26.2.3 Tracer equations

To account for a generalized variation in cell thickness, a factor of dht_i,k,j must also be included in horizontal diffusive and advective operators for tracers. The following equations should be compared with the corresponding set in Section 21.3.2:

  

$$ADV\_Tx_{i,k,j} \frac{1}{2\;\cstj\; dht_{i,k,j}}\delx(adv\_fe_{i-1,k,j})$$ = (26.29)

$$ADV\_Ty_{i,k,j} \frac{1}{2\;\cstj\; dht_{i,k,j}}\dely(adv\_fn_{i,k,j-1})$$ = (26.30)

$$DIFF\_Tx_{i,k,j} \frac{1}{\cstj\; dht_{i,k,j}}\delx(\diff\_fe_{i-1,k,j})$$ = (26.31)

$$DIFF\_Ty_{i,k,j} \frac{1}{\cstj\; dht_{i,k,j}}\dely(\diff\_fn_{i,k,j-1})$$ = (26.32)

The diffusive fluxes on T-cell faces are

$$\diff\_fe_{i,k,j} \frac{\diff\_cet_{i,k,j}}{\cstj}\;min(dht_{i,k,j},dht_{i+1,k,j})\;\delx(t^{int}_{i,k,j,n,\tau-1})$$ = (26.33)

$$\diff\_fn_{i,k,j} \diff\_cnt_{i,k,j}\cdot\csuj \;min(dht_{i,k,j},dht_{i,k,j+1})\; \dely(t^{int}_{i,k,j,n,\tau-1})$$ = (26.34)

The superscript ``int'' in the tracer quantity denotes a linear interpolation in the vertical to the minimum depth of the two points being considered in the horizontal derivative. The form of the linear interpolation is

$$\delx(t^{int}_{i,k,j,n,\tau-1}) = \bigl( t_{i+1,k,j,n,\tau-1} - min(dhwt_{i+1,k,j},\; dhwt_{i,k,j})\cdot \frac{t_{i+1,k-1,j,n,\tau-1}-t_{i+1,k,j,n,\tau-1}}{dhwt_{i+1,k-1,j}}$$

$$(t_{i,k,j,n,\tau-1} - min(dhwt_{i+1,k,j},\; dhwt_{i,k,j})\cdot \frac{t_{i,k-1,j,n,\tau-1}-t_{i,k,j,n,\tau-1}}{dhwt_{i,k-1,j}} )$$ -

$$\bigr)$$ /dxu_i (26.35)

where dhwt_i,k,j is the vertical distance between the grid point in cell T_i,k,j and cell T_i,k+1,j.

If the above linear interpolation is not done, horizontal diffusion leads to an enhanced vertical diffusion in the vicinity of partial cells. This can easily be demonstrated by defining a temperature and salinity stratification which is only a linear function of depth. Horizontal diffusion should be zero. However, in the vicinity of partial cells, it will not be zero unless the above linear interpolation is used.

The form of the advective fluxes given by Equations (22.43) and (22.44) does not change. However, horizontal advective velocities $adv\_vet_{i,k,j}$ and $adv\_vet_{i,k,j}$ absorb a factor of dhu_i,k,j to become horizontal advective transports given by Equations (26.40) and (26.41).