subroutine dh (sal1d, Fx, Fmix, Fneti, ultnt, condb, Tw, Ts, Ti &, Tbot, Tf, qi, hi, hs, dhib, subi, subs, dhit, dhs &, dhif, dhsf, ni) #if defined O_ice && defined O_ice_cpts && defined O_embm ! Energy-conserving sea ice model ! Routines to grow/melt ice and adjust temperature profile ! See Bitz, C.M., and W.H. Lipscomb, 1999: ! A new energy-conserving sea ice model for climate study, ! J. Geophys. Res., in press. ! The author grants permission to the public to copy and use this ! software without charge, provided that this Notice and any statement ! of authorship are reproduced on all copies. ! Computes the thickness changes at the top and bottom ! and adjusts layer specific enthalpy ! This algorithm does not allow h<0 ! Fx= actual flux of heat from the mixed layer under sea ice ! (equal to Fmix unless all the ice melts away) ! parameterized according to the ! Tw as described in papers by McPhee/Steel ! compensates for rare case of melting entire slab through implicit none include "cpts.h" include "thermo.h" include "cembm.h" ! input variables real sal1d(NMAX) ! ice salinity (ppt) real Ts ! top srf temp (C) real Ti(0:NMAX) ! snow/ice internal temp (C) real Tbot ! ice bottom in (C) real Tw ! water temperature below ice (C) real Tf ! freezing temperature of sea water real hi ! initial ice thickness (m) real hs ! initial snow thickness (m) real Fmix ! Flux from mixed layer from relaxation (W/m**2) real Fneti ! net flux at upper surface (W/m**2) real condb ! conductive flux at bottom (w/m**2) real ultnt ! latent heat flux, for sublimation (w/m**2) integer ni ! number of layers ! output variables ! thickness changes from grow/melt (default) or sublimate/flooding real dhib ! ice bot, dhib<0 if melt (m) real dhit ! ice top, dhit<=0 (m) real dhs ! snow top, dhit<=0 (m) real subi ! ice top, subi<0 if sublimating (m) real subs ! snow, subs<0 if sublimating (m) real dhif ! ice top from flooding, dhif>0 (m) real dhsf ! snow from flooding, dhsf>0 (m) real qi(NMAX) ! energy of melt of ice per unit vol. (J/m**3) real Fx ! actual flx of heat used from ocn (w/m**2) ! local variables real delti(NMAX) ! evolving ice layer thickness real delts ! evolving snow thickness real sumr ! dummy for adjusting delti real rmvi ! another dummy for adjusting delti real dtop ! dhit+subi for adjust real qigrow ! energy of melt of ice that grows real Ebot ! heat available to grow/melt at bottom real Etop ! heat available to melt at top real qiflood ! energy of melt of flooded ice (W/m**2) real zintfc ! height of snow/ice interf. wrt ocn (m) real qs ! energy of melting of snow per unit volume real Enet ! sum of the energy of melting relative to melt real Evnet ! sum of the energy of melting relative to vapour real energ ! function integer layer logical alarm2,alarm alarm2=.false. alarm=.false. dhib = 0. dhit = 0. dhs = 0. dhif = 0. dhsf = 0. subi = 0. subs = 0. ! thickness of snow and each ice layer delti(1) = hi/ni do layer = 2,ni delti(layer) = delti(1) enddo delts = hs ! energy of melt per unit vol for snow and ice for each layer and sum qs = -RFLSNO enet = 0. do layer = 1,ni qi(layer) = energ(Ti(layer),sal1d(layer)) enet = enet + qi(layer) enddo enet = enet*delti(1) + qs*hs if (ultnt.gt.0.0) then ! etop is positive is sublimating and negative if condensing Etop = ultnt*dtau Evnet = -RVLSNO*hs-RVLICE*hi+Enet+Etop if ( Evnet .ge. 0.0 ) then subi=-hi subs=-hs Fx=condb-Evnet/dtau ! maybe wrong, irrelevant though # if defined O_ice_cpts_warnings print*,'sublimated though all snow and ice',Fx,hi,hs stop 'sublimate too much is not allowed, model will stop' ! return # endif endif call srfsub( qi, qs, delti, delts, ni, $ subi, subs, etop, enet, alarm2 ) ! adjust the layer thickness to reflect subl/cond changes delts = hs + subs rmvi = subi do layer = 1,ni sumr = max( -delti(layer), rmvi ) rmvi = rmvi-sumr delti(layer) = delti(layer) + sumr enddo endif if ( Fneti .gt. 0.0 ) then ! it is possible to melt at srf when Ts is below melting ! but it never melts more than a negligible amount Etop=Fneti*dtau Enet=Enet+Etop if (Enet.ge.0.0) then dhit=-(hi+subi) dhs=-(hs+subs) Fx=condb-Enet/dtau return endif call srfmelt (qi, qs, delti, delts, ni, $ dhit, dhs, etop, alarm2) ! adjust the layer thickness to reflect melt changes delts = delts + dhs rmvi = dhit do layer = 1,ni sumr = max( -delti(layer), rmvi ) rmvi = rmvi - sumr delti(layer) = delti(layer) + sumr enddo endif dtop=dhit+subi Fx=Fmix Ebot=dtau*(Fmix-condb) if (Ebot.le.0.0) then qigrow=energ(Tbot,SALNEW) dhib=Ebot/qigrow else ! melt at bottom, melting temperature = Tbot qigrow=0. ! on purpose if ((Enet+Ebot).ge.0.0) then dhib=-(hi+dtop) dhs=-(hs+subs) Fx=condb-Enet/dtau return endif call botmelt (qi, qs, delti, delts, ni, $ dhib, dhs, ebot, alarm2) endif ! print out helpful variables if something went wrong if (alarm2) then print*,'Ti ',(Ti(layer),layer=1,ni) print*,'Tw ',Tw,condb,Fx,Fneti print*,'Enet, Ebot',Enet,Ebot,Fneti*dtau,dtau*(Fmix-condb) Enet=0. do layer=1,ni Enet=Enet+qi(layer) enddo Enet=Enet*hi/ni+qs*hs print*,'orig Enet',Enet print*,'delti',delts, (delti(layer),layer=1,ni) print*,dhib,dhit,subs,subi,dhs alarm=.true. endif ! flooding zintfc = hi - (rhosno*hs + rhoice*hi)/rhoocn qiflood = 0. if ((zintfc .lt. 0.) .and. (hs+dhs.gt.0.)) then dhsf = -rhoice/rhosno*zintfc dhsf = min(dhsf,hs+dhs) dhif = rhosno/rhoice*dhsf if (dhif .gt. 0) qiflood = qs*dhsf/dhif endif call adjust(hi,dhib,dtop,dhif,dhsf,qiflood,qigrow,ni,qi) return end subroutine srfsub( qi ,qs ,delti ,delts ,ni , $ subi ,subs ,etop ,enet ,alarm ) ! compute the sea ice and snow thickness changes from ! sublimation/condensation ! Etop>0 qi<0 qs<=0 and Enet<0 implicit none include "cpts.h" include "thermo.h" include "cembm.h" ! input variables real qi (1:nmax), qs ! energy of melt per vol (J/m**3) real delti(NMAX), delts ! thickness of ice/snow layer (m) integer ni ! number of layers ! output variables real subi, subs ! subl/cond. amount for ice/snow (m) ! input/output variables real etop ! energy avail to sub/cond ice/snow (J/m**2) real enet ! energy needed to melt all ice/snow (J/m**2) ! local variables real u ! energy of melt + vapour for ice/snow (J/m**3) real subil ! subl from this layer integer layer logical alarm if ( delts .gt. 0. ) then ! convert etop into snow thickness ! positive if cond/negative if subl u = qs - rvlsno subs = etop/u if ( (delts + subs) .ge. 0. ) then ! either all condensation becomes snow ! or sublimate some snow but no ice etop = 0.0 enet = enet + subs * qs return else ! sublimate all of the snow and some ice too subs = -delts etop = etop + delts * u enet = enet + subs * qs endif endif do layer = 1,ni ! convert etop into ice thickness ! positive if cond/negative if subl u = qi(layer) - rvlice subil = etop/u if ( (delti(layer)+subil) .ge. 0. ) then ! either all condensation becomes ice ! or sublimate some ice from this layer subi = subi + subil etop = 0.0 enet = enet + subil * qi(layer) return else ! sublimate all of the layer subi = subi - delti(layer) etop = etop + delti(layer) * u enet = enet - delti(layer) * qi(layer) endif enddo ! model should never get here, probably a only minor error if it does print*, 'ERROR in srfsub',Etop alarm=.true. return end subroutine srfmelt( qi, qs, delti, delts, ni, $ dhit, dhs, etop, alarm ) ! melt ice/snow from the top srf ! Etop>0 qi<0 and qs<=0 implicit none include "cpts.h" include "thermo.h" include "cembm.h" ! input variables real qi (1:nmax), qs ! energy of melt per vol (J/m**3) real delti(NMAX), delts ! thickness of ice/snow layer (m) integer ni ! number of layers ! output variables real dhit, dhs ! ice/snow thickness change(m) ! input/output variables real etop ! energy avail to melt ice and snow (J/m**2) ! local variables real dhitl ! melt from this layer (m) integer layer logical alarm if ( delts .gt. 0. ) then ! convert etop into snow thickness dhs = etop/qs if ( (delts + dhs) .ge. 0. ) then ! melt only some of the snow etop = 0.0 return else ! melt all of the snow and some ice too dhs = -delts etop = etop+qs*delts endif endif do layer = 1,ni ! convert etop into ice thickness dhitl = etop/qi(layer) if ( (delti(layer)+dhitl) .ge. 0. ) then ! melt some ice from this layer dhit = dhit + dhitl etop = 0.0 return else ! melt all of the ice in this layer dhit = dhit - delti(layer) etop = etop + delti(layer) * qi(layer) endif enddo ! model should never get here, probably a only minor error if it does print*, 'ERROR in srfmelt',Etop alarm=.true. return end subroutine botmelt( qi, qs, delti, delts, ni, $ dhib, dhs, ebot, alarm ) ! melt from bottom ! Ebot>0 qi<0 and qs<=0 implicit none include "cpts.h" include "thermo.h" include "cembm.h" ! input variables real qi (1:nmax), qs ! energy of melt per vol (J/m**3) real delti(NMAX), delts ! thickness of ice/snow layer (m) integer ni ! number of layers ! output variables real dhib ! ice thickness change (m) ! input/output variables real ebot ! energy avail to melt ice and snow (J/m**2) real dhs ! snow thickness change (m) ! local variables real dhibl ! melt from this ice layer (m) real dhsl ! melt from this snow layer (m) integer layer logical alarm do layer = ni,1,-1 dhibl = ebot/qi(layer) if ( (delti(layer)+dhibl) .ge. 0. ) then dhib = dhib + dhibl ebot = 0.0 return else dhib = dhib - delti(layer) ebot = ebot + delti(layer) * qi(layer) endif enddo ! finally melt snow if necessary dhsl = ebot/qs if ( (delts + dhsl) .ge. 0. ) then dhs = dhs + dhsl ebot = 0.0 return endif ! model should never get here, probably a only minor error if it does print*,'melted completely through all ice and snow layer',delts print*,'some diagnostics',(delti(layer),layer=1,ni),(qi(layer) $ ,layer=1,ni) print*,'ERROR in botmelt',Ebot alarm=.true. return end subroutine adjust(hi,dhib,dhit,dhif,dhsf,qiflood,qigrow,ni,qi_tw) ! Adjusts temperature profile after melting/growing. ! E_tw = E at start ! E is the energy density after updating Ti from ! the heat equation without regard dhit and dhib ! h is the thickness from previous time step ! h_tw is the NEW thickness h(i,j) ! dhib is negative if there is melt at the bottom ! dhit is negative if there is melt at the top ! generally _tw is a suffix to label the new layer spacing variables implicit none include "cpts.h" include "thermo.h" include "cembm.h" ! input variables real hi ! initial ice thickness (m) real dhib ! ice bot, dhib<0 if melt (m) real dhit ! ice top, dhit<=0 (m) real dhif ! ice top from flooding, dhif>0 (m) real dhsf ! snow from flooding, dhsf>0 (m) real qiflood ! qi for flooded ice (J/m**3) real qigrow ! qi for ice growing on bot (J/m**3) integer ni ! input and output variables real qi_tw(nmax) ! qi for adjusted layers (J/m**3) ! local variables ! the following pairs are before & after adjusting real h, h_tw ! ice thickness (m) real z(nmax+1) ! vertical layer position (m) real z_tw(nmax+1) ! vertical layer position (m) real D, D_tw ! layer thickness (m) integer k, k_tw ! index of layer real qi(nmax) ! qi of initial layers (J/m**3) real fract(nmax,nmax) ! fract of layer k that makes up layer k_tw integer layer ! first check if there is any ice left after top/bottom melt/growth h = hi h_tw = hi+dhib+dhit if ( h_tw .le. 0. ) then do layer = 1,ni qi_tw(layer) = 0. enddo dhsf = 0. dhif = 0. return endif ! there is ice left so allow snow-ice conversion from flooding h_tw = h_tw+dhif if ( (abs(dhib) .gt. tiny) .or. (abs(dhit) .gt. tiny) $ .or. (dhif.gt.0)) then do layer = 1,ni qi(layer) = qi_tw(layer) enddo ! layer thickness D = h/ni D_tw = h_tw/ni ! z is positive down and zero is relative to the top ! of the ice from the old time step z(1) = -dhit ! necessary z_tw(1) = -dhit-dhif do layer = 2,ni z(layer) = D*(layer-1) z_tw(layer) = z_tw(1)+D_tw*(layer-1) enddo z(ni+1) = h + min(dhib,0.) z_tw(ni+1) = z_tw(1)+h_tw do k_tw = 1,ni qi_tw(k_tw) = 0. do k = 1,ni fract(k,k_tw) = ( min( z_tw(k_tw+1), z(k+1)) - $ max( z_tw(k_tw) , z(k) ) ) if (fract(k,k_tw).gt.0.) $ qi_tw(k_tw) = qi_tw(k_tw)+fract(k,k_tw)*qi(k) enddo enddo if (dhif.gt.D_tw) then do k = 1,ni qi_tw(k) = qi_tw(k) + qiflood*max(0., $ min(D_tw,dhif-D_tw*(k-1))) enddo else qi_tw(1) = qi_tw(1) + dhif*qiflood endif if (dhib.gt.D_tw) then z(ni+2) = h + dhib do k = 1,ni qi_tw(k) = qi_tw(k) + qigrow*max(0., $ (min(z(ni+2),Z_tw(k+1))-max(h,Z_tw(k)))) enddo else qi_tw(ni) = qi_tw(ni) + max(dhib,0.)*qigrow endif do k_tw = 1,ni qi_tw(k_tw) = qi_tw(k_tw)/D_tw enddo endif return end subroutine init_ice_cpts (is, ie, js, je) ! if the cpts ice option is used without the embm, figure out where to ! call this routine implicit none include "size.h" include "param.h" include "pconst.h" include "stdunits.h" include "cembm.h" include "csbc.h" include "cpts.h" include "ice.h" integer i, ie, is, j, je, js, nc, layers ! use double/single Euler steps dtau=dts idx=1 if (mod(nats,2).eq.0) idx=2 if (nats.eq.1) then ! mixing step, mode 2 is kept do nc=1,ncat do j=js,je do i=is,ie A(i,j,1,nc)=A(i,j,2,nc) heff(i,j,1,nc)=heff(i,j,2,nc) hseff(i,j,1,nc)=hseff(i,j,2,nc) Ts(i,j,1,nc)=Ts(i,j,2,nc) enddo enddo enddo do layers=1,ntilay do j=js,je do i=is,ie E(i,j,1,layers)=E(i,j,2,layers) enddo enddo enddo endif return end subroutine adv_ridge_cpts (is, ie, js, je) implicit none # if defined O_ice_evp include "size.h" include "param.h" include "pconst.h" include "stdunits.h" include "cembm.h" include "cpts.h" include "ice.h" integer i, ie, is, j, je, js, layer, nc real A0(is:ie,js:je), ah, A_Ts(is:ie,js:je,ncat) do nc=1,ncat call embmbc (heff(1,1,idx,nc)) call embmbc (A(1,1,idx,nc)) call embmbc (Ts(1,1,idx,nc)) call embmbc (hseff(1,1,idx,nc)) enddo do layer=1,ntilay call embmbc (E(1,1,idx,layer)) enddo ! compute open water area so it can be advected do j=js,je do i=is,ie A0(i,j)=1.0 do nc=1,ncat A0(i,j)=A0(i,j)-A(i,j,idx,nc) enddo A0(i,j)=max(0.,A0(i,j)) enddo enddo ! proper surface temperature advection is done by advecting ! the area-temperature product do nc=1,ncat do j=js,je do i=is,ie A_Ts(i,j,nc) = min(A(i,j,idx,nc)*Ts(i,j,idx,nc),0.) enddo enddo enddo ! At this time mechanical redistribution for this model assumes ! you are using upstream advection. call advupb(uice, vice ,A0, is, ie, js, je) do nc=1,ncat call advupb(uice, vice, A(1,1,idx,nc), is, ie, js, je) enddo do nc=1,ncat call advupb(uice, vice, A_Ts(1,1,nc), is, ie, js, je) enddo do nc=1,ncat call advupb(uice, vice, heff(1,1,idx,nc), is, ie, js, je) enddo do nc=1,ncat call advupb(uice, vice, hseff(1,1,idx,nc), is, ie, js, je) enddo do layer=1,ntilay call advupb(uice, vice, E(1,1,idx,layer), is, ie, js, je) enddo do nc=1,ncat do j=js,je do i=is,ie Ts(i,j,idx,nc)=frzpt(i,j) if (A(i,j,idx,nc).gt.tiny) then Ts(i,j,idx,nc)=A_Ts(i,j,nc)/A(i,j,idx,nc) Ts(i,j,idx,nc)=min(Ts(i,j,idx,nc),0.) endif enddo enddo enddo call mechred(A0, is, ie, js, je) # endif return end ! Energy-conserving sea ice model ! Routines to do basic energy/temperature conversion ! See Bitz, C.M., and W.H. Lipscomb, 1999: ! A new energy-conserving sea ice model for climate study, ! J. Geophys. Res., in press. ! The author grants permission to the public to copy and use this ! software without charge, provided that this Notice and any statement ! of authorship are reproduced on all copies. function energ(Tmp,sal) ! compute the energy of melting for non-new ice ! relative to melting (negative quantity) ! assuming Tmp is in Celsius implicit none include "cpts.h" include "thermo.h" real Tmp,energ,sal energ=-rflice $ -rcpice*(-alpha*sal-Tmp)-gamma*sal/Tmp return end subroutine getTmp(Tmp,E,sal1d,ni) ! compute the midpoint temperature from the ! energy of melting for each layer implicit none include "cpts.h" include "thermo.h" real Tmp(nmax),E(nmax),sal1d(nmax),quad integer ni,i real(kind=8) q, B, C do i=1,ni q=E(i)+rflice-rcpice*alpha*sal1d(i) B=-q/rcpice C=-gamma*sal1d(i)/rcpice Tmp(i)= quad(B,C) enddo return end function quad(B,C) implicit none include "cpts.h" include "thermo.h" real quad,root,T1,T2 real(kind=8) B, C, B_2 B_2=B/2.0 root=sqrt(B_2*B_2-C) ! T1=-B_2+root T2=-B_2-root quad=T2 ! print*,'quad T1=',T1,T2 ! if (T1.le.(TMELT-0.1)) print*,'quad T1=',T1,T2 return end subroutine grownew(fsh_new,Io_new,flo,focean,Fbot, $ hnew,tnew,tair,qair,ug,dtp,fice,frzpt,evp, $ dswr,ultnt,usens,ulwr) ! Uses a Newton_Raphson (non-linear) ! method to compute new ice surface temp ! and thickness. ! This algorithm tends to fail when the correct solution is ! to grow a very thin layer of ice. In this case I just set ! the new ice surface temperature to 0.5*(Tair+frzpt) and ! grow accordingly. It still conserves energy. implicit none include "cpts.h" include "thermo.h" include "cembm.h" include "calendar.h" real coltnt,cosens real fsh_new,flo,Io_new real Fbot,focean,hnew,tnew,frzpt real dswr,ultnt,usens,ulwr real tair,qair,evp,qnew,rday real ug,dt real dtp real b,fice,e1,e2,e3,e4,e5,terror,herror real u real J11,J12,J21,J22,F1,F2,detJ real rand,energ,kthin real ult_new,use_new,ulw_new,evp_new integer iter kthin=kappai+beta*salnew/(frzpt-5.) Tnew=0.5*(frzpt+Tair) ! initial guess Tnew=min(frzpt,Tnew) # if defined O_ice_cpts_simple_growth fice=focean u=0.5*(frzpt+Tnew) u=energ(u,salnew) hnew=(focean+Fbot)*dtp/u # else dt=Tnew-Tair cosens=0.94*rcpatm*dalt_i*ug ult_new=ultnt e1=fsh_new-ult_new+flo+cosens*tair e2=-cosens*Tnew-ESICE*(Tnew+TFFRESH)**4 fice=e1+e2 if ((fice+focean+2*Fbot).lt.0.0) then rday=daylen/dtau hnew=centi*0.0133/rday*(frzpt-Tnew) iter=1 100 continue e2=-cosens*Tnew-esice*Tnew**4 e3=-cosens-4.*ESICE*(Tnew+TFFRESH)**3 if (Tnew.gt.frzpt) then u=frzpt else u=0.5*(frzpt+Tnew) endif e4=-0.5*(rcpice-gamma*salnew/(u**2)) u=-energ(u,salnew) J11=hnew*e3-kthin J12=e1+e2 J21=0.5*e3+hnew/dtp*e4 J22=u/dtp F1=(J12-Io_new)*hnew+kthin*(frzpt-Tnew) F2=0.5*(focean+J12)+Fbot+hnew*J22 detJ=J11*J22-J12*J21 terror=(J22*F1-J12*F2)/detJ herror=(-J21*F1+J11*F2)/detJ Tnew= Tnew-terror hnew=hnew-herror iter=iter+1 if ((iter.gt.10).or.(hnew.lt.0).or.(Tnew.gt.0.)) then ! when cannot converge, assume a temperature and continue Tnew=0.5*(frzpt+Tair) Tnew=min(frzpt,Tnew) go to 200 endif if (abs(terror).gt.0.005) go to 100 200 continue u=0.5*(frzpt+Tnew) u=energ(u,salnew) ulw_new=esice*(Tnew+TFFRESH)**4-Flo use_new=cosens*(Tnew-tair) fice=fsh_new-ulw_new-use_new-ult_new hnew=(0.5*(fice+focean)+Fbot)*dtp/u dswr = 0.5*(dswr+fsh_new) ulwr = 0.5*(ulwr +ulw_new) usens = 0.5*(usens+use_new) else ! it is not clear what to do with the negative ! ocean energy balance if in the presence of ! open water the energy balance is negative and ! would allow ice to grow, yet once it grows a little ! ice the energy balance swings to positive ! I am going to set h=0 in this case and balance the fluxes ! by messing with the sensible heat usens = dswr-ultnt-ulwr+Fbot fice=-focean-2*Fbot Tnew=frzpt hnew=0 endif # endif return end subroutine mechred (A0, is, ie, js, je) ! it is perfectly fine to call this routine when NCAT=1 ! in fact, it will provide a source of open water from shear ! deformation. See Stern et al, 1995 for information about ! how and why this is done implicit none include "size.h" include "param.h" include "pconst.h" include "stdunits.h" include "evp.h" include "atm.h" include "cpts.h" include "ice.h" include "grdvar.h" ! input variables real A0(is:ie,js:je) ! local variables integer i, ie, iem1, is, isp1, j, je, jem1, js, jsp1, n, nc integer layer, l real En(NMAX,ncat) real Esn(ncat) real heffn(ncat) real hseffn(ncat) real An(ncat) real Tsn(ncat) real closing ! lead closing rate real M(ncat,ncat) ! area fraction of ice from i into j real HN(ncat,ncat) ! H*volume fraction of ice from i into j real rpress isp1 = is + 1 iem1 = ie - 1 jsp1 = js + 1 jem1 = je - 1 # if defined O_ice_cpts_roth_press do j=js,je do i=isp1,ie strength(i,j,idx)=0. enddo enddo # endif do n=1,nseg do j=jsi(n),jei(n)+1 do i=isp1,iem1 if (tmsk(i,j) .ge. 0.5) then ! ice velocity divergence and lead closing rate, following FH95 closing=0.25*(del(i,j)-abs(eI(i,j)))-min(eI(i,j),0.) closing=max(closing,-eI(i,j)) l=1 do nc=1,ncat do layer=1,nilay(nc) En(layer,nc)=E(i,j,idx,l) l=l+1 enddo Tsn(nc)=Ts(i,j,idx,nc) An(nc)=A(i,j,idx,nc) heffn(nc)=heff(i,j,idx,nc) hseffn(nc)=hseff(i,j,idx,nc) enddo call ridge(heffn,hseffn,A0(i,j),An,En,Esn,Tsn, $ closing,eI(i,j),M,HN,frzpt(i,j)) # if defined O_ice_cpts_roth_press strength(i,j,idx)=rpress(heffn,A0(i,j),An,HN,M) # endif l=1 do nc=1,ncat do layer=1,nilay(nc) E(i,j,idx,l)=En(layer,nc) l=l+1 enddo Ts(i,j,idx,nc)=Tsn(nc) A(i,j,idx,nc)=An(nc) heff(i,j,idx,nc)=heffn(nc) hseff(i,j,idx,nc)=hseffn(nc) enddo endif enddo enddo enddo do nc=1,ncat call embmbc (heff(1,1,idx,nc)) call embmbc (A(1,1,idx,nc)) call embmbc (Ts(1,1,idx,nc)) call embmbc (hseff(1,1,idx,nc)) enddo do layer=1,ntilay call embmbc (E(1,1,idx,layer)) enddo return end subroutine ridge(heffn,hseffn,A0,An,En,Esn,Tsn,closed,divu, $ M,HN,frzpt) ! note the time available for ridging between the cats is ! dtau, however have to subcycle if a category runs ! out of ice then dtn is less than dtau ! M(nc,k) is the fraction of ice that participates in ridging implicit none include "cpts.h" include "thermo.h" real heffn(ncat) real hseffn(ncat) real An(ncat) real Tsn(ncat),dtf(ncat),Atsn(ncat) real En(nmax,ncat) real Esn(ncat) real closed ! the closing as defined by Stern et al real divu, A0 real frzpt real strg ! ice strength real Asum real W(0:ncat) ! ridging for each category real Wa(0:ncat) ! the area that participates real M(ncat,ncat) ! area fraction of ice from i into j real N(ncat,ncat) ! volume fraction of ice from i into j real HN(ncat,ncat) ! dummy real Hmean(ncat) ! Hi+sqrt(cK*Hi) real dtn,dtnn,dtp integer k,layer,index,isub,nc,layk,lay,i,j real Qi(nmax,ncat) ! energy of melting ice per unit volume real Qs(ncat) ! energy of melting of snow per unit area real delt(ncat),delts(ncat),deltEs(ncat) real deltE(nmax,ncat) real Psi, closdtn real dsum,ssum,esum,fsum real Hi(ncat),Hs(ncat), eta dtn=dtau dtp=0 A0=A0 Asum=A0 do nc=1,ncat Asum = Asum + An(nc) enddo ! unfortunately the second order accurate advection ! schemes do not have Asum=1-divu*dt after advection ! owing to diffusion. ! Adjust opening rate here to compensate. divu=(1-Asum)/dtn closed=-min(divu,0.) ! from Bill Hibler suggestions closed=max(closed,-divu) A0=A0+dtn*(divu+closed) Asum=Asum+dtn*(divu+closed) if (Asum.le.1.0) then A0=A0+1.-Asum # if defined O_ice_cpts_roth_press call ridge_matrices(heffn,An,Hi,Hmean,M,N,HN) # endif return endif 50 continue call ridge_matrices(heffn,An,Hi,Hmean,M,N,HN) call ridging_mode(A0,An,W,Wa,M) # if defined O_ice_cpts3 || defined O_ice_cpts5 || defined O_ice_cpts10 do nc=1,ncat Hs(nc)=0.0 Qs(nc)=0.0 do layer=1,nilay(nc) Qi(layer,nc)=0.0 enddo if ((An(nc).gt.TINY) .and. (heffn(nc).gt.0.)) then Hs(nc)=hseffn(nc)/An(nc) Qs(nc)=Esn(nc)/An(nc) do layer=1,nilay(nc) Qi(layer,nc)=En(layer,nc)*nilay(nc)/heffn(nc) enddo endif enddo dsum=0. ssum=0. esum=0. do nc=1,ncat-1 delt(nc) =-Hi(nc)*Wa(nc) delts(nc) =-Hs(nc)*Wa(nc) deltEs(nc)=-Qs(nc)*Wa(nc) do k=1,nc delt(nc) =delt(nc)+Wa(k)*N(k,nc) delts(nc) =delts(nc)+Wa(k)*M(k,nc)*Hs(k)*Hmean(k)/hi(k) deltEs(nc)=deltEs(nc)+Wa(k)*M(k,nc)*Qs(k)*Hmean(k)/hi(k) enddo dsum=dsum+delt(nc) ssum=ssum+delts(nc) esum=esum+deltEs(nc) enddo delt(ncat) =-dsum delts(ncat) =-ssum deltEs(ncat)=-esum fsum=0. do nc=1,ncat-1 do layer=1,nilay(nc) deltE(layer,nc)=-Wa(nc)*Hi(nc)*Qi(layer,nc) /nilay(nc) do k=1,nc layk=(layer+ncrel(k,nc)-1)/ncrel(k,nc) deltE(layer,nc)=deltE(layer,nc)+Qi(layk,k)*Wa(k)*N(k,nc) $ /nilay(nc) enddo fsum=fsum+deltE(layer,nc) enddo enddo do layer=1,nilay(ncat) deltE(layer,ncat)=0. do k=1,ncat-1 layk=(layer+ncrel(k,ncat)-1)/ncrel(k,ncat) deltE(layer,ncat)=deltE(layer,ncat)+Qi(layk,k)*Wa(k) $ *N(k,ncat)/nilay(ncat) enddo fsum=fsum+deltE(layer,ncat) enddo ! geometrical factor for treating ice/snow surface temp when ridging do nc = 1,ncat dtf(nc) = - tsn(nc)*Wa(nc) do k=1,nc dtf(nc) = dtf(nc)+ tsn(k)*Wa(k)*M(k,nc) enddo enddo # endif ! make sure that only ridge as much ice as is available, otherwise ! reduce timestep Psi=closed*Wa(0) if ((Psi*dtn.gt.A0) .and. (Psi.gt.0.)) then dtnn=A0/(Psi) dtn=min(dtn,dtnn) endif do nc=1,ncat Psi=closed*Wa(nc) if ((Psi*dtn.gt.An(nc)) .and. (Psi.gt.0.)) then dtnn=An(nc)/(Psi) dtn=min(dtn,dtnn) endif enddo dtp=dtp+dtn closdtn=closed*dtn A0=A0+closdtn*W(0) Asum=A0 do nc=1,ncat Atsn(nc) = An(nc)*tsn(nc) + closdtn*dtf(nc) An(nc)=An(nc)+closdtn*W(nc) Asum=Asum+An(nc) # if defined O_ice_cpts3 || defined O_ice_cpts5 || defined O_ice_cpts10 heffn(nc)=heffn(nc)+closdtn*delt(nc) hseffn(nc)=hseffn(nc)+closdtn*delts(nc) Esn(nc)=Esn(nc)+closdtn*deltEs(nc) do layer=1,nilay(nc) En(layer,nc)=En(layer,nc)+closdtn*deltE(layer,nc) enddo # endif enddo do nc = 1,ncat if (An(nc).gt.tiny) then tsn(nc) = Atsn(nc)/An(nc) else tsn(nc) = frzpt endif enddo dtn=dtau-dtp if (dtn.gt.TINY) go to 50 return end subroutine ridging_mode(A0,An,W,Wa,M) ! compute W and Wa ! M(nc,k) is the fraction of ice that participates in ridging ! from category nc that goes into category k implicit none include "cpts.h" real An(ncat),A0 real W(0:ncat) ! ridging for each category real Wa(0:ncat) ! the area that participates real M(ncat,ncat) ! area fraction of ice from i into j real V(1:ncat+1) ! the area that participates real Asum,eta,Wsum integer k,layer,index,isub,nc,layk,lay,i,j Asum=0. Wa(0)=min(gstar-Asum,A0) Asum=Asum+Wa(0) ! print*,0,Wa(0),Asum,(1.0 + 0.5*Wa(0)/gstar - Asum/gstar) Wa(0)=Wa(0)*(1.0 + 0.5*Wa(0)/gstar - Asum/gstar) Wa(0)=max(Wa(0),0.) do nc=1,ncat if (An(nc).gt.0.) then Wa(nc)=min(gstar-Asum,An(nc)) Asum=Asum+Wa(nc) Wa(nc)=Wa(nc)*(1.0 + 0.5*Wa(nc)/gstar - Asum/gstar) Wa(nc)=max(Wa(nc),0.) else Wa(nc)=0. endif enddo Wsum=-Wa(0) do nc=1,ncat W(nc)=-Wa(nc) do k=1,nc W(nc)=W(nc)+Wa(k)*M(k,nc) enddo Wsum=Wsum+W(nc) enddo eta=-Wsum do nc=1,ncat W(nc)=W(nc)/eta Wa(nc)=Wa(nc)/eta enddo Wa(0)=Wa(0)/eta W(0)=-Wa(0) return end subroutine ridge_matrices(heffn,An,Hi,Hmean,M,N,HN) ! compute M,N,HN for ridge and press for true thicknesses implicit none include "cpts.h" real heffn(ncat) real An(ncat) real Hi(ncat) real M(ncat,ncat) ! area fraction of ice from i into j ! N is used in ridge and HN is used in rpress real N(ncat,ncat) ! volume fraction of ice from i into j real HN(ncat,ncat) ! H*volume fraction of ice from i into j real rcKH ! sqrt(cK*Hi) real Hmean(ncat) ! Hi+sqrt(cK*Hi) real bet ! 2*(K-Hi) integer n1,n2 ! see above integer k,nc logical row_flag(ncat) do nc=1,ncat if ((An(nc).gt.TINY) .and. (heffn(nc).gt.0.)) then Hi(nc)=heffn(nc)/An(nc) row_flag(nc)=.true. do k=1,ncat M(nc,k)=0. N(nc,k)=0. HN(nc,k)=0. enddo else Hi(nc)=hstar(nc-1) row_flag(nc)=.false. do k=1,ncat M(nc,k)=M_def(nc,k) N(nc,k)=N_def(nc,k) HN(nc,k)=HN_def(nc,k) enddo endif enddo call comp_matrices(row_flag,Hi,Hmean,M,N,HN) return end subroutine comp_matrices(row_flag,Hi,Hmean,M,N,HN) ! compute M,N,HN for ridge and rpress for true thicknesses ! N is used in ridge and HN is used in rpress implicit none include "cpts.h" logical row_flag(ncat) real Hi(ncat) real M(ncat,ncat) ! area fraction of ice from i into j real N(ncat,ncat) ! volume fraction of ice from i into j real HN(ncat,ncat) ! H*volume fraction of ice from i into j real rcKH ! sqrt(cK*Hi) real Hmean(ncat) ! Hi+sqrt(cK*Hi) real bet ! 2*(K-Hi) integer n1,n2 ! see above integer k,layer,index,isub,nc,layk,lay,i,j do nc=1,ncat rcKH=sqrt(cK*Hi(nc)) Hmean(nc)=Hi(nc)+rcKH bet=2*(cK-Hi(nc)) if (row_flag(nc)) then ! ice from category nc is ridging into a distribution from n1 to n2 n1=nc ! first guess n2=n1 ! first guess do k=nc+1,ncat if (2*Hi(nc).ge.hstar(k-1)) n1=k enddo if (n1.eq.ncat) then ! case when all ice ridges into thickest cat M(nc,n1)=Hi(nc)/Hmean(nc) N(nc,n1)=Hi(nc) HN(nc,n1)=Hi(nc)*Hmean(nc) elseif (2*rcKH.le.hstar(n1)) then ! case when all ice ridges into just one category M(nc,n1)=Hi(nc)/Hmean(nc) N(nc,n1)=Hi(nc) HN(nc,n1)=Hi(nc)*Hmean(nc) else n2=ncat do k=ncat-1,n1+1,-1 if (2*rcKH.lt.hstar(k)) n2=k enddo M(nc,n1)=(hstar(n1)-2*Hi(nc))/bet N(nc,n1)=M(nc,n1)*(Hi(nc)+hstar(n1)/2.) HN(nc,n1)=N(nc,n1)*(Hi(nc)+hstar(n1)/2.) M(nc,n2)=(2.*rcKH-hstar(n2-1))/bet N(nc,n2)=M(nc,n2)*(hstar(n2-1)/2.+rcKH) HN(nc,n2)=N(nc,n2)*(hstar(n2-1)/2.+rcKH) do k=n1+1,n2-1 M(nc,k)=(hstar(k)-hstar(k-1))/bet N(nc,k)=M(nc,k)*(hstar(k)+hstar(k-1))/2. HN(nc,k)=N(nc,k)*(hstar(k)+hstar(k-1))/2. enddo endif endif enddo return end function rpress(heffn,A0,An,HN,M) ! Compute the ice strength based on Rothrock, 1975 and also see FH95 ! does not make sense to do this unless the ice that participates in ! ridging is well resolved -- must have about 5 categories, 10 would ! be better ! M(nc,k) is the fraction of ice that participates in ridging ! from category nc that goes into category k ! n1 is the smallest k with nonzero fraction ! n2 is the largest k with nonzero fraction ! the ice that participates has thickness Hi(nc) ! and it ridges up to linear distribution between ! 2*Hi(nc) and 2*sqrt(cK*Hi(nc)) implicit none include "cpts.h" real rpress real heffn(ncat) real An(ncat) real W(0:ncat) ! ridging for each category real Wa(0:ncat) ! the area that participates real M(ncat,ncat) ! area fraction of ice from i into j real HN(ncat,ncat) ! H*volume fraction of ice from i into j real A0 integer n1,n2 ! see above integer k,nc,i,j real Hi(ncat),Hs(ncat) real peterm, frterm ! ideally cpe should be a function of density (a very sensitive one) ! see FH95, I set it equal to a number so you are more likely to get ! a reasonable ice strength. ! It has units (kg/m**3)*(m/s**2) = (kg/m**2/s) rpress=0. # if defined O_ice_cpts_roth_press if (A0.ge.gstar) return do nc=1,ncat if ((An(nc).gt.TINY) .and. (heffn(nc).gt.0.)) then Hi(nc)=heffn(nc)/An(nc) else Hi(nc)=hstar(nc-1) endif enddo ! strictly speaking, should recompute the M and HN matrices here, ! but they would be only very slightly different from what they were ! before ridging and it does not affect the mass conservation call ridging_mode(A0,An,W,Wa,M) peterm=0. do nc=1,ncat peterm=peterm-Wa(nc)*Hi(nc)*Hi(nc) do k=1,nc peterm=peterm+Wa(k)*HN(k,nc) enddo enddo peterm=max(0.,cpe*peterm) rpress=peterm*17. # endif return end subroutine movedown(frzpt,heff1,hseff1,A1,hsnow1,hice1,h1min,Es1, $ ts1,nn1,heff2,hseff2,A2,hsnow2,hice2,h2min,Es2,ts2,nn2,E1,E2) ! hice and hsnow are outputs (not needed for input) implicit none include "cpts.h" real heff1,hseff1,A1,hsnow1,hice1,h1min,Es1,E1(NMAX),ts1 real heff2,hseff2,A2,hsnow2,hice2,h2min,Es2,E2(NMAX),ts2 real frzpt integer nn1,nn2,M,layer,index,isub real x(NMAX) real Atemp Atemp=A1+A2 if (Atemp.le.0.0) return heff1=heff1+heff2 hseff1=hseff1+hseff2 ts1=ts1*A1+ts2*A2 A1=Atemp hsnow1=hseff1/A1 hice1=heff1/A1 ts1=ts1/A1 Es1=Es1+Es2 M=nn2/nn1 ! combine the N2 layers of energy in category 2 ! into N1 portions do 1000 layer=1,nn1 x(layer)=0. do 1010 isub=1,M index=(layer-1)*M+isub x(layer)=x(layer)+E2(index) 1010 continue 1000 continue ! sum the energies do layer=1,nn1 E1(layer)=E1(layer)+x(layer) enddo call zerocat(heff2,hseff2,A2,hsnow2,hice2,h2min, $ Es2,E2,ts2,frzpt,nn2) return end subroutine moveup(frzpt,heff1,hseff1,A1,hsnow1,hice1,h1min,Es1, $ ts1,nn1,heff2,hseff2,A2,hsnow2,hice2,h2min,Es2,ts2,nn2,E1,E2) ! hice and hsnow are outputs (not needed for input) implicit none include "cpts.h" real heff1,hseff1,A1,hsnow1,hice1,h1min,Es1,E1(NMAX),ts1 real heff2,hseff2,A2,hsnow2,hice2,h2min,Es2,E2(NMAX),ts2 real frzpt integer nn1,nn2,M,layer,index,isub real x(NMAX) real Atemp Atemp=A1+A2 if (Atemp.le.0.0) return heff2=heff1+heff2 hseff2=hseff1+hseff2 ts2=ts1*A1+ts2*A2 A2=Atemp hsnow2=hseff2/A2 hice2=heff2/A2 ts2=ts2/A2 Es2=Es1+Es2 M=nn2/nn1 ! move energy of nn1 layers from 1 into the nn2 of 2 do index=1,nn1 do isub =1,M layer=M*(index-1)+isub x(layer)=E1(index)/M enddo enddo ! sum the energies do layer=1,nn2 E2(layer)=E2(layer)+x(layer) enddo call zerocat(heff1,hseff1,A1,hsnow1,hice1,h1min,Es1,E1,ts1,frzpt, $ nn1) return end subroutine zerocat(heff,hseff,A,hsnow,hice,hmin,Es,E,ts,frzpt,nng) implicit none include "cpts.h" real heff,hseff,A,hsnow,hice,hmin,Es,E(NMAX) real ts, frzpt integer nng,layer heff=0.0 hseff=0.0 A=0.0 hsnow=0.0 hice=hmin Es=0.0 do layer=1,nng E(layer)=0.0 enddo ts=frzpt return end subroutine thermo (is, ie, js, je) ! heat budget ! compute fluxes, temp and thickness over open water and ice ! flux is positive toward the atm/ice or atm/ocean surface even if ! the flux is below the surface implicit none include "size.h" include "param.h" include "pconst.h" include "stdunits.h" include "csbc.h" include "cembm.h" include "atm.h" include "cpts.h" include "ice.h" include "thermo.h" include "coord.h" include "grdvar.h" real tocn ! surface ocean temperature in Celsius real tair ! surface air temperature in celcius real qair ! surface air humidity real ug ! geostrophic wind speed real evp(0:ncat) ! evapourative water flux for each category real wff ! ice-ocean water flux for each category real dswr(0:ncat) ! downward shortwave for each category real ultnt(0:ncat) ! upward latent heat for each category real usens(0:ncat) ! upward sensible heat for each category real ulwr(0:ncat) ! upward longwave for each category real Io ! solar transmitted through top surface real Ib ! solar transmitted through bottom surface real absorb ! sum of Ib*A for each category real Focean ! net flux into ocean from atmosphere real Flwatm ! downward longwave from atmosphere (not net) real Fbot ! flux from ocean due to thermal relaxation real Flat ! flux from ocean due to lateral melt real Fice ! net flux into ice from atmosphere real Fneti ! Fice + conductive flux real Fx ! adjusted Fbot for when ice melts away real condb ! conductive flux in ice at bottom surface real A0 ! open water fraction real Fsh_new ! solar incident on new ice (over open water) real Io_new ! solar transmitted through top of new ice real hnew ! thickness of new ice real Tnew ! surface temperature of new ice real hi(ncat) ! ice thickness for each category real hs(ncat) ! snow thickness for each category real En(1:NMAX,ncat)! energy of melt of each layer and category real qi(1:NMAX,ncat)! like Ei but per unit volume real Ti(0:NMAX,ncat)! Temperature of each layer and category (C) real Tbot(ncat) ! Temperature of ice bottom and category (C) real delhib ! ice thickness change at bottom real delhit ! ice thickness change (no sublimation) real delhs ! snow thickness change (no sublimation) real subi ! ice thickness change from sublimation real subs ! snow thickness change from sublimation real delf ! ice thickness change from flooding real delfs ! snow thickness change from flooding integer i,j,l,layer,nc,nr ! counters real sblmt ! heat to ice-snow from sublimation (J/m**2) real htatm ! heat to ice-snow from atm (J/m**2) real wtatm ! water to ice-snow from atm (m/s) real htocn ! heat to ice-snow from ocn (J/m**2) real wtocn ! waterto ice-snow from ocn (m/s) real einit ! initial energy in ice/snow (J/m**2) real winit ! initial water equiv. in ice/snow (m) real efinl ! final energy in ice/snow (J/m**2) real wfinl ! final water equiv. in ice/snow (m) real ediff ! energy difference in ice/snow (J/m**2) real wdiff ! water difference in ice/snow (m) logical alarm ! vars for small stuff check and computing Ti, hi, hs and albedo real Aleft, tio, Enet ! lateral melt vars real Esum ! sum of energy in ice and snow (J/m**2) real mrate ! lateral melt rate (m/s) real arate ! rate ice concentration is melting laterally (s) real DAice ! change in concentration from lateral melt real Vfrac ! ratio of volume after to before melting real Dhlatm, Dslatm, sla, fracsnow ! redistribute the ice categories vars real h0prime,A0prime,h1prime,A1prime,E0,Ad0 real energ,Tmp,epsilon real ca, as, aicet, coaloff, C2K integer ie, iem1, is, isp1, je, jem1, js, jsp1 ! dummy energy of melting in snow for calls to movedown/up real esdum1, esdum2 !----------------------------------------------------------------------- sla = zw(1)*secday/dampice/2.389e-8 isp1 = is + 1 iem1 = ie - 1 jsp1 = js + 1 jem1 = je - 1 C2K = 273.15 do j=jsp1,jem1 do i=isp1,iem1 if (tmsk(i,j) .ge. 0.5) then Io_new=0.0 ! adjust albedo for new ice growth # if defined O_sulphate_data_transient fsh_new = solins(i,j)*sbc(i,j,iaca)*pass & *max(c0, sbc(i,j,isca) - sulph(i,j,2)) # else fsh_new = solins(i,j)*sbc(i,j,iaca)*pass*sbc(i,j,isca) # endif tocn = sbc(i,j,isst) tair = at(i,j,2,isat) # if defined O_sealev || defined O_sealev_data & - elev_sealev(i,j)*rlapse # endif qair = at(i,j,2,ishum) Fbot = -sla*(frzpt(i,j) - tocn) ug = sbc(i,j,iws) Flwatm = esatm*(tair + C2K)**4 Focean = dnswr(i,j) - uplwr(i,j) - upsens(i,j) - upltnt(i,j) dswr(0) = dnswr(i,j) ultnt(0) = upltnt(i,j) usens(0) = upsens(i,j) ulwr(0) = uplwr(i,j) evp(0) = evap(i,j) !----------------------------------------------------------------------- ! make simple energy of melt matrix for speed !----------------------------------------------------------------------- do nc=1,ncat do layer=1,nilay(nc) En(layer,nc)=E(i,j,idx,layer1(nc)+layer-1) enddo enddo !----------------------------------------------------------------------- ! initial energy and water diagnostics ! could be eliminated to save time !----------------------------------------------------------------------- einit = 0. winit = 0. do nc=1,ncat do layer=1,nilay(nc) einit=einit+En(layer,nc) enddo einit = einit-hseff(i,j,idx,nc)*rflsno winit = winit+(heff(i,j,idx,nc)*rhoice $ +hseff(i,j,idx,nc)*rhosno) enddo # if defined O_ice_cpts3 || defined O_ice_cpts5 || defined O_ice_cpts10 !----------------------------------------------------------------------- ! get rid of small area by moving to other category (if possible) ! does not fix negative ice area/volumes (that comes later) ! push small bit of ice to thicker category ! hi and hs are outputs of moveup (not inputs) !----------------------------------------------------------------------- do nc=1,(ncat-1) if ((A(i,j,idx,nc) .le. ASMALL(nc)) $ .and.(A(i,j,idx,nc) .gt. 0.0) ) then A(i,j,idx,nc)=heff(i,j,idx,nc)/hstar(nc) call moveup(frzpt(i,j),heff(i,j,idx,nc), $ hseff(i,j,idx,nc), $ A(i,j,idx,nc),hs(nc),hi(nc),hstar(nc-1), $ esdum1,Ts(i,j,idx,nc),nilay(nc), $ heff(i,j,idx,nc+1),hseff(i,j,idx,nc+1), $ A(i,j,idx,nc+1),hs(nc+1),hi(nc+1),hstar(nc), $ esdum2,Ts(i,j,idx,nc+1),nilay(nc+1), $ En(1,nc),En(1,nc+1)) endif enddo ! push small bit of ice to thinner category do nc=ncat,2,-1 if ((A(i,j,idx,nc) .le. ASMALL(nc)) $ .and.(A(i,j,idx,nc) .gt. 0.0) ) then ! reshape the thicker ice to thickness of thinner category ! while making certain not to make area > 1 Aleft=1.0 do nr=1,(nc-1) Aleft=Aleft-A(i,j,idx,nr) enddo A(i,j,idx,nc)=min(heff(i,j,idx,nc)/hstar(nc-1),Aleft) call movedown(frzpt(i,j),heff(i,j,idx,nc-1), $ hseff(i,j,idx,nc-1), $ A(i,j,idx,nc-1),hs(nc-1),hi(nc-1),hstar(nc-2), $ esdum2,Ts(i,j,idx,nc-1),nilay(nc-1), $ heff(i,j,idx,nc),hseff(i,j,idx,nc),A(i,j,idx,nc), $ hs(nc),hi(nc),hstar(nc-1), $ esdum1,Ts(i,j,idx,nc),nilay(nc), $ En(1,nc-1),En(1,nc)) endif enddo # endif !----------------------------------------------------------------------- ! get rid of small open water area !----------------------------------------------------------------------- A0=1.0 do nc=1,ncat A0=A0-max(A(i,j,idx,nc),0.0) enddo if (A0.lt.A0SMALL) then ! put tiny open water areas out of misery by reshaping ! the ice starting with the thin category if (A0.ge.0.) then do nc=1,ncat if (A(i,j,idx,nc).gt.0.) then A(i,j,idx,nc)=A(i,j,idx,nc)+A0 A0=0.0 endif enddo else ! if the open water fraction is negative then reduce the area ! of the category with the thin ice first nr=1 do nc=2,ncat if ((A(i,j,idx,nr)+A0).lt.ASMALL(nr)) nr=nc enddo A(i,j,idx,nr)=A(i,j,idx,nr)+A0 endif endif !----------------------------------------------------------------------- ! fix negative ice/snow area/volumes and compute hi, hs, ti ! if ice volume is too small ! energy and water in ice-snow will be transferred into open water ! negative ice or snow volume will increase the temperature ! of the open water !----------------------------------------------------------------------- htatm = 0. wtatm = 0. htocn = 0. wtocn = 0. Aicet=0.0 do nc=1,ncat if ((A(i,j,idx,nc) .le. ASMALL(nc)) $ .or.(heff(i,j,idx,nc) .le. 0.0)) then Enet=0.0 do layer=1,nilay(nc) Enet=Enet-En(layer,nc) enddo htocn=htocn+(Enet+hseff(i,j,idx,nc)*rflsno) wtocn=wtocn+(heff(i,j,idx,nc)*rhoice $ +hseff(i,j,idx,nc)*rhosno) groice(i,j,nc)=-heff(i,j,idx,nc) A(i,j,idx,nc)=0.0 heff(i,j,idx,nc)=0.0 hseff(i,j,idx,nc)=0.0 Ts(i,j,idx,nc)=frzpt(i,j) do layer=0,nilay(nc) Ti(layer,nc)=frzpt(i,j) enddo hi(nc)=HSTAR(nc-1) hs(nc)=0.0 else groice(i,j,nc)=0. hi(nc)=heff(i,j,idx,nc)/A(i,j,idx,nc) hi(nc)=max(HSTAR(nc-1),hi(nc)) A(i,j,idx,nc)=heff(i,j,idx,nc)/hi(nc) do layer=1,nilay(nc) qi(layer,nc)=En(layer,nc)*nilay(nc)/heff(i,j,idx,nc) enddo call getTmp(Ti(1,nc),qi(1,nc),saltz(1,nc),nilay(nc)) Ti(0,nc)=Ti(1,nc) ! need a starting value for tstm if (hseff(i,j,idx,nc) .le. HSSTAR ) then htocn=htocn+hseff(i,j,idx,nc)*rflsno wtocn=wtocn+hseff(i,j,idx,nc)*rhosno hseff(i,j,idx,nc)=0.0 hs(nc)=0.0 else hs(nc)=hseff(i,j,idx,nc)/A(i,j,idx,nc) endif endif Aicet=Aicet+A(i,j,idx,nc) enddo !----------------------------------------------------------------------- ! finally see if aicet is greater than 1, it never should ! be after all the trouble above !----------------------------------------------------------------------- if (Aicet.gt.(1.0+TINY)) then # if defined O_ice_cpts_warnings print*,'warning too much ice area by',Aicet print*,nats,i,j,(A(i,j,idx,nc),nc=1,ncat) # endif A(i,j,idx,ncat)=min(1.0,A(i,j,idx,ncat)) Aleft=1.0-A(i,j,idx,ncat) do nc=(ncat-1),1,-1 A(i,j,idx,nc)=min(A(i,j,idx,nc),Aleft) Aleft=Aleft-A(i,j,idx,nc) enddo endif A0=1.0 do nc=1,ncat A0=A0-A(i,j,idx,nc) enddo !----------------------------------------------------------------------- ! heat conduction in ice and change thickness from growth/melt ! this is only section that needs Ti !----------------------------------------------------------------------- sblmt = 0. do nc=1,ncat if (A(i,j,idx,nc).ge.TINY) then ! if snow is less than 25 cm linearly reduce to ice albedo as = min(hseff(i,j,idx,nc)*0.04, c1) ca = ice_calb*(c1 - as) + sno_calb*as # if defined O_sulphate_data_transient dswr = solins(i,j)*sbc(i,j,iaca)*pass & *max(c0, ca - sulph(i,j,2)) # else dswr = solins(i,j)*sbc(i,j,iaca)*pass*ca # endif delhit=0. delhib=0. delhs=0. subi=0. subs=0. delf=0. delfs=0. fracsnow = hs(nc)/(hs(nc) + 0.1*centi) ! ~ snow param Io=dswr(nc)*0.3*(1-fracsnow) ! solar penetrating surf call tstm(tmelz(1,nc),saltz(1,nc), $ ts(i,j,idx,nc),ti(0,nc),tocn,tbot(nc),frzpt(i,j), $ hi(nc),hs(nc),tair,Qair,Flwatm, $ Io,Ib,ug,A(i,j,idx,nc), $ dswr(nc),ultnt(nc),usens(nc),ulwr(nc), $ nilay(nc),Fneti,condb,i,j) call dh(saltz(1,nc),Fx,Fbot,Fneti,ultnt(nc),condb, $ tocn,ts(i,j,idx,nc),ti(0,nc),tbot(nc),frzpt(i,j), $ qi(1,nc),hi(nc),hs(nc),delhib, $ subi,subs, $ delhit,delhs,delf,delfs,nilay(nc)) ! note that Ti is no longer current because of layer adjustment hi(nc)=hi(nc)+delhit+delhib+subi+delf hs(nc)=hs(nc)+delhs+subs-delfs groice(i,j,nc)=groice(i,j,nc)+(delhit+delhib) $ *A(i,j,idx,nc) ! atm gains lwe from sublimation evp(nc) = -(rhosno*subs+rhoice*subi) ! ocn gains lwe from melt/growth # if defined O_convect_brine if (delhib .le. 0.) then wff = -(rhosno*delhs+rhoice*(delhit+delhib)) else wff = -(rhosno*delhs + rhoice*delhit) if (addflxa) then cbf(i,j,nc) = cbf(i,j,nc) $ - A(i,j,idx,nc)*rhoice*delhib cba(i,j,nc) = cba(i,j,nc) + A(i,j,idx,nc)*dtau endif endif # else wff = -(rhosno*delhs+rhoice*(delhit+delhib)) # endif ! heat flux from atmosphere to ice Fice = dswr(nc)-ultnt(nc)-usens(nc)-ulwr(nc) htatm = htatm + Fice * dtau * A(i,j,idx,nc) sblmt = sblmt + (ultnt(nc) * dtau + $ (rvlsno*subs+rvlice*subi) ) * A(i,j,idx,nc) wtatm = wtatm + evp(nc) * A(i,j,idx,nc) htocn = htocn + (Fx-Ib)* dtau * A(i,j,idx,nc) wtocn = wtocn + wff * A(i,j,idx,nc) else ! give a reasonable value for ts, even if there is no ice ts(i,j,idx,nc)=frzpt(i,j) dswr(nc) = 0. ultnt(nc) = 0. usens(nc) = 0. ulwr(nc) = 0. evp(nc) = 0. endif enddo ! loop over bins dnswr(i,j) = 0. upltnt(i,j) = 0. upsens(i,j) = 0. uplwr(i,j) = 0. evap(i,j) = 0. do nc=1,ncat dnswr(i,j) = dnswr(i,j) + A(i,j,idx,nc)*dswr(nc) upltnt(i,j) = upltnt(i,j) + A(i,j,idx,nc)*ultnt(nc) upsens(i,j) = upsens(i,j) + A(i,j,idx,nc)*usens(nc) uplwr(i,j) = uplwr(i,j) + A(i,j,idx,nc)*ulwr(nc) evap(i,j) = evap(i,j) + A(i,j,idx,nc)*evp(nc)/dtau enddo !----------------------------------------------------------------------- ! recalc heff hseff and energy of melting for each layer !----------------------------------------------------------------------- do nc=1,ncat heff(I,J,idx,nc)=hi(nc)*A(I,J,idx,nc) hseff(I,J,idx,nc)=hs(nc)*A(I,J,idx,nc) do layer=1,nilay(nc) En(layer,nc)=qi(layer,nc)*heff(i,j,idx,nc)/nilay(nc) enddo enddo !----------------------------------------------------------------------- ! ice growth in lead and upper layer water temp !----------------------------------------------------------------------- hnew=0.0 Tnew=frzpt(i,j) if (A0.le.TINY) then ! do nothing elseif ((Focean+Fbot).lt.0.0) then ! grow new ice call grownew(fsh_new,Io_new,Flwatm,Focean,Fbot, $ hnew,Tnew,tair,Qair,ug, $ dtau,fice,frzpt(i,j),evp(0), $ dswr(0),ultnt(0),usens(0),ulwr(0)) Focean=0.5*(Focean+Fice) ! avg. flx w/ and w/o new ice htatm = htatm + Focean*A0*dtau htocn = htocn + A0*Fbot*dtau # if defined O_convect_brine if (hnew .le. 0.) then wtocn = wtocn - A0*hnew*rhoice else if (addflxa) then cbf(i,j,0) = cbf(i,j,0) - A0*hnew*rhoice cba(i,j,0) = cba(i,j,0) + A0*dtau endif endif # else wtocn = wtocn - A0*hnew*rhoice # endif groice(i,j,1)=groice(i,j,1)+hnew*A0 elseif ((A0.lt.1.0)) then ! lateral ice melt ! melt ice lateral mrate=m1*(max(tocn-frzpt(i,j),0.0))**m2 arate=pi_eta*mrate/wclead arate=min(arate,1./dtau) ! concentration that melts laterally per ice conc DAice=dtau*arate Vfrac=1.0-DAice Flat=0. Dhlatm=0. Dslatm=0. do nc=1,ncat if (A(i,j,idx,nc).ge.TINY) then ! diagnostic for heat flux, negative when lateral melt Esum=0. do layer=1,nilay(nc) Esum=Esum+En(layer,nc) enddo Esum=Esum-rflsno*hseff(i,j,idx,nc) Flat=Flat+Esum*arate ! diagnostic for water flux, positive when lateral melt Dhlatm=Dhlatm+heff(i,j,idx,nc) Dslatm=Dslatm+hseff(i,j,idx,nc) ! update the ice state heff(i,j,idx,nc)=heff(i,j,idx,nc)*Vfrac hseff(i,j,idx,nc)=hseff(i,j,idx,nc)*Vfrac A(i,j,idx,nc)=A(i,j,idx,nc)*Vfrac do layer=1,nilay(nc) En(layer,nc)=En(layer,nc)*Vfrac enddo groice(i,j,nc)=groice(i,j,nc)-heff(i,j,idx,nc)*DAice endif enddo Dhlatm=Dhlatm*DAice Dslatm=Dslatm*DAice htocn = htocn - Flat*dtau wtocn = wtocn + (Dhlatm*rhoice+Dslatm*rhosno) endif !----------------------------------------------------------------------- ! budgets: ice-ocn fluxes of heat/water for coupled model ! subtract htatm and add wtatm to compensate ! for adding them back later in embm !----------------------------------------------------------------------- ! do not move this block if (addflxa) then flux(i,j,isat) = flux(i,j,isat) - (htocn + htatm) flux(i,j,ishum) = flux(i,j,ishum) + (wtatm + wtocn) endif dnswr(i,j) = dnswr(i,j) + dswr(0)*A0 upltnt(i,j) = upltnt(i,j) + ultnt(0)*A0 upsens(i,j) = upsens(i,j) + usens(0)*A0 uplwr(i,j) = uplwr(i,j) + ulwr(0)*A0 evap(i,j) = evap(i,j) + evp(0)*A0 !----------------------------------------------------------------------- ! add lead ice to first category !----------------------------------------------------------------------- ! compute the energy in the new ice, ! divide it into nilay(1) equal portions A0=1 ! must recompute after meltlat do nc=1,ncat A0=A0-A(i,j,idx,nc) enddo H0prime=max(hstar(0),hnew) ! new ice h, made >=HOSTAR A0prime=A0*hnew/H0prime ! area covered by H0prime ice A1prime=A0prime+A(i,j,idx,1) h1prime=0.0 if (A1prime.gt.0.0) then h1prime=(hnew*A0+heff(i,j,idx,1))/A1prime Ts(i,j,idx,1)=(Ts(i,j,idx,1)*A(i,j,idx,1)+Tnew*A0prime) $ /A1prime A(i,j,idx,1)=A1prime hi(1)=h1prime heff(i,j,idx,1)=hi(1)*A(i,j,idx,1) Ad0=A0*hnew/nilay(1) Tmp=0.5*(Tnew+frzpt(i,j)) E0=Ad0*energ(Tmp,salnew) epsilon=Ad0*(energ(Tnew,salnew)-energ(frzpt(i,j),salnew)) epsilon=0.0 E0=E0+epsilon/2. ! sum the energies do layer=1,nilay(1) En(layer,1)=E0+En(layer,1)-epsilon*(layer-0.5)/nilay(1) enddo endif ! finished adding new ice to category 1 ! make sure thickness of category 1 is at least hstar(0) heff(i,j,idx,1)=A(i,j,idx,1)*hi(1) if (hi(1).gt.hstar(0)) then A(i,j,idx,1)=heff(i,j,idx,1)/hi(1) else A(i,j,idx,1)=heff(i,j,idx,1)/hstar(0) hi(1)=hstar(0) endif # if defined O_ice_cpts3 || defined O_ice_cpts5 || defined O_ice_cpts10 !----------------------------------------------------------------------- ! redistribute the ice categories !----------------------------------------------------------------------- ! see if category 1 has outgrown its boundary ! if so, add it to category 2, and so forth do nc=1,(ncat-1) ! only does loop if ncat>1 if (hi(nc).ge.hstar(nc)) then call moveup(frzpt(i,j),heff(i,j,idx,nc), $ hseff(i,j,idx,nc), $ A(i,j,idx,nc),hs(nc),hi(nc),hstar(nc-1), $ esdum1,Ts(i,j,idx,nc),nilay(nc), $ heff(i,j,idx,nc+1),hseff(i,j,idx,nc+1), $ A(i,j,idx,nc+1),hs(nc+1),hi(nc+1),hstar(nc), $ esdum2,Ts(i,j,idx,nc+1),nilay(nc+1),En(1,nc), $ En(1,nc+1)) endif enddo ! see if the thickest category has become too thin (and so forth) do nc=ncat,2,-1 if (hi(nc).lt.hstar(nc-1)) then call movedown(frzpt(i,j),heff(i,j,idx,nc-1), $ hseff(i,j,idx,nc-1), $ A(i,j,idx,nc-1),hs(nc-1),hi(nc-1),hstar(nc-2), $ esdum2,Ts(i,j,idx,nc-1),nilay(nc-1), $ heff(i,j,idx,nc),hseff(i,j,idx,nc),A(i,j,idx,nc), $ hs(nc),hi(nc),hstar(nc-1), $ esdum1,Ts(i,j,idx,nc),nilay(nc),En(1,nc-1), $ En(1,nc)) endif enddo ! cannot hurt to do this again heff(i,j,idx,1)=A(i,j,idx,1)*hi(1) hi(1)=max(hstar(0),hi(1)) A(i,j,idx,1)=heff(i,j,idx,1)/hi(1) # endif !----------------------------------------------------------------------- ! reconstruct 3d energy of melting matrix !----------------------------------------------------------------------- l=1 do nc=1,ncat do layer=1,nilay(nc) E(i,j,idx,l)=En(layer,nc) l=l+1 enddo enddo !----------------------------------------------------------------------- ! final energy and water diagnostic ! could be eliminated to save time !----------------------------------------------------------------------- efinl = 0. wfinl = 0. do nc = 1,ncat do layer = 1,nilay(nc) efinl = efinl+En(layer,nc) enddo efinl = efinl-hseff(i,j,idx,nc)*rflsno wfinl = wfinl+(heff(i,j,idx,nc)*rhoice $ +hseff(i,j,idx,nc)*rhosno) enddo ediff = (efinl-einit)-(htatm+htocn+sblmt) wdiff = (wfinl-winit)+(wtatm+wtocn) # if defined O_ice_cpts_warnings if (abs(ediff).gt.maxedif) $ print*,'big energy difference in ice model at',i,j, $ ediff*1.e-9, $ (htatm+htocn+sblmt)*1.e-9, (efinl-einit)*1.e-9 # if !defined O_convect_brine if (abs(wdiff).gt.maxwdif) $ print*,'big water difference in ice model at',i,j, $ wdiff,(wtatm+wtocn), (wfinl-winit) # endif # endif endif enddo enddo do nc=1,ncat call embmbc (heff(1,1,idx,nc)) call embmbc (A(1,1,idx,nc)) call embmbc (Ts(1,1,idx,nc)) call embmbc (hseff(1,1,idx,nc)) enddo do layer=1,ntilay call embmbc (E(1,1,idx,layer)) enddo return end subroutine tstm(tmz1d, sal1d, ts,ti,tw,tbot,tfrz, $ h,hs,Tair,qair,Flo, $ Io,Ib,ug,area, $ dswr,ultnt,usens,ulwr, $ ni,F,condb,i,j) implicit none ! This subroutine calculates the evolution of the ice interior and ! surface temperature from the heat equation and surface energy ! balance ! the albedo is fixed for this calculation ! if you think you want it to be computed implicitly, then Iabs ! will have to depend on ts too and that could get nasty ! I think it would be alot of trouble because the albedo affects ! boundary conditions too ! solves the heat equation which is non-linear for saline ice ! by linearizing and then iterating a set of equations. ! Scheme is backward Euler giving a tridiagonal ! set of equations implicit in ti ! (tried crank-nicholson but it behaves poorly when ! thin ice has a weird initial temperature profile. This can happen ! when I redistribute between cats so I always use backward ! Must have a sufficient amount of snow to solve heat equation in ! snow hsmin is the minimum depth of snow in order to solve for ! ti(0) if snow thickness < hsmin then do not change ti(0) ! The number of equations that must be solved by the tridiagonal ! solver depends on whether the surface is melting and whether ! there is snow. Four cases are possible (see variable case below): ! 1 = freezing w/ snow, 2 = freezing w/ no snow, ! 3 = melting w/ snow, and 4 = melting w/ no snow include "cpts.h" include "thermo.h" include "cembm.h" include "tmngr.h" ! input variables integer ni ! number of vertical layers in the ice integer i,j ! grid points real tmz1d(NMAX) ! melting temp of each layer (C) real sal1d(NMAX) ! salinity of each layer (ppt) real tfrz ! freezing temperature of ocean (C) real h,hs ! ice and snow thickness (m) real area ! area of the ice/snow real Tw ! water temp below ice (C) real ug ! geostrophic wind speed (m) real qice,qair ! vapour at ice/snow srfc and in air above real tair ! air temperature (C) real Flo ! long wave down at srfc real dswr ! above srfc net dnwd shortwave, + down (W/m**2) real Io ! solar penetrating top srfc, + down (W/m**2) ! output variables real Tbot ! bottom ice temp (C) real condb ! conductive flux at bottom srfc, + up (W/m**2) real Ib ! solar passing through bottom of ice (W/m**2) real F ! net flux at top srfc including conductive flux ! in ice/snow, + down (W/m**2) real usens ! upward sensible flux, + up (W/m**2) real ultnt ! upward latent flux, + up (W/m**2) real ulwr ! upward longwave flux, + up (W/m**2) ! input/output variables real ts ! srfc temp of snow or ice (C) real ti(0:NMAX) ! snow(0) and ice(1:NMAX) interior temp (C) ! local variables real ts_old ! initial temperature of the ice/snow srfc (C) real ti_old(0:NMAX) ! initial temperature in the ice and snow (C) real tinext(-1:NMAX) ! incremented ts and ti (C) real dti(-1:NMAX) ! incremental changes to ts and ti (C) real Ts_kelv ! srfc temperature in kelvin (K) real z ! vertical coordinate (m) real coltnt,cosens ! bulk coef for latent and sensible heat flux real condt ! conductive flux at top srfc, + down (W/m**2) real Fo ! net flux at top srfc excluding conductive flux ! in ice/snow (F-condt), + down (W/m**2) real dFo_dt ! derivative of Fo wrt temperature (W/m**2/K) real iru ! net upward longwave flux, + up (W/m**2) real dultnt ! derivative of ultnt wrt temperature (W/m**2/K) real ultnti ! latent heat from initial temperature (W/m**2) real dultnti ! derivative of ultnti wrt temperature (W/m**2/K) logical zult ! flag that is true if latent heat is zero real Iabs(NMAX+1) ! solar absorbed in each layer (W/m**2) real absorb ! sum of Iabs (W/m**2) real melts ! the srfc melting temp (either TSMELT or TMELT) real alph, bet ! parameters for maintaining 2nd order accurate ! diff at boundary real tinterf ! snow and ice interf temp (C) ! a,b,c are vectors that describe the diagonal and off-diagonal elements ! of the matrix [A], such that [A] ti = r real a(-1:NMAX),b(-1:NMAX),c(-1:NMAX),d(-1:1),r(-1:NMAX) real ki(0:NMAX+1) ! layer conductivity divided by layer thickness real zeta(0:NMAX) ! the terms in heat equation independent of ti real eta(0:NMAX) ! time step / ice layer thickness / fresh ice cp real cpi(1:NMAX) ! time step / ice layer thickness / fresh ice cp real dh ! ice layer thickness real dt_dh,dt_hs ! time step / ice or snow thickness integer N ! number of equations solved by tridiag solver integer layer ! counter for ice layers integer ipc ! counter for iterations of dti real errit ! the absolute value of the maximum dti integer case ! indicates the case of snow vs. no snow and ! melting vs. freezing real cond2i,dheat logical convrg ! flag that is true if temperature converges logical complt ! flag that is true if a layer melts internally real hsmin parameter ( hsmin = 0.01*centi ) real EXPMAX parameter(EXPMAX=10.) ! ------------------------------------------------------------------------- ! setup helpful parameters ! ------------------------------------------------------------------------- dh = h/(ni + 1.e-20) dt_dh = dtau/(dh + 1.e-20) dt_hs = 0.0 if (hs.gt.hsmin) dt_hs=dtau/(hs + 1.e-20) ts_old = ts do layer = 0,ni ti_old(layer) = ti(layer) enddo tbot = tfrz ! min(tw,TMELT) call conductiv(sal1d,ki,ti,tbot,hs,dh,ni) ! the solar radiation absorbed internally z = 0.0 Iabs(1) = 1.0 Iabs(1) = 1.0 do layer = 1,ni z = z+dh if (z.lt.0.10*centi) then ! no absorption in the first 10 cm Iabs(layer+1) = 1.0 elseif (z.lt.0.20*centi) then ! ramp up the absorption Iabs(layer+1) = 1.0-(1.0-exp(-kappa*0.20*centi))* $ (z/(0.10*centi)-1.0)**2 else Iabs(layer+1) = exp(-kappa*z) endif Iabs(layer) = Io*(Iabs(layer)-Iabs(layer+1)) enddo coltnt = rslatm*dalt_i*ug cosens = 0.94*rcpatm*dalt_i*ug qice = QS1*exp(min(AI*ts/(ts-BI+1.e-20),EXPMAX)) if (qice.lt.qair) then zult = .true. qice = qair dultnti = 0. ultnti = 0. else zult = .false. dultnti = -coltnt*qice*ai*bi/(ts-bi+1.e-20)**2 ultnti = coltnt*(qice-qair) endif ultnt = ultnti dultnt = dultnti usens = cosens*(ts-tair) ts_kelv = ts+TFFRESH iru = ESICE*(Ts_kelv)**4 dFo_dt = -cosens-dultnt-4*ESICE*Ts_kelv**3 Fo = dswr-Io+Flo-ultnt-usens-iru if (hs.gt.hsmin) then alph = 2.0*(2.*hs + dh)/(hs+dh+1.e-20) bet = -2.0*hs*hs/(2.*hs+dh+1.e-20)/(hs+dh+1.e-20) melts = tsmelt else alph = 3. bet = -1./3. melts = tmelt endif ts_old = ts ts = min(ts,melts-0.01) ! absolutely necessary ! ------------------------------------------------------------------------- ! beginning of iterative procedure ! ------------------------------------------------------------------------- ipc = 1 1000 continue ! ------------------------------------------------------------------------- ! setup terms that depend on the iterated temperature ! ------------------------------------------------------------------------- call get_Cpice(sal1d,tmz1d,Cpi,Ti(1),Ti_old(1),ni) eta(0) = 0.0 if (hs.gt.hsmin) eta(0) = dtau/(hs+1.e-20) do layer = 1,ni eta(layer) = dt_dh/(cpi(layer)+1.e-20) enddo zeta(0) = rcpsno*Ti_old(0) do layer = 1,ni zeta(layer) = Ti_old(layer)+eta(layer)*Iabs(layer) enddo if (ts.lt.melts-tiny) then ! solve heat equation for ice/snow using flux BC if (hs.gt.hsmin) then ! ------------------------------------------------------------------------- case = 1 ! case of freezing with snow layer ! ------------------------------------------------------------------------- call getabc(a,b,c,r,ti,tbot,zeta,ki,eta,ni,1) a(0) = -eta(0)*ki(0)*(alph+bet) c(0) = eta(0)*(bet*ki(0)-ki(1)) b(0) = rcpsno+eta(0)*(alph*ki(0)+ki(1)) r(0) = zeta(0) a(-1) = 0. c(-1) = ki(0)*alph d(-1) = ki(0)*bet b(-1) = dFo_dt-c(-1)-d(-1) r(-1) = -Fo+dFo_dt*ts_old ! row operation to get rid of d(-1) b(-1) = c(0)*b(-1)-d(-1)*a(0) c(-1) = c(0)*c(-1)-d(-1)*b(0) r(-1) = c(0)*r(-1)-d(-1)*r(0) N = ni+2 call tridag(a(-1),b(-1),c(-1),r(-1),tinext(-1),N) dti(-1) = tinext(-1)-ts ts = tinext(-1) do layer = 0,ni dti(layer) = tinext(layer)-ti(layer) ti(layer) = tinext(layer) enddo ! write(*,2000) 'I. iterations',ipc*1.0,ts,(ti(layer),layer = 0,4) ! $ ,(dti(layer),layer = -1,4) else ! ------------------------------------------------------------------------- case = 2 ! case of freezing with no snow layer ! ------------------------------------------------------------------------- call getabc(a,b,c,r,ti,tbot,zeta,ki,eta,ni,2) a(1) = -eta(1)*ki(1)*(alph+bet) c(1) = -eta(1)*(ki(2)-bet*ki(1)) b(1) = 1+eta(1)*(ki(2)+alph*ki(1)) r(1) = zeta(1) a(0) = 0. c(0) = ki(1)*alph d(0) = ki(1)*bet b(0) = dFo_dt-c(0)-d(0) r(0) = -Fo+dFo_dt*ts_old ! row operation to get rid of d(0) b(0) = c(1)*b(0)-d(0)*a(1) c(0) = c(1)*c(0)-d(0)*b(1) r(0) = c(1)*r(0)-d(0)*r(1) N = ni+1 call tridag(a(0),b(0),c(0),r(0),tinext(0),N) dti(-1) = 0. dti(0) = tinext(0)-ts ts = tinext(0) do layer = 1,ni dti(layer) = tinext(layer)-ti(layer) ti(layer) = tinext(layer) enddo ti(0) = ts endif ts = min(ts,melts) else ts = melts if (hs.gt.hsmin) then ! ------------------------------------------------------------------------- case = 3 ! case of melting with snow layer ! ------------------------------------------------------------------------- a(0) = -eta(0)*ki(0)*(alph+bet) c(0) = eta(0)*(bet*ki(0)-ki(1)) b(0) = rcpsno+eta(0)*(alph*ki(0)+ki(1)) r(0) = zeta(0)-a(0)*ts a(0) = 0.0 call getabc(a,b,c,r,ti,tbot,zeta,ki,eta,ni,1) N = ni+1 call tridag(a(0),b(0),c(0),r(0),tinext(0),N) dti(-1) = 0. do layer = 0,ni dti(layer) = tinext(layer)-ti(layer) ti(layer) = tinext(layer) enddo ! write(*,2000)'III. T = ', ipc*1.0,(ti(layer),layer = 0,4) ! $ ,(dti(layer),layer = 1,4) else ! ------------------------------------------------------------------------- case = 4 ! case of melting with no snow layer ! ------------------------------------------------------------------------- call getabc(a,b,c,r,ti,tbot,zeta,ki,eta,ni,2) a(1) = -eta(1)*ki(1)*(alph+bet) c(1) = -eta(1)*(ki(2)-bet*ki(1)) b(1) = 1+eta(1)*(ki(2)+alph*ki(1)) r(1) = zeta(1)-a(1)*Ts a(1) = 0.0 N = ni call tridag(a(1),b(1),c(1),r(1),tinext(1),N) dti(-1) = 0. dti(0) = 0. do layer = 1,ni dti(layer) = tinext(layer)-ti(layer) ti(layer) = tinext(layer) enddo ti(0) = ts endif endif ! ------------------------------------------------------------------------- ! end iterative procedure, see if need to reiterate ! ------------------------------------------------------------------------- ! ------------------------------------------------------------------------- ! make certain that latent heat is always positive ! be sure to iterate one more time if latent heat becomes ! less than zero during the iteration (and was positive before) ! ------------------------------------------------------------------------- ultnt = ultnti + dultnti*(ts-ts_old) if (ultnt.lt.0. .and. .not.zult) then zult = .true. ultnt = 0. ultnti = 0. dultnti = 0. dFo_dt = -cosens-4*ESICE*Ts_kelv**3 Fo = dswr-Io+Flo-usens-iru ipc = ipc+1 go to 1000 endif errit = 0.0 do layer = -1,ni errit = max(errit,abs(dti(layer))) enddo if ((errit.lt.errmax).or.(ipc.gt.20)) go to 6000 ! done iterating ipc = ipc+1 go to 1000 ! to beginning of iterative routine ! ------------------------------------------------------------------------- ! continue from here when done iterative ! begin with error checking ! ------------------------------------------------------------------------- 6000 continue complt = .false. do layer = 1,ni if (ti(layer).gt.tmz1d(layer)) complt = .true. enddo convrg = .true. if (errit.gt.errmax) then if (errit.gt.0.005) then print*,'WARNING No CONVERGENCE in icemodel',errit,i,j convrg = .false. else if (errit.gt.5.e-4) $ print*,'WARNING POOR CONVERGENCE in icemodel',errit,i,j endif endif if (complt .or. .not.convrg) then if (complt .and. convrg) $ print*,'WARNING CONVERGES to ti>TMELT in icemodel',i,j if (complt .and. .not.convrg) $ print*,'& final iteration has ti>TMELT' print*,'diagnostics useful for debugging' print*,area,h,hs,dswr,Flo print*,'tstmnew computed T(z):' print*,ts,(ti(layer),layer = 0,ni) print*,'tstmnew started with T(z):' print*,ts_old,(ti_old(layer),layer = 0,ni),tbot ts = min(ts_old,melts) tinterf=(h*kappas*Ts+hs*kappai*tbot)/(h*kappas+hs*kappai+1.e-20) Ti(0) = 0.5*(Ts+tinterf) do layer = 1,ni Ti(layer) = tinterf+(layer-0.5)*(tbot-tinterf)/(ni+1.e-20) enddo print*,'setting the temperature profile to be linear T(z):' print*,ts,(ti(layer),layer = 0,ni) elseif (ti(0).gt.(TSMELT+TINY)) then print*,'WARNING SNOW has ti>TSMELT in icemodel',i,j print*,' likely no problem if hs or area is tiny' print*,' area=',area,' hs=',hs print*,'setting snow temp to be TSMELT' ti(0) = tsmelt endif ! ------------------------------------------------------------------------- ! finish up by updating the surface fluxes, etc. ! ------------------------------------------------------------------------- if (hs.gt.hsmin) then condt = ki(0)*(alph*(ti(0)-ts)+bet*(ti(1)-ts)) else condt = ki(1)*(alph*(ti(1)-ts)+bet*(ti(2)-ts)) endif condb = ki(ni+1)*( 3.*(tbot-ti(ni)) - (tbot-ti(ni-1))/3.0 ) iru = iru + (4*ESICE*(ts_old+tffresh)**3)*(Ts-Ts_old) ulwr = -Flo + iru usens = usens + cosens*(Ts-Ts_old) Fo = Fo+dFo_dt*(Ts-Ts_old) F = Fo + condt ! in the rare event that F<0 set it equal to 0, adjust sensible heat if (F .lt. 0.) then Fo = -condt F = 0. usens = condt+dswr-Io-ulwr-ultnt endif absorb = 0. do layer = 1,ni absorb = absorb+Iabs(layer) enddo Ib = Io-absorb ! solar passing through the bottom of the ice ! 2000 format(A15, 50(f8.3)) return end subroutine getabc(a,b,c,r,ti,tbot,zeta,k,eta,ni,lfirst) ! compute elements of tridiagonal matrix implicit none include "cpts.h" include "thermo.h" ! parameters for maintaining 2nd order accurate diff at bottom boundary real alph, bet parameter( alph = 3.0 ,bet = -1.0/3.0 ) ! input variables integer ni ! number of layers $ ,lfirst ! start with this layer real ti (0:nmax) ! temperature of ice-snow layers $ ,tbot ! temperature of ice bottom surface $ ,zeta (0:nmax) $ ,k (0:nmax+1) ! ice-snow conductivity $ ,eta (0:nmax) ! output variables real a (-1:nmax) ! sub-diagonal elements $ ,b (-1:nmax) ! diagonal elements $ ,c (-1:nmax) ! super-diagonal elements $ ,r (-1:nmax) ! constants (indep. of ti) ! local variable integer layer ! if there is snow lfirst = 1 otherwise it is 2 do layer = lfirst,(ni-1) a(layer) = -eta(layer)*k(layer) c(layer) = -eta(layer)*k(layer+1) b(layer) = 1-c(layer)-a(layer) r(layer) = zeta(layer) enddo a(ni) = -eta(ni)*(k(ni)-bet*k(ni+1)) c(ni) = 0.0 b(ni) = 1+eta(ni)*(k(ni)+alph*k(ni+1)) r(ni) = zeta(ni)+eta(ni)*(alph+bet)*k(ni+1)*tbot return end subroutine tridag(a,b,c,r,u,n) implicit none include "cpts.h" integer nnmax parameter (nnmax = nmax+2) ! don't change this ! input variables integer n ! number of layers real a (nnmax) ! sub-diagonal elements $ ,b (nnmax) ! diagonal elements $ ,c (nnmax) ! super-diagonal elements $ ,r (nnmax) ! constants (indep. of ti) ! output variables real u (nnmax) ! solution ! local variables integer layer real bet ! work variable $ ,gam (nnmax) ! work array bet = b(1) u(1) = r(1)/bet do layer = 2,n gam(layer) = c(layer-1)/bet bet = b(layer)-a(layer)*gam(layer) u(layer) = (r(layer)-a(layer)*u(layer-1))/bet enddo do layer = n-1,1,-1 u(layer) = u(layer)-gam(layer+1)*u(layer+1) enddo return end subroutine conductiv(sal1d,ki,ti,tbot,hs,dh,ni) implicit none include "cpts.h" include "thermo.h" real sal1d (NMAX) ! salinity of each layer real ki(0:nmax+1),ti(0:nmax),tbot,hs,dh real tmax parameter (tmax = -0.1) integer ni,layer real hsmin parameter ( hsmin = 0.01*centi ) do layer = 2,ni ki(layer) = kappai+BETA*(sal1d(layer)+sal1d(layer+1))/2. $ /min(tmax,(ti(layer-1)+ti(layer))/2.) ki(layer) = max(ki(layer),KIMIN) ki(layer) = ki(layer)/dh enddo ki(ni+1) = kappai+BETA*sal1d(ni+1)/tbot ki(ni+1) = max(ki(ni+1),KIMIN) ki(ni+1) = ki(ni+1)/dh ki(1) = kappai+BETA*sal1d(1)/min(tmax,ti(1)) ki(1) = max(ki(1),KIMIN) ki(1) = ki(1)/dh !cc ki(1) = kappai/dh if (hs.gt.hsmin) then ki(0) = kappas/hs ki(1) = 2.0*ki(1)*ki(0)/(ki(1)+ki(0)) else ki(0) = 0.0 endif return end subroutine get_Cpice(sal1d,tmz1d,cpi,Tnew,Told,ni) ! compute the heat capacity of each layer in the ICE ! Tnew is assumed to be the midpoint temperature of each layer implicit none include "cpts.h" include "thermo.h" real sal1d (NMAX) ! salinity of each layer real tmz1d (NMAX) ! melting temp of each layer real cpi(nmax),Tnew(nmax),Told(nmax) integer ni,layer do layer = 1,ni cpi(layer) = rcpice+gamma*sal1d(layer)/Told(layer)/ $ min(Tnew(layer),tmz1d(layer)) enddo #endif return end