# Interpretation of the radius of curvature in atmospheric dynamics

I'm studying atmospheric dynamics and one concept confuses me a little bit. The circulation of wind flow can be characterised by a radius of curvature, $$R$$ that is taken positive for cyclonic flow (anti clockwise in the northern hemisphere) and negative for anti cyclonic flow. But then I came across an explanation of why high pressure areas are "flat", which stated the following: the condition to have circular flow (one of the idealized solutions of the momentum equations) is $$\mid\frac{\partial \Phi}{\partial n}\mid < \frac{\mid R \mid f^2}{4}$$ where $$\Phi$$ is the geopotential, $$n$$ is a direction perpendicular to the flow and $$f$$ is the Coriolis parameter. The geopotential can be used in stead of pressure, it contains the same information. So this would imply that if the radius of curvature is small (tends to zero) the gradient of the potential (or pressure) must also be small. I think I understand that a small gradient implies a flat potential (the isobars would be far apart), but I rather associate a small radius of curvature to strong curvature and thus a low pressure area? On the other hand, if you would transition from cyclonic to anticyclonic flow, I can sort of imagine you need to pass through zero curvature radius, which might also mean flow in a straight line.

How do I interpret $$R$$? What is the meaning of $$R = 0$$ or $$R = \pm \infty$$ in relation to high or low pressure areas?

For reference, I'm studying from Holton's book "An introduction to dynamic meteorology".

• I will take a closer look on your question soon, but in geophysical fluid dynamics $R$ is often refering to the spatial order of magnitude, $R \to 0$ would then mean neglectable, $R \to \infty$ would refer to system size, e.g. size of the Earth. Jan 20, 2021 at 17:18