4.2.3.4 Nonlinear advective ``metric'' term

The term

$$a^{-1} \, u \, ({\bf u} \wedge \hat{z}) \, \tan\phi$$ (4.15)

arises from the curvature of the earth. Since the longitudinal and latitudinal unit vectors $(\hat{\lambda},\hat{\phi})$ are not material constants, they contribute to the material time derivative of the velocity vector

$$\frac{D {\bf u}_{h}}{Dt} \frac{D (\hat{\lambda} u + \hat{\phi} v)}{Dt}$$ =

$$\hat{\lambda} \, \frac{D u}{Dt} + \hat{\phi} \frac{D v}{Dt} + u \, \frac{D \hat{\lambda}}{Dt} + v \, \frac{D \hat{\phi}}{Dt}.$$ = (4.16)

For a derivation of the material derivatives of the unit vectors, see Section 2.3 of Holton (1992). These these extra ``metric'' terms vanish when working on a plane, such as the f-plane or $\beta$-plane, since for this case the unit vectors $\hat{x}$ and $\hat{y}$ are constant. Additionally, this term, just as the Coriolis force, does zero work on the fluid.