namespace mom_meke¶

Overview¶

Implements the Mesoscale Eddy Kinetic Energy framework with topographic beta effect included in computing beta in Rhines scale. More…

namespace mom_meke {

// global functions

subroutine, public 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
    );

logical function, public meke_init(
    Time Time,
    G G,
    US US,
    param_file param_file,
    diag diag,
    CS CS,
    MEKE MEKE,
    restart_CS restart_CS
    );

subroutine, public meke_alloc_register_restart(
    HI HI,
    param_file param_file,
    MEKE MEKE,
    restart_CS restart_CS
    );

subroutine, public meke_end(MEKE MEKE, CS CS);

} // namespace mom_meke

Detailed Documentation¶

Implements the Mesoscale Eddy Kinetic Energy framework with topographic beta effect included in computing beta in Rhines scale.

The Mesoscale Eddy Kinetic Energy (MEKE) framework¶

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.

MEKE equations¶

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 m 2 s -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.

MEKE source terms¶

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 \(\gamma_v \in [0,1]\).

MEKE dissipation terms¶

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:

\[\gamma_b^2 = \frac{E_b}{E} = \gamma_{d0} + \left( 1 + c_{b} \frac{L_d}{L_f} \right)^{-\frac{4}{5}} ,\]

\[\gamma_b^2 \leftarrow \max{\left( \gamma_b^2, \gamma_{min}^2 \right)} .\]

MEKE smoothing terms¶

\(E\) 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.

Diffusivity derived from MEKE¶

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

\[\begin{split}\begin{eqnarray*} L_d & = & \sqrt{\frac{c_g^2}{f^2+2\beta c_g}} \sim \frac{ c_g }{f} \\\\ L_R & = & \sqrt{\frac{U_e}{\beta^*}} \\\\ L_e & = & \frac{U_e}{|S| N} \\\\ L_f & = & \frac{H}{c_d} \\\\ L_\Delta & = & \sqrt{A_\Delta} . \end{eqnarray*}\end{split}\]

\(L_c\) 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.

Viscosity derived from MEKE¶

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,

\[\kappa_4 = \gamma_4 \sqrt{ U_e^2 A_\Delta^3 }\]

Limit cases for local source-dissipative balance¶

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 \(\overline{E} \approx \left( \frac{ \dot{E}_b + \gamma_\eta \dot{E}_\eta + \gamma_v \dot{E}_v }{ \sqrt{2} C_d \gamma_b^3 } \right)^\frac{2}{3}\).

MEKE module parameters¶

Symbol

Model parameter