# How does one determine which signs to take for the Gradient Wind Equations?

Under geostrophic balance, one can write

$$\frac{V^2}{R}+fV-fV_g=0$$ where $$V:$$wind speed, $$V_g:$$ geostrophic wind speed, $$f:$$ Coriolis parameter, and $$R:$$ radius of curvature.

Solving for $$V$$, we can get a relationship between $$V$$ and $$V_g$$ as $$V=-\frac{fR}{2}\pm\frac{\sqrt{f^2R^2+4fRV_g}}{2}$$

How does one then determine when to use $$\pm$$ for cyclonic and anti-cyclonic flows?

I was able to find a solution from these slides 40-41 and I am also aware that the $$R$$ can be positive and negative, and plays a part in being physically meaningful as seen here. However, I fail to understand why and how the $$\pm$$ signs come into play. To be clear, I am referring to the $$\pm$$ between the two terms on the RHS.

In this case we can proceed as follows: When $$|R|$$ becomes very large (either positive or negative) then the centripetal acceleration becomes negligible and $$V\to V_g$$. The sign has to be chosen to make this happen. So (assuming that $$f$$ is positive) for $$R$$ positive we must take the postive square root and for $$R$$ negative we must take the negative root. To see that this is so it helps to write your quadrtic solution as $$V=-\frac{fR}{2} \pm |fR| \frac {\sqrt{1+4V_g/fR}}{2}\\\approx -\frac{fR}{2} \pm |fR|(1+V_g/fR+\ldots)$$ The last expansion is valid when $$|R|$$ is large.
• Ah, since anti-cyclonic flows have $R<0$, thus we will use the $-\frac{fR}{2}-...$ version as explained. Vice-versa, cyclonic flows have $R>0$ and $-\frac{fR}{2}+...$ version. – fromzero Oct 29 '20 at 22:32