## Q. 4.6

Derive a suitable approximation for the Coriolis parameter for regional scale motions centred on a latitude of $θ_{0}$.

## Verified Solution

We expand the Coriolis parameter in a Taylor series about the latitude $θ_{0}$ as f = $f_{0} + β_{y}$ + … where $β =\left(df/dy\right) _{\theta _{0} }$  and y = 0 at $θ_{0}$ . Thus, β = 2Ωcos(θ)/s and $f_{0}= 2Ωsin(θ_{0})$ . If the Taylor series is truncated after the first term, then we have what is termed the ‘f-plane’ approximation which includes for the Coriolis effect but not its variation with latitude. Retaining the first two terms yields the ‘β-plane approximation’, which includes a simplified form of the variation of Coriolis parameter with latitude. Note that in equatorial regions the second term becomes proportionately more important as $f_{0}$ → 0 as $θ_{0}$ → 0.