(b) Horizontal divergence of geostrophic wind vanishes

In two dimensions the geostrophic velocity is obtained from the streamfunction $\psi=\phi/f$:

$$u_g = -\frac{1}{f}\frac{\partial\phi}{\partial y},\qquad v_g = \frac{1}{f}\frac{\partial\phi}{\partial x}.$$

Compute divergence:

$$\nabla\cdot\mathbf{u}_g = \frac{\partial u_g}{\partial x} + \frac... ...ial x\partial y} + \frac{1}{f}\frac{\partial^2\phi}{\partial y\partial x} = 0.$$

Hence geostrophic flow is nondivergent. Consequently, a naive scaling for vertical velocity obtained from continuity $\partial_x u + \partial_z w = 0$ with $\partial_x u \sim U/L$ would give $w\sim U H/L$, but this is an overestimate because the leading-order geostrophic horizontal flow has zero divergence. The true vertical velocity arises from the small ageostrophic divergence (ageostrophic velocity $\sim Ro\,U$, where $Ro=U/(fL)\ll1$ is the Rossby number), so a more realistic scaling is

$$w\sim ($$divergence$$)\times H \sim (Ro\,U/L)\,H = \frac{U H}{L}Ro = \frac{U H}{L}\frac{U}{fL} = \frac{U^2 H}{fL^2}.$$

A cleaner statement is: the leading-order vertical velocity in a rapidly rotating flow scales like $w\sim Ro\,U H/L$, i.e. smaller than the naive $U H/L$ by a factor $Ro\ll1$. Thus $U H/L$ overestimates $w$.