35.1.6.2 gm_advect

In this approach, the GM90 parameterization is implemented in terms of an eddy-induced transport velocity. The implementation of the eddy-induced velocity in MOM is different than what is described in Danabasoglu and McWilliams (1996). Most notably, a computational mode (see Appendix E for a disucssion of computational modes), which was related to the original Cox (1987) implementation of the isoneutral diffusion, has been removed. Additionally, MOM employs a reference level for every depth level, rather than the reduced number of reference levels originally employed in the Danabasoglu and McWilliams code. As described in Section 34.1.6.1, the advection velocity approach is not as computationally efficient as the skew-flux approach. Therefore, gm_advect is retained solely for the comparative and diagnostic purposes. Consequently, the code for gm_advect is basically frozen, and future implementations of eddy stirring processes (e.g., biharmonic_rm) will be made using the skew-flux approach.

The eddy-induced velocities, as with the regular advection velocities in MOM, are computed at the centers of the eastern, northern, and bottom faces of the cells. The velocities are given by $adv\_vetiso_{i,k,j}$, $adv\_vntiso_{i,k,j}$, and $adv\_vbtiso_{i,k,j}$ respectively. In MOM 2 version 1, the eddy induced transport velocities were discretized based on the notes of Gokhan Danabasoglu as

$$adv\_vetiso_{i,k,j} -\delta_z(\overline{\frac{\kappa}{A_I}[K^{13}]_{i,k-1,j}}^z)$$ = (35.11)

$$adv\_vntiso_{i,k,j} -\delta_z(\overline{\frac{\kappa}{A_I}[K^{23}]_{i,k-1,j}}^z)$$ = (35.12)

However, the above form contains a null mode and has been replaced by the following

$$adv\_vetiso_{i,k,j} -\delta_z(\kappa\cdot S^{xb}_{i,k-1,j})$$ = (35.13)

$$adv\_vntiso_{i,k,j} -\delta_z(\kappa\cdot S^{yb}_{i,k-1,j})$$ = (35.14)

where the isoneutral slope in the zonal direction at the bottom of the eastern face of a T grid cell is given by

$$S^{xb}_{i,k,j} = -\frac{\overline{\alpha_{i,k,j}}^{\lambda,z}... ...}}^{\lambda,z}\overline{\delta_z(t_{i,k,j,2,\tau-1})}^\lambda}$$ (35.15)

and the neutral slope in the meridional direction at the bottom of the northern face of a T grid cell is given by

$$S^{yb}_{i,k,j} = -\frac{\overline{\alpha_{i,k,j}}^{\phi,z}\ov... ...{i,k,j}}^{\phi,z}\overline{\delta_z(t_{i,k,j,2,\tau-1})}^\phi}$$ (35.16)

where the $\alpha_{i,k,j}$ and $\beta_{i,k,j}$ are defined as in Section .

The vertical component of the eddy-advection velocity is obtained by vertically integrating the divergence of the horizontal eddy-advection velocities as is done in the notes of Gokhan Danabasoglu.

$$adv\_vbtiso_{i,k,j} = \frac{1}{\cstj}\sum_{m=1}^{k}\Bigl[\delx(adv\_vetiso_{i-1,m,j}) + \dely(adv\_vntiso_{i,m,j-1})\Bigr]\cdot dzt_m$$ (35.17)

Note that traditionally there is a zero vertical eddy-advection velocity at the top face of cell_i,k=1,j and bottom face of cell_i,k=bottom,j. This boundary condition on the velocity effectively places a boundary condition on the diffusivity $\kappa$(e.g., see discussion in Treguier et al 1997).

noindent The eddy-induced advection terms are discretized as:

 

$${\cal L}^{gm}(\gamma_{i,k,j}) = \frac{1}{\cstj}\Bigl[\!\!\!\!\!\!\!\!\!\! \delx(adv\_vetiso_{i-1,k,j}\overline{\gamma_{i-1,k,j}}^\lambda) + \dely( adv\_vntiso_{i,k,j-1}\overline{\gamma_{i,k,j-1}}^\phi)\Bigr]$$

$$\delta_z(adv\_vbtiso_{i,k-1,j}\overline{\gamma_{i,k-1,j}}^z)$$ - (35.18)

where $adv\_vntiso_{i,k,j-1}$ contains an embedded cosine factor as does $adv\_vnt_{i,k,j}$. Refer to Section 22.8.7 for a definition of the advective operator.