Gradient wind

Remarks

The three terms are the centripetal acceleration, the Coriolis acceleration and the pressure gradient force per unit mass. With the geostrophic speed $V_g = -(\rho f)^{-1} \partial p / \partial n$ of geostrophic balance, the balance reads $V_g / V = 1 + V / (f R)$. Solving the quadratic gives $$V = -\frac{f R}{2} \pm \sqrt{\frac{f^2 R^2}{4} - \frac{R}{\rho} \frac{\partial p}{\partial n}},$$ and the physical root is the one that is non-negative and tends to $V_g$ as $\abs{R} \to \infty$.

Example: in the northern hemisphere ($f > 0$), around a low the flow is anticlockwise, $R > 0$, and $V < V_g$: the wind is subgeostrophic. With $f = 10^{-4}$ s$^{-1}$, $R = 500$ km and $V_g = 20$ m/s, the balance $V^2 + 50 V - 1000 = 0$ gives $V \approx 15.3$ m/s.

Non-example: around a high in the northern hemisphere $R < 0$, and a real solution needs $f^2 R^2 / 4 \ge (R / \rho) \, \partial p / \partial n$. If the pressure gradient of a high is steeper than this bound, no gradient wind exists; this is why the pressure gradient near the centre of a high is weak.

When the Rossby number $V / (\abs{f} \abs{R})$ is small, the centripetal term is negligible and the gradient wind reduces to the geostrophic wind.