mom_meke mom_meke::meke_cs subroutine, public subroutine, public mom_meke::step_forward_meke (MEKE, h, SN_u, SN_v, visc, dt, G, GV, US, CS, hu, hv) step_forward_meke MEKE MEKE h h SN_u SN_u SN_v SN_v visc visc dt dt G G GV GV US US CS CS hu hu hv hv Integrates forward-in-time the MEKE eddy energy equation. See MEKE equations. meke MEKE data. g Ocean grid. gv Ocean vertical grid structure. us A dimensional unit scaling type h Layer thickness [H ~> m or kg m-2]. sn_u Eady growth rate at u-points [T-1 ~> s-1]. sn_v Eady growth rate at v-points [T-1 ~> s-1]. visc The vertical viscosity type. dt Model(baroclinic) time-step [T ~> s]. cs MEKE control structure. hu Accumlated zonal mass flux [H L2 ~> m3 or kg]. hv Accumlated meridional mass flux [H L2 ~> m3 or kg] mom_domains::do_group_pass meke_equilibrium meke_equilibrium_restoring meke_lengthscales mom_error_handler::mom_error mom::step_mom_dynamics subroutine subroutine mom_meke::meke_equilibrium (CS, MEKE, G, GV, US, SN_u, SN_v, drag_rate_visc, I_mass) meke_equilibrium CS CS MEKE MEKE G G GV GV US US SN_u SN_u SN_v SN_v drag_rate_visc drag_rate_visc I_mass I_mass Calculates the equilibrium solutino where the source depends only on MEKE diffusivity and there is no lateral diffusion of MEKE. Results is in MEKEMEKE. g Ocean grid. gv Ocean vertical grid structure. us A dimensional unit scaling type cs MEKE control structure. meke A structure with MEKE data. sn_u Eady growth rate at u-points [T-1 ~> s-1]. sn_v Eady growth rate at v-points [T-1 ~> s-1]. drag_rate_visc Mean flow velocity contribution to the MEKE drag rate [L T-1 ~> m s-1] i_mass Inverse of column mass [R-1 Z-1 ~> m2 kg-1]. meke_lengthscales_0d step_forward_meke subroutine subroutine mom_meke::meke_equilibrium_restoring (CS, G, US, SN_u, SN_v) meke_equilibrium_restoring CS CS G G US US SN_u SN_u SN_v SN_v g Ocean grid. us A dimensional unit scaling type. cs MEKE control structure. sn_u Eady growth rate at u-points [T-1 ~> s-1]. sn_v Eady growth rate at v-points [T-1 ~> s-1]. step_forward_meke subroutine subroutine mom_meke::meke_lengthscales (CS, MEKE, G, GV, US, SN_u, SN_v, EKE, bottomFac2, barotrFac2, LmixScale) meke_lengthscales CS CS MEKE MEKE G G GV GV US US SN_u SN_u SN_v SN_v EKE EKE bottomFac2 bottomFac2 barotrFac2 barotrFac2 LmixScale LmixScale Calculates the eddy mixing length scale and $\gamma_b$ and $\gamma_t$ functions that are ratios of either bottom or barotropic eddy energy to the column eddy energy, respectively. See MEKE equations. cs MEKE control structure. meke MEKE data. g Ocean grid. gv Ocean vertical grid structure. us A dimensional unit scaling type sn_u Eady growth rate at u-points [T-1 ~> s-1]. sn_v Eady growth rate at v-points [T-1 ~> s-1]. eke Eddy kinetic energy [L2 T-2 ~> m2 s-2]. bottomfac2 gamma_b^2 barotrfac2 gamma_t^2 lmixscale Eddy mixing length [L ~> m]. meke_lengthscales_0d step_forward_meke subroutine subroutine mom_meke::meke_lengthscales_0d (CS, US, area, beta, depth, Rd_dx, SN, EKE, bottomFac2, barotrFac2, LmixScale, Lrhines, Leady) meke_lengthscales_0d CS CS US US area area beta beta depth depth Rd_dx Rd_dx SN SN EKE EKE bottomFac2 bottomFac2 barotrFac2 barotrFac2 LmixScale LmixScale Lrhines Lrhines Leady Leady Calculates the eddy mixing length scale and $\gamma_b$ and $\gamma_t$ functions that are ratios of either bottom or barotropic eddy energy to the column eddy energy, respectively. See MEKE equations. cs MEKE control structure. us A dimensional unit scaling type area Grid cell area [L2 ~> m2] beta Planetary beta = $ \nabla f$ [T-1 L-1 ~> s-1 m-1] depth Ocean depth [Z ~> m] rd_dx Resolution Ld/dx [nondim]. sn Eady growth rate [T-1 ~> s-1]. eke Eddy kinetic energy [L2 T-2 ~> m2 s-2]. bottomfac2 gamma_b^2 barotrfac2 gamma_t^2 lmixscale Eddy mixing length [L ~> m]. lrhines Rhines length scale [L ~> m]. leady Eady length scale [L ~> m]. meke_equilibrium meke_lengthscales logical function, public logical function, public mom_meke::meke_init (Time, G, US, param_file, diag, CS, MEKE, restart_CS) meke_init Time Time G G US US param_file param_file diag diag CS CS MEKE MEKE restart_CS restart_CS Initializes the MOM_MEKE module and reads parameters. Returns True if module is to be used, otherwise returns False. time The current model time. g The ocean's grid structure. us A dimensional unit scaling type param_file Parameter file parser structure. diag Diagnostics structure. cs MEKE control structure. meke MEKE-related fields. restart_cs Restart control structure for MOM_MEKE. mom_cpu_clock::cpu_clock_id mom_domains::do_group_pass mom_error_handler::mom_error mom_error_handler::mom_mesg mom::initialize_mom subroutine, public subroutine, public mom_meke::meke_alloc_register_restart (HI, param_file, MEKE, restart_CS) meke_alloc_register_restart HI HI param_file param_file MEKE MEKE restart_CS restart_CS Allocates memory and register restart fields for the MOM_MEKE module. hi Horizontal index structure param_file Parameter file parser structure. meke A structure with MEKE-related fields. restart_cs Restart control structure for MOM_MEKE. mom_error_handler::mom_error mom_error_handler::mom_mesg mom_io::var_desc mom::initialize_mom subroutine, public subroutine, public mom_meke::meke_end (MEKE, CS) meke_end MEKE MEKE CS CS Deallocates any variables allocated in MEKE_init or MEKE_alloc_register_restart. meke A structure with MEKE-related fields. cs The control structure for MOM_MEKE. Implements the Mesoscale Eddy Kinetic Energy framework with topographic beta effect included in computing beta in Rhines scale. The MEKE framework accounts for the mean potential energy removed by the first order closures used to parameterize mesoscale eddies. It requires closure at the second order, namely dissipation and transport of eddy energy. Monitoring the sub-grid scale eddy energy budget provides a means to predict a sub-grid eddy-velocity scale which can be used in the lower order closures. The eddy kinetic energy equation is: \[ \partial_{\tilde{t}} E = \overbrace{ \dot{E}_b + \gamma_\eta \dot{E}_\eta + \gamma_v \dot{E}_v }^\text{sources} - \overbrace{ ( \lambda + C_d | U_d | \gamma_b^2 ) E }^\text{local dissipation} + \overbrace{ \nabla \cdot ( ( \kappa_E + \gamma_M \kappa_M ) \nabla E - \kappa_4 \nabla^3 E ) }^\text{smoothing} \] where $ E $ is the eddy kinetic energy (variable MEKE) with units of m2s-2, and $\tilde{t} = a t$ is a scaled time. The non-dimensional factor $ a\geq 1 $ is used to accelerate towards equilibrium. The MEKE equation is two-dimensional and obtained by depth averaging the the three-dimensional eddy energy equation. In the following expressions $ \left< \phi \right> = \frac{1}{H} \int^\eta_{-D} \phi \, dz $ maps three dimensional terms into the two-dimensional quantities needed. The source term $ \dot{E}_b $ is a constant background source of energy intended to avoid the limit $E\rightarrow 0$. The "GM" source term \[ \dot{E}_\eta = - \left< \overline{w^\prime b^\prime} \right> = \left< \kappa_h N^2S^2 \right> \approx \left< \kappa_h g\prime |\nabla_\sigma \eta|^2 \right>\] equals the mean potential energy removed by the Gent-McWilliams closure, and is excluded/included in the MEKE budget by the efficiency parameter $ \gamma_\eta \in [0,1] $. The "frictional" source term \[ \dot{E}_{v} = \left< \partial_i u_j \tau_{ij} \right> \] equals the mean kinetic energy removed by lateral viscous fluxes, and is excluded/included in the MEKE budget by the efficiency parameter $ \gamma_v \in [0,1] $. The local dissipation of $ E $ is parameterized through a linear damping, $\lambda$, and bottom drag, $ C_d | U_d | \gamma_b^2 $. The $ \gamma_b $ accounts for the weak projection of the column-mean eddy velocty to the bottom. In other words, the bottom velocity is estimated as $ \gamma_b U_e $. The bottom drag coefficient, $ C_d $ is the same as that used in the bottom friction in the mean model equations. The bottom drag velocity scale, $ U_d $, has contributions from the resolved state and $ E $: \[ U_d = \sqrt{ U_b^2 + |u|^2_{z=-D} + |\gamma_b U_e|^2 } .\] where the eddy velocity scale, $ U_e $, is given by: \[ U_e = \sqrt{ 2 E } .\] $ U_b $ is a constant background bottom velocity scale and is typically not used (i.e. set to zero). Following Jansen et al., 2015, the projection of eddy energy on to the bottom is given by the ratio of bottom energy to column mean energy: \[ \gamma_b^2 = \frac{E_b}{E} = \gamma_{d0} + \left( 1 + c_{b} \frac{L_d}{L_f} \right)^{-\frac{4}{5}} , \] \[ \gamma_b^2 \leftarrow \max{\left( \gamma_b^2, \gamma_{min}^2 \right)} . \] $ E $ is laterally diffused by a diffusivity $ \kappa_E + \gamma_M \kappa_M $ where $ \kappa_E $ is a constant diffusivity and the term $ \gamma_M \kappa_M $ is a "self diffusion" using the diffusivity calculated in the section Diffusivity derived from MEKE. $ \kappa_4 $ is a constant bi-harmonic diffusivity. The predicted eddy velocity scale, $ U_e $, can be combined with a mixing length scale to form a diffusivity. The primary use of a MEKE derived diffusivity is for use in thickness diffusion (module mom_thickness_diffuse) and optionally in along isopycnal mixing of tracers (module mom_tracer_hor_diff). The original form used (enabled with MEKE_OLD_LSCALE=True): \[ \kappa_M = \gamma_\kappa \sqrt{ \gamma_t^2 U_e^2 A_\Delta } \] where $ A_\Delta $ is the area of the grid cell. Following Jansen et al., 2015, we now use \[ \kappa_M = \gamma_\kappa l_M \sqrt{ \gamma_t^2 U_e^2 } \] where $ \gamma_\kappa \in [0,1] $ is a non-dimensional factor and, following Jansen et al., 2015, $\gamma_t^2$ is the ratio of barotropic eddy energy to column mean eddy energy given by \[ \gamma_t^2 = \frac{E_t}{E} = \left( 1 + c_{t} \frac{L_d}{L_f} \right)^{-\frac{1}{4}} , \] \[ \gamma_t^2 \leftarrow \max{\left( \gamma_t^2, \gamma_{min}^2 \right)} . \] The length-scale is a configurable combination of multiple length scales: \[ l_M = \left( \frac{\alpha_d}{L_d} + \frac{\alpha_f}{L_f} + \frac{\alpha_R}{L_R} + \frac{\alpha_e}{L_e} + \frac{\alpha_\Delta}{L_\Delta} + \frac{\delta[L_c]}{L_c} \right)^{-1} \] where \begin{eqnarray*} L_d & = & \sqrt{\frac{c_g^2}{f^2+2\beta c_g}} \sim \frac{ c_g }{f} \\\\ L_R & = & \sqrt{\frac{U_e}{\beta^*}} \\\\ L_e & = & \frac{U_e}{|S| N} \\\\ L_f & = & \frac{H}{c_d} \\\\ L_\Delta & = & \sqrt{A_\Delta} . \end{eqnarray*} $L_c$ is a constant and $\delta[L_c]$ is the impulse function so that the term $\frac{\delta[L_c]}{L_c}$ evaluates to $\frac{1}{L_c}$ when $L_c$ is non-zero but is dropped if $L_c=0$. $\beta^*$ is the effective $\beta$ that combines both the planetary vorticity gradient (i.e. $\beta=\nabla f$) and the topographic $\beta$ effect, with the latter weighed by a weighting constant, $c_\beta$, that varies from 0 to 1, so that $c_\beta=0$ means the topographic $\beta$ effect is ignored, while $c_\beta=1$ means it is fully considered. The new $\beta^*$ therefore takes the form of \[ \beta^* = \sqrt{( \partial_xf - c_\beta\frac{f}{D}\partial_xD )^2 + ( \partial_yf - c_\beta\frac{f}{D}\partial_yD )^2} \] where $D$ is water column depth at T points. As for $ \kappa_M $, the predicted eddy velocity scale can be used to form a harmonic eddy viscosity, \[ \kappa_u = \gamma_u \sqrt{ U_e^2 A_\Delta } \] as well as a biharmonic eddy viscosity, \[ \kappa_4 = \gamma_4 \sqrt{ U_e^2 A_\Delta^3 } \] Note that in steady-state (or when $ a>>1 $) and there is no diffusion of $ E $ then \[ \overline{E} \approx \frac{ \dot{E}_b + \gamma_\eta \dot{E}_\eta + \gamma_v \dot{E}_v }{ \lambda + C_d|U_d|\gamma_b^2 } . \] In the linear drag limit, where $ U_e << \min(U_b, |u|_{z=-D}, C_d^{-1}\lambda) $, the equilibrium becomes $ \overline{E} \approx \frac{ \dot{E}_b + \gamma_\eta \dot{E}_\eta + \gamma_v \dot{E}_v }{ \lambda + C_d \sqrt{ U_b^2 + |u|^2_{z=-D} } } $. In the nonlinear drag limit, where $ U_e >> \max(U_b, |u|_{z=-D}, C_d^{-1}\lambda) $, the equilibrium becomes $ \overline{E} \approx \left( \frac{ \dot{E}_b + \gamma_\eta \dot{E}_\eta + \gamma_v \dot{E}_v }{ \sqrt{2} C_d \gamma_b^3 } \right)^\frac{2}{3} $. Symbol Module parameter - USE_MEKE $ a $ MEKE_DTSCALE $ \dot{E}_b $ MEKE_BGSRC $ \gamma_\eta $ MEKE_GMCOEFF $ \gamma_v $ MEKE_FrCOEFF $ \lambda $ MEKE_DAMPING $ U_b $ MEKE_USCALE $ \gamma_{d0} $ MEKE_CD_SCALE $ c_{b} $ MEKE_CB $ c_{t} $ MEKE_CT $ \kappa_E $ MEKE_KH $ \kappa_4 $ MEKE_K4 $ \gamma_\kappa $ MEKE_KHCOEFF $ \gamma_M $ MEKE_KHMEKE_FAC $ \gamma_u $ MEKE_VISCOSITY_COEFF_KU $ \gamma_4 $ MEKE_VISCOSITY_COEFF_AU $ \gamma_{min}^2 $ MEKE_MIN_GAMMA2 $ \alpha_d $ MEKE_ALPHA_DEFORM $ \alpha_f $ MEKE_ALPHA_FRICT $ \alpha_R $ MEKE_ALPHA_RHINES $ \alpha_e $ MEKE_ALPHA_EADY $ \alpha_\Delta $ MEKE_ALPHA_GRID $ L_c $ MEKE_FIXED_MIXING_LENGTH $ c_\beta $ MEKE_TOPOGRAPHIC_BETA - MEKE_KHTH_FAC - MEKE_KHTR_FAC Symbol Model parameter $ C_d $ CDRAG Jansen, M. F., A. J. Adcroft, R. Hallberg, and I. M. Held, 2015: Parameterization of eddy fluxes based on a mesoscale energy budget. Ocean Modelling, 92, 2841, http://doi.org/10.1016/j.ocemod.2015.05.007 . Marshall, D. P., and A. J. Adcroft, 2010: Parameterization of ocean eddies: Potential vorticity mixing, energetics and Arnold first stability theorem. Ocean Modelling, 32, 188204, http://doi.org/10.1016/j.ocemod.2010.02.001 .