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advance_windm_edsclrm_module.F90
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advance_windm_edsclrm_module.F90
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!------------------------------------------------------------------------
! $Id$
!===============================================================================
module advance_windm_edsclrm_module
implicit none
private ! Set Default Scope
public :: advance_windm_edsclrm, xpwp_fnc
private :: windm_edsclrm_solve, &
compute_uv_tndcy, &
windm_edsclrm_lhs, &
windm_edsclrm_rhs
! Private named constants to avoid string comparisons
integer, parameter, private :: &
windm_edsclrm_um = 1, & ! Named constant to handle um solves
windm_edsclrm_vm = 2, & ! Named constant to handle vm solves
windm_edsclrm_scalar = 3, & ! Named constant to handle scalar solves
clip_upwp = 10, & ! Named constant for upwp clipping
! NOTE: This must be the same as the clip_upwp
! declared in clip_explicit!
clip_vpwp = 11 ! Named constant for vpwp clipping
! NOTE: This must be the same as the clip_vpwp
! declared in clip_explicit!
contains
!=============================================================================
subroutine advance_windm_edsclrm &
( dt, wm_zt, Km_zm, Kmh_zm, ug, vg, um_ref, vm_ref, &
wp2, up2, vp2, um_forcing, vm_forcing, &
edsclrm_forcing, &
rho_ds_zm, invrs_rho_ds_zt, &
fcor, l_implemented, &
l_predict_upwp_vpwp, &
l_upwind_xm_ma, &
l_uv_nudge, &
l_tke_aniso, &
um, vm, edsclrm, &
upwp, vpwp, wpedsclrp )
! Description:
! Solves for both mean horizontal wind components, um and vm, and for the
! eddy-scalars (passive scalars that don't use the high-order closure).
! Uses the LAPACK tridiagonal solver subroutine with 2 + # of scalar(s)
! back substitutions (since the left hand side matrix is the same for all
! input variables).
! References:
! Eqn. 8 & 9 on p. 3545 of
! ``A PDF-Based Model for Boundary Layer Clouds. Part I:
! Method and Model Description'' Golaz, et al. (2002)
! JAS, Vol. 59, pp. 3540--3551.
!-----------------------------------------------------------------------
use grid_class, only: &
gr ! Variables(s)
use parameters_model, only: &
ts_nudge, & ! Variable(s)
edsclr_dim
use parameters_tunable, only: &
nu10_vert_res_dep ! Constant
use clubb_precision, only: &
core_rknd ! Variable(s)
use stats_type_utilities, only: &
stat_begin_update, & ! Subroutines
stat_end_update, &
stat_update_var
use stats_variables, only: &
ium_ref, & ! Variables
ivm_ref, &
ium_sdmp, &
ivm_sdmp, &
ium_ndg, &
ivm_ndg, &
iwindm_matrix_condt_num, &
stats_zt, &
l_stats_samp
use clip_explicit, only: &
clip_covar ! Procedure(s)
use error_code, only: &
clubb_at_least_debug_level, & ! Procedure
err_code, & ! Error Indicator
clubb_fatal_error ! Constant
use constants_clubb, only: &
one_half, & ! Constant(s)
zero, &
fstderr, &
eps
use sponge_layer_damping, only: &
uv_sponge_damp_settings, &
uv_sponge_damp_profile, &
sponge_damp_xm ! Procedure(s)
implicit none
! External
intrinsic :: real
! Constant Parameters
real( kind = core_rknd ), dimension(gr%nz) :: &
dummy_nu ! Used to feed zero values into function calls
! Input Variables
real( kind = core_rknd ), intent(in) :: &
dt ! Model timestep [s]
real( kind = core_rknd ), dimension(gr%nz), intent(in) :: &
wm_zt, & ! w wind component on thermodynamic levels [m/s]
Km_zm, & ! Eddy diffusivity of winds on momentum levs. [m^2/s]
Kmh_zm, & ! Eddy diffusivity of themo on momentum levs. [m^s/s]
ug, & ! u (west-to-east) geostrophic wind comp. [m/s]
vg, & ! v (south-to-north) geostrophic wind comp. [m/s]
um_ref, & ! Reference u wind component for nudging [m/s]
vm_ref, & ! Reference v wind component for nudging [m/s]
wp2, & ! w'^2 (momentum levels) [m^2/s^2]
up2, & ! u'^2 (momentum levels) [m^2/s^2]
vp2, & ! v'^2 (momentum levels) [m^2/s^2]
um_forcing, & ! u forcing [m/s/s]
vm_forcing, & ! v forcing [m/s/s]
rho_ds_zm, & ! Dry, static density on momentum levels [kg/m^3]
invrs_rho_ds_zt ! Inv. dry, static density at thermo. levels [m^3/kg]
real( kind = core_rknd ), dimension(gr%nz,edsclr_dim), intent(in) :: &
edsclrm_forcing ! Eddy scalar large-scale forcing [{units vary}/s]
real( kind = core_rknd ), intent(in) :: &
fcor ! Coriolis parameter [s^-1]
logical, intent(in) :: &
l_implemented ! Flag for CLUBB being implemented in a larger model.
logical, intent(in) :: &
l_predict_upwp_vpwp, & ! Flag to predict <u'w'> and <v'w'> along with <u> and <v> alongside
! the advancement of <rt>, <w'rt'>, <thl>, <wpthlp>, <sclr>, and
! <w'sclr'> in subroutine advance_xm_wpxp. Otherwise, <u'w'> and
! <v'w'> are still approximated by eddy diffusivity when <u> and <v>
! are advanced in subroutine advance_windm_edsclrm.
l_upwind_xm_ma, & ! This flag determines whether we want to use an upwind differencing
! approximation rather than a centered differencing for turbulent or
! mean advection terms. It affects rtm, thlm, sclrm, um and vm.
l_uv_nudge, & ! For wind speed nudging
l_tke_aniso ! For anisotropic turbulent kinetic energy, i.e. TKE = 1/2
! (u'^2 + v'^2 + w'^2)
! Input/Output Variables
real( kind = core_rknd ), dimension(gr%nz), intent(inout) :: &
um, & ! Mean u (west-to-east) wind component [m/s]
vm, & ! Mean v (south-to-north) wind component [m/s]
upwp, & ! <u'w'> (momentum levels) [m^2/s^2]
vpwp ! <v'w'> (momentum levels) [m^2/s^2]
! Input/Output Variable for eddy-scalars
real( kind = core_rknd ), dimension(gr%nz,edsclr_dim), intent(inout) :: &
edsclrm ! Mean eddy scalar quantity [units vary]
! Output Variable for eddy-scalars
real( kind = core_rknd ), dimension(gr%nz,edsclr_dim), intent(inout) :: &
wpedsclrp ! w'edsclr' (momentum levels) [m/s {units vary}]
! Local Variables
real( kind = core_rknd ), dimension(gr%nz) :: &
um_tndcy, & ! u wind component tendency [m/s^2]
vm_tndcy ! v wind component tendency [m/s^2]
real( kind = core_rknd ), dimension(gr%nz) :: &
upwp_chnge, & ! Net change of u'w' due to clipping [m^2/s^2]
vpwp_chnge ! Net change of v'w' due to clipping [m^2/s^2]
real( kind = core_rknd ), dimension(3,gr%nz) :: &
lhs ! The implicit part of the tridiagonal matrix [units vary]
real( kind = core_rknd ), dimension(gr%nz,max(2,edsclr_dim)) :: &
rhs, &! The explicit part of the tridiagonal matrix [units vary]
solution ! The solution to the tridiagonal matrix [units vary]
real( kind = core_rknd ), dimension(gr%nz) :: &
wind_speed ! wind speed; sqrt(u^2 + v^2) [m/s]
real( kind = core_rknd ) :: &
u_star_sqd ! Surface friction velocity, u_star, squared [m/s]
logical :: &
l_imp_sfc_momentum_flux ! Flag for implicit momentum surface fluxes.
integer :: nrhs ! Number of right hand side terms
integer :: i ! Array index
logical :: l_first_clip_ts, l_last_clip_ts ! flags for clip_covar
!--------------------------- Begin Code ------------------------------------
dummy_nu = 0._core_rknd
if ( .not. l_predict_upwp_vpwp ) then
!----------------------------------------------------------------
! Prepare tridiagonal system for horizontal winds, um and vm
!----------------------------------------------------------------
! Compute Coriolis, geostrophic, and other prescribed wind forcings
! for um.
call compute_uv_tndcy( windm_edsclrm_um, fcor, vm, vg, & ! In
um_forcing, l_implemented, & ! In
um_tndcy ) ! Out
! Compute Coriolis, geostrophic, and other prescribed wind forcings
! for vm.
call compute_uv_tndcy( windm_edsclrm_vm, fcor, um, ug, & ! In
vm_forcing, l_implemented, & ! In
vm_tndcy ) ! Out
! Momentum surface fluxes, u'w'|_sfc and v'w'|_sfc, are applied through
! an implicit method, such that:
! x'w'|_sfc = - ( u_star(t)^2 / wind_speed(t) ) * xm(t+1).
l_imp_sfc_momentum_flux = .true.
! Compute wind speed (use threshold "eps" to prevent divide-by-zero
! error).
wind_speed = max( sqrt( um**2 + vm**2 ), eps )
! Compute u_star_sqd according to the definition of u_star.
u_star_sqd = sqrt( upwp(1)**2 + vpwp(1)**2 )
! Compute the explicit portion of the um equation.
! Build the right-hand side vector.
call windm_edsclrm_rhs( windm_edsclrm_um, dt, nu10_vert_res_dep, & ! In
Km_zm, um, um_tndcy, & ! In
rho_ds_zm, invrs_rho_ds_zt, & ! In
l_imp_sfc_momentum_flux, upwp(1), & ! In
rhs(:,windm_edsclrm_um) ) ! Out
! Compute the explicit portion of the vm equation.
! Build the right-hand side vector.
call windm_edsclrm_rhs( windm_edsclrm_vm, dt, nu10_vert_res_dep, & ! In
Km_zm, vm, vm_tndcy, & ! In
rho_ds_zm, invrs_rho_ds_zt, & ! In
l_imp_sfc_momentum_flux, vpwp(1), & ! In
rhs(:,windm_edsclrm_vm) ) ! Out
! Store momentum flux (explicit component)
! The surface flux, x'w'(1) = x'w'|_sfc, is set elsewhere in the model.
! upwp(1) = upwp_sfc
! vpwp(1) = vpwp_sfc
! Solve for x'w' at all intermediate model levels.
! A Crank-Nicholson timestep is used.
upwp(2:gr%nz-1) &
= -one_half &
* xpwp_fnc( Km_zm(2:gr%nz-1) + nu10_vert_res_dep(2:gr%nz-1), & ! in
um(2:gr%nz-1), um(3:gr%nz), & ! in
gr%invrs_dzm(2:gr%nz-1) )
vpwp(2:gr%nz-1) &
= -one_half &
* xpwp_fnc( Km_zm(2:gr%nz-1) + nu10_vert_res_dep(2:gr%nz-1), & ! in
vm(2:gr%nz-1), vm(3:gr%nz), & ! in
gr%invrs_dzm(2:gr%nz-1) )
! A zero-flux boundary condition at the top of the model, d(xm)/dz = 0,
! means that x'w' at the top model level is 0,
! since x'w' = - K_zm * d(xm)/dz.
upwp(gr%nz) = zero
vpwp(gr%nz) = zero
! Compute the implicit portion of the um and vm equations.
! Build the left-hand side matrix.
call windm_edsclrm_lhs( dt, nu10_vert_res_dep, wm_zt, Km_zm, & ! In
wind_speed, u_star_sqd, & ! In
rho_ds_zm, invrs_rho_ds_zt, & ! In
l_implemented, l_imp_sfc_momentum_flux, & ! In
l_upwind_xm_ma, & ! In
lhs ) ! Out
! Decompose and back substitute for um and vm
nrhs = 2
call windm_edsclrm_solve( nrhs, iwindm_matrix_condt_num, & ! In
lhs, rhs, & ! In/out
solution ) ! Out
! Check for singular matrices and bad LAPACK arguments
if ( clubb_at_least_debug_level( 0 ) ) then
if ( err_code == clubb_fatal_error ) then
write(fstderr,*) "Fatal error solving for um/vm"
return
endif
endif
!----------------------------------------------------------------
! Update zonal (west-to-east) component of mean wind, um
!----------------------------------------------------------------
um(1:gr%nz) = solution(1:gr%nz,windm_edsclrm_um)
!----------------------------------------------------------------
! Update meridional (south-to-north) component of mean wind, vm
!----------------------------------------------------------------
vm(1:gr%nz) = solution(1:gr%nz,windm_edsclrm_vm)
if ( l_stats_samp ) then
! Implicit contributions to um and vm
call windm_edsclrm_implicit_stats( windm_edsclrm_um, um ) ! in
call windm_edsclrm_implicit_stats( windm_edsclrm_vm, vm ) ! in
endif ! l_stats_samp
! The values of um(1) and vm(1) are located below the model surface and
! do not affect the rest of the model. The values of um(1) or vm(1) are
! simply set to the values of um(2) and vm(2), respectively, after the
! equation matrices has been solved. Even though um and vm would sharply
! decrease to a value of 0 at the surface, this is done to avoid
! confusion on plots of the vertical profiles of um and vm.
um(1) = um(2)
vm(1) = vm(2)
if ( uv_sponge_damp_settings%l_sponge_damping ) then
if ( l_stats_samp ) then
call stat_begin_update( ium_sdmp, um / dt, stats_zt )
call stat_begin_update( ivm_sdmp, vm / dt, stats_zt )
endif
um(1:gr%nz) = sponge_damp_xm( dt, gr%zt, um_ref(1:gr%nz), &
um(1:gr%nz), uv_sponge_damp_profile )
vm(1:gr%nz) = sponge_damp_xm( dt, gr%zt, vm_ref(1:gr%nz), &
vm(1:gr%nz), uv_sponge_damp_profile )
if ( l_stats_samp ) then
call stat_end_update( ium_sdmp, um / dt, stats_zt )
call stat_end_update( ivm_sdmp, vm / dt, stats_zt )
endif
endif ! uv_sponge_damp_settings%l_sponge_damping
! Second part of momentum (implicit component)
! Solve for x'w' at all intermediate model levels.
! A Crank-Nicholson timestep is used.
upwp(2:gr%nz-1) &
= upwp(2:gr%nz-1) &
- one_half * xpwp_fnc( Km_zm(2:gr%nz-1)+nu10_vert_res_dep(2:gr%nz-1), &
um(2:gr%nz-1), um(3:gr%nz), &
gr%invrs_dzm(2:gr%nz-1) )
vpwp(2:gr%nz-1) &
= vpwp(2:gr%nz-1) &
- one_half * xpwp_fnc( Km_zm(2:gr%nz-1)+nu10_vert_res_dep(2:gr%nz-1), &
vm(2:gr%nz-1), vm(3:gr%nz), &
gr%invrs_dzm(2:gr%nz-1) )
! Adjust um and vm if nudging is turned on.
if ( l_uv_nudge ) then
! Reflect nudging in budget
if ( l_stats_samp ) then
call stat_begin_update( ium_ndg, um / dt, stats_zt )
call stat_begin_update( ivm_ndg, vm / dt, stats_zt )
endif
um(1:gr%nz) &
= um(1:gr%nz) - ( ( um(1:gr%nz) - um_ref(1:gr%nz) ) * (dt/ts_nudge) )
vm(1:gr%nz) &
= vm(1:gr%nz) - ( ( vm(1:gr%nz) - vm_ref(1:gr%nz) ) * (dt/ts_nudge) )
if ( l_stats_samp ) then
call stat_end_update( ium_ndg, um / dt, stats_zt )
call stat_end_update( ivm_ndg, vm / dt, stats_zt )
endif
endif ! l_uv_nudge
if ( l_stats_samp ) then
call stat_update_var( ium_ref, um_ref, stats_zt )
call stat_update_var( ivm_ref, vm_ref, stats_zt )
endif
if ( l_tke_aniso ) then
! Clipping for u'w'
!
! Clipping u'w' at each vertical level, based on the
! correlation of u and w at each vertical level, such that:
! corr_(u,w) = u'w' / [ sqrt(u'^2) * sqrt(w'^2) ];
! -1 <= corr_(u,w) <= 1.
!
! Since u'^2, w'^2, and u'w' are each advanced in different
! subroutines from each other in advance_clubb_core, clipping for u'w'
! has to be done three times during each timestep (once after each
! variable has been updated).
! This is the third instance of u'w' clipping.
l_first_clip_ts = .false.
l_last_clip_ts = .true.
call clip_covar( clip_upwp, l_first_clip_ts, & ! intent(in)
l_last_clip_ts, dt, wp2, up2, & ! intent(in)
l_predict_upwp_vpwp, & ! intent(in)
upwp, upwp_chnge ) ! intent(inout)
! Clipping for v'w'
!
! Clipping v'w' at each vertical level, based on the
! correlation of v and w at each vertical level, such that:
! corr_(v,w) = v'w' / [ sqrt(v'^2) * sqrt(w'^2) ];
! -1 <= corr_(v,w) <= 1.
!
! Since v'^2, w'^2, and v'w' are each advanced in different
! subroutines from each other in advance_clubb_core, clipping for v'w'
! has to be done three times during each timestep (once after each
! variable has been updated).
! This is the third instance of v'w' clipping.
l_first_clip_ts = .false.
l_last_clip_ts = .true.
call clip_covar( clip_vpwp, l_first_clip_ts, & ! intent(in)
l_last_clip_ts, dt, wp2, vp2, & ! intent(in)
l_predict_upwp_vpwp, & ! intent(in)
vpwp, vpwp_chnge ) ! intent(inout)
else
! In this case, it is assumed that
! u'^2 == v'^2 == w'^2, and the variables `up2' and `vp2' do not
! interact with any other variables.
l_first_clip_ts = .false.
l_last_clip_ts = .true.
call clip_covar( clip_upwp, l_first_clip_ts, & ! intent(in)
l_last_clip_ts, dt, wp2, wp2, & ! intent(in)
l_predict_upwp_vpwp, & ! intent(in)
upwp, upwp_chnge ) ! intent(inout)
call clip_covar( clip_vpwp, l_first_clip_ts, & ! intent(in)
l_last_clip_ts, dt, wp2, wp2, & ! intent(in)
l_predict_upwp_vpwp, & ! intent(in)
vpwp, vpwp_chnge ) ! intent(inout)
endif ! l_tke_aniso
endif ! .not. l_predict_upwp_vpwp
!----------------------------------------------------------------
! Prepare tridiagonal system for eddy-scalars
!----------------------------------------------------------------
if ( edsclr_dim > 0 ) then
! Eddy-scalar surface fluxes, x'w'|_sfc, are applied through an explicit
! method.
l_imp_sfc_momentum_flux = .false.
! Compute the explicit portion of eddy scalar equation.
! Build the right-hand side vector.
! Because of statistics, we have to use a DO rather than a FORALL here
! -dschanen 7 Oct 2008
!HPF$ INDEPENDENT
do i = 1, edsclr_dim
call windm_edsclrm_rhs( windm_edsclrm_scalar, dt, dummy_nu, & ! In
Kmh_zm, edsclrm(:,i), edsclrm_forcing, & ! In
rho_ds_zm, invrs_rho_ds_zt, & ! In
l_imp_sfc_momentum_flux, wpedsclrp(1,i), & ! In
rhs(:,i) ) ! Out
enddo
! Store momentum flux (explicit component)
! The surface flux, x'w'(1) = x'w'|_sfc, is set elsewhere in the model.
! wpedsclrp(1,1:edsclr_dim) = wpedsclrp_sfc(1:edsclr_dim)
! Solve for x'w' at all intermediate model levels.
! A Crank-Nicholson timestep is used.
! Here we use a forall and high performance fortran directive to try to
! parallelize this computation. Note that FORALL is more restrictive than DO.
!HPF$ INDEPENDENT, REDUCTION(wpedsclrp)
forall( i = 1:edsclr_dim )
wpedsclrp(2:gr%nz-1,i) &
= -one_half * xpwp_fnc( Kmh_zm(2:gr%nz-1), edsclrm(2:gr%nz-1,i), &
edsclrm(3:gr%nz,i), gr%invrs_dzm(2:gr%nz-1) )
end forall
! A zero-flux boundary condition at the top of the model, d(xm)/dz = 0,
! means that x'w' at the top model level is 0,
! since x'w' = - K_zm * d(xm)/dz.
wpedsclrp(gr%nz,1:edsclr_dim) = zero
! Compute the implicit portion of the xm (eddy-scalar) equations.
! Build the left-hand side matrix.
call windm_edsclrm_lhs( dt, dummy_nu, wm_zt, Kmh_zm, & ! In
wind_speed, u_star_sqd, & ! In
rho_ds_zm, invrs_rho_ds_zt, & ! In
l_implemented, l_imp_sfc_momentum_flux, & ! In
l_upwind_xm_ma, & ! In
lhs ) ! Out
! Decompose and back substitute for all eddy-scalar variables
call windm_edsclrm_solve( edsclr_dim, 0, & ! in
lhs, rhs, & ! in/out
solution ) ! out
if ( clubb_at_least_debug_level( 0 ) ) then
if ( err_code == clubb_fatal_error ) then
write(fstderr,*) "Fatal error solving for eddsclrm"
end if
end if
!----------------------------------------------------------------
! Update Eddy-diff. Passive Scalars
!----------------------------------------------------------------
edsclrm(1:gr%nz,1:edsclr_dim) = solution(1:gr%nz,1:edsclr_dim)
! The value of edsclrm(1) is located below the model surface and does not
! effect the rest of the model. The value of edsclrm(1) is simply set to
! the value of edsclrm(2) after the equation matrix has been solved.
forall( i=1:edsclr_dim )
edsclrm(1,i) = edsclrm(2,i)
end forall
! Second part of momentum (implicit component)
! Solve for x'w' at all intermediate model levels.
! A Crank-Nicholson timestep is used.
!HPF$ INDEPENDENT, REDUCTION(wpedsclrp)
forall( i = 1:edsclr_dim )
wpedsclrp(2:gr%nz-1,i) &
= wpedsclrp(2:gr%nz-1,i) &
- one_half * xpwp_fnc( Kmh_zm(2:gr%nz-1), edsclrm(2:gr%nz-1,i), &
edsclrm(3:gr%nz,i), gr%invrs_dzm(2:gr%nz-1) )
end forall
! Note that the w'edsclr' terms are not clipped, since we don't compute
! the variance of edsclr anywhere. -dschanen 7 Oct 2008
endif
if ( clubb_at_least_debug_level( 0 ) ) then
if ( err_code == clubb_fatal_error ) then
write(fstderr,*) "Error in advance_windm_edsclrm"
write(fstderr,*) "Intent(in)"
write(fstderr,*) "dt = ", dt
write(fstderr,*) "wm_zt = ", wm_zt
write(fstderr,*) "Km_zm = ", Km_zm
write(fstderr,*) "ug = ", ug
write(fstderr,*) "vg = ", vg
write(fstderr,*) "um_ref = ", um_ref
write(fstderr,*) "vm_ref = ", vm_ref
write(fstderr,*) "wp2 = ", wp2
write(fstderr,*) "up2 = ", up2
write(fstderr,*) "vp2 = ", vp2
write(fstderr,*) "um_forcing = ", um_forcing
write(fstderr,*) "vm_forcing = ", vm_forcing
do i = 1, edsclr_dim
write(fstderr,*) "edsclrm_forcing # = ", i, edsclrm_forcing
end do
write(fstderr,*) "fcor = ", fcor
write(fstderr,*) "l_implemented = ", l_implemented
write(fstderr,*) "Intent(inout)"
write(fstderr,*) "um = ", um
write(fstderr,*) "vm = ", vm
do i = 1, edsclr_dim
write(fstderr,*) "edsclrm # ", i, "=", edsclrm(:,i)
end do
write(fstderr,*) "upwp = ", upwp
write(fstderr,*) "vpwp = ", vpwp
write(fstderr,*) "wpedsclrp = ", wpedsclrp
return
end if
end if
return
end subroutine advance_windm_edsclrm
!=============================================================================
subroutine windm_edsclrm_solve( nrhs, ixm_matrix_condt_num, &
lhs, rhs, solution )
! Note: In the "Description" section of this subroutine, the variable
! "invrs_dzm" will be written as simply "dzm", and the variable
! "invrs_dzt" will be written as simply "dzt". This is being done as
! as device to save space and to make some parts of the description
! more readable. This change does not pertain to the actual code.
! Description:
! Solves the horizontal wind or eddy-scalar time-tendency equation, and
! diagnoses the turbulent flux. A Crank-Nicholson time-stepping algorithm
! is used in solving the turbulent advection term and in diagnosing the
! turbulent flux.
!
! The rate of change of an eddy-scalar quantity, xm, is:
!
! d(xm)/dt = - w * d(xm)/dz - (1/rho_ds) * d( rho_ds * x'w' )/dz
! + xm_forcings.
!
!
! The Turbulent Advection Term
! ----------------------------
!
! The above equation contains a turbulent advection term:
!
! - (1/rho_ds) * d( rho_ds * x'w' )/dz;
!
! where the momentum flux, x'w', is closed using a down gradient approach:
!
! x'w' = - K_zm * d(xm)/dz.
!
! The turbulent advection term becomes:
!
! + (1/rho_ds) * d [ rho_ds * K_zm * d(xm)/dz ] / dz;
!
! which is the same as a standard eddy-diffusion term (if "rho_ds * K_zm" in
! the term above is substituted for "K_zm" in a standard eddy-diffusion
! term, and if the standard eddy-diffusion term is multiplied by
! "1/rho_ds"). Thus, the turbulent advection term is treated and solved in
! the same way that a standard eddy-diffusion term would be solved. The
! term is discretized as follows:
!
! The values of xm are found on the thermodynamic levels, while the values
! of K_zm are found on the momentum levels. Additionally, the values of
! rho_ds_zm are found on the momentum levels, and the values of
! invrs_rho_ds_zt are found on the thermodynamic levels. The
! derivatives (d/dz) of xm are taken over the intermediate momentum levels.
! At the intermediate momentum levels, d(xm)/dz is multiplied by K_zm and by
! rho_ds_zm. Then, the derivative of the whole mathematical expression is
! taken over the central thermodynamic level, where it is multiplied by
! invrs_rho_ds_zt, which yields the desired result.
!
! ---xm(kp1)----------------------------------------------------- t(k+1)
!
! ===========d(xm)/dz===K_zm(k)=====rho_ds_zm(k)================= m(k)
!
! ---xm(k)---invrs_rho_ds_zt---d[rho_ds_zm*K_zm*d(xm)/dz]/dz----- t(k)
!
! ===========d(xm)/dz===K_zm(km1)===rho_ds_zm(km1)=============== m(k-1)
!
! ---xm(km1)----------------------------------------------------- t(k-1)
!
! The vertical indices t(k+1), m(k), t(k), m(k-1), and t(k-1) correspond
! with altitudes zt(k+1), zm(k), zt(k), zm(k-1), and zt(k-1), respectively.
! The letter "t" is used for thermodynamic levels and the letter "m" is used
! for momentum levels.
!
! dzt(k) = 1 / ( zm(k) - zm(k-1) )
! dzm(k) = 1 / ( zt(k+1) - zt(k) )
! dzm(k-1) = 1 / ( zt(k) - zt(k-1) )
!
! The vertically discretized form of the turbulent advection term (treated
! as an eddy diffusion term) is written out as:
!
! + invrs_rho_ds_zt(k)
! * dzt(k)
! * [ rho_ds_zm(k) * K_zm(k) * dzm(k) * ( xm(k+1) - xm(k) )
! - rho_ds_zm(k-1) * K_zm(k-1) * dzm(k-1) * ( xm(k) - xm(k-1) ) ].
!
! For this equation, a Crank-Nicholson (semi-implicit) diffusion scheme is
! used to solve the (1/rho_ds) * d [ rho_ds * K_zm * d(xm)/dz ] / dz
! eddy-diffusion term. The discretized implicit portion of the term is
! written out as:
!
! + (1/2) * invrs_rho_ds_zt(k)
! * dzt(k)
! * [ rho_ds_zm(k) * K_zm(k)
! * dzm(k) * ( xm(k+1,<t+1>) - xm(k,<t+1>) )
! - rho_ds_zm(k-1) * K_zm(k-1)
! * dzm(k-1) * ( xm(k,<t+1>) - xm(k-1,<t+1>) ) ].
!
! Note: When the implicit term is brought over to the left-hand side,
! the sign is reversed and the leading "+" in front of the term
! is changed to a "-".
!
! The discretized explicit portion of the term is written out as:
!
! + (1/2) * invrs_rho_ds_zt(k)
! * dzt(k)
! * [ rho_ds_zm(k) * K_zm(k)
! * dzm(k) * ( xm(k+1,<t>) - xm(k,<t>) )
! - rho_ds_zm(k-1) * K_zm(k-1)
! * dzm(k-1) * ( xm(k,<t>) - xm(k-1,<t>) ) ].
!
! Timestep index (t) stands for the index of the current timestep, while
! timestep index (t+1) stands for the index of the next timestep, which is
! being advanced to in solving the d(xm)/dt equation.
!
!
! Boundary Conditions:
!
! An eddy-scalar quantity is not allowed to flux out the upper boundary.
! Thus, a zero-flux boundary condition is used for the upper boundary in the
! eddy-diffusion equation.
!
! The lower boundary condition is much more complicated. It is neither a
! zero-flux nor a fixed-point boundary condition. Rather, it is a
! fixed-flux boundary condition. This term is a turbulent advection term,
! but with the eddy-scalars, the only value of x'w' relevant in solving the
! d(xm)/dt equation is the value of x'w' at the surface (the first momentum
! level), which is written as x'w'|_sfc.
!
! 1) x'w' surface flux; generalized explicit form
!
! The x'w' surface flux is applied to the d(xm)/dt equation through the
! turbulent advection term, which is:
!
! - (1/rho_ds) * d( rho_ds * x'w' )/dz.
!
! At most vertical levels, a substitution can be made for x'w', such
! that:
!
! x'w' = - K_zm * d(xm)/dz.
!
! However, the same substitution cannot be made at the surface (momentum
! level 1), as x'w'|_sfc is a surface flux that is explicitly computed
! elsewhere in the model code.
!
! The lower boundary condition, which in this case needs to be applied to
! the d(xm)/dt equation at level 2, is discretized as follows:
!
! --xm(3)------------------------------------------------------- t(3)
!
! ========[x'w'(2) = -K_zm(2)*d(xm)/dz]===rho_ds_zm(2)========== m(2)
!
! --xm(2)---invrs_rho_ds_zt(2)---d[rho_ds_zm*K_zm*d(xm)/dz]/dz-- t(2)
!
! ========[x'w'|_sfc]=====================rho_ds_zm(1)========== m(1) sfc
!
! --xm(1)-------(below surface; not applicable)----------------- t(1)
!
! where "sfc" is the level of the model surface or lower boundary.
!
! The vertically discretized form of the turbulent advection term
! (treated as an eddy diffusion term), with the explicit surface flux,
! x'w'|_sfc, in place, is written out as:
!
! - invrs_rho_ds_zt(2)
! * dzt(2) * [ rho_ds_zm(2) * x'w'(2) - rho_ds_zm(1) * x'w'|_sfc ];
!
! which can be re-written as:
!
! + invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2) * K_zm(2) * dzm(2) * ( xm(3) - xm(2) )
! + rho_ds_zm(1) * x'w'|_sfc ];
!
! which can be re-written again as:
!
! + invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(2) * K_zm(2) * dzm(2) * ( xm(3) - xm(2) )
! + invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(1) * x'w'|_sfc.
!
! For this equation, a Crank-Nicholson (semi-implicit) diffusion scheme
! is used to solve the (1/rho_ds) * d [ rho_ds * K_zm * d(xm)/dz ] / dz
! eddy-diffusion term. The discretized implicit portion of the term is
! written out as:
!
! + (1/2) * invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2) * K_zm(2)
! * dzm(2) * ( xm(3,<t+1>) - xm(2,<t+1>) ) ].
!
! Note: When the implicit term is brought over to the left-hand side,
! the sign is reversed and the leading "+" in front of the term
! is changed to a "-".
!
! The discretized explicit portion of the term is written out as:
!
! + (1/2) * invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2) * K_zm(2)
! * dzm(2) * ( xm(3,<t>) - xm(2,<t>) ) ]
! + invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(1) * x'w'|_sfc.
!
! Note: The x'w'|_sfc portion of the term written above has been pulled
! away from the rest of the explicit form written above because
! the (1/2) factor due to Crank-Nicholson time_stepping does not
! apply to it, as there isn't an implicit portion for x'w'|_sfc.
!
! Timestep index (t) stands for the index of the current timestep, while
! timestep index (t+1) stands for the index of the next timestep, which
! is being advanced to in solving the d(xm)/dt equation.
!
! 2) x'w' surface flux; implicit form for momentum fluxes u'w' and v'w'
!
! The x'w' surface flux is applied to the d(xm)/dt equation through the
! turbulent advection term, which is:
!
! - (1/rho_ds) * d( rho_ds * x'w' )/dz.
!
! At most vertical levels, a substitution can be made for x'w', such
! that:
!
! x'w' = - K_zm * d(xm)/dz.
!
! However, the same substitution cannot be made at the surface (momentum
! level 1), as x'w'|_sfc is a surface momentum flux that is found by the
! following equation:
!
! x'w'|_sfc = - [ u_star^2 / sqrt( um^2 + vm^2 ) ] * xm;
!
! where x'w'|_sfc and xm are either u'w'|_sfc and um, respectively, or
! v'w'|_sfc and vm, respectively (um and vm are located at the first
! thermodynamic level above the surface, which is thermodynamic level 2),
! sqrt( um^2 + vm^2 ) is the wind speed (also at thermodynamic level 2),
! and u_star is defined as:
!
! u_star = ( u'w'|_sfc^2 + v'w'|_sfc^2 )^(1/4);
!
! and thus u_star^2 is defined as:
!
! u_star^2 = sqrt( u'w'|_sfc^2 + v'w'|_sfc^2 ).
!
! The value of u_star is either set to a constant value or computed
! (through function diag_ustar) based on the surface wind speed, the
! height above surface of the surface wind speed (as compared to the
! roughness height), and the buoyancy flux at the surface. Either way,
! u_star is computed elsewhere in the model, and the values of u'w'|_sfc
! and v'w'|_sfc are based on it and computed along with it. The values
! of u'w'|_sfc and v'w'|_sfc are then passed into advance_clubb_core,
! and are eventually passed into advance_windm_edsclrm. In subroutine
! advance_windm_edsclrm, the value of u_star_sqd is then recomputed
! based on u'w'|_sfc and v'w'|_sfc. The value of sqrt( u_star_sqd ) is
! consistent with the value of the original computation of u_star.
!
! The equation listed above is substituted for x'w'|_sfc. The lower
! boundary condition, which in this case needs to be applied to the
! d(xm)/dt equation at level 2, is discretized as follows:
!
! --xm(3)------------------------------------------------------- t(3)
!
! ===[x'w'(2) = -K_zm(2)*d(xm)/dz]=================rho_ds_zm(2)= m(2)
!
! --xm(2)---invrs_rho_ds_zt(2)---d[rho_ds_zm*K_zm*d(xm)/dz]/dz-- t(2)
!
! ===[x'w'|_sfc = -[u_star^2/sqrt(um^2+vm^2)]*xm]==rho_ds_zm(1)= m(1) sfc
!
! --xm(1)-------(below surface; not applicable)----------------- t(1)
!
! where "sfc" is the level of the model surface or lower boundary.
!
! The vertically discretized form of the turbulent advection term
! (treated as an eddy diffusion term), with the implicit surface momentum
! flux in place, is written out as:
!
! - invrs_rho_ds_zt(2)
! * dzt(2) * [ rho_ds_zm(2) * x'w'(2) - rho_ds_zm(1) * x'w'|_sfc ];
!
! which can be re-written as:
!
! - invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2)
! * { - K_zm(2) * dzm(2) * ( xm(3) - xm(2) ) }
! - rho_ds_zm(1)
! * { - [ u_star^2 / sqrt( um(2)^2 + vm(2)^2 ) ] * xm(2) } ];
!
! which can be re-written as:
!
! + invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(2) * K_zm(2) * dzm(2) * ( xm(3) - xm(2) )
! - invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(1) * [ u_star^2 / sqrt( um(2)^2 + vm(2)^2 ) ] * xm(2).
!
! For this equation, a Crank-Nicholson (semi-implicit) diffusion scheme
! is used to solve the (1/rho_ds) * d [ rho_ds * K_zm * d(xm)/dz ] / dz
! eddy-diffusion term. The discretized implicit portion of the term is
! written out as:
!
! + (1/2) * invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2) * K_zm(2)
! * dzm(2) * ( xm(3,<t+1>) - xm(2,<t+1>) ) ]
! - invrs_rho_ds_zt(2)
! * dzt(2)
! * rho_ds_zm(1)
! * [u_star^2/sqrt( um(2,<t>)^2 + vm(2,<t>)^2 )] * xm(2,<t+1>).
!
! Note: When the implicit term is brought over to the left-hand side,
! the signs are reversed and the leading "+" in front of the first
! part of the term is changed to a "-", while the leading "-" in
! front of the second part of the term is changed to a "+".
!
! Note: The x'w'|_sfc portion of the term written above has been pulled
! away from the rest of the implicit form written above because
! the (1/2) factor due to Crank-Nicholson time_stepping does not
! apply to it. The x'w'|_sfc portion of the term is treated
! completely implicitly in order to enhance numerical stability.
!
! The discretized explicit portion of the term is written out as:
!
! + (1/2) * invrs_rho_ds_zt(2)
! * dzt(2)
! * [ rho_ds_zm(2) * K_zm(2)
! * dzm(2) * ( xm(3,<t>) - xm(2,<t>) ) ].
!
! Timestep index (t) stands for the index of the current timestep, while
! timestep index (t+1) stands for the index of the next timestep, which
! is being advanced to in solving the d(xm)/dt equation.
!
!
! The lower boundary condition for the implicit and explicit portions of the
! turbulent advection term, without the x'w'|_sfc portion of the term, can
! easily be invoked by using the zero-flux boundary conditions found in the
! generalized diffusion function (function diffusion_zt_lhs), which is used
! for many other equations in this model. Either the generalized explicit
! surface flux needs to be added onto the explicit term after the diffusion
! function has been called from subroutine windm_edsclrm_rhs, or the
! implicit momentum surface flux needs to be added onto the implicit term
! after the diffusion function has been called from subroutine
! windm_edsclrm_lhs. However, all other equations in this model that use
! zero-flux diffusion have level 1 as the level to which the lower boundary
! condition needs to be applied. Thus, an adjuster will have to be used at
! level 2 to call diffusion_zt_lhs with level 1 as the input level (the last
! variable being passed in during the function call). However, the other
! variables passed in (rho_ds_zm*K_zm, gr%dzt, and gr%dzm variables) will
! have to be passed in as solving for level 2.
!
! The value of xm(1) is located below the model surface and does not effect
! the rest of the model. Since xm can be either a horizontal wind component
! or a generic eddy scalar quantity, the value of xm(1) is simply set to the
! value of xm(2) after the equation matrix has been solved.
!
!
! Conservation Properties:
!
! When a fixed-flux lower boundary condition is used (combined with a
! zero-flux upper boundary condition), this technique of discretizing the
! turbulent advection term (treated as an eddy-diffusion term) leads to
! conservative differencing. When the implicit momentum surface flux is
! either zero or not used, the column totals for each column in the
! left-hand side matrix (for the turbulent advection term) should be equal
! to 0. Otherwise, the column total for the second column will be equal to
! rho_ds_zm(1) * x'w'|_sfc<t+1>. When the generalized explicit surface
! flux is either zero or not used, the column total for the right-hand side
! vector (for the turbulent advection term) should be equal to 0.
! Otherwise, the column total for the right-hand side vector (for the
! turbulent advection term) will be equal to rho_ds_zm(1) * x'w'|_sfc<t>.
! This ensures that the total amount of quantity xm over the entire vertical
! domain is only changed by the surface flux (neglecting any forcing terms).
! The total amount of change is equal to rho_ds_zm(1) * x'w'|_sfc.
!
! To see that this conservation law is satisfied by the left-hand side
! matrix, compute the turbulent advection (treated as eddy diffusion) of xm,
! neglecting any implicit momentum surface flux, multiply by rho_ds_zt, and
! integrate vertically. In discretized matrix notation (where "i" stands
! for the matrix column and "j" stands for the matrix row):
!
! 0 = Sum_j Sum_i
! (rho_ds_zt)_i ( 1/dzt )_i
! ( 0.5_core_rknd * (1/rho_ds_zt) * dzt * (rho_ds_zm*K_zm*dzm) )_ij (xm<t+1>)_j.
!
! The left-hand side matrix,
! ( 0.5_core_rknd * (1/rho_ds_zt) * dzt * (rho_ds_zm*K_zm*dzm) )_ij, is partially
! written below. The sum over i in the above equation removes (1/rho_ds_zt)
! and dzt everywhere from the matrix below. The sum over j leaves the
! column totals that are desired, which are 0.
!
! Left-hand side matrix contributions from the turbulent advection term
! (treated as an eddy-diffusion term using a Crank-Nicholson timestep);
! first five vertical levels:
!
! ------------------------------------------------------------------------------->
!k=1 | 0 0 0 0
! |
!k=2 | 0 +0.5* -0.5* 0
! | (1/rho_ds_zt(k))* (1/rho_ds_zt(k))*
! | dzt(k)* dzt(k)*
! | rho_ds_zm(k)* rho_ds_zm(k)*
! | K_zm(k)*dzm(k) K_zm(k)*dzm(k)
! |