42.3.0.3 z-level frame

The orthogonal isopycnal frame allowed for a simple prescription of the isopycnal diffusion process. In order to describe such a diffusion process in other frames of reference, the components of Kmust be transformed. Such rules of transformation are standard (e.g., Aris 1962). The diagonal components $K^{\overline{m} \overline{n}}$from the orthogonal isopycnal frame are transformed to the z-level frame through

$$K^{m n} = \Lambda^{m}_{\;\; \overline{m}} K^{\overline{m}\overline{n}} \Lambda^{n}_{\;\; \overline{n}},$$ (42.48)

where the transformation matrix is given by equation (B.44). Written as matrices with $\Lambda$ having components $\Lambda^{n}_{\;\; \overline{n}}$, this transformation takes the form $K = \Lambda {\overline K} \Lambda^{T}$, where ${\overline K}$ has the diagonal form given in equation (B.45). Performing the matrix multiplication

$$K^{m n} = \left( \begin{array}{ccc} {S_{y} \over S} & ... ...+S^{2}}} & {1 \over \sqrt{1+S^{2}}} \end{array} \right),$$ (42.49)

yields the components of the diffusion tensor in the z-level system given by Redi (1982)

$$K^{m n} = {A_{I} \over (1+S^{2})} \left( \begin{array}{ccc... ... \over \rho_{z}} & \epsilon + S^{2} \end{array} \right),$$ (42.50)

which can also be written

$$K^{m n} = {A_{I} \over (1+S^{2})} \left( \begin{array}{ccc}... ...1-\epsilon)S_{y} & \epsilon + S^{2} \end{array} \right).$$ (42.51)

Note that Redi uses the symbol $\delta$ instead of S in her expression. The ratio of the diapycnal to isopycnal diffusivities defines the typically small number $\epsilon = A_{D}/A_{I} \approx 10^{-7}$. An equivalent form is suggested by writing the tensor in the orthonormal isopycnal frame as

$$K^{\overline{m} \overline{n}} = A_{I} \left( \begin{array}{c... ...& 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right),$$ (42.52)

which separates the effects of the anisotropy between the along and across isopycnal directions. This expression can be written in the compact form

$$K^{ \overline{m} \overline{n} } = K_{ \overline{m} \overline... ...})_{\overline{m}}(\vec{e}_{\overline{3}})_{\overline{n}} \; ,$$ (42.53)

as noted in equation (B.46). Transforming to the z-level frame yields

$$K^{m n} = A_{I} \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 ... ...{z} & \rho_{y}\rho_{z} & \rho_{z}^{2} \end{array} \right),$$ (42.54)

which can also be written

$$K^{m n} = K_{m n} = A_{I} \delta_{m n} + A_{I}(\epsilon-1) ... ...\rho}_{m}\hat{\rho}_{n}) +A_{D}\hat{\rho}_{m}\hat{\rho}_{n},$$ (42.55)

where $\hat{\rho}_{m}=\partial_{m}\rho/\vert\nabla \rho\vert$ are the components of the diapycnal unit vector ${\vec{e}}_{\overline{3}}$written in the z-level frame [see equation (B.43)]. Note that equation (B.55) could have been written down immediately once equation (B.53) was hypothesized, and the unit vectors (B.41), (B.42), and (B.43) were determined.