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

Question

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?

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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.

Answer 1

It's a general problem that quadratic equations have two roots when only one is expected. One will then be unphysical and we have to decide which is the one we want.

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.