Update MOM_mixed_layer_restrat.F90

Adding and calculating diagnostic front length scale (lf_bodner) to
mixed layer restratification code

Co-authored-by: Kate Hedstrom <kshedstrom@alaska.edu>
Co-authored-by: Marshall Ward <marshall.ward@noaa.gov>

src/parameterizations/lateral/MOM_mixed_layer_restrat.F90

@@ -14,6 +14,7 @@ module MOM_mixed_layer_restrat
 use MOM_forcing_type,  only : mech_forcing, find_ustar
 use MOM_grid,          only : ocean_grid_type
 use MOM_hor_index,     only : hor_index_type
+use MOM_intrinsic_functions, only : cuberoot
 use MOM_lateral_mixing_coeffs, only : VarMix_CS
 use MOM_restart,       only : register_restart_field, query_initialized, MOM_restart_CS
 use MOM_unit_scaling,  only : unit_scale_type
@@ -114,6 +115,7 @@ module MOM_mixed_layer_restrat
   integer :: id_wpup = -1
   integer :: id_ustar = -1
   integer :: id_bflux = -1
+  integer :: id_lfbod = -1
   !>@}
 
 end type mixedlayer_restrat_CS
@@ -801,6 +803,7 @@ subroutine mixedlayer_restrat_Bodner(CS, G, GV, US, h, uhtr, vhtr, tv, forces, d
     wpup                  ! Turbulent vertical momentum [L H T-2 ~> m2 s-2 or kg m-1 s-2]
   real :: uDml_diag(SZIB_(G),SZJ_(G))  ! A 2D copy of uDml for diagnostics [H L2 T-1 ~> m3 s-1 or kg s-1]
   real :: vDml_diag(SZI_(G),SZJB_(G))  ! A 2D copy of vDml for diagnostics [H L2 T-1 ~> m3 s-1 or kg s-1]
+  real :: lf_bodner_diag(SZI_(G),SZJ_(G)) ! Front width as in Bodner et al., 2023 (B22), eq 24 [L ~> m]
   real :: U_star_2d(SZI_(G),SZJ_(G))   ! The wind friction velocity, calculated using the Boussinesq
                           ! reference density or the time-evolving surface density in non-Boussinesq
                           ! mode [Z T-1 ~> m s-1]
@@ -823,6 +826,8 @@ subroutine mixedlayer_restrat_Bodner(CS, G, GV, US, h, uhtr, vhtr, tv, forces, d
   real :: u_star3         ! Cube of surface fruction velocity [m3 s-3]
   real :: r_wpup          ! reciprocal of vertical momentum flux [T2 L-1 H-1 ~> s2 m-2 or m s2 kg-1]
   real :: absf            ! absolute value of f, interpolated to velocity points [T-1 ~> s-1]
+  real :: f2_h            ! Coriolis parameter at h-points squared [T-2 ~> s-2]
+  real :: absurdly_small_freq2 ! Frequency squared used to avoid division by 0 [T-2 ~> s-2]
   real :: grid_dsd        ! combination of grid scales [L2 ~> m2]
   real :: h_sml           ! "Little h", the active mixing depth with diurnal cycle removed [H ~> m or kg m-2]
   real :: h_big           ! "Big H", the mixed layer depth based on a time filtered "little h" [H ~> m or kg m-2]
@@ -852,6 +857,9 @@ subroutine mixedlayer_restrat_Bodner(CS, G, GV, US, h, uhtr, vhtr, tv, forces, d
   covTS(:) = 0.0 ! Might be in tv% in the future. Not implemented for the time being.
   varS(:) = 0.0  ! Ditto.
 
+ ! This value is roughly (pi / (the age of the universe) )^2.
+  absurdly_small_freq2 = 1e-34*US%T_to_s**2
+
   if (.not.associated(tv%eqn_of_state)) call MOM_error(FATAL, "mixedlayer_restrat_Bodner: "// &
          "An equation of state must be used with this module.")
   if (.not.CS%MLE_use_PBL_MLD) call MOM_error(FATAL, "mixedlayer_restrat_Bodner: "// &
@@ -930,6 +938,21 @@ subroutine mixedlayer_restrat_Bodner(CS, G, GV, US, h, uhtr, vhtr, tv, forces, d
     wpup(i,j) = rmean2ts(wpup(i,j), CS%wpup_filtered(i,j), &
                          CS%BLD_growing_Tfilt, CS%BLD_decaying_Tfilt, dt)
     CS%wpup_filtered(i,j) = wpup(i,j)
+
+    if (CS%id_lfbod > 0) then
+      ! Calculate front length used in B22 formula (eq 24).
+
+      ! Include an absurdly_small_freq2 to prevent division by zero.
+      f2_h = max( &
+          (0.25*((G%CoriolisBu(I,J)  + G%CoriolisBu(I-1,J-1)) &
+            + (G%CoriolisBu(I-1,J) + G%CoriolisBu(I,J-1))))**2, &
+          absurdly_small_freq2)
+
+      lf_bodner_diag(i,j) = &
+          0.25 * (cuberoot(CS%mstar * u_star3 + CS%nstar * w_star3)**2 &
+            * (US%m_to_L * GV%m_to_H * US%T_to_s**2) &
+          ) / (f2_h * max(little_h(i,j), GV%Angstrom_H))
+    endif
   enddo ; enddo
 
   if (CS%debug) then
@@ -1117,6 +1140,7 @@ subroutine mixedlayer_restrat_Bodner(CS, G, GV, US, h, uhtr, vhtr, tv, forces, d
     if (CS%id_vhml  > 0) call post_data(CS%id_vhml, vhml, CS%diag)
     if (CS%id_uDml  > 0) call post_data(CS%id_uDml, uDml_diag, CS%diag)
     if (CS%id_vDml  > 0) call post_data(CS%id_vDml, vDml_diag, CS%diag)
+    if (CS%id_lfbod  > 0) call post_data(CS%id_lfbod, lf_bodner_diag, CS%diag)
 
     if (CS%id_uml > 0) then
       do J=js,je ; do i=is-1,ie
@@ -1727,14 +1751,17 @@ logical function mixedlayer_restrat_init(Time, G, GV, US, param_file, diag, CS,
       'm s-1', conversion=US%L_T_to_m_s)
   if (CS%use_Bodner) then
     CS%id_wpup = register_diag_field('ocean_model', 'MLE_wpup', diag%axesT1, Time, &
-        'Vertical turbulent momentum flux in Bodner mixed layer restratificiation parameterization', &
+        'Vertical turbulent momentum flux in Bodner mixed layer restratification parameterization', &
         'm2 s-2', conversion=US%L_to_m*GV%H_to_m*US%s_to_T**2)
     CS%id_ustar = register_diag_field('ocean_model', 'MLE_ustar', diag%axesT1, Time, &
-        'Surface turbulent friction velicity, u*, in Bodner mixed layer restratificiation parameterization', &
+        'Surface turbulent friction velocity, u*, in Bodner mixed layer restratification parameterization', &
         'm s-1', conversion=(US%Z_to_m*US%s_to_T))
     CS%id_bflux = register_diag_field('ocean_model', 'MLE_bflux', diag%axesT1, Time, &
-        'Surface buoyancy flux, B0, in Bodner mixed layer restratificiation parameterization', &
+        'Surface buoyancy flux, B0, in Bodner mixed layer restratification parameterization', &
         'm2 s-3', conversion=(US%Z_to_m**2*US%s_to_T**3))
+    CS%id_lfbod = register_diag_field('ocean_model', 'lf_bodner', diag%axesT1, Time, &
+        'Front length in Bodner mixed layer restratificiation parameterization', &
+        'm', conversion=(US%L_to_m))
   endif
 
   ! If MLD_filtered is being used, we need to update halo regions after a restart