-1 DO i=Istr,Iend Hz_inv2(i,k)=1.0_r8/(Hz(i,j,k)+Hz(i,j,k+1)) tl_Hz_inv2(i,k)=-Hz_inv2(i,k)*Hz_inv2(i,k)* & & (tl_Hz(i,j,k)+tl_Hz(i,j,k+1)) END DO END DO DO k=2,N(ng)-1 DO i=Istr,Iend Hz_inv3(i,k)=1.0_r8/(Hz(i,j,k-1)+Hz(i,j,k)+Hz(i,j,k+1)) tl_Hz_inv3(i,k)=-Hz_inv3(i,k)*Hz_inv3(i,k)* & & (tl_Hz(i,j,k-1)+tl_Hz(i,j,k)+ & & tl_Hz(i,j,k+1)) END DO END DO ! ! Clear tl_Bio and Bio arrays. ! DO itrc=1,NBT ibio=idbio(itrc) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,ibio)=0.0_r8 Bio1(i,k,ibio)=0.0_r8 tl_Bio(i,k,ibio)=0.0_r8 END DO END DO END DO ! ! Restrict biological tracer to be positive definite. If a negative ! concentration is detected, nitrogen is drawn from the most abundant ! pool to supplement the negative pools to a lower limit of MinVal ! which is set to 1E-6 above. ! DO k=1,N(ng) DO i=Istr,Iend ! ! At input, all tracers (index nnew) from predictor step have ! transport units (m Tunits) since we do not have yet the new ! values for zeta and Hz. These are known after the 2D barotropic ! time-stepping. ! ! NOTE: In the following code, t(:,:,:,nnew,:) should be in units of ! tracer times depth. However the basic state (nstp and nnew ! indices) that is read from the forward file is in units of ! tracer. Since BioTrc(ibio,nnew) is in tracer units, we simply ! use t instead of t*Hz_inv. ! DO itrc=1,NBT ibio=idbio(itrc) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) tl_BioTrc(ibio,nstp)=tl_t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio)*Hz_inv(i,k) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio) tl_BioTrc(ibio,nnew)=tl_t(i,j,k,nnew,ibio)* & & Hz_inv(i,k)+ & & t(i,j,k,nnew,ibio)*Hz(i,j,k)* & & tl_Hz_inv(i,k) END DO ! ! Impose positive definite concentrations. ! cff2=0.0_r8 DO itime=1,2 cff1=0.0_r8 tl_cff1=0.0_r8 iTrcMax=idbio(1) DO itrc=1,NBT ibio=idbio(itrc) cff1=cff1+MAX(0.0_r8,MinVal-BioTrc(ibio,itime)) tl_cff1=tl_cff1- & & (0.5_r8-SIGN(0.5_r8, & & BioTrc(ibio,itime)-MinVal))* & & tl_BioTrc(ibio,itime) IF (BioTrc(ibio,itime).gt.BioTrc(iTrcMax,itime)) THEN iTrcMax=ibio END IF BioTrc1(ibio,itime)=BioTrc(ibio,itime) BioTrc(ibio,itime)=MAX(MinVal,BioTrc1(ibio,itime)) tl_BioTrc(ibio,itime)=(0.5_r8- & & SIGN(0.5_r8, & & MinVal- & & BioTrc1(ibio,itime)))* & & tl_BioTrc(ibio,itime) END DO IF (BioTrc(iTrcMax,itime).gt.cff1) THEN BioTrc(iTrcMax,itime)=BioTrc(iTrcMax,itime)-cff1 tl_BioTrc(iTrcMax,itime)=tl_BioTrc(iTrcMax,itime)- & & tl_cff1 END IF END DO ! ! Load biological tracers into local arrays. ! DO itrc=1,NBT ibio=idbio(itrc) Bio_old(i,k,ibio)=BioTrc(ibio,nstp) tl_Bio_old(i,k,ibio)=tl_BioTrc(ibio,nstp) Bio(i,k,ibio)=BioTrc(ibio,nstp) tl_Bio(i,k,ibio)=tl_BioTrc(ibio,nstp) END DO END DO END DO ! ! Calculate surface Photosynthetically Available Radiation (PAR). The ! net shortwave radiation is scaled back to Watts/m2 and multiplied by ! the fraction that is photosynthetically available, PARfrac. ! DO i=Istr,Iend #ifdef CONST_PAR ! ! Specify constant surface irradiance a la Powell and Spitz. ! PARsur(i)=158.075_r8 tl_PARsur(i)=0.0_r8 #else PARsur(i)=PARfrac(ng)*srflx(i,j)*rho0*Cp tl_PARsur(i)=(tl_PARfrac(ng)*srflx(i,j)+ & & PARfrac(ng)*tl_srflx(i,j))*rho0*Cp #endif END DO ! !======================================================================= ! Start internal iterations to achieve convergence of the nonlinear ! backward-implicit solution. !======================================================================= ! ! During the iterative procedure a series of fractional time steps are ! performed in a chained mode (splitting by different biological ! conversion processes) in sequence of the main food chain. In all ! stages the concentration of the component being consumed is treated ! in a fully implicit manner, so the algorithm guarantees non-negative ! values, no matter how strong the concentration of active consuming ! component (Phytoplankton or Zooplankton). The overall algorithm, ! as well as any stage of it, is formulated in conservative form ! (except explicit sinking) in sense that the sum of concentration of ! all components is conserved. ! ! In the implicit algorithm, we have for example (N: nutrient, ! P: phytoplankton), ! ! N(new) = N(old) - uptake * P(old) uptake = mu * N / (Kn + N) ! {Michaelis-Menten} ! below, we set ! The N in the numerator of ! cff = mu * P(old) / (Kn + N(old)) uptake is treated implicitly ! as N(new) ! ! so the time-stepping of the equations becomes: ! ! N(new) = N(old) / (1 + cff) (1) when substracting a sink term, ! consuming, divide by (1 + cff) ! and ! ! P(new) = P(old) + cff * N(new) (2) when adding a source term, ! growing, add (cff * source) ! ! Notice that if you substitute (1) in (2), you will get: ! ! P(new) = P(old) + cff * N(old) / (1 + cff) (3) ! ! If you add (1) and (3), you get ! ! N(new) + P(new) = N(old) + P(old) ! ! implying conservation regardless how "cff" is computed. Therefore, ! this scheme is unconditionally stable regardless of the conversion ! rate. It does not generate negative values since the constituent ! to be consumed is always treated implicitly. It is also biased ! toward damping oscillations. ! ! The iterative loop below is to iterate toward an universal Backward- ! Euler treatment of all terms. So if there are oscillations in the ! system, they are only physical oscillations. These iterations, ! however, do not improve the accuaracy of the solution. ! ITER_LOOP: DO Iter=1,BioIter(ng) ! ! Compute appropriate basic state arrays I. ! DO k=1,N(ng) DO i=Istr,Iend ! ! At input, all tracers (index nnew) from predictor step have ! transport units (m Tunits) since we do not have yet the new ! values for zeta and Hz. These are known after the 2D barotropic ! time-stepping. ! ! NOTE: In the following code, t(:,:,:,nnew,:) should be in units of ! tracer times depth. However the basic state (nstp and nnew ! indices) that is read from the forward file is in units of ! tracer. Since BioTrc(ibio,nnew) is in tracer units, we simply ! use t instead of t*Hz_inv. ! DO itrc=1,NBT ibio=idbio(itrc) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio)*Hz_inv(i,k) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio) END DO ! ! Impose positive definite concentrations. ! cff2=0.0_r8 DO itime=1,2 cff1=0.0_r8 iTrcMax=idbio(1) DO itrc=1,NBT ibio=idbio(itrc) cff1=cff1+MAX(0.0_r8,MinVal-BioTrc(ibio,itime)) IF (BioTrc(ibio,itime).gt.BioTrc(iTrcMax,itime)) THEN iTrcMax=ibio END IF BioTrc(ibio,itime)=MAX(MinVal,BioTrc(ibio,itime)) END DO IF (BioTrc(iTrcMax,itime).gt.cff1) THEN BioTrc(iTrcMax,itime)=BioTrc(iTrcMax,itime)-cff1 END IF END DO ! ! Load biological tracers into local arrays. ! DO itrc=1,NBT ibio=idbio(itrc) Bio_old(i,k,ibio)=BioTrc(ibio,nnew) Bio(i,k,ibio)=BioTrc(ibio,nnew) END DO END DO END DO ! ! Calculate surface Photosynthetically Available Radiation (PAR). The ! net shortwave radiation is scaled back to Watts/m2 and multiplied by ! the fraction that is photosynthetically available, PARfrac. ! DO i=Istr,Iend #ifdef CONST_PAR ! ! Specify constant surface irradiance a la Powell and Spitz. ! PARsur(i)=158.075_r8 #else PARsur(i)=PARfrac(ng)*srflx(i,j)*rho0*Cp #endif END DO ! !======================================================================= ! Start internal iterations to achieve convergence of the nonlinear ! backward-implicit solution. !======================================================================= ! DO Iteradj=1,Iter ! ! Compute light attenuation as function of depth. ! DO i=Istr,Iend PAR=PARsur(i) IF (PARsur(i).gt.0.0_r8) THEN ! day time DO k=N(ng),1,-1 ! ! Compute average light attenuation for each grid cell. Here, AttSW is ! the light attenuation due to seawater and AttPhy is the attenuation ! due to phytoplankton (self-shading coefficient). ! Att=(AttSW(ng)+AttPhy(ng)*Bio(i,k,iPhyt))* & & (z_w(i,j,k)-z_w(i,j,k-1)) ExpAtt=EXP(-Att) Itop=PAR PAR=Itop*(1.0_r8-ExpAtt)/Att ! average at cell center Light(i,k)=PAR ! ! Light attenuation at the bottom of the grid cell. It is the starting ! PAR value for the next (deeper) vertical grid cell. ! PAR=Itop*ExpAtt END DO ELSE ! night time DO k=1,N(ng) Light(i,k)=0.0_r8 END DO END IF END DO ! ! Phytoplankton photosynthetic growth and nitrate uptake (Vm_NO3 rate). ! The Michaelis-Menten curve is used to describe the change in uptake ! rate as a function of nitrate concentration. Here, PhyIS is the ! initial slope of the P-I curve and K_NO3 is the half saturation of ! phytoplankton nitrate uptake. ! cff1=dtdays*Vm_NO3(ng)*PhyIS(ng) cff2=Vm_NO3(ng)*Vm_NO3(ng) cff3=PhyIS(ng)*PhyIS(ng) DO k=1,N(ng) DO i=Istr,Iend cff4=1.0_r8/SQRT(cff2+cff3*Light(i,k)*Light(i,k)) cff=Bio(i,k,iPhyt)* & & cff1*cff4*Light(i,k)/ & & (K_NO3(ng)+Bio(i,k,iNO3_)) Bio1(i,k,iNO3_)=Bio(i,k,iNO3_) Bio(i,k,iNO3_)=Bio(i,k,iNO3_)/(1.0_r8+cff) Bio1(i,k,iPhyt)=Bio(i,k,iPhyt) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)+ & & Bio(i,k,iNO3_)*cff END DO END DO ! IF (Iteradj.ne.Iter) THEN ! ! Grazing on phytoplankton by zooplankton (ZooGR rate) using the Ivlev ! formulation (Ivlev, 1955) and lost of phytoplankton to the nitrate ! pool as function of "sloppy feeding" and metabolic processes ! (ZooEEN and ZooEED fractions). ! cff1=dtdays*ZooGR(ng) cff2=1.0_r8-ZooEEN(ng)-ZooEED(ng) DO k=1,N(ng) DO i=Istr,Iend cff=Bio(i,k,iZoop)* & & cff1*(1.0_r8-EXP(-Ivlev(ng)*Bio(i,k,iPhyt)))/ & & Bio(i,k,iPhyt) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)/(1.0_r8+cff) Bio(i,k,iZoop)=Bio(i,k,iZoop)+ & & Bio(i,k,iPhyt)*cff2*cff Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*ZooEEN(ng)*cff Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*ZooEED(ng)*cff END DO END DO ! ! Phytoplankton mortality to nutrients (PhyMRN rate) and detritus ! (PhyMRD rate). ! cff3=dtdays*PhyMRD(ng) cff2=dtdays*PhyMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iPhyt)=Bio(i,k,iPhyt)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*cff3 END DO END DO ! ! Zooplankton mortality to nutrients (ZooMRN rate) and Detritus ! (ZooMRD rate). ! cff3=dtdays*ZooMRD(ng) cff2=dtdays*ZooMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iZoop)=Bio(i,k,iZoop)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iZoop)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iZoop)*cff3 END DO END DO ! ! Detritus breakdown to nutrients: remineralization (DetRR rate). ! cff2=dtdays*DetRR(ng) cff1=1.0_r8/(1.0_r8+cff2) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iSDet)=Bio(i,k,iSDet)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iSDet)*cff2 END DO END DO ! !----------------------------------------------------------------------- ! Vertical sinking terms: Phytoplankton and Detritus !----------------------------------------------------------------------- ! ! Reconstruct vertical profile of selected biological constituents ! "Bio(:,:,isink)" in terms of a set of parabolic segments within each ! grid box. Then, compute semi-Lagrangian flux due to sinking. ! DO isink=1,Nsink ibio=idsink(isink) ! ! Copy concentration of biological particulates into scratch array ! "qc" (q-central, restrict it to be positive) which is hereafter ! interpreted as a set of grid-box averaged values for biogeochemical ! constituent concentration. ! DO k=1,N(ng) DO i=Istr,Iend qc(i,k)=Bio(i,k,ibio) END DO END DO ! DO k=N(ng)-1,1,-1 DO i=Istr,Iend FC(i,k)=(qc(i,k+1)-qc(i,k))*Hz_inv2(i,k) END DO END DO DO k=2,N(ng)-1 DO i=Istr,Iend dltR=Hz(i,j,k)*FC(i,k) dltL=Hz(i,j,k)*FC(i,k-1) cff=Hz(i,j,k-1)+2.0_r8*Hz(i,j,k)+Hz(i,j,k+1) cffR=cff*FC(i,k) cffL=cff*FC(i,k-1) ! ! Apply PPM monotonicity constraint to prevent oscillations within the ! grid box. ! IF ((dltR*dltL).le.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF ! ! Compute right and left side values (bR,bL) of parabolic segments ! within grid box Hz(k); (WR,WL) are measures of quadratic variations. ! ! NOTE: Although each parabolic segment is monotonic within its grid ! box, monotonicity of the whole profile is not guaranteed, ! because bL(k+1)-bR(k) may still have different sign than ! qc(i,k+1)-qc(i,k). This possibility is excluded, ! after bL and bR are reconciled using WENO procedure. ! cff=(dltR-dltL)*Hz_inv3(i,k) dltR=dltR-cff*Hz(i,j,k+1) dltL=dltL+cff*Hz(i,j,k-1) bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL WR(i,k)=(2.0_r8*dltR-dltL)**2 WL(i,k)=(dltR-2.0_r8*dltL)**2 END DO END DO cff=1.0E-14_r8 DO k=2,N(ng)-2 DO i=Istr,Iend dltL=MAX(cff,WL(i,k )) dltR=MAX(cff,WR(i,k+1)) bR(i,k)=(dltR*bR(i,k)+dltL*bL(i,k+1))/(dltR+dltL) bL(i,k+1)=bR(i,k) END DO END DO DO i=Istr,Iend FC(i,N(ng))=0.0_r8 ! NO-flux boundary condition #if defined LINEAR_CONTINUATION bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=2.0_r8*qc(i,N(ng))-bL(i,N(ng)) #elif defined NEUMANN bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=1.5*qc(i,N(ng))-0.5_r8*bL(i,N(ng)) #else bR(i,N(ng))=qc(i,N(ng)) ! default strictly monotonic bL(i,N(ng))=qc(i,N(ng)) ! conditions bR(i,N(ng)-1)=qc(i,N(ng)) #endif #if defined LINEAR_CONTINUATION bR(i,1)=bL(i,2) bL(i,1)=2.0_r8*qc(i,1)-bR(i,1) #elif defined NEUMANN bR(i,1)=bL(i,2) bL(i,1)=1.5_r8*qc(i,1)-0.5_r8*bR(i,1) #else bL(i,2)=qc(i,1) ! bottom grid boxes are bR(i,1)=qc(i,1) ! re-assumed to be bL(i,1)=qc(i,1) ! piecewise constant. #endif END DO ! ! Apply monotonicity constraint again, since the reconciled interfacial ! values may cause a non-monotonic behavior of the parabolic segments ! inside the grid box. ! DO k=1,N(ng) DO i=Istr,Iend dltR=bR(i,k)-qc(i,k) dltL=qc(i,k)-bL(i,k) cffR=2.0_r8*dltR cffL=2.0_r8*dltL IF ((dltR*dltL).lt.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL END DO END DO ! ! After this moment reconstruction is considered complete. The next ! stage is to compute vertical advective fluxes, FC. It is expected ! that sinking may occurs relatively fast, the algorithm is designed ! to be free of CFL criterion, which is achieved by allowing ! integration bounds for semi-Lagrangian advective flux to use as ! many grid boxes in upstream direction as necessary. ! ! In the two code segments below, WL is the z-coordinate of the ! departure point for grid box interface z_w with the same indices; ! FC is the finite volume flux; ksource(:,k) is index of vertical ! grid box which contains the departure point (restricted by N(ng)). ! During the search: also add in content of whole grid boxes ! participating in FC. ! cff=dtdays*ABS(Wbio(isink)) DO k=1,N(ng) DO i=Istr,Iend FC(i,k-1)=0.0_r8 WL(i,k)=z_w(i,j,k-1)+cff WR(i,k)=Hz(i,j,k)*qc(i,k) ksource(i,k)=k END DO END DO DO k=1,N(ng) DO ks=k,N(ng)-1 DO i=Istr,Iend IF (WL(i,k).gt.z_w(i,j,ks)) THEN ksource(i,k)=ks+1 FC(i,k-1)=FC(i,k-1)+WR(i,ks) END IF END DO END DO END DO ! ! Finalize computation of flux: add fractional part. ! DO k=1,N(ng) DO i=Istr,Iend ks=ksource(i,k) cu=MIN(1.0_r8,(WL(i,k)-z_w(i,j,ks-1))*Hz_inv(i,ks)) FC(i,k-1)=FC(i,k-1)+ & & Hz(i,j,ks)*cu* & & (bL(i,ks)+ & & cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks)))) END DO END DO DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,ibio)=qc(i,k)+ & & (FC(i,k)-FC(i,k-1))*Hz_inv(i,k) END DO END DO END DO END IF END DO ! ! End of compute basic state arrays I. ! ! Compute light attenuation as function of depth. ! DO i=Istr,Iend PAR=PARsur(i) tl_PAR=tl_PARsur(i) IF (PARsur(i).gt.0.0_r8) THEN ! day time DO k=N(ng),1,-1 ! ! Compute average light attenuation for each grid cell. Here, AttSW is ! the light attenuation due to seawater and AttPhy is the attenuation ! due to phytoplankton (self-shading coefficient). ! Att=(AttSW(ng)+AttPhy(ng)*Bio1(i,k,iPhyt))* & & (z_w(i,j,k)-z_w(i,j,k-1)) tl_Att=AttPhy(ng)*tl_Bio(i,k,iPhyt)* & & (z_w(i,j,k)-z_w(i,j,k-1))+ & & (AttSW(ng)+AttPhy(ng)*Bio1(i,k,iPhyt))* & & (tl_z_w(i,j,k)-tl_z_w(i,j,k-1)) ExpAtt=EXP(-Att) tl_ExpAtt=-ExpAtt*tl_Att Itop=PAR tl_Itop=tl_PAR PAR=Itop*(1.0_r8-ExpAtt)/Att ! average at cell center tl_PAR=(-tl_Att*PAR+tl_Itop*(1.0_r8-ExpAtt)- & & Itop*tl_ExpAtt)/Att !^ Light(i,k)=PAR !^ tl_Light(i,k)=tl_PAR ! ! Light attenuation at the bottom of the grid cell. It is the starting ! PAR value for the next (deeper) vertical grid cell. ! PAR=Itop*ExpAtt tl_PAR=tl_Itop*ExpAtt+Itop*tl_ExpAtt END DO ELSE ! night time DO k=1,N(ng) !^ Light(i,k)=0.0_r8 !^ tl_Light(i,k)=0.0_r8 END DO END IF END DO ! ! Phytoplankton photosynthetic growth and nitrate uptake (Vm_NO3 rate). ! The Michaelis-Menten curve is used to describe the change in uptake ! rate as a function of nitrate concentration. Here, PhyIS is the ! initial slope of the P-I curve and K_NO3 is the half saturation of ! phytoplankton nitrate uptake. ! cff1=dtdays*Vm_NO3(ng)*PhyIS(ng) cff2=Vm_NO3(ng)*Vm_NO3(ng) cff3=PhyIS(ng)*PhyIS(ng) DO k=1,N(ng) DO i=Istr,Iend cff4=1.0_r8/SQRT(cff2+cff3*Light(i,k)*Light(i,k)) tl_cff4=-cff3*tl_Light(i,k)*Light(i,k)*cff4*cff4*cff4 cff=Bio1(i,k,iPhyt)* & & cff1*cff4*Light(i,k)/ & & (K_NO3(ng)+Bio1(i,k,iNO3_)) tl_cff=(tl_Bio(i,k,iPhyt)*cff1*cff4*Light(i,k)+ & & Bio1(i,k,iPhyt)*cff1* & & (tl_cff4*Light(i,k)+cff4*tl_Light(i,k))- & & tl_Bio(i,k,iNO3_)*cff)/ & & (K_NO3(ng)+Bio1(i,k,iNO3_)) !^ Bio(i,k,iNO3_)=Bio(i,k,iNO3_)/(1.0_r8+cff) !^ tl_Bio(i,k,iNO3_)=(tl_Bio(i,k,iNO3_)- & & tl_cff*Bio(i,k,iNO3_))/ & & (1.0_r8+cff) !^ Bio(i,k,iPhyt)=Bio(i,k,iPhyt)+ & !^ & Bio(i,k,iNO3_)*cff !^ tl_Bio(i,k,iPhyt)=tl_Bio(i,k,iPhyt)+ & & tl_Bio(i,k,iNO3_)*cff+ & & Bio(i,k,iNO3_)*tl_cff END DO END DO ! ! Compute appropriate basic state arrays II. ! DO k=1,N(ng) DO i=Istr,Iend ! ! At input, all tracers (index nnew) from predictor step have ! transport units (m Tunits) since we do not have yet the new ! values for zeta and Hz. These are known after the 2D barotropic ! time-stepping. ! ! NOTE: In the following code, t(:,:,:,nnew,:) should be in units of ! tracer times depth. However the basic state (nstp and nnew ! indices) that is read from the forward file is in units of ! tracer. Since BioTrc(ibio,nnew) is in tracer units, we simply ! use t instead of t*Hz_inv. ! DO itrc=1,NBT ibio=idbio(itrc) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio)*Hz_inv(i,k) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio) END DO ! ! Impose positive definite concentrations. ! cff2=0.0_r8 DO itime=1,2 cff1=0.0_r8 iTrcMax=idbio(1) DO itrc=1,NBT ibio=idbio(itrc) cff1=cff1+MAX(0.0_r8,MinVal-BioTrc(ibio,itime)) IF (BioTrc(ibio,itime).gt.BioTrc(iTrcMax,itime)) THEN iTrcMax=ibio END IF BioTrc(ibio,itime)=MAX(MinVal,BioTrc(ibio,itime)) END DO IF (BioTrc(iTrcMax,itime).gt.cff1) THEN BioTrc(iTrcMax,itime)=BioTrc(iTrcMax,itime)-cff1 END IF END DO ! ! Load biological tracers into local arrays. ! DO itrc=1,NBT ibio=idbio(itrc) Bio_old(i,k,ibio)=BioTrc(ibio,nnew) Bio(i,k,ibio)=BioTrc(ibio,nnew) END DO END DO END DO ! ! Calculate surface Photosynthetically Available Radiation (PAR). The ! net shortwave radiation is scaled back to Watts/m2 and multiplied by ! the fraction that is photosynthetically available, PARfrac. ! DO i=Istr,Iend #ifdef CONST_PAR ! ! Specify constant surface irradiance a la Powell and Spitz. ! PARsur(i)=158.075_r8 #else PARsur(i)=PARfrac(ng)*srflx(i,j)*rho0*Cp #endif END DO ! !======================================================================= ! Start internal iterations to achieve convergence of the nonlinear ! backward-implicit solution. !======================================================================= ! DO Iteradj=1,Iter ! ! Compute light attenuation as function of depth. ! DO i=Istr,Iend PAR=PARsur(i) IF (PARsur(i).gt.0.0_r8) THEN ! day time DO k=N(ng),1,-1 ! ! Compute average light attenuation for each grid cell. Here, AttSW is ! the light attenuation due to seawater and AttPhy is the attenuation ! due to phytoplankton (self-shading coefficient). ! Att=(AttSW(ng)+AttPhy(ng)*Bio(i,k,iPhyt))* & & (z_w(i,j,k)-z_w(i,j,k-1)) ExpAtt=EXP(-Att) Itop=PAR PAR=Itop*(1.0_r8-ExpAtt)/Att ! average at cell center Light(i,k)=PAR ! ! Light attenuation at the bottom of the grid cell. It is the starting ! PAR value for the next (deeper) vertical grid cell. ! PAR=Itop*ExpAtt END DO ELSE ! night time DO k=1,N(ng) Light(i,k)=0.0_r8 END DO END IF END DO ! ! Phytoplankton photosynthetic growth and nitrate uptake (Vm_NO3 rate). ! The Michaelis-Menten curve is used to describe the change in uptake ! rate as a function of nitrate concentration. Here, PhyIS is the ! initial slope of the P-I curve and K_NO3 is the half saturation of ! phytoplankton nitrate uptake. ! cff1=dtdays*Vm_NO3(ng)*PhyIS(ng) cff2=Vm_NO3(ng)*Vm_NO3(ng) cff3=PhyIS(ng)*PhyIS(ng) DO k=1,N(ng) DO i=Istr,Iend cff4=1.0_r8/SQRT(cff2+cff3*Light(i,k)*Light(i,k)) cff=Bio(i,k,iPhyt)* & & cff1*cff4*Light(i,k)/ & & (K_NO3(ng)+Bio(i,k,iNO3_)) Bio(i,k,iNO3_)=Bio(i,k,iNO3_)/(1.0_r8+cff) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)+ & & Bio(i,k,iNO3_)*cff END DO END DO ! ! Grazing on phytoplankton by zooplankton (ZooGR rate) using the Ivlev ! formulation (Ivlev, 1955) and lost of phytoplankton to the nitrate ! pool as function of "sloppy feeding" and metabolic processes ! (ZooEEN and ZooEED fractions). ! cff1=dtdays*ZooGR(ng) cff2=1.0_r8-ZooEEN(ng)-ZooEED(ng) DO k=1,N(ng) DO i=Istr,Iend cff=Bio(i,k,iZoop)* & & cff1*(1.0_r8-EXP(-Ivlev(ng)*Bio(i,k,iPhyt)))/ & & Bio(i,k,iPhyt) Bio1(i,k,iPhyt)=Bio(i,k,iPhyt) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)/(1.0_r8+cff) Bio1(i,k,iZoop)=Bio(i,k,iZoop) Bio(i,k,iZoop)=Bio(i,k,iZoop)+ & & Bio(i,k,iPhyt)*cff2*cff Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*ZooEEN(ng)*cff Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*ZooEED(ng)*cff END DO END DO ! IF (Iteradj.ne.Iter) THEN ! ! Phytoplankton mortality to nutrients (PhyMRN rate) and detritus ! (PhyMRD rate). ! cff3=dtdays*PhyMRD(ng) cff2=dtdays*PhyMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iPhyt)=Bio(i,k,iPhyt)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*cff3 END DO END DO ! ! Zooplankton mortality to nutrients (ZooMRN rate) and Detritus ! (ZooMRD rate). ! cff3=dtdays*ZooMRD(ng) cff2=dtdays*ZooMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iZoop)=Bio(i,k,iZoop)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iZoop)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iZoop)*cff3 END DO END DO ! ! Detritus breakdown to nutrients: remineralization (DetRR rate). ! cff2=dtdays*DetRR(ng) cff1=1.0_r8/(1.0_r8+cff2) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iSDet)=Bio(i,k,iSDet)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iSDet)*cff2 END DO END DO ! !----------------------------------------------------------------------- ! Vertical sinking terms: Phytoplankton and Detritus !----------------------------------------------------------------------- ! ! Reconstruct vertical profile of selected biological constituents ! "Bio(:,:,isink)" in terms of a set of parabolic segments within each ! grid box. Then, compute semi-Lagrangian flux due to sinking. ! DO isink=1,Nsink ibio=idsink(isink) ! ! Copy concentration of biological particulates into scratch array ! "qc" (q-central, restrict it to be positive) which is hereafter ! interpreted as a set of grid-box averaged values for biogeochemical ! constituent concentration. ! DO k=1,N(ng) DO i=Istr,Iend qc(i,k)=Bio(i,k,ibio) END DO END DO ! DO k=N(ng)-1,1,-1 DO i=Istr,Iend FC(i,k)=(qc(i,k+1)-qc(i,k))*Hz_inv2(i,k) END DO END DO DO k=2,N(ng)-1 DO i=Istr,Iend dltR=Hz(i,j,k)*FC(i,k) dltL=Hz(i,j,k)*FC(i,k-1) cff=Hz(i,j,k-1)+2.0_r8*Hz(i,j,k)+Hz(i,j,k+1) cffR=cff*FC(i,k) cffL=cff*FC(i,k-1) ! ! Apply PPM monotonicity constraint to prevent oscillations within the ! grid box. ! IF ((dltR*dltL).le.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF ! ! Compute right and left side values (bR,bL) of parabolic segments ! within grid box Hz(k); (WR,WL) are measures of quadratic variations. ! ! NOTE: Although each parabolic segment is monotonic within its grid ! box, monotonicity of the whole profile is not guaranteed, ! because bL(k+1)-bR(k) may still have different sign than ! qc(i,k+1)-qc(i,k). This possibility is excluded, ! after bL and bR are reconciled using WENO procedure. ! cff=(dltR-dltL)*Hz_inv3(i,k) dltR=dltR-cff*Hz(i,j,k+1) dltL=dltL+cff*Hz(i,j,k-1) bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL WR(i,k)=(2.0_r8*dltR-dltL)**2 WL(i,k)=(dltR-2.0_r8*dltL)**2 END DO END DO cff=1.0E-14_r8 DO k=2,N(ng)-2 DO i=Istr,Iend dltL=MAX(cff,WL(i,k )) dltR=MAX(cff,WR(i,k+1)) bR(i,k)=(dltR*bR(i,k)+dltL*bL(i,k+1))/(dltR+dltL) bL(i,k+1)=bR(i,k) END DO END DO DO i=Istr,Iend FC(i,N(ng))=0.0_r8 ! NO-flux boundary condition #if defined LINEAR_CONTINUATION bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=2.0_r8*qc(i,N(ng))-bL(i,N(ng)) #elif defined NEUMANN bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=1.5*qc(i,N(ng))-0.5_r8*bL(i,N(ng)) #else bR(i,N(ng))=qc(i,N(ng)) ! default strictly monotonic bL(i,N(ng))=qc(i,N(ng)) ! conditions bR(i,N(ng)-1)=qc(i,N(ng)) #endif #if defined LINEAR_CONTINUATION bR(i,1)=bL(i,2) bL(i,1)=2.0_r8*qc(i,1)-bR(i,1) #elif defined NEUMANN bR(i,1)=bL(i,2) bL(i,1)=1.5_r8*qc(i,1)-0.5_r8*bR(i,1) #else bL(i,2)=qc(i,1) ! bottom grid boxes are bR(i,1)=qc(i,1) ! re-assumed to be bL(i,1)=qc(i,1) ! piecewise constant. #endif END DO ! ! Apply monotonicity constraint again, since the reconciled interfacial ! values may cause a non-monotonic behavior of the parabolic segments ! inside the grid box. ! DO k=1,N(ng) DO i=Istr,Iend dltR=bR(i,k)-qc(i,k) dltL=qc(i,k)-bL(i,k) cffR=2.0_r8*dltR cffL=2.0_r8*dltL IF ((dltR*dltL).lt.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL END DO END DO ! ! After this moment reconstruction is considered complete. The next ! stage is to compute vertical advective fluxes, FC. It is expected ! that sinking may occurs relatively fast, the algorithm is designed ! to be free of CFL criterion, which is achieved by allowing ! integration bounds for semi-Lagrangian advective flux to use as ! many grid boxes in upstream direction as necessary. ! ! In the two code segments below, WL is the z-coordinate of the ! departure point for grid box interface z_w with the same indices; ! FC is the finite volume flux; ksource(:,k) is index of vertical ! grid box which contains the departure point (restricted by N(ng)). ! During the search: also add in content of whole grid boxes ! participating in FC. ! cff=dtdays*ABS(Wbio(isink)) DO k=1,N(ng) DO i=Istr,Iend FC(i,k-1)=0.0_r8 WL(i,k)=z_w(i,j,k-1)+cff WR(i,k)=Hz(i,j,k)*qc(i,k) ksource(i,k)=k END DO END DO DO k=1,N(ng) DO ks=k,N(ng)-1 DO i=Istr,Iend IF (WL(i,k).gt.z_w(i,j,ks)) THEN ksource(i,k)=ks+1 FC(i,k-1)=FC(i,k-1)+WR(i,ks) END IF END DO END DO END DO ! ! Finalize computation of flux: add fractional part. ! DO k=1,N(ng) DO i=Istr,Iend ks=ksource(i,k) cu=MIN(1.0_r8,(WL(i,k)-z_w(i,j,ks-1))*Hz_inv(i,ks)) FC(i,k-1)=FC(i,k-1)+ & & Hz(i,j,ks)*cu* & & (bL(i,ks)+ & & cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks)))) END DO END DO DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,ibio)=qc(i,k)+ & & (FC(i,k)-FC(i,k-1))*Hz_inv(i,k) END DO END DO END DO END IF END DO ! ! End of compute basic state arrays II. ! ! Grazing on phytoplankton by zooplankton (ZooGR rate) using the Ivlev ! formulation (Ivlev, 1955) and lost of phytoplankton to the nitrate ! pool as function of "sloppy feeding" and metabolic processes ! (ZooEEN and ZooEED fractions). ! cff1=dtdays*ZooGR(ng) cff2=1.0_r8-ZooEEN(ng)-ZooEED(ng) DO k=1,N(ng) DO i=Istr,Iend cff=Bio1(i,k,iZoop)* & & cff1*(1.0_r8-EXP(-Ivlev(ng)*Bio1(i,k,iPhyt)))/ & & Bio1(i,k,iPhyt) tl_cff=(tl_Bio(i,k,iZoop)* & & cff1*(1.0_r8-EXP(-Ivlev(ng)*Bio1(i,k,iPhyt)))+ & & Bio1(i,k,iZoop)*Ivlev(ng)*tl_Bio(i,k,iPhyt)*cff1* & & EXP(-Ivlev(ng)*Bio1(i,k,iPhyt))- & & tl_Bio(i,k,iPhyt)*cff)/ & & Bio1(i,k,iPhyt) !^ Bio(i,k,iPhyt)=Bio(i,k,iPhyt)/(1.0_r8+cff) !^ tl_Bio(i,k,iPhyt)=(tl_Bio(i,k,iPhyt)- & & tl_cff*Bio(i,k,iPhyt))/ & & (1.0_r8+cff) !^ Bio(i,k,iZoop)=Bio(i,k,iZoop)+ & !^ & Bio(i,k,iPhyt)*cff2*cff !^ tl_Bio(i,k,iZoop)=tl_Bio(i,k,iZoop)+ & & cff2*(tl_Bio(i,k,iPhyt)*cff+ & & Bio(i,k,iPhyt)*tl_cff) !^ Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & !^ & Bio(i,k,iPhyt)*ZooEEN(ng)*cff !^ tl_Bio(i,k,iNO3_)=tl_Bio(i,k,iNO3_)+ & & ZooEEN(ng)*(tl_Bio(i,k,iPhyt)*cff+ & & Bio(i,k,iPhyt)*tl_cff) !^ Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & !^ & Bio(i,k,iPhyt)*ZooEED(ng)*cff !^ tl_Bio(i,k,iSDet)=tl_Bio(i,k,iSDet)+ & & ZooEED(ng)*(tl_Bio(i,k,iPhyt)*cff+ & & Bio(i,k,iPhyt)*tl_cff) END DO END DO ! ! Phytoplankton mortality to nutrients (PhyMRN rate) and detritus ! (PhyMRD rate). ! cff3=dtdays*PhyMRD(ng) cff2=dtdays*PhyMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend !^ Bio(i,k,iPhyt)=Bio(i,k,iPhyt)*cff1 !^ tl_Bio(i,k,iPhyt)=tl_Bio(i,k,iPhyt)*cff1 !^ Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & !^ & Bio(i,k,iPhyt)*cff2 !^ tl_Bio(i,k,iNO3_)=tl_Bio(i,k,iNO3_)+ & & tl_Bio(i,k,iPhyt)*cff2 !^ Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & !^ & Bio(i,k,iPhyt)*cff3 !^ tl_Bio(i,k,iSDet)=tl_Bio(i,k,iSDet)+ & & tl_Bio(i,k,iPhyt)*cff3 END DO END DO ! ! Zooplankton mortality to nutrients (ZooMRN rate) and Detritus ! (ZooMRD rate). ! cff3=dtdays*ZooMRD(ng) cff2=dtdays*ZooMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend !^ Bio(i,k,iZoop)=Bio(i,k,iZoop)*cff1 !^ tl_Bio(i,k,iZoop)=tl_Bio(i,k,iZoop)*cff1 !^ Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & !^ & Bio(i,k,iZoop)*cff2 !^ tl_Bio(i,k,iNO3_)=tl_Bio(i,k,iNO3_)+ & & tl_Bio(i,k,iZoop)*cff2 !^ Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & !^ & Bio(i,k,iZoop)*cff3 !^ tl_Bio(i,k,iSDet)=tl_Bio(i,k,iSDet)+ & & tl_Bio(i,k,iZoop)*cff3 END DO END DO ! ! Detritus breakdown to nutrients: remineralization (DetRR rate). ! cff2=dtdays*DetRR(ng) cff1=1.0_r8/(1.0_r8+cff2) DO k=1,N(ng) DO i=Istr,Iend !^ Bio(i,k,iSDet)=Bio(i,k,iSDet)*cff1 !^ tl_Bio(i,k,iSDet)=tl_Bio(i,k,iSDet)*cff1 !^ Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & !^ & Bio(i,k,iSDet)*cff2 !^ tl_Bio(i,k,iNO3_)=tl_Bio(i,k,iNO3_)+ & & tl_Bio(i,k,iSDet)*cff2 END DO END DO ! ! Compute appropriate basic state arrays III. ! DO k=1,N(ng) DO i=Istr,Iend ! ! At input, all tracers (index nnew) from predictor step have ! transport units (m Tunits) since we do not have yet the new ! values for zeta and Hz. These are known after the 2D barotropic ! time-stepping. ! ! NOTE: In the following code, t(:,:,:,nnew,:) should be in units of ! tracer times depth. However the basic state (nstp and nnew ! indices) that is read from the forward file is in units of ! tracer. Since BioTrc(ibio,nnew) is in tracer units, we simply ! use t instead of t*Hz_inv. ! DO itrc=1,NBT ibio=idbio(itrc) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nstp)=t(i,j,k,nstp,ibio) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio)*Hz_inv(i,k) !^ BioTrc(ibio,nnew)=t(i,j,k,nnew,ibio) END DO ! ! Impose positive definite concentrations. ! cff2=0.0_r8 DO itime=1,2 cff1=0.0_r8 iTrcMax=idbio(1) DO itrc=1,NBT ibio=idbio(itrc) cff1=cff1+MAX(0.0_r8,MinVal-BioTrc(ibio,itime)) IF (BioTrc(ibio,itime).gt.BioTrc(iTrcMax,itime)) THEN iTrcMax=ibio END IF BioTrc(ibio,itime)=MAX(MinVal,BioTrc(ibio,itime)) END DO IF (BioTrc(iTrcMax,itime).gt.cff1) THEN BioTrc(iTrcMax,itime)=BioTrc(iTrcMax,itime)-cff1 END IF END DO ! ! Load biological tracers into local arrays. ! DO itrc=1,NBT ibio=idbio(itrc) Bio_old(i,k,ibio)=BioTrc(ibio,nstp) Bio(i,k,ibio)=BioTrc(ibio,nstp) END DO END DO END DO ! ! Calculate surface Photosynthetically Available Radiation (PAR). The ! net shortwave radiation is scaled back to Watts/m2 and multiplied by ! the fraction that is photosynthetically available, PARfrac. ! DO i=Istr,Iend #ifdef CONST_PAR ! ! Specify constant surface irradiance a la Powell and Spitz. ! PARsur(i)=158.075_r8 #else PARsur(i)=PARfrac(ng)*srflx(i,j)*rho0*Cp #endif END DO ! !======================================================================= ! Start internal iterations to achieve convergence of the nonlinear ! backward-implicit solution. !======================================================================= ! DO Iteradj=1,Iter ! ! Compute light attenuation as function of depth. ! DO i=Istr,Iend PAR=PARsur(i) IF (PARsur(i).gt.0.0_r8) THEN ! day time DO k=N(ng),1,-1 ! ! Compute average light attenuation for each grid cell. Here, AttSW is ! the light attenuation due to seawater and AttPhy is the attenuation ! due to phytoplankton (self-shading coefficient). ! Att=(AttSW(ng)+AttPhy(ng)*Bio(i,k,iPhyt))* & & (z_w(i,j,k)-z_w(i,j,k-1)) ExpAtt=EXP(-Att) Itop=PAR PAR=Itop*(1.0_r8-ExpAtt)/Att ! average at cell center Light(i,k)=PAR ! ! Light attenuation at the bottom of the grid cell. It is the starting ! PAR value for the next (deeper) vertical grid cell. ! PAR=Itop*ExpAtt END DO ELSE ! night time DO k=1,N(ng) Light(i,k)=0.0_r8 END DO END IF END DO ! ! Phytoplankton photosynthetic growth and nitrate uptake (Vm_NO3 rate). ! The Michaelis-Menten curve is used to describe the change in uptake ! rate as a function of nitrate concentration. Here, PhyIS is the ! initial slope of the P-I curve and K_NO3 is the half saturation of ! phytoplankton nitrate uptake. ! cff1=dtdays*Vm_NO3(ng)*PhyIS(ng) cff2=Vm_NO3(ng)*Vm_NO3(ng) cff3=PhyIS(ng)*PhyIS(ng) DO k=1,N(ng) DO i=Istr,Iend cff4=1.0_r8/SQRT(cff2+cff3*Light(i,k)*Light(i,k)) cff=Bio(i,k,iPhyt)* & & cff1*cff4*Light(i,k)/ & & (K_NO3(ng)+Bio(i,k,iNO3_)) Bio(i,k,iNO3_)=Bio(i,k,iNO3_)/(1.0_r8+cff) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)+ & & Bio(i,k,iNO3_)*cff END DO END DO ! ! Grazing on phytoplankton by zooplankton (ZooGR rate) using the Ivlev ! formulation (Ivlev, 1955) and lost of phytoplankton to the nitrate ! pool as function of "sloppy feeding" and metabolic processes ! (ZooEEN and ZooEED fractions). ! cff1=dtdays*ZooGR(ng) cff2=1.0_r8-ZooEEN(ng)-ZooEED(ng) DO k=1,N(ng) DO i=Istr,Iend cff=Bio(i,k,iZoop)* & & cff1*(1.0_r8-EXP(-Ivlev(ng)*Bio(i,k,iPhyt)))/ & & Bio(i,k,iPhyt) Bio(i,k,iPhyt)=Bio(i,k,iPhyt)/(1.0_r8+cff) Bio(i,k,iZoop)=Bio(i,k,iZoop)+ & & Bio(i,k,iPhyt)*cff2*cff Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*ZooEEN(ng)*cff Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*ZooEED(ng)*cff END DO END DO ! ! Phytoplankton mortality to nutrients (PhyMRN rate) and detritus ! (PhyMRD rate). ! cff3=dtdays*PhyMRD(ng) cff2=dtdays*PhyMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iPhyt)=Bio(i,k,iPhyt)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iPhyt)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iPhyt)*cff3 END DO END DO ! ! Zooplankton mortality to nutrients (ZooMRN rate) and Detritus ! (ZooMRD rate). ! cff3=dtdays*ZooMRD(ng) cff2=dtdays*ZooMRN(ng) cff1=1.0_r8/(1.0_r8+cff2+cff3) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iZoop)=Bio(i,k,iZoop)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iZoop)*cff2 Bio(i,k,iSDet)=Bio(i,k,iSDet)+ & & Bio(i,k,iZoop)*cff3 END DO END DO ! ! Detritus breakdown to nutrients: remineralization (DetRR rate). ! cff2=dtdays*DetRR(ng) cff1=1.0_r8/(1.0_r8+cff2) DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,iSDet)=Bio(i,k,iSDet)*cff1 Bio(i,k,iNO3_)=Bio(i,k,iNO3_)+ & & Bio(i,k,iSDet)*cff2 END DO END DO ! IF (Iteradj.ne.Iter) THEN ! !----------------------------------------------------------------------- ! Vertical sinking terms: Phytoplankton and Detritus !----------------------------------------------------------------------- ! ! Reconstruct vertical profile of selected biological constituents ! "Bio(:,:,isink)" in terms of a set of parabolic segments within each ! grid box. Then, compute semi-Lagrangian flux due to sinking. ! DO isink=1,Nsink ibio=idsink(isink) ! ! Copy concentration of biological particulates into scratch array ! "qc" (q-central, restrict it to be positive) which is hereafter ! interpreted as a set of grid-box averaged values for biogeochemical ! constituent concentration. ! DO k=1,N(ng) DO i=Istr,Iend qc(i,k)=Bio(i,k,ibio) END DO END DO ! DO k=N(ng)-1,1,-1 DO i=Istr,Iend FC(i,k)=(qc(i,k+1)-qc(i,k))*Hz_inv2(i,k) END DO END DO DO k=2,N(ng)-1 DO i=Istr,Iend dltR=Hz(i,j,k)*FC(i,k) dltL=Hz(i,j,k)*FC(i,k-1) cff=Hz(i,j,k-1)+2.0_r8*Hz(i,j,k)+Hz(i,j,k+1) cffR=cff*FC(i,k) cffL=cff*FC(i,k-1) ! ! Apply PPM monotonicity constraint to prevent oscillations within the ! grid box. ! IF ((dltR*dltL).le.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF ! ! Compute right and left side values (bR,bL) of parabolic segments ! within grid box Hz(k); (WR,WL) are measures of quadratic variations. ! ! NOTE: Although each parabolic segment is monotonic within its grid ! box, monotonicity of the whole profile is not guaranteed, ! because bL(k+1)-bR(k) may still have different sign than ! qc(i,k+1)-qc(i,k). This possibility is excluded, ! after bL and bR are reconciled using WENO procedure. ! cff=(dltR-dltL)*Hz_inv3(i,k) dltR=dltR-cff*Hz(i,j,k+1) dltL=dltL+cff*Hz(i,j,k-1) bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL WR(i,k)=(2.0_r8*dltR-dltL)**2 WL(i,k)=(dltR-2.0_r8*dltL)**2 END DO END DO cff=1.0E-14_r8 DO k=2,N(ng)-2 DO i=Istr,Iend dltL=MAX(cff,WL(i,k )) dltR=MAX(cff,WR(i,k+1)) bR(i,k)=(dltR*bR(i,k)+dltL*bL(i,k+1))/(dltR+dltL) bL(i,k+1)=bR(i,k) END DO END DO DO i=Istr,Iend FC(i,N(ng))=0.0_r8 ! NO-flux boundary condition #if defined LINEAR_CONTINUATION bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=2.0_r8*qc(i,N(ng))-bL(i,N(ng)) #elif defined NEUMANN bL(i,N(ng))=bR(i,N(ng)-1) bR(i,N(ng))=1.5*qc(i,N(ng))-0.5_r8*bL(i,N(ng)) #else bR(i,N(ng))=qc(i,N(ng)) ! default strictly monotonic bL(i,N(ng))=qc(i,N(ng)) ! conditions bR(i,N(ng)-1)=qc(i,N(ng)) #endif #if defined LINEAR_CONTINUATION bR(i,1)=bL(i,2) bL(i,1)=2.0_r8*qc(i,1)-bR(i,1) #elif defined NEUMANN bR(i,1)=bL(i,2) bL(i,1)=1.5_r8*qc(i,1)-0.5_r8*bR(i,1) #else bL(i,2)=qc(i,1) ! bottom grid boxes are bR(i,1)=qc(i,1) ! re-assumed to be bL(i,1)=qc(i,1) ! piecewise constant. #endif END DO ! ! Apply monotonicity constraint again, since the reconciled interfacial ! values may cause a non-monotonic behavior of the parabolic segments ! inside the grid box. ! DO k=1,N(ng) DO i=Istr,Iend dltR=bR(i,k)-qc(i,k) dltL=qc(i,k)-bL(i,k) cffR=2.0_r8*dltR cffL=2.0_r8*dltL IF ((dltR*dltL).lt.0.0_r8) THEN dltR=0.0_r8 dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR END IF bR(i,k)=qc(i,k)+dltR bL(i,k)=qc(i,k)-dltL END DO END DO ! ! After this moment reconstruction is considered complete. The next ! stage is to compute vertical advective fluxes, FC. It is expected ! that sinking may occurs relatively fast, the algorithm is designed ! to be free of CFL criterion, which is achieved by allowing ! integration bounds for semi-Lagrangian advective flux to use as ! many grid boxes in upstream direction as necessary. ! ! In the two code segments below, WL is the z-coordinate of the ! departure point for grid box interface z_w with the same indices; ! FC is the finite volume flux; ksource(:,k) is index of vertical ! grid box which contains the departure point (restricted by N(ng)). ! During the search: also add in content of whole grid boxes ! participating in FC. ! cff=dtdays*ABS(Wbio(isink)) DO k=1,N(ng) DO i=Istr,Iend FC(i,k-1)=0.0_r8 WL(i,k)=z_w(i,j,k-1)+cff WR(i,k)=Hz(i,j,k)*qc(i,k) ksource(i,k)=k END DO END DO DO k=1,N(ng) DO ks=k,N(ng)-1 DO i=Istr,Iend IF (WL(i,k).gt.z_w(i,j,ks)) THEN ksource(i,k)=ks+1 FC(i,k-1)=FC(i,k-1)+WR(i,ks) END IF END DO END DO END DO ! ! Finalize computation of flux: add fractional part. ! DO k=1,N(ng) DO i=Istr,Iend ks=ksource(i,k) cu=MIN(1.0_r8,(WL(i,k)-z_w(i,j,ks-1))*Hz_inv(i,ks)) FC(i,k-1)=FC(i,k-1)+ & & Hz(i,j,ks)*cu* & & (bL(i,ks)+ & & cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks)))) END DO END DO DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,ibio)=qc(i,k)+ & & (FC(i,k)-FC(i,k-1))*Hz_inv(i,k) END DO END DO END DO END IF END DO ! ! End of compute basic state arrays III. ! !----------------------------------------------------------------------- ! Tangent linear vertical sinking terms. !----------------------------------------------------------------------- ! ! Reconstruct vertical profile of selected biological constituents ! "Bio(:,:,isink)" in terms of a set of parabolic segments within each ! grid box. Then, compute semi-Lagrangian flux due to sinking. ! SINK_LOOP: DO isink=1,Nsink ibio=idsink(isink) ! ! Copy concentration of biological particulates into scratch array ! "qc" (q-central, restrict it to be positive) which is hereafter ! interpreted as a set of grid-box averaged values for biogeochemical ! constituent concentration. ! DO k=1,N(ng) DO i=Istr,Iend qc(i,k)=Bio(i,k,ibio) tl_qc(i,k)=tl_Bio(i,k,ibio) END DO END DO ! DO k=N(ng)-1,1,-1 DO i=Istr,Iend FC(i,k)=(qc(i,k+1)-qc(i,k))*Hz_inv2(i,k) tl_FC(i,k)=(tl_qc(i,k+1)-tl_qc(i,k))*Hz_inv2(i,k)+ & & (qc(i,k+1)-qc(i,k))*tl_Hz_inv2(i,k) END DO END DO DO k=2,N(ng)-1 DO i=Istr,Iend dltR=Hz(i,j,k)*FC(i,k) tl_dltR=tl_Hz(i,j,k)*FC(i,k)+Hz(i,j,k)*tl_FC(i,k) dltL=Hz(i,j,k)*FC(i,k-1) tl_dltL=tl_Hz(i,j,k)*FC(i,k-1)+Hz(i,j,k)*tl_FC(i,k-1) cff=Hz(i,j,k-1)+2.0_r8*Hz(i,j,k)+Hz(i,j,k+1) tl_cff=tl_Hz(i,j,k-1)+2.0_r8*tl_Hz(i,j,k)+tl_Hz(i,j,k+1) cffR=cff*FC(i,k) tl_cffR=tl_cff*FC(i,k)+cff*tl_FC(i,k) cffL=cff*FC(i,k-1) tl_cffL=tl_cff*FC(i,k-1)+cff*tl_FC(i,k-1) ! ! Apply PPM monotonicity constraint to prevent oscillations within the ! grid box. ! IF ((dltR*dltL).le.0.0_r8) THEN dltR=0.0_r8 tl_dltR=0.0_r8 dltL=0.0_r8 tl_dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL tl_dltR=tl_cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR tl_dltL=tl_cffR END IF ! ! Compute right and left side values (bR,bL) of parabolic segments ! within grid box Hz(k); (WR,WL) are measures of quadratic variations. ! ! NOTE: Although each parabolic segment is monotonic within its grid ! box, monotonicity of the whole profile is not guaranteed, ! because bL(k+1)-bR(k) may still have different sign than ! qc(i,k+1)-qc(i,k). This possibility is excluded, ! after bL and bR are reconciled using WENO procedure. ! cff=(dltR-dltL)*Hz_inv3(i,k) tl_cff=(tl_dltR-tl_dltL)*Hz_inv3(i,k)+ & & (dltR-dltL)*tl_Hz_inv3(i,k) dltR=dltR-cff*Hz(i,j,k+1) tl_dltR=tl_dltR-tl_cff*Hz(i,j,k+1)-cff*tl_Hz(i,j,k+1) dltL=dltL+cff*Hz(i,j,k-1) tl_dltL=tl_dltL+tl_cff*Hz(i,j,k-1)+cff*tl_Hz(i,j,k-1) bR(i,k)=qc(i,k)+dltR tl_bR(i,k)=tl_qc(i,k)+tl_dltR bL(i,k)=qc(i,k)-dltL tl_bL(i,k)=tl_qc(i,k)-tl_dltL WR(i,k)=(2.0_r8*dltR-dltL)**2 tl_WR(i,k)=2.0_r8*(2.0_r8*dltR-dltL)* & & (2.0_r8*tl_dltR-tl_dltL) WL(i,k)=(dltR-2.0_r8*dltL)**2 tl_WL(i,k)=2.0_r8*(dltR-2.0_r8*dltL)* & & (tl_dltR-2.0_r8*tl_dltL) END DO END DO cff=1.0E-14_r8 DO k=2,N(ng)-2 DO i=Istr,Iend dltL=MAX(cff,WL(i,k )) tl_dltL=(0.5_r8-SIGN(0.5_r8,cff-WL(i,k )))* & & tl_WL(i,k ) dltR=MAX(cff,WR(i,k+1)) tl_dltR=(0.5_r8-SIGN(0.5_r8,cff-WR(i,k+1)))* & & tl_WR(i,k+1) bR1(i,k)=bR(i,k) bL1(i,k+1)=bL(i,k+1) bR(i,k)=(dltR*bR(i,k)+dltL*bL(i,k+1))/(dltR+dltL) tl_bR(i,k)=(tl_dltR*bR1(i,k )+dltR*tl_bR(i,k )+ & & tl_dltL*bL1(i,k+1)+dltL*tl_bL(i,k+1))/ & & (dltR+dltL)- & & (tl_dltR+tl_dltL)*bR(i,k)/(dltR+dltL) bL(i,k+1)=bR(i,k) tl_bL(i,k+1)=tl_bR(i,k) END DO END DO DO i=Istr,Iend FC(i,N(ng))=0.0_r8 ! NO-flux boundary condition tl_FC(i,N(ng))=0.0_r8 ! NO-flux boundary condition #if defined LINEAR_CONTINUATION bL(i,N(ng))=bR(i,N(ng)-1) tl_bL(i,N(ng))=tl_bR(i,N(ng)-1) bR(i,N(ng))=2.0_r8*qc(i,N(ng))-bL(i,N(ng)) tl_bR(i,N(ng))=2.0_r8*tl_qc(i,N(ng))-tl_bL(i,N(ng)) #elif defined NEUMANN bL(i,N(ng))=bR(i,N(ng)-1) tl_bL(i,N(ng))=tl_bR(i,N(ng)-1) bR(i,N(ng))=1.5_r8*qc(i,N(ng))-0.5_r8*bL(i,N(ng)) tl_bR(i,N(ng))=1.5_r8*tl_qc(i,N(ng))-0.5_r8*tl_bL(i,N(ng)) #else bR(i,N(ng))=qc(i,N(ng)) ! default strictly monotonic bL(i,N(ng))=qc(i,N(ng)) ! conditions bR(i,N(ng)-1)=qc(i,N(ng)) tl_bR(i,N(ng))=tl_qc(i,N(ng)) ! default strictly monotonic tl_bL(i,N(ng))=tl_qc(i,N(ng)) ! conditions tl_bR(i,N(ng)-1)=tl_qc(i,N(ng)) #endif #if defined LINEAR_CONTINUATION bR(i,1)=bL(i,2) tl_bR(i,1)=tl_bL(i,2) bL(i,1)=2.0_r8*qc(i,1)-bR(i,1) tl_bL(i,1)=2.0_r8*tl_qc(i,1)-tl_bR(i,1) #elif defined NEUMANN bR(i,1)=bL(i,2) tl_bR(i,1)=tl_bL(i,2) bL(i,1)=1.5_r8*qc(i,1)-0.5_r8*bR(i,1) tl_bL(i,1)=1.5_r8*tl_qc(i,1)-0.5_r8*tl_bR(i,1) #else bL(i,2)=qc(i,1) ! bottom grid boxes are bR(i,1)=qc(i,1) ! re-assumed to be bL(i,1)=qc(i,1) ! piecewise constant. tl_bL(i,2)=tl_qc(i,1) ! bottom grid boxes are tl_bR(i,1)=tl_qc(i,1) ! re-assumed to be tl_bL(i,1)=tl_qc(i,1) ! piecewise constant. #endif END DO ! ! Apply monotonicity constraint again, since the reconciled interfacial ! values may cause a non-monotonic behavior of the parabolic segments ! inside the grid box. ! DO k=1,N(ng) DO i=Istr,Iend dltR=bR(i,k)-qc(i,k) tl_dltR=tl_bR(i,k)-tl_qc(i,k) dltL=qc(i,k)-bL(i,k) tl_dltL=tl_qc(i,k)-tl_bL(i,k) cffR=2.0_r8*dltR tl_cffR=2.0_r8*tl_dltR cffL=2.0_r8*dltL tl_cffL=2.0_r8*tl_dltL IF ((dltR*dltL).lt.0.0_r8) THEN dltR=0.0_r8 tl_dltR=0.0_r8 dltL=0.0_r8 tl_dltL=0.0_r8 ELSE IF (ABS(dltR).gt.ABS(cffL)) THEN dltR=cffL tl_dltR=tl_cffL ELSE IF (ABS(dltL).gt.ABS(cffR)) THEN dltL=cffR tl_dltL=tl_cffR END IF bR(i,k)=qc(i,k)+dltR tl_bR(i,k)=tl_qc(i,k)+tl_dltR bL(i,k)=qc(i,k)-dltL tl_bL(i,k)=tl_qc(i,k)-tl_dltL END DO END DO ! ! After this moment reconstruction is considered complete. The next ! stage is to compute vertical advective fluxes, FC. It is expected ! that sinking may occurs relatively fast, the algorithm is designed ! to be free of CFL criterion, which is achieved by allowing ! integration bounds for semi-Lagrangian advective flux to use as ! many grid boxes in upstream direction as necessary. ! ! In the two code segments below, WL is the z-coordinate of the ! departure point for grid box interface z_w with the same indices; ! FC is the finite volume flux; ksource(:,k) is index of vertical ! grid box which contains the departure point (restricted by N(ng)). ! During the search: also add in content of whole grid boxes ! participating in FC. ! cff=dtdays*ABS(Wbio(isink)) tl_cff=dtdays*SIGN(1.0_r8,Wbio(isink))*tl_Wbio(isink) DO k=1,N(ng) DO i=Istr,Iend FC(i,k-1)=0.0_r8 tl_FC(i,k-1)=0.0_r8 WL(i,k)=z_w(i,j,k-1)+cff tl_WL(i,k)=tl_z_w(i,j,k-1)+tl_cff WR(i,k)=Hz(i,j,k)*qc(i,k) tl_WR(i,k)=tl_Hz(i,j,k)*qc(i,k)+Hz(i,j,k)*tl_qc(i,k) ksource(i,k)=k END DO END DO DO k=1,N(ng) DO ks=k,N(ng)-1 DO i=Istr,Iend IF (WL(i,k).gt.z_w(i,j,ks)) THEN ksource(i,k)=ks+1 FC(i,k-1)=FC(i,k-1)+WR(i,ks) tl_FC(i,k-1)=tl_FC(i,k-1)+tl_WR(i,ks) END IF END DO END DO END DO ! ! Finalize computation of flux: add fractional part. ! DO k=1,N(ng) DO i=Istr,Iend ks=ksource(i,k) cu=MIN(1.0_r8,(WL(i,k)-z_w(i,j,ks-1))*Hz_inv(i,ks)) tl_cu=(0.5_r8+SIGN(0.5_r8, & & (1.0_r8-(WL(i,k)-z_w(i,j,ks-1))* & & Hz_inv(i,ks))))* & & ((tl_WL(i,k)-tl_z_w(i,j,ks-1))*Hz_inv(i,ks)+ & & (WL(i,k)-z_w(i,j,ks-1))*tl_Hz_inv(i,ks)) FC(i,k-1)=FC(i,k-1)+ & & Hz(i,j,ks)*cu* & & (bL(i,ks)+ & & cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks)))) tl_FC(i,k-1)=tl_FC(i,k-1)+ & & (tl_Hz(i,j,ks)*cu+Hz(i,j,ks)*tl_cu)* & & (bL(i,ks)+ & & cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks))))+ & & Hz(i,j,ks)*cu* & & (tl_bL(i,ks)+ & & tl_cu*(0.5_r8*(bR(i,ks)-bL(i,ks))- & & (1.5_r8-cu)* & & (bR(i,ks)+bL(i,ks)- & & 2.0_r8*qc(i,ks)))+ & & cu*(0.5_r8*(tl_bR(i,ks)-tl_bL(i,ks))+ & & tl_cu* & & (bR(i,ks)+bL(i,ks)-2.0_r8*qc(i,ks))- & & (1.5_r8-cu)* & & (tl_bR(i,ks)+tl_bL(i,ks)- & & 2.0_r8*tl_qc(i,ks)))) END DO END DO DO k=1,N(ng) DO i=Istr,Iend Bio(i,k,ibio)=qc(i,k)+(FC(i,k)-FC(i,k-1))*Hz_inv(i,k) tl_Bio(i,k,ibio)=tl_qc(i,k)+ & & (tl_FC(i,k)-tl_FC(i,k-1))*Hz_inv(i,k)+ & & (FC(i,k)-FC(i,k-1))*tl_Hz_inv(i,k) END DO END DO END DO SINK_LOOP END DO ITER_LOOP ! !----------------------------------------------------------------------- ! Update global tracer variables: Add increment due to BGC processes ! to tracer array in time index "nnew". Index "nnew" is solution after ! advection and mixing and has transport units (m Tunits) hence the ! increment is multiplied by Hz. Notice that we need to subtract ! original values "Bio_old" at the top of the routine to just account ! for the concentractions affected by BGC processes. This also takes ! into account any constraints (non-negative concentrations, carbon ! concentration range) specified before entering BGC kernel. If "Bio" ! were unchanged by BGC processes, the increment would be exactly ! zero. Notice that final tracer values, t(:,:,:,nnew,:) are not ! bounded >=0 so that we can preserve total inventory of nutrients ! when advection causes tracer concentration to go negative. !----------------------------------------------------------------------- ! DO itrc=1,NBT ibio=idbio(itrc) DO k=1,N(ng) DO i=Istr,Iend cff=Bio(i,k,ibio)-Bio_old(i,k,ibio) tl_cff=tl_Bio(i,k,ibio)-tl_Bio_old(i,k,ibio) !^ t(i,j,k,nnew,ibio)=t(i,j,k,nnew,ibio)+cff*Hz(i,j,k) !^ tl_t(i,j,k,nnew,ibio)=tl_t(i,j,k,nnew,ibio)+ & & tl_cff*Hz(i,j,k)+cff*tl_Hz(i,j,k) END DO END DO END DO END DO J_LOOP ! RETURN END SUBROUTINE tl_npzd_powell_tile END MODULE tl_biology_mod