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& rn_wdmin1 + rn_wdmin2
ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. &
& ( MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
IF(ll_tmp1) THEN
zcpx(ji,jj) = 1.0_wp
ELSE IF(ll_tmp2) THEN
! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
zcpx(ji,jj) = MAX(MIN( zcpx(ji,jj) , 1.0_wp),0.0_wp)
ELSE
zcpx(ji,jj) = 0._wp
END IF
ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) > &
& rn_wdmin1 + rn_wdmin2
ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. &
& ( MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
IF(ll_tmp1) THEN
zcpy(ji,jj) = 1.0_wp
ELSE IF(ll_tmp2) THEN
! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
zcpy(ji,jj) = MAX(MIN( zcpy(ji,jj) , 1.0_wp),0.0_wp)
ELSE
zcpy(ji,jj) = 0._wp
ENDIF
END_2D
ENDIF
! Clean 3-D work arrays
zhpi(:,:,:) = 0._wp
zrhh(:,:,:) = rhd(A2D(nn_hls),:)
! Preparing vertical density profile "zrhh(:,:,:)" for hybrid-sco coordinate
DO_2D( 1, 1, 1, 1 )
jk = mbkt(ji,jj)
IF( jk <= 1 ) THEN ; zrhh(ji,jj, : ) = 0._wp
ELSEIF( jk == 2 ) THEN ; zrhh(ji,jj,jk+1:jpk) = rhd(ji,jj,jk)
ELSEIF( jk < jpkm1 ) THEN
DO jkk = jk+1, jpk
zrhh(ji,jj,jkk) = interp1(gde3w(ji,jj,jkk ), gde3w(ji,jj,jkk-1), &
& gde3w(ji,jj,jkk-2), zrhh (ji,jj,jkk-1), zrhh(ji,jj,jkk-2))
END DO
ENDIF
END_2D
! Transfer the depth of "T(:,:,:)" to vertical coordinate "zdept(:,:,:)"
DO_2D( 1, 1, 1, 1 )
zdept(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) - ssh(ji,jj,Kmm)
END_2D
DO_3D( 1, 1, 1, 1, 2, jpk )
zdept(ji,jj,jk) = zdept(ji,jj,jk-1) + e3w(ji,jj,jk,Kmm)
END_3D
fsp(:,:,:) = zrhh (:,:,:)
xsp(:,:,:) = zdept(:,:,:)
! Construct the vertical density profile with the
! constrained cubic spline interpolation
! rho(z) = asp + bsp*z + csp*z^2 + dsp*z^3
CALL cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
! Integrate the hydrostatic pressure "zhpi(:,:,:)" at "T(ji,jj,1)"
DO_2D( 0, 1, 0, 1 )
zrhdt1 = zrhh(ji,jj,1) - interp3( zdept(ji,jj,1), asp(ji,jj,1), bsp(ji,jj,1), &
& csp(ji,jj,1), dsp(ji,jj,1) ) * 0.25_wp * e3w(ji,jj,1,Kmm)
! assuming linear profile across the top half surface layer
zhpi(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) * zrhdt1
END_2D
! Calculate the pressure "zhpi(:,:,:)" at "T(ji,jj,2:jpkm1)"
DO_3D( 0, 1, 0, 1, 2, jpkm1 )
zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + &
& integ_spline( zdept(ji,jj,jk-1), zdept(ji,jj,jk), &
& asp (ji,jj,jk-1), bsp (ji,jj,jk-1), &
& csp (ji,jj,jk-1), dsp (ji,jj,jk-1) )
END_3D
! Z coordinate of U(ji,jj,1:jpkm1) and V(ji,jj,1:jpkm1)
! Prepare zsshu_n and zsshv_n
DO_2D( 0, 0, 0, 0 )
!!gm BUG ? if it is ssh at u- & v-point then it should be:
! zsshu_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji+1,jj) * ssh(ji+1,jj,Kmm)) * &
! & r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
! zsshv_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji,jj+1) * ssh(ji,jj+1,Kmm)) * &
! & r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
!!gm not this:
zsshu_n(ji,jj) = (e1e2u(ji,jj) * ssh(ji,jj,Kmm) + e1e2u(ji+1, jj) * ssh(ji+1,jj,Kmm)) * &
& r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
zsshv_n(ji,jj) = (e1e2v(ji,jj) * ssh(ji,jj,Kmm) + e1e2v(ji+1, jj) * ssh(ji,jj+1,Kmm)) * &
& r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
END_2D
DO_2D( 0, 0, 0, 0 )
zu(ji,jj,1) = - ( e3u(ji,jj,1,Kmm) - zsshu_n(ji,jj) )
zv(ji,jj,1) = - ( e3v(ji,jj,1,Kmm) - zsshv_n(ji,jj) )
END_2D
DO_3D( 0, 0, 0, 0, 2, jpkm1 )
zu(ji,jj,jk) = zu(ji,jj,jk-1) - e3u(ji,jj,jk,Kmm)
zv(ji,jj,jk) = zv(ji,jj,jk-1) - e3v(ji,jj,jk,Kmm)
END_3D
DO_3D( 0, 0, 0, 0, 1, jpkm1 )
zu(ji,jj,jk) = zu(ji,jj,jk) + 0.5_wp * e3u(ji,jj,jk,Kmm)
zv(ji,jj,jk) = zv(ji,jj,jk) + 0.5_wp * e3v(ji,jj,jk,Kmm)
END_3D
DO_3D( 0, 0, 0, 0, 1, jpkm1 )
zu(ji,jj,jk) = MIN( zu(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
zu(ji,jj,jk) = MAX( zu(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
zv(ji,jj,jk) = MIN( zv(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
zv(ji,jj,jk) = MAX( zv(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
END_3D
DO_3D( 0, 0, 0, 0, 1, jpkm1 )
zpwes = 0._wp; zpwed = 0._wp
zpnss = 0._wp; zpnsd = 0._wp
zuijk = zu(ji,jj,jk)
zvijk = zv(ji,jj,jk)
!!!!! for u equation
IF( jk <= mbku(ji,jj) ) THEN
IF( -zdept(ji+1,jj,jk) >= -zdept(ji,jj,jk) ) THEN
jis = ji + 1; jid = ji
ELSE
jis = ji; jid = ji +1
ENDIF
! integrate the pressure on the shallow side
jk1 = jk
DO WHILE ( -zdept(jis,jj,jk1) > zuijk )
IF( jk1 == mbku(ji,jj) ) THEN
zuijk = -zdept(jis,jj,jk1)
EXIT
ENDIF
zdeps = MIN(zdept(jis,jj,jk1+1), -zuijk)
zpwes = zpwes + &
integ_spline(zdept(jis,jj,jk1), zdeps, &
asp(jis,jj,jk1), bsp(jis,jj,jk1), &
csp(jis,jj,jk1), dsp(jis,jj,jk1))
jk1 = jk1 + 1
END DO
! integrate the pressure on the deep side
jk1 = jk
DO WHILE ( -zdept(jid,jj,jk1) < zuijk )
IF( jk1 == 1 ) THEN
zdeps = zdept(jid,jj,1) + MIN(zuijk, ssh(jid,jj,Kmm)*znad)
zrhdt1 = zrhh(jid,jj,1) - interp3(zdept(jid,jj,1), asp(jid,jj,1), &
bsp(jid,jj,1) , csp(jid,jj,1), &
dsp(jid,jj,1)) * zdeps
zpwed = zpwed + 0.5_wp * (zrhh(jid,jj,1) + zrhdt1) * zdeps
EXIT
ENDIF
zdeps = MAX(zdept(jid,jj,jk1-1), -zuijk)
zpwed = zpwed + &
integ_spline(zdeps, zdept(jid,jj,jk1), &
asp(jid,jj,jk1-1), bsp(jid,jj,jk1-1), &
csp(jid,jj,jk1-1), dsp(jid,jj,jk1-1) )
jk1 = jk1 - 1
END DO
! update the momentum trends in u direction
zdpdx1 = zcoef0 * r1_e1u(ji,jj) * ( zhpi(ji+1,jj,jk) - zhpi(ji,jj,jk) )
IF( .NOT.ln_linssh ) THEN
zdpdx2 = zcoef0 * r1_e1u(ji,jj) * &
& ( REAL(jis-jid, wp) * (zpwes + zpwed) + (ssh(ji+1,jj,Kmm)-ssh(ji,jj,Kmm)) )
ELSE
zdpdx2 = zcoef0 * r1_e1u(ji,jj) * REAL(jis-jid, wp) * (zpwes + zpwed)
ENDIF
IF( ln_wd_il ) THEN
zdpdx1 = zdpdx1 * zcpx(ji,jj) * wdrampu(ji,jj)
zdpdx2 = zdpdx2 * zcpx(ji,jj) * wdrampu(ji,jj)
ENDIF
puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + (zdpdx1 + zdpdx2 - zpgu(ji,jj)) * umask(ji,jj,jk)
ENDIF
!!!!! for v equation
IF( jk <= mbkv(ji,jj) ) THEN
IF( -zdept(ji,jj+1,jk) >= -zdept(ji,jj,jk) ) THEN
jjs = jj + 1; jjd = jj
ELSE
jjs = jj ; jjd = jj + 1
ENDIF
! integrate the pressure on the shallow side
jk1 = jk
DO WHILE ( -zdept(ji,jjs,jk1) > zvijk )
IF( jk1 == mbkv(ji,jj) ) THEN
zvijk = -zdept(ji,jjs,jk1)
EXIT
ENDIF
zdeps = MIN(zdept(ji,jjs,jk1+1), -zvijk)
zpnss = zpnss + &
integ_spline(zdept(ji,jjs,jk1), zdeps, &
asp(ji,jjs,jk1), bsp(ji,jjs,jk1), &
csp(ji,jjs,jk1), dsp(ji,jjs,jk1) )
jk1 = jk1 + 1
END DO
! integrate the pressure on the deep side
jk1 = jk
DO WHILE ( -zdept(ji,jjd,jk1) < zvijk )
IF( jk1 == 1 ) THEN
zdeps = zdept(ji,jjd,1) + MIN(zvijk, ssh(ji,jjd,Kmm)*znad)
zrhdt1 = zrhh(ji,jjd,1) - interp3(zdept(ji,jjd,1), asp(ji,jjd,1), &
bsp(ji,jjd,1) , csp(ji,jjd,1), &
dsp(ji,jjd,1) ) * zdeps
zpnsd = zpnsd + 0.5_wp * (zrhh(ji,jjd,1) + zrhdt1) * zdeps
EXIT
ENDIF
zdeps = MAX(zdept(ji,jjd,jk1-1), -zvijk)
zpnsd = zpnsd + &
integ_spline(zdeps, zdept(ji,jjd,jk1), &
asp(ji,jjd,jk1-1), bsp(ji,jjd,jk1-1), &
csp(ji,jjd,jk1-1), dsp(ji,jjd,jk1-1) )
jk1 = jk1 - 1
END DO
! update the momentum trends in v direction
zdpdy1 = zcoef0 * r1_e2v(ji,jj) * ( zhpi(ji,jj+1,jk) - zhpi(ji,jj,jk) )
IF( .NOT.ln_linssh ) THEN
zdpdy2 = zcoef0 * r1_e2v(ji,jj) * &
( REAL(jjs-jjd, wp) * (zpnss + zpnsd) + (ssh(ji,jj+1,Kmm)-ssh(ji,jj,Kmm)) )
ELSE
zdpdy2 = zcoef0 * r1_e2v(ji,jj) * REAL(jjs-jjd, wp) * (zpnss + zpnsd )
ENDIF
IF( ln_wd_il ) THEN
zdpdy1 = zdpdy1 * zcpy(ji,jj) * wdrampv(ji,jj)
zdpdy2 = zdpdy2 * zcpy(ji,jj) * wdrampv(ji,jj)
ENDIF
pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + (zdpdy1 + zdpdy2 - zpgv(ji,jj)) * vmask(ji,jj,jk)
ENDIF
!
END_3D
!
IF( ln_wd_il ) DEALLOCATE( zcpx, zcpy )
!
END SUBROUTINE hpg_prj
SUBROUTINE cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
!!----------------------------------------------------------------------
!! *** ROUTINE cspline ***
!!
!! ** Purpose : constrained cubic spline interpolation
!!
!! ** Method : f(x) = asp + bsp*x + csp*x^2 + dsp*x^3
!!
!! Reference: CJC Kruger, Constrained Cubic Spline Interpoltation
!!----------------------------------------------------------------------
REAL(wp), DIMENSION(A2D(nn_hls),jpk), INTENT(in ) :: fsp, xsp ! value and coordinate
REAL(wp), DIMENSION(A2D(nn_hls),jpk), INTENT( out) :: asp, bsp, csp, dsp ! coefficients of the interpoated function
INTEGER , INTENT(in ) :: polynomial_type ! 1: cubic spline ; 2: Linear
!
INTEGER :: ji, jj, jk ! dummy loop indices
REAL(wp) :: zdf1, zdf2, zddf1, zddf2, ztmp1, ztmp2, zdxtmp
REAL(wp) :: zdxtmp1, zdxtmp2, zalpha
REAL(wp) :: zdf(jpk)
!!----------------------------------------------------------------------
!
IF (polynomial_type == 1) THEN ! Constrained Cubic Spline
DO_2D( 1, 1, 1, 1 )
!!Fritsch&Butland's method, 1984 (preferred, but more computation)
! DO jk = 2, jpkm1-1
! zdxtmp1 = xsp(ji,jj,jk) - xsp(ji,jj,jk-1)
! zdxtmp2 = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
! zdf1 = ( fsp(ji,jj,jk) - fsp(ji,jj,jk-1) ) / zdxtmp1
! zdf2 = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp2
!
! zalpha = ( zdxtmp1 + 2._wp * zdxtmp2 ) / ( zdxtmp1 + zdxtmp2 ) / 3._wp
!
! IF(zdf1 * zdf2 <= 0._wp) THEN
! zdf(jk) = 0._wp
! ELSE
! zdf(jk) = zdf1 * zdf2 / ( ( 1._wp - zalpha ) * zdf1 + zalpha * zdf2 )
! ENDIF
! END DO
!!Simply geometric average
DO jk = 2, jpk-2
zdf1 = (fsp(ji,jj,jk ) - fsp(ji,jj,jk-1)) / (xsp(ji,jj,jk ) - xsp(ji,jj,jk-1))
zdf2 = (fsp(ji,jj,jk+1) - fsp(ji,jj,jk )) / (xsp(ji,jj,jk+1) - xsp(ji,jj,jk ))
IF(zdf1 * zdf2 <= 0._wp) THEN
zdf(jk) = 0._wp
ELSE
zdf(jk) = 2._wp * zdf1 * zdf2 / (zdf1 + zdf2)
ENDIF
END DO
zdf(1) = 1.5_wp * ( fsp(ji,jj,2) - fsp(ji,jj,1) ) / &
& ( xsp(ji,jj,2) - xsp(ji,jj,1) ) - 0.5_wp * zdf(2)
zdf(jpkm1) = 1.5_wp * ( fsp(ji,jj,jpkm1) - fsp(ji,jj,jpkm1-1) ) / &
& ( xsp(ji,jj,jpkm1) - xsp(ji,jj,jpkm1-1) ) - 0.5_wp * zdf(jpk - 2)
DO jk = 1, jpk-2
zdxtmp = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
ztmp1 = (zdf(jk+1) + 2._wp * zdf(jk)) / zdxtmp
ztmp2 = 6._wp * (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / zdxtmp / zdxtmp
zddf1 = -2._wp * ztmp1 + ztmp2
ztmp1 = (2._wp * zdf(jk+1) + zdf(jk)) / zdxtmp
zddf2 = 2._wp * ztmp1 - ztmp2
dsp(ji,jj,jk) = (zddf2 - zddf1) / 6._wp / zdxtmp
csp(ji,jj,jk) = ( xsp(ji,jj,jk+1) * zddf1 - xsp(ji,jj,jk)*zddf2 ) / 2._wp / zdxtmp
bsp(ji,jj,jk) = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp - &
& csp(ji,jj,jk) * ( xsp(ji,jj,jk+1) + xsp(ji,jj,jk) ) - &
& dsp(ji,jj,jk) * ((xsp(ji,jj,jk+1) + xsp(ji,jj,jk))**2 - &
& xsp(ji,jj,jk+1) * xsp(ji,jj,jk))
asp(ji,jj,jk) = fsp(ji,jj,jk) - xsp(ji,jj,jk) * (bsp(ji,jj,jk) + &
& (xsp(ji,jj,jk) * (csp(ji,jj,jk) + &
& dsp(ji,jj,jk) * xsp(ji,jj,jk))))
END DO
END_2D
ELSEIF ( polynomial_type == 2 ) THEN ! Linear
DO_3D( 1, 1, 1, 1, 1, jpk-2 )
zdxtmp =xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
ztmp1 = fsp(ji,jj,jk+1) - fsp(ji,jj,jk)
dsp(ji,jj,jk) = 0._wp
csp(ji,jj,jk) = 0._wp
bsp(ji,jj,jk) = ztmp1 / zdxtmp
asp(ji,jj,jk) = fsp(ji,jj,jk) - bsp(ji,jj,jk) * xsp(ji,jj,jk)
END_3D
!
ELSE
CALL ctl_stop( 'invalid polynomial type in cspline' )
ENDIF
!
END SUBROUTINE cspline
FUNCTION interp1(x, xl, xr, fl, fr) RESULT(f)
!!----------------------------------------------------------------------
!! *** ROUTINE interp1 ***
!!
!! ** Purpose : 1-d linear interpolation
!!
!! ** Method : interpolation is straight forward
!! extrapolation is also permitted (no value limit)
!!----------------------------------------------------------------------
REAL(wp), INTENT(in) :: x, xl, xr, fl, fr
REAL(wp) :: f ! result of the interpolation (extrapolation)
REAL(wp) :: zdeltx
!!----------------------------------------------------------------------
!
zdeltx = xr - xl
IF( abs(zdeltx) <= 10._wp * EPSILON(x) ) THEN
f = 0.5_wp * (fl + fr)
ELSE
f = ( (x - xl ) * fr - ( x - xr ) * fl ) / zdeltx
ENDIF
!
END FUNCTION interp1
FUNCTION interp2( x, a, b, c, d ) RESULT(f)
!!----------------------------------------------------------------------
!! *** ROUTINE interp1 ***
!!
!! ** Purpose : 1-d constrained cubic spline interpolation
!!
!! ** Method : cubic spline interpolation
!!
!!----------------------------------------------------------------------
REAL(wp), INTENT(in) :: x, a, b, c, d
REAL(wp) :: f ! value from the interpolation
!!----------------------------------------------------------------------
!
f = a + x* ( b + x * ( c + d * x ) )
!
END FUNCTION interp2
FUNCTION interp3( x, a, b, c, d ) RESULT(f)
!!----------------------------------------------------------------------
!! *** ROUTINE interp1 ***
!!
!! ** Purpose : Calculate the first order of derivative of
!! a cubic spline function y=a+b*x+c*x^2+d*x^3
!!
!! ** Method : f=dy/dx=b+2*c*x+3*d*x^2
!!
!!----------------------------------------------------------------------
REAL(wp), INTENT(in) :: x, a, b, c, d
REAL(wp) :: f ! value from the interpolation
!!----------------------------------------------------------------------
!
f = b + x * ( 2._wp * c + 3._wp * d * x)
!
END FUNCTION interp3
FUNCTION integ_spline( xl, xr, a, b, c, d ) RESULT(f)
!!----------------------------------------------------------------------
!! *** ROUTINE interp1 ***
!!
!! ** Purpose : 1-d constrained cubic spline integration
!!
!! ** Method : integrate polynomial a+bx+cx^2+dx^3 from xl to xr
!!
!!----------------------------------------------------------------------
REAL(wp), INTENT(in) :: xl, xr, a, b, c, d
REAL(wp) :: za1, za2, za3
REAL(wp) :: f ! integration result
!!----------------------------------------------------------------------
!
za1 = 0.5_wp * b
za2 = c / 3.0_wp
za3 = 0.25_wp * d
!
f = xr * ( a + xr * ( za1 + xr * ( za2 + za3 * xr ) ) ) - &
& xl * ( a + xl * ( za1 + xl * ( za2 + za3 * xl ) ) )
!
END FUNCTION integ_spline
!!======================================================================
END MODULE dynhpg