Rossby radius as horizontal scale
The Rossby radius of deformation tells you the distance when the Coriolis effect becomes just as important as the buoyancy or gravity.

Let the Rossby radius of deformation be denoted as $L_R$. According to Holton's Fifth Edition of "An introduction to Dynamic Meteorology,"  $$L_R=\frac{NH}{f_0}$$ where $N$ is the Brunt-Vaisala frequency, $H$ is the scale height, and $f_0$ is the Coriolis parameter.

For phenomena with length scale $L$, buoyancy can be neglected if $L&lt;&lt;L_R$.

In statically stable equatorial flows, $$\lim_{f_0\to 0} L_R=\infty$$

In certain coastal flows, $L \approx L_R$

So the author is saying that in equatorial flows, buoyancy can't be neglected, since the Rossby radius "blows up" and for coastal flows $L\approx L_R$