35.1.6.1 gm_skew

The addition of a divergence-free eddy-induced velocity to the tracer equation is equivalent to adding an anti-symmetric component to the tracer mixing tensor. This equivalence is well known. Griff ies (1998) argued that for the purpose of numerically implementing GM90, the use of its anti-symmetric tensor is preferred to the use of eddy-induced velocities. The advantages over the use of the advective formulation are basically the following:

1.

One less spatial differentiation of the slope and diffusivity is required with the skew-flux approach. Spatial differentiation of these objects can be numerically problematical, especially in regions of steep isoneutral slopes where both the slope, and the possibly spatially dependent diffusivity, are rapidly changing.

2.

It exploits the work of Griff ies et al (1998) for implementing isoneutral diffusion. That framework makes for a trivial implementation of GM90 in which the tracer variance is conserved.

3.

The skew-flux is faster computationally than the advective flux.

4.

When setting the diffusivities A_I and $\kappa$ equal, the off-diagonal terms in the horizontal flux drop out identically, unless option biharmonic_rm is also enabled. This relative setting of the diffusivities is commonly employed in the literature (e.g., Danabasoglu and McWilliams 1996, Hirst and McDougall 1996). It should be noted that researchers are currently suggesting that the GM90 diffusivity be set to roughly 1/2 to 1/4 that of the along isoneutral diffusivity. Regardless of the relative setting of the diffusivities, the skew-flux approach is preferable.

The implementation of option gm_skew is trivial once the implementation of the small slope isoneutral diffusion process has been done. All that is necessary is to modify the diffusivities appearing in the off-diagonal terms already used in the isoneutral diffusion scheme. Note that all skew-flux components are integrated explicitly in time, since there is no diagonal contribution to the vertical tracer flux component.

Regardless of the numerical preference for the skew-flux approach, it is useful diagnostically to map the GM90 eddy induced velocities. For this purpose, anytime option snapshots is enabled, these GM90 velocities are computed using the algorithm from option gm_advect and placed in the snapshots file.