35.2.1.1 The thermal wind Richardson number and the depth range

Fundamental to the mesoscale eddy closure theories (e.g., Held and Larichev, Treguier et al., Visbeck et al., Killworth) is the assumption that the mesoscale eddy field is quasi-geostrophic. As such, the Richardson number, Ri, is a large-scale Richardson number based on vertical shears under thermal-wind balance with the buoyancy field. This assumption renders

 

$$\frac{N^{2}}{U_{z}^{2} + V_{z}^{2}}$$ Ri =

$$\frac{- g \, \rho_{z}/\rho_{o}}{ (g/f\rho_{o})^{2}(\rho_{x}^{2} + \rho_{y}^{2})}$$ =

$$-\left(\frac{f^{2} \, \rho_{o}}{g \, \rho_{z}} \right) S^{-2}$$ =

= (f/NS)² (35.87)

where $S^{2} = (\rho_{x}^{2} + \rho_{y}^{2})/\rho_{z}^{2}$ is the squared isoneutral slope vector and $N^{2} = -g\, \rho_{z}/\rho_{o}$is the squared buoyancy frequency based on the vertical gradient of locally referenced potential density.

The integration depth range, D = D_b - D_t, corresponds to the depth over which the baroclinic eddies predominanty occur. Treguier et. al. (1996) use the values D_t = 100m and D_b = 2000m. This depth range is also taken in the Hadley Centre ocean model in which they implement the Visbeck et al. scheme (Section 34.2.2), and it is also the default for MOM. This depth range is not fundamental, and sensitivity of the results to the details of this range has not been documented. In order to avoid problems with unstratified lowest model levels, as might occur with bottom boundary layers, the bottom level of the depth range is set to at least two depth levels above the ocean bottom. In this way, the computed Eady growth rate is taken over interior model levels. Pragmatically, this restriction also avoids the distinction between partial and full bottom cells (Chapter 26). In regions where the ocean is shallower than D_t, the diffusivities default to the background values A_I = ahisop and $\kappa = ahthk$used in the constant diffusivity case, which are set in the namelist.

Using the thermal wind Richardson number (34.87) brings the squared inverse time scale to the form

 

$$\frac{1}{D} \int^{-D_{t}}_{-D_{b}} \, N^{2} \, S^{2} \, dz$$ T^-2 =

$$\frac{g}{\rho_{o} \, D} \int^{-D_{t}}_{-D_{b}} \, \vert\rho_{z}\vert \, S^{2} \, dz.$$ = (35.88)

Note how the explicit dependence on the Coriolis parameter f has cancelled. Again, the source of this cancellation is the use of thermal wind balance for computing the Richardson number. Consequently, the time scale is an explicit function only of the vertically averaged horizontal and vertical stratification. Notably, the inverse time scale, or the growth rate, vanishes when the vertically integrated horizontal stratification vanishes; i.e., when there is zero baroclinicity. As such, the diffusivity vanishes when the neutral directions are flat, as one would expect from theories of baroclinic instability. Relatedly, the explicit cancelation of the Coriolis parameter allows for the time scale calculation to be naively applied globally, including at the equator, where geostrophy is irrelevant.