(a) Show $\phi\sim f U L$

Geostrophic balance (horizontal) is

$$f\,\mathbf{\hat{z}}\times\mathbf{u}_g = -\nabla\phi.$$

Taking magnitudes and using characteristic scales $U$ for horizontal velocity and $L$ for horizontal length, we have

$$f U \sim \frac{\Phi}{L},\qquad \Rightarrow\qquad \Phi\equiv \phi \sim f U L.$$

This is the desired scaling.