Normalize wt_[uv] in MOM_barotropic

Add an option to normalize wt_[uv] in the barotropic solver. This is
fix step 1 of 2 to address the issue that visc_rem is applied
incorrectly to the barotropic solver.

A runtime flag BAROTROPIC_VERTICAL_WEIGHT_FIX is added to control the
behavior of this commit. The current default is false (not applying the
fix).

src/core/MOM_barotropic.F90

@@ -282,6 +282,7 @@ module MOM_barotropic
   logical :: tidal_sal_flather !< Apply adjustment to external gravity wave speed
                              !! consistent with tidal self-attraction and loading
                              !! used within the barotropic solver
+  logical :: wt_uv_fix       !< If true, use a normalized wt_[uv] for vertical averages.
   type(time_type), pointer :: Time  => NULL() !< A pointer to the ocean models clock.
   type(diag_ctrl), pointer :: diag => NULL()  !< A structure that is used to regulate
                              !! the timing of diagnostic output.
@@ -506,6 +507,9 @@ subroutine btstep(U_in, V_in, eta_in, dt, bc_accel_u, bc_accel_v, forces, pbce,
   real :: wt_v(SZI_(G),SZJB_(G),SZK_(GV)) ! normalized weights to
                 ! be used in calculating barotropic velocities, possibly with
                 ! sums less than one due to viscous losses [nondim]
+  real :: Iwt_u_tot(SZIB_(G),SZJ_(G)) ! Iwt_u_tot and Iwt_v_tot are the
+  real :: Iwt_v_tot(SZI_(G),SZJB_(G)) ! inverses of wt_u and wt_v vertical integrals,
+                ! used to normalize wt_u and wt_v [nondim]
   real, dimension(SZIB_(G),SZJ_(G)) :: &
     av_rem_u, &   ! The weighted average of visc_rem_u [nondim]
     tmp_u, &      ! A temporary array at u points [L T-2 ~> m s-2] or [nondim]
@@ -1052,6 +1056,30 @@ subroutine btstep(U_in, V_in, eta_in, dt, bc_accel_u, bc_accel_v, forces, pbce,
     wt_v(i,J,k) = CS%frhatv(i,J,k) * visc_rem
   enddo ; enddo ; enddo
 
+  if (CS%wt_uv_fix) then
+    do j=js,je ; do I=is-1,ie ; Iwt_u_tot(I,j) = wt_u(I,j,1) ; enddo ; enddo
+    do k=2,nz ; do j=js,je ; do I=is-1,ie
+      Iwt_u_tot(I,j) = Iwt_u_tot(I,j) + wt_u(I,j,k)
+    enddo ; enddo ; enddo
+    do j=js,je ; do I=is-1,ie
+      Iwt_u_tot(I,j) = G%mask2dCu(I,j) / (Iwt_u_tot(I,j) + h_neglect)
+    enddo ; enddo
+    do k=1,nz ; do j=js,je ; do I=is-1,ie
+      wt_u(I,j,k) = wt_u(I,j,k) * Iwt_u_tot(I,j)
+    enddo ; enddo ; enddo
+
+    do J=js-1,je ; do i=is,ie ; Iwt_v_tot(i,J) = wt_v(i,J,1) ; enddo ; enddo
+    do k=2,nz ; do J=js-1,je ; do i=is,ie
+      Iwt_v_tot(i,J) = Iwt_v_tot(i,J) + wt_v(i,J,k)
+    enddo ; enddo ; enddo
+    do J=js-1,je ; do i=is,ie
+      Iwt_v_tot(i,J) = G%mask2dCv(i,J) / (Iwt_v_tot(i,J) + h_neglect)
+    enddo ; enddo
+    do k=1,nz ; do J=js-1,je ; do i=is,ie
+      wt_v(i,J,k) = wt_v(i,J,k) * Iwt_v_tot(i,J)
+    enddo ; enddo ; enddo
+  endif
+
   !   Use u_Cor and v_Cor as the reference values for the Coriolis terms,
   ! including the viscous remnant.
   !$OMP parallel do default(shared)
@@ -4580,10 +4608,14 @@ subroutine barotropic_init(u, v, h, eta, Time, G, GV, US, param_file, diag, CS,
   call get_param(param_file, mdl, "BAROTROPIC_ANSWER_DATE", CS%answer_date, &
                  "The vintage of the expressions in the barotropic solver. "//&
                  "Values below 20190101 recover the answers from the end of 2018, "//&
-                 "while higher values uuse more efficient or general expressions.", &
+                 "while higher values use more efficient or general expressions.", &
                  default=default_answer_date, do_not_log=.not.GV%Boussinesq)
   if (.not.GV%Boussinesq) CS%answer_date = max(CS%answer_date, 20230701)
 
+  call get_param(param_file, mdl, "BAROTROPIC_VERTICAL_WEIGHT_FIX", CS%wt_uv_fix, &
+                 "If true, use a normalized weight function for vertical averages of "//&
+                 "baroclinic velocity and forcing.", &
+                 default=.false.)
   call get_param(param_file, mdl, "TIDES", use_tides, &
                  "If true, apply tidal momentum forcing.", default=.false.)
   call get_param(param_file, mdl, "CALCULATE_SAL", CS%calculate_SAL, &