35.2.2.1 Time scale same as Held and Larichev

For the VMHS scheme, the time scale T is written

$$\frac{1}{D} \int^{-D_{t}}_{-D_{b}} \, (f^{2} /Ri)^{1/2} \, dz.$$ T^-1_vmhs = (35.94)

The square of this expression is not identical to equation (34.84) from the Held and Larichev scheme

$$\frac{1}{D} \int^{-D_{t}}_{-D_{b}} \, (f^{2} /Ri) \, dz.$$ T^-2_hl = (35.95)

However, in both approaches the time scale is determined by scaling arguments rather than from a fundamental theory. Therefore, consistency and simplicity motivate using an identical expression in MOM. Note that the expression from Held and Larichev was also used by Wright (1997) in the VMHS scheme implemented in the Hadley Centre ocean model. Hence, due partly to historical reasons (the Held and Larichev scheme was implemented first), and the desire to be consistent with the Hadley Centre implementation, the Held and Larichev expression for the time scale is implemented in MOM for both the hl_diffusivity and vmhs_diffusivity schemes.

As mentioned in Section 34.2.1.1, the thermal wind relation is used to compute the above Richardson number (equation (34.87)). The reason is that the theories used to define the diffusivities are based on quasi-geostrophic scaling. For computational reasons, the Hadley Centre uses the vertical shears of the full velocity field, rather than the thermal wind shears, in their Richardson number computation. This difference in Richardson number calculation represents the central difference between the MOM implementation of VMHS and that of the Hadley Centre. It is unclear how much difference this approach will make for the resulting time scale.