C $Header: /u/gcmpack/MITgcm/pkg/gmredi/gmredi_calc_eigs.F,v 1.7 2014/11/24 10:23:50 dfer Exp $ C $Name: $ #include "GMREDI_OPTIONS.h" C !ROUTINE: EIGENVAL C !INTERFACE: SUBROUTINE GMREDI_CALC_EIGS( I iMin, iMax, jMin, jMax, I bi, bj, N2, myThid, kLow, I mask, hfac, recip_hfac, I rlow, nmodes, writediag, O Rmid, vec) C !DESCRIPTION: \bv C *==========================================================* C | SUBROUTINE GMREDI_CALC_URMS C | o Calculate the vertical structure of the rms eddy C | velocity based on baroclinic modal decomposition C *==========================================================* C \ev IMPLICIT NONE C == Global variables == #include "SIZE.h" #include "GRID.h" #include "EEPARAMS.h" #include "PARAMS.h" #include "GMREDI.h" C !INPUT/OUTPUT PARAMETERS: C == Routine arguments == C bi, bj :: tile indices LOGICAL writediag INTEGER iMin,iMax,jMin,jMax INTEGER bi, bj INTEGER myThid INTEGER nmodes INTEGER kLow(1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL mask(1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL hfac(1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL recip_hfac(1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL rlow(1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL N2( 1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL Rmid(1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL vec(nmodes,1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) #ifdef GM_K3D C !LOCAL VARIABLES: C == Local variables == INTEGER i,j,k,kk,m # ifdef HAVE_LAPACK C info :: error code from LAPACK C idx :: index used for sorting the eigenvalues C a3d :: lower diagonal of eigenvalue problem C b3d :: diagonal of eigenvalue problem C c3d :: upper diagonal of eigenvalue problem C val :: Eigenvalue (wavenumber) of the first baroclinic mode C eigR :: Real component of all the eigenvalues in a water column C eigI :: Imaginary component of all the eigenvalues in a water column C vecs :: All the eigenvectors of a water column C dummy :: Redundant variable for calling lapack C work :: Work array for lapack C array :: Array containing the matrix with a,b,c C eigval :: Vector containing the eigenvalues INTEGER info INTEGER idx _RL a3d( 1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL b3d( 1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL c3d( 1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL val( 1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL eigR(Nr),eigI(Nr),vecs(Nr,Nr),dummy(1,Nr),work(Nr*Nr) _RL array(Nr,Nr) _RL eigval(0:nmodes) else C drNr :: distance from bottom of cell to cell centre at Nr C BuoyFreq :: buoyancy frequency, SQRT(N2) C intN :: Vertical integral of BuoyFreq to each grid cell centre C intN0 :: Vertical integral of BuoyFreq to z=0 C c1 :: intN0/pi C nEigs :: number of eigenvalues/vectors to calculate _RL drNr _RL BuoyFreq(1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr+1) _RL intN(1-Olx:sNx+Olx,1-Oly:sNy+Oly,Nr) _RL intN0(1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL c1(1-Olx:sNx+Olx,1-Oly:sNy+Oly) INTEGER nEigs(1-Olx:sNx+Olx,1-Oly:sNy+Oly) # endif C small :: a small number (used to avoid floating point exceptions) C vecint :: vertical integral of eigenvector and/or square of eigenvector C fCori2 :: square of the Coriolis parameter _RL small _RL vecint(nmodes,1-Olx:sNx+Olx,1-Oly:sNy+Oly) _RL fCori2(1-Olx:sNx+Olx,1-Oly:sNy+Oly) CHARACTER*(MAX_LEN_MBUF) msgBuf small = TINY(zeroRL) C Square of the Coriolis parameter DO i=1-OLx,sNx+OLx DO j=1-OLy,sNy+OLy fCori2(i,j) = fCori(i,j,bi,bj)*fCori(i,j,bi,bj) ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nmodes vec(m,i,j,k) = zeroRL ENDDO ENDDO ENDDO ENDDO # ifdef HAVE_LAPACK C Calculate the tridiagonal operator matrix for C f^2 d/dz 1/N^2 d/dz C a3d is the lower off-diagonal, b3d is the diagonal C and c3d is the upper off-diagonal DO k=1,Nr DO j=1-OLy,sNy+OLy DO i=1-OLx,sNx+OLx IF (kLow(i,j) .GT. 0) THEN IF (k.EQ.1) THEN a3d(i,j,k) = zeroRL c3d(i,j,k) = fCori2(i,j)*recip_hFac(i,j,k) & *recip_drC(k+1)*recip_drF(k)/N2(i,j,k+1) b3d(i,j,k) = -c3d(i,j,k) ELSEIF (k.LT.kLow(i,j)) THEN a3d(i,j,k) = fCori2(i,j)*recip_hFac(i,j,k) & *recip_drF(k)*recip_drC(k)/N2(i,j,k) c3d(i,j,k) = fCori2(i,j)*recip_hFac(i,j,k) & *recip_drF(k)*recip_drC(k+1)/N2(i,j,k+1) b3d(i,j,k) = -a3d(i,j,k)-c3d(i,j,k) ELSEIF (k.EQ.kLow(i,j)) THEN a3d(i,j,k) = fCori2(i,j)*recip_hFac(i,j,k) & *recip_drF(k)*recip_drC(k)/N2(i,j,k) c3d(i,j,k) = zeroRL b3d(i,j,k) = -a3d(i,j,k) ELSE a3d(i,j,k) = zeroRL b3d(i,j,k) = zeroRL c3d(i,j,k) = zeroRL ENDIF ELSE a3d(i,j,k) = zeroRL b3d(i,j,k) = zeroRL c3d(i,j,k) = zeroRL ENDIF ENDDO ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx IF (kLow(i,j).GT.0) THEN DO kk=1,Nr DO k=1,Nr array(k,kk) = zeroRL ENDDO ENDDO k=1 array(k,k) = b3d(i,j,k) array(k,k+1) = c3d(i,j,k) DO k=2,Nr-1 array(k,k-1) = a3d(i,j,k) array(k,k) = b3d(i,j,k) array(k,k+1) = c3d(i,j,k) ENDDO k=Nr array(k,k-1) = a3d(i,j,k) array(k,k) = b3d(i,j,k) CALL DGEEV('N','V',Nr,array,Nr,eigR,eigI,dummy,1, & vecs,Nr,work,Nr*Nr,info) IF( info.LT.0 ) THEN WRITE(msgBuf,'(A,x,2(A1,I2),A1,x,A,I4)') & 'GMREDI_CALC_EIGS problem with arguments for DGEEV at', & '(',i,',',j,')', 'error code =',info CALL PRINT_ERROR( msgBuf , myThid ) ELSEIF(info.GT.0 ) THEN WRITE(msgBuf,'(A,x,2(A1,I2),A1,x,A,I4)') & 'GMREDI_CALC_EIGS problems with eigensolver DGEEV at', & '(',i,',',j,')', 'error code =',info CALL PRINT_ERROR( msgBuf , myThid ) ENDIF C Find the second largest eigenvalue (the Rossby radius) C and the first M baroclinic modes (eigenvectors) DO m=0,nmodes eigval(m) = -HUGE(zeroRL) ENDDO DO k=1,kLow(i,j) eigval(0) = MAX(eigval(0),eigR(k)) ENDDO DO m=1,MIN(nmodes,klow(i,j)-1) DO k=1,kLow(i,j) IF (eigR(k).LT.eigval(m-1)) THEN eigval(m) = MAX(eigval(m),eigR(k)) IF (eigval(m).EQ.eigR(k)) idx=k ENDIF ENDDO IF(vecs(1,idx).LT.zeroRL) THEN DO k=1,Nr vec(m,i,j,k) = -vecs(k,idx) ENDDO ELSE DO k=1,Nr vec(m,i,j,k) = vecs(k,idx) ENDDO ENDIF ENDDO val(i,j) = eigval(1) ELSE val(i,j)=zeroRL DO k=1,Nr DO m=1,nmodes vec(m,i,j,k)=zeroRL ENDDO ENDDO ENDIF ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx IF (kLow(i,j).GT.2 .AND. val(i,j).NE.zeroRL) THEN Rmid(i,j) = 1.0/(SQRT(ABS(val(i,j)))+small) ELSE Rmid(i,j) = zeroRL ENDIF ENDDO ENDDO else DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx BuoyFreq(i,j,k) = mask(i,j,k)*SQRT(N2(i,j,k)) ENDDO ENDDO ENDDO k=Nr+1 DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx BuoyFreq(i,j,k) = zeroRL ENDDO ENDDO C integrate N using something like Simpson s rule (but not quite) C drC*( (N(k+1)+N(k+2))/2 + (N(k)+N(k+1))/2 )/2 C when k=Nr, say that N(k+2)=0 and N(k+1)=0 k=Nr C drNr is like drC(Nr+1)/2 drNr = rC(Nr)-rF(Nr+1) DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx intN(i,j,k) = op5*BuoyFreq(i,j,k)*drNr ENDDO ENDDO DO k=Nr-1,1,-1 DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx intN(i,j,k) = intN(i,j,k+1) & + drC(k)*( op25*BuoyFreq(i,j,k+2) + op5*BuoyFreq(i,j,k) & + op25*BuoyFreq(i,j,k+1) ) ENDDO ENDDO ENDDO C intN integrates to z=rC(1). We want to integrate to z=0. C Assume that N(0)=0 and N(1)=0. C drC(1) is like half a grid cell. DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx intN0(i,j) = intN(i,j,1) & + drC(1)*op5*BuoyFreq(i,j,2) ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx c1(i,j) = intN0(i,j)/pi Rmid(i,j) = c1(i,j)/ABS(fCori(i,j,bi,bj)) ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx nEigs(i,j) = MIN(klow(i,j),nmodes) ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx IF (mask(i,j,k).NE.0.0) THEN DO m=1,nEigs(i,j) vec(m,i,j,k) = -COS(intN(i,j,k)/(m*c1(i,j))) ENDDO ENDIF ENDDO ENDDO ENDDO C The WKB approximation for the baroclinic mode does not always C integrate to zero so we adjust it. DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nEigs(i,j) vecint(m,i,j) = zeroRL ENDDO ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nEigs(i,j) vecint(m,i,j) = vecint(m,i,j) + hfac(i,j,k)*vec(m,i,j,k)*drF(k) ENDDO ENDDO ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nEigs(i,j) vecint(m,i,j) = vecint(m,i,j)/(-rlow(i,j)+small) ENDDO ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nEigs(i,j) vec(m,i,j,k) = vec(m,i,j,k) - vecint(m,i,j) ENDDO ENDDO ENDDO ENDDO # endif C Normalise the eigenvectors DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nmodes vecint(m,i,j) = zeroRL ENDDO ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nmodes vecint(m,i,j) = vecint(m,i,j) + & mask(i,j,k)*drF(k)*hfac(i,j,k) & *vec(m,i,j,k)*vec(m,i,j,k) ENDDO ENDDO ENDDO ENDDO DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nmodes vecint(m,i,j) = SQRT(vecint(m,i,j)) ENDDO ENDDO ENDDO DO k=1,Nr DO j=1-Oly,sNy+Oly DO i=1-Olx,sNx+Olx DO m=1,nmodes vec(m,i,j,k) = vec(m,i,j,k)/(vecint(m,i,j)+small) ENDDO ENDDO ENDDO ENDDO # ifdef ALLOW_DIAGNOSTICS C Diagnostics IF ( useDiagnostics.AND.writediag ) THEN # ifdef HAVE_LAPACK CALL DIAGNOSTICS_FILL(a3d, 'GM_A3D ',0,Nr,0,1,1,myThid) CALL DIAGNOSTICS_FILL(b3d, 'GM_B3D ',0,Nr,0,1,1,myThid) CALL DIAGNOSTICS_FILL(c3d, 'GM_C3D ',0,Nr,0,1,1,myThid) # endif ENDIF # endif #endif /* GM_K3D */ RETURN END