42.3.0.5 Errors with z-level mixing

Consider the traditional diffusive mixing (Section B.2) with some tensor ${\bf I}$ that is diagonal in the z-level frame. Diffusive mixing with ${\bf I}$ is quite different than diffusive mixing with ${\bf K}$, as can be seen clearly by transforming ${\bf I}$ to the orthonormal isopycnal frame

$$I^{\overline{m} \overline{n}} = \left( \begin{array}{ccc} ... ...S^{2}}} & {1 \over \sqrt{1+S^{2}}} \end{array} \right),$$ (42.60)

which yields

$$I^{\overline{m} \overline{n}} = {A_{H} \over 1+S^{2}} \le... ...epsilon}) & \tilde{\epsilon} + S^{2} \end{array} \right),$$ (42.61)

where $\tilde{\epsilon} = A_{V}/A_{H}$ is the ratio of the vertical to horizontal diffusion coefficient.

Therefore, diffusive mixing with ${\bf I}$ in the z-level frame introduces first order in slope errors in the off diagonal terms, whereas the diagonal terms contain second order in slope errors. The error in the (3,3) component, however, is the most relevant as it represents an added source of diapycnal mixing (i.e., a false diapycnal mixing). For example, with the usual diffusivity ratio $\tilde{\epsilon} \approx 10^{-7}$, modest slopes $S \approx 3 \times 10^{-4}$ are sufficient to add diapycnal mixing through the S²term which is on the same order as $\tilde{\epsilon}$. It is for this reason that horizontal mixing, especially in regions of the ocean with larger than modest slopes but still within the small slope approximation, is incompatible with the hypothesis that mixing is predominantly along the isopycnal directions.

As seen above, the distinction between horizontal and along isopycnal mixing is quite important. However, the distinction between vertical mixing and diapycnal mixing is not generally important except for extremely large slopes. The reason for this ambiguity can be easily understood by looking at the small angle approximation to the isopycnal tensor [equation (B.56], and setting A_I to zero in order to focus on the diapycnal piece

$$K^{m n}(A_{I} =0)_{\mbox{small}} = A_{D} \left( \begin{arr... ... -S_{y} \\ -S_{x} & -S_{y} & 1 \end{array} \right).$$ (42.62)

The off diagonal terms represent the difference between vertical mixing and diapycnal mixing. These terms are down by a single factor of the slope. To completely determine the error introduced by neglecting these terms, it is useful to write the diffusion operator R(T) arising from this tensor and performing a scaling within the small slope approximation. For this purpose, consider the horizontal direction of steepest slope to be the x-direction. The y-direction will therefore be ignored. Also assume a constant diapycnal mixing coefficient A_D. With these approximations, the diffusion operator takes the form

$$A_{D} [ \partial_{z}(S_{x} \partial_{x}T) + \partial_{z}\partial_{z}T]$$ R(T) =

$$\approx A_{D}\left( {\Delta T \over (\Delta x)^{2} } +{\Delta T \over (\Delta z)^{2} } \right)$$

$$\approx A_{D} {\Delta T \over (\Delta z)^{2} },$$ (42.63)

where the last approximation assumed $\Delta z << \Delta x$. Hence, the off-diagonal pieces in the diffusion tensor are quite negligible in the small slope approximation. Therefore, with this scaling, the distinction between vertical and diapycnal mixing is negligible as well.