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I understand that it is possible for the coriolis effect and the pressure gradient force (PGF) to cancel each other out, but it appears to be a tendency (rather than just a possibility). What is the mechanism behind this?

I understand that as the wind is bent, the part/vector of the coriolis effect that acts in direct opposition of the PGF increases. But once the wind is bent so that it blows parallel to the isobars, why would the coriolis effect (have the tendency to) be similar to the PGF at that point?

Is it because 1) the wind picks up speed as long as it moves down the gradient at angle less than 90°, and 2) an increase in wind speed increases the coriolis effect? So that there will always be a moment where the coriolis force has grown equal to the PGF (at which point the wind will stop speeding up, because it moves paralel to the isobars)?

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Yes it is because of that:

1 - The wind must pick up speed as long its angle is less than 90° to the pressure gradient. This is because it has a net force in its direction of motion, and $F=ma$, so it has acceleration towards the low pressure. Whereas Coriolis "force" being always 90° from movement means it cannot accelerate/affect the magnitude, only change the direction.

2 - And Coriolis is $f\times v$, meaning Coriolis Force indeed increases as velocity increases.

As the process continues, the change in direction means less and less of the PGF is a component that changes the magnitude of velocity, and more and more of it is normal to motion... and so opposing the Coriolis.

If they didn't cancel each other out when the parcel reached 90° to the PGF, then the parcel would still have a directional acceleration... meaning it then again becomes not 90° to the PGF.
If PGF were stronger/it is less than 90° to PGF, it would further accelerate in velocity, and so thus further increase Coriolis... and be directed back to perpendicular. If Coriolis were somehow stronger/it somehow got beyond 90°, the PGF would then start to have a component that decelerates the velocity. Which would decrease the Coriolis counteracting it, and so pull it back (left in the NH) towards perpendicular.

So the situation is entirely restorative/stable. It can only be at balance when they cancel each other out, and the motion is at that 90° angle (unless there are additional forces). When it somehow isn't in that balanced situation, the forces work to accelerate/push it back towards that situation.

With a "sudden" low pressure, nearby parcels would not have the time/distance to undergo this process and accelerate/change direction enough before reaching the low. But typically atmospheric pressure changes are gradual changes, so basically the change in motion are so gradual that the motion at any time is essentially geostrophic at any moment.

Rapid pressure changes can and do still lead to local ageostrphic motion, where the change is of such speed and magnitude that the Coriolis Force hasn't responded yet and so there is a more meaningful component of motion out of balance (and so the motion is more non-perpendicular to the PGF).

And other forces... in particular friction at lower levels of the atmosphere... do alter the balance. Friction continually decelerating motion leads to a result where the path is consistently a bit less than 90°, towards the low pressure, which leads (apart from other factors) to filling of the low pressure over time. Particularly strong persistent low pressure can also result in a flow nearby that must turn so rapidly that there is a significant centrifugal "force" as well... this extra force isn't degenerative like friction is, but leads to a different balance instead, the gradient wind balance, with a somewhat lower velocity compared to geostrophic balance.

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Page 556 of "Earth Science" by Tarbuck & Lutgens (2015) says the following:

"The only force acting on a stationary parcel of air is the pressure gradient force. Once the air begins to accelerate, the Coriolis effect deflects it to the right in the Northern Hemisphere. Greater wind speeds result in a stronger Coriolis effect (deflection) until the flow is parallel to the isobars. At this point, the pressure gradient force and Coriolis effect are in balance, and the flow is called a geostrophic wind."

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