42.3.0.1 Basis vectors

Consider a first order tensor, or a vector $\vec{V}$. This object has any number of representations determined by the particular frame of reference. For example, a basis for two frames of interest yield the representations

  

$$\vec{V} V^{m}\vec{e}_{m},$$ = (42.36)

$$V^{\overline{m}}\vec{e}_{\overline{m}},$$ = (42.37)

where the space-time dependent functions V^m and $V^{\overline{m}}$ are the coordinates for the vector as represented in the respective frame. There are two sets of basis vectors which define the frames considered here:

$$\vec{e}_{1} \hat{x}$$ = (42.38)

$$\vec{e}_{2} \hat{y}$$ = (42.39)

$$\vec{e}_{3} \hat{z},$$ = (42.40)

which is the familiar Cartesian unit basis for the z-level frame, and

   

$$\vec{e}_{\overline{1}} { {\hat z} \times \nabla \rho \over \vert{\hat z} \times \nabla \rho\vert}$$ = (42.41)

$$\vec{e}_{\overline{2}} \vec{e}_{\overline{3}} \times \vec{e}_{\overline{1}},$$ = (42.42)

$$\vec{e}_{\overline{3}} {\nabla \rho \over \vert\nabla \rho\vert},$$ = (42.43)

which defines the orthonormal isopycnal frame determined by the fluctuating geometry of a locally referenced isopycnal surface.

Transforming from the z-level frame to the orthonormal isopycnal frame requires a linear transformation. As a tensor equation, the transformation is written $\vec{e}_{\overline{m}} = \Lambda^{m}_{\;\; \overline{m}}{\vec{e}_{m}}$. For the purposes of organizing the components of the transformation, this equation can be written as

$$(\vec{e}_{\overline{1}} \; \; \vec{e}_{\overline{2}} \; \; \... ...1+S^{2}}} & {1 \over \sqrt{1+S^{2}}} \end{array} \right)$$ (42.44)

where $\vec{S} = \nabla_{\rho} z = -z_{\rho}\nabla_{z}\rho = (S_{x}, S_{y},0) = (-\rho_{x}/\rho_{z}, -\rho_{y}/\rho_{z},0)$ is the isopycnal slope vector with magnitude S. Since the transformation is between two orthonormal frames, this transformation matrix is a rotation (unit determinant and inverse given by the transpose) and so can be interpreted in terms of Euler angles (e.g., Redi 1982).