From f6547daad4ff04d822ef138f9f42b6e5e6a9c599 Mon Sep 17 00:00:00 2001 From: Robert Hallberg Date: Fri, 22 Dec 2023 09:34:36 -0500 Subject: [PATCH] Refactor set_viscous_BBL channel drag calculations Refactored set_viscous_BBL to separate out the routines setting the open interface lengths used for the channel drag, shortening a 1070 line long routine to 915 lines and reducing the scope of a number of temporary variables. A number of logical branch points have been moved outside of the innermost do loops. This refactoring will also make it easier to provide alternatives to some of the solvers that do not use the trigonometric functions to solve for the roots of a cubic expression and avoiding the issues noted at NOAA-GFDL/MOM6/issues/483. All answers are bitwise identical and public interfaces are unchanged. --- .../vertical/MOM_set_viscosity.F90 | 511 +++++++++++------- 1 file changed, 322 insertions(+), 189 deletions(-) diff --git a/src/parameterizations/vertical/MOM_set_viscosity.F90 b/src/parameterizations/vertical/MOM_set_viscosity.F90 index b207b1ff1c..bf0db51a44 100644 --- a/src/parameterizations/vertical/MOM_set_viscosity.F90 +++ b/src/parameterizations/vertical/MOM_set_viscosity.F90 @@ -20,6 +20,7 @@ module MOM_set_visc use MOM_grid, only : ocean_grid_type use MOM_hor_index, only : hor_index_type use MOM_interface_heights, only : thickness_to_dz +! use MOM_intrinsic_functions, only : cuberoot use MOM_io, only : slasher, MOM_read_data, vardesc, var_desc use MOM_kappa_shear, only : kappa_shear_is_used, kappa_shear_at_vertex use MOM_open_boundary, only : ocean_OBC_type, OBC_segment_type, OBC_NONE, OBC_DIRECTION_E @@ -258,40 +259,15 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) real :: crv ! crv is the curvature of the bottom depth across a ! cell, times the cell width squared [Z ~> m]. real :: crv_3 ! crv/3 [Z ~> m]. - real :: C24_crv ! 24/crv [Z-1 ~> m-1]. real :: slope ! The absolute value of the bottom depth slope across ! a cell times the cell width [Z ~> m]. - real :: apb_4a, ax2_3apb ! Various nondimensional ratios of crv and slope [nondim]. - real :: a2x48_apb3, Iapb, Ibma_2 ! Combinations of crv (a) and slope (b) [Z-1 ~> m-1] - ! All of the following "volumes" have units of vertical heights because they are normalized - ! by the full horizontal area of a velocity cell. real :: Vol_bbl_chan ! The volume of the bottom boundary layer as used in the channel ! drag parameterization, normalized by the full horizontal area ! of the velocity cell [Z ~> m]. - real :: Vol_open ! The cell volume above which it is open [Z ~> m]. - real :: Vol_direct ! With less than Vol_direct [Z ~> m], there is a direct - ! solution of a cubic equation for L. - real :: Vol_2_reg ! The cell volume above which there are two separate - ! open areas that must be integrated [Z ~> m]. - real :: vol ! The volume below the interface whose normalized - ! width is being sought [Z ~> m]. - real :: vol_below ! The volume below the interface below the one that - ! is currently under consideration [Z ~> m]. - real :: Vol_err ! The error in the volume with the latest estimate of - ! L, or the error for the interface below [Z ~> m]. - real :: Vol_quit ! The volume error below which to quit iterating [Z ~> m]. - real :: Vol_tol ! A volume error tolerance [Z ~> m]. + real :: vol_below(SZK_(GV)+1) ! The volume below each interface, normalized by the full + ! horizontal area of a velocity cell [Z ~> m]. real :: L(SZK_(GV)+1) ! The fraction of the full cell width that is open at ! the depth of each interface [nondim]. - real :: L_direct ! The value of L above volume Vol_direct [nondim]. - real :: L_max, L_min ! Upper and lower bounds on the correct value for L [nondim]. - real :: Vol_err_max ! The volume error for the upper bound on the correct value for L [Z ~> m] - real :: Vol_err_min ! The volume error for the lower bound on the correct value for L [Z ~> m] - real :: Vol_0 ! A deeper volume with known width L0 [Z ~> m]. - real :: L0 ! The value of L above volume Vol_0 [nondim]. - real :: dVol ! vol - Vol_0 [Z ~> m]. - real :: dV_dL2 ! The partial derivative of volume with L squared - ! evaluated at L=L0 [Z ~> m]. real :: h_neglect ! A thickness that is so small it is usually lost ! in roundoff and can be neglected [H ~> m or kg m-2]. real :: dz_neglect ! A vertical distance that is so small it is usually lost @@ -315,14 +291,9 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) real, parameter :: C1_3 = 1.0/3.0, C1_6 = 1.0/6.0, C1_12 = 1.0/12.0 ! Rational constants [nondim] real :: C2pi_3 ! An irrational constant, 2/3 pi. [nondim] real :: tmp ! A temporary variable, sometimes in [Z ~> m] - real :: tmp_val_m1_to_p1 ! A temporary variable [nondim] - real :: curv_tol ! Numerator of curvature cubed, used to estimate - ! accuracy of a single L(:) Newton iteration [Z5 ~> m5] - logical :: use_L0, do_one_L_iter ! Control flags for L(:) Newton iteration logical :: use_BBL_EOS, do_i(SZIB_(G)) integer, dimension(2) :: EOSdom ! The computational domain for the equation of state integer :: i, j, k, is, ie, js, je, Isq, Ieq, Jsq, Jeq, nz, m, n, K2, nkmb, nkml - integer :: itt, maxitt=20 type(ocean_OBC_type), pointer :: OBC => NULL() is = G%isc ; ie = G%iec ; js = G%jsc ; je = G%jec ; nz = GV%ke @@ -332,7 +303,6 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) dz_neglect = GV%dZ_subroundoff Rho0x400_G = 400.0*(GV%H_to_RZ / (US%L_to_Z**2 * GV%g_Earth)) - Vol_quit = (0.9*GV%Angstrom_Z + dz_neglect) C2pi_3 = 8.0*atan(1.0)/3.0 if (.not.CS%initialized) call MOM_error(FATAL,"MOM_set_viscosity(BBL): "//& @@ -442,8 +412,7 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) !$OMP Isq,Ieq,Jsq,Jeq,h_neglect,dz_neglect,Rho0x400_G,C2pi_3, & !$OMP U_bg_sq,cdrag_sqrt,cdrag_sqrt_H,cdrag_sqrt_H_RL, & !$OMP cdrag_L_to_H,cdrag_RL_to_H,use_BBL_EOS,BBL_thick_max, & - !$OMP OBC,maxitt,D_u,D_v,mask_u,mask_v,pbv) & - !$OMP firstprivate(Vol_quit) + !$OMP OBC,D_u,D_v,mask_u,mask_v,pbv) do j=Jsq,Jeq ; do m=1,2 if (m==1) then @@ -865,12 +834,11 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) if (CS%body_force_drag) bbl_thick = dz_bbl_drag(i) if (CS%Channel_drag) then - ! The drag within the bottommost Vol_bbl_chan is applied as a part of - ! an enhanced bottom viscosity, while above this the drag is applied - ! directly to the layers in question as a Rayleigh drag term. - ! Restrict the volume over which the channel drag is applied. - if (CS%Chan_drag_max_vol >= 0.0) Vol_bbl_chan = min(Vol_bbl_chan, CS%Chan_drag_max_vol) + vol_below(nz+1) = 0.0 + do K=nz,1,-1 + vol_below(K) = vol_below(K+1) + dz_vel(i,k) + enddo !### The harmonic mean edge depths here are not invariant to offsets! if (m==1) then @@ -887,162 +855,33 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) Dm = 2.0 * D_vel * tmp / (D_vel + tmp) endif if (Dm > Dp) then ; tmp = Dp ; Dp = Dm ; Dm = tmp ; endif - crv_3 = (Dp + Dm - 2.0*D_vel) ; crv = 3.0*crv_3 slope = Dp - Dm + ! If the curvature is small enough, there is no reason not to assume ! a uniformly sloping or flat bottom. if (abs(crv) < 1e-2*(slope + CS%BBL_thick_min)) crv = 0.0 - ! Each cell extends from x=-1/2 to 1/2, and has a topography - ! given by D(x) = crv*x^2 + slope*x + D - crv/12. - - ! Calculate the volume above which the entire cell is open and the - ! other volumes at which the equation that is solved for L changes. - if (crv > 0.0) then - if (slope >= crv) then - Vol_open = D_vel - Dm ; Vol_2_reg = Vol_open - else - tmp = slope/crv - Vol_open = 0.25*slope*tmp + C1_12*crv - Vol_2_reg = 0.5*tmp**2 * (crv - C1_3*slope) - endif - ! Define some combinations of crv & slope for later use. - C24_crv = 24.0/crv ; Iapb = 1.0/(crv+slope) - apb_4a = (slope+crv)/(4.0*crv) ; a2x48_apb3 = (48.0*(crv*crv))*(Iapb**3) - ax2_3apb = 2.0*C1_3*crv*Iapb - elseif (crv == 0.0) then - Vol_open = 0.5*slope - if (slope > 0) Iapb = 1.0/slope - else ! crv < 0.0 - Vol_open = D_vel - Dm - if (slope >= -crv) then - Iapb = 1.0e30*US%Z_to_m ; if (slope+crv /= 0.0) Iapb = 1.0/(crv+slope) - Vol_direct = 0.0 ; L_direct = 0.0 ; C24_crv = 0.0 - else - C24_crv = 24.0/crv ; Iapb = 1.0/(crv+slope) - L_direct = 1.0 + slope/crv ! L_direct < 1 because crv < 0 - Vol_direct = -C1_6*crv*L_direct**3 - endif - Ibma_2 = 2.0 / (slope - crv) - endif - L(nz+1) = 0.0 ; vol = 0.0 ; Vol_err = 0.0 ; BBL_visc_frac = 0.0 - ! Determine the normalized open length at each interface. - do K=nz,1,-1 - vol_below = vol - - vol = vol + dz_vel(i,k) - h_vel_pos = h_vel(i,k) + h_neglect - - if (vol >= Vol_open) then ; L(K) = 1.0 - elseif (crv == 0) then ! The bottom has no curvature. - L(K) = sqrt(2.0*vol*Iapb) - elseif (crv > 0) then - ! There may be a minimum depth, and there are - ! analytic expressions for L for all cases. - if (vol < Vol_2_reg) then - ! In this case, there is a contiguous open region and - ! vol = 0.5*L^2*(slope + crv/3*(3-4L)). - if (a2x48_apb3*vol < 1e-8) then ! Could be 1e-7? - ! There is a very good approximation here for massless layers. - L0 = sqrt(2.0*vol*Iapb) ; L(K) = L0*(1.0 + ax2_3apb*L0) - else - L(K) = apb_4a * (1.0 - & - 2.0 * cos(C1_3*acos(a2x48_apb3*vol - 1.0) - C2pi_3)) - endif - ! To check the answers. - ! Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol - else ! There are two separate open regions. - ! vol = slope^2/4crv + crv/12 - (crv/12)*(1-L)^2*(1+2L) - ! At the deepest volume, L = slope/crv, at the top L = 1. - !L(K) = 0.5 - cos(C1_3*acos(1.0 - C24_crv*(Vol_open - vol)) - C2pi_3) - tmp_val_m1_to_p1 = 1.0 - C24_crv*(Vol_open - vol) - tmp_val_m1_to_p1 = max(-1., min(1., tmp_val_m1_to_p1)) - L(K) = 0.5 - cos(C1_3*acos(tmp_val_m1_to_p1) - C2pi_3) - ! To check the answers. - ! Vol_err = Vol_open - 0.25*crv_3*(1.0+2.0*L(K)) * (1.0-L(K))**2 - vol - endif - else ! a < 0. - if (vol <= Vol_direct) then - ! Both edges of the cell are bounded by walls. - L(K) = (-0.25*C24_crv*vol)**C1_3 - else - ! x_R is at 1/2 but x_L is in the interior & L is found by solving - ! vol = 0.5*L^2*(slope + crv/3*(3-4L)) - - ! Vol_err = 0.5*(L(K+1)*L(K+1))*(slope + crv_3*(3.0-4.0*L(K+1))) - vol_below - ! Change to ... - ! if (min(vol_below + Vol_err, vol) <= Vol_direct) then ? - if (vol_below + Vol_err <= Vol_direct) then - L0 = L_direct ; Vol_0 = Vol_direct - else - L0 = L(K+1) ; Vol_0 = vol_below + Vol_err - ! Change to Vol_0 = min(vol_below + Vol_err, vol) ? - endif - - ! Try a relatively simple solution that usually works well - ! for massless layers. - dV_dL2 = 0.5*(slope+crv) - crv*L0 ; dVol = (vol-Vol_0) - ! dV_dL2 = 0.5*(slope+crv) - crv*L0 ; dVol = max(vol-Vol_0, 0.0) - - use_L0 = .false. - do_one_L_iter = .false. - if (CS%answer_date < 20190101) then - curv_tol = GV%Angstrom_Z*dV_dL2**2 & - * (0.25 * dV_dL2 * GV%Angstrom_Z - crv * L0 * dVol) - do_one_L_iter = (crv * crv * dVol**3) < curv_tol - else - ! The following code is more robust when GV%Angstrom_H=0, but - ! it changes answers. - use_L0 = (dVol <= 0.) - - Vol_tol = max(0.5 * GV%Angstrom_Z + dz_neglect, 1e-14 * vol) - Vol_quit = max(0.9 * GV%Angstrom_Z + dz_neglect, 1e-14 * vol) + ! Determine the normalized open length (L) at each interface. + if (crv == 0.0) then + call find_L_open_uniform_slope(vol_below, Dp, Dm, L, GV) + elseif (crv > 0.0) then + call find_L_open_concave_analytic(vol_below, D_vel, Dp, Dm, C2pi_3, L, GV) + else ! crv < 0.0 + call find_L_open_convex(vol_below, D_vel, Dp, Dm, L, GV, US, CS) + endif ! end of crv<0 cases. - curv_tol = Vol_tol * dV_dL2**2 & - * (dV_dL2 * Vol_tol - 2.0 * crv * L0 * dVol) - do_one_L_iter = (crv * crv * dVol**3) < curv_tol - endif + ! Determine the Rayliegh drag contributions. - if (use_L0) then - L(K) = L0 - Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol - elseif (do_one_L_iter) then - ! One iteration of Newton's method should give an estimate - ! that is accurate to within Vol_tol. - L(K) = sqrt(L0*L0 + dVol / dV_dL2) - Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol - else - if (dV_dL2*(1.0-L0*L0) < dVol + & - dV_dL2 * (Vol_open - Vol)*Ibma_2) then - L_max = sqrt(1.0 - (Vol_open - Vol)*Ibma_2) - else - L_max = sqrt(L0*L0 + dVol / dV_dL2) - endif - L_min = sqrt(L0*L0 + dVol / (0.5*(slope+crv) - crv*L_max)) + ! The drag within the bottommost Vol_bbl_chan is applied as a part of an enhanced bottom + ! viscosity, while above this the drag is applied directly to the layers in question as a + ! Rayleigh drag term. - Vol_err_min = 0.5*(L_min**2)*(slope + crv_3*(3.0-4.0*L_min)) - vol - Vol_err_max = 0.5*(L_max**2)*(slope + crv_3*(3.0-4.0*L_max)) - vol - ! if ((abs(Vol_err_min) <= Vol_quit) .or. (Vol_err_min >= Vol_err_max)) then - if (abs(Vol_err_min) <= Vol_quit) then - L(K) = L_min ; Vol_err = Vol_err_min - else - L(K) = sqrt((L_min**2*Vol_err_max - L_max**2*Vol_err_min) / & - (Vol_err_max - Vol_err_min)) - do itt=1,maxitt - Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol - if (abs(Vol_err) <= Vol_quit) exit - ! Take a Newton's method iteration. This equation has proven - ! robust enough not to need bracketing. - L(K) = L(K) - Vol_err / (L(K)* (slope + crv - 2.0*crv*L(K))) - ! This would be a Newton's method iteration for L^2: - ! L(K) = sqrt(L(K)*L(K) - Vol_err / (0.5*(slope+crv) - crv*L(K))) - enddo - endif ! end of iterative solver - endif ! end of 1-boundary alternatives. - endif ! end of a<0 cases. - endif + ! Restrict the volume over which the channel drag is applied from the previously determined valu.e + if (CS%Chan_drag_max_vol >= 0.0) Vol_bbl_chan = min(Vol_bbl_chan, CS%Chan_drag_max_vol) + BBL_visc_frac = 0.0 + do K=nz,1,-1 !modify L(K) for porous barrier parameterization if (m==1) then ; L(K) = L(K)*pbv%por_layer_widthU(I,j,K) else ; L(K) = L(K)*pbv%por_layer_widthV(i,J,K); endif @@ -1050,8 +889,8 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) ! Determine the drag contributing to the bottom boundary layer ! and the Rayleigh drag that acts on each layer. if (L(K) > L(K+1)) then - if (vol_below < Vol_bbl_chan) then - BBL_frac = (1.0-vol_below/Vol_bbl_chan)**2 + if (vol_below(K+1) < Vol_bbl_chan) then + BBL_frac = (1.0-vol_below(K+1)/Vol_bbl_chan)**2 BBL_visc_frac = BBL_visc_frac + BBL_frac*(L(K) - L(K+1)) else BBL_frac = 0.0 @@ -1063,6 +902,7 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) cdrag_conv = cdrag_L_to_H endif + h_vel_pos = h_vel(i,k) + h_neglect if (m==1) then ; Cell_width = G%dy_Cu(I,j)*pbv%por_face_areaU(I,j,k) else ; Cell_width = G%dx_Cv(i,J)*pbv%por_face_areaV(i,J,k) ; endif gam = 1.0 - L(K+1)/L(K) @@ -1085,7 +925,7 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) else ; visc%Ray_v(i,J,k) = 0.0 ; endif endif - enddo ! k loop to determine L(K). + enddo ! k loop to determine visc%Ray_[uv]. ! Set the near-bottom viscosity to a value which will give ! the correct stress when the shear occurs over bbl_thick. @@ -1202,6 +1042,299 @@ subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, pbv) end subroutine set_viscous_BBL +!> Determine the normalized open length of each interface, given the edge depths and normalized +!! volumes below each interface. +subroutine find_L_open_uniform_slope(vol_below, Dp, Dm, L, GV) + type(verticalGrid_type), intent(in) :: GV !< The ocean's vertical grid structure. + real, dimension(SZK_(GV)+1), intent(in) :: vol_below !< The volume below each interface, normalized by + !! the full horizontal area of a velocity cell [Z ~> m] + real, intent(in) :: Dp !< The larger of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, intent(in) :: Dm !< The smaller of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, dimension(SZK_(GV)+1), intent(out) :: L !< The fraction of the full cell width that is open at + !! the depth of each interface [nondim] + + ! Local variables + real :: slope ! The absolute value of the bottom depth slope across a cell times the cell width [Z ~> m]. + real :: I_slope ! The inverse of the normalized slope [Z-1 ~> m-1] + real :: Vol_open ! The cell volume above which it is open [Z ~> m]. + integer :: K, nz + + nz = GV%ke + + slope = abs(Dp - Dm) + if (slope == 0.0) then + L(1:nz) = 1.0 ; L(nz+1) = 0.0 + else + Vol_open = 0.5*slope + I_slope = 1.0 / slope + + L(nz+1) = 0.0 + do K=nz,1,-1 + if (vol_below(K) >= Vol_open) then ; L(K) = 1.0 + else + ! With a uniformly sloping bottom, the calculation of L(K) is the solution of a simple quadratic equation. + L(K) = sqrt(2.0*vol_below(K)*I_slope) + endif + enddo + endif + +end subroutine find_L_open_uniform_slope + +!> Determine the normalized open length of each interface for concave bathymetry (from the ocean perspective) using +!! analytic expressions. In this case there can be two separate open regions. +subroutine find_L_open_concave_analytic(vol_below, D_vel, Dp, Dm, C2pi_3, L, GV) + type(verticalGrid_type), intent(in) :: GV !< The ocean's vertical grid structure. + real, dimension(SZK_(GV)+1), intent(in) :: vol_below !< The volume below each interface, normalized by + !! the full horizontal area of a velocity cell [Z ~> m] + real, intent(in) :: D_vel !< The average bottom depth at a velocity point [Z ~> m] + real, intent(in) :: Dp !< The larger of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, intent(in) :: Dm !< The smaller of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, intent(in) :: C2pi_3 !< An irrational constant, 2/3 pi [nondim] + real, dimension(SZK_(GV)+1), intent(out) :: L !< The fraction of the full cell width that is open at + !! the depth of each interface [nondim] + + ! Local variables + real :: crv ! crv is the curvature of the bottom depth across a + ! cell, times the cell width squared [Z ~> m]. + real :: crv_3 ! crv/3 [Z ~> m]. + real :: slope ! The absolute value of the bottom depth slope across + ! a cell times the cell width [Z ~> m]. + ! The following "volumes" have units of vertical heights because they are normalized + ! by the full horizontal area of a velocity cell. + real :: Vol_open ! The cell volume above which the face is fully is open [Z ~> m]. + real :: Vol_2_reg ! The cell volume above which there are two separate + ! open areas that must be integrated [Z ~> m]. + real :: C24_crv ! 24/crv [Z-1 ~> m-1]. + real :: apb_4a, ax2_3apb ! Various nondimensional ratios of crv and slope [nondim]. + real :: a2x48_apb3, Iapb ! Combinations of crv (a) and slope (b) [Z-1 ~> m-1] + real :: L0 ! A linear estimate of L approprate for tiny volumes [nondim]. + real :: slope_crv ! The slope divided by the curvature [nondim] + real :: tmp_val_m1_to_p1 ! A temporary variable [nondim] + real, parameter :: C1_3 = 1.0/3.0, C1_12 = 1.0/12.0 ! Rational constants [nondim] + integer :: K, nz + + nz = GV%ke + + ! Each cell extends from x=-1/2 to 1/2, and has a topography + ! given by D(x) = crv*x^2 + slope*x + D_vel - crv/12. + crv_3 = (Dp + Dm - 2.0*D_vel) ; crv = 3.0*crv_3 + if (crv < 0.0) call MOM_error(FATAL, "find_L_open_concave should only be called with a positive curvature.") + slope = Dp - Dm + + ! Calculate the volume above which the entire cell is open and the volume at which the + ! equation that is solved for L changes because there are two separate open regions. + if (slope >= crv) then + Vol_open = D_vel - Dm ; Vol_2_reg = Vol_open + else + slope_crv = slope / crv + Vol_open = 0.25*slope*slope_crv + C1_12*crv + Vol_2_reg = 0.5*slope_crv**2 * (crv - C1_3*slope) + endif + ! Define some combinations of crv & slope for later use. + C24_crv = 24.0/crv ; Iapb = 1.0/(crv+slope) + apb_4a = (slope+crv)/(4.0*crv) ; a2x48_apb3 = (48.0*(crv*crv))*(Iapb**3) + ax2_3apb = 2.0*C1_3*crv*Iapb + + L(nz+1) = 0.0 + ! Determine the normalized open length (L) at each interface. + do K=nz,1,-1 + if (vol_below(K) >= Vol_open) then ! The whole cell is open. + L(K) = 1.0 + elseif (vol_below(K) < Vol_2_reg) then + ! In this case, there is a contiguous open region and + ! vol_below(K) = 0.5*L^2*(slope + crv/3*(3-4L)). + if (a2x48_apb3*vol_below(K) < 1e-8) then ! Could be 1e-7? + ! There is a very good approximation here for massless layers. + L0 = sqrt(2.0*vol_below(K)*Iapb) ; L(K) = L0*(1.0 + ax2_3apb*L0) + else + L(K) = apb_4a * (1.0 - & + 2.0 * cos(C1_3*acos(a2x48_apb3*vol_below(K) - 1.0) - C2pi_3)) + endif + ! To check the answers. + ! Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol_below(K) + else ! There are two separate open regions. + ! vol_below(K) = slope^2/4crv + crv/12 - (crv/12)*(1-L)^2*(1+2L) + ! At the deepest volume, L = slope/crv, at the top L = 1. + ! L(K) = 0.5 - cos(C1_3*acos(1.0 - C24_crv*(Vol_open - vol_below(K))) - C2pi_3) + tmp_val_m1_to_p1 = 1.0 - C24_crv*(Vol_open - vol_below(K)) + tmp_val_m1_to_p1 = max(-1., min(1., tmp_val_m1_to_p1)) + L(K) = 0.5 - cos(C1_3*acos(tmp_val_m1_to_p1) - C2pi_3) + ! To check the answers. + ! Vol_err = Vol_open - 0.25*crv_3*(1.0+2.0*L(K)) * (1.0-L(K))**2 - vol_below(K) + endif + enddo ! k loop to determine L(K) in the concave case + +end subroutine find_L_open_concave_analytic + + +!> Determine the normalized open length of each interface for convex bathymetry (from the ocean perspective) using +!! Newton's method iterations. In this case there is a single open region with the minimum depth +!! at one edge of the cell. +subroutine find_L_open_convex(vol_below, D_vel, Dp, Dm, L, GV, US, CS) + type(verticalGrid_type), intent(in) :: GV !< The ocean's vertical grid structure. + real, dimension(SZK_(GV)+1), intent(in) :: vol_below !< The volume below each interface, normalized by + !! the full horizontal area of a velocity cell [Z ~> m] + real, intent(in) :: D_vel !< The average bottom depth at a velocity point [Z ~> m] + real, intent(in) :: Dp !< The larger of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, intent(in) :: Dm !< The smaller of the two depths at the edge + !! of a velocity cell [Z ~> m] + real, dimension(SZK_(GV)+1), intent(out) :: L !< The fraction of the full cell width that is open at + !! the depth of each interface [nondim] + type(unit_scale_type), intent(in) :: US !< A dimensional unit scaling type + type(set_visc_CS), intent(in) :: CS !< The control structure returned by a previous + !! call to set_visc_init. + + ! Local variables + real :: crv ! crv is the curvature of the bottom depth across a + ! cell, times the cell width squared [Z ~> m]. + real :: crv_3 ! crv/3 [Z ~> m]. + real :: slope ! The absolute value of the bottom depth slope across + ! a cell times the cell width [Z ~> m]. + ! All of the following "volumes" have units of vertical heights because they are normalized + ! by the full horizontal area of a velocity cell. + real :: Vol_err ! The error in the volume with the latest estimate of + ! L, or the error for the interface below [Z ~> m]. + real :: Vol_quit ! The volume error below which to quit iterating [Z ~> m]. + real :: Vol_tol ! A volume error tolerance [Z ~> m]. + real :: Vol_open ! The cell volume above which the face is fully open [Z ~> m]. + real :: Vol_direct ! With less than Vol_direct [Z ~> m], there is a direct + ! solution of a cubic equation for L. + real :: Vol_err_max ! The volume error for the upper bound on the correct value for L [Z ~> m] + real :: Vol_err_min ! The volume error for the lower bound on the correct value for L [Z ~> m] + real :: Vol_0 ! A deeper volume with known width L0 [Z ~> m]. + real :: dVol ! vol - Vol_0 [Z ~> m]. + real :: dV_dL2 ! The partial derivative of volume with L squared + ! evaluated at L=L0 [Z ~> m]. + real :: L_direct ! The value of L above volume Vol_direct [nondim]. + real :: L_max, L_min ! Upper and lower bounds on the correct value for L [nondim]. + real :: L0 ! The value of L above volume Vol_0 [nondim]. + real :: Iapb, Ibma_2 ! Combinations of crv (a) and slope (b) [Z-1 ~> m-1] + real :: C24_crv ! 24/crv [Z-1 ~> m-1]. + real :: curv_tol ! Numerator of curvature cubed, used to estimate + ! accuracy of a single L(:) Newton iteration [Z5 ~> m5] + real, parameter :: C1_3 = 1.0/3.0, C1_6 = 1.0/6.0 ! Rational constants [nondim] + logical :: use_L0, do_one_L_iter ! Control flags for L(:) Newton iteration + integer :: K, nz, itt, maxitt=20 + + nz = GV%ke + + ! Each cell extends from x=-1/2 to 1/2, and has a topography + ! given by D(x) = crv*x^2 + slope*x + D_vel - crv/12. + crv_3 = (Dp + Dm - 2.0*D_vel) ; crv = 3.0*crv_3 + if (crv > 0.0) call MOM_error(FATAL, "find_L_open_convex should only be called with a negative curvature.") + slope = Dp - Dm + + ! Calculate the volume above which the entire cell is open and the volume at which the + ! equation that is solved for L changes because there is a direct solution. + Vol_open = D_vel - Dm + if (slope >= -crv) then + Iapb = 1.0e30*US%Z_to_m ; if (slope+crv /= 0.0) Iapb = 1.0/(crv+slope) + Vol_direct = 0.0 ; L_direct = 0.0 ; C24_crv = 0.0 + else + C24_crv = 24.0/crv ; Iapb = 1.0/(crv+slope) + L_direct = 1.0 + slope/crv ! L_direct < 1 because crv < 0 + Vol_direct = -C1_6*crv*L_direct**3 + endif + Ibma_2 = 2.0 / (slope - crv) + + if (CS%answer_date < 20190101) Vol_quit = (0.9*GV%Angstrom_Z + GV%dZ_subroundoff) + + L(nz+1) = 0.0 ; Vol_err = 0.0 + ! Determine the normalized open length (L) at each interface. + do K=nz,1,-1 + if (vol_below(K) >= Vol_open) then + L(K) = 1.0 + elseif (vol_below(K) <= Vol_direct) then + ! Both edges of the cell are bounded by walls. + ! if (CS%answer_date < 20240101)) then + L(K) = (-0.25*C24_crv*vol_below(K))**C1_3 + ! else + ! L(K) = cuberoot(-0.25*C24_crv*vol_below(K)) + ! endif + else + ! x_R is at 1/2 but x_L is in the interior & L is found by iteratively solving + ! vol_below(K) = 0.5*L^2*(slope + crv/3*(3-4L)) + + ! Vol_err = 0.5*(L(K+1)*L(K+1))*(slope + crv_3*(3.0-4.0*L(K+1))) - vol_below(K+1) + ! Change to ... + ! if (min(vol_below(K+1) + Vol_err, vol_below(K)) <= Vol_direct) then ? + if (vol_below(K+1) + Vol_err <= Vol_direct) then + L0 = L_direct ; Vol_0 = Vol_direct + else + L0 = L(K+1) ; Vol_0 = vol_below(K+1) + Vol_err + ! Change to Vol_0 = min(vol_below(K+1) + Vol_err, vol_below(K)) ? + endif + + ! Try a relatively simple solution that usually works well + ! for massless layers. + dV_dL2 = 0.5*(slope+crv) - crv*L0 ; dVol = (vol_below(K)-Vol_0) + ! dV_dL2 = 0.5*(slope+crv) - crv*L0 ; dVol = max(vol_below(K)-Vol_0, 0.0) + + use_L0 = .false. + do_one_L_iter = .false. + if (CS%answer_date < 20190101) then + curv_tol = GV%Angstrom_Z*dV_dL2**2 & + * (0.25 * dV_dL2 * GV%Angstrom_Z - crv * L0 * dVol) + do_one_L_iter = (crv * crv * dVol**3) < curv_tol + else + ! The following code is more robust when GV%Angstrom_H=0, but + ! it changes answers. + use_L0 = (dVol <= 0.) + + Vol_tol = max(0.5 * GV%Angstrom_Z + GV%dZ_subroundoff, 1e-14 * vol_below(K)) + Vol_quit = max(0.9 * GV%Angstrom_Z + GV%dZ_subroundoff, 1e-14 * vol_below(K)) + + curv_tol = Vol_tol * dV_dL2**2 & + * (dV_dL2 * Vol_tol - 2.0 * crv * L0 * dVol) + do_one_L_iter = (crv * crv * dVol**3) < curv_tol + endif + + if (use_L0) then + L(K) = L0 + Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol_below(K) + elseif (do_one_L_iter) then + ! One iteration of Newton's method should give an estimate + ! that is accurate to within Vol_tol. + L(K) = sqrt(L0*L0 + dVol / dV_dL2) + Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol_below(K) + else + if (dV_dL2*(1.0-L0*L0) < dVol + & + dV_dL2 * (Vol_open - vol_below(K))*Ibma_2) then + L_max = sqrt(1.0 - (Vol_open - vol_below(K))*Ibma_2) + else + L_max = sqrt(L0*L0 + dVol / dV_dL2) + endif + L_min = sqrt(L0*L0 + dVol / (0.5*(slope+crv) - crv*L_max)) + + Vol_err_min = 0.5*(L_min**2)*(slope + crv_3*(3.0-4.0*L_min)) - vol_below(K) + Vol_err_max = 0.5*(L_max**2)*(slope + crv_3*(3.0-4.0*L_max)) - vol_below(K) + ! if ((abs(Vol_err_min) <= Vol_quit) .or. (Vol_err_min >= Vol_err_max)) then + if (abs(Vol_err_min) <= Vol_quit) then + L(K) = L_min ; Vol_err = Vol_err_min + else + L(K) = sqrt((L_min**2*Vol_err_max - L_max**2*Vol_err_min) / & + (Vol_err_max - Vol_err_min)) + do itt=1,maxitt + Vol_err = 0.5*(L(K)*L(K))*(slope + crv_3*(3.0-4.0*L(K))) - vol_below(K) + if (abs(Vol_err) <= Vol_quit) exit + ! Take a Newton's method iteration. This equation has proven + ! robust enough not to need bracketing. + L(K) = L(K) - Vol_err / (L(K)* (slope + crv - 2.0*crv*L(K))) + ! This would be a Newton's method iteration for L^2: + ! L(K) = sqrt(L(K)*L(K) - Vol_err / (0.5*(slope+crv) - crv*L(K))) + enddo + endif ! end of iterative solver + endif ! end of 1-boundary alternatives. + endif ! end of 0, 1- and 2- boundary cases. + enddo ! k loop to determine L(K) in the convex case + +end subroutine find_L_open_convex + !> This subroutine finds a thickness-weighted value of v at the u-points. function set_v_at_u(v, h, G, GV, i, j, k, mask2dCv, OBC) type(ocean_grid_type), intent(in) :: G !< The ocean's grid structure