42.3.0.4 Small angle approximation

There have been no small angle approximations (i.e., S << 1) made in computing the representation K^{m n}. Therefore, for modeling the physical process of diffusion employing the isopycnal diffusive mixing hypothesis in the z-level frame, K^{m n} given here is the form of the diffusion tensor to be used. In so doing, the physics of diapycnal diffusion is isolated in the parameter $\epsilon$. The small angle approximation of Cox (1987) is an additional statement regarding the behavior of the bulk of the ocean (that isopycnals have only a rather small slope except in convection regions). This approximation recovers the form quoted in Gent and McWilliams (1990)

$$K^{m n}_{\mbox{small}} = A_{I} \left( \begin{array}{ccc} 1 ... ...(1-\epsilon)S_{y} & \epsilon + S^{2} \end{array} \right)$$ (42.56)

Note that Cox (1987) originally retained the (1,2) = (2,1) elements equal to -S_xS_y. In the small angle approximation, however, this term is negligible. Additionally, and most crucially, if this term is retained, the small angle tensor will diffuse locally referenced potential density whereas the full tensor will not. Hence, it is important to use the physically consistent form of the small angle approximation which drops the (1,2) and (2,1) elements.

As the small angle approximation is commonly used, it is perhaps interesting to ask the following question: Given the small angle approximation representation of the diffusion tensor in the z-level frame, what is the representation of this tensor in the orthonormal isopycnal frame? This tensor cannot be the diagonal form given in equation (B.45) since that form transformed into the full non-small angle representation of equation (B.50). Using the full transformation back to the orthonormal isopycnal frame, $K^{\overline{m} \overline{n}}_{\mbox{small}} = \Lambda^{\overline{m}}_{\;\; m} K^{m n}_{\mbox{small}} \Lambda^{\overline{n}}_{\;\; n}$, or in matrix form

$$K^{\overline{m} \overline{n}}_{\mbox{small}} = A_{I} \left( \begin{array}{ccc} {S_{y} \over S} & -{S_{x} \over S} & 0 \\ ... ...} & -{S_{y} \over \sqrt{1+S^{2}}} & {1 \over \sqrt{1+S^{2}}} \end{array}\right) \times$$ (42.57)

$$\left( \begin{array}{ccc} 1 & 0 & (1-\epsilon)S_{x} \\ 0 & 1 & (1-\... ...\\ (1-\epsilon)S_{x} & (1-\epsilon)S_{y} & \epsilon + S^{2} \end{array}\right) \left( \begin{array}{ccc} {S_{y} \over S} & {S_{x} \over S \sqrt{... ...\\ 0 & {S \over \sqrt{1+S^{2}}} & {1 \over \sqrt{1+S^{2}}} \end{array}\right),$$ (42.58)

yields the orthonormal isopycnal frame representation of the small angle approximated tensor^42.4

$$K^{\overline{m} \overline{n}}_{\mbox{small}} = \left( \begi... ...0 & 0 \\ 0 & -1 & S \\ 0 & S & 1 \end{array} \right).$$ (42.59)

The small angle approximation is seen to add a small amount of along isopycnal mixing (the (2,2) term A_I(1+S²)) as well as a term proportional to the generally small number A_D S². Dropping these terms is consistent with the small angle approximation, which then recovers the purely diagonal mixing tensor $K^{\overline{m}\overline{n}} = A_{I} (\delta^{\overline{m}\overline{n}} - e^{\o... ...line{3}}) + A_{D}e^{\overline{m}}_{\overline{3}}e^{\overline{n}}_{\overline{3}}$.