There are two common aspects of each mixing scheme. Firstly, each involves the computation of neutral directions, or more precisely, the slopes at each grid point of the tangent to the locally defined potential density surface. These isoneutral slopes provide the fundamental orientation for the tracer mixing schemes described here. As such, all schemes which compute the slopes are referred to in the following as isoneutral mixing schemes, and the central model option associated with these schemes is isoneutralmix. In MOM, the isoneutral slopes are computed using the best available (within the chosen grid stencil) discrete approximation to the locally referenced potential density. A great deal of effort has gone into refining the discretization of these slopes. The reason for such effort is that inconsistently discretized slopes can lead to instabilities in the isoneutral diffusion scheme (Griff ies et al. 1998). The isoneutral slopes used in all of the following mixing schemes are computed as discussed in Griff ies et al. (see also Appendix C).
Secondly, Griff ies et al. motivate a discretization of the diffusive flux components consisting of groups of four ``triads'' of slopes. These triads build up each of the isoneutral diffusion flux components on the side of the tracer cells. The use of this discretization ensures that isoneutral diffusion will reduce the tracer variance. Section 34.1.5 provides details of the isoneutral diffusion scheme in MOM.
In addition to using the slope computation and triad structure for isoneutral diffusion, Griff ies (1998) showed how this technology can be exploited for the discretization of the Gent and McWilliams (1990; GM90) eddy stirring process. The results provide strong motivation for using a skew-diffusion, or skew-flux, approach for discretizing parameterized adiabatic eddy stirring. As a result, this approach is the default option in MOM for the adiabatic stirring schemes. The alternative approach, which is more cumbersome and inefficient computationally, involves the use of an extra term in the tracer advection velocity. This ``eddy-induced advection velocity'' approach is retained for comparative and diagnostic purposes.
Roberts and D. Marshall (1998; RM98) provide motivation for employing an adiabatic biharmonic operator. RM98 show how such an operator can provide an adiabatic means to damp grid-scale structures. Their scheme is implemented in MOM using the skew-flux approach. Section 34.1.8 provides details of the RM98 scheme in MOM.
Note that the continous versions of the GM90 and RM98 schemes are adiabatic in the sense that they preserve all moments of a tracer. On the discrete lattice, however, this property can at best be emulated by conserving the first (mean) and second (variance) moments. The skew flux approach, when implemented in terms of the triads of Griff ies et al. (1998), conserves the first and second tracer moments for the GM90 and RM98 schemes.
In a level model, the isoneutral slopes can become steep, if not vertical. For example, isoneutral slopes are effectively vertical in regions of free convection. The relative steepness of a slope is determined by the model's grid aspect ratio, the time step, and the diffusivity. For slopes that are greater in magnitude than some maximum slope, it is necessary to implement a numerical stabilization for each of the isoneutral mixing schemes. Otherwise, the schemes will become linearly unstable (Cox 1987, Griff ies et al. 1998). The form for stabilization in the past was ``slope clipping.'' As discussed by Griff ies et al., slope clipping can introduce a substantial amount of unphysical dianeutral fluxes. Consequently, slope clipping is not available in MOM. Rather, either one of two means of diffusivity tapering is now employed. This approach ensures numerical stability without introducing spurious dianeutral fluxes (see Gerdes et al. 1991, Danabasoglu and McWilliams 1996, or Griff ies et al. 1998). Note that when tapering, all of the isoneutral mixing processes are turned down to zero as the slope increases. The only nonzero mixing, besides that from advection or convection, arises from any nonzero vertical diffusivity. For various reasons, such as those discussed by Treguier et al. (1997), it might be desirable to maintain an additional nonzero horizontal diffusivity in these steep sloped regions. Such diffusion will be the only dissipative source of density mixing in regions with vertical density isolines. This option is available in MOM. Section 34.1.9 provides details of steep slope options.
Within the framework of any of the above mixing schemes, particular values for the eddy diffusivities must be chosen. Currently, most large-scale modelers use constant diffusivities. However, recent papers (e.g., Held and Larichev 1996, Treguier et al. 1997, Visbeck et al. 1997, Killworth 1997, and others) have criticized this approach from various perspectives. In MOM, the schemes of Held and Larichev (1996) and Visbeck et al. (1997) have been implemented. Section 34.2 provides details of these nonconstant diffusivity schemes. The remainder of this section is devoted to the constant diffusivity schemes.