34.9 tracer_horz_biharmonic

With option tracer_horz_biharmonic, the general form of the horizontal piece of the tracer mixing operator is determined by iterating the Laplacian mixing. That is, first one computes minus the Laplacian mixing operator

$$del2\_tracer_{i,k,j,n} = -DIFF\_Tx_{i,k,j} -DIFF\_Ty_{i,k,j},$$ (34.112)

where the fluxes used to construct $DIFF\_Tx_{i,k,j}$ and $DIFF\_Ty_{i,k,j}$ are the usual diffusive fluxes described in Section 33.8. Note that the full biharmonic diffusivity is used to compute these diffusive fluxes. The n label is for the tracers, with n=1,nt. The biharmonic diffusivity will be a constant if option tracer_horz_mix_const is enabled, and an array if option tracer_horz_mix_var is enabled. Next, fluxes of the scalar quantity $del2\_tracer_{i,k,j,n}$ on the eastern and northern sides of T cells are computed using

  

$$\diff\_fe_{i,k,j} \frac{\delx(del2\_tracer_{i,k,j,n})}{\cstj}$$ =

$$\frac{del2\_tracer_{i+1,k,j,n} - del2\_tracer_{i,k,j,n}} {\cstj dxu_{i}} \; , j=2,jmw-1$$ = (34.113)

$$\diff\_fn_{i,k,j} \csuj \dely(del2\_tracer_{i,k,j,n})$$ =

$$\csuj\frac{del2\_tracer_{i,k,j+1,n} - del2\_tracer_{i,k,j,n}}{dyu_{jrow}} \;, j=1,jmw-1.$$ =

These fluxes are of the same form as fluxes of the tracers, with the important difference that fluxes of $del2\_tracer$ require no diffusivity. The reason is that the full biharmonic diffusivity has already been used to compute the scalar $del2\_tracer$. The fluxes of $del2\_tracer$ are then used in the operators $DIFF\_Tx_{i,k,j}$ and $DIFF\_Ty_{i,k,j}$. The result is the tracer biharmonic mixing operator. Two extra boundary conditions are required beyond the usual Laplacian operator, and they are prescribed to be a vanishing of fluxes of del2 at land boundaries. Since option tracer_horz_biharmonic is a fourth order scheme, the minimum size of the memory window is jmw=4 instead of jmw=3 required for second order differences. As with all fourth order schemes, option fourth_order_memory_window must be enabled to handle this option. This is automatically done when either option tracer_horz_biharmonic or option velocity_horz_biharmonic is enabled.