42.3.0.2 Orthonormal isopycnal frame

The relevance of the orthonormal isopycnal frame arises from the diagonal nature of diffusion within this frame, as assumed by Redi (1982). In general, diffusion is thought to occur predominantly along the two orthogonal directions ${\vec{e}}_{\overline{1}}$ and ${\vec{e}}_{\overline{2}}$, which define the neutral directions that are tangent to the locally referenced isopycnal surface. Diffusion in the diapycnal direction ${\vec{e}}_{\overline{3}}$ typically occurs with a diffusion coefficient which is on the order of 10^-7 times smaller. Therefore, in this frame the symmetric diffusion tensor takes on the diagonal form

$$K^{\overline{m}\overline{n}} = \left( \begin{array}{ccc} ... ... 0 & A_{I} & 0 \\ 0 & 0 & A_{D} \\ \end{array} \right),$$ (42.45)

where A_I are the along isopycnal diffusion coefficients and $A_{D} \approx 10^{-7}A_{I}$ is the diapycnal diffusion coefficient ^42.3. The diffusion tensor written in terms of projection operators takes the form

$$K^{\overline{m}\overline{n}} = A_{I} (\delta^{\overline{m}\ov... ...^{\overline{m}}_{\overline{3}}e^{\overline{n}}_{\overline{3}}$$ (42.46)

with ${\vec{e}}_{\overline{3}}$ having the components (0,0,1)^T in the orthonormal isopycnal frame. Explicitly, the diffusion operator in these coordinates is

$$R(T) = \partial_{\vec{e}_{\overline{1}}} (A_{I} \partial_{\v... ..._{\overline{3}}}(A_{D} \partial_{\vec{e}_{\overline{3}}} T)$$ (42.47)

where the diffusion coefficients are generally nonconstant.