mom_thickness_diffuse mom_thickness_diffuse::thickness_diffuse_cs subroutine, public subroutine, public mom_thickness_diffuse::thickness_diffuse (h, uhtr, vhtr, tv, dt, G, GV, US, MEKE, VarMix, CDp, CS) thickness_diffuse h h uhtr uhtr vhtr vhtr tv tv dt dt G G GV GV US US MEKE MEKE VarMix VarMix CDp CDp CS CS Calculates thickness diffusion coefficients and applies thickness diffusion to layer thicknesses, h. Diffusivities are limited to ensure stability. Also returns along-layer mass fluxes used in the continuity equation. g Ocean grid structure gv Vertical grid structure us A dimensional unit scaling type h Layer thickness [H ~> m or kg m-2] uhtr Accumulated zonal mass flux [L2 H ~> m3 or kg] vhtr Accumulated meridional mass flux [L2 H ~> m3 or kg] tv Thermodynamics structure dt Time increment [T ~> s] meke MEKE control structure varmix Variable mixing coefficients cdp Diagnostics for the continuity equation cs Control structure for thickness diffusion add_detangling_kh mom_diag_mediator::diag_update_remap_grids mom_error_handler::mom_error thickness_diffuse_full mom::step_mom_dynamics subroutine subroutine mom_thickness_diffuse::thickness_diffuse_full (h, e, Kh_u, Kh_v, tv, uhD, vhD, cg1, dt, G, GV, US, MEKE, CS, int_slope_u, int_slope_v, slope_x, slope_y) thickness_diffuse_full h h e e Kh_u Kh_u Kh_v Kh_v tv tv uhD uhD vhD vhD cg1 cg1 dt dt G G GV GV US US MEKE MEKE CS CS int_slope_u int_slope_u int_slope_v int_slope_v slope_x slope_x slope_y slope_y Calculates parameterized layer transports for use in the continuity equation. Fluxes are limited to give positive definite thicknesses. Called by thickness_diffuse(). g Ocean grid structure gv Vertical grid structure us A dimensional unit scaling type h Layer thickness [H ~> m or kg m-2] e Interface positions [Z ~> m] kh_u Thickness diffusivity on interfaces at u points [L2 T-1 ~> m2 s-1] kh_v Thickness diffusivity on interfaces at v points [L2 T-1 ~> m2 s-1] tv Thermodynamics structure uhd Zonal mass fluxes [H L2 T-1 ~> m3 s-1 or kg s-1] vhd Meridional mass fluxes [H L2 T-1 ~> m3 s-1 or kg s-1] cg1 Wave speed [L T-1 ~> m s-1] dt Time increment [T ~> s] meke MEKE control structure cs Control structure for thickness diffusion int_slope_u Ratio that determine how much of the isopycnal slopes are taken directly from the interface slopes without consideration of density gradients [nondim]. int_slope_v Ratio that determine how much of the isopycnal slopes are taken directly from the interface slopes without consideration of density gradients [nondim]. slope_x Isopycnal slope at u-points slope_y Isopycnal slope at v-points mom_error_handler::mom_error streamfn_solver mom_isopycnal_slopes::vert_fill_ts thickness_diffuse subroutine subroutine mom_thickness_diffuse::streamfn_solver (nk, c2_h, hN2, sfn) streamfn_solver nk nk c2_h c2_h hN2 hN2 sfn sfn Tridiagonal solver for streamfunction at interfaces. nk Number of layers c2_h Wave speed squared over thickness in layers [L2 Z-1 T-2 ~> m s-2] hn2 Thickness times N2 at interfaces [L2 Z-1 T-2 ~> m s-2] sfn Streamfunction [Z L2 T-1 ~> m3 s-1] or arbitrary units On entry, equals diffusivity times slope. On exit, equals the streamfunction. thickness_diffuse_full subroutine subroutine mom_thickness_diffuse::add_detangling_kh (h, e, Kh_u, Kh_v, KH_u_CFL, KH_v_CFL, tv, dt, G, GV, US, CS, int_slope_u, int_slope_v) add_detangling_kh h h e e Kh_u Kh_u Kh_v Kh_v KH_u_CFL KH_u_CFL KH_v_CFL KH_v_CFL tv tv dt dt G G GV GV US US CS CS int_slope_u int_slope_u int_slope_v int_slope_v Modifies thickness diffusivities to untangle layer structures. g Ocean grid structure gv Vertical grid structure us A dimensional unit scaling type h Layer thickness [H ~> m or kg m-2] e Interface positions [Z ~> m] kh_u Thickness diffusivity on interfaces at u points [L2 T-1 ~> m2 s-1] kh_v Thickness diffusivity on interfaces at v points [L2 T-1 ~> m2 s-1] kh_u_cfl Maximum stable thickness diffusivity at u points [L2 T-1 ~> m2 s-1] kh_v_cfl Maximum stable thickness diffusivity at v points [L2 T-1 ~> m2 s-1] tv Thermodynamics structure dt Time increment [T ~> s] cs Control structure for thickness diffusion int_slope_u Ratio that determine how much of the isopycnal slopes are taken directly from the interface slopes without consideration of density gradients. int_slope_v Ratio that determine how much of the isopycnal slopes are taken directly from the interface slopes without consideration of density gradients. thickness_diffuse subroutine, public subroutine, public mom_thickness_diffuse::thickness_diffuse_init (Time, G, GV, US, param_file, diag, CDp, CS) thickness_diffuse_init Time Time G G GV GV US US param_file param_file diag diag CDp CDp CS CS Initialize the thickness diffusion module/structure. time Current model time g Ocean grid structure gv Vertical grid structure us A dimensional unit scaling type param_file Parameter file handles diag Diagnostics control structure cdp Continuity equation diagnostics cs Control structure for thickness diffusion mom_error_handler::mom_error mom::initialize_mom subroutine, public subroutine, public mom_thickness_diffuse::thickness_diffuse_get_kh (CS, KH_u_GME, KH_v_GME, G) thickness_diffuse_get_kh CS CS KH_u_GME KH_u_GME KH_v_GME KH_v_GME G G Copies ubtav and vbtav from private type into arrays. cs Control structure for this module g Grid structure kh_u_gme interface height diffusivities at u-faces [L2 T-1 ~> m2 s-1] kh_v_gme interface height diffusivities at v-faces [L2 T-1 ~> m2 s-1] mom_hor_visc::horizontal_viscosity subroutine, public subroutine, public mom_thickness_diffuse::thickness_diffuse_end (CS) thickness_diffuse_end CS CS Deallocate the thickness diffusion control structure. cs Control structure for thickness diffusion Thickness diffusion (or Gent McWilliams) Thickness diffusion is implemented via along-layer mass fluxes \[ h^\dagger \leftarrow h^n - \Delta t \nabla \cdot ( \vec{uh}^* ) \] where the mass fluxes are cast as the difference in vector streamfunction \[ \vec{uh}^* = \delta_k \vec{\psi} . \] The GM implementation of thickness diffusion made the streamfunction proportional to the potential density slope \[ \vec{\psi} = - \kappa_h \frac{\nabla_z \rho}{\partial_z \rho} = \frac{g\kappa_h}{\rho_o} \frac{\nabla \rho}{N^2} = \kappa_h \frac{M^2}{N^2} \] but for robustness the scheme is implemented as \[ \vec{\psi} = \kappa_h \frac{M^2}{\sqrt{N^4 + M^4}} \] since the quantity $\frac{M^2}{\sqrt{N^2 + M^2}}$ is bounded between $-1$ and $1$ and does not change sign if $N^2<0$. Optionally, the method of Ferrari et al, 2010, can be used to obtain the streamfunction which solves the vertically elliptic equation: \[ \gamma_F \partial_z c^2 \partial_z \psi - N_*^2 \psi = ( 1 + \gamma_F ) \kappa_h N_*^2 \frac{M^2}{\sqrt{N^4+M^4}} \] which recovers the previous streamfunction relation in the limit that $ c \rightarrow 0 $. Here, $c=\max(c_{min},c_g)$ is the maximum of either $c_{min}$ and either the first baroclinic mode wave-speed or the equivalent barotropic mode wave-speed. $N_*^2 = \max(N^2,0)$ is a non-negative form of the square of the Brunt-Vaisala frequency. The parameter $\gamma_F$ is used to reduce the vertical smoothing length scale. \[ \kappa_h = \left( \kappa_o + \alpha_{s} L_{s}^2 < S N > + \alpha_{M} \kappa_{M} \right) r(\Delta x,L_d) \] where $ S $ is the isoneutral slope magnitude, $ N $ is the square root of Brunt-Vaisala frequency, $\kappa_{M}$ is the diffusivity calculated by the MEKE parameterization (mom_meke module) and $ r(\Delta x,L_d) $ is a function of the local resolution (ratio of grid-spacing, $\Delta x$, to deformation radius, $L_d$). The length $L_s$ is provided by the mom_lateral_mixing_coeffs module (enabled with USE_VARIABLE_MIXING=True and the term $<SN>$ is the vertical average slope times the Brunt-Vaisala frequency prescribed by Visbeck et al., 1996. The result of the above expression is subsequently bounded by minimum and maximum values, including an upper diffusivity consistent with numerical stability ( $ \kappa_{cfl} $ is calculated internally). \[ \kappa_h \leftarrow \min{\left( \kappa_{max}, \kappa_{cfl}, \max{\left( \kappa_{min}, \kappa_h \right)} \right)} f(c_g,z) \] where $f(c_g,z)$ is a vertical structure function. $f(c_g,z)$ is calculated in module mom_lateral_mixing_coeffs. If KHTH_USE_EBT_STRUCT=True then $f(c_g,z)$ is set to look like the equivalent barotropic modal velocity structure. Otherwise $f(c_g,z)=1$ and the diffusivity is independent of depth. In order to calculate meaningful slopes in vanished layers, temporary copies of the thermodynamic variables are passed through a vertical smoother, function vert_fill_ts(): \begin{eqnarray*} \left[ 1 + \Delta t \kappa_{smth} \frac{\partial^2}{\partial_z^2} \right] \theta & \leftarrow & \theta \\ \left[ 1 + \Delta t \kappa_{smth} \frac{\partial^2}{\partial_z^2} \right] s & \leftarrow & s \end{eqnarray*} Symbol Module parameter - THICKNESSDIFFUSE $ \kappa_o $ KHTH $ \alpha_{s} $ KHTH_SLOPE_CFF $ \kappa_{min} $ KHTH_MIN $ \kappa_{max} $ KHTH_MAX - KHTH_MAX_CFL $ \kappa_{smth} $ KD_SMOOTH $ \alpha_{M} $ MEKE_KHTH_FAC (from mom_meke module) - KHTH_USE_EBT_STRUCT (from mom_lateral_mixing_coeffs module) - KHTH_USE_FGNV_STREAMFUNCTION $ \gamma_F $ FGNV_FILTER_SCALE $ c_{min} $ FGNV_C_MIN Ferrari, R., S.M. Griffies, A.J.G. Nurser and G.K. Vallis, 2010: A boundary-value problem for the parameterized mesoscale eddy transport. Ocean Modelling, 32, 143-156. http://doi.org/10.1016/j.ocemod.2010.01.004 Viscbeck, M., J.C. Marshall, H. Jones, 1996: Dynamics of isolated convective regions in the ocean. J. Phys. Oceangr., 26, 1721-1734. http://dx.doi.org/10.1175/1520-0485(1996)026%3C1721:DOICRI%3E2.0.CO;2