35.2.1.3 Smoothing and temporal frequency of computation

As the Held and Larichev mixing coefficients are derived under the assumptions of quasi-geostrophy, it is sensible to impose on the diagnosed diffusivity some smoothness whose scales reflect the large-scale geostrophic flow. To ensure this smoothness, the following filtering is performed. The researcher can impose more or less filtering as desired.

For evaluating $\beta_{eff}$, the model's stepped bottom topography is first smoothed with a 2-dimensional finite impulse response filter

$$\tilde{H}_{i,j} \sum_{ip=-1}^{1}\sum_{jq=-1}^{1} \, H_{i+ip,j+jq} \, M_{ip,jq}$$ = (35.92)

where the smoothing matrix has components

 

$$\left( \begin{array}{ccc} 1/16 & 1/8 & 1/16 \\ 1/8 & 1/4 & 1/8 \\ 1/16 & 1/8 & 1/16 \end{array}\right).$$ M_ip,jq = (35.93)

Multiple passes through this filter will further smooth the topography. The researcher can choose the number of passes through changing the parameter numfltrtopog inside of the fortran routine topog.F. The default is one pass. The spatial derivatives of the topography are then computed using the filtered topography field.

Implicit in the formalism is some smoothing in both time and space due to the use of the thermal wind relation when computing the time scale in equation (34.88). A further smoothing can be performed on this time scale through the use of the two dimensional finite impulse response filter. The parameter numfltrgrth, set inside of nonconstdiff.F, determines the number of passes through the FIR for the growth rate. The default is numfltrgrth = 0 for zero filtering. To avoid problems with overly huge growth rates computed in regions of very low vertical stratification, a minimum time scale for the growth is taken to be 1/4 day, which leads to a maximum squared growth rate of $2.14 \times 10^{-9} sec^{-1}$. This limit can be changed through altering the parameter growth2max inside of nonconstdiff.F. Temporal smoothing of the growth rate in the form of a Robert filter has been suggested by Visbeck et al. (1997). This smoothing has not been implemented in MOM.

The time scale T is computed in MOM based on density fields one time step previous to the present time step. T is accumulated as a vertical average within the isopyc.F routine using the same code that computes the isoneutral slopes (inside subroutines $ai\_east$, $ai\_north$, and $ai\_bottom$), where the slopes are computed as described in Griff ies et al. (1998), Section 34.1.5, and Appendix C. Conveniently, no extra slope computations are needed beyond that already required by the constant diffusivity isoneutral schemes. A minimum time scale of T=1day is imposed on the computation; this value is set by the parameter growthmax inside of nonconstdiff.F.

When finished computing the vertically averaged Richardson number, a two-dimensional time scale field $T(\lambda,\phi)$ is retained and then used to define the Rhines' length and diffusivity. The time scale field is saved inside of a restart file as well as the nonconstant diffusivity, thus allowing for a smooth evolution of the nonconstant diffusivity across model restarts.

To facilitate those cases in which one wishes to turn on a nonconstant diffusivity after running for some time with constant diffusivities, the option $nonconst\_diffusivity\_initial$ will initialize $T(\lambda,\phi)$ to zero and it will override the attempt to read in $T(\lambda,\phi)$ from the restart file. Conversely, for those wishing to change from nonconstant to constant diffusivity in the middle of an ongoing experiment, turning on $nonconst\_diffusivity\_final$ will mean that $T(\lambda,\phi)$ will not be written to the restart file.

The namelist parameter diffint sets the temporal frequency used to update the diffusivities. The idea is that the diffusivities should change only over time scales determined by the eddy time scale T, which is a few days. Indeed, for the extreme example of a static model exhibiting no internal or forced varibility, the diffusivities are unchanging. For more dynamic models, for example with a seasonal cycle and/or realistic atmospheric forcing, more frequent diffusivity calculations are prudent. The parameter diffint determines the number of time steps skipped before computing a new value for the diffusivity. Hence, the frequency of computing the diffusivity is dependent on the model time step. The current implementation was found useful in order to ensure results agree across restart files. The MOM default is diffint=5, but this should be adjusted according to the time steps used for the particular experiment.