Chapter 2

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(This information is a reproduction of Appendix A from PL07)

In Section 2 of PL07 the flux-form multidimensional transport scheme is discretized in general non-orthogonal curvilinear coordinates. The covariant and contra-variant wind vector components are presented in Eqs. (4) and (5) (of Section 2 of PL07) based on the local unit vectors \( (\vec{e_1},\vec{e_2}) \) of the coordinate system. Given the angle \((\alpha)\) between the two unit vectors

\[ \cos \alpha = \vec{e_1} \cdot \vec{e_2} \\ \tag {2.1} \]

the covariant and contravariant components are related by the following relationships:

\[ u = \tilde{u} + \tilde{v} \cos \alpha \\ \tag {2.2} \]

\[ v = \tilde{v} + \tilde{u} \cos \alpha \\ \tag {2.3} \]

or (solving for the contravaraint components)

\[ \tilde{u} = \frac{1}{\sin^2 \alpha} [u - v \cos \alpha] \\ \tag {2.4} \]

\[ \tilde{v} = \frac{1}{\sin^2 \alpha} [v - u \cos \alpha] \\ \tag {2.5} \]

The winds on the cubed-sphere can be oriented to/from local coordinate orientation to a spherical latitude–longitude component form using the local unit vectors of the curvilinear coordinate system \((\vec{e_1},\vec{e_2})\) and the unit vector from the center of the sphere to the surface at the point of the vector location \((\vec{e_\lambda},\vec{e_\theta})\). Eqs.(2.6) and (2.7) represent the transformation from the spherical orientation \((u_{\lambda \theta},v_{\lambda \theta})\) to the local cubed-sphere form \((u,v)\) and the reverse transformation is presented in Eqs. (2.8) and (2.9).

\[ u = (\vec{e_1} \cdot \vec{e_\lambda}) u_{\lambda \theta} + (\vec{e_1} \cdot \vec{e_\theta}) v_{\lambda \theta} \ \tag {2.6} \]

\[ v = (\vec{e_2} \cdot \vec{e_\lambda}) u_{\lambda \theta} + (\vec{e_2} \cdot \vec{e_\theta}) v_{\lambda \theta} \ \tag {2.7} \]

\[ u_{\lambda \theta} = \frac{ (\vec{e_2} \cdot \vec{e_\theta}) u - (\vec{e_1} \cdot \vec{e_\theta}) v }{ (\vec{e_1} \cdot \vec{e_\lambda}) (\vec{e_2} \cdot \vec{e_\theta}) - (\vec{e_2} \cdot \vec{e_\lambda}) (\vec{e_1} \cdot \vec{e_\theta}) } \\ \tag {2.8} \]

\[ v_{\lambda \theta} = \frac{ (\vec{e_2} \cdot \vec{e_\lambda}) u - (\vec{e_1} \cdot \vec{e_\lambda}) v }{ (\vec{e_1} \cdot \vec{e_\lambda}) (\vec{e_2} \cdot \vec{e_\theta}) - (\vec{e_2} \cdot \vec{e_\lambda}) (\vec{e_1} \cdot \vec{e_\theta}) } \\ \tag {2.9} \]