w3sbt4md Module Reference

SHOWEX bottom friction source term (Ardhuin et al. More...

Functions/Subroutines
subroutine	insbt4
	Initialization for bottom friction source term routine. More...
subroutine	tabu_erf
	Tabulation of ERF function, which is used in bottom friction subgrid modeling. More...
subroutine	w3sbt4 (A, CG, WN, DEPTH, D50, PSIC, TAUBBL, BEDFORM, S, D, IX, IY)
	Computes the SHOWEX bottom friction with movable bed effects. More...

Variables
integer, parameter	sizeerftable =300
real, dimension(0:sizeerftable)	erftable
real	delxerf
real, parameter	xerfmax = 4.

Detailed Description

SHOWEX bottom friction source term (Ardhuin et al.

2003).

Using a subgrid depth parameterization based on Tolman (CE 1995).

Author

F. Ardhuin

J. Lepesqueur

Date

14-Mar-2012

Copyright

Copyright 2009-2022 National Weather Service (NWS), National Oceanic and Atmospheric Administration. All rights reserved. WAVEWATCH III is a trademark of the NWS. No unauthorized use without permission.

Function/Subroutine Documentation

insbt4()

subroutine w3sbt4md::insbt4

Initialization for bottom friction source term routine.

Author

F. Ardhuin

Date

14-Mar-2012

Definition at line 146 of file w3sbt4md.F90.

    !/
    !/                  +-----------------------------------+
    !/                  | WAVEWATCH III           NOAA/NCEP |
    !/                  |                         SHOM      |
    !/                  |            F. Ardhuin             |
    !/                  |                        FORTRAN 90 |
    !/                  | Last update :         14-Mar-2012 |
    !/                  +-----------------------------------+
    !/
    !/    14-Mar-2012 : Origination.                        ( version 4.05 )
    !
    !  1. Purpose :
    !
    !     Initialization for bottom friction source term routine.
    !
    !  2. Method :
    !
    !  3. Parameters :
    !
    !     Parameter list
    !     ----------------------------------------------------------------
    !     ----------------------------------------------------------------
    !
    !  4. Subroutines used :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      STRACE    Subr. W3SERVMD Subroutine tracing.
    !     ----------------------------------------------------------------
    !
    !  5. Called by :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      W3SBT4    Subr. W3SRC3MD Corresponding source term.
    !     ----------------------------------------------------------------
    !
    !  6. Error messages :
    !
    !       None.
    !
    !  7. Remarks :
    !
    !  8. Structure :
    !
    !     See source code.
    !
    !  9. Switches :
    !
    !     !/S  Enable subroutine tracing.
    !
    ! 10. Source code :
    !
    !/ ------------------------------------------------------------------- /
    !
#ifdef W3_S
    USE w3servmd, ONLY: strace
#endif
    !/
    IMPLICIT NONE
    !/
    !/ ------------------------------------------------------------------- /
    !/ Parameter list
    !/
    !      NONE
    !/
    !/ ------------------------------------------------------------------- /
    !/ Local parameters
    !/
#ifdef W3_S
    INTEGER, SAVE           :: IENT = 0
#endif
    !/
    !/ ------------------------------------------------------------------- /
    !/
#ifdef W3_S
    CALL strace (ient, 'INSIN3')
#endif
    !
    ! 1.  .... ----------------------------------------------------------- *
    !
    CALL tabu_erf   !tabulates ERF function
    !/
    !/ End of INSBT4 ----------------------------------------------------- /
    !/

References w3servmd::strace(), and tabu_erf().

Referenced by w3sbt4().

tabu_erf()

subroutine w3sbt4md::tabu_erf

Tabulation of ERF function, which is used in bottom friction subgrid modeling.

Initialization for source term routine.

Author

J. Lepesqueur

Date

14-Mar-2012

Definition at line 243 of file w3sbt4md.F90.

    !/
    !/                  +-----------------------------------+
    !/                  | WAVEWATCH III           NOAA/NCEP |
    !/                  |        J. Lepesqueur              |
    !/                  |                        FORTRAN 90 |
    !/                  | Last update :         14-Mar-2012 |
    !/                  +-----------------------------------+
    !/
    !/    14-Mar-2012 : Origination.                        ( version 3.13 )
    !/
    !  1. Purpose :
    !     Tabulation of ERF function, which is used in bottom friction subgrid modelling
    !
    !     Initialization for source term routine.
    !
    !  2. Method :
    !
    !  3. Parameters :
    !
    !     Parameter list
    !     ----------------------------------------------------------------
    !     ----------------------------------------------------------------
    !
    !  4. Subroutines used :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      STRACE    Subr. W3SERVMD Subroutine tracing.
    !     ----------------------------------------------------------------
    !
    !  5. Called by :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      W3SIN3    Subr. W3SRC3MD Corresponding source term.
    !     ----------------------------------------------------------------
    !
    !  6. Error messages :
    !
    !       None.
    !
    !  7. Remarks :
    !
    !  8. Structure :
    !
    !     See source code.
    !
    !  9. Switches :
    !
    !     !/S  Enable subroutine tracing.
    !
    ! 10. Source code :
    !
    IMPLICIT NONE
    INTEGER :: I
    REAL :: x,y
 
    delxerf   = (2*xerfmax)/real(sizeerftable)
    DO i=0,sizeerftable
      x=-1.*xerfmax+i*delxerf
      if(x.lt.0.)then
        y=2**(1/2)*(1-abs(erf(x)))/2
      else
        y=2**(1/2)*(1+erf(x))/2
      end if
      erftable(i)=y
    END DO
    RETURN
    !/ ------------------------------------------------------------------- /

References delxerf, erftable, sizeerftable, and xerfmax.

Referenced by insbt4().

w3sbt4()

subroutine w3sbt4md::w3sbt4	(	real, dimension(nspec), intent(in)	A,
		real, dimension(nk), intent(in)	CG,
		real, dimension(nk), intent(in)	WN,
		real, intent(in)	DEPTH,
		real, intent(in)	D50,
		real, intent(in)	PSIC,
		real, dimension(2), intent(out)	TAUBBL,
		real, dimension(3), intent(inout)	BEDFORM,
		real, dimension(nspec), intent(out)	S,
		real, dimension(nspec), intent(out)	D,
		integer, intent(in)	IX,
		integer, intent(in)	IY
	)

Computes the SHOWEX bottom friction with movable bed effects.

Uses a Gaussian distribution for friction factors, and estimates the contribution of rippled and non-rippled fractions based on the bayesian approach of Tolman (1995).

Parameters

[in]	A	Action density spectrum.
[in]	CG	Group velocities.
[in]	WN	Wavenumbers.
[in]	DEPTH	Water depth.
[in]	D50	Median grain size.
[in]	PSIC	Critical Shields parameter.
[out]	TAUBBL	Components of stress leaking to the bottom.
[in,out]	BEDFORM	Ripple parameters (roughness and wavelength).
[out]	S	Source term (1-D version).
[out]	D	Diagonal term of derivative.
[in]	IX	Spatial grid index.
[in]	IY	Spatial grid index.

Author

F. Ardhuin

J. Lepesqueur

Date

15-Mar-2012

Definition at line 341 of file w3sbt4md.F90.

    !/
    !/                  +-----------------------------------+
    !/                  | WAVEWATCH III           NOAA/NCEP |
    !/                  |            F. Ardhuin             |
    !/                  !            J. Lepesqueur          !
    !/                  |                        FORTRAN 90 |
    !/                  | Last update :         15-Mar-2012 |
    !/                  +-----------------------------------+
    !/
    !/    23-Jun-2011 : Origination.                        ( version 4.04 )
    !/    04-Jul-2011 : Adding momentum flux TAUBBL         ( version 4.05 )
    !/    15-Mar-2012 : Adding subgrid treatment for depth  ( version 4.05 )
    !/
    !  1. Purpose :
    !
    !     Computes the SHOWEX bottom friction with movable bed effects
    !
    !  2. Method :
    !     Uses a Gaussian distribution for friction factors, and estimates
    !     the contribution of rippled and non-rippled fractions based on
    !     the bayesian approach of Tolman (1995).
    !
    !  3. Parameters :
    !
    !     Parameter list
    !     ----------------------------------------------------------------
    !       A       R.A.  I   Action density spectrum.
    !       CG      R.A.  I   Group velocities.
    !       WN      R.A.  I   Wavenumbers.
    !       DEPTH   Real  I   Water depth.
    !       D50     Real  I   Median grain size.
    !       PSIC    Real  I   Critical Shields parameter
    !       BEFORMS Real I/O  Ripple parameters (roughness and wavelength).
    !       TAUBBL  Real  O   Components of stress leaking to the bottom.
    !       S       R.A.  O   Source term (1-D version).
    !       D       R.A.  O   Diagonal term of derivative.             *)
    !       IX,IY   Int. I   Spatial grid indices
    !     ----------------------------------------------------------------
    !
    !  4. Subroutines used :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      STRACE    Subr. W3SERVMD Subroutine tracing.
    !     ----------------------------------------------------------------
    !
    !  5. Called by :
    !
    !      Name      Type  Module   Description
    !     ----------------------------------------------------------------
    !      W3SRCE    Subr. W3SRCEMD Source term integration.
    !      W3EXPO    Subr.   N/A    Point output post-processor.
    !      GXEXPO    Subr.   N/A    GrADS point output post-processor.
    !     ----------------------------------------------------------------
    !
    !  6. Error messages :
    !
    !       None.
    !
    !  7. Remarks :
    !
    !  8. Structure :
    !
    !     See source code.
    !
    !  9. Switches :
    !
    !     !/S  Enable subroutine tracing.
    !
    ! 10. Source code :
    !
    !/ ------------------------------------------------------------------- /
 
    USE constants
    USE w3odatmd, ONLY: ndse
    USE w3servmd, ONLY: extcde
    USE w3gdatmd, ONLY: nk, nth, nspec, sig, dden,  &
         sbtcx, ecos, esin, dth
 
#ifdef W3_S
    USE w3servmd, ONLY: strace
#endif
    IMPLICIT NONE
    !/
    !/ ------------------------------------------------------------------- /
    !/ Parameter list
    !/
    !/
    !/ ------------------------------------------------------------------- /
    !/ Local parameters
    !/
#ifdef W3_S
    INTEGER, SAVE           :: IENT = 0
#endif
    !/
    LOGICAL, SAVE           :: FIRST = .true.
    REAL, INTENT(IN)        :: CG(NK), WN(NK), DEPTH, A(NSPEC), D50
    REAL, INTENT(IN)        :: PSIC
    INTEGER, INTENT(IN)     :: IX, IY
    REAL, INTENT(OUT)       :: S(NSPEC), D(NSPEC), TAUBBL(2)
    REAL, INTENT(INOUT)     :: BEDFORM(3)
    REAL                    :: CBETA(NK)
    REAL :: UORB2,UORB,AORB, EBX, EBY, AX, AY, LX, LY
    REAL :: CONST2, TEMP2
    REAL :: FW, KSUBN, KSUBS, KSUBR, MINADIM
    REAL :: SHIELDS(3), PSI, DELI1, DELI2, EB, XI, VARU, DD50
    INTEGER :: IK, ITH, IS, IND, INDE, ISUB
 
    REAL :: KRR, DSUB
    REAL DSUM(NK)
    ! These are the 3-point Gauss-Hermitte quadrature coefficients
    REAL, PARAMETER :: WSUB(3) = (/ 0.1666667,   0.1666666  , 0.6666667/)
    REAL, PARAMETER :: XSUB(3) = (/ -0.001,  0.001 , 0. /)
 
    REAL :: PROBA1, PROBA2, PSIX, PSIXT, PSIN2, DPSI , FACTOR
    REAL :: BACKGROUND
 
    !/
    !/ ------------------------------------------------------------------- /
    !/
#ifdef W3_S
    CALL strace (ient, 'W3SBT4')
#endif
    !
    ! 0.  Initializations ------------------------------------------------ *
    IF ( first ) THEN
      CALL insbt4
      first  = .false.
    END IF
 
    !
    ! 1.  Min / Max settings for grain size D50---------------------------- *
    !
    dd50=max(d50,1e-5)
    dd50=min(dd50,1.)
    !
    ! 1.1 Set background roughness when ripples are not active
    !
    background=max(sbtcx(6),sbtcx(7)*dd50)
    !
    ! 2. Subgrid loop
    !
    dsum(:)=0.
    taubbl(:)=0.
    !
    DO isub=1,3
      !
      ! 2.a  Computes bulk parameters : E, Uorb, Aorb------------------------- *
      !
      dsub=depth*(1.+xsub(isub))
      uorb=0.
      aorb=0.
      ax  =0.
      ay  =0.
 
      DO ik=1, nk
        IF ( wn(ik)*dsub .LT. 6. ) THEN
          eb  = 0.
          ebx = 0.
          eby = 0.
          DO ith=1, nth
            is=ith+(ik-1)*nth
            eb  = eb  + a(is)
            ebx = ebx +a(is)*ecos(ith)
            eby = eby +a(is)*esin(ith)
          END DO
          !
          ! U_bot=sigma * Zeta / sinh(KD)  and CBETA = 0.5*sigma^2 /(g*sinh^(kD))
          ! therefore variance(u_bot)= variance(elevation)*2*CBETA/D
          !
          !            CBETA(IK) = MAX(0., (CG(IK)*WN(IK)/SIG(IK)-0.5) )/DSUB
          cbeta(ik) = 0.5*sig(ik)**2 /(grav*(sinh(wn(ik)*dsub))**2)
          !  N.B.:  could also include shoaling effect on EB ...
          factor= (dden(ik) / cg(ik))*2*cbeta(ik)*grav
          varu= eb * factor
          uorb = uorb + varu
          aorb = aorb + varu/(sig(ik)**2)
          ax   = ax   + (ebx * factor)
          ay   = ay   + (eby * factor)
        ELSE
          cbeta(ik) = 0.
        END IF
      END DO
      !
      ! Computes RMS orbital amplitudes
      !
      uorb2 = 2*uorb
      uorb = sqrt(max(1.0e-7,uorb2))
      aorb = sqrt(max(1.0e-7,2*aorb))
      !
      ! Computes potential ripple wavelength, 1.7 = 2 * sqrt(2) * 0.6
      ! Based on Ardhuin et al. (JGR 2002): lambda = 0.6 * d_1/3
      !
      lx = aorb*1.7*ax/sqrt(ax**2+ay**2+1e-12)
      ly = aorb*1.7*ay/sqrt(ax**2+ay**2+1e-12)
      !
      ! 2.b First use of FWTABLE to get skin roughness and estimate Shields parameter
      !
      xi=max((alog10(max(aorb/dd50,0.3))-abmin)/delab,1.)
      ind  = min(sizefwtable-1, int(xi))
      deli1= min(1. ,xi-float(ind))
      deli2= 1. - deli1
      fw =fwtable(ind)*deli2+fwtable(ind+1)*deli1
 
      psi=fw*uorb2/(2.*grav*(sed_sg-1)*dd50)
      !
      ! Normalized Shields parameter
      !
      shields(isub)=psi/psic
      !
    END DO ! end of loop on ISUB
    dpsi=(shields(2)-shields(1))/(xsub(2)-xsub(1))*sbtcx(5)
    !
    ! Tests if the variation in psi is large enough to use subgrid
    !
    IF (abs(dpsi).LT.0.0001*shields(3).OR.abs(dpsi).LT.1.e-8) THEN
      !
      ! no subgrid in this case
      !
      IF(shields(3).GT.sbtcx(3)) THEN
 
        !  ripple roughness, see Ardhuin et al. (2003)
        ksubr=aorb*sbtcx(1)*shields(3)**sbtcx(2)
        !  Sheet flow roughness, see Wilson (1989)
        ksubs=aorb*0.0655*(uorb2/((sed_sg-1)*grav*aorb))**1.4
        ksubn = ksubr + ksubs
        bedform(2)=lx
        bedform(3)=ly
      ELSE
        !  relict roughness, see Ardhuin et al. (2003)
        ksubn=max(background,aorb*sbtcx(4))
        bedform(2)=-lx
        bedform(3)=-ly
      END IF
 
      bedform(1)=ksubn
 
    ELSE
      !
      ! subgrid in this case
      !
      psix=(sbtcx(3)-shields(3))/dpsi
 
      psixt=max((psix + xerfmax)/delxerf,0.)
      inde  = max(min(sizeerftable-1, int(psixt)),0)
      deli1  = min(1. ,psixt-float(inde))
      deli2  = 1. - deli1
      proba2=max(min(erftable(inde)*deli2+erftable(inde+1)*deli1,1.),0.)
      proba1 = 1. - proba2
      ! Mean psi with ripples (Tolman 1995, eq. XX)
      psin2=max(shields(3)+exp(-(0.5*psix**2))/sqrt(tpi)*dpsi/(proba2+0.0001),sbtcx(3))
      ! Sum of relict, ripple and sheet flow roughnesses
      ksubn = proba1*max(background,aorb*sbtcx(4))       &
           +proba2*aorb*(sbtcx(1)*psin2**sbtcx(2)+                   &
           0.0655*(uorb2/((sed_sg-1)*grav*aorb))**1.4)
      !
      IF (proba2.GT.0.5) THEN
        bedform(2)=lx
        bedform(3)=ly
      ELSE
        bedform(2)=-lx
        bedform(3)=-ly
      END IF
      !
    END IF
    bedform(1)=ksubn
 
    !
    ! 2.c second use of FWTABLE to get FW from the full roughness
    !
    xi=max((alog10(max(aorb/ksubn,0.3))-abmin)/delab,1.)
    ind  = min(sizefwtable-1, int(xi))
    deli1= min(1. ,xi-float(ind))
    deli2= 1. - deli1
    fw =fwtable(ind)*deli2+fwtable(ind+1)*deli1
    !
    ! 5. Fills output arrays and estimates the energy and momentum loss
    !
    DO ik=1, nk
      const2=dden(ik)/cg(ik) &         !Jacobian to get energy in band
           *grav/(sig(ik)/wn(ik))    ! coefficient to get momentum
      dsum(ik)=-fw*uorb*cbeta(ik)      !*WSUB(ISUB)
      DO ith=1,nth
        is=ith+(ik-1)*nth
        d(is)=dsum(ik)
        temp2=const2*d(is)*a(is)
        taubbl(1) = taubbl(1) - temp2*ecos(is)
        taubbl(2) = taubbl(2) - temp2*esin(is)
        s(is)=d(is)*a(is)
      END DO
    END DO
    !
    RETURN
    !/
    !/ End of W3SBT4 ----------------------------------------------------- /
    !/

References constants::abmin, w3gdatmd::dden, constants::delab, delxerf, w3gdatmd::dth, w3gdatmd::ecos, erftable, w3gdatmd::esin, w3servmd::extcde(), constants::fwtable, constants::grav, insbt4(), w3odatmd::ndse, w3gdatmd::nk, w3gdatmd::nspec, w3gdatmd::nth, w3gdatmd::sbtcx, constants::sed_sg, w3gdatmd::sig, sizeerftable, constants::sizefwtable, w3servmd::strace(), constants::tpi, and xerfmax.

Referenced by gxexpo(), w3exnc(), w3expo(), and w3srcemd::w3srce().

Variable Documentation

delxerf

real w3sbt4md::delxerf

Definition at line 132 of file w3sbt4md.F90.

  REAL                    :: DELXERF

Referenced by tabu_erf(), and w3sbt4().

erftable

real, dimension(0:sizeerftable) w3sbt4md::erftable

Definition at line 131 of file w3sbt4md.F90.

  REAL                    :: ERFTABLE(0:SIZEERFTABLE)

Referenced by tabu_erf(), and w3sbt4().

sizeerftable

integer, parameter w3sbt4md::sizeerftable =300

Definition at line 130 of file w3sbt4md.F90.

  INTEGER, PARAMETER      :: SIZEERFTABLE=300

Referenced by tabu_erf(), and w3sbt4().

xerfmax

real, parameter w3sbt4md::xerfmax = 4.

Definition at line 133 of file w3sbt4md.F90.

  REAL,    PARAMETER      :: XERFMAX =  4. ! number of stdev

Referenced by tabu_erf(), and w3sbt4().